computational methods in multiphase flow v (wit transactions on the engineering sciences) (wit...

COMPUTATIONAL METHODS IN

MULTIPHASE FLOW V

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FIFTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL AND

EXPERIMENTAL METHODS IN MULTIPHASE AND COMPLEX FLOW

MULTIPHASE FLOW V

CONFERENCE CHAIRMEN

A.A. MammoliUniversity of New Mexico, USA

C.A. BrebbiaWessex Institute of Technology, UK

INTERNATIONAL SCIENTIFIC ADVISORY COMMITTEE

Organised byWessex Institute of Technology, UK

andUniversity of New Mexico, USA

Sponsored byWIT Transactions on Engineering Sciences

J. Adilson de CastroM. Asuaje

A. DoinikovR. Groll

R. KlasincC. Koenig

N. MahinpeyJ. Mls

P. MontgomeryA. RychkovL. Skerget

R. van der SmanY. Yan

M. Zadravec

WIT Transactions

Editorial Board

Transactions Editor

Carlos BrebbiaWessex Institute of Technology

Ashurst Lodge, AshurstSouthampton SO40 7AA, UKEmail: [email protected]

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COMPUTATIONAL METHODS IN

MULTIPHASE FLOW V

EDITORS



Editors:



Published by

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British Library Cataloguing-in-Publication Data

A Catalogue record for this book is availablefrom the British Library

ISBN: 978-1-84564-188-7ISSN: 1746-4471 (print)ISSN: 1743-3533 (on-line)

The texts of the papers in this volume were set individually by the authors or under their supervision. Only minor corrections to the text may have been carried out by the publisher.

No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/ordamage to persons or property as a matter of products liability, negligence or otherwise, orfrom any use or operation of any methods, products, instructions or ideas contained in thematerial herein.

© WIT Press 2009

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All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted in any form or by any means, electronic, mechanical, photocopying,recording, or otherwise, without the prior written permission of the Publisher.

Preface

Professionals in the energy, automotive, manufacturing, construction, foodprocessing, and pharmaceuticals industries, to name a few, are frequently facedwith problems associated with multiphase flow. Scientists in environmental sciences,biology, and medicine also encounter multiphase flows regularly. Despite the effortsof several generations of fluid mechanicsists who tackled multiphase flows,including Einstein in the early 20th century, a complete description of the behaviorof multiphase materials still eludes us. Constitutive equations have been developedfor some cases, and numerical models based on these have been successful atpredicting phenomena such as particle migration in dense suspensions. Much ofthe inspiration for the development of models comes from experimental observation.In some cases, the phenomena of interest are recorded directly, while in other cases,simplified models of the flows are necessary to restrict and control the parameterspace. Direct numerical simulation is becoming increasingly useful for providinginsight to constitutive modelers, and in some cases even for calibrating models.However, it remains true that very specialized models are developed for individualareas.

In the first part of this book, several examples of modeling approaches, each wellsuited to a particular application, are grouped together, to present a view of thebreadth of the field. The second part of the book contains papers describingexperimental observation of multiphase flows at scales ranging from rivers andlakes to laboratory experiments. Topics include turbulence, interfaces and fluidizedbeds. Experimental techniques, such as PIV and other imaging methods, are alsorepresented.

We are confident that conference participants will find a fruitful exchange of ideas,and that readers of this book will find many insights. The contents of this bookreflect the quality of the submissions and the diligence of the reviewers, whom wewish to thank.

The EditorsNew Forest, 2009

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Contents

Section 1: Multiphase flow simulation 3D Eulerian simulation of a gas-solid bubbling fluidized bed: assessment of drag coefficient correlations E. Esmaili & N. Mahinpey ................................................................................... 3 Computational and experimental methods for the on-line measurement of the apparent viscosity of a crystal suspension C. Herman, F. Debaste, V. Halloin, T. Leyssens, A. Line & B. Haut ................ 15 Two-phase flow modelling within expansion and contraction singularities V. G. Kourakos, P. Rambaud, S. Chabane, D. Pierrat & J. M. Buchlin............ 27 Numerical simulation of gas-solid flow in ducts by CFD techniques R. K. Decker, D. Noriler, H. F. Meier & M. Mori ............................................. 45 Modelling of solidification of binary fluids with non-linear viscosity models T. Wacławczyk, D. Sternel & M. Schäfer........................................................... 55 A simulation of the coupled problem of magnetohydrodynamics and a free surface for liquid metals S. Golak & R. Przyłucki ..................................................................................... 67 Ammonia concentration analysis for the steam condenser by combining two phase flow CFD simulation with condensation and process simulation K. Karube, M. Maekawa, S. Lo & K. Mimura ................................................... 77

An approach to the prediction of wax and asphaltene deposition in a pipeline based on Couette device experimental data D. Eskin, J. Ratulowski, K. Akbarzadeh & T. Lindvig ....................................... 85 Large amplitude waves in a slug tracking scheme A. De Leebeeck & O. J. Nydal ........................................................................... 99 Numerical simulation of an intermediate sized bubble rising in a vertical pipe J. Hua, S. Quan & J. Nossen ........................................................................... 111 Modelling of stratified two phase flows using an interfacial area density model T. Höhne & C. Vallée ...................................................................................... 123 Multi-phase mixture modelling of nucleate boiling applied to engine coolant flows V. Pržulj & M. Shala........................................................................................ 135 On the application of Mesoscopic Eulerian Formalism to modulation of turbulence by solid phase Z. Zeren & B. Bédat......................................................................................... 147 VOF-based simulation of conjugate mass transfer from freely moving fluid particles A. Alke, D. Bothe, M. Kroeger, & H.-J.Warnecke ........................................... 157 Computational fluid dynamic as a feature to understand the heat and mass transfer in a vacuum tower K. Ropelato, A. V. Castro, W. O. Geraldelli & M. Mori.................................. 169 Understanding segregation and mixing effects in a riser using the quadrature method of moments A. Dutta, J. Raeckelboom, G. J. Heynderickx & G. B. Marin.......................... 181 Numerical simulation of heavy oil flows in pipes using the core-annular flow technique K. C. O. Crivelaro, Y. T. Damacena, T. H. F. Andrade, A. G. B. Lima & S. R. Farias Neto................................................................... 193 Simulation of flow and modelling the residence time distribution in a continuous two impinging liquid-liquid streams reactor using the Monte Carlo Technique M. Sohrabi & E. Rajaie ................................................................................... 205

Section 2: Interaction of gas, liquids and solids Vortex study on a hydraulic model of Godar-e-Landar Dam and Hydropower Plant R. Roshan, H. Sarkardeh & A. R. Zarrati ........................................................ 217 Minimum fluidization velocity, bubble behaviour and pressure drop in fluidized beds with a range of particle sizes B. M. Halvorsen & B. Arvoh............................................................................ 227 Section 3: Turbulent flow Mathematical modelling on particle diffusion in fluidised beds and dense turbulent two-phase flows R. Groll ............................................................................................................ 241 A numerical study of the scale effects affecting the evolution and sediment entrainment capacity of a gravity current, propagating over a loose bed containing large-scale roughness elements T. Tokyay & G. Constantinescu ....................................................................... 251 Effect of relative motion between bubbles and surrounding liquid on the Reynolds stress as a mechanism controlling the radial gas holdup distribution K. Ueyama ....................................................................................................... 263 Velocity and turbulence measurements of oil-water flow in horizontal and slightly inclined pipes using PIV W. A. S. Kumara, B. M. Halvorsen & M. C. Melaaen ..................................... 277 Section 4: Environmental multiphase flow Meandering of a particle-laden rivulet P. Vorobieff, A. Mammoli, J. Coonrod, V. Putkaradze & K. Mertens ............. 295 Experimental study on the rheological behaviour of debris flow material in the Campania region A. Scotto di Santolo, A. M. Pellegrino & A. Evangelista ................................. 305 Experimental and numerical investigation of mixed flow in a gallery S. Erpicum, F. Kerger, P. Archambeau, B. J. Dewals & M. Pirotton.............. 317

Sediment transport via dam-break flows over sloping erodible beds M. Emmett & T. B. Moodie.............................................................................. 329 Section 5: Bubble and drop dynamics Hydrodynamic drag and velocity of micro-bubbles in dilute paper machine suspensions A. Haapala, M. Honkanen, H. Liimatainen, T. Stoor & J. Niinimäki .............. 343 Effects of physical properties on the behaviour of Taylor bubbles V. Hernandez-Perez, L. A. Abdulkareem & B. J. Azzopardi............................ 355 Numerical simulation of one-dimensional mixed flow with air/water interaction F. Kerger, S. Erpicum, P. Archambeau, B. J. Dewals & M. Pirotton.............. 367 Geometry effects on the interaction of two equal-sized drops in simple shear flow at finite Reynolds numbers S. Mortazavi & M. Bayareh ............................................................................. 379 Section 6: Flow in porous media Modelling the tide effects in groundwater J. Mls ............................................................................................................... 391 Modelling of diffusion in porous structures E. du Plessis & S. Woudberg ........................................................................... 399 Measurement and prediction for air flow drag in different packing materials C. Rautenbach, B. M. Halvorsen, E. du Plessis, S. Woudberg & J. P. du Plessis ....................................................................... 409 CFD simulation with multiphase flows in porous media and open mineral storage pile S. Torno, J. Toraño, I. Diego, M. Menéndez, M. Gent & J. Velasco ............... 421 Powered addition applied to the fluidisation of a packed bed P. D. de Wet, B. M. Halvorsen & J. P. du Plessis............................................ 431 Section 7: Heat transfer Tube bundle’s cooling by aqueous foam J. Gylys, S. Sinkunas, T. Zdankus, M. Gylys & R. Maladauskas ..................... 445

Desulfurization of heavy crude oil by microwave irradiation A. Miadonye, S. Snow, D. J. G. Irwin, M. Rashid Khan & A. J. Britten .......... 455 Section 8: Image processing Reconstruction of a three-dimensional bubble surface from high-speed orthogonal imaging of dilute bubbly flow M. Honkanen ................................................................................................... 469 Experimental investigation on air entrainment below impinging jets by means of video observations and image processing D. V. Danciu, M. J. da Silva, M. Schmidtke, D. Lucas & U. Hampel.............. 481 Section 9: Interfacial behaviour LBM simulation of interfacial behaviour of bubbles flow at low Reynolds number in a square microchannel Y. Y. Yan & Y. Q. Zu ........................................................................................ 495 Experimental investigation of a 2d impinging jet on a liquid surface R. Berger, S. Depardon, P. Rambaud & J. M. Buchlin.................................... 507 Author Index .................................................................................................. 521

Section 1 Multiphase flow simulation

3D Eulerian simulation of a gas-solid bubbling fluidized bed: assessment of drag coefficient correlations

E. Esmaili & N. Mahinpey

Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, Canada

Abstract

Fluidized beds have been widely used in power generation and in the chemical, biochemical, and petroleum industries. The 3D simulation of commercial scale fluidized beds has been computationally impractical due to the required memory and processor speeds. However, in this study, 3D Computational Fluid Dynamics simulation of a gas-solid bubbling fluidized bed is performed to investigate the effect of using different inter-phase drag models. The drag correlations of Syamlal–O’Brien, Gidaspow, and Wen–Yu are reviewed using a multiphase Eulerian–Eulerian model to simulate the momentum transfer between phases. Comparisons are made with both a 2D Cartesian simulation and experimental data. The experiments are performed on a Plexiglas rectangular fluidized bed consisting of spherical glass beads and ambient air as the gas phase. The aim of this work is to present an optimum drag model to simulate the momentum transfer between phases and compare the results using 3D versus 2D simulation of gas-solid bubbling fluidized beds. Comparisons were made based on solid volume fractions, expansion height, and pressure drop inside the fluidized bed at different superficial gas velocities. The results were found to agree well with experimental data. Keywords: multiphase flow, fluidized bed, computational fluid dynamics, interphase drag model.

1 Introduction

Two approaches are typically used for Computational Fluid Dynamics (CFD) modeling of gas–solid fluidized beds. The first one is Lagrangian–Eulerian

© 2009 WIT PressWIT Transactions on Engineering Sciences, Vol 63, www.witpress.com, ISSN 1743-3533 (on-line) doi:10.2495/MPF090011

Computational Methods in Multiphase Flow V 3

modeling [1,2,7,10–12], which solves the equations of motion for each particle individually and uses a continuous interpenetrating model (Eulerian framework) for modeling the gas phase. As a consequence, the Lagrangian–Eulerian model requires large computational resources for large systems of particles. The second method is Eulerian–Eulerian modeling [4,9,12–16], which assumes both phases can be considered as fluid and also considers the interpenetrating effect of each phase by using drag models. Therefore, applying a proper drag model in Eulerian–Eulerian modeling is of a great importance. Many researchers have applied 2D Cartesian simulations to model pseudo-2D beds [1,4,6,15]. Peirano et al. [5] have investigated the importance of three dimensionality in the Eulerian approach simulations of stationary bubbling fluidized beds. The results of their simulations show that two-dimensional simulations should be used with caution and only for sensitivity analysis, whereas three-dimensional simulations are able to reproduce both the statics (bed height and spatial distribution of particles) and the dynamics (power spectrum of pressure fluctuations) of the bed. In addition, they believe that the issue of accurate prediction of the drag force (the force exerted by the gas on a single particle in a suspension) does not seem to be the most important problem when dealing with bubbling beds where accurate empirical correlations are available. In contrast, in the present work, it is found that using a proper drag model can increase the accuracy of results in 3D simulation of bubbling fluidized beds. Goldschmidt et al. [6] applied a two-dimensional multi-fluid Eulerian CFD model to study the influence of the coefficient of restitution on the hydrodynamics of dense gas-solid fluidized beds. They showed that, in order to obtain realistic bed dynamics from fundamental hydrodynamic models, it is of prime importance to correctly take the effect of energy dissipation due to non-ideal particle-particle encounters into account. van Wachem et al. [8] implemented a CFD model for a free bubbling fluidized bed in the commercial code CFX of AEA Technology to verify experimentally Eulerian-Eulerian gas-solid model simulations of bubbling fluidized beds with existing correlations for bubble size and bubble velocity. They concluded that smaller bubbles and a lower rise velocity are usually observed with 2D beds. Cammarata et al. [9] compared the bubbling behavior predicted by 2D and 3D simulations of a rectangular fluidized bed using commercial software, CFX. The bed expansion, bubble hold-up, and bubble size calculated from the 2D and 3D simulations were compared with the predictions obtained from the Darton equation [3]. A more realistic physical behavior model of fluidization was obtained using 3D simulations. They also indicate that 2D simulations could be used to conduct sensitivity analyses. Xie et al. [14] compared the results of 2D and 3D simulation of slugging, bubbling, and turbulent gas-solid fluidized beds. They also investigated the effect of using different coordinate systems. Their results show that there is a significant difference between 2D and 3D simulations, and only 3D simulations could predict the correct bed height and pressure spectra. Behjat et al. [15] applied a two dimensional CFD technique in order to investigate hydrodynamic and heat transfer phenomena. They conclude that a Eulerian-Eulerian model is

© 2009 WIT PressWIT Transactions on Engineering Sciences, Vol 63, www.witpress.com, ISSN 1743-3533 (on-line)

4 Computational Methods in Multiphase Flow V

suitable for modeling industrial fluidized bed reactors. Their results indicate that considering two solid phases, particles with smaller diameters have lower volume fraction at the bottom of the bed and higher volume fraction at the top of the bed and also that the gas temperature increases as it moves upward in the reactor due to the heat of polymerization reaction leading to the higher temperatures at the top of the bed. Li et al. [16] conducted a three-dimensional numerical simulation of a single horizontal gas jet into a laboratory-scale cylindrical gas–solid fluidized bed. They proposed a scaled drag model and implemented it into the simulation of a fluidized bed of FCC particles. They also obtained the jet penetration lengths of different jet velocities and compared them with published experimental data, as well as with predictions of empirical correlations. No previous works in the literature have investigated the effect of using different drag models in 3D simulation of fluidized beds to present an optimum drag model for simulation of bubbling gas-solid fluidized beds. In this respect, the underlying objective of this study is to present an optimum drag model to simulate the momentum transfer between phases and compare the results using 3D versus 2D simulation of gas-solid bubbling fluidized beds.

Figure 1: Geometry of 3D Plexiglas fluidized bed.

2 Experimental setup

Experiments were carried out in the Department of Chemical and Biological Engineering at the University of British Columbia. The Column is a psudo-2D Plexiglas of 1.2 m height, 0.28 m width, and 0.025 m thickness. Spherical glass beads of 250–300 μm diameter and density 2500 kg/m3 were fluidized with air at ambient conditions. Pressure drops were measured using three differential pressure transducers located at elevation 0.03, 0.3, and 0.6 m above the gas distributor, respectively. The static bed height of 0.4 m with a solid volume fraction of 0.6 was used in all the experiments. Pressure drop and bed expansion were monitored at different superficial gas velocities ranging from 0 to 0.8 (m/s).



3 Governing equations

Table 1.

4 Numerical simulation

Governing equations of mass and momentum conservation are solved using finite volume method employing the Semi Implicit Method for Pressure Linked Equations (PC-SIMPLE) algorithm, which is an extension of the SIMPLE algorithm to multiphase flow. A multi fluid Eulerian-Eulerian model, which considers the conservation of mass and momentum for the gas and solid phases, was applied. The kinetic theory of granular flow, which considers the conservation of solid fluctuation energy, was used for closure of the solids stress terms. The three-dimensional (3D) geometry has been meshed using 336,000 structured rectangular cells. Volume fraction, density, and pressure are stored at the main grid points that are placed in the center of each control volume. A staggered grid arrangement is used, and the velocity components are solved at the control volume surfaces. A pressure correction equation is built based on total volume continuity. Pressure and velocities are then corrected so as to satisfy the continuity constraint. A grid sensitivity analysis is performed using different mesh sizes and 2 mm mesh interval spacing was chosen for all the simulation runs. Second-order upwind discretization schemes were used for discretizing the governing equations. Based on the estimation of the truncation error, an adaptive time-stepping algorithm with 100 iterations per each time step and a minimum value of order 10-5 for the lower domain of time step was used to ensure a stable convergence. The convergence criteria for other residual components associated with the relative error between two successive iterations has been specified in the order of 10-5. Three different drag models are studied in this work to simulate the momentum transfer between phases (Gidaspow, Syamlal–O’Brien, and Wen–Yu.). FLUENT 6.3 on a 20 AMD/Opteron 64bit processor Sun Grid Microsystems workstation W2100Z with 4 GB RAM is employed to solve the governing equations. Computational model parameters are listed in Table.2.

5 Results and discussion

Simulation results were compared with the experimental data in order to validate the model. Figure 2 shows the time average pressure drop inside the bed between two specific elevations (i.e. 0.03 m and 0.3 m as demonstrated in Fig. 1) for different studied cases and experimental results. In order to calculate the pressure at each pressure sensor (i.e. y=0.03 m), two kinds of averaging have been applied. The first one is the spatial averaging, which is the average value of pressure for all nodes in the plane of first pressure sensor (plane y=0.03). The second one is the time averaging of spatial-averaged pressure values in the period of 3-10 sec real time. As indicated in Fig. 2, the pressure drop for all the



The proposed model’s equations, which are solved numerically, are presented in

Table 1: Governing equations.

Continuity:

. 0

. 0

Momentum equations:

. .

. .

Solid pressure 1 2 1

Radial distribution function

1,

Granular temperature

2 1

√1

2√

3 31 1 3 1

8

5√1

12 1

√

Stress tensors

,

. .

. .

. .

45

1

5√48 √

11 1



Table 1: Continued.

43

1

Inter-phase momentum exchange Gidaspow [17]

150 . .

. 0.2

24.

1 0.15 ..

. 1000

0.44 . 1000

Syamlal and O’Brien [18]

34

0.63 4.8

0.5 0.06 0.06 0.12 2 .

. 0.85

0.85

0.8 2.65

Wen Yu [19]

34

.

24.

1 0.15 ..

. 1000

0.44 . 1000



Table 2: Computational model parameters.

Parameter Value Particle density 2500 ⁄ Gas density 1.225 ⁄ Mean particle diameter 275Initial solid packing 0.6Superficial gas velocity 11.7,21,38,46 ⁄ Bed dimension 0.28 1.2 0.025 Static bed height 0.4 Grid interval spacing 0.002 Inlet boundary condition type Outlet boundary condition type

Under-relaxation factors

Pressure 0.6 Momentum 0.4 Volume fraction 0.3 Granular temperature 0.2

Figure 2: Pressure drop inside the bed mZmZ PPP 3.003.01 .

models showed a declining trend with increase of the superficial gas velocity, which is in good qualitative agreement with the experimental data. It can be easily seen that 3D simulations show their superiority in predicting the pressure drop inside the bed compared with 2D simulation. Also, it can be concluded that the Gidaspow drag model in a 3D simulation will give results closer to the experimental data.



The experimental data of the time-average bed expansion ratio were compared with corresponding values predicted by the model, using Syamlal–O'Brien, Gidaspow, and Wen–Yu drag functions for various velocities as depicted in Figure 3. All drag models demonstrate a consistent increase in bed expansion with gas velocity and predict the bed expansion reasonably well. Figure 3 shows the considerable relative increase in bed expansion as the fluidizing velocity increases; a 5% increase was obtained at 0.11 m/s, a 20% increase at 0.21 m/s, and 42% at 0.38 m/s, and up to a 50% increase in bed height was measured at 0.46 m/s, the highest fluidized velocity investigated. It can be seen that using a 3D simulation, especially for lower superficial gas velocities, will increase the accuracy of the results. The reason can be the effect of participating governing equations in the z direction (depth of the bed) in simulating the fluid flow when the gas velocity increases. It is also seen that using the Gidaspow drag model in 3D simulation of a gas-solid fluidized bed will give better results for predicting the bed expansion ratio than the other two drag models.

Figure 3: Comparison of simulated bed expansion ratio with experimental data.

Figure 4 shows a contour plot of solid volume fraction for the three drag models studied in this work for a superficial gas velocity of 0.38 m/sec at 10 sec real-time simulations. As can be observed from the plots, the Syamlal model represents the lowest bed expansion and gas void fraction. This fact could have been foreseen from the minimum fluidization velocity prediction of this model, which is almost five times larger than experimental data reported in the literature. The rest of the models showed approximately the same range of bed expansion. Expansion of the bed started with the formation of bubbles for all the



Figure 4: Contours of solid volume fraction at t=10s and u=0.38m/s (a) Syamlal (b) Wen Yu (c) Gidaspow.

models and eventually reached a statistically steady-state bed height. After this point, an unsteady chaotic generation of bubbles was observed after almost 3 seconds of real time simulation.

6 Conclusion

Numerical simulation of a bubbling gas-solid fluidized bed have been performed in a three dimensional solution domain using the Eulerian-Eulerian approach to investigate the effect of using three dimensional analysis versus two dimensional simulation of fluidized beds. FLUENT 6.3 was used to perform the calculations. The results show that although three-dimensional simulation takes more time and computing processors than two-dimensional simulation, it gives more accurate results when the models are compared with experimental data. Also, a comparison between three common drag models, Syamlal, Wen Yu, Gidaspow, was performed to develop an optimized drag model for simulation of momentum transfer between phases in a 3D bubbling gas-solid fluidized bed. It is concluded that the Gidaspow drag model achieved better results in predicting the bed expansion ratio and pressure drop inside the bed than Syamlal drag correlations. However, further modeling efforts are required to study the influence of using other drag models, which have not been studied, and optimizing existing drag models based on minimum fluidization velocity in three dimensional simulations will be performed in future work.

Nomenclature

Single particle drag function, dimensionless Rate of strain tensor

(b)(a) (c)



Solid diameter, m Restitution coefficient, dimensionless Acceleration due to gravity, ⁄ Radial distribution coefficient, dimensionless Gas/solid momentum exchange coefficient, dimensionless

Pressure, Pa Solid pressure, Pa Velocity, m/s Reynolds number, dimensionless

Greek symbols

Density, ⁄ Granular temperature, ⁄

Stress tensor, Pa Shear viscosity, ⁄ Bulk viscosity, ⁄ Volume fraction, dimensionless

Subscripts

Gas Solid

References

[1] B.P.B. Hoomans, J.A.M. Kuipers, W.J. Briels, V.W.P.M. Swaaij, Discrete particle simulation of bubble and slug formulation in a two-dimensional gas-fluidized bed: a hard-sphere approach, Chem. Eng. Sci. 51 (1996) 99.

[2] B. Xu, A. Yu, Numerical simulation of the gas–solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics, Chem. Eng. Sci. 52 (1997) 2785.

[3] R.C. Darton, R.D. LaNauZe, J.F. Davidson, D. Harrison, Bubble growth due to coalescence in fluidized beds, Trans. Inst. Chem. Eng. 55 (1977) 274.

[4] B.G.M. van Wachem, J.C. Schouterf, R. Krishnab, and C.M. van den Bleek, Eulerian Simulations of Bubbling Behavior in Gas-Solid Fluidized Beds, Computers chem. Engng Vol. 22, Suppl., pp. S 299-S306. 1998.

[5] E. Peiranoa, V. Delloumea, B. Lecknera, Two- or three-dimensional simulations of turbulent gas–solid flows applied to fluidization, Chemical Engineering Science 56 (2001) 4787–4799.

[6] M. J. V. Goldschmidt, J. A. M. Kuipers, W. P. M. van Swaaij, Hydrodynamic modeling of dense gas-fluidized beds using the kinetic theory of granular flow: effect of coefficient of restitution on bed dynamics, Chemical Engineering Science 56 (2001) 571-578.



[7] K.D. Kafui, C. Thornton, M.J. Adams, Discrete particle-continuum fluid modeling of gas–solid fluidized beds, Chem. Eng. Sci. 57 (2002) 2395.

[8] W.B.G.W. Wachem, A.E. Almstedt, Methods for multiphase computational fluid dynamics, Chem. Eng. J. 96 (2003) 81.

[9] L. Cammarata, P. Lettieri, G.D.M. Micale, D. Colman, 2d and 3d cfd simulations of bubbling fluidized beds using Eulerian–Eulerian models, Int. J. Chem. Reactor Eng. 1 (2003) (Article A48).

[10] M.J.V. Goldschmidt, R. Beetstra, J.A.M. Kuipers, Hydrodynamic modeling of dense gas-fluidized beds: comparison and validation of 3d discrete particle and continuum models, Powder Technol. 142 (2004) 23.

[11] J.S. Curtis, V.B. Wachem, Modeling particle-laden flows: a research outlook, AIChE J. 50 (2004) 2638.

[12] M. Chiesa, V. Mathiesen, J. A. Melheim, B. Halvorsen, Numerical simulation of particulate flow by the Eulerian–Lagrangian and the Eulerian–Eulerian approach with application to a fluidized bed. Comput. Chem. Eng. 29 (2005) 291.

[13] J. Sun, F. Battaglia, Hydrodynamic modeling of particle rotation for segregation in bubbling gas-fluidized beds, Chem. Eng. Sci. 61 (2006) 1470.

[14] N. Xie, F. Battaglia, S. Pannala, Effects of using two-versus three-dimensional computational modeling of fluidized beds Part I, hydrodynamics, Powder Technology 182 (2008).

[15] Y. Behjat, S. Shahhosseini, S. H. Hashemabadi, CFD modeling of hydrodynamic and heat transfer in fluidized bed reactors, International Communications in Heat and Mass Transfer 35 (2008) 357–368.

[16] T. Li, K. Pougatch, M. Salcudean, D. Grecov, Numerical simulation of horizontal jet penetration in a three-dimensional fluidized bed, Powder Technology 184 (2008) 89–99.

[17] D. Gidaspow, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions, Academic Press, Boston, 1994.

[18] M. Syamlal, T.J. O'Brien, Computer simulation of bubbles in a fluidized bed, AIChE Symposium Series 85 (1989) 22–31.

[19] C.Y. Wen, Y.H. Yu, Mechanics of fluidization, Chemical Engineering Progress Symposium Series, 1966, pp. 100–111.



Computational and experimental methods for the on-line measurement of the apparent viscosity of a crystal suspension

C. Herman1, F. Debaste1, V. Halloin1, T. Leyssens2, A. Line3 & B. Haut1

1Transferts, Interfaces and Processes (TIPs), Chemical Engineering Unit, Université Libre de Bruxelles, Bruxelles, Belgium 2Automation and Process Control Laboratory (APC), UCB Pharma, Braine l’Alleud, Belgium 3Laboratoire de l’Ingénierie des Systèmes Biologiques et des Procédés (LISBP), Institut National des Sciences Appliquées (INSA) de Toulouse, Toulouse, France

Abstract

This paper proposes an experimental method, based on the Metzner and Otto concept, for the on-line measurement of the apparent viscosity of a crystal slurry during a crystallization process. The first step of this procedure consists of the determination of the Np–Re–Fr relation for Newtonian liquids, for two impeller-tank configurations, chosen such that this relation is a bijective one. This is achieved both experimentally and numerically, using Computational Fluid Dynamics. In the second step of the procedure, the same impeller-tank configurations are used for the determination of the evolution of the apparent viscosity of the crystal slurry (non-Newtonian liquid) during a reference pharmaceutical crystallization process (Etiracetam – UCB). The paper concludes on the influence of the Particle Size Distribution of the crystals on the apparent viscosity of the suspension. For a given crystal mass fraction, the bigger the crystals are – and then the less abundant they are – and the smaller the span is, the smaller the apparent viscosity of the suspension is. Keywords: apparent viscosity, non-Newtonian liquid, suspension, on-line measurement, Computational Fluid Dynamics, crystallization process, Metzner and Otto concept, process rheometer.



1 Introduction

The follow up of a crystallization process can be achieved using several probes. These probes often give information about the Particle Size Distribution (PSD) and form, or the solution concentration, but do not allow estimating any physical characteristics of the suspension such as its apparent viscosity. This physico-chemical parameter can be used to characterize and control a crystallization process. Very few on-line experimental techniques exist to directly determine the evolution of the apparent viscosity of a crystal slurry during a crystallization process. Moreover, very few theoretical or empirical correlations of the apparent viscosity of a suspension can be found in the literature. Actually, the correlations proposed are based on the crystal volume fraction [1]. More recent studies show the influence of the PSD on the apparent viscosity of a suspension [2, 3]. Therefore, this paper proposes an experimental method to follow the apparent viscosity of a crystal suspension during a crystallization process, based on the Metzner and Otto [4] concept. Using an agitation system as a process rheometer, the method consists of determining the apparent viscosity of a non-Newtonian liquid, as equal to the one of a Newtonian liquid whose agitation, in the same impeller-tank configuration and operational condition, leads to an equal dissipated power. As presented in Fig.1, the link between the dissipated power, P, and the apparent viscosity, µ, consists of a relation between three dimensionless numbers, the Reynolds number, Re, the Froude number, Fr, and the Power number, Np, which are respectively defined as follows:

µ

ρ 2

Re Nd= (1)

g

dNFr2

= (2)

35 NdPN p ρ

= (3)

where ρ is the density of the suspension, d the diameter of the impeller, N the rotational speed of the impeller and g the gravitational acceleration. For a given impeller-tank configuration, the first step of this procedure consists of the determination of the Np – Re – Fr relation for Newtonian liquids. It is achieved using several reference Newtonian liquids of known density, ρ, and viscosity, µ. The determination of the power dissipated in the liquid when it is agitated with an impeller of diameter d, at a fixed rotational speed, N, allows determining the three dimensionless numbers (Fig.1). These determinations are realized experimentally and numerically, using Computational Fluid Dynamics (CFD). The relation between Re, Fr and Np, in the laminar and intermediate Reynolds range, is proposed such as [1, 5]:

yxp FrcN Re= (4)



The unknown parameters, c, x and y, which depend on the impeller-tank configuration, are identified by a least square optimisation. In the turbulent Reynolds range, the Power number is constant (independent on Re and Fr).

µρ 2

Re Nd=

gdNFr

2

=

53dNPN p ρ

=

yxp FrcN Re=

µ

'PTP 'PPP T −=

N

dρ

x y

gdNc

NdP

Nd

=

2

35

2

ρ

ρµ

g 'PTP'PPP T −=

N

dρ

g

NumericallyExperimentallyDetermination of the Np – Re – Fr relationfor Newtonian liquids

Determination of the apparent viscosityof a non-Newtonian liquid (crystal slurry)

µρ 2

Re Nd=

gdNFr

2

=

53dNPN p ρ

=

yxp FrcN Re=

µ

'PTP 'PPP T −=

N

dρ

x y

gdNc

NdP

Nd

=

2

35

2

ρ

ρµ

g 'PTP'PPP T −=

N

dρ

g

NumericallyExperimentallyDetermination of the Np – Re – Fr relationfor Newtonian liquids

Determination of the apparent viscosityof a non-Newtonian liquid (crystal slurry)

Figure 1: Schematical presentation of method to determine the apparent viscosity of a suspension, based on the Metzner and Otto [4] concept.

In the second step of the procedure, the same impeller-tank configuration is used for the determination of the apparent viscosity of a non-Newtonian liquid. For this purpose, the power dissipated in the liquid is measured when it is agitated with the impeller of diameter d, rotating at a fixed rotational speed, N. Knowing the density, ρ, of the suspension, the inversion of the Np – Re – Fr relation allows determining its apparent viscosity. This method can only be used in the Reynolds range in which the Np – Re – Fr relation is a bijective one.

x y

gNdc

NdP

Nd

=

2

35

2

ρ

ρµ (5)

The method is then applied in the second part of the paper for the determination of the evolution of the apparent viscosity of the crystal slurry during the crystallization process of a reference pharmaceutical compound.



2 Materials and methods

2.1 Introduction

The experimental device consists of an unbaffled double–jacketed glass tank of 1l (Fig.2). The mixing of the liquid is ensured by a stirrer equipped with two impellers types: an Anchor or a 4-arms Flat Blade Turbine. The characteristics about the two impeller-tank configurations are reported in Tab.1.

Figure 2: Schematical presentation of the experimental installation.

Table 1: Impeller-tank configurations.

As the two impeller-tank configurations used are not standard ones, very few results about the Np – Re – Fr relation of these systems can be found in the literature. Therefore, this relation (Eq.(4)) is here determined, both experimentally and numerically, for each of the two impeller-tank configurations used.

Impeller Anchor 4-arms Flat Blade Turbine

D (cm) 10 10 H (cm) 10 10 d (cm) 8.9 7.9 b (cm) 2.5 3.1

ammeter

D

d

H

b

ammeter

D

d

H

b



2.2 Experimental determination of the Np – Re – Fr relation for Newtonian liquids

In order to determine this relation for the two impeller-tank configurations in an extended range of Reynolds number (Re), different Newtonian liquids are used:

1. In order to investigate the low Re range, a mixture of glycerine and water (97%) at different temperatures (10°C, 15°C, 20°C, 25°C, 30°C, 35°C, 40°C, 50°C and 60°C) is used. Several rotational speeds (100 rpm, 200 rpm, 300 rpm, 400 rpm and 500 rpm) are used. The density and the viscosity of the mixture are determined experimentally with a falling sphere viscosimeter (Tab.2).

2. A mixture of salt and water (300 g of salt per kg of water) at different temperatures (5°C, 10°C, 15°C and 20°C) is used to investigate the intermediate Re range. 300 rpm is used as the rotational speed of the impeller device. The density and the viscosity of the mixture are provided by Solvay S.A. (Tab.2).

3. Information for the high Re range is obtained using the methanol solvent at ambient temperature (20°C). The rotational speeds used extend between 100 rpm and 450 rpm by 50 rpm steps. The density and the viscosity of methanol are obtained from literature (Tab.2).

For each experiment, the tank is filled with 10 cm of the reference Newtonian liquid. The power dissipated in the liquid is determined using an on-line ammeter (APPA 350) (Fig.2). The dissipated power is obtained deducing the value measured at no load (P’) from the observed total power (PT) value (Fig.1).

2.3 Numerically determination of the Np – Re – Fr relation for Newtonian liquid in the turbulent range

The CFD simulations are realized with the commercial codes Gambit 2.3 and Fluent 6.3. The geometries of the two impeller-tank configurations are created. They are meshed with approximately 500.000 tetrahedral elements: the size of the elements in the rotor zone and the stator zone are 1.5 mm and 3 mm, respectively. The Sliding Meshes (SM) model is preferred to the Moving Reference Frame (MRF) model as it has been shown that it determines the dissipated power with more accuracy. As the tank is unbaffled, in order to take up the effect of the free surface deformation, the Volume Of Fluid (VOF) multiphase approach is selected. The standard k-ε approach is selected to model the turbulence. The tank is filled with 10 cm of methanol. The first order-upwind discretization scheme is selected for the momentum, the turbulent kinetic energy and the turbulent dissipation range. The discretization scheme selected for the pressure and the volume fraction are the PRESTO! and the geo-reconstruct, respectively. The pressure-velocity coupling scheme recommended for this kind of simulation is the PISO. In order to reach the convergence and conserve a courant number inferior to 0.25, the time-step used is 5.10-5 sec. For each simulation, the dissipated power is determined by calculating the total moment forces on the impeller shaft.



Table 2: Density and viscosity of the reference Newtonian liquids used.

Substance Temperature °C

Density, ρ kg/m3

Viscosity, µ mPa.s

Glycerine – Water 10 1230 1372 Glycerine – Water 15 1230 1015 Glycerine – Water 20 1230 751 Glycerine – Water 25 1220 556 Glycerine – Water 30 1220 411 Glycerine – Water 35 1220 304 Glycerine – Water 40 1220 225 Glycerine – Water 50 1210 123 Glycerine – Water 60 1210 67

Salt – Water 5 1239 3.09 Salt – Water 10 1237 2.01 Salt – Water 15 1234 2.32 Salt – Water 20 1231 1.53

Methanol 20 780 0.55


The experimental results of the Power number as a function of the Reynolds number and the Froude number are obtained for the two impeller-tank configurations in the extended Reynolds number range. The results indicate that there is not a significant influence of the Froude number on the Power number. Therefore, in the intermediate range, the relation links the Power number to the Reynolds number only. The least square optimisation in the intermediate range leads to two relations. The first one is valid for Re < 500, and the second one for 500 < Re < 104 – 105. The Power number in the turbulent range (Re > 104 – 105) is constant. The relations Np – Re are reported in Tab.3.

Table 3: Np –Re – Fr relation for the two impeller-tank configurations.

Impeller Range c x Anchor Intermediate (Re < 500) 0.499 -0.541 Intermediate (500 < Re < 104 – 105) 83.811 -1.369 Turbulent ( Re > 104 – 105) 0.0025 0 4-arms FBT Intermediate (Re < 500) 0.465 -0.465 Intermediate (500 < Re < 104 – 105) 58.343 -1.275 Turbulent ( Re > 104 – 105) 0.005 0

The experimental results of the Power number as a function of the Reynolds number are presented in Fig.3 for the 4-arms Flat Blade Turbine. The curve adjusting on the experimental results related to the Anchor is also presented. It can be observed that the curves of both impeller-tank configurations in the intermediate range are quite similar. However, as it can also be seen in the



literature [5], the Power number in the turbulent range is bigger for the 4-arms Flat Blade Turbine than for the Anchor.

Figure 3: Power number as a function of the Reynolds number for the two impeller-tank configurations.

Fig.4 (left) presents the comparison of the Power number as a function of the Reynolds number in the turbulent range obtained both experimentally and numerically by CFD. Fig.4 (right) shows that at least 5 revolutions of the impeller are required to reach the numerical stationary conditions.

Figure 4: (Left) Comparison between the experimental and the numerical results of the Np vs. Re in the turbulent range. (Right) Numerical evolution of the Power number with the increase of the number of the rotational tours.

The Power numbers obtained are smaller that those generally found in the literature. Indeed, the important deformation of the free surface leads to a limitation of the dissipated power [5]. Moreover, with these impeller-tank configurations, the turbulent range is reduced. This characteristic is useful for the

1,E-03

1,E-02

1,E-01

1,E+00

1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06

Reynolds Number

Pow

er N

umbe

r

Glycerine-Water Salt-Water Methanol Num

Anchor

0

0,002

0,004

0,006

0,008

0 2 4 6 8 10Number of rotational tours

Pow

er n

umbe

r

4-arms Flat Blade Turbine - 150 rpm Anchor -350 rpm

1,E-03

1,E-02

1,E+04 1,E+05 1,E+06Reynolds Number

Pow

er N

umbe

r

Salt-Water - Turbine Methanol - Turbine Num -TurbineSalt-Water - Anchor Methanol - Anchor Num - Anchor



method proposed. Indeed, Eq.(5) can be used to determine the apparent viscosity of a suspension assuming that the relation between the Power number and the Reynolds number is a bijective one.

4 Application

The method is then applied for the determination of the evolution of the apparent viscosity of the crystal slurry during a crystallization process. The process chosen consists of the re-crystallization of the Etiracetam product (racemic compound (R-S product)): an intermediate product of the Levetiracetam (UCB Keppra). In this crystallization process, the drug can crystallize into two crystallographic forms, called morph I and morph II. The morph I is the stable crystallographic form below the transition temperature (30°C) while the morph II is the stable crystallographic form beyond this temperature. The reference pharmaceutical crude compound is initially dissolved (0.6 gproduct/gsolution) in the methanol solvent at high temperature (~ 60°C) and stirred in a tank. The cooling of this solution (15.5°C/h, started at time t = 0) induces the crystallization of both morphs. Nevertheless, the non-desired crystallographic form (morph II) nucleation kinetics is dominant. The massive primary nucleation of morph II is observed at the so-called induction time t = t1 by the first exothermic peak on the temperature curve (curve 1) in Fig.5. In order to obtain the pharmaceutical desired crystallographic form (morph I), the temperature is further lowered and the system is kept at a constant ripening temperature (Trip). At the so called latency time t = t2, a polymorphic transition from the unstable morph II to the stable morph I is observed. The transition is characterized by a second exothermic peak on the temperature curve (curve 1) in Fig.5. This transition, mediated by the solvent, is controlled by the morph I nucleation.

Figure 5: Evolution of the temperature (curve 1) and the power dissipated (curve 2) in the suspension during the polymorphic crystallization process.

1 2 3 … … … Time (h)

Trip

60

Temperature (°C)

10

Power (W)

0

phase a phase b phase c

t1t2 = latency time

1

2

1 2 3 … … … Time (h)

Trip

60

Temperature (°C)

10

Power (W)

0

phase a phase b phase c

t1t2 = latency time

1

2



In order to determine the evolution of the apparent viscosity of the suspension during the crystallization process (curve 2 in Fig.5), it is realized in the impeller-tank configurations presented in Tab.1 (H = 10 cm). Several rotational speeds and ripening temperatures are investigated in order to discuss the evolution of the apparent viscosity of the suspension during the crystallization process as a function of the operational conditions. On the one hand, the influence of the mixing is investigated. For this purpose, two rotational speeds are investigated for both configurations (250 rpm and 350 rpm) fixing the ripening temperature equal to -2°C. On the other hand, the influence of the ripening temperature is investigated. For this purpose, three temperatures are investigated (-10°C, -2°C and 10°C) fixing the agitation (Anchor – 250 rpm). The power dissipated in the suspension during the crystallization process (curve 2 in Fig.5) is recorded for each experimental condition. The evolution of the apparent viscosity of the suspension is determined using the method proposed (Eq.(5)) assuming that the density of the suspension is constant (960 kg/m3). For each experiment, the morph II and morph I PSD, recorded at the end of the phase a and c, respectively, are off-line determined on dry powder with a MasterSizer. The average particle size (D[4,3]), the median (D(v,0.5)) and the span distribution ([D(v,0.9) – D(v,0.1)] / D(v,0.5)) are reported in this paper.

4.1 Influence of the Particle Size Distribution of the crystals of morph I and morph II

Fig.6 (left) presents the dissipated power measured during the crystallization process using the Anchor rotating at 350 rpm. The evolution of the apparent viscosity of the suspension, determined by the method, is also presented. The curves present a drastic change between the end of phase a and the end of phase c. In both cases, there are approximately 50% in mass of solid crystals in the suspension. Therefore, this change may be explained by the difference in the PSD and number of morph II and morph I crystals in suspension. At the end of the phase a, the density and the size of the morph II solid particles are large and they quickly settle down after stopping the rotation of the impeller. In this case, the flow is in a intermediate state and the suspension is a Newtonian liquid. At the end of the phase c, the density and the size of the morph I solid particles are smaller than those of the morph II crystals (Fig.6 (right)) and the amount of solid suspended is more abundant, so that the particles do not settle down quickly. In this case, the flow is in a laminar state, and the suspension is a non-Newtonian liquid. The analysis of both Fig.6 seems to indicate that, for a same crystal mass fraction in the suspension, the smaller the median is (and then the smaller and the more abundant the crystals are) and the bigger the span is, the bigger the apparent viscosity of the suspension is.



0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

50

,00

0,7

2

1,4

3

2,1

5

2,8

7

3,5

8

4,3

0

5,0

2

5,7

3

6,4

5

7,1

7

7,8

8

8,6

0

9,3

2

Time (h)

Dis

sip

ate

d P

ow

er

(W)

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

Ap

pare

nt

vis

co

sit

y (

Pa

.s)

Dissipated power Apparent viscosity

Latency time

Induction time

phase a phase cphase b

0

50

100

150

200

250

300

350

µm

D[4,3] 304,23 85,77

D(v,0,5) 274,17 39,60

Span * 10 20,82 59,47

morph II morph I

Figure 6: (Left) Evolution of the dissipated power and the apparent viscosity of the suspension (Anchor – 350rpm). (Right) Average morph II (end of phase a) and morph I (end of phase c) PSD.

4.2 Influence of the mixing

The influence of the mixing on the evolution of the apparent viscosity of the crystal slurry is investigated by 4 experimental conditions. Both impeller-tank configurations (Tab.1) are used with two rotational speeds (250 rpm and 350 rpm). Fig.7 (left) presents the apparent viscosity at the end of the phase a (morph II) and at the end of the phase c (morph I) for the four mixing experimental conditions. For each of them, the total mass of morph II crystals and morph I crystals at the end of the phase a and c, respectively, is the same. Moreover, they are approximately 50% in mass of crystals in the suspension. It can be observed that the apparent viscosity at the end of each phase depends on the impeller-tank configuration but is quite independent on the rotational speed investigated. These results can be discussed with the results of the morph I PSD (Fig.7 (right)) for the four experimental conditions. It can be seen that, for the two impeller-tank configurations, D[4,3] and D(v,0.5) are reduced when the rotational speed is increased while the span does not present a significant change. Fig.7 also indicate that, for a given rotational speed, the morph I PSD related to the Turbine is more extended (bigger span) than those related to the Anchor while they present a same median. This induces a bigger heterogeneity of the particle size of the morph I crystals in the suspension. As previously, this may explain the difference in the apparent viscosity of the suspension obtained when agitating with the Anchor or the Turbine. For a same crystal mass fraction and a same median, the bigger the span is, the bigger the apparent viscosity of the suspension is.



0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

Anchor -

250rpm

Anchor -

350rpm

Turbine -

250rpm

Turbine -

350rpm

Experimental conditions

Ap

pa

ren

t vis

co

sit

y (

Pa.s

)

End phase a - morph II End phase c - morph I

0

10

20

30

40

50

60

70

80

90

100

µm

D[4,3] Anchor 90,07 77,25

D(v,0,5) Anchor 43,40 38,11

Span * 10 Anchor 54,65 52,15

D[4,3] Turbine 99,98 92,32

D(v;0,5) Turbine 44,71 37,50

Span * 10 Turbine 63,15 72,11

250 rpm 350 rpm

Figure 7: For the four experimental conditions investigated. (Left) Apparent viscosity at the end of phase a and phase c. (Right) Morph I PSD.

4.3 Influence of the ripening temperature

Three ripening temperatures are investigated in order to study their influence on the time evolution of the apparent viscosity of the crystal slurry. Fig.8 (left) presents the dissipated power in the suspension at the end of the phase a and at the end of the phase c for the three ripening temperatures investigated. Fig.8 (right) present the morph I PSD at the end of the phase c.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

-10 -2 10

Ripening Temperature (°C)

Ap

pare

nt

vis

co

sit

y (

Pa.s

)

End phase a - morph II End phase c - morph I

0

10

20

30

40

50

60

70

80

90

µm

D[4,3] 88,35 85,77 80,02

D(v,0,5) 32,56 39,60 59,30

Span * 10 83,16 59,47 29,43

-10 °C -2 °C 10 °C

Figure 8: For the three experimental conditions investigated. (Left) Apparent viscosity at the end of phase a and phase c. (Right) Morph I PSD.

As previously, these Fig.8 allow one to conclude that the bigger the crystals are and the smaller the span is, the smaller the apparent viscosity of the suspension is. Moreover, when reducing the ripening temperature from 10°C to -10°C, as the solubility decreases, the mass and number of the crystals in suspension increase leading to an augmentation of the apparent viscosity .



5 Conclusion

This paper presents an experimental method to follow the apparent viscosity of a crystal suspension during a crystallization process, based on the Metzner and Otto concept. The first part of this paper focuses on the experimental and numerical (by CFD) determination of the Np – Re – Fr relation for Newtonian liquid for two impeller-tank configurations (unbaffled tank, Anchor or 4-arms Flat Blade Turbine). These one are chosen in order to obtain a bijective relation between the Power number and the Reynolds number in an extended Reynolds range. The method proposed is then described and applied to on reference pharmaceutical crystallization process. In order to analyse the influence of the solid phase characteristics on the apparent viscosity of the suspension, several operational conditions (mixing and temperature) are investigated. The paper concludes on the influence of the Particle Size Distribution of the crystals on the apparent viscosity of the suspension. For a given crystal mass fraction, the bigger the crystals are - and then the less abundant they are - and/or the smaller the span is, the smaller the apparent viscosity of the suspension is.

Acknowledgements

Christelle Herman acknowledges the Fonds National de la Recherche Scientifique (FNRS), UCB Pharma and the Hubert Currien (Tournesol) partnership for their financial support.

References

[1] Mullin, J.W., Crystallization, Elsevier, Oxford, UK, 2001 [2] Roscoe, R., The viscosity of suspension of rigid spheres, J. Appl. Phys., 3,

pp. 267-269, 1952 [3] Snabre, P. and Mills, P., Rheology of concentrated suspensions of

viscoelastic particles, Colloïd and Surfaces A., Physicochemical and engineering aspects, 152(1-2), pp. 79-88, 1999

[4] Metzner, A.B. and Otto, R.E., Agitation of Non-Newtonian Fluids, AIChE Journal, 3(1), pp. 3-10, 1957

[5] Nagata, S., Mixing: Principles and applications, A. Halsted press book, Tokyo, Japan, 1975



Two-phase flow modelling within expansionand contraction singularities

V. G. Kourakos1, P. Rambaud1, S. Chabane2, D. Pierrat2

& J. M. Buchlin1

1von Karman Institute for Fluid Dynamics, Belgium2Centre Technique des Industries Mecaniques, France

Abstract

An experimental study is performed in order to describe the single- and two-phase (air-water) horizontal flow in the presence of pipe expansion and contraction.Three types of singularities are investigated; smooth convergence and sudden andprogressive enlargement. The opening angles for progressive singularities are 5, 8,9 and 15 degrees. The surface area ratios tested are σ = 0.43, 0.64, 0.65 and 1.56.Bubbly flow is the dominant flow regime that is investigated for volumetric qualityup to 30%. The pressure distribution for both single and two-phase horizontalflow is examined versus axial position. For expansion geometries, it is found thatthe smaller the enlargement angle, the larger the recovery pressure for the sameflow conditions; the pressure drop caused by the singularity is higher in the caseof a sharper expansion. The comparison of the experimental results to publishedmodels leads to a proposed corrective coefficient for Jannsen’s correlation. Flowvisualization is also performed; the flow patterns downstream from the differentsingularities are identified in each configuration and plotted in Baker’s map forhorizontal flow.Keywords: two-phase flow, singularity, sudden expansion, contraction, pressuredrop, bubbly flow, flow visualization.

1 Introduction

Two-phase flow can be frequently met in nuclear, chemical or mechanical engi-neering where gas-liquid reactors, boilers, condensers, evaporators and combus-tion systems are often used. The presence of geometrical singularities in pipesmay significantly affect the behaviour of two-phase flow and subsequently the



resulting pressure drop. Therefore, it is an important subject of investigation inparticular when the application concerns industrial safety valves. The studies oftwo-phase flow in straight pipes existing in the literature are numerous. However,investigations of two-phase flow in divergence, convergence, bends and other typesof singularities are rather sparse. The aim of studying these geometries is to findhow these geometrical accidents influence the two-phase flow pattern and pressuredistribution. In particular, the understanding of the flow in such basic geometriescan lead to a better design of safety systems.

Some of the authors that have analyzed two-phase flow in expansion geometriesare Jannsen and Kervinen [1], McGee [2], Chisholm [3], Chisholm [4] and Lottes[5]. Correlations for estimating the pressure change in two-phase flow in thistype of piping geometry are reported by these authors. These correlations canbe extracted from the conservation equations applied downstream of the suddenexpansion. The equations used take into account different parameters of thegeometry and the flow such as surface area ratio σ , mass quality x and massvelocity G. More recently, Aloui and Souhar [6], Aloui et al. [7], Schmidt andFriedel [8], Hwang and Pal [9], et al. Ahmed et al. [10] and Ahmed et al. [11]have evaluated the pressure change in a sudden expansion duct. Moreover, someof them (Aloui and Souhar [6]; Ahmed et al. [10]) have measured the bubblevelocities and local void fraction to characterize the flow regime downstreamfrom the singularity. The lack of studies in progressive enlargements in two-phaseflow in the literature makes such an investigation more appealing. In this paper,progressive contraction and divergence geometry of different opening angles isconsidered. The latter is compared to the case of sudden expansion. The two fluidsare air and water in isothermal conditions. The volumetric quality of the air variesfrom 0–30% and bubbly flow is the dominant regime. Four surface area ratios,σ = 0.43, 0.64, 0.65 and 1.56, are tested. The opening angles for the case ofprogressive singularities are 5, 8, 9 and 15 degrees. The Reynolds number Reof the liquid ranges from 8 · 104 to 23 · 104. The determination of the recoverypressure for each of the aforementioned geometries is one of the main objectivesof this investigation.

Flowd D

Reattachment length-L/d

dD

Reattachment length-L/d

B)

Flow

A)

Figure 1: A) Progressive expansion of different opening angles-reattachmentlength L/d. B) Sudden expansion-reattachment length L/d.



In Figure 1, the two different types of expansion geometries tested in this paperare presented. Figure 1A shows the divergent pipe with the opening angle α andFigure 1B the sudden expansion. The normalized reattachment length L/d, noticedin Figure 1, denotes the eventual recirculation zone. In the case of convergencegeometry, a contraction region can be observed; a vena contracta is formed in thepipe downstream from the singularity.

2 Experimental facility and conditions

2.1 Experimental facility

A schematic of the horizontal air-water flow facility used for the present study isshown in Figure 2. A centrifugal pump (1) with a maximum flow rate of 65 m3/h issucking water from a reservoir and is controlled with a frequency inverter. Duringthe experiments, an air release valve (11) connected to the tank is kept continuouslyopen to the atmosphere to avoid bubbles entering the circuit. A by pass valve (12)is used to prevent facility from the water hammer phenomenon. A temperaturesensor is placed in the reservoir to monitor the temperature for each measurement.Two electronic flow meters are used to measure the water flow rate (2 and 3); theirmaximum capacity is 12 m3/h (3) and 32 m3/h (2), respectively. In the case ofthe desired maximum flow rate, which is 40 m3/h, the two flow meters are usedin series. A bourdon tube pressure gauge (4) is placed upstream in the pipe toobtain the wall static pressure relative to the atmosphere. This indication helped toprevent excessive pressure that could lead to a breaking of the test section (made inPolymethyl Methacrylate, PMMA). Moreover, the pressure has to be high enoughto allow the necessary purging of the pressure transducers. Therefore, the pressureis held constant at around 200 kPa. The setup has an upstream calming section (5)consisting of a stainless steel pipe length of 50 diameters (50d). This ensures a fullydeveloped flow after the bend. Close to the test section, the injection of the air isperformed through a gas injector (6) as indicated in Figure 2. A regulation valve (7)

1 Pump

2 Big electronic water flow meter

3 Small electronic water flow meter

4 Bourdon tube pressure gauge

5 Calming length

6 Air injector

7 Regulation valve

8 Electronic air mass flow meter

9 Heat exchanger

10 Pressure regulation valve

11 Air release valve

12 By pass valve

T Temperature measurement

Water tank

Test section

InverterT

PP

Water discharge

Compressed air

1111

1212

Figure 2: Schematic of the experimental facility.



controls the air that is supplied from a compressor. The air flow rate is measuredby an electronic mass flow meter (8). The design and the positioning of the airinjection devices are such that uniform bubbly flow is produced at the inlet of thetest section. It is found that the most suitable distance for the air injection is 20 pipediameters upstream from the singularity. After the test section, a heat exchanger(9) is placed for maintaining the temperature constant at around 21C during theexperiments. A draining valve is also located at the bottom of the reservoir. Finally,a pressure regulation valve (10) controls the pressure of the system.

A detailed view of the test section is presented in Figures 3, 4 and 5. The case ofa DN 40/65 (σ=0.43) divergent section with an opening angle of 8 is exemplified.At each section of measurement, four pressure taps are placed with an angle of45 between them as shown in Figure 3. Thus, any three dimensionality of the flowcould be identified from pressure measurement. The four taps are named as A, B, Cand D according to Figure 3. Figure 4 depicts an overview of the test section. Thesetup is built in PMMA to allow optical access. Pressure taps are placed along thetube in several points as is shown in Figure 5. The distance between pressure holesis normally equal to one tube diameter but becomes smaller when approaching thesingularity. The pressure taps are also more dense inside and downstream fromthe singularity. This allows better tracking of the flow behaviour in the singularity.

Aluminum table

4 pressure taps

(45° angle between them)

Aluminum table

4 pressure taps

(45° angle between them)

A B

CD

Figure 3: Pressure taps placed in fourdifferent points of the tubewith 45 between them.

Figure 4: Overview of the test section.

Figure 5: Detailed view of the test section with the pressure taps and their position.



Table 1: Different test cases studied.

Singularity d1 [m] D2 [m] σ [-] Ls [m] Ls/d1 Angle α []Smooth contraction 0.04 0.032 1.56 0.025 0.63 9

Divergence 0.032 0.04 0.64 0.025 0.78 9Divergence 0.041 0.0627 0.43 0.041 1 15Divergence 0.041 0.0627 0.43 0.07503 1.83 8Divergence 0.041 0.0627 0.43 0.1238 3 5Divergence 0.0627 0.078 0.65 0.0529 0.8 8

Sudden expansion 0.041 0.0627 0.43 - - 90Sudden expansion 0.0627 0.078 0.65 - - 90

Table 2: Upstream conditions for pressure measurements and flow visualization.

d1[m] Fluid Q [l/s] J [m/s] β [%] G [kg/m2s] ReL1·104 Flow regime

0.032

Water 2 2.5

1-40

2500 9 Laminar MinWater 4.7 5.8 5850 20 Turbulent Max

Air 0.017 0.02 0.03 0.005 Laminar MinAir 1.8 2.2 2.61 0.46 Turbulent Max

0.041

Water 2.3 1.8

5-30

1750 8 Turbulent MinWater 7 5.4 5300 23 Turbulent Max


0.0627

Water 6 1.9

5-25

1950 13 Turbulent MinWater 10.5 3.4 3400 23.5 Turbulent Max


Pressure distribution is measured upstream and downstream from the divergence.The test matrix is summarized in Table 1.

2.2 Flow conditions and measurement devices

The flow conditions of the experimental campaigns are listed in Table 2. Table 2presents the test conditions for the pressure measurements and for flow visual-ization. It should be pointed out that the ReL1 number of the liquid is based onthe upstream pipe diameter d. For the comparison between single and two-phaseflow, ReL1 is kept constant. This is obtained by adjusting the water flow rate whenincreasing the air to reach a higher volumetric quality β. Consequently, we canassume that the total mass flux is constant, since the mass of the air comparedto that of water is negligible. Differential pressure transducers of the Rosemounttype are used. The uncertainty associated with the pressure transducers varies froma minimum of 0.35% to a maximum of 0.75%, depending on the range of the



measurement (100-20% of the scale of the range respectively). To obtain the bestaccuracy possible, four different pressure transducers are selected:

1. Calibrated at 0–1.6 kPa2. Calibrated at 0–4 kPa3. Calibrated at 0–8 kPa4. Calibrated at 0–16 kPa

Every transducer is used in a range that gives the best accuracy in all theconditions covered. Prior to the measurements, predictions of regular pressuredrop are performed by means of Blasius and Colebrook-White formulas forsingle-phase and Lockhart and Martinelli [12] for two-phase flow. Thus, this Pestimation allows the selection of the appropriate pressure transducers for eachtest. Additionally, for the prediction of the singular pressure change in single-phase, the coefficients given by Idel’cik [13] are used. The uncertainty relatedto the flow rate measurements varies from a minimum of 0.5% to a maximum of1.10%. The temperature variation during the experiments is of the order of ±4Cwith an average value of 21C. Although a heat exchanger is used for reducingthis variation, a small fluctuation of the temperature could not be avoided. Avariation of ±5C will change ρ and ν by 0.1% and 11% respectively. Therefore, acorrection of the liquid density and viscosity is performed. The sampling frequencyof the measurements is fsampling = 2 Hz and the acquisition time for eachmeasurement point is tacq. = 1 minute with the aim of assuring a more accurateaverage. In some cases (for sudden and progressive enlargement of σ = 0.65), ahigher fluctuation of the signal is observed; in this occurrence an acquisition timeof 2 minutes is chosen.


3.1 Pressure measurements

One of the main objectives of the study is the determination of the pressuredistribution through the different singularities. Figure 6 indicates how the mea-surements are performed and how the singular single and two-phase pressurechange is determined (the case of divergence is chosen). As the graph of Figure6 shows, following a normal decrease upstream from the geometrical accident,the pressure will increase to a maximum value inside the divergent section andwill start decreasing after a certain length in a regular way. We can split thewhole phenomenon into three regions; the upstream fully developed flow, thetransitional region with a recirculation zone and the downstream fully developedflow. The length of the transitional region varies with ReL1, σ , and the type ofthe singularity. In all the tests, the measurement of the regular and singular staticpressure changes refers to the pressure measured at ≈ 10d upstream from thesingularity (Figure 6). The singular pressure changeP can be finally determinedby extrapolating the regular static pressure drop from the start of the singularityto the reattachment point. Since the points downstream from the singularityare not enough to obtain fully established flow, the regular pressure drop is



Singularity

Preference

Pmax

P [mbar]

Axial position [z/d]

Pregular

(measured)

Pregular (calculated)

z/d=0

Psingular

(measured)PSINGULAR-FINAL

P=0

Flow

Fully developed flowInlet

Transitional flow Fully developed flow

Outlet

Figure 6: Explanation of the way to determine the singular pressure change inexpansion geometry.

calculated by means of the Blasius and Colebrook-White formulas for single-phase and the model of Lockhart and Martinelli [12] for two-phase. The finalsingular pressure change is calculated by a simple summation of these three terms(|Pregular-measured|+|Psingular-measured|+|Pregular-calculated|). The reattachmentlength is determined as the location of the maximum recovery pressure.

3.1.1 Expansion singularities3.1.1.1 Sudden expansion In Figure 7, the two-phase pressure change alongthe pipe and the singularity is plotted for sudden expansion of σ = 0.43 andat ReL1 = 1.82 · 105. The single-phase result is also drawn on the same graph.The pressure is measured at the four peripheral taps on the tube sections closeto the singularity (points A, B, C and D) as well as their average (point M).The two-phase experimental data are compared with prediction of the singularpressure change for axisymmetric sudden expansion geometry obtained from thetwo following models:

Jannsen and Kervinen [1]:

Ptot = − G21

2ρL(1 − σ)2

[1 + x

(ρL

ρG− 1

)], (1)

whereG1 is the mass flux upstream from the singularity, ρL is the density of water,σ is the area ratio, x is the mass quality of air and ρG is the density of air.



-20

-15

-10

-5

0

5

10

15

20

25

30

35

40

45

-10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

z/d [-]

P [

mb

ar]

Single-phase-Experimental

L-M&Chisholm (1969)

L-M&Jannsen(1966)

Point M

Point A

Point B

Point C

Point D

Sudden enargement =0.43

Two-phase-20%air-ReL1=1.82E5

A B

CD

A B

CD

Figure 7: Two-phase static pressure change versus axial position for suddenenlargement of σ = 0.43 and for ReL1 = 1.82 · 105-comparison withexperimental single-phase and with the models of Jannsen and Kervinen[1] and Chisholm [4].

Chisholm [4]:

Pst = − G21

2ρLσ (1 − σ) (1 − x)2

(1 + C

X+ 1

X2

), (2)

where

X2 (

1 − x

x

2)ρG

ρL,

C =[

1 + 0.5

(ρL − ρG

ρL

)0.5][(

ρL

ρG

)0.5

+(ρG

ρL

)0.5].

Both models rely on the assumption of a homogeneous flow. Figure 7 showsthat Jannsen’s model [1] fits satisfactorily with the experimental results whileChisholm’s [4] model overestimates the pressure change. This was also reportedby Velasco [14].

To better emphasize the effect of two-phase flow we define the dimensionlesspressure changeL as follows:

L = P TPSingular

P SPSingular

, (3)

where P TPSingular is the singular two-phase pressure change as explained in

Figure 6 and P SPSingular the single-phase one. Figure 8 displays the evolution of

the experimental L versus volumetric quality at ReL1 = 2.0 · 105. The data arecompared to the model of Jannsen and Kervinen [1] and Chisholm [4], respec-tively. As it was previously mentioned, Jannsen’s [1] correlation agrees betterthan Chisholm’s [4] correlation with the experimental results. The comparativegraphs given in Figures 9 and 10 indicate that the maximum deviation from the



0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

0 5 10 15 20 25 30 35

Volumetric quality [%]

L [

-]

Experimental

Chisholm (1969)

Jannsen (1966)


ReL1=2.0E5

Figure 8: Dimensionless singular pressure change L versus volumetric quality.Comparison with the models of Jannsen and Kervinen [1] and Chisholm[4].

5%

10%

40

45

50

55

60

65

70

75

80

40 45 50 55 60 65 70 75 80

Psingular experimental [mbar]

Ps

ing

ula

r Jan

nsen

[m

bar]

Single-phase

Air 5%

Air 10%

Air 15%

Air 20%

Air 25%

Air 30%

Air 35%


ReL1=2.0E5

Figure 9: Deviation of Jannsen andKervinen [1] model fromexperimental results.

5%

10%

40

45

50

55

60

65

70

75

80

40 45 50 55 60 65 70 75 80

Psingular experimental [mbar]

Ps

ing

ula

r C

his

ho

lm [

mb

ar]

Single-phase

Air 5%

Air 10%

Air 15%

Air 20%

Air 25%

Air 30%

Air 35%


ReL1=2.0E5

Figure 10: Deviation of Chisholm [4]model from experimentalresults.

experimental data for the model of Jannsen and Kervinen [1] is limited to 5%while it reaches 10% for Chisholm [4] model.

Measurements with the same flow conditions are repeated for a sudden enlarge-ment of surface area ratio σ = 0.65. A summarizing graph of the static pressurerecovery measured in both geometries of σ = 0.43 and 0.65 for different ReL1and for volumetric quality, β, varying from 0 to 35% is presented in Figure 11.The singular pressure change is increasing for higher β and ReL1. Furthermore,for the same ReL1 lower σ results in a lower P (up to three times smaller).

3.1.1.2 Progressive and sudden enlargement: comparison Compared to sud-den expansion, a progressive enlargement will create for the same flow conditions,less pressure loss and accordingly will exhibit a higher pressure recovery asdepicted in Figures 12 and 13. Figure 12 shows a single-phase P diagram alongsudden expansion and divergent of angles 5, 8 and 15, of surface area ratioσ = 0.43 and at ReL1 = 1.8 · 105. In Figure 13, the same type of plot is builtfor β = 20% of air. It can be seen that, for single-phase, the pressure drops17% passing from divergent section of 5 to 15 and 29% from 5 to sudden



0

10

20

30

40

50

60

70

80

90

80000 100000 120000 140000 160000 180000 200000 220000 240000

ReL1 [-]

PS

ing

ula

r [m

bar]

Single-phaseAir 5%Air 10%Air 15%Air 20%Air 25%Air 30%Air 35%



Figure 11:Psingular for several ReL1 from 0–35% of air for sudden enlargementof surface areas σ = 0.43 and σ = 0.65.

-20

-15

-10

-5

0

5

10

15

20

25

30

35

40

45

50

-10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16

z/d [-]

P [

mb

ar]

Sudden enlargement

Divergent-angle 5

Divergent-angle 8

Divergent-angle 15

Singularity =0.43

Single-phase-ReL1=1.8E5

Figure 12: Pressure recovery diagram for single-phase flow and the same ReL1, asingularity of σ = 0.43 and for sudden enlargement and divergence ofangles 5, 8 and 15.

expansion. For two-phase flow, the pressure drop is 11% and 21% respectively.Additionally, we can notice that all the curves in Figure 13 are shifted to theright, meaning that the flow becomes fully developed further downstream from thesingularity and thus the recirculation zone is longer in two-phase flow. In the caseof sudden enlargement, contrary to smooth divergence, the pressure before startingto increase slightly decreases at 1d and starts increasing again at 2d upstream ofthe singularity. This is due to the presence of a secondary recirculation zone.

3.1.1.3 Proposed correlation for expansion singularities The proposed cor-relation relies upon Jannsen [1] formulation. By fitting this model to the experi-mental values, a corrective coefficient is defined. It turns out that this parameterC is a function of the opening angle α and ReL1 as shown by the 3D repre-sentation proposed in Figure 14. Although Jannsen’s [1] model is chosen as themost accurate, attempts are made with Chisholm [4] model as well. Hence, thecorrective coefficient C for Chisholm’s [4] correlation is represented in a 3D plot



-20

-15

-10

-5

0

5

10

15

20

25

30

35

40

45

50

-10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16

z/d [-]

P [

mb

ar]

Sudden enlargement

Divergent-angle 5

Divergent-angle 8

Divergent-angle 15

Singularity =0.43

Two-phase 20 % air-ReL1=1.8E5

Figure 13: Pressure recovery diagram for two-phase flow (20% of air) and thesame ReL1, a singularity of σ = 0.43 and for sudden enlargement anddivergence of angles 5, 8 and 15.

Angle [°]

68

1012

14

Re

[-]

1.8x10+05

2.0x10+05

2.2x10+05

C[-

]

0.3

0.4

0.5

0.6

0.7

0.3

0.35

0.4

0.45

0.4

0.45

0.5

0.55

0.6

0.65

XY

Z

C [-]

0.650.60.550.50.450.40.350.3

Jannsen (1966) model

Figure 14: CoefficientC in function ofα and ReL1 for Jannsen andKervinen [1] model.

Angle [°]

68

1012

14

Re [-]

1.8x10+05

2.0x10+05

2.2x10+05

C[-

]

1.2

1.3

1.4

1.3

2

1.3

4

1.36

1.38

1.4

1.42

1.18

1.2

1.22

1.24

1.26

1.28

1.3

1.321.34

1.36

XY

Z

C [-]

1.421.41.381.361.341.321.31.281.261.241.221.21.18

Chisholm (1969) model

Figure 15: Coefficient C in functionof α and ReL1 for theChisholm [4] model.

in Figure 15. In Table 3, the coefficients that are calculated for both models andfor the different parameters tested in progressive expansion are given.

The corrective coefficient C for Jannsen and Kervinen [1] formulation can bemodelled as follows:

C = 0.061 · α0.8917 − 10717 · Re−0.8283L1 + 0.378. (4)

This coefficient when applied to Jannsen’s [1] model gives a maximum deviationfrom the model fit of 58% for the case of σ = 0.43, ReL1 = 1.84 · 105 and 5and minimum of 1.4% for ReL1 = 2.36 · 105 and 15. It should be stressed thatfurther experimental data are needed to refine the C modelling and improve thevalidation.



Table 3: Coefficient for adaption of the Jannsen and Kervinen [1] and Chisholm [4]models to fit to the experimental results for divergence geometry and forseveral α, σ and ReL1.

Jannsen [1] ReL1·105 α [] C [-] Chisholm [4] ReL1·105 α [] C [-]

σ=0.43

1.84 5 0.4

σ=0.43

1.84 5 1.342.3 5 0.26 2.3 5 1.445

1.78 8 0.48 1.78 8 1.32.36 8 0.38 2.36 8 1.3651.76 15 0.7 1.76 15 1.1552.36 15 0.69 2.36 15 1.163

σ=0.651.79 8 0.3

σ=0.651.79 8 1.187

2.26 8 0.24 2.26 8 1.365

Finally, the modified Jannsen’s [1] correlation can be written as:

Ptot = −[0.061 · α0.8917 − 10717 · Re−0.8283

L1 + 0.378]

· G21

2ρL(1 − σ)2

[1 + x

(ρL

ρG− 1

)].

(5)

3.1.2 Contraction singularity3.1.2.1 Measurements in progressive contraction Convergence geometry ofσ = 1.56 and angle 9 is studied. The geometry is identical to the test sectionshown in Figure 5 with a scaling factor of 1/2 (DN40/32). The experimentalfacility and flow conditions are described in section 2.2. Pressure transducers oftype Validyne are used for this experimental campaign with the same acquisitiontime (tacq. = 1 min) and sampling frequency (fsampling = 2 Hz). The differentmembranes that cover all the range of the pressure measurements are:

1. Calibrated at 0–2.2 kPa2. Calibrated at 0–8.6 kPa3. Calibrated at 0–35 kPa

Additionally, numerical simulations are carried out with the commercial CFDcode Fluent. The test parameters and conditions are: 2D axisymmetric computa-tion, realizable k − ε turbulence model with enhanced wall treatment and secondorder discretization scheme. Convergence criterion is set at 10−7. In Figure 16, theexperimental and numerical static pressure drop is plotted against axial position forseveral ReL1 in single and two-phase flow. The pressure is decreasing in a regularway before the singularity; the contraction creates a high pressure drop step andthen starts decreasing regularly downstream.

The flow is observed fully developed close to the singularity (at ≈2d upstreamand downstream) contrary to the case of divergence for which the reattachmentlength is detected at ≈10d. Therefore, the singular pressure change Psingular forconvergence geometry is determined by measuring the static pressure at equal



distance upstream and downstream from the singularity (2d). A summarizinggraph of all experimental and numerical results obtained for single and two-phaseflow is shown in Figure 17. The results concerning the case of sudden contractionfor several σ and G (Guglielmini et al. [15]) are compared to the experimentaldata. The experimental results for smooth contraction are plotted in terms of thedimensionless pressure change L, defined by eqn. (3). In Figure 17 Jannsen’s[1] correlation for sudden contraction is adapted with a correction coefficient ofC = 0.81 to fit with the results (G = 1990 kg/m2s).

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

L [m]

P [

mb

ar]

Single-phase-Exp-Re=136000Single-phase-Exp-Re=79300Single-phase-CFD-Re=739000Two-phase-11% air-Exp-Re=95100Two-phase-10%air-CFD-Re=66500

Smooth convergence =1.56,

angle 9°

Figure 16: Experimental and numerical single and two-phase static pressurechange versus axial position for convergence of σ = 1.56 and angle9 for several ReL1.

G=1990 kg/m^2s

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

0 5 10 15 20 25 30 35

Volumetric quality [%]

L [

-]

Exp-G=1990 kg/m^2s

Exp-G=2786 kg/m^2s

Exp-G=1990-3424 kg/m^2s

CFD-G=1300-1700 kg/m^2s

Guglielmini et al.-G and varying

Janssen(1966) correlation-C=0.81

Smooth convergence =1.56,

angle 9°

Figure 17: Experimental and numerical dimensionless singular pressure changeL versus volumetric quality. Comparison to literature (Guglielminiet al. [15]) and to the adapted (C = 0.81) Jannsen and Kervinen [1]model.



Table 4: Coefficient for adaption of Jannsen’s [1] formulation to fit to the experi-mental and numerical results for progressive contraction for severalG.

Jannsen [1] correlation GL1 [kg/m2s] Correction coefficient C [-]

Convergence-angle 9, σ=1.56

1592 0.8351990 0.8002786 0.7704378 0.754

3.1.2.2 Proposed correlation for progressive contraction The correlation forsudden convergence, as described from Jannsen [1], is recalled:

PTP = G22

2ρL

((1

CC− 1

)2

+ 1 − 1

σ 2

)[1 + x

(ρL

ρG

)]. (6)

where Cc is the contraction coefficient defined as Cc = Ac/A1 where Ac theflow area in the vena contracta. A typical value of this parameter equal to 0.64 isconsidered for this investigation. This correlation can be modified and then appliedfor the case of smooth contraction. The parameter varying is the mass flux ofwater upstream of the singularity GL1. A fit to the present results is made andthe resulting corrective coefficients are listed in Table 4.

A correlation to calculate the correction coefficient C is obtained as a functionof GL1.

C = 2 · 10−8G2L1 − 0.0001 ·GL1 + 0.9913. (7)

The relative discrepancy between experimental-numerical data and model fit,when eqn. (7) is applied, varies from 5.72% to a maximum of 24.25%. The finalcorrected correlation for the case of smooth convergence of angle 9 is:

PTP =[2 · 10−8G2

L1 − 0.0001 ·GL1 + 0.9913]

· G22

2ρL

((1

CC− 1

)2

+ 1 − 1

σ 2

)[1 + x

(ρL

ρG

)].

(8)

3.2 Flow pattern maps and visualization

Flow regime maps are often considered in two-phase flow. A common chart is theone proposed by Baker [16]. It has been established for horizontal flows in pipesof constant cross section. In the present study, the flow is visualized both upstreamand downstream from the singularity. As it is illustrated in Figure 18, four differentflow patterns are identified downstream of the divergence; Bubbly, Plug, Disperseand Annular flow.

For sudden and progressive enlargement (angles 5 and 8) with σ = 0.43 andσ = 0.65, a normal video camera is used to determine the condition for transition



from bubbly flow to other types of flow just after the singularity. The results areplotted on Baker’s map and are reported in Figure 19. However, since the departurefrom bubbly flow is decided on visual information, the transition criterion remainsrather subjective and the results given in Figure 19 are only indicative.

The second campaign of visualization is performed, using a high-speed camera,in a fully transparent setup that allows better optical access (without pressuretaps). Consequently, distinction between flow regimes is more straightforward.In this facility, a progressive enlargement of σ = 0.64 for an opening angle ofα = 9 is tested. The flow conditions for which these regimes are visualized arereported in Figure 20. Finally, we should draw attention to the fact that all flowconditions calculated refer to the upstream position. Indeed, for these test cases,the flow regime upstream from the singularity corresponds to bubbly flow (Bakermap) while downstream three additional flow patterns occur (plug, disperse andannular).

Bubbly Plug

AnnularDisperse

Figure 18: Flow patterns identified downstream of the divergence geometry of α =9 and σ = 0.64.

p p

10 100 1000 10000 100000

GL1 [kg.m-2

s-1

]

Sud.enl.- =0.43

Sud.enl.- =0.65

Div.angle 5 =0.43

Div.angle 8 =0.43

Div. angle 8 =0.65

Slug

Bubbly

Plug

WavyAnnular

Stratified

Singularities =0.43

and =0.65

0.1

1

10

100

1

GG

1 /

[

kg

.m-2

s-1

]

Figure 19: Modified Baker [16] mapfor progressive and suddenexpansion of σ = 0.43 and0.65.

10 100 1000 10000 100000

GL1 [kg.m-2

s-1

]

Bubbly

Disperse

Plug

Annular

Slug

Bubbly

Plug

Wavy Annular

Stratified

Divergence =0.64

Angle 9°

0.1

1

10

100

1

GG

1 /

[

kg

.m-2

s-1

]

Figure 20: Modified Baker [16] mapfor progressive expansion ofσ = 0.64 and α = 9.



4 Conclusions

An investigation of horizontal air-water flow in sudden and progressive enlarge-ments and smooth contraction is performed. The static pressure evolution alongthese geometrical accidents is measured and flow visualization is performed.The results are expressed in terms of the dimensionless singular pressure changeL. Compared to literature, a deviation of 5% is found with Jannsen’s [1]model and 10% with Chisholm’s [4] model for axisymmetric sudden expansion.For progressive enlargement of the same surface area ratio σ , the smallest theopening angle, the highest the pressure recovery. For the same flow conditions, theminimum pressure recovery occurs for sudden enlargement geometry. A modifiedversion of Jannsen’s [1] correlation is suggested for both progressive expansionand contraction. A corrective parameter taking into account the different effects ofthe divergent angle and the liquid Reynolds number of the divergent section and theupstream mass flux for convergence, is introduced. The proposed correlation givessatisfactory results but needs further validation. In the convergence configuration,the single and two-phase static pressure drop along the pipe is compared withliterature and CFD simulations; a satisfactory agreement is found. Finally, flowvisualization shows that departure from bubbly flow to plug, disperse or annularflow may occur in the downstream section of a divergent singularity.

Acknowledgements

The support of the French company CETIM (Centre Technique des Industries

Delgado-Tardaguila are thanked for their contribution to the numerical and exper-imental study in this paper.

Nomenclature

A surface area [m2]C correction coefficient [-]d upstream diameter [m]D downstream diameter [m]f frequency [Hz]G mass velocity [kg/m2s]J superficial velocity [m/s]L length of the pipe [m]P pressure [Pa]Q volumetric flow rate [l/s]ReReynolds number [-]t time [s]x mass quality [-]z axial position [m]

Greek symbolsα opening angle []β volumetric quality [-]ν kinematic viscosity [m2/s]ρ density [kg/m3]σ surface area A1/A2 [-] dimensionless P [-]

Sub-Superscripts Abbreviationsc contraction SP single-phase

G gaseous phase st staticL liquid phase tot total1 upstream TP two-phase2 downstreams singularity



Mecaniques) is gratefully acknowledged. Mr. E.C. Bacharoudis and Miss R.

References

[1] Jannsen, E. & Kervinen, J.A., Two-phase pressure drop across contractionsand expansions of water-steam mixture at 600 to 1400 psia. Technical ReportGeap 4622-US, 1966.

[2] McGee, J., Two-phase flow through abrupt expansion and contraction. Ph.D.thesis, North Carolina State University, Raleigh, 1966.

[3] Chisholm, D., Prediction of pressure losses at changes of sections, bends andthrottling devices. Technical Report NEL rept. 388, 1968.

[4] Chisholm, D., Theoretical aspects of pressure changes at changes of sectionduring steam-water flow. Technical Report NEL rept. 418, 1969.

[5] Lottes, P., Expansion losses in two-phase flow. Nucl Sci Eng, 9, pp. 26–31,1960.

[6] Aloui, F. & Souhar, M., Experimental study of a two-phase bubbly flow ina flat duct symmetric sudden expansion. Part I: Visualization, pressure andvoid fraction. Int J Multiphase Flow, 4, pp. 651–665, 1996.

[7] Aloui, F., Doubliez, L., Legrand, J. & Souhar, M., Bubbly flow in anaxisymmetric sudden expansion: pressure drop, void fraction, wall shearstress, bubble velocities and sizes. Exp Therm Fluid Sci, 18, pp. 118–130,1999.

[8] Schmidt, J. & Friedel, L., Two-phase pressure change across sudden expan-sions in duct areas. Chem Eng Commun, 141, pp. 175–190, 1996.

[9] Hwang, C.Y. & Pal, R., Flow of two-phase oil/water mixtures through suddenexpansions and contractions. Chem Eng J, 68, pp. 157–163, 1997.

[10] Ahmed, W., Ching, C. & Shoukri, M., Pressure recovery of two-phase flowacross sudden expansions. Int J Multiphase Flow, 33, pp. 579–594, 2008.

[11] Ahmed, W., Ching, C. & Shoukri, M., Development of two-phase flowdownstream of a horizontal sudden expansion. Int J Heat and Fluid Flow,29, pp. 194–206, 2008.

[12] Lockhart, R.W. & Martinelli, R.C., Proposed correlation of data for isother-mal two-phase two-component flow in pipes. Chem Eng Prog, 45, pp. 39–48,1949.

[13] Idel’cik, I.E., Memento des pertes de charge. Editions Eyrolles: 61 Bd Saint-Germain Paris, 5th edition, 1986.

[14] Velasco, I., L’ ecoulement diphasique a travers un elargissement brusque,1975.

[15] Guglielmini, G., Muzzio, A. & Sotgia, G., The structure of two-phase flowin ducts with sudden contractions and its effects on the pressure drop.Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, 1997.

[16] Baker, O. Oil Gas J, 53, p. 185, 1954.



Numerical simulation of gas-solid flow in ducts by CFD techniques

R. K. Decker1, D. Noriler2, H. F. Meier2 & M. Mori1

1Chemical Process Department, State University of Campinas, Brazil 2Chemical Engineering Department, Regional University of Blumenau, Brazil

Abstract

Numerical models are nowadays used on a large scale for the simulation of two-phase flow processes in the chemical industry, due mainly to their low implementation cost and important results. However, to find the best numerical models in such processes, validation with experimental data is required. In this sense, the main objective of this work is to apply a CFD model under Eulerian-Eulerian framework for the gas-particle flow, with the capability to predict the fluid dynamics of the two-phase flows in vertical and horizontal ducts separated by 90° elbows. A three-dimensional and transient model has been applied for predictions of volume fractions, pressure, velocities and turbulence properties fields. For the momentum transfer between phases a drag model based on the application of “Churchill” asymptotes techniques was used to obtain a continuous function for all flow regimes. Furthermore, the wall effects over the particulate flow have been investigated. The mathematical model was applied in CFD commercial codes for numerical studies and compared with experimental data obtained by the research group at Lehigh University. The model is solved using the finite-volume method with variables located in a generalized co-ordinate system. The main results present the volume fraction and velocities profiles as a function of the time and position. In addition, the mean velocity and particle concentration of the radial profiles were compared with experimental data in different axial positions of the vertical duct. The results showed a good agreement for the mean particle velocity with the increment of the axial position. Keywords: CFD, gas-particle flow, ducts, elbows, numerical simulation.



1 Introduction

In the chemical industry several processes occur in the presence of particulate flows, such as the up flow of catalytic particles in riser reactors in the petroleum industry and particle conveying systems found in the cement industry. From one first point of view the particulate flow in ducts seems to be not so difficult to study in comparison with other geometries where the particulate flow also exists. However, the flow phenomena existent in such flows is not simple. There are a lot of variables that may change the characteristics of the gas-solid flow in ducts, such as particle diameter distribution, particle-wall, particle-particle and particle-gas interactions, operational conditions, geometry design and so on. In order to analyze such variables, several research groups are dedicating efforts to obtaining experimental data for the gas-particle flow in different geometries and operational conditions. Tsuji and Morikawa [9] and Tsuji et al. [10] for instance developed a complete analysis of the gas-particle flow in the developed flow region of horizontal and vertical ducts by means of Laser Doppler Velocimeter (LDV). Some years later the research group at Lehigh University (Yilmaz and Levy [12]; Akilli et al [1]; Bilirgen and Levy [2]) analyzed the gas-particle flow by means of a fiber optic probe in the presence of different geometrical configurations and devices. The studied geometry was constituted of a vertical duct connected to the first by a 90° elbow and a second horizontal duct connected with the vertical one with another 90° elbow. In order to understand the particle rope formation, mixture and dispersion phenomena in both sections, the authors evaluated devices for rope dispersion at the beginning of the vertical section just after the horizontal-to-vertical elbow and analyzed the parameters of the geometry and the flow operational conditions as well. As a chronological sequence of works showing the behavior of the two-phase flow in ducts after the analysis of the developed and developing regions in horizontal and vertical ducts, as previously reported, Yang and Kuan [11] investigated the behavior of dilute turbulent particle flows inside a curved 90° bend using a 2D Laser Doppler Anemometry (LDA). As a result the authors observed the rope formation phenomena inside the bend. Even so, experimental measurements can bring information about the gas-particle conveying as previously described, although simulations of the two-phase flow system are also nowadays required. Basically, two phenomenological approaches are employed in the representation of the two-phase flows: the Eulerian-Lagragian (E-L) and the Eulerian-Eulerian (E-E) models. Several numerical studies showing the two-phase flow behavior in ducts were developed in the literature using the E-L approach, [1, 2, 5, 6, 12], and the E-E approach [4, 7]. It is known that the E-L approach requires a great amount of tracking particles to describe efficiently the two-phase flow in ducts, which increases the computational time. Furthermore, the particle-particle and particle-wall interactions are very sensitive to the attributed restitution coefficient, which makes its application difficult for others flow and geometrical conditions. On the other hand, the E-E approach can be applied in a basic methodology by the adoption of an inviscid model, which does not have viscosity stress for the solid



characterization. Furthermore, in such a model the solid agglomeration behave as a hypothetic fluid and interact with the gas phase by the drag force model (Decker [4]). In this sense, the intention of this study is to compare the drag force model proposed by Coelho and Massarani [3] based on the application of “Churchill” asymptotes techniques with the model developed by Shiller and Neuman [8], as well as to compare the inviscid model proposed in this study with the model based on the kinetic theory of granular flow (KTGF) for the two-phase flow in a vertical section of the test facility proposed by Yilmaz and Levy [12]. For a better understanding of the flow phenomena, particle velocity and concentration maps are also presented for the measuring sections.

2 Mathematical modeling

The mathematical model applied in this work is based on two-fluid, three-dimensional and transient flow approach in Eulerian-Eulerian framework. The main assumptions of the model are:

Phases interpenetrating; Hypothetical fluid; The viscous stress and pressure of the particle phase is negligible; Particles are considered to be uniform spheres of same mean diameter; The momentum transfers is due only the drag force; The flow is isothermal.

2.1 Conservation equations

The equations of mass conservation for gas and solid phases are expressed by the eqn. (1) and eqn. (2), as follows:

0f.ft ggggg

v , (1)

0f.ft sssss

v . (2)

Whereas the momentum equations for the gas and solid phases are expressed by eqn (3) and eqn (4), as follows

resFgΤvvv

pf.ff.ft gg

efggggggggg , (3)

resFgvvv

sssssssss f f.ft

. (4)

2.2 Constitutive equations

The effective tensor on the gas phase ( efgΤ ) establishes a similar relation to the

model for a general newtonian fluid, where the stress is directly proportional to

the deformation rate ( gefef

g 2 DΤ ) and is giving by the expression:



Tgg

efefg vvΤ , (5)

where )t(g

ef . (6)

In eqn (6) the turbulent viscosity, )t( is obtained from an isotropic turbulence

model known as standard k- model:

g

2g

g)t( k

C

. (7)

There are two additional transport equations, one for the turbulent kinetic energy (k) and another for the rate of dissipation of turbulent kinetic energy ():

gggggk

ef

gggggggg Gfk.fkf.kft

v , (8)

.k

CGCf.ff.ft g

ggg2g1gg

ef

gggggggg

v (9)

The turbulent kinetic energy generation (Gg) for the gas phase is obtained by the tensorial product, which is denominated as double dot product, between the

Reynolds tensor ( tgΤ ) and the velocity gradient for the gas phase (vg), as

follows:

g)t(

ggG v:Τ . (10)

The resistive force between the phases ( resF ) is modeled by the equation:

sgs,g vvFres . (11)

The interface coefficient s,g can be predicted by dilute flow (fg > 0,8), as:

p

sggsDs,g d

fC

4

3 vv . (12)

2.2.1 Two correlations for the drag coefficient 2.2.1.1 Coelho and Massarani correlation The drag coefficient (Cd) prediction is made for the studied flows by means of a Coelho and Massarani [3] correlation, valid for all Reynolds number range avoiding possible discontinuities provoked by the flow change, typical of a standard discontinue model where its adopted for each Reynolds number range. The correlation is:

18.1

85.02

85.0

p1

KReK

24Cd

, (13)

with

065.0log.843.0K p

101 and . 88.431.5K p2

The particle Reynolds number (Rep) is:

g

psg

p

dRe

vvg. (14)



2.2.1.2 Shiller and Nauman correlation Three regimes are considered:

Stokes Regime:

.200Re0for Re/24Cd pp

Viscous Regime:

.2500Re002for Re15.01Re

24Cd p

687.0p

p

Inertial Regime:

.Re5002for 44.0Cd p

2.2.2 Kinetic theory of granular material model With the aim to verify the inviscid approach, simulations applied the kinetic theory to predict the stress in the solid phase. So, the solid stress sΓ can be

written as follow:

2.

3T

s s s s s s sp Τ v v v

(15)

The solid pressure, ps, and the bulk, s, and shear, µs, viscosities are given in terms of granular temperature, s, the radial distribution function, g0, the coefficient restitution, e, and the particle diameter, i.e.,

01 2 1 ,s s s ssp f e g f

(16)

0.5

0

41 ,

3s

s s s pf d g e

(17)

0.520.5

0 00

10 4 41 1 1 .

96 1 5 5s p s

s s s s ps

de g f f d g e

e g f

(18)

The radial distribution function is given by: 11

3

0,max

0.6 1 .s

s

fg

f

.

(19)

and the granular temperature is obtained from an equation: .s s s sp v

(20)

Where the dissipation of fluctuation energy takes the form:

0.5

2 20

43 1 .s

s s s s sp

e g fd

v

(21)

2.3 Boundary conditions

The boundary conditions for physical frontiers of the gas-solid flow in ducts are as follows.



2.3.1 Inlet It is adopted a uniform and constant gas-solid flow with an initial velocity for both phase (U0=29m/s). All other flow properties are considered uniform.

2.3.2 Outlet Constant pressure with continuity conditions considered for all flow properties.

2.3.3 Wall It is considered null derivates for volumetric fractions (impermeable frontiers), and wall logarithm function for the turbulent properties. The wall condition for the gas velocity for all numerical studies was considered to be no-slip and for the particle velocity, no-slip and slip, depending on the desired simulation condition.

3 Numerical modeling

The numerical simulations where conducted by a CFD commercial package, the CFX 4.4. The pressure-velocity coupling applied was SIMPLEC, and the interpolation scheme of first order was the UPWIND due to the strong convective component of the gas-solid transport. The algorithm was applied with the AMG and ICCG proceeding to solve the discrete algebraic equations. The convergence criteria for all studied cases was 1.10-4 for the Euclidian norm in the mass source. The relaxation factors were used only for the turbulent kinetic energy (k) and for its dissipation rate () in the turbulence model with values equal to 0.7 to guarantee stability and convergence. The interactions were solved for a time step equal to 5.10-4 using one time typical implicit integration.

4 Case study

Fig. 1 presents the geometrical system proposed by Yilmaz and Levy [12] for the analysis of the gas-solid flow and details of the numerical grid used for the simulation process. The system is composed of three different duct sections, two horizontal and one vertical (L1, H and L2), connected among them by 90º elbows with bend ratio equal 1.5. The mass load ratio A/F=1 was used for all numerical simulations. The physical and geometrical properties are showed in Table 1.

5 Results

For the analysis of the gas-solid flow in a vertical duct a sequence of studies was developed. The first study was the investigation of the drag force model

Table 1: Physical and geometrical properties.

Physical and Geometrical Properties

g (Kg/m3) 1.225 p 1.000

g (Kg/m.s) 1.850E-05 D (m) 0.154 s (Kg/m

3) 1680.000 L1 and L2 (m) 6.100 dp (m) 75E-06 H (m) 3.400



proposed by Coelho and Massarani [3] in comparison with the default model implemented in the CFD software developed by Schiller and Neuman [8]. The criteria adopted to define each model are the comparison with the experimental data obtained by Yilmaz and Levy [12] as well as the convergence rate. The results presented in fig. 2 in terms of particle velocity shows that both models are equally far from the experimental data. However the Coelho and Massarani model showed much better convergence stability than the Shiller and Neuman model which hardly reduced the computational time for the mass flow balance convergence. Due to this criteria and not just the comparison with the experimental data the Coelho and Massarani model was chosen to be used in this study for other developments.

Figure 1: Details of geometry and numerical grid.

Once the drag model is defined the analysis of the inviscid model can be carried out and compared with a model based on the theory of granular materials (KTGF). For both developments the drag force model of Coelho and Massarani [3] has been applied. Fig. 3 shows for L/D = 17 that the inviscid model seems to disperse the particulate phase in the transversal section of the duct much better than the KTGF model and more likely the experimental data of Yilmaz and Levy [12]. In this way, the inviscid model showed to be a relevant tool to be applied for the gas-solid flow modeling due to its simple characteristics when compared with E-L models and some others E-E models in the presence of several functions for the solid pressure and viscosity such as KTGF. The results previously presented considered the no-slip condition at the wall for the particulate phase. However, owing the flow direction change promoted by the 90° elbows an analysis of the wall treatment has been developed and



presented in fig. 4. As result it is observed that the wall condition just interfere in the region near the wall. However when compared with the experimental data the slip condition showed to be apparently consistent near the wall. This comparison can just be done in the x/D region near to zero due to the better experimental data acquisition at this region near the wall. Considering that the slip condition at the wall showed a reasonable adjustment in these previous numerical experiments combined with the inviscid proposed model, and considering that the Coelho and Massarani model showed to be faster in the convergence of the mass balance between inlet and outlet, a complete analysis of the two-phase flow in a vertical section is developed. For this analysis snapshots of particle concentration, particle velocity and tangential vectors are showed for L/D = 1, 5, 9 and 17 and presented in fig. 5. As can be seen in fig. 5a and fig. 5b for L/D = 1 the particle flow is concentrated in the outer wall of the duct (x/D = 0 and Cp = 2Kg/m³) due to the tangential forces acting in the particulate flow by the presence of a 90° elbow. At the same instant that the ropes are formed, secondary flows in the order of 5m/s start to act, as showed in fig. 5c, redirecting and dispersing the ropes in direction to the opposite wall (x/D = 1) of the vertical section. Owing to the

Figure 2: Drag force model analysis.

Figure 3: Inviscid model analysis.

Figure 4: Wall treatment condition.



L/D = 1 L/D = 5 L/D = 9 L/D = 17

a) Particle Concentration

b) Particle Axial Velocity

c) Tangencial Vector Plot

Figure 5: Snapshots in different positions (L/D = 1, 5, 9 and 17) for the vertical measuring section. a) Particle concentration ranging from 0,5 Kg/m³ to 2 Kg/m³; b) particle axial velocity ranging from 16 m/s to 32 m/s; c) tangential vector plot for the particle phase ranging from 0 to 5 m/s.

presence of secondary flows the axial particle velocity is also redirect in direction to the inner wall (x/D = 1) with the increment of the L/D region of the vertical section as seen for L/D = 5 and herewith the particle concentration. After its redirection the secondary flows becomes weak and particle axial velocity and concentration are dispersed along the transversal section with the increment of axial position. These observations are showed in fig. 5a and 5b for L/D regions equal to 9 and 17. Fig. 5c also shows that secondary flows also becomes weak and more homogeneous with the increment of axial position. These observations were also reported by Yilmaz and Levy [12] in their studies.



6 Conclusions

The CFD tools presented and discussed in this work makes possible to know better the gas-solid flow conditions in a vertical duct and how the secondary flow interact with the axial flow dispersing ropes. It is possible to verify the rope formation in the outer wall formed just after the elbow due to the centrifugal forces and the dispersion toward with the increment of the axial position. Furthermore, the inviscid model showed to be a relevant tool to be applied for the two-phase flow modeling due to its simple characteristics when compared with E-L models and some others E-E models in the presence of several functions for the solid pressure and viscosity such as KTGF.

References

[1] Akilli, H., Levy, E. K. & Sahin, B., Gas-solid behavior in a horizontal pipe after a 90º vertical-to-horizontal elbow. Powder Technology, 116, pp. 43-52, 2001.

[2] Bilirgen, H. & Levy, E. K., Mixing and dispersion of particle ropes in lean phase pneumatic conveying. Powder Technology, 119, pp. 134-152, 2001.

[3] Coelho, R. M. L. & Massarani, G. Fluidodinâmica de partículas: ainda sobre correlações em base aos dados experimentais de Pettyjohn e Christiansen. Relatório LSP/COPPE 1/96, 1996.

[4] Decker, R. K.; Modelagem e simulação tridimensional transiente do escoamento gás-sólido, Campinas: Faculdade de Engenharia Química da Unicamp, Tese, 2003.

[5] Huber, N. & Sommerfeld, M., Modeling and numerical calculation of dilute-phase pneumatic conveying in pipe systems. Powder Technology, 99, pp. 90-101, 1998.

[6] Kuan, B., Yang, W. & Schwarz, M. P., Dilute gas-solid two-phase flows in a curved 90° duct bend: CFD simulation with experimental validation. Chemical Engineering Science, 62, pp. 2068-2088, 2007.

[7] Mohanarangam, K., Tian, Z. F. & Tu, J. Y., Numerical simulation of turbulent gás-particle flow in a 90° Bend: Eulerian-Eulerian approach. Computers & Chemical Engineering, 32, pp. 561-571, 2008.

[8] Shiller, L. & Neuman, A. Z., A drag coefficient correlation, Verein Deutschen Ingenieure Zeitung, 77, pp. 318–320, 1933.

[9] Tsuji, Y. & Morikawa, Y., LDV measurements of an air-solid two-phase flow in a horizontal pipe. J. Fluid Mech., 120, pp. 385-409, 1982.

[10] Tsuji, Y., Morikawa, Y. & Shiomi, H., LDV measurements of an air-solid two-phase flow in a vertical pipe. J. Fluid Mech., 139, pp. 417-434, 1984.

[11] Yang, W. & Kuan, B., Experimental investigation of dilute turbulent particulate flow inside a curved 90° bend. Chemical Engineering Science, 61, pp. 3593-3601, 2006.

[12] Yilmaz, A. & Levy, E. K., Formation and dispersion of ropes in pneumatic conveying. Powder Technology, 114, pp. 168-185, 2001.



Modelling of solidification of binary fluidswith non-linear viscosity models

T. Wacławczyk, D. Sternel & M. SchaferDepartment of Numerical Methods in Mechanical Engineering,Technische Universitat Darmstadt, Germany

Abstract

This paper deals with the numerical modelling of multiphase flows with phasetransition during solidification of binary alloys. First, verification of the effectiveviscosity assumption in the regime of moving solid (equiaxed crystals) and liquid(melt) for large solid mass fractions is presented. In order to extend the effectiveviscosity model to the region of stationary solid (columnar crystals); non-lineardependence of the viscosity on the solid mass fraction and the shear velocity isintroduced based on the experimental evidence. The proposed formulation is usedin a numerical study of the metal alloy solidification in a rectangular cavity.Keywords: multiphase flow, solidification, mushy zone, phase change.

1 Introduction

During solidification of binary fluids, e.g. metal alloy Al–Si, Al–Cu, for a certainrange of temperatures and compositions a mushy zone is created (cf. Refs. [1, 2]).The influence of the mushy zone morphology on the flow field can be modelled bytwo approaches: porous media model, where the Carman-Kozeny relation coupleslocal porosity of the medium with the local liquid fraction, or direct modificationof the local fluid viscosity relatively to the local solid fraction, see e.g. Refs. [3,4],respectively. These two physical models are valid in different regions of the mushylayer: the region of stationary, columnar crystals where the solid velocity is us = 0and the region of the equiaxed crystals where the velocity of solid is assumed tobe equal to the velocity of the melt us = ul , see Fig. 1. The difficulties in themodelling of the mushy zone by a one-field model arise during the transition fromthe mixture velocity (for fluid and solid) to the interstitial velocity in the porouszone, see Refs. [5, 6].



Figure 1: Schematic presentation of the mushy zone with two characteristicregions: columnar dendrites us = 0 and advected equiaxed crystalsus ≈ ul .

The disadvantage of the first method is the necessity of the permeabilitycoefficient estimation by experimental or theoretical investigations. This issue isnot straightforward because the mushy zone has a complex morphology dependenton the material and external conditions. A porous media model should be used onlyin the region where the solid phase is stationary.

In the case of the second approach, an assumption about the continuous changeof the material properties across the solid, the mushy layer and the liquid is used. Acommon approach employs a lever rule to approximate density and viscosity in themushy zone together with a linear dependence of the solid fraction on temperatureinherited from the linearised phase change diagram, see Fig. 2 and Refs. [5, 6].The linear dependence of solid and liquid viscosities based on the lever rule is notappropriate since the solid viscosity μs can not be defined. An alternative for thelinear viscosity approximation was given in Ref. [7], however, it does not take intoaccount the dependence of the viscosity on the shear velocity.

In this paper, based on the experimental evidence from Ref. [8] and theparametrisation study from Ref. [9], we postulate an alternative non-linear depen-dence of the viscosity on the solid fraction and the shear velocity. The viscosityvalues used for partial validation of the model were obtained during measurementsin a cylindrical rheometer, see Ref. [8]. In section 3 experimentally obtained data,i.e. viscosities as a function of solid fraction and the shear velocity, were usedto confirm the possibility of a modelling of multiphase systems by the effectiveviscosity model.

Numerical simulations presented in this paper were carried out with the com-mercial software Star-CD where the user coding was used for implementation of



the physical models. The proposed viscosity data parametrisation is tested in thecase of the binary alloy solidification in a rectangular cavity, see Ref. [3].

2 Description of the solidification model

A solidification model implemented in the Star-CD (ver. 4.06) commercial soft-ware is employed. The description of the model is given in supplementary notesdistributed together with the program, for this reason here only a short comparisonof the model used and other models presented in the literature, see Refs. [1, 3, 5],is given.

The set of the conservation equations that describe the mixture medium isobtained by the volume averaging under additional assumptions: equal solid/liquidvelocity us = ul , equal solid/liquid density ρs = ρl = ρ and the mechanicalequilibrium ps = pl = p. The set of conservation equations consists of: themomentum, the continuity, the energy and the species mass fraction transportequations:

∂ (ρui)

∂t+ ∂

(ρujui

)∂xj

= − ∂p

∂xi+ ∂

∂xj

[μ

(∂ui

∂xj+ ∂uj

∂xi

)− 2

3μ∂ul

∂xlδij

]

+ ρref gi[1 − β(T − Tref )

],

(1)

∂ρ

∂t+ ∂

(ρuj

)∂xj

= 0, (2)

∂ (ρh)

∂t+ ∂

(ρujh

)∂xj

= ∂p

∂t+ uj

∂p

∂xj+ τij

∂ui

∂xj+ k

∂2T

∂x2j

, (3)

∂ (ρC)

∂t+ ∂

(ρujC

)∂xj

= D∂2C

∂x2j

. (4)

One can notice that the mathematical model in the Star-CD allows for simulationof the material compressibility, therefore, modelling of the shrinkage effects ispossible. In the mushy zone the variables and the material properties in the aboveequations represent quantities obtained by volume averaging (see Ref. [5, 6]).Hence, the velocity ui , the density ρ, the viscosity μ, the enthalpy h, the thermalconductivity k, the species mass fraction C and the species diffusion coefficientD represent mass averaged quantities of the solidified alloy (solid) and the melt(fluid) mixture:

φ = Csφ + Clφ, (5)

where Cs = ρscs/ρ and Cl = ρlcl/ρ are solid and liquid mass fractions, cs andcl denote solid and liquid volume fractions and φ represents the aforementionedvariables and the material properties of the mixture. The local thermodynamicequilibrium assumption allows to define the temperature T as the equilibrium



a) b)

Figure 2: The phase change diagram a) typical for binary fluids of α + β com-position, b) approximation of the phase diagram currently used in theStar-CD valid for the constant melt composition.

temperature. Therefore the buoyancy effects can be approximated using theBoussinesq assumption, see Eq. (1). The enthalpy in the solid, liquid and mushyzone are calculated as follows, see Fig. 2b:

hs =cpsT : T ≤ Ts

cpsTs : Ts < T < Tl, (6)

hl =cplT + (cps − cpl)Tm + L : T ≥ Tl

cplTl + (cps − cpl)Tm + L : Ts < T < Tl, (7)

where L is the latent heat of fusion, cpl, cps are specific heats of the liquid andsolid, respectively, and Tm = (Ts + Tl)/2 where Ts , Tl are solidus and liquidustemperatures.

The Star-CD code uses a simplified phase change diagram where the enthalpyis calculated for constant (initial) composition of the binary fluid, see Fig. 2b. Thiskind of simplification is the source of the main difference between solidificationmodel known in literature and the procedure applied in Star-CD. The equationused for the determination of the liquid volume fraction cl is deduced from thesimplified phase change diagram, cf. Fig. 2b:

cl = T − Ts

Tl − Ts, cs = 1 − cl, (8)

where solidus Ts and liquidus Tl temperatures are constant and must be suppliedby the user, whereas the calculation of the liquid mass fractions from the phasechange diagram in Fig. 2a requires information about the local composition C



since it influences its solidus and liquidus temperatures, see Refs. [5, 10]:

Cl = C − Cαs

Cαl − Cαs, Cl = ρlcl

ρ. (9)

The approximation in Star-CD is based on the assumption that the compositionof the binary fluid remains constant. The main consequence of this simplificationis a direct dependence of the liquid volume fraction distribution on the temperaturesince Ts , Tl are set constant in Eq. (8). Hence, one can expect that according to Eq.(8), the cl distribution must follow isotherms. Thus, modelling of the real shapeof the solidification front is largely an approximation. One can also notice anotherimplication of the simplified model. The conservation equation (4) is no more aspecies mass conservation equation but only a solid and liquid mass conservationequation since the composition of the binary liquid is constant.

3 Verification of the effective viscosity model Cs ≤ 0.45

To verify the hypothesis about the applicability of the effective viscosity modelin the case of large solid fractions, experimental data obtained from viscositymeasurements of aAl−Si metal alloy carried out in Ref. [8] were used. During theexperiment, a cylindrical rheometer was placed in an electrical thermostat allowingto sustain constant temperature around it. Measurements of the torque M on thegrooved rod rotating inside the cylinder allowed to calculate value of the tensionacting at the surface of the rod, see Eq. (10). The number of the revolutions perminute n was used to calculate the shear velocity γ , cf. Eq. (11). The ratio of thetension τ and the shear velocity γ gives the viscosity of the multiphase fluid, seeEq. (12):

τ =(

1

2πLR2i CL

1 + δ2

2δ2

)M, (10)

γ =(

1 + δ2

δ2 − 1

π

30

)n, (11)

μ = τ

γ, (12)

where δ = Ra/Ri is the ratio of the cylinder radius Ra and the rod radius Ri . Itis important to notice that Eqs. (10–12) are valid only under the assumption of alinear velocity profile between the external cylinder surface and the rotating rod,i.e. a Couette flow assumption.

Since the whole cylindrical rheometer was placed inside of the thermostat, itwas possible to assume that the temperature and thus the solid mass fractionare constant during the simulation. This simplification allows to employ onlythe momentum and continuity equations, see Eqs. (1–2) respectively, where allvariables and material properties are defined for the multiphase mixture. Theisothermal assumption allow for relatively straightforward simulation of this



multiphase flow. The single simulation point in Fig. 4 (left) corresponds to thesingle viscosity measurement for given shear velocity γ see Eq. (11). Experimentaldata for two temperatures 594C, 605C were used, density ρ = 2700 kg/m3 wasassumed constant during all simulations. After the convergence criterion for thetorque calculated on the grooved rod is obtained (variation of the z-componentof the total torque was monitored) the new values of the tension, see Eq. (10)and then viscosity, see Eq. (12), are calculated and compared with experimentalfindings, see Fig. 4. Obtained results show that when using exact experimentaldata, the flow in the multiphase system is accurately modelled with the effectiveviscosity assumption. The difference between the experimental value and thenumerical solution ε = 1 −μcf d/μexp can be defined due to the knowledge aboutthe measured viscosities μexp . One can notice that the value of ε grows withincreasing shear velocity γ , see Fig. 4 (left). In the case of the solid mass fractionCs ≈ 45%, the error ε starts to grow from the value ε = 3.9%, γ ≈ 490 1/suntil ε ≈ 22% for γ ≈ 520 1/s the last computational point in Fig. 4 (left,top). For the solid mass fraction Cs = 33% the error ε grows from ε ≈ 3.9%,γ ≈ 500 1/s until the solution with the experimentally obtained viscosities doesno more follow the experimental data, i.e., ε ≈ 44%, γ ≈ 600 1/s. The sourceof the error variation has a twofold nature. First of all, in the case of large solidfractions, for larger shear velocities γ the non-slip condition at the wall of thecylindrical rheometer and the wall of the grooved rod (rod is grooved to avoid slipeffect) can be no more satisfied. Secondly, the important factor that limits the rangeof the measurements in the cylindrical rheometer is the development of the Taylorinstability. This phenomenon occurs for large values of the revolutions per minute

a) b)

Figure 3: The cylindrical rheometer a) characteristic dimensions of the groovedrod and the cylinderRa = 13mm,Ri = 10mm, b) cross section throughthe numerical model build from about 8 × 105 CV’s.



V[m/s]

V[m/s]

Figure 4: The viscosity obtained from simulations in the cylindrical rheometercompared with experimental data (left) and velocity magnitude for thelast computational point (right) in the case of the two solid mass fractionsCs = 45% (top), Cs = 33% (bottom). In the case of smaller solid massfraction Cs = 33% vortices developed due to the Taylor instability arevisible.

n and prevents accurate viscosity measurements since the Couette flow assumptionis no more satisfied, see Fig. 4. The accurate prediction of this effect in multiphasesystems is difficult, since it is directly connected with the variable viscosity of themixture fluid. The numerical modelling of the flow in the cylindrical rheometershould be further investigated since it might become a valuable verification toolfor the experimental investigations.

4 Extension of the viscosity model for Cs > 0.45

The key problem during modelling of the mushy zone by the effective viscositymodel is the extension of this assumption for large solid fractions Cs > 0.45since in this case viscosity measurements in the rheometer are not possible. In fact,



α = 0.999

α = 0.987

α = 0.95

Figure 5: Two parametrisations of the effective viscosity: error function erf andpiecewise functional pwf viscosity parametrisations. In the case of theerf parametrisation three differentα values were used 0.95, 0.987, 0.999,coefficients β = 1e − 4, γ = 9.81 are the same for all three cases. Thesecond pwf parametrisation uses μl = 1Pa · s, μs = 9e + 4Pa · sand Cs,cr = 0.6. Notice that the relative viscosity erf parametrisationμr = μ/μl is almost independent of the α value when Cs ≤ 0.5.

viscosity of the solid μs can be considered only as an auxiliary parameter that doesnot possess physical meaning. For this reason the commonly used approximation:

μ = Csμs + (1 − Cs)μl, (13)

where μl is the liquid viscosity, is not valid since μs can be an arbitrarily largenumber. Alternatives for this approach are rarely presented in the literature, twoexamples given in Refs. [4, 9] will be shortly discussed below.

In the case of the first parametrisation, the mixture viscosity μ is approximatedby a piecewise functional approach, cf. Fig. 5:

μ (Cs) =

⎧⎪⎨⎪⎩

μle4.5Cs : Cs ≤ Cs,cr

b1Cs + b2 : Cs,cr + 0.1 > Cs > Cs,cr

(1 − Cs)μl + Csμs : Cs ≥ Cs,cr + 0.1

(14)

where Cs,cr (here after Ref. [4] Cs,cr = 0.6) is the critical solid fraction value thatdefines the highly viscous zone interpreted as the columnar crystals region, seeFig. 1; b1 and b2 are two constants determined by the solution of the two equationsystem in point Cs,cr , Cs,cr + 0.1. One needs to notice that Eq. (14) employs alsothe linear dependence given by equation Eq. (13). The main disadvantage of thisapproach is its discontinuity, cf. Fig. 5, and its lack of physical justification.



The second effective viscosity parametrisation uses error function to approxi-mate variation of the effective viscosity μ (Cs):

μ (Cs) = μl

1 − α · erf

(√π

2Cs

[1 + β

(1 − Cs)γ

])−B/α(15)

where α, β, γ are parameters to set, B = 2.5, cf. Ref. [9]. The second approach isnon-linear and the error function is known to be a solution of the heat transportequation, when the initial condition is given by the Heaviside function. In thecase of solidification in the mushy zone, the aforementioned condition can beinterpreted as the jump of the enthalpy caused by the latent heat rejected from thesolid to the liquid phase, cf. Fig. 2b. Since the solid fraction variation across themushy zone is expressed by this function, see Ref. [1], and the effective viscositydepends on the solid fraction, it should also be possible to express it in terms ofthe error function.

4.1 Dependence on the shear velocity γ

To obtain a formula for the effective viscosity μ as a function of the solid fractionCs and the shear velocity γ , additional normalised variables are introduced: thenormalised viscosity μr = μ/μl and the normalised shear velocity γr = γ /γC .The values of μl , μs and γC were set to 0.02Pa · s, 500Pa · s and 1000 1/s,respectively, based on the available experimental data for five temperatures T :594C, 600C, 605C, 610C, 615C and corresponding solid fractions Cs : 0.45,0.39, 0.33, 0.25, 0.17, see Fig. 6. First, fitting of the continuous functions to thenormalised viscosities μr(γr) obtained for each solid fraction was carried out, inorder to obtainμr(γr , Cs = const.). Afterwards, the obtained functions were usedto calculate formula for the B coefficient. The remaining coefficients α = 0.988,β = 1e − 4, γ = 9.66 were set only once and are constant in the whole domain.The new, effective viscosity parametrisation μr(Cs, γr ) is given by the followingequations:

μr (Cs, γr ) =

1 − α · erf(√

π

2Cs

[1 + β

(1 − Cs)γ

])−B(γr )/α, (16)

B = γ−0.52 + 1.1. (17)



Figure 6: Comparison of the two effective viscosity parametrisations a) linearviscosity see Eq. (13), b) non-linear viscosity cf. Eqs. (16–17).

In Fig. 6 the proposed viscosity parametrisation is compared with experimentaldata and the old parametrisation given by Eq. (13). One can notice that the chosenapproach allows for relatively accurate approximation of the experimental dataunlike the original linear approach given by Eq. (13) that over-predicts viscosityvalues for Cs ≤ 0.5.

Trun = 66s T C cl

Trun = 36s T C cl

659650.6

654.8

0.990.85

0.92

Trun = 22s659

650.3

654.8

0.99

0.85

0.92

T C cl

Figure 7: Comparison of the isotherms (left) and the liquid volume fractionisolines (right) in the case of solidification in the rectangular cavity. Thetwo top figures come from the Ref. [3] at Trun = 66 s, the two figuresin the middle present the Star-CD result with the linear viscosity modelat Trun = 36 s and the two bottom figures depict the Star-CD solutionobtained with the proposed effective viscosity model at Trun = 22 s.



5 Solidification of the metal alloy in the rectangular cavity

In order to access the properties of the proposed effective viscosity parametrisationsolidification of an Al–Cu alloy (μl = 0.003 Pa · s, Ts =548C, Tl =660C andρ = 2525 kg/m3) in a rectangular cavity (20 mm × 67 mm) is chosen. A detaileddescription of the test case is given in Ref. [3].

In order to implement the viscosity model given by Eqs. (16–17) an approxi-mation of the error function by elementary function is used (see Ref. [11]) sinceerf() is not supplied by the compiler intrinsic functions library. The relative shearvelocity γr is approximated by the second invariant of the strain rate tensor thatcan be considered as the mean shear rate.

Initially, the binary alloy has constant temperature Ti =660C in the wholecavity. The convective boundary condition q = −hconv (T − Tamb) (hconv =1 kW/(m2C), Tamb =20C and T is temperature in the domain) causes a dropof the temperature close to the left side of the cavity and initialise solidificationprocess. The obtained results in Fig. 7 are compared for different run times Trunbecause it was noticed that due to the Courant number restriction a restart isrequired with a ten times smaller time step (t = 1e − 5 s). Despite this factit is possible to compare the main features of the reference solution and theobtained results. From Fig. 7 it is clear that the over prediction of the viscosityby the first model, see also Fig. 6, is responsible for the lack of deformation inthe mushy zone visible in the case of the new viscosity parametrisation given byEqs. (16–17). The temperature distribution and the isotherms obtained with thenew parametrisation are closer to the reference solution. As mentioned before, inthe case of the solidification model used in Star-CD the front of the solidificationalways follows isotherms unlike in the reference solution. Moreover, because thefirst order upwind scheme is the only available discretization for the convectiveterm in Eq. (4), smearing due to the numerical diffusion influences the frontof solidification in Fig. 7. Finally, the magnitude of the velocity generated bybuoyancy effects is closer to the reference solution in the case of the non-linearviscosity model, however, final quantitative comparison can only be performedwhen the final result is obtained.

6 Conclusions

The paper concerns the binary alloy solidification with the effective viscositymodel. It was shown that the effective viscosity assumption can be used in themodelling of the multiphase flow when the solid mass fraction Cs ≤ 0.45 showinggood agreement with experimental evidence from cylindrical rheometer. More-over, the influence of the Taylor instability on the measurements in the cylindricalrheometer was emphasised. The proposed viscosity parametrisation covers moreaccurately the distribution of the experimental data and gives a realistic solution.Further work on this subject should be devoted to the determination of reliableverification test case based on the experimental evidence.



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[9] Costa, A., Viscosity of high crystal content melts: depencence on solidfraction. arXiv:physics, 0510191v1, 2005.

[10] Le Bars, M. & Worster, M.G., Solidification of a binary alloy: Finite-element,single-domain simulation and new benchmark solutions. J Comp Phys, 216,pp. 247–253, 2006.

[11] Error function, approximation with elementary functions. Wikipedia The FreeEncyklopedia.



A simulation of the coupled problem of magnetohydrodynamics and a free surface for liquid metals

S. Golak & R. Przyłucki

Faculty of Materials Science and Metallurgy, Department of Electrotechnology, Silesian University of Technology, Poland

Abstract

The simulations of induction melting and stirring of the molten metal have focused on the flows inside the bath, while the meniscus forming on the surface of the bath has been neglected. The fact of ignoring the meniscus results in ignoring a whole range of problems caused by the phenomenon, and in consequence all quantitative analyses of the processes occurring on the metal-gas interface are significantly distorted because of the underestimation of the real free surface of the metal. A tendency to ignore the meniscus in the studies on the processes of induction melting results from the complexity of the problem. In order to correctly simulate the phenomenon we are forced to make allowances for a triple coupling of magnetic field, flow velocity field of the molten metal, and shape of the bath surface. The paper presents the methodology for conducting a simulation in which such allowances are made. Keywords: magnetohydrodynamics, meniscus, induction melting and stirring.

1 Introduction

Today, the processes of induction melting and stirring are commonly used in metallurgy. At the same time constant development of the devices utilised in these processes can be observed. In the case of induction furnaces applied for metal melting the researchers focus on increasing their unit power, with the aim to reduce the charge melting time, which in turn means higher efficiency of the furnace. The increased power of the furnace results in enhanced electrodynamic



field affecting the already molten metal. Consequently, it is hydrodynamic phenomena that acquire significance in the processes of induction melting. In effect, the research concerning the induction furnace design, which previously concentrated only on the analyses of electromagnetic and temperature fields, now must be extended to the issue of modeling the hydrodynamics of the metal bath. So far it has usually been assumed in the research that the area of the liquid metal is determined after it has been melted. In such a case there is a possibility to run a simulation of the heating process of the liquid metal in two separate stages. The first step is to determine the distribution of electromagnetic forces affecting the metal, which can be done when metal geometry, materials parameters, inductor geometry, and supply parameters are known. With the assumption that the geometry of the liquid metal is fixed the obtained distribution of the field forces is constant, too. The second step includes hydrodynamic calculations for the known geometry of the metal, for the material parameters which can influence hydrodynamics, and for the distribution of forces determined in the first step (Adler and Schwarze [1]). Owing to this most widely used methodology, the problem is boiled down to a single-phase flow, which on the one hand simplifies the calculations but at the same time is far from being realistic. In a real process of metal melting a convex meniscus occurs. It is initially dependent on the resultant of the surface energy of the crucible walls and the surface tension of the liquid metal which is depending on the liquid surface area. However, in case of induction melting the ultimate shape of the meniscus and the degree of the phenomenon are strongly influenced by hydrodynamics inside the metal induced by electromagnetic field. The main difficulty with such simulation of the induction melting process is the fact that the shape of the meniscus is affected (through the hydrodynamic phenomena caused by it) by electromagnetic field, and electromagnetic filed is in turn dependent on the geometry of the liquid metal. When one wants to run a simulation of the melting process taking into account the meniscus, the coupling between the shape of the metal bath and electromagnetic field must not be neglected. In this case the hydrodynamic simulation will concern a two-phase system of liquid metal-atmosphere. Any neglect of the meniscus in induction melting will significantly distort the principal quantitative information concerning the process, that is the power output in the charge. Besides, the information about the real shape of the meniscus is of practical use for this process as it allows a prediction of the potential problems with the barrier protection of metal, oxidation of its surface and wearing off of the crucible lining. A process similar to induction melting is induction stirring, which is often an immediate continuation of the former one. Both processes can also take place simultaneously. The goal of induction stirring is homogenizing of the liquid metal and supporting the processes occurring at the liquid metal-gas interface through providing suitable substrates near the area and transporting the products. An example of such a process might be elimination of impurities from the metal



through evaporation, as it takes place in the process of removing lead from copper (Blacha et al. [2]). Also in this process, neglecting the occurrence of the meniscus or imprecise estimation of its shape might distort the obtained results completely. First, it must be remembered that estimating the effectiveness of the metal homogenizing process is based on the determination of metal velocity field, which may be calculated correctly only in case of precisely assigned metal geometry and the distribution of electromagnetic forces. Ignoring the meniscus will determine the velocity field that is far from the real one. Still, this distortion may be seen as relatively small in comparison with the error made when estimating the phenomena occurring at the liquid metal-gas interface. The reason is practically linear dependence of the intensity of these phenomena (e.g. evaporation of lead from copper) on the surface area of contact between metal and gas. An assumption of an unreal shape of the metal surface may totally distort the quantitative measure of the reactions occurring on it. For the above reasons it can be said that precise determination of the meniscus is essential for the accurate calculation of most quantitative measures representing utility values of induction furnaces and stirrers.

2 Methodology

The software available on the market enables to run the simulations of the induction melting and stirring processes in which the coupling between electromagnetic field and flow field is taken into account. Examples of such applications packages are Ansys Fluent MHD. However, in case of induction melting and stirring velocities of the liquid metal are so small that their influence on the electromagnetic field can be ignored. That is why this model of coupling between metal hydrodynamics and electromagnetics is of no use in case of the devices discussed in this paper. When analyzing the functioning of electromagnetic furnace and stirrer, the simulation performed should allow for the coupling among electromagnetic field, velocity field induced by it, and the changes in the shape of the liquid metal free surface caused by the metal flow. Unfortunately, to the authors’ knowledge no commercial software available on the market allows such a possibility. Therefore it proved necessary to seek for another solution. The authors of this paper decided to create their own system on the basis of the existing software devoted to hydrodynamic and electromagnetic calculations. The main reason for this decision is a high level of development of these instruments, significant optimisation of their codes and their high efficiency, which are extremely difficult to obtain in one’s own solutions created by a small team. The operation of the induction furnace and stirrer was run in 2D space, which was possible thanks to the fact that the configuration considered here both from electromagnetic and hydrodynamic perspective can be regarded as axially symmetrical. The assumption was that the simulation started with the formation



of the meniscus, whose shape was dependent only on the resultant of the surface energy of the crucible walls and the total surface tension of the liquid metal. The first stage entailed the analysis of hydrostatic field through which the initial geometry of the liquid metal was determined. The meniscus shape was determined by VOF (volume of fluid) method allowing a simulation of behaviour of separate, immiscible phases (Hirt and Nichols [3]). In the study discussed here the interaction occurred between liquid metal phase, gas phase, and the crucible wall. The simulation yielded a set of points defining a curve that describes the shape of the free surface of the metal. During the next stage of the simulation the obtained geometry was used to determine electromagnetic field. The result was the distribution of electromagnetic forces acting on the liquid metal. The meshes applied in the programmes for electromagnetic and hydrodynamic calculations do not overlap. The main reason is different methods of solving the differential equations describing electromagnetic and hydrodynamic fields. In the former the finite element method is widely used, while in the latter the finite volume method is preferred because of greater universality. Additionally, since the electromagnetic and hydrodynamic phenomena are different from each other in nature, the optimal mesh for electromagnetic calculations is not optimal in case of hydrodynamic calculations. For this reason the exchange of data between the programmes for electromagnetic and hydrodynamic calculations was based on a rectangular mesh of density twice as high as the density of the meshes used in electromagnetic and hydrodynamic simulations. The decision concerning such density was taken on the basis of Kotielnikow-Shannon law of signal sampling. The values of the force-field components in the nodes of the mesh used for electromagnetic calculations were recalculated to the nodes of the rectangular mesh by bilinear interpolation. The known distribution of electromagnetic forces made it possible to proceed to the simulation of liquid metal hydrodynamics aimed at the determination of the current distribution of liquid metal velocity and the change in the original shape of the meniscus caused by electromagnetic forces. Certainly, the rectangular mesh containing the distribution of forces had to be transformed into the node mesh applied in hydrodynamic calculations. Once more bilinear interpolation was used. Unfortunately, the change in the shape of the meniscus induced by electromagnetic field causes some changes of the field itself. That is why it was necessary to go back to electromagnetic calculations in order to update the field distribution. The set of points defining the curvature of the free surface are the input data for the electromagnetic calculations programme. The main problem encountered at this point is the rate of the above updating. The update rate of field distribution during the hydrodynamic simulation is closely connected with the communication rate between the two separate programmes and with the need to conduct relatively time-consuming electromagnetic calculations.



The hydrodynamic simulation is run in the unsteady mode, meaning that it analyses the change in time. An important element is here to adjust an adequate time step. When VOF method is used to trace the free surface of the metal, a time step below 10-4 second is often required in order to ensure the convergence of solution. Since this is the updating rate of the free surface shape, the optimal solution would be to update the distribution of electromagnetic forces with the same frequency, that is 10000 times a second. However, the exchange of such amount of data between programmes and the necessity to conduct time-consuming electromagnetic calculations make the whole issue very difficult because the simulation time of the stabilization of free surface shape may even be as long as a few dozen seconds. Total time of simulation would then become excessively time-consuming. The only way out seems to be less frequent updating of the electromagnetic field distribution. In such a case another problem occurs, and namely the decision concerning the moment of calculating the forces. This decision may depend either on the selected time step or the change in the shape of free surface. Because the distribution of electromagnetic field does not depend directly on time but on the actual geometry of the metal, the method based on the monitoring of the surface shape seems to be better justified. For the sake of the simulation a scalar measure was defined which reflects the change in the shape of free surface compared with the surface for which the previous electromagnetic calculations were conducted. Exceeding the threshold value (adjusted experimentally) means the necessity to repeat the calculations.

Figure 1: Relative radial coordinate.

ρ=1.00ρ=0.75

ρ=0.50 ρ=0.25

ρ=0.00

r

z



Before the definition of the above measure is provided first the notion of relative radial coordinate ρ should be introduced. It denotes the ratio of the absolute coordinate r to the crucible radius R ratio at a given height, eqn. (1). Figure 1 shows graphic interpretation of the relative radial coordinator.

( )zRr

=ρ (1)

where: r, z – radial and axial coordinates of the point R(z) – function representing the dependence of crucible inner radius on radial coordinate On the basis of the measurements are obtained the dependencies of the distance of bath surface from the crucible bottom as function of relative radius for the original surface Ho(ρ) and the current surface Hc(ρ). Knowing these values the measure of the surface change SC can be determined:

( ) ( )∫ −=1

o

nco dHHSC ρρρ (2)

where: ρ - relative radial coordinate Ho(ρ) – dependency of the distance of original metal surface from the bottom in function of relative radial coordinate ρ Hc(ρ) – dependency of the distance of current metal surface from the bottom in function of relative radial coordinate ρ n – empirical coefficient Unfortunately, the experiment proved that the assumption that the distributions of electromagnetic forces between the consecutive updates are constant caused that the threshold value of the shape change measure had to be so low that the updating rate of the force field was only slightly lower than the time step in the hydrodynamic simulation. In any other case the convergence of solutions was not possible to obtain. This situation is easily explained by the physics of electromagnetic field, whose penetration is limited only to small distances from the upper, lower and side surfaces of the liquid metal. That is why the strongest electromagnetic forces occur in the upper and lower areas of the metal, nearby the crucible walls. Even the slightest shifts of the free surface cause significant changes in the distribution of the electromagnetic forces in its vicinity. It proved necessary to apply at least an approximate extrapolation of the field distribution based on the changes in the bath shape between the consecutive updating steps. In the simulation considered a simple mathematical extrapolation was chosen. The process of extrapolation entails a comparison of the free surface curvature for which the electromagnetic calculations were conducted with the current curvature, and on the basis of this a re-scaling of the electromagnetic field determined for the original curvature, eqn. (3).



( ) ( )( )

⋅= z

HH

FzFc

ooc ρ

ρρρ ,, (3)

where:

cF (ρ,z) – extrapolated vector of electromagnetic force at the point with the coordinates (ρ,z)

oF (ρ,z) – vector of electromagnetic force at the point with the coordinates (ρ,z) calculated from the magnetic field equations Hc(ρ), Ho(ρ) – dependency of the distance of current and original metal surface from the bottom in function of relative radial coordinate ρ The above formula is a heuristic one and it is not derived from any physical equations of electromagnetic field. However, this strictly mathematical spatial transformation observes the rule that the highest values of electromagnetic forces are encountered at the boundaries of the metal charge, and it does not change their directions. Its application resulted in a surprisingly efficient hundredfold reduction of the frequency of electromagnetic calculations, limiting the total simulation time to an acceptable amount. Figure 2 presents a diagram of the simulation methodology discussed here.

Figure 2: Schema of the method.

3 Example

An experimental simulation was run as part of the presentation of the method. The stirring process of the liquid aluminium was performed in a cone-shaped crucible in order to prove the high degree of the deformation of the liquid metal surface. Figure 3 presents a diagram of the modeled object. Table 1 presents the materials, supply and geometric parameters of the simulation.

HYDRODYNAMIC SIMULATION

ELECTROMAGNETICSIMULATION

STOP

START



A 176 percent change was recorded in the free surface area of the liquid metal (table 2). It means that if the meniscus is not taken into account in the calculations, a similar degree of error should be expected in case of all quantitative analyses of the phenomena occurring at the metal-gas interface.

`

Figure 3: Modeled object.

Table 1: Parameters of the simulation.

Quantity name Value Frequency 3 kHz

Current of source 2903 A Resistivity of aluminium 8.8 10 -8 Ωm

Volume of aluminium 5.31·10-4 m3

Density of aluminium 2375 kg/m3

Viscosity of aluminium 9,5·10-4 kg/m⋅s Surface tension 1.0 N/m

Contact angle 120° Height of crucible 0.15 m

Bottom radius of crucible 0.036 m Top radius of crucible 0.051 m

Table 2: Free surface area for different meniscus types.

Meniscus Free surface area [m2] not taken into account 0.006617

natural 0.006675 intensified by electromagnetic field 0.017046

coil

liquid

free surface

crucible



aluminium

a)

b)

Figure 4: Geometry of the liquid aluminium: a) natural meniscus, b) meniscus intensified by electromagnetic field.

Figure 5: Velocity distribution.

4 Conclusion

The methodology described here is now used by the authors in many simulations concerning the influence of various inductor designs, supply parameters and crucible shapes on the process of induction heating and stirring of metals (Golak [4, 5]). Its main advantage, that is the fact that the deformation of the liquid metal surface is taken into account in calculations, offers new possibilities of a more precise optimisation of the devices for induction heating and stirring of liquid metals. Knowing the shape of the free surface of the liquid metal and being able determine its actual size allow a more precise determination of these utility parameters of the devices that are somehow connected with the surface or generally with the shape of the molten metal. The most important of them are the power emitted in the charge and exchange of the elements between the liquid metal and the surrounding atmosphere.



Our meta-system for the type of simulation described above consists of two commercial packages, which are (a) programme for electromagnetic calculations Cedrad 2D and (b) programme for hydrodynamic calculations Ansys Fluent. However, the presented methodology of triple coupling of electromagnetic field, flow field and liquid metal geometry supported by mathematical extrapolation of force field may be based on any software created for this kind of calculations.

Acknowledgement

This research work was carried out within project No. N508 034 31/1889, financially sponsored by the Polish Ministry of Science and Higher Education.

References

[1] Adler K., Schwarze, R.: Numerical Modelling of the Evaporation Process of an Electromagnetically Stirred Copper Melt, FLUENT CFD Forum 2005 Bad Nauheim, 2005

[2] Blacha L., Fornalczyk A., Przyłucki R., Golak S.: Kinetics of the evaporation process of the volatile component in induction stirred melts, 2nd International Conference Simulation and Modelling of Metallurgical Processes in Steelmaking STEELSIM 2007, Graz, Austria, pp. 389-395, 2007

[3] Hirt C. W., B. D. Nichols: Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J. Comput. Phys., 39, pp. 201-225, 1981

[4] Golak S., Przyłucki R.: Oxidation of the surface of a liquid metal in the induction furnaces., Acta Metallurgica Slovaca 13, pp. 256-259, 2007

[5] Golak S., Przyłucki R.: The optimization of an inductor position for minimization of a liquid metal free surface, Electrotechnical Review, 11/2008, SIGMA-NOT, pp. 163–164, 2008



Ammonia concentration analysis for the steam condenser by combining two phase flow CFD simulation with condensation and process simulation

K. Karube1, M. Maekawa1, S. Lo2 & K. Mimura1 1Chiyoda Advanced Solutions Corporation, Yokohama, Japan

2CD-adapco, UK

Abstract

Ammonia corrosion in alumi-brass tubes in steam condensers can be a serious problem. It generally occurs in the high ammonia concentration area. In this case, it is planned to replace the alumi-brass tubes by higher grade material, such as cupronickel. Therefore, it is important to minimize the area to be replaced in order to keep the equipment cost down. It is known that the ammonia concentration is related to the degree of corrosion damage. We propose a hybrid analysis method to obtain ammonia concentration by combining two-phase flow Computational Fluid Dynamics (CFD) analysis and network analysis by a chemical engineering process simulator for the tube bundle. Ammonia concentration obtained by the simulation is therefore used to determine the area to be replaced by the higher grade material tubes. Keywords: condensation, process simulation, CFD, ammonia concentration distribution, network analysis.

1 Introduction

A steam condenser is installed downstream of the steam turbine to recover the exhaust steam. It is known that ammonia corrosion of the alumi-brass tubes in areas of high ammonia concentration often occurs in steam condensers. Ammonia is used in water treatment agents, such as pH adjusters and boiler compounds. Usually, the condenser has thousands of tubes that consist of both higher grade material (such as cupronickel) and lower grade material (such as



alumi-brass) for reducing equipment cost. The higher grade tube is used in the severe corrosion area. However, ammonia corrosion often occurs in the lower grade tube area when the ammonia concentration is high. It is known that the ammonia concentration is related to the actual corrosion damage. Therefore, it is very important to determine the higher grade tube area quantitatively. High ammonia concentration is caused by steam condensation. Ammonia concentration in the steam is usually very low. However, high ammonia concentration occurs during condensation governed by ammonia-water vapor-liquid equilibrium. In order to obtain ammonia concentration, coupled analysis is required for two-phase flow dynamics with condensation and ammonia-water vapor-liquid equilibrium. However, direct coupled analysis is very difficult and needs a lot of calculation time. The Computational Fluid Dynamics (CFD) method was used for simulating two-phase flow with condensation [1, 2]. However, it is very difficult to consider phase equilibrium in CFD. A chemical engineering process simulator can simulate rigorous ammonia-water vapor-liquid equilibrium but cannot simulate complicated two-phase flow patterns in the condenser. In order to solve the above problems, we proposed a hybrid analysis method to simulate ammonia concentration in the steam condenser. In this method, two-phase flow CFD analysis was used to obtain the steam-water flow in the condenser and the flow distribution was passed to the chemical engineering process simulator for network analysis. Some researchers have proposed techniques in combining CFD and process simulation [3, 4]. Although these calculation techniques were applied to the stirred tank etc, only gas and liquid flows were considered. Few simulation examples were proposed for two-phase flow with condensation, such as steam condensers. The proposed method was very useful and was used to determine the area to be replaced by the higher grade material tubes to achieve an optimum condenser design while keeping the cost of manufacturing or modification to a minimum.

2 Numerical method

For analyzing ammonia concentration, we proposed a hybrid analysis method to simulate ammonia concentration in the steam condenser. In this method, two-phase flow CFD analysis was used to obtain the mass flow distribution of steam and water in the tube bundle of the condenser and the computed mass flow distribution was passed to the chemical engineering process simulator for network analysis.

2.1 CFD method

In the CFD analysis, we tried two methods of analysis. One was a single-phase flow analysis and the other was a two-phase flow analysis. In the single-phase flow analysis only the steam flow was calculated and the steam condensation was treated as mass sink in the mass and momentum equations. In the two-phase flow analysis both the steam and water flows were calculated.



The Navier-Stokes equations were solved for the single-phase flow analysis. The Eulerian multiphase flow model was applied to the two-phase flow analysis [5]. In the Eulerian multiphase flow model, mass and momentum conservation equations were solved for both the gas and liquid phases. Since this analysis assumed isothermal condition, the energy equations were neglected. The gas phase was treated as the continuous phase and the liquid phase as the dispersed phase. The mass and momentum conservation equations used in present CFD method are given by

kmasskkk S uρα・ (1)

kpk

tkkkkkk

kkkk

SMp

α・ραα

ρα・

g

uu (2)

where and u are the phase fraction, the density, and the velocity, respectively. and t represent the viscosity stress and the turbulence stress, respectively. In the present study, the standard k - model is used as the turbulent model to estimate turbulence stress t. p and g are pressure and gravity acceleration, respectively. The subscript k denotes the phase in the two-phase flow model, where k = c represents the continuous phase, k = d represents the dispersed phase. In the single-phase flow model, subscript .k in the above equation is eliminated. And the phase fractions satisfies the following equation.

1 dc (3)

In the equation (1), Smass represents the interphase mass transfer due to the condensation of steam in the steam condenser, which is considered as a constant (Smass = 1.407 kg m-2 s-1) in the present study, simply. In the equation (2), M, which is eliminated in the single-phase flow model, represents the momentum exchange through the interface or a force per unit volume acting between phases. In the present study, only drag and lift force are considered. Hence,

LD MMM (4)

The drag force MD is estimated using following equation.

cdcdDcdD Cd

F uuuu 4

3 (5)

where d is the diameter of dispersed phase. The drag coefficient CD is estimated using the modified Schiller and Naumann [6] correlation, shown in following equation.

1000Re;44.0

1000Re0;Re15.01Re

24 687.0

d

dddDC (6)

where the particle Reynolds number is defined as below.



c

cdcd

d

uu

Re (7)

Outlet of steam

Inlet of steam

Tube bundle Area

Outlet of condensing liquid

Figure 1: CFD analysis model.

Table 1: Analysis condition summary.

Steam mass flow rate kg/h 100,000Density of steam kg/m3 0.050Density of condensing liquid kg/m3 1000.0

The lift force ML is defined as below.

)( ccdcdLL CM uuu (8)

where CL is the lift coefficient. In these analyses, the general-purpose CFD software Star-CD from CD-adapco was used, and steady state and isothermal calculation was conducted. Figure 1 shows the CFD analysis model. Around 7,000 regular fine cells were used in the model. A summary of the analysis condition is provided in Table 1. The steam flow in the steam turbine was assumed as the uniform flow distribution for the longitudinal direction of the tubes. Therefore, the two



dimensional axi-symmetry model was applied to the sectional plane of the tube bundle. Inlet was located at the top of model and steam flows towards the outlet at the center of the tube bundle. Condensing liquid flows towards the outlet located at the bottom of the model. Steam was condensed as it moves from outside to inside of the tube bundle. The gray part in Figure 1 shows the tube bundle area. Thousands of tubes contained in the tube bundle area of the steam condenser are simply represented by using porous media model. The pressure drop for the tube bundle was calculated by the following empirical correlation, which is counted as the continuous phase momentum source term Sp c of momentum equation (2) [7].

Ndb

uST

kp

08.10

16.00

2cc

)1/(

1175.025.0Re4 (9)

where bT, d0, and N are the pitch of tubes, outer-diameter of tube, and tube number per meter, respectively. And Re0 is the Reynolds number based on tube diameter, which is defined as below.

c

cc d

0

0Reu

(10)

Figure 2: Network model.

2.2 Network analysis method

In the network analysis, the tube bundle area was divided into relatively coarse cells shown as Figure 2. Each cell was modeled by the heat exchanger and flash drum module in the chemical engineering process simulator. Non-uniform cells were used (coarse cells outside the tube bundle and fine cells inside) for simplifying the model. A commercial process simulator Pro/2 (Invensys SIMSCI) was used for this analysis. In the case of single-phase flow analysis, only the steam mass flow



distribution was transferred to the network analysis. Condensed liquid flow was assumed to flow vertically downwards. In the case of two-phase flow analysis, steam and condensing liquid mass flow distributions were passed to the network analysis. The two-phase flow model was considered to have a higher accuracy than the single-phase flow model. Ammonia concentration in the condensation liquid was calculated by the network analysis. The Non Random Two Liquids (NRTL) activity coefficient model was used for the ammonia-water equilibrium relationship.

3 Results

Figure 3(a) and (b) show the steam velocity distribution in the condenser obtained by the single-phase flow analysis and the two-phase flow analysis, respectively. The calculated steam flow pattern was similar in Figure 3(a) and (b), although the highest velocity was a little bit different.

Velocity[m/s]

High

Low

Outlet of steam

Outlet of steam

(a) single-phase flow analysis (b) two-phase flow analysis

Figure 3: Steam velocity distribution.

Figure 4(a) and (b) show the distribution of the liquid phase ammonia concentration calculated by the network analysis. According to these results, it was confirmed that the ammonia concentration became higher near the steam outlet. It was considered that steam condensation started immediately as the steam entered into the tube bundle. However, ammonia condensed much later because volatility of the ammonia was much higher than steam. Therefore,



ammonia was condensed near the outlet where total steam condensation occurred. As we can see in Figure 4(a) and (b), the ammonia concentrations looked very similar, so it was concluded that the difference of steam velocity between both results did not have any significant effect on the ammonia concentration distribution.

Ammonia Concentration ratio

[-]

Outlet of steam

Outlet of steam

High

Low

(a) single-phase flow analysis (b) two-phase flow analysis

Figure 4: Ammonia concentration distribution in the tube bundle.

4 Conclusions

We proposed a hybrid analysis method to obtain ammonia concentration by combining two-phase flow analysis and network analysis for the tube bundle of a condenser. In this method, two-phase flow CFD analysis was used to obtain mass flow distribution of steam and water in the tube bundle. The calculated mass flow distribution was passed to the chemical engineering process simulator for network analysis. It was confirmed that there was little differences between single-phase and two-phase flow CFD analysis results. Therefore, single-phase flow analysis was more economical to obtain the flow distribution and made simulation easier without losing accuracy. The proposed method was very useful in condenser design or modification to determine the area where higher grade material tubes should be used to avoid corrosion problems.

References

[1] S. S. Gulawania, S. K. Dahikara, J. B. Joshia, M. S. Shahb, C. S. amaPrasadb, D. S. Shuklab, CFD simulation of flow pattern and plume dimensions in submerged condensation and reactive gas jets into a liquid bath, Chemical Engineering Science ,63, pp. 2420 – 2435, 2008

[2] S. S. Gulawania, J. B. Joshia, M. S. Shahb, C.S. RamaPrasadb, D. S. Shuklab, CFD analysis of flowpattern and heat transfer in direct contact



steam condensation , Chemical Engineering Science ,61, pp. 5204 – 5220, 2006

[3] F. Bezzo, S. Macchietto & C.C.Pantelides, A general methodology for hybrid multizonal/CFD models Part I. Theoretical framework, Computers & Chemical Engineering, 28, pp.501-511, 2004

[4] [4] F. Bezzo, S. Macchietto & C.C.Pantelides, A general framework for the integration of computational fluid dynamics and process simulation, Computers & Chemical Engineering, 24, pp. 653-658,2000

[5] M. Yasuhara & H. Daiguji, Numerical Fluid Dynamics-Basic and Applications-, Tokyo University Inc.: Tokyo, pp. 325-328, 1992

[6] Schiller, L., and Naumann, A., Ŭber die grundlegenden berechungen bei der schwerkraftbereitung. Z. Vereins deutcher Ing., 77(12), pp.318-320,1933

[7] N. Suzuki, Chemical Engineering Standard five edition , Maruzen Inc.: Tokyo, pp. 276-277, 1995



An approach to the prediction of wax and asphaltene deposition in a pipeline based on Couette device experimental data

D. Eskin, J. Ratulowski, K. Akbarzadeh & T. Lindvig DBR Technology Center, Schlumberger, Canada

Abstract

An analysis of similarities of turbulent flows in a pipeline and a Couette device is performed. The transport processes in both cases are determined mainly by the boundary layer structure. A wax deposition model requires a single parameter to be determined from the Couette flow experiments. The asphaltene deposition modeling is more complicated and presented by the model framework only. The effect of the centrifugal force on asphaltene particle transport in a Couette device is investigated numerically. An approach to modeling wax deposition in a pipe based on Couette device experimental results is illustrated by a numerical example. The approaches developed can be straightforwardly applied for the transport pipeline design. Keywords: asphaltene, Couette device, deposition, pipe, precipitation, scaling, turbulent transport, wax.

1 Introduction

Prevention of both wax and asphaltene deposition are important problems of oil transport in pipelines. The mechanisms of wax and asphaltene deposition are different. The major mechanism of wax deposition is a molecular diffusion. Due to intensive heat exchange between the outer pipeline wall and the cold environment (for example, sea water) the oil temperature in a pipeline wall vicinity may fall below the “wax appearance temperature” (WAT). Then, wax crystals precipitate from the fluid phase. On one hand the wax crystal concentration gradient directed towards the wall is highest near the wall where the temperature gradient oriented oppositely is biggest. On the other hand the wax particle concentration gradient causes the opposite gradient of wax



doi:10.2495/MPF090081

molecules of the same absolute value. The latter causes the diffusion of wax molecules to the wall leading to the deposit layer formation. The growth of the deposit layer thickness is slowed down by the partial deposit removal caused by the shear flow in the vicinity of the deposit surface. There are a number of papers on wax deposition modeling. Some models describing both the diffusion and the shear removal demonstrated fairly good performance (for example, [1]). The physics of the asphaltene deposition phenomenon is fundamentally different from wax deposition. The asphaltene particles precipitate from oil when due to the friction losses the pressure in the pipeline drops below the “asphaltene precipitation onset pressure”. Precipitated particles grow due to the molecular diffusion and the particle-particle aggregation, and eventually reach the wall forming the deposit layer. The asphaltene particles are usually characterized by a wide size distribution because they are prone to forming agglomerates, sizes of which may reach tens of microns. The major mechanisms of particle transport to the wall are (for example, [2]): 1) turbulent and Brownian diffusion; 2) turbophoresis. When a particle collides with the wall it sticks to the wall if the van der Waals attractive force is sufficient to prevent the particle removal by the shear flow. There are many papers devoted to modeling particle deposition in pipes. Most of them are concentrated on the particle transport to the wall only [2, 3]. There are no physically justified models for asphaltene deposition available in literature. Since modeling both wax and asphaltene deposition is associated with a number of difficulties and uncertainties we employed a Couette Device for imitation of the complex deposition processes. In this device the inner cylinder is rotating while the outer one is immobile. The deposit layer is formed on the outer wall.

2 Hydrodynamic similarity

For providing similarity of the deposition process on the wall in a Couette device to that in a pipe the hydrodynamic conditions in the wall vicinity should be similar. For wax deposition the hydrodynamic similarity provides similarity of the shear removal process. For asphaltene deposition the hydrodynamic similarity guaranties the similarities of both particle transport and the probability of a particle sticking to the wall. Note that in the case of wax deposition the temperature gradient at the wall should also be the same as that at the pipe wall. The majority of transport pipelines operate under turbulent flow conditions. A Couette device imitating such a flow should also be run under similar turbulent conditions. Both pipe and Couette flows are relatively simple shear flows. The flow structure in such geometries can be considered as composed of a boundary layer flow and a core flow. Conventionally, the boundary layer is considered as consisting of a laminar sub-layer, a buffer layer and a turbulent boundary layer [4]. Within the laminar sub-layer the momentum transport is mainly controlled by

the molecular viscosity. The thickness of this layer is evaluated as 5L [4],



where *f u is the conventional non-dimensional thickness calculated

assuming that the Reynolds number, based on the layer thickness and the

velocity on its boundary, equals unity; fw*u is the friction velocity; f

is the fluid kinematic viscosity; f is the fluid density; w is the shear stress at

the wall. The velocity distribution within the laminar sub-layer is linear. Within the buffer layer the momentum transport is controlled by both the molecular viscosity and the turbulence. The buffer layer thickness is usually

evaluated as bB . Different authors use different values of the constant b.

Schlichting and Gersten [4] employed 65b . The velocity distribution within the buffer layer can be described by either a turbulence model or by a so called wall function. The wall function is a unique normalized velocity distribution valid for a wide range of flow parameters. The wall function for the buffer layer

in a Couette device can be found in [4] formulated as yfu , where

*uuu ; u is the circumferential flow velocity, yy is the normalized

coordinate; y is the coordinate determining a position within the boundary layer

(y=0 at the wall). This normalized velocity distribution in the buffer layer can be employed for a pipe flow as well. The momentum transport within the turbulent boundary layer is controlled by turbulence only. Maintaining high accuracy it can be assumed that the turbulent boundary layer extends to the channel center (this assumption is valid for both Couette and pipe flows). Then the velocity distribution in a core flow can be calculated by applying the Prandtl mixing length model. The velocity distribution in a pipe can be also accurately described by a power law function (see [4]). As one can see from the above analysis the thicknesses of the laminar sub-layer and the buffer layer are functions of the shear stress at the wall, w , and the

fluid viscosity, f . Let us assume that the particle transport to the wall is not

affected by inertial forces (particles are relatively small). Then the hydrodynamic similarity of two near-wall flows is obtained if the shear stress at the wall, w ,

and the wall temperature, determining the fluid kinematic viscosity, f , are the

same. The shear stress at the pipe wall is calculated as [4]:

8

Uf

2

fw (1)

where f is the Fanning friction factor that is a function of the pipe Reynolds number and the surface roughness and U is the superficial flow velocity. Usually, the maximum roughness of the transport pipe walls is below m50 .

Our calculations show that in the majority of flow regimes the transport pipelines are hydraulically smooth. Moreover, at the initial stage of the deposition process the cavities between asperities forming roughness are filled with deposit material, i.e., after a relatively short time the pipe surface is covered with



deposit. A newly formed surface is hydraulically smooth therefore for calculation of the Fanning friction factor the Blausius correlation for a smooth pipe can be employed [4]. Based on the experimental particle velocity distribution in the Couette device buffer layer [4] and the velocity distribution in the core flow obtained on the basis of the Prandtl mixing length approach [4] we derived the analytical expression relating the non-dimensional torque G applied to the Couette device rotor and the Reynolds number cRe :

GlnG

Rec (2)

where L/TG 2ff is the non-dimensional torque, L is the Couette device

height, 0r , R are the inner and outer radii of a Couette device respectively,

f00c rRrRe is the Couette device Reynolds number, LR2T 2w

is the torque, Rr0 is the Couette device radius ratio,

1

21

,

1

1ln

12 , is the inner cylinder angular velocity, 406.0 .

Equation (2) is in a good agreement with the experimental data for 13000Rec [5].

Thus, the rotation speed of a Couette device providing the shear stress at the outer wall that is equal to the stress at the pipe wall (Eq. (1)) is easily calculated by Eq. (2). For asphaltene deposition it is important to consider the mechanism of a particle interaction with the wall. As it was mentioned above the main mechanisms of particle transport to the wall are turbulent and Brownian diffusions, and turbophoresis. Brownian diffusion is the dominating mechanism within the diffusive boundary layer only. The thickness of the diffusive layer can be evaluated by assuming that the Peclet number, calculated on the basis of the diffusive layer thickness and the velocity at the layer boundary, equals unity ( 1DuPe BddB ):

B

Ld

Sc~

(3)

where BfB DSc is the Schmidt number, BD is the Brownian diffusivity.

Since for regular hydrocarbons and asphaltene particles the Schmidt numbers

are usually very large (up to 910 ) the diffusive boundary layer is much thinner than the hydrodynamic boundary layer. In the majority of practical cases the Brownian diffusion can be ignored. The turbulent diffusion leads to the particle dispersion over a flow domain and plays an important role in particle transport. The turbulent diffusion decreases with increasing the particle size [2]. The turbophoresis is a



phenomenon caused by the sharp decrease in turbulent kinetic energy in the wall vicinity. Due to this phenomenon particles acquire velocities directed to the wall. The turbophoretic effect increases with increasing the particle size [2]. Fortunately, as it will be shown below employing a Couette device for imitation of the deposition in a pipe allows simplified modeling the complicated transport processes.

3 Influence of the centrifugal force on particle transport in a Couette device

The centrifugal force in a Couette flow may lead to particle stratification in the Couette device, i.e., an uneven distribution of particles across the gap. This effect must be understood and quantified to properly imitate the deposition in a pipe by using a Couette device. Let us calculate the particle concentration distribution in a turbulent Couette flow assuming the concentration distribution is steady-state. We will neglect the turbophoresis effect because 1) it reveals itself only in the wall vicinity and 2) it occurs in both pipe and Couette flows. Then the convection-diffusion equation takes the form:

0vcdr

cdD trP (4)

where pD is the particle diffusivity in a turbulent flow, mccc ; mc is the

mean particle concentration; trv is the particle drift velocity in a fluid under

action of the centrifugal force. Since the amount of particles deposited is small the total volume of suspended particles in a Couette Device is assumed to be constant. Thus, the boundary condition for Eq. (4) is a volume conservation equation for solids:

20

2R

r

rR5.0rdrc

0

(5)

Thus, the distribution of the relative concentration c does not depend on mc .

Assuming that the particle circumferential velocity equals that of the fluid (the assumption is valid for relatively small particles) the particle drift velocity in radial direction can be evaluated by the Stokes settling equation in a centrifugal force field:

f

fs

22s

tr 18r

rud

v

(6)

where sd is the particle size, f is the fluid dynamic viscosity, s is the particle

density. Equation (6) was derived at ignoring the particle fluctuation velocity due to turbulence. However, it provides reasonable accuracy for relatively small particles, such as asphaltene particles, which do not exceed a few tens of



micrometers. These are characterized by a relatively low density

( 3s mkg1200 ). Our evaluations show that the particle diffusivity pD in

this case is close to the fluid turbulent diffusivity (the eddy diffusivity tD ). The

latter is numerically close to the turbulent kinematic viscosity t because the

transport of momentum, mass and heat in a turbulent flow has the same mechanism [4]. Usually, it is assumed that ttt DSc is in the range 0.8 – 1.0.

Note that the eddy and the turbulent thermal diffusivities are equal to each other.

0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.0280

1

2

3

4

5

6

Radial Coordinate, m

C/C

m

Particle Size = 30 micronsParticle Size = 50 micronsParticle Size = 100 microns

Rotation Speed = 2948 rpmParticle Density = 1200 kg/m3

Fluid Density = 800 kg/m3

Fluid Viscosity = 1 cp

Figure 1: The normalized distributions of the volume concentration of asphaltene particles of different sizes along the radius of a Couette device.

Within the present research we will employ the empirical equation for the

eddy diffusivity in the wall vicinity ( 45y ) that can be found in [6]. The eddy

diffusivity in the core flow can be calculated by the Prandtl mixing length model with reasonable accuracy (for example, [4]). In Figure 1 we showed the normalized particle concentration distribution by volume vs. the Couette device radius calculated for the different particle sizes:

50,30ds and 100 m . The dimensions of the Couette device

were mm14r0 , mm28R and mm70L . The fluid viscosity was

cp1f , the fluid density, 3f mkg800 . The particle density was



3s mkg1200 . The relatively high rotation speed ( rpm2948 ) was

selected for calculations. One can see that the concentration distribution is relatively uniform for the m30 particles, but an increase in the particle size

leads to rapid strengthening of particle stratification. Since we expect that not very big asphaltene particles (probably smaller than m20 ) mainly contribute into the deposition there is a high possibility that

the centrifugal stratification can be ignored in a deposition study by a Couette device.

4 Calculation principles of the deposition process

As it will be shown below employing a Couette device for deposition enables us to simplify modeling the complicated mechanisms of the particle transport and the deposit layer formation.

4.1 Wax deposition calculation

Let us characterize wax deposition in a pipeline of a given diameter D at the known superficial velocity U . We employ a hypothesis that in the wall vicinity the precipitated particles are in thermodynamic equilibrium with the fluid. In this case the concentration distribution of precipitated wax particles near the wall is determined by the temperature distribution. Then the wax molecule flux to the wall can be evaluated as [1]:

RrRTTm

Rrmw dr

dT

dT

dcD

dr

dcDq

(7)

where mD is the wax molecular diffusivity; drdc is the gradient of the wax

molecular concentration; dTdc is the rate of the wax molecular concentration

change with temperature that can be measured or calculated. Not the whole wax amount transported to the wall by the molecular diffusion will deposit due to shear removal. The removal rate depends on the shear stress at the wall and the rheological properties of the deposit layer. Currently, there is no clear understanding of mechanism of this phenomenon. The shear removal effect can approximately be taken into account by replacing the molecular diffusivity with an effective diffusivity, , that can be identified from a Couette device experiment. This approach is plausible since it is difficult to separately quantify the oil molecular diffusivity and the shear removal rate. The wax flux contributing to the deposit growth is calculated as:

RrRTTRrd dr

dT

dT

dc

dr

dcq

(8)

where is the effective diffusivity of wax molecules. Note that for this application the Couette device is equipped with a heater incorporated into the inner cylinder, and a cooling jacket mounted over the outer



cylinder. This design allows maintaining the temperature of the outer wall constant and below WAT as well as controlling the temperature gradient at the wall to imitate deposition conditions in a pipe. The amount of wax deposited in the Couette device during the time t, at neglecting the wax depletion effect, is calculated as:

tRL2

dr

dT

dT

dctRL2qM s

RrRTTsdd

(9)

Thus, if the deposit amount in a Couette device is accurately measured, the effective diffusivity can be straightforwardly calculated from Eq. (9). Let us illustrate how this technique can be applied for calculating the deposition layer thickness in pipe. To determine we need to run a Couette device experiment. For providing the shear stress at the outer wall equal to that at the pipe wall Eqs. (1) and (2) have to be used to calculate the required inner cylinder rotation speed. For providing the equality of the temperature gradients at equivalent hydrodynamic conditions in the wall vicinity the heat flux through the wall of the Couette device should be equal to that through the pipe wall. The heat balance equation for pipe flow can be written as follows:

UDc

TTk4

dx

dT

pf

wbbwb

(10)

where pc is the isobaric heat capacity of a fluid, D is the pipe diameter, bwk is

the heat transfer coefficient from the fluid to the pipe wall, bT is the temperature

in the central area of a pipe, wT is the temperature at the deposit surface

(initially, at the wall surface), x is the coordinate along a pipeline. The temperature at the deposit surface is below the wax appearance temperature. The initial temperatures 0bT and 0wT at 0xx are given.

For the illustrative purpose only we use a simplified approach. To avoid the routine calculation of the heat transfer through the growing deposit layer we assume that the temperature at the deposit layer surface wT is constant along a

pipe. This assumption to some extent takes into account an observation that the low conductivity of the growing deposit layer prevents the deposit surface from cooling. The same assumption means also that the deposit layer should be relatively thin and we can ignore an increase in the temperature wT at a fixed

pipe cross-section in time due to the deposit layer insulation effect. Then we can also assume that the physical parameters of a fluid remain constant along a pipeline. Thus, the heat transfer coefficient bwk is constant. Then Eq. (10) is

solved analytically. After performing a routine math we obtain the distribution of the temperature difference wb TTT along a pipe as:

D

x

PrRe

Nu

0 eTT

(10)



where fbwDkNu is the Nusselt number, f is the fluid heat conductivity,

ff aPr is the Prandtl number, fpff ca is the fluid thermal diffusivity.

The Nusselt number can be evaluated by the empirical equation as follows [7]:

33.08.0 PrRe027.0Nu (11) The temperature gradient at the wall is calculated as:

f

bw

f

bw

Rr

Tkq

dr

dT

(12)

where bwq is the heat flux to the wall.

Substituting this equation into Eq. (9), we get the following deposit flux to the wall:

D

x

PrRe

Nu

0RTTf

bw

Rrd eT

dT

dck

dr

dcxq

(13)

The deposit layer thickness is calculated from the deposit volume balance as:

1

txqx d (14)

where t is the time, is the wax deposit layer porosity.

4.1.1 Calculation examples Let us now consider a pipeline with diameter m1778.0D and length

m3200Lp in which oil of a certain chemical composition flows at a

superficial velocity sm1.3U . The initial oil temperature is set to

K350Tb and the wall temperature K322Tw . The oil dynamic viscosity

is assumed to be sPa107 3f , the density 3

f mkg843 , the heat

conductivity KmW15.0f , the heat capacity KkgJ2020cp .

The rate of the wax molecular concentration change with temperature at the wall, calculated for the known oil chemical composition by the DBRSolids

commercial software, is K11028.1dTdc 4K322T

. The wax particle

density is 3s mkg900 . The deposition experiment in the Couette device

(the dimensions were presented above) was conducted at a rotation speed of the inner cylinder set to rpm3900 . The wall temperature and the temperature

gradient were maintained the same as those in the pipeline. The two hour experiment produced mg270M of wax deposited on the outer wall. The

effective diffusivity, calculated by Eq. (9) was sm1066.1 210 that is in

line with the literature data on the molecular diffusivity [1]. In Fig.2 we showed the calculated thicknesses of the wax deposit obtained at various flow times: h100,50,10t . The wax deposit thickness linearly



increases with time (Eq. (14)) and sharply decreases along the pipeline due to rapid reduction of the temperature gradient at the wall caused by fluid cooling. Note that the effect of the deposit layer growth rate reduction in time due to the deposit insulation effect is not taken into account by the employed simplified model.

Figure 2: Distribution of the deposit thickness along a pipeline for the

different operation times.

4.2 Asphaltene deposition modeling

The forecasting of asphaltene deposition is a more complicated problem than that of wax deposition. In this paper we present the model framework only. Guha [2] suggested a robust convection-diffusion model, according to which the volume flux of particles depositing on the wall is determined as:

pyBt cVdy

dcDDJ (15)

where pyV is the particle drift velocity caused by turbophoresis, that can be

calculated for a given particle size [2]. The first right-hand side term determines the particle flux due to Brownian and turbulent diffusion, the second term due to the turbophoresis. Equation (15) requires a boundary condition at the wall that should be set in dependence on the probability of a single particle deposition after a contact with the wall. Note that no clear approach to the boundary condition formulation is available in the literature. The same author [2] showed that the deposition mechanism (diffusion or turbophoresis) is determined by the particle inertia. The particle velocity

relaxation time f2ss 18d is employed as the measure of the inertia. It is

0 500 1000 1500 2000 2500 3000 35000

1

2

3

4

5

6

7

x, m

Dep

osi

t Thi

ckne

ss, m

m

t=10 ht=50 ht=100 h

D=0.1778 mU=3.1 m/sFluid Viscosity = 7 cpEffective Wax Diffusivity=1.66 e-10 m2/s



convenient to use the dimensionless particle relaxation time: f2*u . Guha

[2] compared the deposition rate results obtained by the convection-diffusion model with the experimental data for deposition of aerosol droplets in a pipe. It was assumed in the calculations that all particles reaching the wall deposit on it.

The author [2] demonstrated that small particles ( 1.0 ) move to the wall

mainly due to the diffusion, whereas for large particles ( 1 ) turbophoresis dominates. Thus, there is a significant size range where both diffusion and turbophoresis are important. It is should be noted that according to Guha [2] in the medium particle size range ( 11.0 ) the calculated results deviate

noticeably from the experimental data. An interpretation of the Couette device experimental data for asphaltene deposition is complicated because both the particle size distribution and the deposition mechanism are not known a priori. Let us assume that only particles smaller than a certain (critical) size crd stick to the wall as a result of a particle-

wall collision. This assumption is explained by considering the force balance for a particle attached to the wall. A drag force tending to remove the particle from the wall increases with increasing the particle size, while the particle - wall van der Waals attraction force per unit mass is reduced. The flux of “small” particles to the wall can be determined as (see Eq. (15)):

5ysmallpy

small

5y

small

pd Vcdy

dcDJ (16)

where smallc is the volume concentration of “small” particles, i.e. particles

smaller than the critical size crd , smallpyV is the mean turbophoretic velocity of

small particles. For the sake of convenience we consider the particle flux at the laminar

boundary sub-layer surface ( 5y ). This flux is practically equal to the flux of

particles depositing on the wall under steady-state conditions (the continuity equation for particles within the boundary layer is 0dydJd ). Since the

particles are small the particle diffusivity is approximately equal to the eddy diffusivity ( tp DD ). According to Notter and Sleicher [6] the eddy diffusivity

at the laminar sub-layer surface is: ft 104.05yD .

Assuming that the rate of establishing the concentration distribution profile along a Couette device radius is much higher than the rate of changing the concentration of small particles in the Couette device, we obtain an equation that describes the evolution of the mean concentration of small particles in time:

c

dsmallv

smallm

V

RL2Jtq

dt

tdc (17)

where smallvq is the volumetric rate of generation of small particles, cV is the

volume of a Couette chamber.



On the basis of Eqs. (17) and (16) we formulate the convection-diffusion equation describing the deposition process:

2smallm

smallv

25ysmallpy

small

5y

small

fsmallm

smallv

smallm

1R

2c)t(q

1R

2Vc

dy

cd104.0ctq

dt

dc

(18)

where smallm

smallsmall ccc is the relative volume concentration of small

particles, is the effective deposition velocity, which is constant for given flow parameters. The initial condition for this equation is that the initial volume concentration

of small particles is zero: 00csmallm .

Since the parameter does not depend on the concentration it can be

identified from a constant pressure Couette device experiment where asphaltene particles are pre-generated. The particle generation term in Eq. (18) is zero in

this case. Assuming that the initial condition is small0m

smallm c0c we obtain the

analytical solution as:

)1(R

t2

small0m

smallm

2

ec)t(c

(19)

The deposit mass for the time t is then calculated as:

21R

t2

small0m

22s

t

0

smallms e1cL)1(RdtcRL2tM

(20)

There are two unknowns in this equation: and small0mc . To exclude the

variable small0mc we can use the results of two deposition experiments of two

different durations ( 1t and 2t ). Then the effective deposition velocity can be

determined from the following equation:

22

21

1R

t2

1R

t2

2

1

e1

e1

tM

tM

(21)

One of the most difficult problems is evaluating the rate of small particle generation. It is expected that this generation rate should change from maximum to zero due to the fluid depletion. The evaluation of this rate is possible if monitor the deposit mass while conducting the deposition experiments at a gradual pressure reduction with the same rate as that observed in the pipeline ( dxdpUdtdp ).

The effective deposition velocity and the rate of small particle generation determined by the Couette device experiment can be straightforwardly employed for predicting the asphaltene deposit thickness in a pipeline.



5 Conclusions

An analysis of similarities of turbulent flows in a Pipeline and a Couette device has been performed. It has been shown that the transport processes in both cases are determined mainly by the boundary layer structure. The wax deposition model requires a single parameter to be determined from the Couette experiments. The asphaltene deposition modeling is more complicated and only the model framework has been presented here. The effect of the centrifugal force on asphaltene particle transport in a Couette device has been investigated numerically. It has been also demonstrated how experimental data obtained in the Couette device can be used to predict the wax deposition thickness along a pipeline at different production times.

References

[1] Akbarzadeh K., Zougari M., Introduction to a Novel Approach for Modeling Wax Deposition in Fluid Flows. 1. Taylor−Couette System Ind. Eng. Chem. Res., 47(3), pp 953–963, 2008.

[2] Guha A, Transport and Deposition of Particles in Turbulent and Laminar Flow, Annual Review of Fluid Mechanics, 40, pp. 311-341, 2008.

[3] Johansen S.T., The Deposition of Particles on Vertical Walls, International Journal of Multiphase Flow, 17(3), pp. 355-376, 1991.

[4] Schlichting H., Gersten K., Boundary-Layer Theory, Springer-Verlag, Berlin, Heidelberg, New York, 2000.

[5] Lewis G.S. and Swinney H.L., Velocity Structure Functions, Scaling and Transitions in High-Reynolds-Number Couette-Taylor Flow, Physical Review, 59(5), pp. 5457-5467, 1999

[6] Notter R.H., Sleicher C.A., Eddy Diffusivity in the Turbulent Boundary Layer Near a Wall, Chemical Engineering Science, 26(1), pp. 161-171, 1971.

[7] Rohsenow W.M., Hartnett J. and Cho Y.I., Handbook of Heat Transfer, McGraw-Hill, 3-rd ed., 1998.



Large amplitude waves in a slug tracking scheme

A. De Leebeeck & O. J. Nydal Department of Energy and Process Engineering, Norwegian University of Science and Technology (NTNU), Norway

Abstract

Large amplitude roll waves are incorporated into a previously developed slug tracking scheme for two phase gas-liquid pipe flow. The applicability of the tracking scheme to large amplitude waves is demonstrated with a simplified model for the waves. The waves are modelled analogous to slugs on a moving grid with corresponding wave velocities and a pressure variation determined using an orifice type relation. Slugs and waves in the tracking scheme are separated by regions of stratified flow, which are modelled on a stationary grid using the two-fluid model. The computational scheme is described, compared to experimental data on roll waves, and some wave dynamics such as waves developing to slugs and slugs decaying to waves are demonstrated. Keywords: roll waves, tracking, two phase pipe flow, modelling.

1 Introduction

In two phase gas-liquid pipe flow, different flow regimes occur depending on gas and liquid phase velocities, fluid properties, and pipe geometries. Various numerical strategies exist for the different flow regimes in dynamic models. Slug flow, for example, can be treated with unit cell models (Bendiksen et al. [2]), in slug capturing (Bonizzi and Issa [3], Issa and Kempf [7], Renault [12]), or in tracking schemes [Taitel and Barnea [13], Nydal and Banerjee [11]). Although capturing schemes (Issa and Kempf [7], Bonizzi and Issa [3]) can model the initiation of slugs and roll waves, they require the use of fine grids which are computationally expensive and the large computational times are prohibitive for simulation in long pipelines. Tracking schemes, however, use



orders of magnitude fewer grid points. Tracking schemes can also be suitable for plug simulations (Kjølaas [9]). A combination of capturing and tracking has also been tested by Renault [12]. Large roll waves can have similar scales and behavior to slug flow in that they transport liquid and have a propagating front. A simple model treating waves as moving objects in a similar way as for slugs is therefore desired in the tracking scheme. Similarities between slugs and roll waves such as a propagating front and a sloping tail have been observed experimentally in, for example, Johnson [8]. Pressure variations across wave fronts similar to slugs have also been measured (De Leebeeck et al. [4]). These experiments are used to develop a wave model including the observed pressure variation. The slug tracking scheme of Kjølaas [9] is the starting point for incorporating wave tracking capabilities into a slug tracking scheme. Slug flow is modelled as alternating liquid slugs and bubbles with stratified flow. The two-fluid model is solved on a stationary staggered grid in bubbles, while integral momentum equations are solved in slugs on a moving grid. Before the addition of wave tracking, decaying slugs were replaced immediately with stratified flow. With the addition of wave tracking, slugs can decay into waves, modelled analogous to slugs with integral momentum equations and their own front and tail velocities. The tracking scheme of Hu et al. [6] includes wave tracking with a liquid height profile behind waves and slugs, solving the two-fluid model in combination with modelling the wave front as a hydraulic jump (Hu et al. [6]). Other models for roll waves include, for example, Johnson [8] and Holmås [5] who solve the two-fluid model with modified friction terms in roll waves. Johnson [8] assumes a sequence of repeating “maximum amplitude” waves with a sharp front and includes a unique interfacial friction factor as part of a steady state solution. The model of Holmås [5] model is dynamic and includes increased turbulence at the wave front using a modified Biberg friction model. In our scheme, we simplify waves and slugs as square objects that can be modelled dynamically on a coarse grid. The dynamics of the waves and slugs are determined from mass and momentum balances. The stratified gas regions between waves and slugs are solved with a two fluid model. A large grid gives square shaped bubbles. Slug and wave tails can be reproduced by refining the grid in the bubble region.

2 Description of the model

The wave tracking model builds on a slug tracking scheme (Kjølaas [9]) which is coded in C++ using object oriented programming techniques. Slug flow is represented in one dimension with alternating slug objects that completely fill the pipe and stratified sections including both phases as shown in figure 1A. In stratified sections, the two-fluid model is solved on a stationary staggered grid where phase velocities are determined at section borders while pressure and masses are determined at section centers. Slug sections are modelled as moving



objects where liquid phase velocity, slug length, front and tail velocities are determined from mass and momentum balances. Waves, shown in figure 1B, are modelled in a similar way to slugs as moving objects and include a pressure variation due to liquid acceleration at the wave front. In the gas phase, the pressure variation across the wave front is modelled with an orifice type relation. Assuming that the pressure variation is the same in both phases, the phase velocities in the wave can be determined from the momentum balance equations. Front and tail velocities and liquid holdup are also determined in the wave assuming a fixed length as opposed to slugs which have a variable length.

Figure 1: Schematic of models for A. slug flow and B. wave flow in the tracking scheme. The arrows indicate direction of flow. Dashed lines represent section borders. Gray – liquid phase. White – gas phase.

Gas flow in a slug is modelled using a slip relation, however, in a wave the gas phase flows through a gap between the liquid phase and the upper pipe wall. In this way, gas flow over a large wave can be thought of as similar to gas flow through an orifice, and therefore an orifice type relation is used in the gas momentum balance eqn (1) for waves. The orifice type relation, the second term in eqn (1), replaces the gas wall and interfacial friction terms. Eqn (1) is then used to determine the gas velocity in a wave by relating it to the pressure variation across the wave front.

θ

ρ

sin)()1(

)(1111

21)1( 1

2

2

gMPPAH

UUUUHH

CAH

Ut

M

gRL

frontngfront

ng

Rg

d

gg

−−−=

−−⋅

−−

−−+

∆∆

+ (1)

Using the same pressure variation across a wave front as in eqn (1), the liquid momentum balance eqn (2) is used to determine the liquid phase velocity in a wave. In the liquid phase, the main component giving pressure variation is the



acceleration of the liquid at the wave front. There is also a contribution from liquid wall friction and gravity.

θρλ

ρ

sin)()(

))((

181

1

gMUULSPPHA

UUUUAHUt

M

lnl

nllllRL

nl

nlRfront

nlll

l

−−+−=

−−+∆∆

+

+

(2)

Slugs are modelled as objects with moving boundaries, the front of a slug moves with a front velocity determined from a mass balance across the front while the tail moves with a bubble nose velocity. If the front velocity is greater than the tail velocity, the slug will grow in length, otherwise its length will decrease. Similarly, waves are modelled as moving objects but they have a fixed length of one to two pipe diameters and move with the wave front velocity. The front velocity of a wave is determined from the mass balance across the front, eqn (3), and given in eqn (4).

)()( 1front

nlRRfront

nl UUHUUH −=−+ (3)

nlR

R

Rnl

Rfront U

HHH

UHH

HU−

−−

= +1 (4)

One of the aims of the wave tracking scheme was to have a simplified model, therefore a simple wave tail speed relationship was desired. The wave tail speed is given in eqn (5).

ltail UU 2.1= (5) The factor of 1.2 allows for continuous transition between wave and slug flow. When the liquid holdup in a wave approaches unity, the liquid phase velocity in the wave approaches the mixture velocity. The bubble nose velocity or wave tail velocity is commonly related to the mixture velocity by a factor of 1.2. The mass balance equations in a wave or slug are the same, where the change in mass in a given time step is the difference in mass flux in and out. Eqns (6) and (7) are the liquid phase and gas phase mass balance equations respectively. The liquid holdup is given in eqn (8).

( )LUUMLUUMt

Mfront

nl

nltail

nl

nl

l /)(/)( 11 −−−=∆∆ ++ (6)

( )LUUMLUUMt

Mfront

ng

ngtail

ng

ng

g /)(/)( 11 −−−=∆∆ ++ (7)

l

nl

ALMH

ρ

1+

= (8)

Since wave fronts are modelled with a fixed length and they move at the wave front velocity, the wave tail velocity only appears in the mass balance eqns (6) and (7). If the front speed is larger than the tail speed the liquid mass in the wave will increase and vice versa. Therefore waves can grow or decay in amplitude.



2.1 Transitions and wave insertion

Waves can be formed from stratified flow or they can form from decayed slugs. At the transition from stratified flow waves are inserted according to the inviscous Kelvin Helmholtz stability criteria, eqn (9). Neglecting surface tension and viscous effects, stratified flow is stable if (Barnea and Taitel [1], Lin and Hanratty [10]):

0)(cos)( 2 >−−−

+ lggl

ig

g

il

l UUgS

AS

Aθρρ

ρρ (9)

On the other hand, if a wave is dying, it will be removed when the holdup in the wave approaches the holdup in the stratified section in front of it. A decaying slug will be converted to a wave when its length goes below a user defined minimum, i.e. one or two pipe diameters in length. In the reverse case where a wave grows to a slug, a wave will be converted to a slug if its holdup goes above a user defined maximum, e.g. a holdup of 0.99. Slugs can also form if two waves merge or a slug overtakes a slower moving wave.


The roll wave tracking model has been compared with experiments that were conducted in the multiphase flow laboratory at NTNU in a 16 m long, 0.06 m I.D. pipe using air and water at atmospheric pressure (De Leebeeck et al. [4]). The experiments included pipe inclinations from -1 to 3 degrees, Usg from 2 to 11.5 m/s, and Usl from 0.04 to 0.52 m/s where a mixture of waves and slugs occurred. Data for comparison include liquid holdup and pressure time traces, average wave velocities from cross correlation between holdup time traces, and pressure drop. One purpose of doing the experiments was to measure the pressure variation over a wave, as this is an assumption in the tracking model. This was confirmed in the experiments, and led to an estimate of the discharge coefficient in the orifice relation in eqn (1) of dC = 0.2 to 0.4. The simulations discussed here used a fixed grid size of 2 pipe diameters in wave fronts, a minimum of 20 and maximum of 100 pipe diameters in stratified sections. Pipe length, diameter, fluid properties, and a simulation time of 80 sec were as in the experiments. A discharge coefficient giving the best approximation of experimental wave speed and pressure variation with value

dC = 0.2 was used in the gas momentum equation for waves. The waves were inserted at a similar frequency to the experiments. Figure 2 shows a plot of wave or slug velocity for all experiments and wave velocity from the model plotted against mixture velocity. The largest velocities are associated with slugs which have larger velocities than waves. The experiments with lower velocities contained more waves than slugs. Looking at the data qualitatively, the model gives wave velocities in the same range as the experiments.



0 2 4 6 8 10 120

1

2

3

4

5

6

7

Mixture velocity (m/s)

Wav

e or

slug

vel

ocity

(m/s

)

modelexperimental

Figure 2: Experimental velocities compared with wave velocities from dynamic wave tracking simulations. For Umix < 4 m/s: mainly slug flow. Umix > 4 m/s: a mixture of waves and slugs.

0 2 4 6 8 100

100

200

300

400

500

600

Mixture velocity (Usl + Usg) (m/s)

Pres

sure

dro

p (P

a/m

)

modelexperimental

Figure 3: Experimental pressure drop compared with averaged pressure drop from tracking simulations.

In figure 3, the experimental and modelled pressure drops are compared. Although all of the simulations reproduced a pressure variation in waves, some of the pressure drops tended to be low compared to the experimental values. The



magnitude of the pressure drop depends on the number of waves and slugs in the experiment or simulation. An experiment with more slugs will have a larger pressure drop than one with fewer. All the experiments contained a mixture of waves and slugs but some of the simulations, especially downward inclined, reproduced waves which did not grow to slugs resulting in low pressure drops. One advantage of modelling waves in a tracking scheme is that coarse grids can be used allowing for longer pipe systems to be modelled. Using a coarse grid means that waves and slugs are modelled as square objects corresponding to the plots figure 4A, using a maximum stratified section length of 100 pipe diameters. Finer grids can be used, however, allowing for waves with more apparent tails, shown in figure 4B, as occurs in the experimental time traces, for example figure 4C.

Figure 4: Pressure, holdup and liquid velocity time traces where a wave

passes at a given location in the pipe. A. A coarse grid with maximum length 100 pipe diameters. B. A fine grid with maximum length 10 pipe diameters. C. Experimental. Usg = 8.0 m/s, Usl = 0.1 m/s, θ = 1 degree.

Figure 5 shows how an experimental holdup time trace compares to a simulated time trace on a fine grid at the same Usg, Usl and pipe inclination. The model time trace in figure 5B shows a mixture of slugs and waves of various sizes occurring in the pipe as well as the shape of the waves. Wave dynamics such as waves growing to slugs, or slugs decaying to waves is inherent in the tracking model. Examples of a wave growing to a slug and a

1.0e5

1.1e5

Pres

sure

(Pa)

0

0.5

Hol

dup

(-)

22 230

6

Liqu

id v

eloc

ity (m

/s)

Time (sec)

1.0e5

1.1e5

Pres

sure

(Pa)

0

0.5

Hol

dup

(-)

32 33 0

6

Liqu

id v

eloc

ity (m

/s)

A B C

1.0e5

1.1e5

16 170

0.5

Hol

dup

(-)



Pres

sure

(Pa)

Time (sec)

Time (sec)

Figure 5: Holdup time traces A. experimental and B. simulation using a fine

grid with maximum length 10 pipe diameters. Usg = 6.09 m/s, Usl = 0.18 m/s, θ = 2 degrees.

Figure 6: Pressure variation, liquid holdup and liquid velocity in a wave vs.

time for A. a wave growing to B. a slug. Usg = 6.01 m/s, Usl = 0.2 m/s, horizontal pipe.

0

10000

Pres

sure

varia

tion

(Pa)

0

1

Hol

dup

(-)

32 32.2 32.4 32.6 32.8 33 33.2 33.40

6

Liqu

id v

eloc

ity (m

/s)

Time (sec)

A B

20 25 30 35 40 45 50 55 60 65 700

0.5

1H

oldu

p (-

)

Time (sec)

20 25 30 35 40 45 50 55 60 65 700

0.5

1

Hol

dup

(-)

Time (sec)

A

B



Figure 7: Pressure variation, liquid holdup, and liquid velocity in a wave vs. time for A. a wave growing to B. a slug and then decaying to C. a wave again. Usg = 5.87 m/s, Usl = 0.13 m/s, θ = 1 degrees.

wave which becomes a slug and then decays into a wave again are shown in figures 6 and 7 respectively. The pressure variation across the wave, liquid holdup and velocity in the wave object as it moves are plotted against time in both figures. When a wave becomes a slug the pressure variation across it and the liquid velocity increase, and the holdup approaches one. In figure 7, when the slug decays again, pressure variation, liquid velocity and holdup decrease. The time traces are cut off when the wave or slug exits the pipe.

4 Conclusions

A model for large roll waves has been implemented and tested in a slug tracking scheme. The model introduces an orifice type relation for pressure variation across the wave front and a simplified relationship for wave speed in a similar way as for slug flow. Computations have been demonstrated in comparison to experimental data on roll waves in two-phase air-water pipe flow at atmospheric pressure. The model gives a reasonable approximation of wave speed and pressure variations in waves. Looking at pressure drops, modelled pressure drops

A B C



are sometimes low compared to experiments due to a difference in the number of waves and slugs in the pipe. The tracking scheme can run with a coarse grid which allows simulation in longer pipes but means that waves and slugs are modelled as square objects without tails. A finer grid allows a more physical representation of waves with tails. The model includes wave dynamics such as a wave growing to a slug or a slug decaying to a wave.

5 List of Symbols

A area, m2

dC discharge coefficient g gravity, 9.81 m/s2

H liquid holdup I.D. internal diameter L length of section, m M mass, kg P pressure, Pa S wetted perimeter, m t time, sec U velocity, m/s Umix mixture velocity, m/s Usg superficial gas velocity, m/s Usl superficial liquid velocity, m/s Greek symbols ∆ change in a given quantity λ friction factor θ angle of pipe inclination, degrees ρ density, kg/m3

Superscripts n current time step n+1 next time step Subscripts front front of a wave or slug g gas phase i interface l liquid phase L left section R right section tail tail of a wave of slug



Acknowledgement

Financial support from Total E&P Norge is gratefully acknowledged by A. De Leebeeck.

References

[1] Barnea, D. & Taitel, Y., Kelvin-Helmholtz stability criteria for stratified flow: Viscous versus non-viscous (inviscid) approaches, International Journal of Multiphase flow, 19, pp. 639-649, 1993.

[2] Bendiksen, K.H., Malnes, D. & Nydal, O.J., On the modelling of slug flow, Chemical Engineering Communications, 141, pp. 71-102, 1996.

[3] Bonizzi, M. & Issa, R.I., A model for simulating gas bubble entrainment in two-phase horizontal slug flow, International Journal of Multiphase flow, 29, pp. 1685-1717, 2003.

[4] De Leebeeck, A., Gaarder, A.H. & Nydal, O.J., Experiments on Roll Waves in Air-Water Pipe Flow, 16th Australasian Fluid Mechanics Conference, Gold Coast, Australia, 2007.

[5] Holmås, H., Numerical simulation of waves in two phase pipe flow using 1D two-fluid models, Doctoral dissertation, University of Oslo, 2008.

[6] Hu, B., Stewart, C., Manfield, P.D., Ujang, P.M., Hale, C.P., Lawrence, C.J. & Hewitt, G.F., A Model for Tracking the Evolution of Slugs and Waves in Straight Pipelines, 6th International Conference on Mulitphase Flow, Leipzig, Germany, 2007.

[7] Issa, R.I. & Kempf, M.H.W., Simulation of slug flow in horizontal and nearly horizontal pipes with the two-fluid model, International Journal of Multiphase flow, 29, pp. 69-95, 2003.

[8] Johnson, G.W., A Study of Stratified Gas-Liquid Pipe Flow, Doctoral dissertation, University of Oslo, 2005.

[9] Kjølaas, J., Plug propagation in multiphase flow, Doctoral thesis, Norwegian University of Science and Technology, 2007.

[10] Lin, P.Y. & Hanratty, T.J., Prediction of the initiation of slugs with linear stability theory, International Journal of Multiphase flow, 12, pp. 79-98, 1986.

[11] Nydal, O.J. & Banerjee, S., Dynamic slug tracking simulation for gas-liquid flow in pipes, Chemical Engineering Communications, 141-142, pp. 13-39, 1996.

[12] Renault, F., A Lagrangian slug capturing scheme for gas-liquid flows in pipes, Doctoral thesis, Norwegian University of Science and Technology, 2007.

[13] Taitel, Y. & Barnea D., Effect of gas compressibility on a slug tracking model, Chemical Engineering Science, 53(11), pp. 2089-2097, 1998.



Numerical simulation of an intermediate sized bubble rising in a vertical pipe

J. Hua1, S. Quan2 & J. Nossen1 1Department of Process and Fluid Flow Technology, Institute for Energy Technology, Kjeller, Norway 2Institute of High Performance Computing, Connexis, Singapore

Abstract

A Computational Fluid Dynamics (CFD) based front tracking algorithm is applied to investigate the rising behaviour of a single bubble in a vertical pipe with a stagnant or flowing viscous liquid. The ratio of the pipe diameter to the bubble equivalent diameter ( bDD / ) is varied within the range of 1.0∼10.0. The wall effects on the terminal bubble rising speed (U ) and shape are investigated under various flow conditions, which are characterised by the parameters Archimedes number ( Ar ), Bond number ( Bo ), and bulk liquid flow speed ( lU ). It is found that the terminal bubble rising speed (U ) relates to the bubble rising speed in an infinite domain ( ∞U ) and the pipe diameter by the formula

α)/(/)( DDUUU b∝− ∞∞ , whereα is an exponent relating to the bubble deformability, and it is found to be in the range of 1.0∼0.7 in this study. In addition, the effects of flowing liquid on the terminal bubble rising speed and shape are also investigated for different sized pipes. It is demonstrated that the bubble rising behaviour is significantly affected by the flowing liquid in the pipe with a small diameter. Moreover, the detailed flow field around the bubble is presented to understand the physics of bubble rising behaviour in a vertical pipe under various flow conditions. Keywords: bubble rising, wall effect, front tracking method, Taylor bubble.

1 Introduction

The dynamics of a gas bubble rising in a vertical pipe filled with a viscous fluid is of great importance in nuclear and process industries, e.g. petroleum, refining,



bubble columns and boiling flows. The terminal rising speed and shape of a gas bubble in a large domain without wall effects have been studied by many researchers [1–2]. The correlations obtained from such studies have been widely used in the modelling of bubbly flows [3]. However, in reality, most of the bubbly flow problems occur in a container with walls, e.g. a pipe. Unfortunately, the effects from the nearby stationary wall (pipe wall) and moving wall (neighbouring bubble) are usually neglected due to the lack of the knowledge in this aspect. Extensive experimental and numerical studies have been conducted to investigate the rising of a Taylor bubble (elongated bubble compared to pipe diameter) [4], however, studies of intermediate sized bubble rising in a vertical pipe are quite limited [5], and it is of great importance to understand the physics of the wall effects on bubble rising behaviour in pipes. It is believed that the wall plays critical roles in the regime transitions of multiphase pipe flow. For example, the transition between bubbly flow and slug flow is resulted by the changes of the relative sizes of bubbles in the liquid flows [6]. In order to qualitatively justify the wall effect on the bubble rising dynamics, such as the terminal velocity and shape, detailed studies of the wall proximity are necessary. Uno and Kintner [7] experimentally studied the effect of wall proximity on the velocity of a single air bubble rising in a quiescent liquid contained in a cylinder. Four different liquids were used, namely, distilled water, 61% glycerine, diethylene glycol, and a solution of a surface-active agent. On the basis of the experimental results, a single equation was proposed to express a velocity-correction factor in term of the ratio of the bubble diameter to pipe diameter and an empirical constant, i.e. α)/1(/ DDUU b−∝∞ , where bD is the equivalent bubble diameter, D is the pipe diameter, andα is the exponent with a value of 0.765. Numerical simulations of a bubble rising freely in viscous liquid have been performed by a number of researchers. Chen et al. [8] used a modified Volume of Fluid method to simulate a bubble rising in a stationary liquid contained in a closed, vertical cylinder. The effect of density and viscosity ratios on the bubble rising is investigated. Hua and Lou [9] proposed an improved front tracking method to simulate the bubble rising in viscous liquid. The simulation results were compared with the available experimental data. Recently, Mukundakrishnan et al. [5] studied the wall effects on a buoyant gas-bubble rising in a liquid-filled finite cylinder using a front tracking finite difference method coupled with a level contour reconstruction of the front. They made detailed simulations on the flow pattern around the bubble and the bubble shape. They presented preliminary discussions about the wall effects on the terminal bubble rising speed. Since it is of great importance to both the scientific researches and engineering applications, further studies are performed here. In this paper, a Computational Fluid Dynamics based front tracking algorithm is applied to examine the rising behaviour of a single bubble in a vertical pipe with a stagnant or flowing viscous liquid. The ratio of the bubble equivalent diameter to the pipe diameter is varied in the range from 0.1 to 1.0. The simulation is started with the basic cases of single bubble rising in a large domain without wall effects in different regimes, where different terminal bubble



shapes such as spherical, oblate ellipsoidal, dimpled ellipsoidal, skirted bubble and spherical cap, can be obtained. Then, further simulations are performed to investigate the wall effects by reducing the pipe diameter. The improved front tracking method by Hua and Lou [9] is employed for this study, as this method is extensively validated against a number of experiments on bubble rising in various flow regimes. Our simulation results about the wall effects on the bubble terminal velocity agree reasonably with Uno and Kintner’s [7] correlation when the Archimedes number ( Ar ) and Bond number ( Bo ) are relatively higher. However, we found that the exponent varies with the flow regimes. And a new correlation for bubble terminal rising speed is proposed to take into account the wall effect α)/(/)( DDUUU b∝− ∞∞ . The effects of pipe wall on the bubble shape are also qualitatively investigated. The proximity of the cylinder wall tends to elongate the bubble in the pipe axial direction. But in the surface tension dominated flow regime, the bubble shape will remain spherical.

2 Mathematical formulation and computational method

2.1 Physical problem

Figure 1: Schematic illustration of a gas bubble rising in a vertical pipe with a flowing viscous liquid.

The physical problem of a gas bubble rising in a pipe with a flowing viscous liquid is illustrated in Figure 1. The gas bubble has an equivalent diameter 3 6

πVDb = , density bρ , viscosity gµ and rising speedU . The bubble

volume is assumed to beV . The pipe has diameter D or radius R and height H . The liquid inside the pipe has density lρ , viscosity lµ , and the bulk speed of liquid flow is assumed to be lU . The flow inside the pipe is assumed to be fully developed laminar flow with a parabolic distribution: [ ].)/(10.2)( 2RrUru ll −⋅⋅= .

H

R

g

ρl , µl

ρb , µb

Db

r

z

rb

zb

Ul

U

Ul

D



The gravitational acceleration is assumed to be g . Two coordinate systems are employed in the current analysis: a stationary cylindrical system ( zr, ) used for solving the governing equations; and a moving cylindrical system ( bb zr , ) with the origin located on the bubble nose used for post-processing and analysing the flow field around the bubble.

2.2 Mathematical formulation and numerical method

In this study, it is reasonable to assume that both the gas and liquid phases are incompressible. The governing equations for the multiphase fluid flow system can be expressed as

0=⋅∇ u , (1)

gxxnuuuuu )()( )]([ ) (lffff

T dspt

ρρδκσµρρ

−+−+∇+∇⋅∇+−∇=⋅∇+∂

∂ ∫ , (2)

where u is the fluid velocity, p denotes pressure, ρ and µ stand for the density and viscosity, respectively, g is the gravitational acceleration, s stands for the arc length measured on the interface, fκ denotes the curvature of the interface, σ is the surface tension coefficient and is assumed to be a constant, fn stands for the unit normal vector on the interface, fx is the position vector on the interface, and )( fxx −δ stands for the delta function that is non-zero only when

fxx = . The governing equations can be further non-dimensionalized using characteristic length (the equivalent bubble diameter bD ) and speed ( bgD ).

0* =⋅∇ u , (3)

*****

*T*******

**

)1()(1

)]([1)(

gxxn

uuuuu

−+−+

∇+∇⋅∇+−∇=⋅∇+∂

∂

∫Γ

ρδκ

µρτρ

dsBo

Arp

f

(4)

where, bD

xx =* , bgD

uu =* , tDg

b=*τ ,

lρρ

ρ =* , bl gD

ppρ

=* , lµµ

µ =* ,

κκ bD=* , ggg =* . And the Archimedes number ( Ar ) and Bond number ( Bo )

are defined as,

l

bl DgAr

µρ 2/32/1

= and σ

ρ 2bl gD

Bo = .

The Archimedes number ( Ar ) denotes the importance of the buoyancy force over viscous force, and the Bond number ( Bo , also called Eotvos number) represents the relative importance of the buoyancy and surface tension forces. Hence, the problem of bubble rising can be specified by the following flow parameters such as density ratio ( lg ρρη /= ), viscosity ratio ( lg µµλ /= ),



Archimedes number, Bond number, and the ratio of the bubble diameter to the pipe diameter ( DDb / ), and the bulk speed of the liquid flow inside the pipe ( lU ).

Table 1: Flow conditions for the simulation cases and the predicted terminal bubble shape.

D / Db Simulation No. Ar Bo Ul 1 2 4 8

1 10 5 0 ----

2 10 50 0

3 50 10 0

4 100 50 0

5 100 100 0

6 10 50 0.2

----

7 10 50 -0.2

----

The numerical method used in this study is based on the finite volume / front tracking method developed by Hua and Lou [9], which has been extensively validated for simulating single bubble rising freely in a quiescent viscous liquid. In the current numerical simulation, a bubble is released at a short distance above the bottom of the pipe (as shown in Fig. 1), and is initially assumed to be spherical or elliptical (depending upon the pipe diameter) with a dimensionless equivalent diameter of one. The pipe is filled with liquid, and the bubble will be accelerated and move upwards due to buoyancy force. The vertical pipe height is long enough to allow the bubble to reach the steady state. In order to study the effects of the pipe wall proximity on the bubble rising, the pipe diameter is varied within the range of bb DD ⋅⋅ 0.10~0.1 , while other parameters and the fluid properties are kept constant for the simulations. Ar and Bo are varied in order to investigate the wall effects on bubble rising in different flow regimes (spherical, dimpled ellipsoidal, ellipsoidal, spherical cap, skirted bubble) while the density ratio and viscosity ratio are kept constant for all simulations ( 001.0/ =lb ρρ ; 01.0/ =lb µµ ) in this study. The flow conditions for the different simulation cases reported in this paper are listed in Table 1.




3.1 Simulation of transient bubble rising in vertical pipe

The temporal variations of bubble shape and position while it is rising in the vertical pipes with different diameters are shown in Figure 2. The liquid is initially stagnant. As the pipe diameter is reduced, the bubble inside the pipe is elongated in shape due to the constraints of the pipe wall, and the bubble rising speed is also reduced significantly. To investigate the effect of pipe wall on the bubble rising speed, the detailed flow field around the terminal bubble is shown in Figures 3(a), (b) and (c). Here, the reference system is located on the nose of the rising bubble. It shows that a falling liquid film is formed between the bubble and the pipe wall when the pipe diameter is decreased to the same order as the bubble equivalent diameter. Flow dynamics in the falling liquid film has significant effect on the bubble rising. The velocity profiles along the radial direction crossing the bubble and liquid film inside the pipes with different diameters of bD0.1 , bD6.1 and bD0.2 are shown in Figure 3(d). Here the reference system is located on the stationary pipe. As the pipe diameter becomes smaller, the liquid film thickness decreases, and the length of the liquid film increases with the bubble being elongated. A stable bubble rising speed can be obtained when the buoyancy force acting on the bubble is balanced by the drag force including the viscous shear force from the flow in the falling liquid film. It can be seen from Figure 3(d) that the liquid film thickness decreases as the pipe diameter is reduced, and the maximum downward speed in the falling liquid film increases. A higher velocity gradient is induced in the liquid film, and results in a higher viscous shear stress. Hence, the terminal bubble rising velocity becomes lower as the pipe size decreases.

Figure 2: Bubble rising behaviour in vertical pipes with different diameters (a) bDD 0.1= ; (b) bDD 6.1= ; (c) bDD 0.4= , 100=Bo , and 0=lU .

(a) (b)



(c)

Figure 3: Detailed flow field around the terminal rising bubble in different sized pipes, (a) bDD 0.1= ; (b) bDD 6.1= ; and (c) bDD 0.4= , and (d) shows the velocity profiles in the bubble and liquid film across the pipe.

3.2 Wall effects on terminal bubble shape and rising speed

It is understood that both terminal bubble shape and rising speed can be significantly affected by the relative size of the bubble to the pipe ( DDb / ). Table 1 shows the terminal bubble shapes under different flow conditions. When the pipe diameter is large enough ( 0.4/ >bDD ), the pipe wall has much less effects on the terminal bubble shape for most simulation cases. Significant effect of pipe wall on the terminal bubble shape occurs when the pipe size is small or the bubble size is large with .0.2/ <bDD Generally, when a pipe wall exists in the proximity of a bubble, the bubble will be elongated along the axial direction of the pipe. The bubble rising speed is reduced as a result of the increase of the resistance from the liquid. The variations of bubble shape and terminal speed with the relative size of bubble to pipe behave differently at different flow regimes. In the surface tension dominated regime ( 0.5<Ar and 0.5<Bo ), the strong surface tension will always keep the bubble in spherical shape. Even when the pipe diameter is of the same order of magnitude of the bubble, the spherical bubble will not deform significantly as it moves very slowly in the pipe. With

(a) (b) (c)

(d)



slightly increase of both Archimedes number ( 0.5>Ar ) and Bond number ( 0.500.5 << Bo ), an oblate ellipsoidal bubble will be observed in liquid without wall effects. When the pipe diameter becomes smaller, the bubble will be elongated and become prolate shaped with spherical cap at both head and tail. With a further increase in the Bond number ( 0.50>Bo ), the role of surface tension becomes less important in determining the bubble shape, and the balance of inertial and viscous force becomes more important, and the bubble wake starts to affect the shape of the bubble bottom. When the Archimedes number is small ( 0.200.5 << Ar and 0.50>Bo ), the bubble bottom becomes dimpled. When the Archimedes number is large ( 0.100>Ar and 0.50>Bo ), the strong bubble wake flattens the bubble bottom, creating a spherical cap bubble. Within the intermediate range Archimedes number ( 0.1000.20 << Ar and 0.50>Bo ), skirted bubbles can be observed. When the bubble is rising in a small pipe ( )1(/ ODD b ∝ ) with 0.50>Bo , the bubble will be elongated, and a falling liquid film will be formed between the bubble and pipe wall. The bubble head normally has a semi-spherical shape, and the bubble bottom may be dimpled, skirted or flattened, depending upon the Archimedes number.

0.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1.0

( D b /D ) 1.05

( U∞ -

U )

/ U∞

0.0

0.2

0.4

0.6

0.8

0.2 0.4 0.6 0.8 1.0

( D b /D ) 0.79

( U∞ -

U )

/ U∞

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.2 0.4 0.6 0.8 1.0( D b /D ) 0.72

( U∞

- U )

/ U∞

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.2 0.4 0.6 0.8 1.0

( D b /D ) 0.71

( U∞ -

U )

/ U∞

Figure 4: The effects of pipe wall on the terminal bubble rising speed under different flow regimes.

The wall effects on the bubble rising is investigated in comparison with the corresponding case of bubble rising in an infinite liquid without a wall. Hence, it is reasonable to express the relative change of the bubble terminal velocity ( ∞∞ − UUU /)( ) as a function of the ratio of bubble size to the pipe diameter

Ar=10 Bo=5

Ar=10 Bo=50

Ar=100

Bo=50 Ar=100 Bo=100

(a) (b)

(c) (d)



( DDb / ). Figure 4 shows the correlation between ∞∞ − UUU /)( and DDb / under different flow regimes. It can be concluded that the wall effects on the terminal bubble rising speed follows the correlation α)/(/)( DDUUU b∝− ∞∞ , where α is an exponent depending on the flow regime. It is found that the exponent α is about 0.7~0.8 when 0.50>Bo .

3.3 Effect of pipe flow on bubble rising

Pipes are generally used to transport gas and liquid. Therefore, investigation of the effect of pipe flow on the bubble rising behaviour is of great engineering interest. Figure 5 shows the detailed flow field around the bubble in different sized pipes with liquid flowing upwards ( 2.0=lU ) or downwards ( 2.0−=lU ). When the pipe diameter is large enough, the terminal bubble shape is not affected by the upward flow or downward flow in the pipe. Figure 6(a) shows the velocity profile along the radial direction crossing the bubble and at the far field away from the bubble (D/Db = 4. 0). From Figure 6(a), it can be concluded that the terminal bubble rising velocity can be estimated by adding the bubble rising speed in stagnant liquid and the pipe flow speed. On the contrary, when the pipe diameter becomes small enough, a liquid film is formed between the bubble and pipe wall. The liquid flow inside the pipe starts to affect the bubble shape and bubble moving speed. As shown in Figure 5, the upward liquid flow enhances the bubble rising speed when D/Db = 2.0, and the dimple at the bubble bottom becomes deeper. On other hand, the downward liquid flow reduces the bubble rising speed significantly and the dimple at the bubble bottom disappears. When the pipe diameter become smaller D/Db = 1.0, the pipe flow affects the bubble shape and movement more significantly. The upward liquid flow helps

Figure 5: The detailed flow field around the rising bubble in different sized pipes with upward or downward flows. 0.10=Ar , 0.50=Bo .

D/Db = 4. 0 D/Db = 2.0 D/Db = 1.0

Ul = 0.2

Ul = -0.2



Figure 6: Comparison of velocity profiles crossing the rising bubble and the pipe flow for different sized pipe (a) D/Db = 4. 0 and (b) D/Db = 1.0.

0.10=Ar , 0.50=Bo .

the rising of the elongated bubble, creating a high velocity gradient in the pipe and builds a dimple at the bubble bottom. On the contrary, the downward flow pushes the “bullet” shaped bubble downwards. A comparison of the velocity profiles across the bubble under different pipe flow conditions in the small pipe (D/Db = 1.0) is shown in Figure 6(b). It is interesting to find that the velocity profile across the liquid film is independent of the pipe flow conditions for the current study cases. This is maybe due to the high viscosity of the liquid when the Archimedes number is low ( 0.10=Ar ), and the boundary layer along the pipe wall determines the velocity profile in the liquid film.

4 Conclusion

A Computational Fluid Dynamics (CFD) based front tracking algorithm has been applied to investigate the rising behaviour of a single bubble in a vertical pipe with a stagnant (in the far field) or flowing viscous liquid. It is found that the

(b)

(a)



relative size of the bubble and pipe has significant effect on the terminal bubble rising velocity (U ) and shape. When 0.4/ >bDD , the wall effect on the terminal bubble shape and rising speed can be neglected. A strong wall effect on bubble shape and terminal velocity can be observed when 0.2/ <bDD . In general, a small pipe diameter elongates the bubble in the axial direction and reduces the bubble rising velocity. It is found that the bubble rising velocity can be affected by the pipe diameter as α)/(/)( DDUUU b∝− ∞∞ , whereα is an exponent relating to the bubble deformability. The exponent α has a value about 0.7~0.8 when 0.50>Bo . The liquid flow may also produce significant effects on the bubble moving behaviour in a vertical pipe. It is found that the pipe flow has an effect on the bubble shape and moving pattern only when the pipe diameter is small. In a large pipe, the bubble shape is not significantly affected by the pipe flow, and the bubble moving speed is the sum of the pipe flow speed and its rising speed in the stagnant liquid.

References

[1] Clift, R., Grace, J.R., & Weber, M.E., Bubbles, drops, and particles, Academic Press: New York, 1978.

[2] Bhaga, D. & Weber, M.E., Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech., 105, pp. 61–85, 1981.

[3] Lucas, D., Krepper, E. & Prasser, H.M., Use of models for lift, wall and turbulent dispersion forces acting on bubbles for poly-disperse flows. Chem. Eng. Sci., 62, pp. 4146-4157, 2007.

[4] Nogueria, S., Riethmuller, M.L., Campos, J.B.L.M. & Pinto, A.M.F.R., Flow patterns in the wake of a Taylor bubble rising through vertical columns of stagnant and flowing Newtonian liquids: An experimental study. Chem. Eng. Sci., 61, pp. 7199-7212, 2006.

[5] Mukundakrishnan, K., Quan, S.P., Eckmann, D.M. and Ayyaswamy P.S., Numerical study of wall effects on buoyant gas-bubble rise in a liquid-filled finite cylinder. Phys. Rev. E, 76, 36308, 2007.

[6] Omebere-Yari, N.K., Azzopardi, B.J., Lucas, D., Beyer M. and Prasser, H-M, The characteristics of gas/liquid flow in large risers at high pressures. Int. J. Multiphase Flow, 34, pp. 461-476, 2008.

[7] Uno, S. and Kintner, R.C., Effect of wall proximity on the rate of rise of single air bubbles in a quiescent liquid. AIChE J., 2, pp. 420–425, 1956.

[8] Chen, L., Garimella, S.V., Reizes, J.A., & Leonardi, E., The development of a bubble rising in a viscous liquid. J. Fluid Mech., 387, pp. 61–96, 1999.

[9] Hua, J.S. & Lou, J., Numerical simulation of bubble rising in viscous liquid. J. Comput. Phys., 222, pp. 769–795, 2007.



Modelling of stratified two phase flows using an interfacial area density model

T. Höhne & C. Vallée Forschungszentrum Dresden-Rossendorf e.V., Dresden, Germany

Abstract

Stratified two-phase flow regimes can occur in chemical plants, nuclear reactors and oil pipelines. A relevant problem is the development of wavy stratified flows which can lead to slug generation. The slug flow regime is characterized by an acceleration of the gaseous phase and by the transition of fast liquid slugs, which carry a significant amount of liquid with high kinetic energy. It is potentially hazardous to the structure of the system due to the strong oscillating pressure levels formed behind the liquid slugs as well as the mechanical momentum of the slugs. Because these flow conditions cannot be predicted with the required accuracy and spatial resolution by the one-dimensional system codes, the stratified flows are increasingly modelled with computational fluid dynamics (CFD) codes. In CFD, closure models are required that must be validated. The recent improvements of the multiphase flow modelling in the ANSYS CFX code make it now possible to simulate these mechanisms in detail. In order to validate existing multiphase flow models and to further develop these, measurement data with a high-resolution in time and also in space are needed. For the experimental investigation of co-current air/water flows, the HAWAC (Horizontal Air/Water Channel) was built. The channel allows in particular the study of air/water slug flow under atmospheric pressure. Parallel to the experiments, CFD calculations were carried out. The two-fluid model was applied with a special turbulence damping procedure at the free surface. An Algebraic Interfacial Area Density (AIAD) model on the basis of the implemented mixture model was introduced, which allows the detection of the morphological form of the two phase flow and the corresponding switching via a blending function of each correlation from one object pair to another. As a result this model can distinguish between bubbles, droplets and the free surface. The behaviour of slug generation and propagation at the experimental setup was qualitatively reproduced by the simulation, while local deviations require a continuation of the work. The creation of small instabilities due to pressure surge or an increase of interfacial momentum should be analysed in the future. Furthermore, experiments with pressure and velocity measurements are planned and will allow quantitative comparisons, also at other superficial velocities. Keywords: CFD, stratified flow, slug flow, HAWAC.



1 Introduction

Stratified two phase flows occur in many industrial applications. The effects of the flow on the quantities (such as flow rate, pressure drop and flow regimes) have been always of engineering interest. Wallis and Dobson [1] analysed the onset of slugging in horizontal and near horizontal gas-liquid flows. A prediction of horizontal flow regime transitions in pipes was introduced by Taitel and Dukler [2]. They explained the formation of slug flow by the Kelvin-Helmholtz instability. They also proposed a model for the frequency of slug initiation [3]. The viscous Kelvin-Helmholtz analysis proposed by Lin and Hanratty [4] generally gives better predictions for the onset of slug flow. A general overview of the phenomenological modelling of slug flow was given by Hewitt [5]. Various multidimensional numerical models were developed to simulate stratified flows: Marker and Cell (Harlow and Welch [6]), Lagrangian grid methods and Volume of Fluid method (Hirt and Nichols [7]) and Level set method (Sussman [8]). These methods are in principle able to accurately capture most of the physics of the stratified flows. However, they cannot capture all the morphological formations like small bubbles and droplets if the grid is not reasonable small enough. One of the first attempts to simulate mixed flows was presented by Cerne et al. [9] who coupled the VOF method with a two-fluid model in order to bring together the advantages of the both analytical formulations. A systematic study of numerical simulation of slug flow in horizontal pipes using the two fluid formulation was carried out by Frank [10]. It was shown that the formation of the slug flow regime strongly depends on the wall friction of the liquid phase. In simulations using inlet/outlet boundary conditions it was found, that the formation of slug flow regimes strongly depends on the agitation or perturbation of the inlet boundary conditions. Furthermore Frank showed that the length of the computational domain plays an important role in slug formation. However, the direct comparison between CFD calculations and measurements of the slug generation mechanisms and its propagation in horizontal pipes was not analysed. For the experimental investigation of air/water flows, HAWAC (Horizontal Air/Water Channel) with rectangular cross-section was built at Forschungszentrum Dresden-Rossendorf (FZD). Its inlet device provides defined inlet boundary conditions. The channel allows in particular the study of air/water slug flow under atmospheric pressure. Parallel to the experiments, CFD calculations were carried out [11]. The aim of the numerical simulations presented in this paper is the validation of prediction of the slug flow with newly developed and implemented multiphase flow models in the code ANSYS CFX [12].

2 HAWAC

The Horizontal Air/Water Channel (HAWAC) (Fig. 1) is devoted to co-current flow experiments. A special inlet device provides defined inlet boundary conditions by a separate injection of water and air into the test-section. A blade separating the phases can be moved up and down to control the free inlet cross-



section for each phase. This allows influencing the evolution of the two-phase flow regime. The cross-section of the channel are 100 x 30 mm² (height x width). The test-section is about 8 m long, and therefore the length-to-height ratio L/h is 80. Alternatively, related to the hydraulic diameter, the dimensionless length of the channel is L/Dh = 173. The inlet device (Fig. 1) is designed for a separate injection of water and air into the channel. The air flows through the upper part and the water through the lower part of this device. Because the inlet geometry produces a lot of perturbations in the flow (bends, transition from pipes to rectangular cross-section), four wire mesh filters are mounted in each part of the inlet device. The filters are made of stainless steel wires with a diameter of 0.63 mm and have a mesh size of 1.06 mm. They aim at providing homogenous velocity profiles at the test-section inlet. Moreover, the filters produce a pressure drop that attenuate the effect of the pressure surge created by slug flow on the fluid supply systems. Air and water come in contact at the final edge of a 500 mm long blade that divides both phases downstream of the filter segment. The free inlet cross-section for each phase can be controlled by inclining this blade up and down. In this way, the perturbation caused by the first contact between gas and liquid can be either minimised or, if required, a perturbation can be introduced (e. g. hydraulic jump). Both, filters and inclinable blade, provide well-defined inlet boundary conditions for the CFD model and therefore offer very good validation possibilities. Optical measurements were performed with a high-speed video camera.

Figure 1: Schematic view of the horizontal channel HAWAC with inlet device for a separate injection of water and air into the test-section.

3 Free surface modelling

The CFD simulation of free surface flows can be performed using the multi-fluid Euler-Euler modelling approach available in ANSYS CFX. Detailed derivation of the two-fluid model can be found in the book of Ishii and Hibiki [13]. However it requires careful treatment of several aspects of the model: The interfacial area density should satisfy the integral volume balance condition. In case if surface waves are present, their contribution to the interfacial area density should be also taken into account.

pump

air outlet air inlet

8 m GV

LV



The turbulence model should address the damping of turbulence near the free surface. The interphase momentum models should take the surface morphology into account.

3.1 Turbulence damping at the free surface

As the goal of the CFD calculation was to induce surface instabilities, which are later generating waves and slugs, the interfacial momentum exchange and also the turbulence parameters had to be modelled correctly. Without any special treatment of the free surface, the high velocity gradients at the free surface, especially in the gaseous phase, generate too high turbulence throughout the two-phase flow when using the differential eddy viscosity models like the k-ε or the k-ω model [12]. Therefore, damping of turbulence is necessary in the interfacial area because the normally in industrial applications the mesh is too coarse to resolve the velocity gradient in the gas at the interface. A few empirical models have been suggested, which address the turbulence anisotropy at the free surface, see among others Celik and Rodi [14]. However, no model is applicable for a wide range of flow conditions, and all of them are non-local: they require for example explicit specification of the liquid layer thickness, of the amplitude and period of surface waves, etc. Menter proposed a simple symmetric damping procedure. This procedure provides for the solid wall-like damping of turbulence in both gas and liquid phases. It is based on the standard ω -equation, formulated by Wilcox [16] as follows:

( ) ( ) ( )[ ]ω∇⋅µ⋅σ+µ∇+ω⋅ρ⋅β−⋅τ⋅ω⋅ρ

⋅α=ω⋅⋅ρ⋅∇+ω⋅ρ∂∂

ω t2

t Sk

Ut

(1)

where α = 0.52 and β = 0.075 are the k-ω model closure coefficients of the generation and the destruction terms in the ω-equation, σω = 0.5 is the inverse of the turbulent Prandtl number for ω, τt is the Reynolds stress tensor, and S is the strain-rate tensor. In order to mimic the turbulence damping near the free surface, Menter [15] introduced a source term in the right hand side of the gas and liquid phase ω-equations. A factor activates a source term only at the free surface, where it cancels the standard ω-destruction term of the ω-equation ( )2

iiir ω⋅ρ⋅β⋅− and enforces the required high value of ωi and thus the turbulence damping.

3.2 Algebraic Interfacial Area Density (AIAD) Model

Fig. 2 shows different morphologies at slug flow conditions. Separate models are necessary for dispersed particles and separated continuous phases (interfacial drag etc.). Two approaches are possible within the Euler-Euler methodology: Four phases: Bubble/Droplet generation and degassing have to be implemented as sources and sinks. Two phases: Momentum exchange coefficients depend on local morphology.



Figure 2: Different morphologies at slug flow conditions.

Four phases: Bubble/Droplet generation and degassing have to be implemented as sources and sinks. Two phases: Momentum exchange coefficients depend on local morphology. For the second approach Yegorov [15] proposed an Algebraic Interfacial Area Density (AIAD) Model. The basic idea of the model is: The interfacial area density allows the detection of the morphological form and the corresponding switching of each correlation from one object pair to another. It provides a law for the interfacial area density and the drag coefficient for full range 0≤rα≤1. The model improves the physical modelling in the asymptotic limits of bubbly and droplet flows. The interfacial area density in the intermediate range is set to the interfacial area density for free surface In an Euler-Euler simulation of horizontal slug flow the air entrainment below the water surface can be caused by the drag force. The magnitude of the force density for the drag is

2

21|| UACD D ρ= (2)

where CD is the drag coefficient, A the interfacial area density and ρ the density of the continuous phase (if the other phase is a dispersed phase). U is the relative velocity between the two phases. The AIAD model applies three different drag coefficients, CD,B for bubbles, CD,D for the droplets and CD,S for free surface (Fig. 3). Non-drag forces (e.g. lift force and turbulent dispersion force) are neglected. The interfacial area density A also depends on the morphology of the phases. For bubbles it is

(3) where dB is the bubble diameter and rG is the gas void fraction. For a free surface the interfacial area density is the gradient of the void fraction

(4) For ρ the average density is applied, i.e.

(5)

where rL and rG are the liquid and the gas phase density respectively. In the bubbly regime, where aG is low, the average density ρ is close to the liquid phase

Droplets

Bubbles

Air (conti.)

Water(conti.)

6rAdα

αβα

=

nrrA L

LFS ∂∂

=∇=

BBGG ραραρ +=



density ρL. According to the flow regime (bubbly flow, droplet flow or stratified flow with a free surface) the corresponding drag coefficients and interfacial area densities have to be applied (Fig. 5). The simplest switching procedure for the interfacial area density, uses the blending function Fd. Introducing void fraction limits, the weights for flow regimes and length scales for bubbly and droplet flow (dB,dD) are the following:

(6)

(7)

(8)

(9) Fig. 5 shows different blending functions fB for different VF limits and blending coefficients. For the simulation of slug flow the void fraction limits of rB,limit=0.3 resp. rD,limit=0.3 and blending coefficients of aB=aD=70 are recommended.

3.3 Modelling the free surface drag

In simulations of free surface flows eq. (2) does not represent a realistic physical model. It is reasonable to expect that the velocities of both fluids in the vicinity of the interface are rather similar. To achieve this result, a shear stress like a wall shear stress is assumed near the surface from both sides to reduce the velocity differences of both phases (Fig. 3).

Figure 3: Air velocity near the free surface.

A viscous fluid moving along a “solid” like boundary will incur a shear stress, the no-slip condition, the morphology region “free surface” is the boundary layer, the shear stress is imparted onto the boundary as a result of this loss of velocity

(10)

( ),limit1

1 B G Ba r rBf e

−− = +

1FS B Df f f= − −

, , ,FS FS B B D DA f A f A f Aαβ αβ αβ αβ= + +

, , ,D FS D FS B D B D D DC f C f C f C= + +

0=∂∂

=yW y

uµτ



The result is a drag coefficient, which is mainly locally dependent on the velocity gradient and the viscosity of both fluids.

∂

∂=

yu

C GLGLSD

,,, ,µ (11)

a) Fluid domain (channel with inlet blade in horizontal position)

b) Air volume fraction, initial state (Zoom) [-]

Figure 4: Model and initial conditions of the volume fractions.


The HAWAC channel with rectangular cross-section was modelled using ANSYS CFX. The model dimensions are 8000 x 100 x 30 mm³ (length x height x width) (Fig. 4a). The grid consists of 1.2x106 hexahedral elements. A slug flow experiment at a superficial water velocity of 1.0 m/s and a superficial air velocity of 5.0 m/s was chosen for the CFD calculations. In the experiment, the inlet blade was in horizontal position. Accordingly, the inlet blade was modelled (Fig. 4a) and the inlet was divided into two parts: in the lower 50% of the inlet cross-section, water was injected and in the upper 50% air. An initial water level of y0 = 50 mm was assumed for the entire model length (Fig. 4b). In the simulation, both phases have been treated as isothermal and incompressible, at 25°C and at a reference pressure of 1 bar. A hydrostatic pressure was assumed for the liquid phase. Buoyancy effects between the two phases are taken into account by the directed gravity term. At the inlet, the turbulence properties were set using the “Medium intensity and Eddy viscosity ratio” option of the flow solver. This is equivalent to a turbulence intensity of 5% in both phases. The inner surface of the channel walls has been defined as hydraulically smooth with a non-slip boundary condition applied to both gaseous and liquid phases. The channel outlet was modelled with a pressure controlled outlet boundary condition. The parallel transient calculation of 15.0 s of simulation time on 4 processors took 10 CPU days. A high-resolution discretization scheme was used. For time integration, the



fully implicit second order backward Euler method was applied with a constant time step of dt = 0.001 s and a maximum of 15 coefficient loops. A convergence in terms of the RMS values of the residuals to be less then 10-4 could be assured most of the time. The implementation of the AIAD model and turbulence damping functions into CFX was done via the command language CCL (CEL, Expressions) and User Fortran Routines.

4 Results: comparison between simulation and experiment

A simulated free surface at the HAWAC channel with small surface instabilities is given in Fig. 6. Fig. 7 shows the resulting Interfacial Area Density variable. The AIAD model uses the following three different drag coefficients: CD,B = 0.44 for bubbles, CD,D = 0.44 for the droplets and CD,S according to eq. 11 for the free surface (see Fig. 8). In the picture sequences (Fig. 9 and 10) a comparison is presented between CFD calculation and experiment: the calculated phase distribution is visualized and comparable camera frames are shown. In both cases, a slug is generated. The sequences show that the qualitative behaviour of the creation and propagation of the slug is similar in the experiment and in the CFD calculation.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Air.vfc [-]

Wei

ghtB

ubb

[-]

CoefBubb=70, LimitVFBubb=0.1



Figure 5: Blending functions fB blending coeff.

Figure 6: Air volume fraction [-].

In the CFD calculation, the slug develops, induced by instabilities. The single effects leading to slug flow that can be simulated are: Instabilities and small waves are generated by the interfacial momentum transfer randomly. As a result bigger waves are generated. The waves can have different velocities and can merge together. Bigger waves roll over and can close the channel cross-section. However, a detailed comparison shows quantitative deviations between simulation and measurement. The needed entrance length for slug generation was defined as the length between the inlet and the location nearest the inlet



Figure 7: Interfacial area density Variable [1/m].

Figure 8: Drag coefficient [-].

Figure 9: Measured picture sequence at JL = 1.0 m/s and JG = 5.0 m/s with ∆t = 50 ms (depicted part of the channel: 0 to 3.2 m after the inlet).

where a wave closes nearly the entire cross-section. This was observed at about 1.5 m in the experiment and 2.5 m in the calculation. These quantitative differences can be explained with the flow regimes observed at the test-section inlet. In fact, the flow pattern has an important influence on the momentum exchange between gas and liquid, especially at high velocity differences between the phases. Small disturbances of the interface provide a more efficient momentum transfer from the air to the water than in a stratified smooth flow. A high momentum transfer induces a rapid wave growth and therefore slug generation. In this case, in the experiment supercritical flow waves were observed from the inlet of the channel. This means that the boundary conditions chosen for the CFD model do not reproduce all small disturbances



Figure 10: Calculated picture sequence at JL = 1.0 m/s and JG = 5.0 m/s

(depicted part of the channel: 1.4 to 6 m after the inlet).

observed in the experiment. In the end, a quite long channel length is needed before waves appear spontaneously in the simulation. Future work should focus on the proper modelling of the small instabilities observed at the channel inlet.

5 Summary

In the HAWAC test facility, a special inlet device provides well defined as well as variable boundary conditions, which allow very good CFD-code validation possibilities. A picture sequence recorded during slug flow was compared with the equivalent CFD simulation made with the code ANSYS CFX. The two-fluid model was applied with a special turbulence damping procedure at the free surface. An Algebraic Interfacial Area Density (AIAD) model on the basis of the implemented mixture model was introduced and implemented. It improves the physical modelling, detection of the morphological form and the corresponding switching of each correlation is now possible. The behaviour of slug generation and propagation at the experimental setup was reproduced, while deviations require a continuation of the work. Experiments like pressure and velocity measurements are planned and will allow quantitative comparisons, also at other superficial velocities.

Acknowledgements

This work is carried out in the frame of a current research project funded by the German Federal Ministry of Economics and Labour, project number 150 1329. Thanks to Yuri Yegorov and Thomas Frank from ANSYS CFX for their fruitful cooperation.



References

[1] Wallis, G. D., and Dobson, J. E. 1973. Onset of slugging in horizontal stratified air-water flow. Int. J. Multiphase Flow 1, 173-193.

[2] Taitel, Y., and Dukler, A. E. 1976. A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow. AIChE J. 22, 47-55.

[3] Taitel, Y., and Dukler, A.E., 1977. A model for slug frequency during gas-liquid flow in horizontal and near horizontal pipes. Int. J. Multiphase Flow, 3, 585-596.

[4] Lin, P. Y., and Hanratty, T. J. 1986. Prediction of the initiation of slugs with linear stability theory. Int. J. Multiphase Flow, 12, 79-98.

[5] Hewitt, G. F. 2003. Phenomenological modelling of slug flow. Short course modelling and computation of multiphase flows, ETH Zurich, Switzerland.

[6] Harlow, F. H., Welch, J. E., 1965. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids 8 (12), 2182-2189.

[7] Hirt, C. W., Nichols, B. D., 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics 39 (1), 201-225.

[8] Sussman, M., 1994. A level set approach for computing solutions to incompressible two-phase ow. Journal of Computational Physics 114 (1), 146-159.

[9] Cerne, G., Petelin, S., Tiselj, I., 2001. Coupling of the interface tracking and the two- fluid models for the simulation of incompressible two-phase flow. Journal of Computational Physics 171 (2), 776-804.

[10] Frank, T. 2003. Numerical simulations of multiphase flows using CFX-5. CFX Users conference, Garmisch-Partenkirchen, Germany.

[11] Vallée, C., Höhne, T., Prasser, H.-M., Sühnel, T. 2008, Experimental investigation and CFD simulation of horizontal stratified two-phase flow phenomena. NED, Volume 238, Issue 3, March 2008, Pages 637-646

[12] ANSYS CFX, 2008. User Manual. Ansys Inc. [13] Ishii, M., Hibiki, T., 2006. Thermo-fluid Dynamics of Two-phase Flow.

Springer-Verlag. [14] Celik, I., and Rodi, W. 1984. A deposition-entrainment model for

suspended sediment transport. Report SFB 210/T/6, Strömungstechnische Bemessungsgrundlagen für Bauwerke, University of Karlsruhe, Germany.

[15] Yegorov, Y. 2004. Contact condensation in stratified steam-water flow, EVOL-ECORA –D 07.

[16] Wilcox, D. C. 1994. Turbulence modelling for CFD. La Cañada, California: DCW Industries Inc.



Multi-phase mixture modelling of nucleateboiling applied to engine coolant flows

V. Pržulj & M. ShalaRicardo Software, Ricardo UK LimitedShoreham-by-Sea, West Sussex, UK

Abstract

The paper reports on the use of the homogeneous multi-phase mixture modellingapproach to simulate nucleate boiling in low pressure flows. A variant of theRPI (Rensselaer Polytechnic Institute) mechanistic nucleate boiling model pro-vides closures for the wall thermal conditions and mass transfer rates due to phasechange in the bulk flow. The bubble departure diameter at the wall and the bubblebulk diameter are identified as the most influential factors and their original modelcoefficients are modified.

The numerical difficulties due to large density variations associated with phasechange are successfully addressed in conjunction with the segregated pressurebased SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm.

The nucleate boiling model has been compared against published data from twoexperiments. The computed and measured values for the wall heat flux and vapourvolume fractions are in broad agreement. The model capability is also demon-strated for the engine coolant flow where the conjugate heat transfer probleminvolving complex engine components is solved.Keywords: CFD, nucleate boiling, multi-phase mixture, RPI model, engine cool-ing.

1 Introduction

Nucleate boiling is the exceedingly effective mode of heat transfer from a heatedwall to a liquid. In many liquid cooling systems, ranging from nuclear reactorsto electronic devices, a change from single-phase convection to nucleate boilingcan effectively provide the desired high rates of heat transfer. The high heat trans-fer rates (thermal loads) also characterise cylinder blocks and heads of internal



combustion engines. Their cooling is achieved by pumping a water-glycol mix-ture through a system of connected coolant passages. Whether nucleate boiling isutilised intentionally or not, the design of efficient liquid cooling systems shouldavoid departure from the nucleate boiling regime, the boiling crisis. This depar-ture, described by the critical (maximum) wall heat flux (CHF) might lead to thehazardous overheating of wall materials.

Practical CFD modelling of multi-phase flows requires built-in mechanistic mod-els for interfacial mass, momentum and energy transfer between phases. In thepast, the majority of published CFD predictions of boiling flows employed the RPImodel of Kurul and Podowski [1] for interfacial transfer closures. It was validatedagainst high pressure flows but later some authors [3, 4, 5] adopted this model forlow pressure boiling flows. Model improvements have been proposed by [6, 7],while the model extension to conditions close to CHF was reported in [8].

In the majority of CFD simulations [3–5, 9, 10] the boiling model has been cou-pled with multi-fluid conservation equations where individual velocity, tempera-ture and other field variables can be solved for each phase. The exception is Bo’swork [11], where the homogeneous multi-phase mixture (single fluid) approachwas used in conjunction with the author’s own mass transfer model.

In the present study, modelling of nucleate boiling based on the RPI model andusing the homogeneous multi-phase mixture approach is used. This choice is moti-vated by uncertainties in modelling interphase transfers and by computational effi-ciency of the mixture approach. The objective is to explore this overall approachfor modelling low pressure coolant flows in internal combustion engines.

2 Mathematical formulation

A homogeneous mixture model is adopted in this work. This implies that relativemotion between phases is neglected. The modelling equations describing mass,momentum and energy conservation of the mixture have the same form as thesingle-phase Reynolds Averaged Navier–Stokes equations. A variant of the con-ventional k − ε model, with imposed bounds on the turbulence time scale is usedto model turbulence. The near-wall region turbulence is modelled by enhancedwall functions. In the viscous sub-layer they satisfy corresponding wall-limitingexpressions and in the fully turbulent region they are identical to the conventionalwall functions. Further details regarding the employed k − ε model and enhancedwall functions can be found in [12].

The transport and thermodynamic properties appearing in the mixture equationsdepend on the properties of constituent fluid phases k and their volume fractionsαk. In case of Nph fluid phases, the mixture properties are calculated as:

φ =Nph∑k=1

αk φk = αlφl + αgφg , φ = ρ, µ, λ (1)

where φk denotes the property value of the constituent phase; in this work of liq-uid k = l and vapour k = g. In the above equation, ρ, µ and λ denote the mixture



density, dynamic viscosity and thermal conductivity,respectively. The mixture spe-cific heat takes a different form:

Cp =Nph∑k=1

XkCp,k = XlCp,l +XgCp,g , Xk =αkρk

ρ(2)

where Xk is the phase mass fraction. Individual phase properties are in generaldependent on the temperature. The vapour can be treated as an ideal gas so itsdensity is calculated from the equation of state ρg = p/(RgT ) where p is thepressure and Rg is the gas constant.

The volume fractions are governed by their own transport equations:

∂αkρk

∂t+

∂

∂xj(αkρkUj) = Γk, k = l, g (3)

where t is the time, xi are the Cartesian space coordinates and Uj are Favre-averaged velocities. The source term, Γk, represents the phase mass generationrate due to evaporation and condensation.

The energy conservation equation is defined in terms of total enthalpy, H =h+0.5U2

i +k, with h being the specific enthalpy of mixture and k is the turbulentkinetic energy. In order to calculate temperature from the specific enthalpy andvice versa, the specific enthalpy h and its relationship with temperature T needsto be defined. Considering enthalpies of a liquid phase, hl = Cp,lTl, and a vapourphase, hg = Cp,lTsat + ∆hgl + Cp,g(Tg − Tsat) (Tsat denotes the saturationtemperature), and setting T = Tl = Tg, the mixture enthalpy can be defined interms of temperature as:

h = Xlhl +Xghg = hl,sat +Xg∆hgl + Cp (T − Tsat) , (4)

where the specific heat of mixture Cp is given by eqn. (2). In the above equation,∆hgl signifies the latent heat of evaporation: ∆hgl = hg,sat − hl,sat, where hl,sat

and hg,sat, are saturated liquid and vapour enthalpies, respectively. The mixturetemperature then becomes

T = Tsat +h− (hl,sat +Xg∆hgl)

Cp(5)

2.1 Mechanistic boiling model

The mechanistic model presented here provides closures for the wall thermal con-ditions for the energy and the mass transfer rate Γk in the volume fractions eqn. (3).



Thermal conditions for the near-wall heat transfer. Considering the wall heatflux qw, a general approach is to apportion it to the liquid and vapour phasesaccording to their wall volume fractions αl/g,w:

qw = αl,wqw,l + αg,wqw,g, αg,w = 1 − αl,w (6)

In the case of sub-cooled nucleate boiling, vapour bubbles, created at the wallnucleation sites and then departed from the wall, are continuously replaced byfresh liquid (quenching) and by new generation of bubbles. Thus the wall liquidfraction is close to one, as the absence of dry wall regions can be assumed. Athigh wall heat fluxes, the boiling process can move towards CHF conditions. Thismeans that heat transfer between dry wall regions (αg,w > 0) and vapour has tobe considered. A phenomenological function fα,l reported in [8]:

fα,l = 1 − 0.5 exp [−20 (αl − αl,cr)] , for αl ≥ αl,cr, (7)

fα,l = 0.5 (αl/αl,cr)20αl,cr for αl < αl,cr, αl,cr = 0.2

is used in this work to define the wall liquid fraction αl,w = fα,l in order to modeltransition from the nucleate boiling to the CHF regime. In the above expression,αl is a value at the near-wall cell.

Kurul and Podowski’s RPI model [1] is used to split the heat flux to the liquidinto the single-phase convection qc, the wall quenching qq and the evaporation partqe:

qw,l = (1 − Ω)qc + Ωqq + qe (8)

where Ω is the fraction of the wall area affected by nucleation sites and quenching,see eqn. (11) below. The wall heat flux expression (6) then reads:

qw = qw,d + αl,w [Ω (qq − qc) + qe] , qw,d = αl,wqc + αg,wqg,w (9)

where contribution of convection to the liquid and gas phase is described by thewall diffusion flux qw,d. This flux is calculated within the solver using the mixtureproperties. The liquid convective heat flux qc is calculated in the same way as thewall diffusion flux qw,d except that properties of liquid are used.

The evaporative and quenching flux are given by:

qe =D3

bwπ

6ρgfnNn∆hgl, qq =

2√πfn

√λlρlCp,ltq (Tw − Tl) (10)

where Dbw is the bubble diameter at departure from the wall, fn and Nn are thefrequency of nucleation and the number of nucleation sites per unit area, respec-tively. In the expression for the quenching flux, tq denotes the waiting time elapsedbetween the departure of a bubble and the nucleation of a new one. It can be esti-mated as tq = 0.8/fn [1]. The fraction of the wall area affected by nucleation isgiven as [4]:

Ω = min(

1,D2

bwπ

4Nnη

), η = 4.8e−Ja/80, Ja =

Cp,lρl (Tsat − Tl)ρg∆hgl

(11)

where Ja is the Jacob number.



The closures for the remaining model parameters have been selected based onthe assessment of the existing empirical closures reported in literature. For thebubble diameter, the correlation of Tolubinsky, see [5], is modified by introducingan adjustable factor Cbw (Cbw = 0.6 in the original correlation):

Dbw = min(Cbw × 10−3e−∆Tsub/45, 1.4 × 10−3

), ∆Tsub = Tsat − Tl (12)

The bubble release frequency and nucleation site density are calculated from Cole’sand Lemmert and Chwala’s correlations (see for example [5]), respectively:

fn =

√4g (ρl − ρg)

3ρlDbw, Nn = 210 (Tw − Tsat)

1.805 (13)

Interfacial mass transfer. The mass transfer in the bulk flow due to evaporationΓe or condensation Γc is calculated with the help of empirical correlations forthe interface heat transfer coefficient around the representative vapour bubble withdiameter Db:

Γe/c =hi,e/cAi (T − Tsat)

∆hgl, Ai =

6αsg(1 − αg)(1 − αsg)Db

, αsg = min (0.25, αg) (14)

where Ai (1/m) is the interfacial area density [1]. For both evaporation andcondensation process, the heat transfer coefficient hi is determined from Ranz-Marshall correlation [3]. The bubble Nusselt number Nub is correlated byReynolds, Reb, and liquid Prandtl, Prl, numbers:

hi =Nubλl

Db, Nub = 2 + 0.6Re

12b Pr

13l , Reb =

ρlUbDb

µl, P rl =

µlCp,l

λl(15)

The characteristic bubble velocity Ub is taken to be the magnitude of the mixturecell velocity. Then, mass transfer rates in eqn. (3) read as Γg = −Γl = Γe + Γc

The bubble mean diameter Db has usually been modelled by either a constant,estimated value [8] or as a linear function of local sub-cooling ∆Tsub = Tsat−Tl,[1, 3, 4] with adjustable reference bubble sizes and corresponding sub-cooling tem-peratures. As shown by [10], these simple approaches are very deficient when com-pared against the measured bubble size profiles. To accommodate some observedbubble behaviour such as the coalescence, Wintterle et al. [9] adopted models ofBasuki et al. and El Jouhary (see [9]):

Db =(

6αg

πN(αg)

) 13

, N =[1 + exp

(αg − 0.015

0.004

)]−1

(N0 −N1)+N1 (16)

where N0 = 1.1 · 107 and N1 = 3.0 · 105 are the number densities (1/m3) corre-sponding to the very low and very high vapour fractions αg.



3 Numerical framework

The boiling model has been implemented into an advanced, commercial CFDsolver, VECTIS-MAX. The governing equations are discretised over generalunstructured (polyhedral) grids employing an advanced collocated Finite VolumeMethod, [12].

The pressure based, segregated SIMPLE-like solution algorithm [2] ensures effi-cient pressure-velocity-density coupling. Considering the compressible gas flow,in addition to the pressure and velocity corrections, the density corrections areintroduced and defined in terms of the pressure corrections as ρ′ ∝ Cρp

′, whereCρ is the density derivative over pressure. In the case of a multi-phase mixture,Cρ

can be calculated from the mixture sonic speed cm:

Cρ =1c2m

= ρ

(αg

κRgTρg+

αl

ρlc2l

)≈ ρ

ρg

αg

RgT. (17)

where κ = 1.4 is the isentropic exponent. The above simple formulation for themixture density derivative over pressure has been effective in handling large den-sity variations associated with phase change.


The nucleate boiling model is first validated against published data from two exper-iments. To assess the model capability for automotive applications, real enginecoolant flow simulations are then presented.

The initial validation results identified the bubble departure diameter Dbw andthe bubble bulk diameter Db as the most influential model parameters. In orderto reproduce the experimental trends, it was necessary to adjust their correlations.Therefore, the adjustable factor in eqn. (12) was set to Cbw = 1.2, while thenumber densities in the expression for Db, eqn. (16), were tuned to the valuesN0 = 1.6 · 107 and N1 = 3.0 · 106.

Horizontal channel flow. Robinson [13] tried to replicate engine flow condi-tions in a cylinder head cooling passages by using a rectangular channel (241mmlong, 16 mm wide and 10 mm high) with a heated section (10 × 50 mm2, posi-tioned 76 mm downstream from the flow inlet) at the bottom wall, see fig. 1. The

Inlet

Outlet

Heated section

Figure 1: Simulated channel geometry and location of the heated section.



heated surface (aluminium alloy) was machine finished and can be considered assmooth. The working fluid was a mixture of 50% water and 50% ethylene glycolby volume.

The present results were obtained for the sets of test data with the inlet liquidtemperature Tinl = 363K and with two mass flow rates corresponding to the inletvelocities of 0.25 m/s and 1.0 m/s. Within these sets the pressure values were1, 2 and 3 bars (the corresponding saturation temperatures are 381 K , 401 K and415 K , respectively). These pressure values were specified as the outlet boundaryconditions. On the heated section, a constant temperature (known from the exper-iments) was specified. Other walls were considered adiabatic.

The calculated wall heat flux values can be compared against the measured onesas shown in fig. 2. Very good agreement has been obtained for the inlet velocity0.25m/s, fig. 2 (a). The heat flux values for the high velocity of 1m/s are signif-icantly under-predicted as shown in fig. 2 (b). This model behaviour can be partlyexplained by evident under-prediction of the single-phase heat fluxes before theonset of nucleate boiling. Thus the effect of turbulence modelling and wall func-tions should be anticipated. Another factor is the dependence of the bubble depar-ture size on the local velocity [3] which is not taken into account by eqn. (12).

370 380 390 400 410 420 430 440Wall temperature (K)

0

0.5

1

1.5

2

Hea

tFlu

x(M

W/m

2 )

Exp, 1 barExp, 2 barExp, 3 barPresent, 1 barPresent, 2 barPresent, 3 bar

(a) Inlet velocity: 0.25 m/s


0

0.5

1

1.5

2

Hea

tFlu

x(M

W/m

2 )

Exp., 1 barExp., 2 barExp., 3 barPresent, 1 barPresent, 2 barPresent, 3 bar

(b) Inlet velocity: 1.0 m/s

Figure 2: Channel flow, inlet temperature 363K . Comparison of the predicted wallheat fluxes against measured data from [13].

An example of the computed flow variable fields is depicted in fig. 3. Theselected case has the high wall temperature and heat flux which resulted in thelarge amount of vapour near the heated wall. Consequently, the velocity profilesare distorted when compared to the profiles upstream of the heated section.

Upward flow in a vertical annulus. Bae et al. [5] benchmarked their full Euler-Euler (two-fluid) multi-phase model with an interfacial area equation using theexperiment from the Seoul National University. The sub-cooled boiling of water



(a) Temperature, from 363 K (blue) to 414 K (red)

(b) Vapour volume fraction, from 0 (blue) to 0.75 (red)

(c) Velocity vectors, from 0.009 m/s to 0.33 m/s

Figure 3: Channel flow, inlet temperature 363 K , velocity 0.25 m/s. Computedvariable fields in the symmetry plane for p = 1 bar, Tw = 418K .

in a vertical upward flow through a concentric annulus was investigated under con-stant wall heat flux conditions along the heated inner tube section. The total lengthof the annulus was 3.06 m. The inner tube consisted of the inflow unheated, centralheated and outflow unheated section, each 0.5, 1.87 and 0.69 m long, respectively.The inner and outer annulus radii were Rin = 9.5 and Rout = 20 mm. The radialdistribution of the vapour fraction was measured at three axial positions along theheated section.

For the present validation, a test case with the following conditions has beenselected: the mass flow rate 342.207 kg/(m2s) (applied as the outlet boundarycondition), the wall heat flux at the heated section qw = 212.706 kW/m2, the pres-sure 1.21 bar (applied as the inlet boundary condition) and the inlet sub-coolingTsat − Tinl = 21.695 K . Steady-state simulations are carried out for quarter ofthe annulus. Apart from the heated inner section, all other walls are assumed adi-abatic. As the heat flux is prescribed at the heated section, the wall temperatureis calculated with an iterative procedure which ensures that the heat flux eqn. (9),expressed as qw = F (Tw), is satisfied.

The predictions of the radial profile of the vapour fraction at the exit of theheated section are depicted in fig. 4(a). The axial profile of the average vapour frac-tion is shown in fig. 4(b). The present profiles can be compared with the measureddata as well as with two-fluid model predictions from [5]. Two present profiles arepresented: the first with the original densities numbers in eqn. (16) for the bubblebulk diameter and the second with modified values of these numbers. The modifieddensities numbers have improved predictions of the average axial profile with ref-erence to the measured data. Considering the local radial predictions, the presentapproach over-predicts the amount of vapour near the wall and under-predicts it inthe bulk flow. Calculated temperature and vapour fraction fields as obtained withadjusted density numbers are plotted in fig. 5 for the whole solution domain.



0 0.2 0.4 0.6 0.8 1(R-R

in)/(R

out-R

in))

0

0.1

0.2

0.3

0.4

0.5

0.6

Vol

ume

frac

tionα g

ExperimentRef. [Bae et al.]Present, N

01.1e 7, N

13e 5

Present, N0

1.6e 7, N1

3e 6

(a) Local radial profile.

0 0.5 1 1.5 2Axial length (m)

0

0.04

0.08

0.12

Vap

our

frac

tionα g

ExperimentRef. [Bae et. al.]Present, N

01.1e 7, N

13e 5

Present, N0

1.6e 7, N1

3e 6

(b) Average axial profile.

Figure 4: Annular flow. Computed and measured vapour volume fraction profiles.(Zero value on the x-axes in (b) denotes beginning of the heated section.)

(a) Temperature, from 356 K (blue) to 376.3 K (red).

(b) Vapour fraction, from 0 (blue) to 0.69 (red).

Figure 5: Vertical annulus (flow direction from the right to the left). Predicted dis-tribution of temperature and vapour fraction.

In the first half of the heated section the wall temperature experiences an oscil-latory pattern which corresponds to the vapour fraction pattern.

Engine coolant flow. The CFD solution domain needs to include the completeengine assembly consisting of typical components such as the cylinder head, headgasket, engine block, cylinder liners and coolant jacket. These components (exceptliners) are shown in fig. 6 for the selected engine type. The coolant jacket rep-resents the fluid domain while other components define separate solid materialdomains, each with their own thermal properties.

To make the numerical approach practical, automatic meshing of the solutiondomain is required because of the very complex geometry. For the present simula-tion, the Cartesian cut-cell grid, having around 3.3 million cells, has been gener-ated by VECTIS–MAX mesher. This mesher delivers fast meshing without resort-ing to the boundary surface grid generation.

As part of the steady Conjugate Heat Transfer (CHT) simulation, the mass,momentum, turbulence and volume fraction equations are solved for the fluiddomain (coolant flow). The energy equation is solved over a global solution domain



Figure 6: Engine cooling. Temperature distribution at surfaces of fluid (coolantjacket) and solid (cylinder head, gasket and cylinder block) materialsparticipating in the CHT simulation with boiling model.

(containing all participating fluid and solid domains) in a fully implicit and conser-vative manner. For this, conformal numerical grids at fluid-solid interfaces are pro-vided. At the coolant inlet the mass flow rate (2.8 kg/s) and temperature (363K)are specified; at the outlet fixed static pressure (1.01bar) is maintained. The exter-nal heat transfer coefficient and temperature describe heat transfer at the solid sur-faces exposed to the environment. At the gas side cylinder surfaces the heat fluxdistribution is prescribed.

Fig. 6 presents temperature field at the surfaces of each material domain as cal-culated with the boiling model. These surfaces are either fluid/solid or solid/solidinterfaces or external boundaries.

The effect of the boiling model on the temperature levels is clearly demon-strated in fig. 7. Here, one can compare temperature on the cylinder head as pre-dicted with and without the nucleate boiling model. Inclusion of the boiling modelreduces temperature levels. In this case, the maximum temperature without theboiling model is 576.8K , while the maximum temperature with the boiling modelreaches 550.5K . The maximum temperature of coolant is also reduced from 466.6to 425.2 K. As expected, the temperature reduction indicates presence of a cer-tain amount of vapour. This amount can be quite large as shown in fig. 8. Thefigure depicts the coolant passage region in which the near-wall vapour fractionhas the high value, around 0.3. The spot with the maximum value of 0.61 is notshown in the above figure. Identification of such local spots where departure from



Figure 7: Engine cooling. Comparison of the surface temperature on the cylinderhead as computed with (left) and without (right) boiling model, valuesfrom 352 K (blue) to 576.8 K (red).

Figure 8: Engine cooling. Near-wall vapour fraction distribution in the coolant pas-sage region with the large amount of vapour, range: from 0 (blue) to 0.3(red).

nucleate boiling might lead to overheating of the wall materials is useful for designof engine cooling systems.

5 Conclusion

The capability of the multi-phase mixture modelling approach, supplemented withthe mechanistic nucleate boiling model, has been assessed for CFD predictions oflow pressure coolant flows in internal combustion engines. The nucleate boilingmodel is based on the popular RPI model. Among a number of modelling param-eters, the bubble departure diameter at the wall and the bubble bulk diameter arethe most influential and required adjustments. In comparison to the original RPImodel, the wall heat flux partitioning is modified to account for the possible heattransfer from the wall to the gas phase.

Two boiling flow experiments are used to benchmark the model. The computedand measured values for the wall heat flux and vapour volume fraction are foundto be in broad agreement. The CHT simulation of an engine cooling system hasshown reasonable model behaviour. Bearing in mind uncertainties in the modellingof nucleate boiling, the present approach can be seen as a good compromise interms of the accuracy and computational efficiency.



References

[1] Kurul, N. & Podowski, M., Multidimensional effects in forced convectionsubcooled boiling. Proceedings of the 9th International Heat Transfer Con-ference, Jerusalem, Israel, 1-BO-04, pp. 21–26, 1990.

[2] Ferziger, J. & Peric, M., Computational Methods for Fluid Dynamics.Springer: Berlin, 1997.

[3] Koncar, B. & Mavko, B., CFD simulation of subcooled flow boiling atlow pressure. Intl. Conf. Nuclear Energy in Central Europe 2001, Portoroz,Slovenia, pp. 208.1–208.8, 2001.

[4] S. Narumanchi, D.B., A. Troshko & Hassani, V., Numerical simulations ofnucleate boiling in impinging jets: Application in power electronics cooling.Int J Heat and Mass Transfer, 51, pp. 1–12, 2008.

[5] B-U. Bae, H.Y.Y., Euh, D.J., Song, C.H. & Park, G.C., Computational anal-ysis of a subcooled boiling flow with a one-group interfacial area transportequation. J Nuclear Science and Technology, 45(4), pp. 341-351, 2008.

[6] N Basu, G.W. & Dhir, V.K., Wall heat flux partitioning during subcooled flowboiling: Part 1 - Model development. ASME J Heat Transfer, 127, pp. 131-140, 2005.

[7] G. Yeoh, M.H. & Tu, Y., Improved wall heat partition for subcooled boilingflows. 6th Int. Conf. Multiphase Flow ICMF 2007, Leipzig, Germany, 2007.

[8] Koncar, B. & Mavko, B., Simulation of boiling flow experiments close toCHF with the Neptune-CFD code. Science and Technology of Nuclear Instal-lations, 2008, p. 8, 2008.

[9] T. Wintterle, Y.E., E. Laurien & Menter, F., Numerical simulation of a sub-cooled boiling flow in a nuclear reactor bundle geometry. 11th Workshop onTwo-Phase Flow Predictions, Merseburg, Germany, 2005.

[10] Yeoh, G. & Tu, J., Two-fluid and population balance models for subcooledboiling flow. Applied Mathematical Modelling, 30, pp. 1370-1391, 2006.

[11] Bo, T., CFD homogeneous mixing flow modelling to simulate subcoolednucleate boiling flow. SAE International, 2004-01-1512, 2004.

[12] Przulj, V., Birkby, P. & Mason, P., Finite volume method for conjugate heattransfer in complex geometries using cartesian cut-cell grids. CHT-08, Mar-rakech, Morocco, 2008.

[13] Robinson, K., IC Engine Coolant Heat Transfer Studies. PhD thesis, Univer-sity of Bath, Department of Mechanical Engineering, 2001.



On the application of Mesoscopic EulerianFormalism to modulation of turbulence bysolid phase

Z. Zeren1,2 & B. Bedat1,2

1 Institut de Mecanique des Fluides de Toulouse, France2 CNRS, France

Abstract

Recently developed method, Mesoscopic Eulerian Formalism, is searched for itsextension to the gas-solid flows where the carrier phase is modified by the solidphase. The possibility is shown to be existing by the introduction of two classesof particles with all the same properties except their initial positions. Classes aredistributed homogeneously in space and only one of them is two-way coupled withthe flow. The others are with ghost particles (particles with one-way coupling).With increased number of ghost particles, the field of source terms’ of classesbecome similar letting the fluid realization become the same for each class. Thenthe conditional one-particle probability density function is definable.Keywords: Mesoscopic Eulerian Formalism, two-way coupling, particle and fluidrealization, initial particle conditions.

1 Introduction

Spatial distribution of inertial particles in a turbulent flow is very important inunderstanding different phenomena occurring in gas-solid flows such as particle-particle interactions, interactions fluid-particle, etc. For example, it is shown bySundaram and Collins [1] that particle spatial distributions can cause significantchanges in the collision rates of particles. On the other hand, particle distributioncan also have specific effect on the modulation of fluid turbulence, Elgobashi andTruesdell [2].

In this regard, Fevrier et al. [3] have proposed the Mesoscopic Eulerian For-malism (MEF) to have comprehensive understanding on the distribution of finite



inertia particles in a turbulent flow. The particle spatial velocity correlations areassumed to be induced only via the interactions with the fluid. Specifically, hydro-dynamic interactions and inter-particle collisions are assumed not to induce anyspatial correlation between the particles. This considers the fact that dispersedphase statistical quantities do not depend on the initial conditions of particles hav-ing chaotic motion. The statistical measures are developed then via the definitionof one-particle probability density function, f (1)

p (cp, x, t,Hf ), conditioned on thesingle fluid realization.

The extension of the method to the gas-solid flows with inter-particle collisionsis proposed by Fevrier et al. [3] for dilute regimes. However, implementation inturbulence modulation regimes is not that direct due to the fact that the single fluidrealization, Hf , is not obvious to be definable.

In this paper, it is shown that Mesoscopic Eulerian Formalism is applicablethrough the introduction of two classes of particles into a turbulent flow whereonly one of the classes is coupled with the flow (two-way coupling) and the other isassumed to be ghost particles (particles with one-way coupling). Particles are pointsources tracked in the Lagrangian frame of reference whereas the fluid is solvedin the Eulerian grid. Particle Source In Cell approximation (PSIC), initialized byCrowe et al. [4], is shown to be not to cause any significant non-physical oscilla-tions on the fluid velocity. Discussion will continue with describing the numericalscheme and the application to two-way coupling will be discussed.

2 Governing equations and numerical configuration

The studied configuration is a cubical domain with a volume of L3b = (2π)3. The

domain is discretized with 1283 grid points with periodic boundary conditions forthe both phases.

Homogeneous isotropic turbulence is generated and kept stationary using astochastic forcing scheme [5, 6]. The code used is finite difference code with 6th

order spectral-like scheme in space and 3rd order Runge-Kutta scheme in time.The code is parallelized with MPI (Message Passing Interface) library.

Governing equations of the fluid including the effect of particles are written as:

∂ρ

∂x+∂ρui

∂xi= 0 (1)

∂ρui

∂t+ ρuj

∂ui

∂xj= − ∂p

∂xi+∂τij∂xj

+ Πi + fi (2)

Equations (1) and (2) are respectively for the conservation of mass and momen-tum.

While solving the Navier-Stokes equations in the Eulerian frame, particles aretracked individually in the Lagrangian frame. The effect of the particles on the fluidis taken into account through the term, Πi, on the right hand side of the momentumequation. Without taking into account the effect of gravity, this term in the contextof point-source approximation is written as:



Πi = −Np∑n=1

F(n)p,i (x(n)

p,i )δ(x− x(n)p,i ) (3)

where F (n)p,i is the force applied by the fluid to the nth particle in i direction defined

as:

F(n)p,i = −m

(n)p

τ(n)p

(u(n)p,i − u

(n)f@p,i) (4)

where uf@p is the fluid velocity at the position of particle and τp is the particlerelaxation time defined as:

τ (n)p =

ρpd2p

18µffD(5)

fD = (1 + 0.15Re0.687p ) is the correction to take into account the effect of par-

ticle Reynolds number, Rep. Fluid velocity at the position of particle, uf@p, iscalculated by 3rd order Lagrangian interpolation scheme.

The term fi in the equation 2 is the stochastic forcing term which keeps theturbulence stationary.

3 Initial conditions of particle phase

Characteristics of turbulence and particles can be found in table 1 and table 2. TE isthe Eulerian time scale calculated by the Eulerian one-point autocorrelation func-tion, Te is eddy turnover time defined as Lf/u

′where Lf is the longitudinal large

scales length and u′

is the characteristic velocity of turbulence. Lg is the transver-sal large scales. TLf

is Lagrangian time scale of large structures calculated by theLagrangian one-point autocorrelation function. Based on these values, turbulentReynolds number is defined as ReL = u

′Lf/νf and Reynolds number based on

the Taylor scales is defined as Reλ = u′λg/νf .

The crucial condition for MEF to be applicable is that the particle spatial corre-lations are induced only by the interaction with the fluid. As mentioned in intro-duction, this dictates the fact that the hydrodynamic interactions and interparticlecollisions are assumed not to induce any spatial correlation. To this end, smalldeviations in the initial conditions are quickly magnified, after several relaxationtime, particle statistics become independent of the initial conditions and controlledby the interactions with the fluid.

Under these assumptions, two populations of particles are introduced into thesame fluid realization with the same mesoscopic fields except their positions. Thecharacteristics of particles and of turbulence are defined in table 1 and table 2.Note that in the table, total number of particles is given. Each populations isthen with 2 particles per grid in DNS1283 application. These two populationsare time-stepped with the fluid realization without two-way coupling so that theysettle to an equilibrium with the turbulent field. Their statistical quantities, particle



Table 1: Turbulence characteristics without two-way coupling.

TE TE/Te TLf/TE Lf/Lb Lf/Lg ηf ReL Reλ

6.2816 1.0267 0.8275 0.0864 1.9797 0.0177 96 49

Table 2: Particle characteristics.

ρp/ρf dp/ηf Nptotal

12000 0.0751 4x1283

Table 3: Fluid and particle statistics.

Population q2p q2f qfp q2f@p

ClassA .4544E-02 .1176E-01 .8883E-02 .1141E-01

ClassB .4546E-02 .1176E-01 .8883E-02 .1141E-01

Inlet

Outlet

Heated section

Figure 1: Spatial distribution of particles, T+/τp = 0. ClassA on the left andClassB on the right.

kinetic energy q2p, turbulent kinetic energy q2f , fluid-particle covariance qfp, turbu-lent kinetic energy seen by the particles q2f@p, in stationarity are shown in table 3and they’re exactly the same. This final field obtained is referred as T+/τp = 0where T+ is the nondimensional time of the simulations. These populations willbe useful not only for the verification of the PSIC approximation but also for thetest that is performed to validate the MEF’s application to two-way coupling.

Initial distributions of the particles are shown in fig. 1. It is clear that the bothclasses are distribution homogeneously in space.

To be more quantitative on the distributions of the both classes, the normalizedspatial distribution function, P (C), which is possible number of particles, C, in anelementary volume and longitudinal spatial correlations of particles, Rpp(r) =<up(x)up(x+ r) >, are plotted. Shown in fig. 2, both populations have exactly thesame correlation and distribution curves validating the equality of the mesoscopic




0

0.5

1

1.5

2

Hea

tFlu

x(M

Wm

2 )

Exp, 1 barExp, 2 barExp, 3 barPresent, 1 barPresent, 2 barPresent, 3 bar

(a) Inlet velocity: 0.25 m/s


0

0.5

1

1.5

2

Hea

tFlu

x(M

Wm

2 )

Exp., 1 barExp., 2 barExp., 3 barPresent, 1 barPresent, 2 barPresent, 3 bar

(b) Inlet velocity: 1.0 m/s

Figure 2: Spatial correlations,Rpp(r) normalized by particle phase kinetic energy,q2p, and distribution functions of particle classes, P (C), at T+/τp = 0.

(a) Temperature, from 363 K (blue) to 414 K (red)

(b) Vapour volume fraction, from 0 (blue) to 0.75 (red)

(c) Velocity vectors, from 0.009 m/s to 0.33 m/s

Figure 3: The error of the PSIC approximation, qpfp is particle-fluid covariance

for the particles effecting the flow and qgfp is covariance for the ghost

particles.

fields of both classes.

4 Validity of point source approximation

Once being sure of the initial conditions of the particle classes, a numerical studyhas been performed to verify the point source approximation (PSIC). It is beenshown by Eaton [7] how the point source approximation fails with increase in theparticle radius. In this paper, particles smaller than the Kolmogorov scale are used(see table 2) so the effect of wake production is negligible. Then all the rest is thevalidation of the approximation.

Simple test is then applied to two-classes of particles introduced into the samefluid realization. One of them is coupled with the fluid and the other is chosen asthe ghost particles (particles with one-way coupling) (see Vermorel et al. [8]). Ifthe approximation is to be valid, then the statistical values of each class should besimilar with a negligible difference between each other so that they see the samefluid field.



0 0.2 0.4 0.6 0.8 1(R-R

in) (R

out-R

in))

0

0.1

0.2

0.3

0.4

0.5

0.6

Vol

ume

frac

tionα g

ExperimentRef. [Bae et al.]Present, N

01.1e 7, N

13e 5

Present, N0

1.6e 7, N1

3e 6

(a) Local radial profile.

0 0.5 1 1.5 2Axial length (m)

0

0.04

0.08

0.12

Vap

our

frac

tionα g

ExperimentRef. [Bae et. al.]Present, N

01.1e 7, N

13e 5

Present, N0

1.6e 7, N1

3e 6

(b) Average axial profile.

Figure 4: Parameter D, the difference between the fluid realizations and statisticsof the two classes, ClassA and ClassB.

As seen in fig. 3, the difference of the fluid-particle covariance, qfp, between thetwo populations is less than 1% which is rather acceptable for the application.

5 Application of MEF to two-way coupling

As explained in introduction, for the MEF’s application to the flows where the two-way coupling cannot be ignored, the definition of the probability density function,f

(1)p (cp, x, t), is not direct forward preventing the definitions of statistical quanti-

ties conditioned on a single realization of fluid turbulence.Initially, the idea was to explore the effect of different initial conditions of par-

ticles on the same fluid field, Hf . Performing number of two-way coupled sim-ulations with particles having different initial positions, it was observed how theturbulence evolves with a response to different initial conditions. Specifically, itwas curious to find a time range where the fluid turbulence does not differ betweendifferent fluid realizations enormously. To compare the fluid fields in these differ-ent simulations, the proper method is the utilization of the normalized parameterD:

D = (< (uA − uB)2 > /uArmsuBrms)1/2 (6)

where uA is the fluid velocity with response to the ClassA and uB is the one toClassB where < . > denotes volume averaging.

Two simulations are performed with 2 classes of particles (see table 2). The par-ticle initial conditions are as explained in section 3. Fig. 4 shows the behavior ofD and the particle statistics in time. As seen in the figure, the flow fields differen-tiate more than 10% and significantly the difference increase in time whereas thestatistical quantities are the same for each class (the figure on the right hand side).

This difference in instantaneous fluid field is due to the small deviations in thefeedback of particles which are quickly amplified in time by the non-linear chaoticnature of turbulence. The conclusion is then the turbulent field does not allow a



(a) Temperature, from 356 K (blue) to 376.3 K (red).

(b) Vapour fraction, from 0 (blue) to 0.69 (red).

Figure 5: Spatial correlations,Rpp(r) normalized by particle phase kinetic energy,q2p, and distribution functions of particle classes, P (C), at T+/τp = 8.

fluid realization rest the same at least for a small time-range when it is coupledwith the particle phase.

To come over the non-linear nature of turbulence, the simulation resumed in thefirst test is considered with only one class being active (two-way coupled to thefluid) and the other class is considered as ghost particles (particles with one-waycoupling). In the same fluid realization, difference between the source terms ofclasses are then expected to be dependent on the number of particles, at least for atime range. Two simulations are performed with the configuration where ClassAandClassB are the ones active in respective simulations to see the interchangabil-ity of the active class. As might be guessed, different classes correspond to differ-ent initial conditions and the fundamentality here is then the ghost particles’ utilitywhich is to increase the precision of the computation of the mesoscopic quantitiesusing the p.d.f. f (1)

p .As seen in fig. 5, at the end of the simulation, spatial distributions and correla-

tions of the two classes rest the same to each other, shown only for one simulation.This is to say that even with two-way coupling, particle field keeps the mesoscopicfield, as in the initial conditions, the same for both classes.

However, the effect on the fluid field is not the same when the active class is AorB. Turbulent field responding to ClassA andClassB in respective simulationsis shown in fig. 6. It is clear that the topology of the flow stays the same. However,analyzing closer the field, the dashed-line regions are shown in fig. 7. The dots inthe fields shows the approximative centers of the vortexes and as seen, they haveslight deviations between each other.

The source terms of active and non-active classes in one of the simulations areshown in fig. 8. As seen in the figure, the difference quantified by the parameterD in time stays constant for a time range more than one particle relaxation time.The initial peak on the graph is due to the transition of both the particles and fluidfield to arrive at a new equilibrium, it is to be reminded that turbulence is forced bythe scheme of Eswaran and Pope [5]. In the stationary period, constant differenceseems like promising and more strictly, it has been found out that the differencedepends on the number of ghost particles, not shown here. As the number of parti-cles in each class, one-way or two-way, increases the parameter D decreases.



Figure 6: Fluid velocity vectors, T+/τp = 8, on the left hand side, ClassA isactive, on the right hand side, ClassB is active in seperate realizations.

Figure 7: Fluid velocity vector zoom field, the dashed-line regions in fig. 6.

So there is a statistical relation between the number of particles and the differ-ence of the source terms among the classes. This sounds a bit like the statisticalconfirmation of PSIC method. Using high number of particles lets the changing



Figure 8: Source term Πx, eq. (3), of particles in x direction at T+/τp = 8, ofClassA (active) on the left and of ClassB (non-active) on the right inthe same fluid realization.

Figure 9: ParameterD, difference between the source terms of particles inClassAand ClassB for the two realizations of fluid.

the active class to any other classes to generate the same fluid field. The definitionof the conditional probability density function, f (1)

p (cp, x, t,Hf ), is then possible.

6 Conclusion and perspectives

The Mesoscopic Eulerian Formalism is shown to be applicable to the flows wherethe carrier phase is modified by the presence of solid particle phase. From statisti-cal point of view, increasing the number of particles reduces the error of the PSICapproximation where non-physical oscillations are not obtained and also highnumber of particles lets the definition of a single fluid realization, Hf , in whichlarge number of particle realizations, Hp, is imaginable. Study can be extendedto different Stokes numbers to see the difference when there is no concentrationof particles. Also classes with different numbers of particles should be very infor-mative on the effects of initial conditions. With the definition of the probability



density function of particles conditioned on a single fluid realization, mesoscopicfield values become measurable.

Acknowledgements

This research project has been supported by a Marie Curie Early Stage ResearchTraining Fellowship of the European Community Sixth Framework Program undercontract number MEST-CT-2005-020426’.

References

[1] Sundaram, S. & Collins, L.R., Collision statistics in an isotropic particle-ladenturbulent suspension. part 1. direct numerical simulations. Journal of FluidMechanics, 335, pp. 75–109, 1997.

[2] Elghobashi, S. & Truesdell, G.C., On the two-way interaction between homo-geneous turbulence and dispersed solid particles. part 1: Turbulence modifica-tion. Phys Fluids A, 5(7), pp. 1790–1801, 1993.

[3] Fevrier, P., Simonin, O. & Squires, K.D., Partitioning of particle velocities ingas-solid turbulent flows into a continuous field and a spatially uncorrelatedrandom distribution: theoretical formalism and numerical study. Journal ofFluid Mechanics, 533, pp. 1–46, 2005.

[4] Crowe, C., Sharm, M. & Stock, D., The particle source in cell (psi-cell) modelfor gas-droplet flows. J Fluids Engineering, 99, pp. 325–332, 1977.

[5] Eswaran, V. & Pope, S., An examination of forcing in direct numerical simu-lations of turbulence. Computers and Fluids, 16, pp. 257–278, 1988.

[6] Zeren, Z. & Bedat, B., Spectral and physical forcing of turbulence. Proceed-ings of the iTi International Conference on Turbulence, Bertinoro, Italy, 2008.

[7] Eaton, J.K., Two-way coupled turbulence simulations of gas-particle flowsusing point particle tracking. International Journal of Multiphase Flow, InPress, Accepted Manuscript.

[8] Vermorel, O., Bedat, B., Simonin, O. & Poinsot, T., Numerical study and mod-elling of turbulence modulation in a particle laden slab flow. Journal of Tur-bulence, 4, p. 25, 2003.



VOF-based simulation of conjugate masstransfer from freely moving fluid particles

A. Alke1, D. Bothe1, M. Kroeger1 & H.-J.Warnecke2

1 Center of Smart Interfaces, Technical University Darmstadt, Germany2 Department of Chemical Engineering, University Paderborn, Germany

Abstract

In this paper two variants of a VOF-based approach for the numerical simulation ofthe molar mass transport of a diluted species in two-phase flows with deformableinterfaces are introduced and compared. The variants differ in the manner of thecomputation of the mass transfer flux across the interface. The method assumeslocal thermodynamical equilibrium at the interface and enables the simulation ofconjugated mass transfer problems across deformable interfaces, where the masstransport resistance lies in both phases. The considered model also allows for arbi-trary distribution coefficients. First numerical simulations show the potential andthe present limits of this method.Keywords: Direct Numerical Simulation, two-phase flow, Volume of Fluid, conju-gate mass transfer, two scalar approach.

1 Introduction

In Process Engineering, mass transfer operations based on dispersed two-phaseflows are frequently applied. Typical examples are gas purification by bubbling ofa gas through a liquid, oxygenation of aqueous systems in biological processes,and solvent extraction as a thermal separation process. Besides the departure fromthe phase equilibrium, the mass transfer depends mainly on the characteristics ofthe dispersed two-phase flow, i.e. on the particle size and shape, slip velocities,internal circulation, swarm behaviour etc., and on the species diffusivities. Besidesexperimental studies, Direct Numerical Simulations of single fluid particles, whichbecome more and more feasible due to the ongoing increase in computationalpower, can be very useful since they can provide local data which usually can-not be accessed by experiments. Contrary to heat transfer, in mass transfer prob-



lems the transported scalar - the molar concentration c, say- is not continuous atthe interface. To handle the interfacial jump discontinuity numerically is a chal-lenging task. Volume of Fluid (VOF)-based simulations of mass transfer acrossdeforming interfaces have been reported in [1] and in [2, 3]. In the latter papers,transfer of oxygen from air bubbles rising in water or aqueous solutions has beensimulated, taking into account the realistic jump discontinuity of the oxygen pro-files at the interface. Darmana et al. [4] performed 3D simulations of mass transferat rising fluid particles for Sc = 1 using the Front Tracking method. There, thetransport resistance inside the fluid particle is neglected, i.e. a constant concentra-tion value inside the bubble is assumed. Radl et al. [5] performed 2D simulationsof deformable bubbles and bubble swarms with mass transfer in non-Newtonianliquids using a semi-Lagrangian advection scheme. To prevent stability problems,a reduced density ratio between gas and liquid is used there. Recently, first paperson numerical simulation of reactive mass transfer appeared. In [6, 7], the impactof single bubble wake dynamics on the reaction-enhanced mass transfer and onthe yield and selectivity of the cyclohexane oxidation reaction is studied numeri-cally for fixed shapes in 2D. In [8], reactive mass transfer at deformable interfacesis examined using a 2D Front Tracking/Front Capturing hybrid method. In [9], aLevel Set based method is used to simulate mass transfer across the interface ofa moving deformable droplet. This method is extended to reactive mass transferin [10], where an instantaneous chemical reaction occurs inside a moving dropletwhich leads to a quasi-stationary problem for the mass transfer. In [11], 2D sim-ulations are performed using a Front-Tracking method to investigate the effect ofdifferent Hatta and Schmidt numbers on the catalytic hydrogenation of nitroarenesfor single bubbles and bubble clusters.

The focus of the present work is on a VOF-based method having the potential tobe used for Direct Numerical Simulations (DNS) of mass transport in two-phaseflows with deformable interfaces, including droplets, bubbles, falling films etc.For the computation of the mass transfer across the interface, two variants areemployed. In the numerical study presented here, single fluid particles rising in aNewtonian fluid with mass transfer from the fluid particle to the surrounding liquidare considered.

2 The governing equations

In the following, we consider a fluid-particle (domain Ωd(t)) which is immersedin a liquid (domain Ωc(t)). The deformable interface between the two phases ispresented as a surface of zero thickness and is denoted by Σ(t). The transfer com-ponent k has a constant initial concentration ck(t0) > 0 inside the fluid particleand a zero concentration inside the surrounding liquid. Furthermore, the followingassumptions are imposed:

• dilute two-phase system,• chemically inert non surface active transfer component,• local thermodynamical equilibrium at the interface,• no phase change,



• isothermal conditions,• incompressible bulk phases.

The present paper employs a continuum mechanical model in which the govern-ing equations are based on the conservation of mass, momentum, and (molar) massof the transfer component. Inside the phases the transport of species k is governedby the local balance equation

∂tck + ∇ · (uck + jk) = Rk in Ωc(t) ∪ Ωd(t). (1)

Here ck is the volume specific molar concentration of the dissolved species k,jk is the area specific diffusive (molecular) flux density, and Rk is the overallreaction rate accounting for all chemical reactions in which species k is involved.In the following, chemical reactions are not more considered, i.e. Rk = 0. Forthe diffusive flux density, a suitable constitutive equation is required. Here, onlydiluted systems are considered. In this case the molecular transport inside the bulkphases can be described by Fick’s law, i.e.

jk = −Dk ∇ck, (2)

with diffusion coefficient Dk. The concentration ck and with it the flux jk as wellas the velocity u are local and time dependent quantities. To solve the parabolicpartial differential equations (1) inside the bulk phases, initial and suitable bound-ary conditions are required. The solutions inside the phases are not independent atthe interface and two jump conditions are required. The first one is a transmissioncondition and comes from the interfacial balance. Since only non surface activetransfer components are considered, the normal component of the diffusive fluxesare equal at the interface, i.e.

[jk] · nΣ = 0 (3)

with the jump notation

[φ] (xΣ) = limh→+0

(φ(xΣ + hnΣ) − φ(xΣ − hnΣ)) . (4)

For the second interfacial condition, local thermodynamical equilibrium isassumed, i.e. the chemical potential µk of component k is continuous at the inter-face:

[µk] = 0, (5)

withµk(T, p) = µ0

k(T, p0) +RT ln ak. (6)

The first term in (6) is the chemical potential of component k in a pure system (con-sisting only of component k) at temperature T and standard pressure p0, whereasthe second term, containing the activity ak of species k in a multicomponent fluid,accounts for mixing effects. For liquid systems, the activity is proportional to the



concentration and for gas mixtures it is proportional to the partial pressure. In caseof a diluted system, the thermodynamical equilibrium condition (5) reduces to

cdk|Σ = cck|Σ /mk, (7)

with the distribution coefficientmk = mk(p, T ) > 0, where cdk|Σ and cck|Σ are theone-sided limits of the concentrations at the interface in the dispersed and contin-uous bulk phase, respectively. Since we consider only isothermal flows with smallpressure gradients, the distribution coefficient mk is assumed to be constant. Thesecond condition (5) is only an approximation since the deviation from the localthermodynamical equilibrium is the driving force of the mass transfer. However,this deviation is very small. Therefore, the equilibrium assumption is commonlyaccepted.

The underlying velocity field is governed by the two-phase Navier-Stokes equa-tions expressing conservation of mass and momentum. Assuming continuity of thevelocity at the interface, a one-field formulation is possible in which the interfacialmomentum jump conditions act as source terms in the momentum equations. Forviscous (Newtonian) fluids of constant density and constant surface tension, thegoverning equations read as

∇ · u = 0, (8)

andρ ∂tu + ρ (u · ∇)u = −∇p+ η∆u + ρg + σκnΣδΣ, (9)

where the momentum jump conditions are incorporated via the interfacial Deltadistribution δΣ. In this interfacial source term, κ = −∇ ·nΣ denotes the curvature(more precisely, the sum of the principal curvatures). In (9) the material propertiesρ and η refer to the phase dependent values which are given as

ρ = fρd + (1 − f)ρc (10)

andη = fηd + (1 − f)ηc, (11)

where f is the phase indicator function of the phase domain Ωd(t).

3 Numerical method

For complex flow situations such as a freely moving fluid particle with a deform-able interface, the mathematical model described in the previous section cannotbe solved analytically but has to be treated numerically. There are several require-ments for an appropriate numerical method. One challenge is the capturing of theinterfacial concentration jump of the transfer component at the interface accordingto (7). Since a Lagrangian fluid particle cannot cross the interface (Lagrangian the-orem), the species transport across the interface is purely diffusive. Therefore, theconvective transport of the discontinuity shall not lead to an artificial mass transfer



across the interface. Furthermore, the continuous-side concentration gradient nor-mal to the interface may be very steep depending on the particle Reynolds number,Re, and the Schmidt number, Sc, of the continuous phase. For practically relevantSc numbers around 500, say, the concentration boundary layer cannot be resolvedwithout specific computational techniques. The numerical scheme presented hereis based on the VOF method [12] using the one-field formulation of the Navier-Stokes equations (9). In comparison with other free surface simulation methods,the VOF method can handle massive deformations and even topology changesas they can appear in case of large bubbles. Furthermore, the VOF method con-serves the phase volume, which is an important issue if chemical reactions shallbe accounted for. The phase indicator f is obtained from its initial distribution bysolving the advection equation

∂tf + u · ∇f = 0. (12)

In the Finite Volume context, f corresponds to the volume fraction of phase Ωd

inside a computational cell V . The employed Finite Volume based VOF-solver,Free Surface 3D (FS3D) developed by Rieber [13], applies a directional as wellas a kind of operating splitting. To avoid systematic errors and unsymmetries,the sequence of processed directions in the splitting scheme is altered in eachtime step. That is, the convective terms of all transport equations are computedfirstly. Than, the forces for the momentum equations are computed and impressedbefore the diffusive transport terms are calculated. For the volumetric surface ten-sion force the conservative continuum surface stress (CSS)-model of Lafaurie etal. [14] is used. The numerical solution of the discrete version of (12) is based ona geometrical based flux calculation. Application of the so called piecewise lin-ear (or planar in case of 3D) interface calculation (PLIC) scheme for the outgoingphase volume fluxes in interfacial cells prevents interface smearing.

3.1 Transport of molar species mass

For the computation of the transport of a transfer species k, the concentration isrepresented by two separate scalar variables according to

φdk(x, t) =

ck(x, t) for x ∈ Ωd(t)0 for x ∈ Ωc(t)

(13)

and

φck(x, t) =

0 for x ∈ Ωd(t)ck(x, t) for x ∈ Ωc(t).

(14)

This allows for capturing the different one-sided limits of the concentrations at theinterface. In the discrete Finite Volume scheme, these scalars are related to the cell



volume V , i.e. in interface-containing cells the cell centered values are given as

φdk(t) =

1|V |

∫V ∩Ωd(t)

ckdV (15)

and

φck(t) =

1|V |

∫V ∩Ωc(t)

ckdV. (16)

3.1.1 Convection

The new variables are similar to the VOF-variable f in that these quantities areall nonnegative. The only difference is that φd

k and φck can take arbitrary positive

values whereas f is always less or equal to one. Therefore, the convective trans-port of φd

k and φck is treated analogously to that of the convective f transport, using

the PLIC algorithm for the outgoing flow in interfacial cells to prevent an (artifi-cial) convective mass transfer and a flux limiter scheme inside the bulk phases tominimize numerical diffusion.

3.1.2 Mass transfer across the interface

After the computation of the convective transport of species mass (and also of theother quantities like phase volume and momentum) mass transfer across the inter-face is calculated. Within the two scalar approach, mass transfer is accounted forby source terms. Inside a computational cell Vi containing a part of the interface,the transferred volume specific molar mass is substracted from and added to thecorresponding values of φd

k,i and φck,i. In this notation, index i stands as a short

form for (i, j, k) which is the full index of a grid cell in a 3D Cartesian mesh. Forbrevity the characters j and k are omitted. In the following, the calculation of one-dimensional fluxes are explained only for the x-direction where index i+1 has themeaning of (i+ 1, j, k). For the calculation of mass transfer, where the two jumpconditions (3) and (7) have to be accounted for, we employ two different variants.

Variant I: Equilibration of interfacial cells

In the first variant, we assume that inside a computational cell Vi, containinga part of the interface, the transfer component k is ideal mixed in the separatephases. After the convective transport both variables φc

k,i and φdk,i have certain

values which are assumed to be constant within the respective phases lying inthe considered interfacial cell. However, the ratio φd

k,i(1 − fi)/(φck,ifi) does not

corresponds to mk, in general. Therefore, the characteristic of this equilibriumapproach is the conservative redistribution of the (molar) species mass accordingto

φdk,i |Vi| + φc

k,i |Vi| = φd,eqk,i |Vi| + φc,eq

k,i |Vi| , (17)



where the values of the variables φd,eqk,i and φc,eq

k,i fulfil the thermodynamical con-dition (7). In this variant, the first transmission condition (3) is not explicitlyaccounted for. But this condition expresses the local (molar) mass balance at theinterface which is inherently fulfilled by this approach.

Variant II: Computation of the one-sided concentration gradient

The second variant is based on the computation of the concentration gradientadjacent to the interface at the continuous side. Within an interfacial cell Vi thetotal (molar) mass flux of species k normal to and across the interface with inter-facial area |AΣi | is given by

jΣi,k · nΣi |AΣi | = (jΣi,k,x nΣi,x + jΣi,k,y nΣi,y + jΣi,k,z nΣi,z) |AΣi | . (18)

Here, only directions are accounted for in which the neighbour cell is completelyfilled with the continuous phase. To calculate the one-dimensional diffusive fluxdensity jΣi,k,x (where Σi denotes the interface in cell Vi, k the component index,and x the direction) it is assumed that within the interfacial cell the dispersed phaseis well mixed. Therefore, the cell centered value of φd

k,i is taken as the concentra-tion value cdk|Σ ,i adjacent to the interface. Then, depending on whether cell Vi+1 orcell Vi−1 lies completely in the continuous phase, the one-dimensional flux densityis computed as

jΣi,k,x = Dck

φdk,i/(fimk) − φc

k,i+1/(1 − fi)xi+1 − xi

(19)

or

jΣi,k,x = Dck

φck,i+1/(1 − f1) − φd

k,i/(fimk)xi+1 − xi

, (20)

respectively. The local interfacial area in an interfacial cell Vi is calculated fromthe cell centered value of the gradient of the VOF-variable f according to

|AΣ,i| = ‖∇f‖i |Vi| . (21)

After calculating the cell specific total molar mass flux normal to the interface (18),the values for φd

k,i and φck,i are updated according to

φdk,i = φd

k,i − qk,i and φck,i = φc

k,i + qk,i (22)

respectively, where the source term is given by

qk,i =jΣi,k · nΣi |AΣi | ∆t

|Vi| . (23)



phase density dynamical diffusion Schmidt number

viscosity coefficient

ρ in kg/m3 η in mPas D in m2/s Sc = ν/D

dispersed 1.2 18 · 10−3 5 · 10−6 3continuous 1000 10 10−6 10

3.1.3 Diffusive transport inside the bulk phases

Finally, diffusive transport in the bulk phases is computed. Here, especially in com-putational cells with a very small f value it might occur that too much species massleaves the cell during time interval ∆t. The directional splitting scheme allows fora limitation of the diffusive fluxes which are calculated, using the forward differ-encing scheme according to

Nk,i,i+1 = Dc/dk

ck,i − ck,i+1

xi+1 − ci|Ai,i+1| , (24)

where Ai,i+1 is the cell face which connects the cells Vi and Vi+1. This one-dimensional flux is limited by the equilibrium criteria cn+1

i+1 ≥ cn+1i if cni+1 ≥

cni and vice versa. Furthermore, diffusive fluxes across cell faces connecting twointerfacial cells are also accounted for.

4 Simulation results and discussion

To compare the two variants, 2D numerical simulations (i.e. with translationalsymmetry) of an air bubble rising in a Newtonian liquid have been performed.The area equivalent diameter of the bubble is 3 mm, the liquid has a dynamicalviscosity of 10 times higher than that of water and a density of 1000 kg/m3.To keep the time step sufficiently high, a reduced surface tension of 36 mN/mhas been used. The physical parameters are given in Table 1. The computationaldomain of 1.2 cm× 2.4 cm is resolved by three different computational grids; cf.Table 2. Figure 1 shows the increase of the molar mass of the transfer componentwithin the continuous phase with time. For all resolution cases the mass transfercalculated with the equilibrium approach is higher than that calculated with thegradient approach. But the results of both variants get closer together with higherresolution. This indicates insufficient resolution of the thin concentration boundarylayer; recall that the latter has a thickness proportional to 1/

√Re Sc. Obviously,

with the gradient approach the mass transfer is underestimated. The reason for thismay be an inaccurate approximation of the interfacial area. The total interfacialarea of a fluid particle is given by

∑all cells ‖∇f‖i |Vi|. But not only interfacial



Table 1: Physical parameters.

Case number of grid cells cell width in µm cells per diameter

A 256 x 512 46.9 64B 512 x 1024 23.4 128C 1024 x 2048 11.7 256

Figure 1: Molar mass of transfer species in continuous phase related to initial

cells have ‖∇f‖ greater than zero but also the neighboring cells. Therefore, equa-tion (21) yields a too low value for the local interfacial area inside a computationalcell. Otherwise, the equilibrium approach inherently overestimates the mass trans-fer since the (local) thermodynamical equilibrium is only valid adjacent to theinterface. The larger the normal distance from the interface the lower is the con-centration. The assumption of ideally mixed interface cells can be interpreted asan infinitely fast molecular transport at the interface leading to a too large masstransfer. Therefore, the equilibrium approach yields an upper bound for the masstransfer. However, from Figure 1 it can be seen that grid independence is alreadyreached. The different mass transfer rates are also noticeable in the concentra-



Table 2: Used numerical grids.

molar mass in fluid particle versus time.

tion profiles. Figure 2 shows the concentration distribution within the continuousphase when stationary hydrodynamical conditions are reached, obtained with thetwo variants (left: equilibrium and right: gradient variant) at the highest resolution(case C). In both cases, species is mainly present in the wake of the fluid parti-

Figure 2: Concentration distribution of transfer component in the continuous phaseyielded at highest resolution (Case C): mass transfer calculated with

cle. However, with the equilibrium approach the region of high concentrations atthe stagnation points is more pronounced than those obtained with the gradientapproach.

5 Conclusions and outlook

A new VOF-based two scalar approach for simulating the transport of chemicalspecies within a two-phase flow is introduced. The method allows for the sim-ulation of conjugate mass transfer problems across deformable interfaces withan arbitrary distribution coefficient. The treatment of the convective transport isanalogous to that of the VOF-variable f , using a geometrical flux calculation.This procedure avoids artificial mass transfer due to convection. First numericalresults in 2D at a moderate Schmidt number of 10 are performed. The resultsshow that the mass transfer rate obtained by equilibration of the interfacial cellsare always higher as those obtained by using the one-sided limit of the concentra-tion gradient in the continuous phase. However, with finer resolution the results ofboth variants get closer together. For the equilibrium variant, grid independencyis reached. However, the finest resolution presented here will not be sufficient for



equilibrium variant (left) and with the gradient variant (right).

higher Schmidt numbers and, moreover, is not suitable for 3D simulations. Themass transfer rate obtained with the gradient variant at the highest resolution isstill lower than the ”true” rate. The reason for this may lie in the underestima-tion of the interfacial area. Therefore, further steps are the use of a more accurateinterfacial area calculation scheme, the development of a subgrid model for theconcentration profile at the interface, use of a moving grid technique to reduce thecomputational domain as well as a local grid refinement around the bubble.

Acknowledgement

We gratefully acknowledge financial support from the Deutsche Forschungsge-meinschaft (DFG) within the DFG-project ”Reactive mass transfer from rising gasbubbles” (PAK-119).

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[10] Deshpande, K.B. & Zimmermann, W.B., Simulations of mass transfer lim-ited reaction in a moving droplet to study transport limited characteristics.Chemical Engineering Science, 61, pp. 6424–6441, 2006.

[11] Radl, S., Koynov, A., Tryggvason, G. & Khinast, J.G., DNS-based predictionof the selectivity of fast multiphase reactions: Hydrogenations of nitroarenes.Chemical Engineering Science, 63, pp. 3279–3291, 2008.

[12] Hirt, C.W. & Nichols, B.D., Volume of fluid (vof) method for the dynamics offree boundaries. Journal of Computational Physics, 39, pp. 201–225, 1981.

[13] Rieber, M., Numerische Modellierung der Dynamik freier Grenzflachen inZweiphasenstromungen. Ph.D. thesis, ITLR Universitat Stuttgart, 2004.

[14] Lafaurie, B., Scardovelli, C.N., Scardovelli, R., Zaleski, S. & G.Zanetti,Modelling merging and fragmentation in multiphase flows with surfer. Jour-nal of Computational Physics, 113, pp. 134–147, 1994.



Computational fluid dynamic as a feature to understand the heat and mass transfer in a vacuum tower

1 2 2 1

1School of Chemical Engineering, State University of Campinas (UNICAMP), Campinas-SP, Brazil 2CENPES – PETROBRAS Research and Development Center, Cidade Universitaria, Rio de Janeiro, Brazil

Abstract

The understanding of fluid dynamic phenomena in industrial equipment is extremely important for new projects and their optimization. Distillation columns have been used for centuries. Since then many advances have been achieved. The present study shows a methodology to study the heat and mass transfer in empty sections of distillation columns considering the Eulerian-Lagrangian approach. A multiphase model is applied to the vacuum tower, with the vapor-liquid flow being modeled with a Eulerian-Lagrangian approach. The Computational Fluid Dynamic (CFD) technique is adopted as the tool to predict vacuum tower dynamics. The CFD results were validated with real operation behaviour. An ideal multicomponent equilibrium model is adopted to represent the thermodynamics in the heat and mass transfer processes. The characteristic time scales are used in the classification and the understanding of the dominant mechanisms in gas-liquid suspensions. This methodology is general, and therefore can be applicable to any turbulent gas-liquid flow. A discussion on the validity of the models is given, as well as an application to distillation vacuum towers. Keywords: CFD, thermodynamic equilibrium, multiphase, Lagrangian, spray.

1 Introduction

The heat and mass transfer takes place in a distillation column by vapor and liquid contact. In the vacuum towers wash zones, this contact is promoted by

K. Ropelato , A. V. Castro , W. O. Geraldelli & M. Mori



spray nozzle distributors. This sort of device is widely used in distillation and absorption columns to provide a uniform liquid distribution. One of the challenges in this column section is to reduce the space available for heat and mass transfer, without decreasing process efficiency. The reduction of space between liquid sprays and vapor inlet can bring a couple of advantages:

• minimize column height; • less maintenance • lower pressure drop

This potential height reduction was observed in experimental measures, indicated by thermal equilibrium near the vapor entrance. The literature presents many studies considering spray simulations, each one having its own modeling approach as presented by Mugele and Evans [1], Meyer et al. [2], Trompiz and Fair [3] and Beck and Watkins [4], where the heat and mass transfer is modeled considering an Eulerian-Eulerian approach. Authors as Lefebvre [5] and Guet et al. [6] argue that an accurate knowledge of drop size distribution as a function of the conditions of the system is an important prerequisite for fundamental analysis of the heat and mass transfer. Kim [7] and Bozorgi [8] considered a Eulerian-Lagrangian model to represent droplet evaporation. Kim considered phase equilibrium using the fugacities of the liquid and gas phases assuming real gas behavior, and its importance in the calculation of the evaporation of a droplet or spray at high pressures. Bozorgi studied variables effects, such as liquid film, total projected surface area of the droplets and velocity profile of the droplets on the performance of the spray scrubber in the aerosol removing process. A Lagrangian model has some advantages in comparison to the Eulerian model, such as:

• easy modeling of droplets diameter distribution; • spatial discretization of spray nozzles (computational mesh) is not required since it is represented by points within the domain

In this study a 4-m diameter vacuum tower was considered with a liquid distributor having 41 spray nozzles. The vapor-liquid thermodynamic equilibrium is modeled with a “gamma-phi” approach. The model is capable of representing the evaporation and condensation of the droplets. Thus, droplets diameter is variable as a function of system properties. The thermodynamic equilibrium was modeled via FORTRAN subroutines coupled to the commercial CFD code, ANSYS-CFX 11. The main advantage of CFD technique is to allow the user to evaluate conceptual changes in equipment in short-term simulations (compared to experimental measurements) at low computational cost. Based on these statements, the main target of this work is to gather information on vacuum tower fluid dynamic using an Eulerian-Lagrangian model using Computational Fluid Dynamics (CFD) techniques that represent a powerful tool for better understanding of physical phenomena involved in vacuum towers.



2 Mathematical modelling

The present work applies simplified multicomponent Fick's Law and gamma-phi approach for liquid-vapor equilibrium in a Eulerian-Lagrangean multiphase flow model together with multicomponent heat and mass transfer. This model was applied successfully before by Paladino et al. [9] and Ropelato et al. [10]. The simplification proposed here is the independence of diffusive mass flux of one component from other components. This implies that each mass flux can be calculated without other fluxes information. In the other hand, thermodynamic equilibrium takes the all components into account. Thus, it is calculated rigorously. With the liquid-vapor equilibrium considering a “gamma-phi” approach, an equation of state is applied to calculate fugacity coefficient (which provides a measure of non-ideality of vapor phase), while an excess Gibbs free energy model is applied to predict the behavior and non-idealities of liquid phase by calculation of an activity coefficient. The considered equilibrium is a traditional approach and can be applied to many different mixtures cases. It is adequate for systems at low or moderate pressure, as the vacuum tower, for example. This model cannot be applied in systems at high pressures. This limitation can be explained because the excess Gibbs free energy models are determined at low pressures. Moreover, the “gamma-phi” equilibrium model is one of the most used equilibrium approach in Petroleum Industry. The conservation equations calculated in CFD code are presented below for the Eulerian phase. The mass conservation of the continuous (Eulerian) phase is:

( ) ( ) ∑=

Γ=ρ⋅∇+ρ∂∂ PN

1DCDCCCt

v (1)

The ΓCD represents each component mass flux through the interface. The subscripts C and D are the continuous and dispersed phase respectively. The momentum equation is:

( ) ( ) gMTvvv

cCCCCCC .p.

tρ++∇+−∇=ρ∇+

∂ρ∂

(2)

The Mc represents the interfacial momentum transfer terms, turbulent stress tensors and T is the viscous stress tensor. For Newtonian fluid, viscous stress can be expressed via Stokes equation:

( )[ ]TCCC vvT ∇+∇µ= (3)

Where µC is the fluid dynamic viscosity. The conservation equation for component “i” in the continuous phase is:

( ) ( ) ( ) mC,iiCiCCiCCiCC )y(Dyyt

Γ=∇ρ⋅∇−⋅∇+ρ∂∂ v (4)

The term ρCDiC represents the mass diffusion coefficient, DiC represents the kinematic diffusivity and Γi,mC the source term due to mass transfer to/from dispersed phase.



The energy equation can be represented as follows:

( ) ( ) ( )

∑=

++Γ

=∇λ⋅∇−ρ⋅∇+ρ∂∂

Ncomp

1icCCmC,i

CCCCCCC

SQh

Thht

v

(5)

where hC is the enthalpy, TC is the temperature and λC is the convective heat transfer from/to liquid phase. Evaporation and condensation rates in the Lagrangian model will be indicated by the droplet mass conservation equation:

( )VeLpmC,iD wKwDShd

dtdm

−ρπ−=Γ= (6)

The dp is the droplet diameter; D represents the mass diffusivity term; Sh, the droplet Sherwood number; Ke, equilibrium ratio; and wL and wV, the component of mass fraction at the interface of the dispersed and the continuous phases, respectively. The simulation considered a two-way coupling. The coupling concept is very important in a multiphase flow. If the flow of one phase affects the other while there is no reverse effect, the flow is said to be one-way coupled. If there is a mutual effect between the flow of both phases, the flow is two-way coupled. Therefore, the effects of the presence of droplets on the turbulent motion of the continuous phase can be observed. A schematic diagram of coupling is shown in fig 1. The carrier phase is described by density, temperature, pressure and velocity field. The droplet phase is described by concentration, size, temperature and velocity field. Coupling can take place through mass, momentum and energy transfer between phases. Mass coupling is the addition of mass through evaporation or the removal of mass from the carrier phase stream by condensation. Momentum coupling is the result of the drag force on the dispersed and continuous phases. Momentum coupling can also occur with momentum addition or depletion due to mass transfer. Energy coupling occurs through heat transfer between phases.

Figure 1: Schematic diagram of coupling effects.



2.1 Characteristics time scales

In the formulation of the transport equations, several characteristic time scales can be defined. These time scales are of fundamental importance in the classification and understanding of the dominant mechanisms in suspensions. (Elgobashi [11]; Peirano and Leckner [12]). Time scales are fundamental to understand the vapor and liquid behavior, and also the effects of phase interaction. The characteristic time scale of the large eddies in the continuous phase ( t

cτ ) is defined in accordance with the k-ε model as eqn (7)

ε=

k09.0τ tc

(7) Where k is the turbulent kinetic energy and ε its dissipation. The Kolmogorov scale dissipative ( kτ ) is the characteristic time scale of the smallest scales.

5.0c

kτ

ευ

= (8)

The time of interaction between particle motion and continuous phase fluctuations is defined by eqn (9)

k2345.01

τ 2

tct

cdrV

+

τ= (9)

tcdτ is the Lagrangian integral time scale seen by the particles (computed along

the trajectory of the particle). This means that the time spent by a virtual fluid particle in an eddy is not the same as the time experienced by a liquid particle, due to the mean relative velocity between the two phases. Eqn (10) represents the droplet relaxation time ( t

dτ ).

rDc

dptd VC3

d4τ

ρ

ρ= (10)

The ratio tdτ / kτ represents the Stokes number in Kolmogorov scale (Stk),

which indicates the probability of coalescence in the region of study. The Stokes number is a very important parameter in fluid-particle flows. If Stk<<1, the response time of the droplets is much less than the characteristic time associated with the flow field. Thus the particle will have ample time to respond to changes in flow velocity. Thus the droplets and fluid velocities will be nearly equal. If Stk>>1, then the droplets will have essentially no time respond to the fluid velocity changes and the droplets velocity will be little affected in the equipment, Crowe et al. [13].



2.2 Numerical method and geometry details

Element based finite volume method (EbFVM) with unstructured is used by ANSYS CFX solver to solve the set of equations. The pressure-velocity coupling formulation is similar to the Rhie-Chow method where the solution of velocity and pressure is in the same node of element in the numerical mesh. A computational mesh study was conducted to obtain mesh independent results. A mesh with approximately 1.106 nodes was considered good for the simulations. The independency of mesh and particle tracking was analyzed considering all characteristics of the model (multiphase, heat and mass transfer). Fig. 2(a) shows the boundary conditions imposed in this study: red color indicates de vapor inlet; yellow indicates the demister pad; blue color, the spray section; and green color represents the bottom of equipment. Fig 2(b) shows the numerical grid details. The vapor inlet doesn’t present symmetrical characteristics, thus the whole geometry was taken into account. The spray distributor with 42 sprays is showed in fig. 3. Each of it with 200 particle tracks being used. Each spray nozzle was modeled as a full cone with 60º. The droplet diameter distribution used in the simulations was obtained from the spray nozzle provider which was characterized for water. A Petrobras internal correlation was applied considering the oil properties. The model has been generated as three-dimensional (3D), steady-state and the continuous phase (vapor) was model using the k-ε turbulence model. The k-ε

Figure 2: Vacuum tower. (a) Physical geometry with and boundary

conditions; (b) Numerical grid details.



Figure 3: Spray section distributor.

turbulence model is the most prominent turbulence models has been implemented in most general purpose CFD codes and is considered the industry standard model. It has proven to be stable and numerically robust and has a well established regime of predictive capability. For general purpose simulations, the k-ε model offers a good compromise in terms of accuracy and robustness (Wilcox [14]). A high resolution scheme (a bounded second order upwind) was used to model the advection terms of the momentum equation.


The vapor flows upwards through the vacuum tower and the light gas oil liquid phase flows downwards counter-courant to the vapor phase. The case analyzed is considered a large vacuum tower with high capacity; the vapor/liquid flow ratio is 0.43.

3 Results and discussions

The analysis of time scales was considered in the whole domain in two orthogonal slice planes. The vapor and liquid contact considering the effects of vapor inlet was evaluated by time scales. The characteristic time scale of the large eddies in the continuous phase (τc), fig. 4(a), considered the rate of effects of turbulent kinetic energy and the rate of dissipation of turbulent kinetic energy. Regions of high values of dissipation energy represent the low values of τc. These regions are close to the spray section. The Kolmogorov scales (τk) indicates regions of smallest vortices in the domain, hence these regions show the highest values of turbulent energy dissipation fig. 4(b). Fig. 5 considered the effects of vapor velocities in droplets behavior. Regions closed to the vapor inlet indicates low values of droplet relaxation time ( t

dτ ), fig. 5(a), which means that these are regions of low droplets inertia. Thus phases are well mixed. Drag values are high in these regions. The time of interaction between particle motion and continuous phase fluctuations is defined by the Lagrangian integral time scale ( t

cdτ ), fig. 5(b). The



Figure 4: Vapor fluid dynamics behavior (a) characteristic time scale of the large eddies in the continuous phase (τc), (b) characteristic time scale of the Kolmogorov scales (τk).

Figure 5: Droplets flow properties, (a) droplet relaxation time ( tdτ ),

(b) Lagrangian integral time scale ( tcdτ ).

red color indicates regions where low values of heat and mass transfer occurs, these regions are close to spray section. The Stokes in Kolmogorov scale is a very important parameter in liquid-vapor flows. If Stokes number approaches to zero, the response time of droplets is much less than characteristic time associated with the flow field. Thus the droplets will have enough time to respond to the changes in flow field. In other words, droplets and vapor will be flowing at very close velocities. If the opposite behavior is noticed the droplets velocity is unaffected by vapor phase. Fig. 6 shows the Stokes number in the vacuum tower and the regions with Stokes number close to one represent regions with risk of coalescence.



The studied region doesn’t show a strong risk to this phenomenon take place. There are no assumptions in the model with this regard. Problems with large coalescence regions should be treated with a more suitable model.

Figure 6: Stokes in Kolmogorov scale (Stk).

Fig. 7(a) and (b) represent the temperature and volume fraction profile respectively. As the time scale analyses showed, the main region of heat transfer is close to the bottom of equipment. The thermodynamic equilibrium is achieved at half of the height of the equipment. The CFD temperature and thermodynamic equilibrium results were compared with values obtained in a Petrobras` Vacuum Tower with good agreement.

Figure 7: (a) Temperature profile, (b) Volume fraction profile.

4 Conclusion

The proposed 3D, steady-state, and multiphase model represented adequately the vacuum tower fluid dynamics. The time scale methodology was presented as a feature for fluid dynamics understanding of a vacuum tower. The continuous phase turbulent behavior and its interaction with the droplets were studied in equipment entrance section. The



more relevant regions of heat transfer and thermodynamic equilibrium was analyzed. CFD tool proved to be also a powerful tool to predict the fluid dynamic of heat and mass transfer in vacuum towers, and it can be used with success in chemical process optimization to predict critical points of scientific investigation. A potential height reduction in this case is possible as long as the assumptions taken in this model are suitable for the present case. Future studies are being carried out, to propose some optimization points in the studied vacuum tower.

5 Nomenclature

CD drag coefficient, [kg.m3/s] dp droplet diameter [m] F diffusion, resistive force g gravity acceleration, [m/s2] k turbulent kinetic energy, [m2/s2] p pressure, [Pa] S source term t time, [s] v velocity vector, [m/s] Vr slip velocity, [m/s] x Cartesian coordinate, [m] y Cartesian coordinate, [m] z Cartesian coordinate, [m] Greek letter Γ interphase mass flux [kg/m2/s] ε rate of dissipation of “k”, [m2/s3] µ dynamic viscosity, [kg/m s] ρ density, [kg/m3] τ characteristic time scale (s) ν kinematic viscosity [m2/s] Subscript c gas phase d liquid phase k Kolmogorov scale x cartesian coordinate y cartesian coordinate z cartesian coordinate φ fluid dynamic property Superscript t turbulent



References

[1] Mugele, R. A and Evans, H. D. Droplet size distribution in sprays. Ind. Eng. Chem., 43, n 6, pp. 1317 – 1324, June 1951.

[2] Meyer, M., Hendou, M., Prevost, M. Simultaneous heat and mass transfer model for spray tower design: application on VOCs removal. Computers Chem. Engng, 19, po. S277 – S282, 1995.

[3] Trompiz, C. J., Fair, J. R. Entrainment from spray distributors for packed columns. Ind. Eng. Chem, 39, pp. 1797 – 1808, 2000.

[4] Beck, J. C., Watkins, A. P. The droplet number moments approach to spray modelling: The development of heat and mass transfer sub-models. Int. Journal of Heat and Fluid Flow, 24, pp. 242 – 259, 2003.

[5] Lefebvre, A. H. Atomization and Sprays. Taylor & Francis. Purdue University, West Lafayette, Indiana, 1989.

[6] Guet, S., Ooms, G., Oliemans, R. V. A., Mudde, R. F. Bubble injector effect on the Gas lift efficiency. Fluid Mechanics and Transport Phenomena. 49, pp. 2242-2252, 2003.

[7] Kim, H., Sung. The effect of ambient pressure on the evaporating of a single droplet and a spray. Combustion and Flame. 135, pp. 261 – 270. 2003.

[8] Bozorgi, Y., Keshavarz, P., Taheri, M., Fathikljahi, J. Simulation of a spray scrubber performance with Eulerian/Lagrangian approach in the aerosol removing process. Journal of Hazardous Materials. 2006 .

[9] Paladino, E. E., Ribeiro, D., Reis, M. V., Geraldelli, W. O., Barros, F. C. C. A CFD model for the washing zone in coker fractionators. AIChE 2005 Annual Meeting.

[10] Ropelato, K., Rangel, L. P., Marins, E. R., Geraldelli, W. O. A CFD Study Comparing Different Feed Nozzle Arrangement within an Empty Spray Section in a Coker Fractionator. AIChE 2007 Spring Meeting.

[11] Elgobashi, S. On predicting particle-laden turbulent flows. Applied Scientific Research , 52, pp. 309-329, 1994.

[12] Peirano, E., Leckner, B. Fundamentals of turbulent gas-solid flows applied to circulating fluidized bed combustion. Department of Energy Conversion. Chalmers University of technology Göteborg, 1998.

[13] Crowe, C., Sommerfeld, M., Tsuji, Y. Multiphase Flows with Droplets and Particles. CRC Press, pp 17-36, 1998.

[14] Wilcox, D.C. Turbulence Modelling for CFD. DCW Industries, pp. 314, 2000.



Understanding segregation and mixing effects in a riser using the quadrature method of moments

A. Dutta, J. Raeckelboom, G. J. Heynderickx & G. B. Marin Laboratorium voor Chemische Technologie, Ghent University, Belgium

Abstract

Segregation and mixing effects of particle diameter distributions are numerically investigated in the riser section of a circulating fluidized bed. A granular kinetic theory based approach, which implements the Eulerian quadrature-based moment method to describe the particle phase, is coupled to an Eulerian multi-fluid solver through a population balance model. The gas-solid multiphase flow is two-dimensional, transient and isothermal. The particle distributions fed from each side inlet of the riser have different variances, but the same mean diameter. The core-annular regime used as a numerical benchmark for riser flows is well predicted. A comparison in the homogeneity of particle mixing is made from the lower-order moments of the particle distribution obtained at various positions and at different axial lengths along the riser. It is seen that the relative standard deviation of the particle distribution varies spatially, indicating dynamic mixing inside the riser. Keywords: mixing, riser, multiphase, quadrature method of moments.

1 Introduction

Multiphase flows involving solid particles in contact with a carrier gas in circulating fluidized beds (CFB) are ubiquitous in most chemical, petrochemical and pharmaceutical industries. Gas-fluidized circulating beds are commonly used in coal combustion, the catalytic cracking of crude oil, pharmaceutical granulation, etc. Quantitative understanding of the hydrodynamics of fluidization is needed for the design and scale-up of these processes. In these processes, the knowledge of particle segregation and mixing is limited. Nevertheless, there are applications in which it is important to avoid particle segregation, e.g. in



catalytic fluidized bed reactors, whereas in other applications it is necessary to achieve complete segregation or mixing, e.g. in fluidized bed combustors [1]. Usually, segregation is referred to as the spatial (both axial and radial) distribution of particles of different size and/or density. Dependence on the size and/or density differences is linked to operating conditions [2]. In a CFB riser, the solids size and/or density distribution gives rise to radial segregation resulting in a core-annulus structure, in which particles flow downwards along the walls in clusters while strands of particles move upwards, together with the dispersed particles in the centre [3]. The single-particle terminal velocity is often a decisive parameter affecting axial segregation in fluidized beds [4], while core-annulus flow is attributed to the radial segregation of the particles. Karri and Knowlton [5] experimentally found that the particle diameter distribution in the core is smaller than in the annulus due to shear and recirculation/backmixing effects generated from the solids downflow near the wall. Numerical studies using computational fluid dynamics (CFD) on solids segregation/mixing have been reported for binary, ternary [6,7] and even quaternary [8] mixtures of particles of varying size/density. To simulate a realistic particle distribution, the approach followed by most researchers [9,10] is the use of discrete approximations of the continuous size distribution function. However, this sectional approach is not feasible when the distribution is broad as it requires a large number of discrete sizes to accurately represent the size distribution leading to high computational costs even for relatively simple situations. Recently, Fan and Fox [11] have implemented a direct quadrature-based moment method to simulate segregation phenomena for a continuous distribution, thus avoiding the need for discrete approximations. In this method, the particle distribution is represented through a finite number of nodes, commonly referred to as abscissas, in the quadrature method. The evolution of these nodes is tracked through the lower-order moments of the distribution. As such, only a few nodes suffice to represent the entire distribution thus representing an attractive alternative to the traditional discrete approaches [11]. In the present study, the mixing effect of particle diameter distributions in a riser is numerically investigated using the quadrature method of moments (QMOM) in a multi-fluid CFD code, based on an Eulerian-Eulerian approach. The influence of polydispersity on the core-annulus flow profile is discussed. The standard deviation and mean determined through the lower-order moments of the distributions are calculated at various positions and at different axial lengths along the riser. These are used to study the influence of the particle distribution on the overall flow behavior and in particular, the mixing and segregation effects in the riser.

2 Numerical approach

A multi-fluid model based on the Eulerian-Eulerian approach is used. The conservation equations are solved for each phase in the Eulerian frame. The gas phase is considered as the primary phase, whereas the solid phase is considered as secondary or dispersed phase. For the gas phase the mass and momentum conservation equations are Reynolds averaged. The effects of turbulence are



taken into account via a dispersed k-ε turbulence model, which is an extension of the single-phase k-ε model [12], adapted for gas–solid interactions. For the solid phase, the transport equations for mass, momentum and granular temperature are obtained via the kinetic theory of granular flow (KTGF) [13].

2.1 Population balance model (PBM)

The general form of a particle population balance conservation equation, as written by Ramkrishna [14] is:

[ ( ; )] .[ ( ; ) ] ( ; )ss s sn x t n x t u S x tt

(1)

where ( ; ) ( ; ) ( ; ).S x t B x t D x t

In eq. 1, the number of particles, n, is distributed with respect to some

intrinsic parameter, x (say particle size). The variables s and su

refer to the

density and velocity vector of the particles respectively. The source terms B and D represent birth and death rates due to aggregation, breakage, etc. Several approaches for solving these equations are available, the most common being the discrete approach. However, a computationally attractive approach is the method of moments in which the population balance is formulated in terms of the lower-order moments in closed form [15]. For a homogeneous system, considering the number density function n(x;t) in terms of the particle size length (i.e. x≡L), the kth moment is defined by:

0( ) ( ; )k

km t L n L t dL

k = 0, 1,…..,2N-1. (2)

It is important to note that, usually, L is referred to as an internal coordinate in contrast to x and t, which are external coordinates. A number of lower-order moments are sufficient to represent the particle number density, total surface area and total volume of the particles. The resulting moment transport equation obtained by applying the above moment transform to eq. 1 is:

( ) .( ) ( ).k ks k s s k sm u m B Dt

(3)

A major drawback of this method is the need for an exact closure model for the source contributions. An exact closure is only available for constant or simple linear forms of the aggregation kernel and size-independent growth [15]. This constraint is, however, avoided by replacing the exact closure by an approximation of the unclosed terms using an ad hoc Gaussian quadrature formula.

2.1.1 The quadrature approximation (or Gaussian quadrature) In order to solve the transport equations (eq. 3) for the moments, a Gaussian quadrature is used to approximate the integral (eq. 2) by a finite summation of the products of weight, and abscissa, L [15] as:

01

( ) ( ; )N

k kk i i

i

m t L n L t dL L

(4)



Thus a quadrature approximation of order N is defined by its N weights j

and abscissas jL and can be calculated from the first 2N moments

0 1 2 1, ..., Nm m m by writing the recursive relationship for the polynomials in terms

of the moments .km From a practical point of view, the system is ill-conditioned

because the direct solution would require a non-linear search. McGraw [16] recommends the use of Product Difference (PD) algorithm, first described by Gordon [17]. The PD algorithm requires only the moments as inputs and gives the weights and abscissas as output. This algorithm is based on the minimization of the error introduced by replacing the integral in eq. (2) with its quadrature approximation by deriving a tridiagonal matrix of rank N/2 and finding its eigenvalues and eigenvectors.

2.2 Coupling CFD and PBM

The population balance model is coupled to the hydrodynamic model through a Sauter mean diameter, 32d , of the particle distribution given as:

3

332 2

2

.i i

i i

N d md

mN d

(5)

Here, 32d is calculated from the ratio of the moments, 3m and 2m . This value

is then replaced with the particle diameter in the drag model and updated with every time step. The use of the Sauter mean diameter causes only a single set of Navier-Stokes equations to be solved. As such, the particle distribution and with it, the moments, will be transported at the same velocity of the solids phase. This implies that particles of different sizes will have the same velocity.

3 System description

The riser set-up used in the present study is similar to the one used in the challenge problem reported by Knowlton et al [18] and explained in detail by Benyahia et al [19,20]. Knowlton et al [18] presented their experimental data obtained from a CFB column riser with FCC particles at the 8th International Fluidization Conference. They measured solids concentration and flux at given locations of the riser and observed core-annulus flow behavior along the riser height. The geometry of the 0.2-m riser with a total height of 14.2 m is similar to the experimental set-up used by Knowlton et al. [18]. The bottom inlet of the riser is a gas distributor yielding a superficial velocity of 5.2 m/s. Gas and solids are fed from two side inlets with a velocity of 0.476 m/s. The total solids volume fraction at the riser inlets is 40%. Although the original geometry of the riser has a single solid inlet [18], the main reason for selecting a two-inlet geometry design is to obtain mixing effects in the riser entrance zone similar to the experimental results [19]. At the inlets, the velocity and volume fraction of both the phases are specified. At the outlet, only the pressure (atmospheric) is specified. At the walls, the gas velocities are set equal to zero by introducing a no-slip boundary condition. The partial-slip boundary condition for the solids at



the wall is specified by applying an equation for the tangential velocity of the particles [21] using a specularity coefficient of 0.0001. The value is justified because an increase in the specularity coefficient would result in a reduction of the solids volume fraction at the walls, causing the core-annulus behavior to disappear [22]. Two lognormal distribution functions with a mean average diameter, aved of 77 μm and a relative standard deviation, / aved of 0.2 and 0.5

are implemented at the side-inlets. The corresponding 32d of the inlet feed

distribution are 83.3 μm and 120.3 μm respectively. This approach is also a check to ensure that the QMOM model returns the correct non-varying average diameter as the statistical mean. In the simulations, only the first six moments of the lognormal distributions are tracked (i.e. 0 1 5, ,..,m m m ), which implies the use

of a 3-node quadrature approximation (i.e. N = 3).


A two-dimensional transient model approach incorporating the kinetic theory for the solid particles is used in the commercial code Fluent, version 6.3, to simulate the gas-solids flow in the riser. A transient two-dimensional approach simulates the dynamic behavior of a multiphase flow in a Circulating Fluidized Bed (CFB) within a reasonable period of time [23]. The simulations are continued for 40 s of real time. The time-averaged distribution of the variables is then computed considering the last 30 s of the simulation. Both log-normal ( / aved = 0.2, 0.5)

and monosize ( d 77 μm) particle distributions at the inlet are simulated for comparison with the experimental observations of Knowlton et al [18]. It is noteworthy to mention that the value of 77 μm for the monosize distribution corresponds to the probabilistic mean diameter of the log-normal distribution to ensure a test-similarity. The flow patterns in the riser show a transient behavior of the solids volume fraction and the solids velocity profiles for both log-normal (fig. 1(i)(a) and 1(ii)(a)) and monosize (fig. 1(i)(b) and 1(ii)(b)) distributions. Note that fig. 1 corresponds to the lower and upper section of the riser respectively. In both the cases, the solids volume fraction is higher near the walls where the solids velocity is lower. Clusters are seen to be formed in the top of the riser and to flow downwards along the walls. In the centre of the riser, the solids velocity is considerably higher giving rise to core-annular flow: a dilute core region and a dense region near the walls. The core-annulus profile appears to be more uniform when the solids are log-normally distributed through their moments. This is probably due to the coupling of the gas-solid hydrodynamics with the population balance model through the Sauter mean diameter, 32d , which

varies with the moments of the distribution. Figure 2 shows a comparison of the time-averaged solids density distribution ( s s ) at a height of 3.9 m from the

bottom of the riser. The flow in the core is better predicted using a log-normal size distribution than a monosize distribution. The core width agrees reasonably well with the experimental data. However, the solids density at the wall is under-predicted, possibly due to inaccuracies in the applied wall boundary condition [20].



(i) Lower section (ii) Upper section

Figure 1: Comparison of the time-averaged (10-40s) solids volume fraction and velocity profiles for (a) log-normal and (b) monosize distribution, for the (i) lower (0-8 m) and (ii) upper (8-14 m) section of the riser.

Although the density of the solid particles used in the experimental set-up [18] is slightly different from the density of the particles used in the present study, a good qualitative fit for the time-averaged profile of the solids mass flux is obtained in the core of the riser when using a log-normal distribution. Similar to the experimental observations [18], the solids mass flux is found to be maximum at the central core of the riser (see fig. 3), although the solids density is at its lowest value there (see fig. 2). This is attributed with the core-annulus behavior i.e. with a high upward solids (and gas) axial velocity in the core and a low downward solids (and gas) axial velocity in the annulus of the riser. Thus a particle distribution captures a more realistic dynamic flow behavior in a gas-solid riser. Figure 4 show a comparison of the time-averaged solids volume fraction profile for the log-normal and monosize distributions respectively. The effect of riser height on solids volume fraction is investigated by studying the profiles at 1m, 3.9 m and 10 m from the bottom of the riser. Although a core-annular flow is observed in both the cases, there is a slight variation in the radial distribution of the solids volume fraction with the height of the riser.



Figure 2: Time-averaged solids density distribution, using both log-normal and monosize distribution at 3.9 m from the bottom of the riser, compared with the numerical simulation [19] and experimental data [18].

Figure 3: Time-averaged solids mass flux distribution using a log-normal distribution at 3.9 m from the bottom of the riser, compared with the numerical simulation [19] and experimental data [18].

The log-normal disribution almost maintains the core-annulus pattern higher up the riser (e.g. at 10 m) whereas this pattern is assymetrically flattened in a monosize distribution. Indeed, the time-averaged profile of solids volume fraction differs remarkably along the height of the riser, as shown in fig. 4(b). A shift in the solids (and gas) axial velocity towards the right wall (see fig. 1(ii)) is indicated by a two-fold decrease in its solids volume fraction compared to the left wall. The difference in the solids flow behaviour in both the cases indicates that the particle distribution does, in fact, influence the gas-solid hydrodynamics in the riser. A similar conclusion can be drawn from the time-averaged axial solids velocity profiles given in fig. 5.



(a)

(b)

Figure 4: Effect of riser height on the time-averaged solids volume fraction profile in the riser for (a) log-normal and (b) monosize distribution.

(a)

(b)

Figure 5: Effect of riser height on the time-averaged solids velocity profile in the riser for (a) log-normal and (b) monosize distribution.



As seen in fig. 5, the peak of the solids axial velocity shifts slowly towards the wall along with the height of the riser, but in different directions. This shift in the peak of the the dilute core zone is slightly more for the monosize distribution (also see fig. 1). The influence of the particle distribution on the hydrodynamic behaviour in the riser can thus be clearly seen through the comparison in fig. 5. At the wall, the solids have a negative axial velocity which indicates the downward movement of clusters along the walls. Remark that the calculated solids axial velocities are higher for the monosize distribution, due to a smaller width of the core. To evaluate mixing of the particle distributions, a user-defined subroutine is written to obtain the values of the moments calculated for various positions and for multiple cross-sectional surfaces at different axial locations along the riser. The mean, , and standard deviation, , is then calculated from the following:

1

0

m

m (6)

2

2 1

0 0

.m m

m m

(7)

Applying these formulas give the values of the mean and the standard deviation of the local and global distributions respectively. The standard deviation gives insight on how the particles are distributed over a surface depending on their size. The smaller the standard deviation (i.e. smaller range of different sizes), the more segregated is the particle distribution in the surface. The larger this value (i.e. larger range of different sizes), the more mixed is the distribution. Based on a standard deviation, it is possible to conclude whether two or more surface distributions are equally mixed or not. Through the calculation of the mean of the distribution for a surface, it is possible to determine differences in the sizes of the particles present in these surfaces. The mean, together with the standard deviation, are thus a measure of the radial segregation along the riser height. The results obtained at 1 m, 3.9 m and 10m from the bottom of the riser are listed in table 1 and table 2 respectively at a simulated time of 40 s. Note that the left and right positions in table 1 refer to 5 mm radial distance from the left and the right wall of the riser respectively, while the middle refers to the mid-point of the cross-sectional surface. The mixing and segregation of the lognormal distributions and the effect on the riser hydrodynamics can be observed from the results of table 1 and 2. From table 1, it can be concluded that the radial mixing of particles over a cross section of the riser along the height of the riser is very dynamic. At a height of 1m, the two distributions are mixing in the middle because the standard deviation of the distribution in the middle lies between the values of those at the walls. Higher up the riser i.e. at 3.9 m and further at 10 m from the bottom, it is observed that the distribution in the middle becomes more segregated, meaning that only a fraction of the particles introduced in the riser resides in the middle part of the riser. This also indicates the formation of a core-annular regime higher up the riser.



Table 1: Estimation of the local particle diameter distribution over a surface for three positions along the riser height.

Height from the bottom of the riser (m)

Radial position Left Middle Right

Average diameter, local

1.0 7.70 10-5 7.70 10-5 7.70 10-5 3.9 7.70 10-5 7.70 10-5 7.70 10-5 10.0 7.70 10-5 7.70 10-5 7.70 10-5 Standard deviation, local

1.0 3.72 10-5 3.41 10-5 3.01 10-5 3.9 3.52 10-5 2.75 10-5 3.02 10-5 10.0 3.25 10-5 2.96 10-5 3.01 10-5 Spread ratio, local local/ '

1.0 1.0 0.92 0.81 3.9 1.0 0.78 0.86 10.0 1.0 0.91 0.93

Table 2: Estimation of the global particle diameter distribution for different radial surfaces along the riser height.

Parameters of distribution Height from the bottom of the riser (m) 1.0 3.9 10.0

Average diameter, global 7.70 10-5 7.70 10-5 7.70 10-5

Standard deviation, global 3.37 10-5 2.93 10-5 3.02 10-5

Spread ratio, global global/ ' 1.0 0.87 0.89

Rel. std. deviation, global global/ 0.438 0.38 0.392

The surface-averaged mean and standard deviation of the lognormal distributions are given in table 2 for surfaces at different radial cross-sections along the height of the riser. It can be concluded that the distribution of particles along the riser height show segregation which depends on the hydrodynamic conditions of the riser. The correspondence of the local and the global mean with the mean average diameter of the distributions indicates the robustness of the QMOM model in returning the correct average diameter for multiple distributions with a varying standard deviation. These observations confirm that the gas-solid simulation in a riser using particle distributions show segregation and mixing effects, which is influenced by the overall riser hydrodynamics.

5 Conclusion

The Quadrature-based Method of Moment (QMOM) has been successfully applied to solve the evolution of a continuous particle distribution in a multi-



fluid CFD model for flows in a gas-solid riser. The core-annulus profile usually observed in riser flow is well predicted by this model. The simulation results considering a log-normal particle diameter distribution give a better qualitative fit with the experimental data as compared to the fit when using a monosize distribution. In comparison with the results for a monosize distribution, the core-annulus profile is much more uniform and maintained higher up in the riser for the log-normal distribution. It can be said that polydispersity has an influence on the gas-solid hydrodynamics of a riser, especially on the core-annulus flow profile. The radial segregation and mixing for the particle distributions, calculated for various positions and for multiple cross-sectional surfaces at different axial locations along the riser, is evaluated from the local and surface-averaged moments of the distribution. It is seen that the radial mixing profile is dynamic as it changes along the height of the riser. Although the distributions are quite mixed in the middle of a cross-sectional surface close to the inlet, it becomes more segregated as it approaches the outlet. The effect of mixing homogeneity due to particles size distribution on the hydrodynamic conditions of a riser is investigated. Spatial segregation of particles can be observed if, according to their size, different velocities of the particles are computed. For this, a modification in the quadrature-based moment method is necessary.

References

[1] Das, M., Banerjee, M. & Saha, R.K., Segregation and mixing effects in the riser of a circulating fluidized bed. Powder Technology, 178(3), pp. 179-186, 2007.

[2] Nakagawa, N., Dingrong Bai, Shibuya, E., Kinoshita H., Takarada, T. & Kato, K., Segregation of particles in binary solids circulating fluidized beds. Journal of Chemical Engineering of Japan, 27(2), pp. 194-198, 1994.

[3] Bader, R., Findlay, J. & Knowlton, T., Gas/solid flow patterns in a 30.5-cm diameter circulating fluidized Bed. Circulating Fluidized Bed Technology: II, Basu, P. and Large, J. F. (eds.), Pergamon Press, p. 123, 1988.

[4] Hirschberg, B. & Werther, J., Factors affecting solids segregation in circulating fluidized-bed riser. AIChE Journal, 44(1), pp. 25–34, 1998.

[5] Karri, S.B.R. & Knowlton, T.M., Flow direction and size segregation of annulus solids in a riser In: L.S. Fan and T.M. Knowlton, Editors, Fluidization IX, Engineering Foundation, New York, pp. 189-194, 1998.

[6] Iddir, H., Arastoopour, H. & Hrenya, C.M., Analysis of binary and ternary granular mixtures behavior using the kinetic theory approach. Powder Technology, 151(1-3), pp.117-125, 2005.

[7] Huilin, L. & Gidaspow, D., Hydrodynamics of binary fluidization in a riser: CFD simulation using two granular temperatures. Chemical Engineering Science, 58(16), pp. 3777-3792, 2003.

[8] Nienow, A.W., Naimer, N.S. & Chiba, T.S., Studies of segregation/mixing in fluidized beds of different size particles. Chemical Engineering Communications 62(1-6), pp. 53-66, 1987.



[9] Mathiesen, V., Solberg, T. & Hjertager, B.H., Predictions of gas/particle flow with an Eulerian model including a realistic particle diameter distribution. Powder Technology 112(1-2) pp. 34-45, 2000.

[10] van Wachem, B.G.M., Schouten, J.C., van den Bleek, C.M., Krishna, R. & Sinclair, J.L., CFD modeling of gas-fluidized beds with a bimodal particle mixture. AIChE Journal, 47(6), pp.1292-1302, 2001.

[11] Fan, R & Fox, R., Segregation in polydisperse fluidized beds: Validation of a multi-fluid model. Chemical Engineering Science 63(1) pp.272 -285, 2008.

[12] Hinze, J., Turbulence: An Introduction to its Mechanism and Theory. McGraw-Hill, New York, 1959.

[13] Gidaspow, D., Multiphase Flow and Fluidization, Continuum and Kinetic Theory Descriptions. Academic Press, New York, 1994.

[14] Ramkrishna, D., Population balances. Theory and applications to particulate systems in engineering. Academic Press, New York, 2000.

[15] Marchisio, D.L., Pikturna, J.T., Fox, R.O., Vigil, R.D. & Barresi, A.A., Quadrature method of moments for population-balance equations. AIChE Journal, 49(5), pp. 1266-1276, 2003.

[16] McGraw, R., Description of Aerosol Dynamics by the Quadrature Method of Moments, Aerosol Sci. Tech. 27(2), pp. 255-265,1997.

[17] Gordon, R. G., Error Bounds in Equilibrium Statistical Mechanics. J. Math. Phys., 9, pp.655-663, 1968.

[18] Knowlton, T., Geldart, D., Matsen, J. & King, D. Comparison of CFB Hydrodynamic Models. PSRI Challenge Problem, 8th International Fluidization Conference, Tour, France, May, 1995.

[19] Benyahia, S., Arastoopour, H. & Knowlton, T. M. Two-dimensional transient numerical simulation of solids and gas flow in the riser section of a circulating fluidized bed. Chemical Engineering Communications, 189(4), pp. 510-527, 2002.

[20] Benyahia, S., Arastoopour, H., Knowlton, T.M. & Massah, H., Simulation of particles and gas flow behavior in the riser section of a circulating fluidized bed using the kinetic theory approach for the particulate phase. Powder Technology, 112(1-2), pp. 24-33, 2000.

[21] Johnson, P.C. & Jackson, R., Frictional-collisional constitutive relations for granular materials, with application to plane shearing. Journal of Fluid Mechanics 176, pp. 67-93, 1987.

[22] Benyahia S., Syamlal M. & O’Brien, T.J. Study of the ability of multiphase continuum models to predict core-annulus flow. AIChE Journal, 53(10), pp.2549-2568, 2007.

[23] Tsuo, Y. P. & Gidaspow, D., Computation of flow patterns in circulating fluidized beds. AIChE Journal, 36(6), pp. 885-896, 1990.



Numerical simulation of heavy oil flows in pipes using the core-annular flow technique

K. C. O. Crivelaro1, Y. T. Damacena1, T. H. F. Andrade2, A. G. B. Lima1 & S. R. Farias Neto2 1Mechanical Engineering Department, Federal University of Campina Grande, Brazil 2Chemical Engineering Department, Federal University of Campina Grande, Brazil

Abstract

The importance of heavy oils in the world market for petroleum has increased very quickly in the last years. The reserves of heavy oils in the world are estimated at 3 trillion barrels, while reserves of light oils have reduced progressively in the last decade. The high oil viscosity creates major problems in the production and transportation of the oil. This situation leads to the high pressure and power required for its flow, overloading and damaging the equipment, increasing the cost of production. Due to the need to develop new alternatives that will make the production and transport of heavy oil economically viable, this work has the objective to study, numerically, the behavior of isothermal multiphase flow (heavy oil and water), type “core flow”, in pipelines, using the software CFX® 3D. The pressure drop was determinated to a core-flow in a pipe with 7 in. diameter, 2.7 Pa.s oil viscosity and water at environment temperature. Results of the pressure, velocity and volume fraction distributions of the phases are presented and analyzed. It was verified that the pressure drop was reduced 58 times when compared to that obtained with oil flow alone in the pipe. Keywords: heavy oil, numerical simulation, CFX®, two-phase flow oil-water.



1 Introduction

The reserves of heavy oils in the world are estimated at 3 trillion barrels, while reserves of light oils have reduced progressively in the last decade. This leads to much economic interest in the reserves of heavy oils and research to make its production economically viable. The heavy oil has a low degree API (between 10º and 22º) and high viscosity (between 100 cp e 10000 cp). Besides, it has a high ratio of carbons/hydrogen, a large amount of residue carbon, asphaltenes, sulfur, nitrogen, heavy metal and paraffin. The major problems in heavy oil production are: high density, which increases the fluid hydrostatic weight, high viscosity, which makes the flow very difficult, resulting in high pressures and therefore high power being required for its flow, increasing the cost of production. In offshore fields, these difficulties become more serious due to the adverse conditions present during production and transportation. Nowadays, the methods currently used for heavy oil production are based on the reduction in the viscosity of the oil within the reservoir and well and, frequently, are associated with a system lift. The core annular flow technique can be called core-flow; different to other techniques, it is based on the lateral injection of relatively small quantities of water into the pipe and is capable of generating an annular pattern of two-phase flow. This annular pattern will be very stable, since the two fluids are immiscible, where the oil is in center of the pipe and a thin layer of water is flowing near the wall surface. This injection of water will lubricate the wall of the pipe, reducing the friction between the wall and the oil along the flow, as reported by Prada and Bannwart [13] and Silva [14]. According to Bensakhria et al. [6], this technique was created by Isaacs and Speed in 1904, patent request nº 759374 in the United States and was the first to mention the ability to transport viscous product through water lubrication. However, only in 1970 was a large-scale industrial pipeline built (length of 30 km and diameter of 15 cm) to transport heavy oil by the Shell Company near Bakersflield in California. For more than ten years, a viscous crude oil has been produced at the flow rate of 24000 bbl/d in a water lubricated regime. The interest in heavy oil production employing the core annular flow technique has increased in recent years because of the large reserves of heavy oil accessible. This technique brings attractive results with regard to energy consumption. This fact is related to the reduction in pressure drop during the water/oil flow type core-annular when compared with the transport of oil alone (Andrade [1]). It has been observed in the literature related to the use of this technique to improve the transport of heavy oils using water as lubricant (Andrade [1]; Bai [3]; Bannwart [4]; Bensakhria et al. [6]; Joseph et al. [7]; Ko et al [8]; Oliemans et al. [9]; Ooms and Poesio [10]; Prada and Bannwart [13]). Bannwart [4] proposed a theory to stabilize the pattern annular when two liquids of density and viscosity flow differently in a horizontal pipe. The theory is based on the analysis of the linear momentum equation in a cross section of the pipe leading to account for the effect of interfacial tension. Bensakhria et al. [6] evaluated the radial position of the annular flow and showed that the position



depends only on the relationships between the contact perimeter of the wall of the pipe and the fluid that forms the core (oil) and the perimeter of the pipe. This ratio depends on the difference in density between the fluids to transport and the fluid to lubricate, as well as of the amount of injected water. The ideal annular flow or perfect core annular flow (PCAF) corresponds to an exact solution of the concentric fluid flow with a different density and viscosity in a pipe of circular cross section. According to Bensakhria et al. [6], the ideal or perfect annular flow appears to be very rare and can exist only for density matched fluid. Prada and Bannwart [13] also study a simplified solution to perfect annular flow, where two immiscible Newtonian fluids (oil and water) are flowing in a vertical pipe in a concentric configuration with a smooth circular interface. According to Prada and Bannwart [13], core flow lift is a new alternative for heavy oil production, because it significantly increases the well productivity by decreasing of the frictional pressure losses, without the addition of heat and without the use of chemical agents or diluents. Hence, artificial lift technology can be applied in either onshore or offshore fields, horizontal or vertical wells, and its installation in the field is relatively simple. Ooms and Poesio [10] analyzed the annular flow in a stationary regime in a horizontal pipe and proposed a theoretical model based on hydrodynamic lubrication theory. According to this model, a harmonic motion was observed in the annular flow. Indeed, the more viscous fluid (heavy oil) is moved to a wave form in the center of the horizontal pipe; this behavior is known as Wavy Core Annular Flow (WCAF). Ooms et al. [11] theoretically investigated the hydrodynamic counterbalancing of buoyancy force on a core of heavy oil flowing in the pipe, by considering the difference in density between two fluids. During the study it was assumed that the fluid that forms the core consists of a solid center surrounded by a high-viscosity liquid layer. To represent the pressure drop correctly it was necessary to model the effects of the WCAF, such as turbulence and fluctuability. The buoyancy term favors the heavy oil flow, but this is affected by the wave interface in the two-phase flow water/heavy oil (Prada and Bannwart [13]). In this sense the aims this work are to study, by numerical simulation, the isothermal behavior of the two-phase flow (oil and water) type core-flow of ultra-viscous heavy oils in pipe, using the finite-volumes method through the software CFX® 3D. Through these simulations, the pressure, velocity and volume fraction distributions of the phases present will be analyzed.

2 Methodology

2.1 Mathematical modeling

The set of equations to be solved by CFX are as follows. Continuity:

PN

MSSUfft 1

(1)



Momentum equation:

TUUfpfUUfUf

t

PN

MMSUU1

(2)

where α and β represent the phases involved (water or oil), f is the volume

fraction, ρ is density, U

is the velocity vector, Np is the number of phases involved, p is the pressure and is viscosity. In addition, the term SMSα

describes user specified mass sources, Гαβ is the mass flow rate per unit volume from phase β to phase α, SMα describes momentum sources due to external body forces (buoyancy force and rotational force), Mα describes the interfacial forces acting on phase α due to the presence of other phases (drag force, lift force, wall lubrication force, virtual mass force and interphase turbulent dispersion force), and the term UU

represents momentum transfer induced by

interphase mass transfer. To simplify the model and the governing equations solution, some consideration were assumed (table 1). The following boundary conditions were used:

a) oil: u = v = 0 and w = Uo in z = 0 to (x,y); u = v = w = 0 to (x,y,z) / x2 + y2 = r2, where “r” represents the radius

of the pipe. Laminar flow regime.

b) water: u = v = 0 e w = Uw em z = 0 para (x,y); u = v = w = 0 para (x,y,z) / x2 + y2 = r2; Turbulent flow regime.

The turbulence model used to water flow was the k-Epsilon model. In this model it is assumed that Reynolds stresses are proportional to the average velocity gradient, with the constant of proportionality characterized by turbulent viscosity (known as the hypothesis of Boussinesq).

Table 1: Considerations adopted for solving the governing equations.

ConsiderationFlow Two-phase (water/oil), tridimensional,

incompressible and isothermal. thermal-physical properties constant Interphase transfer models Mixture model Buoyancy force Not considered Mass convergence criteria 10-7 kg/s Advection scheme High resolution Pressure interpolation type Trilinear Velocity interpolation type Trilinear



The transport equations to turbulent kinetic energy, k, eqn. (3), turbulent dissipation, , eqn. (4), are:

Gf

k

k

tkUft

kf )( (3)

21

)(

CGCk

f

tUft

f

(4)

where G is the generation of turbulent kinetic energy inside the phase α and

1C and 2C are empirical constants. In eqn. (5), is the rate of dissipation of

the turbulent kinetic energy of the phase α, eqn. (6), defined by:

l

qc 3

(5)

and αk is the turbulent kinetic energy to phase α given by:

2

2

q

k (6)

where l is the spatial scale length, q is the scale of velocity and c is an

empirical constant calculated by eqn. (7), given by: 24 cc (7)

In this equation c is an empirical constant and t corresponds to

turbulent viscosity, defined by eqn. (8) as follows:

2k

ct (8)

The constant used in the eqns. (3)–(8) are: 1C = 1,44; 2C = 1,92; c = 0,09;

k = 1,0; = 1,3.

2.2 Numerical solution

The software CFX 10 was used to generate the mesh and numerical solution of the physical problem. This software uses the methodology of the finite volumes to solve its equations. The thermal-physical properties of the fluids and inlet velocity are illustrated in table 2.



Table 2: The thermal-physical properties of the water and oil used in all simulations.

Figure 1: Numerical mesh used in all simulations.

For the initial condition, we considering that the pipe is full with water, and has a null vector velocity (Vx water = Vy water = Vz water = 0 m/s).

2.3 Numerical mesh

All the simulations were developed in the Thermal and Fluids Computational Laboratory, Mechanical Engineering Department, in the Center of Science and Technology of the Federal University of Campina Grande. The development of the mesh was a simplification of a nozzle reported by Prada and Bannwart [12]. The domain of study was created by definition of the points, curves, surfaces and solids and by describing the size and shape (D = 0,1778m e L = 15m) of the nozzle, so we generate the unstructured mesh and after several refinements we obtain a mesh with 105700 elements as illustrated in fig. 1.

3 Numerical results

The numerical results, in the transient state, were obtained for an elapsed time of t = 90 s and time steps of Δt = 0.3 s. In the steady state, we analyze the volume fraction, pressure field and velocity profiles.

Thermal-physical properties Inlet velocity

(m/s) Density (kg/m3) Viscosity (Pa.s) Water

Oil 997,00 989,00

0,0008899 2,7

1,0 0,8



Fig. 2 shows the pressure drop in different positions along the length of the pipe (z axis). A decrease of pressure along of the pipe is verified, with the highest pressure in the inlet and the lowest in the outlet of pipe. A pressure drop Δp = 675.23 Pa was found, this being necessary to dislocate the oil and water. No gravity effects were considered. The mass flow rate of the oil and water were 15.86 kg/s and 4.69 kg/s, respectively. Figs. 3 and 4 show the volume fractions profile of the oil in the center. It is observed that the oil flows almost in central region of the pipe; approximately 95% of the center of the pipe is full of oil.

Figure 2: Pressure drop of the core flow as a fraction of the radial position for different locations along the pipe.

Figure 3: Volumes fractions of the oil in the core-flow along the pipe.



-0.12 -0.08 -0.04 0 0.04 0.08 0.12

Radius (m)

0

0.2

0.4

0.6

0.8

1

Vol

ume

frac

tions

of

oil

Figure 4: Volume fractions of the oil as a function of the radial position in z = 7.5 m.

-0.12 -0.08 -0.04 0 0.04 0.08 0.12

Radius (m)

0

0.2

0.4

0.6

0.8

1

Vol

ume

frac

tion

s of

the

wat

er

Figure 5: Volume fractions of the water as a function of the radial position in z = 7.5 m.

The volume fractions of the water in a steady state in different regions of the pipe are shown in fig. 5. As expected, the water moves near the wall of the pipe in the form of a ring. In addition, in fig. 5, we observe that a small portion of the water moves together with the oil in the center of the pipe ( 5% ). This can be justified by the occurrence of dispersion of the water phase in the center of the pipe due to the turbulence level of the flow. The oil velocity profiles along of the pipe are shown in fig. 6. A small region, the so called hydrodynamic entrance length, where the fluid moves with constant velocity profiles along the pipe, is verified.



Figure 6: Oil velocity profiles in a core-flow along the pipe.

Table 3: Comparison of the pressure drop among water and oil single-phase flow and core annular flow.

Flow Pressure drop (Pa) Reduction factor (Ω)

Core-flow Water single-phase flow

Oil single-phase flow

675.23 539.52

39112.18

57,2

The oil velocity profiles in a pattern core-flow stay almost constant in the center of the pipe until the annular region, where a water fraction of 5% can be seen in fig. 7. In the wall, the water velocity profiles were made null by increasing the distance of the wall until the region is found where the oil flows alone, see fig. 7. To evaluate the efficiency of the core-flow technique, we compare the pressure drop in the two-phase flow with the pressure drop obtained in the water and oil single-phase flow. The water in the core flow is always in contact with the internal wall of the pipe and the pressure drop in the two-phase flow should be close to the pressure drop in the water flow alone at the mass flow rate of the mixture (Barbosa [5]). Table 3 shows this pressure drop. It was necessary for Δp

= 539.52 Pa to move a water flow rate of wm 20.6 kg/s along the pipe, without

considering the effects of gravity. This pressure drop was lowest at the value obtained in the core-flow (Δp = 675.22 Pa). This demonstrates that through this method, we can move ultra-viscous petroleum with a pressure drop near to the value found in water flow. The pressure drop in the case of the single-phase flow



-0.12 -0.08 -0.04 0 0.04 0.08 0.12Radius (m)

0

0.2

0.4

0.6

0.8

1

Vel

ocit

y (m

/s)

água

óleo

Figure 7: Velocity profiles of the water and oil in the center of the pipe as a function of the radial position.

of oil is introduced in fig. 7. It was necessary for Δp = 39112.1 Pa to move an oil flow rate of 20.44 kg/s along the pipe. By comparing the Δp of the cases core-flow, a reduction factor of the pressure drop of approximately 58 times was obtained.

4 Conclusions

In this work a numerical study of the phenomenon associated with water-oil two-phase flow of the core-flow type was developed. Results of the velocity, pressure and volume fraction of the phases were obtained, analyzed and compared, by showing the efficiency of the core-flow technique in the production and transport of heavy oils. In this study, it can be concluded that Δp = 676.17 Pa was necessary to move the heavy oil in a core-flow, in a pipe of 7 inch diameter and 15.0 m length. The mass flow rate of oil and water were 15.85 kg/s and 4.68 kg/s, respectively. By comparing the core-flow technique with a water single-phase flow, we notice that the two-phase pressure drop (Δp = 676.17 Pa) approaches the pressure drop in water single-phase flow (Δp = 501.5 Pa) to oil viscosity of 2.7 Pas. By comparing the core-flow technique with an oil single-phase flow, the reduction factor of the pressure drop was approaching 58 times. Due to these benefits, we observe great interest in the use of the core-flow technique to solve the problems related to the production and transport of ultra-viscous oil in pipes.

References

[1] Andradre, T. R. F. Numerical study of heavy oils transport on pipe lubrificated by water. Master thesis in Chemical Engineering – Federal University of Campina Grande, Paraíba, Brazil, 2008.



[2] ANSYS, CFX-Theory Manual, 2005. [3] Bai, R. Traveling waves in a high viscosity ratio and axisymmetric core

annular flow. PhD Thesis, Faculty of Graduate School of the University of Minnesota, Minnesota-USA, 1995.

[4] Bannwart, A. C. Modeling aspects of oil–water core annular flows, Journal of Petroleum Science and Engineering, vol., 32, pp. 127– 143, 2001.

[5] Barbosa, A. Transient effects in the pressure drop for heavy oil-water core annular flow in metallic pipes. Master thesis in Petroleum Engineering – State University of Campinas, Campinas, 2004

[6] Bensakhria, A.; Peysson, Y. & Antonini, G., Experimental study of the pipeline lubrication for heavy oil transport. Oil & Gas Science and Technology – Rev. IFP, vol. 59, N°. 5, pp. 523-533, 2004.

[7] Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y., Core annular flows. Annual Reviews Fluid Mechanical, vol. 29, pp.65–90 1997.

[8] Ko, T.; Choi, H. G.; Bai, R. & Joseph, D.D. Finite element method simulation of turbulent wavy core annular flows using a k-w turbulence model method. International Journal of Multiphase Flow, vol. 28, Nº 7 pp. 1205–1222, 2002.

[9] Oliemans, R.V.A.; Ooms, G.; Wu, H.L. & Duijvestijn. A., The core annular oil/water flow turbulent-lubricating-film model and measurements in a 5 cm pipe loop. International Journal of Multiphase Flow, vol. 13, Nº. 1, pp. 23-31, 1987.

[10] Ooms, G. & Poesio, P., Stationary core-annular flow through a horizontal Pipe. Physical Review, vol. 68, 2003.

[11] Ooms, G.; Vuik, C. & Poesio, P., core-annular flow through a horizontal Pipe: hydrodynamic counterbalancing of buoyancy force on core. Physics of Fluids, vol. 19, Nº 092103 (2007).

[12] Prada, J. W. V. & Bannwart, A. C. Pressure drop in vertical core annular flow, XV Brazilian Congress f Mechanical Engineering, Água de Lindóia, São Paulo, 1999.

[13] Prada, J. W. V. & Bannwart, A., C., Modeling of vertical core annular flows and application to heavy oil production. Energy for the New Millennium February 14-17, New Orleans, LA. Proceedings of ETCE/OMAE, 2000.

[14] Silva, R.C.R. Wettability alteration of internal surfaces of pipelines for used in the transportation heavy oil via core-flow. Master thesis in Science and Petroleum Engineering – State University of Campinas, São Paulo, 2003.



Simulation of flow and modelling the residence time distribution in a continuous two impinging liquid-liquid streams reactor using the Monte Carlo Technique

M. Sohrabi & E. Rajaie Amirkabir University of Technology, Department of Chemical Engineering, Iran

Abstract

In the present study, a stochastic model for the residence time distribution (RTD) in a coaxial counter current liquid-liquid impinging streams system has been developed. Simulations of droplets movements in the impinging spray systems determine the residence time distribution. Droplets dynamics has been formulated from the Boltzmann equation using direct simulation Monte Carlo (DSMC) method. The data predicted from the model has been correlated with the experimental results obtained from a coaxial counter current two impinging liquid-liquid steams apparatus. The reactor consisted of a cylindrical vessel made of Pyrex glass, length 60 cm and internal diameter 14 cm. The apparatus was equipped with two circular plates placed at the two ends of the reactor. Thus the length of the contact compartment could be varied by moving the plates away from or towards each other. Water and toluene were used as two immiscible liquid streams. These were sprayed into the reactor by applying pressurized nitrogen via special ducts, placed behind the feed nozzles. The degree of agreement between the experimental results and those predicted from the model was 85%. Keywords: impinging streams reactor, residence time distribution, Monte Carlo method, Boltzmann relation, Navier-Stokes equation.



1 Introduction

One of the important operations in chemical engineering is the mass transfer between immiscible phases. In an impinging streams apparatus a unique flow behavior is utilized by which the transfer processes in heterogeneous systems is intensified [l]. In such systems two feed streams, flowing parallel or counter currently collides with each other at a zone in which the two streams impinge. Impinging streams systems, first proposed by Elperin [2], have been applied as some suitable devices for enhancing mass and heat transfer processes in flowing gas-liquid, gas-solid and liquid-liquid emulsions. In such apparatus, two droplet-laden gaseous jets flowing in opposite directions are allowed to impinge. Some of the multiphase reactions carried out in two impinging streams reactors include two-phase mono-nitration of toluene [3], chemical absorption of CO2 gas in both sodium hydroxide [4] and mono ethanolamine solutions [1] and isomerization of glucose to fructose by an immobilized enzyme [5]. Impinging streams systems have been successfully applied to other chemical processes such as dissolution and mixing as well as mass and heat transfer operations [6–9]. In the present study a stochastic model for the residence time distribution (RTD) of the aqueous phase within a coaxial counter current two liquid-liquid impinging streams contactor (TISC) has been carried out. In addition a comparison has been made between the experimentally determined RTD data and those predicted from the model.

2 Contactor system

The experimental apparatus used in the present study is shown in Fig. 1. The contactor consists of a cylindrical vessel made of Pyrex glass, length 60 cm and internal diameter 14 cm. The contactor is equipped with two circular plates made of “Teflon” placed at the two ends of the contactor. Thus the length of the contact compartment can be varied by moving the plates away from or towards each other.

Figure 1: Contactor dimensions (figures are in mm).

The two immiscible liquids (toluene and water) were kept in separate glass containers and fed to the contactor via four identical feed nozzles made of glass, situated at the two ends of the vessel. Liquids were transported to the nozzles



using metering pumps. The liquid streams were sprayed into the contactor by applying pressurized nitrogen via special ducts, placed behind the feed nozzles. In figure 2, the diagram of a feed nozzle is shown. As it may be observed from this figure, there is no mixing of fluids within the nozzle. The mixing is occurred only at the exit of the system. In each end of the contactor two identical feed nozzles were installed spraying two different liquids towards the impingement zone.

Figure 2: Diagram of a feed nozzle.

Operating conditions are shown in Table 1.

Table 1: Operating conditions of the impinging streams contactor.

Temperature (ºC)

Distance between the

two feed nozzles

(cm)

Flow rate of nitrogen

(cm3/min)

Toluene flow rate

(cm3/min)

Water flow rate

(cm3/min)

20.1 25 1300 320 325 In order to determine the suitable positions and angles for the feed nozzles at which stable jets of liquids would be established, a number of experiments were performed. This experimental set up allows consideration of the effects of changing certain pertinent parameters of the system on the residence time distribution of materials and hence on the extent of the heterogeneous liquid-liquid mixing and reaction.


3.1 Measurement of the residence time distribution of aqueous phase

To determine the residence time distribution of the aqueous phase within the contactor, the following experiment was performed. Toluene and distilled water were fed to the contactor via spray nozzles. At a time, a change from the water stream to one of water containing a mineral salt (potassium dichromate) with known concentration and colour intensity (Co) was rapidly performed (step input). Samples at the outlet of the contactor were collected, using a circular vessel divided into 24 segments with equal volumes, placed under the exit port



and rotated at a pre-set speed applying an electric motor. Successive samples were obtained at equal time intervals by this method. Change in the rotating speed altered the time intervals between the sample collections. The content of each segment was transferred to a separating funnel. The aqueous phase was separated from the organic layer and the concentration (colour intensity) of the former was measured (C), using a UV spectrophotometer. From a plot of C/Co versus time, the RTD data were determined.

3.2 Developing a stochastic model for the residence time distribution of the aqueous phase in TISC

3.2.1 Gas flow simulation Gas flow pattern is required to simulate the droplet movement. By the velocity pattern, the drag force which affects the droplet velocity may be estimated. Complete solution of Navier-Stokes equation for such a system can provide the necessary data, for this pressure, although the approximate solution of this equation (analytical relations) for free jet may be also used. In the present study the Navier-stokes equation has been solved, using the SIMPLE method and a mesh system shown in figure 3.

Figure 3: Pyramid mesh for the contactor.

The gas stream is assumed to be an incompressible flow (This assumption is valid for the fluid flows having Mach number lower than 0.3). Calculation of pressure contours is the main problem in this kind of fluid flow and the SIMPLE method seems to be a way by which the pressure profile under these conditions may be estimated [10]. The results are shown in figure (4).



Figure 4: Velocity vectors and streamlines.

3.2.2 Droplet simulation An analogy between the droplet collisions in emulsions and the molecular collisions, described in the kinetic theory of gases, enables the application of the Boltzmann equation to the droplets, as first suggested by Pai [11]. In this work, the direct simulation Monte Carlo (DSMC) method, first proposed by Bird [12] for solving the Boltzmann equation in molecular gas dynamics, is used for modeling the droplet interactions in dense sprays.

3.3 Mathematical formulations

The Boltzmann equation may be written as,

( ) ( ) ( ) ( )4

2 * *1 1 1

0

. F.r

nf V nf nf n f f ff V d dVt v

π

σ+∞

−∞

∂ ∂ ∂+ + = − Ω

∂ ∂ ∂ ∫ ∫ r (1)

In this equation, n is the number density of droplets, f is the velocity probability distribution function of droplet of class having the velocity V, f1 is the velocity distribution function of the droplet of class having the velocityV1, Vr is the velocity of a test droplet in the class of droplets having the velocity V1, *f ,

is the post-collision velocity probability distribution function and *1f is the post-

collision velocity probability distribution function of the droplet of class having the velocity V1, F, is the external force per unit mass and Ω is the angle in the spherical coordinates. The key ideas of the DSMC method are: (a) the uncoupling of droplet motions and collisions during a time step mt∆ i.e. the use of the operator-splitting technique (b) the simulation of droplet collisions by disregarding droplet position coordinates within spatial coils: and (c) the simulation of fewer droplets than those present in the real flow, while normalizing the collision cross-section so that the collisions rate is not changed. Assumption (a) is valid when mt∆ is smaller than the time between collisions, and larger than that of collision duration, and assumption (b) is valid provided that the cell is so small that the spatial variation of flow variables in the cell is negligible. Assumption



(c) may not be necessary for dilute emulsion flow, due to relatively small droplet number densities; in the present work, however, due to the formation of large number of fragments, the implementation of this procedure is required. Under the assumptions listed above the DSMC method for the solution of Boltzmann kinetic equation, describing the flow of gas and droplets emulsions can be formulated as follows. The flow system is divided into equal-volume cells. Simulated droplets are distributed in the system, with their positions, sizes and velocities sampled from the initial distribution function. When a stationary kinetic equation is solved, the initial distribution function is chosen arbitrarily. The droplets population is normalized such that each Kf droplets of identical size in the real system is substituted by a single droplet in the simulation, having the same diameter as these droplets. The collision cross-section for each such simulated droplet is accordingly increased by the factor Kf so as to preserve the true collision rate. Provided that the droplet distribution function at time (n - 1)

mt∆ is determined, the distribution function at time n mt∆ maybe calculated as follows. Droplets are allowed to move in the system, without colliding with each other, for a time interval mt∆ , with each droplet’s subsequent position, velocity and diameter are determined from the droplet equation of motion. Within the time interval mt∆ , a droplet may encounter a boundary either an open boundary, through which it leaves the system or a wall, onto which it sticks. Following the collisionless flow, droplets are allowed to collide with each other. The droplet population is discredited by location and size such that the number of droplets of type k in cell m, ,m kN having a volume mV , , is given as,

, 0( ) ( , , , )

lk

sm k

m k VN t dr dv d f r v t

δ

δδ δ

∞= ∫ ∫ ∫ (2)

Where, skδ and l

kδ are the lower and upper diameter limits, respectively, for

droplets of type k . The total number of droplets in cell m is,

,1

s

m m kk

N N=

= ∑ (3)

In the stochastic model it has been assumed that collision durations are negligible, so that droplet motion can be described as a free motion, disrupted instantaneously by collisions. For high collision velocity, which is characteristic of impinging streams systems, shorter collision durations may be expected. Another assumption made was that droplets are spherical and the aerodynamic break up due to gas shear during free droplet motion is negligible. Borisov et al. [13] investigated experimentally the break up of single droplets moving in a gas flow. Their results show that droplet break up dose not occur when Weber number < 6. When only fluid drag acts upon droplets during mt∆ , droplet trajectory is calculated by integrating the following ordinary differential equation,

( ) ( )2

2 0.75 D g pd r C U v U v gdt

ρ ρ δ = − − + (4)



where, U, is the gas velocity at position r that is determined according to the procedure presented in gas flow simulation section, g, is the gravity acceleration and DC , is the gas drag coefficient for liquid droplet flowing in a gas and may be determined from the following relation proposed by Hestroni [14].

( )

( ) ( ) 0.78 1/ 5

8(3 2) / Re 1 Re 2

14.9 / Re 24 / Re 4 / /( 1) 2 Re 500

Re /

p p

D

p p p

p g g p g

forC

Re for

U vρ δ µ µ µ

Θ + Θ + < = + Θ + Θ + < <

= − Θ =

(5)

In this model the droplet were considered to be spherical and collide with each other at different speeds. In simulation of processes, using the DSMC, two grid systems are normally considered. The first grid system is used to calculate the averages of flow properties. This grid system is chosen to be fine enough in order to increase the computational accuracy. The grid system is refined until the variations of the flow properties are not substantial (the variations of the flow properties should be less than 2%). The second grid system (Fig. 5) is selected to be extremely fine (the mesh size is equal to 0.2 times of the mean free path of the droplets) so that the collision of droplets could be controlled within each mesh with high accuracy. The grid system chosen in this study consisted of 88 divisions in X direction, 54 divisions in Y direction, 54 divisions in Z direction (totally 256608 meshes) and the total number of model droplets was 3,800,000. Each model droplet consists of fK = 70 real droplets. The size of the mesh is in order of the mean free path of droplets and the time step in the simulation process is chosen to be 0.2 times the collision time [12].

Figure 5: Contactor mesh for DSMC simulation.

The droplets are distributed in the mesh system according to the normal distribution. The initial velocity of the droplets is chosen based upon the velocity of gas jets in the flow field. Then as time passes the new position of the droplets is designated. The collisions of the model droplets are occurred based on the pattern put forward by Bird [12]. Passage of time is continued until the statistical fluctuations of the flow properties attain minimum values. To implement the DSMC method, spatial cells have to be produced in the first step. The model droplets are placed in the cells with appropriate distribution.



With regard to local gas velocity, a normal distributed velocity is supposed for droplets. Without any statistical sampling, droplets are moved (NPS times repetition), until the latter attain suitable conditions for statistical sampling. For large domain simulations, this step can be ignored. NPS value definition is dependent on cells conditions; however optimization can not be approached without sufficient experience. At the next time step, according to each droplet velocity and the extended time step, the droplets are moved and so their new coordinates will be attained. If a droplet passes cell boundary, it will be eliminated from the cell, and a new droplet would be entered to the domain. Furthermore, if in this replacement, the droplet collides with a physical surface, it will be reflected from the surface according to the collision of “surface-droplet model”. The index number of cell and sub-cell containing the droplet is defined; and then the probability droplet collision is investigated based on the model and if there is any collision, the velocity and location of the droplet will be modified. After NIS times repetition of movements, indexing, and collision investigations, statistical sampling should be done, the output of the program will be provided after NPT times repetition of sampling. This output is corrected up to NPT times. The expected number of collisions in a cell, during a time interval mt∆ is given as,

20.5c T rN n cσ= (6)

The probability P of collision between two simulated droplets over the time interval mt∆ is equal to the ratio of the volume swept out by their total cross-section moving at the relative speed between the droplets to the volume of the cell, i.e.,

/N T r cP F c t vσ= ∆ (7)

Maximum efficiency is achieved if the fraction is such that the maximum probability becomes unity. The fraction is given by,

( )max max/N T r cP F c t vσ= ∆ (8)

In the above equation Tσ is collision cross section and cv is the relative droplet velocity. The average number of real droplet in the cell is cnv and the average number of simulated droplets is /c NN nv F= , where, n , is the number density in liquid phase. full set of collisions could be calculated by selecting in turn, all

( 1) / 2N N − pairs in the cell and by computing the collision with probability P. Pairs are selected from the cell at the time step, and the collision is computed using the ( )max

/T r t rc cσ σ probability relation given by Bird [12]. In the present work two different liquids (water and toluene) have been used. Therefore, the collisions were between similar or unlike droplets so the method has to be slightly modified. A pair of colliding droplets of type p and q is sampled from the possible pairs of these types with the above probability. This



term has been substituted for ( )max/T r t rc cσ σ in all related equations. Simulation

results for these droplets are shown in figures 6-8.

Figure 6: Droplets velocities in X direction.

Figure 7: Droplets velocities in Y direction.

Figure 8: Droplets velocities in Z direction.

Figure 9: A typical residence time distribution data.



A typical experimental residence time distribution (RTD) curve obtained for the continuous two impinging streams contactor and those predicted applying the Markov chain model and DSMC technique are shown in figure 9.

4 Conclusion

The Direct Simulation Monte Carlo (DSMC) technique was applied to predict the motion of liquid droplets and residence time distribution in a two impinging streams contactor. This method was found to be more accurate and flexible in prediction of RTD data compare to the Markov chain discrete time formulation applied by Sohrabi et al [3,5,15] in some previous studies. The degree of agreement between the data estimated from the present model and those determined experimentally was within 85%. While in case of Markov formulation the degree of agreement was lower than 75%.

References

[1] Sohrabi, M., Jamshidi, A.M., Studies on the behaviour and application of the continuous two impinging streams reactors in gas-liquid reactions, J. Chem. Tech. Biotechnol. 69, pp. 415 420, 1997.

[2] Tamir, A., Impinging Streams Reactors, Fundamentals and Applications, Elsevier B. V., Amsterdam, The Netherlands, 1994.

[3] Sohrabi, M. Kaghazchi, T. & Yazdani, F., Modelling and application of the continuous impinging streams reactors in liquid - liquid heterogeneous reactions, J. Chem. Tech. Biotechnol., 58, pp. 363 370, 1993.

[4] Tamir, A., Herskovitz, D., Absorption of CO2 in a new two-impinging-streams absorber, Chem. Eng. Sci. 40, pp. 2149 2160, 1985.

[5] Sohrabi, M., Ahmadi Marvast, M., Application of a continuous two impinging streams reactor in solid-liquid enzyme reactions, Ind. Eng. Chem. Res., 39, pp.1903 1910, 2000.

[6] Tamir, A., Kirton, Y., Chem. Eng. Comm., 50, pp. 241 252, 1987. [7] Tamir, A., Kirton, Y., Drying Technol., 7, pp. 183 191, 1989. [8] Tamir, A., Luzzatto, K., Mixing of solids in impinging streams reactor, J.

Powder Bulk Solids Technol., 9(15), pp. 15 17, 1989. [9] Tamir, A., Chem. Eng. Progress, 85, pp. 53 67, 1989. [10] Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Elsevier, B.V.,

The Netherlands, pp. 123 144, 1993. [11] Pai, S.I., Sci. Tech. Mech. Appl., 19, pp. 605 621, 1974. [12] Bird, G.A., Molecular Gas Dynamics and the Direct Simulation of Gas

Flows, Oxford University Press, UK, pp. 196 211, 1994. [13] Borisov, A.A., Gelfand, B., Natanzon, M.S. & Kossov, O., J. Eng. Phys.

40, pp. 44 49, 1981. [14] Hetsroni, G., Handbook of Multiphase Systems, Hemisphere Inc.,

Washington, 1982. [15] Sohrabi, M., Zareikar, B., Modeling of the residence time distribution and

application of the continuous two impinging streams reactors in liquid-liquid reactions, Chem.Eng.Technol. 28, pp. 61 66, 2005.



Section 2 Interaction of gas, liquids

and solids

Vortex study on a hydraulic model of Godar-e-Landar Dam and Hydropower Plant R. Roshan1, H. Sarkardeh2 & A. R. Zarrati3 1Water Research Institute, Iran 2Department of Engineering, Tarbiat Moallem University of Sabzevar, Iran 3Department of Civil Engineering, Amirkabir University of Technology, Iran

Abstract

In hydropower plants the kinetic energy of falling water is captured to generate electricity. In this process, the formation of vortices with an air core at the power intake entrances is expected at lower reservoir levels. The entrainment of air and swirl into the power tunnels leads to a reduction of power generation and vibration and damage to the turbine blades. To use the maximum potential of water power at lower reservoir levels when water is scarce, it is necessary to prevent vortex formation. Anti-vortex devices are usually considered as an efficient method for vortex prevention. The Godar-e-Landar Dam and Hydropower Plant is sited on the Karun River in the province of Khuzestan, Iran, with the capacity of 2000 MW. There are four horizontal power intakes where the capacity of each intake is equal to 375 m3/s. The dam is a rock fill type with 170 m height from the foundation. The dam has a gated spillway with an ogee chute and stilling basin. In the present work a physical model was used to study the formation and prevention of air core vortices at power intakes of the dam. Studies showed that vortices form when the reservoir water level decreases from a certain elevation and air enters the power tunnels. The performance of anti-vortex walls was therefore examined to eliminate vortices or reduce their strength and prevent entrainment of air. The anti-vortex walls were constructed on top of each intake to increase the friction stresses within the vortex path. To distinguish the vortex type, light colorful objects were released in the flow. Results of experiments showed Type 4 vortices (which may be a stronger air core Type 6 in prototype) reduced to a weak vortex Type 2 and 1 when an anti-vortex wall was installed. Moreover, the vortices became very unstable. Keywords: hydropower plant, power intake, vortex, anti-vortex wall, vortex prevention, trash rack.



1 Introduction

Power generation is one of the main targets in dam construction. To do so, many hydropower plants have been constructed and put into operation alongside dams all over the world. The formation of vortex at power intakes is an undesirable phenomenon that may cause a number of problems, such as decreasing the efficiency of turbines and their vibration, increasing hydraulic losses at the entrance of power intakes, entraining debris that may block the trash racks, entraining air into the power tunnel and reducing the working life of turbines [1]. The stronger the vortex the greater will be its negative effects on power intake performance. Based on Alden Research Laboratory visual classification, vortices are divided into 6 types (Figure 1).

Figure 1: Different types of vortices [1].

Vortex Type 1 is observed as a weak rotation of flow at water surface. In vortex Type 2 in addition to water surface rotation a drop is also observed in water surface. In vortex Type 3 the rotation of flow is extended down to the intake itself. In vortex Type 4, debris is dragged into the intake. In vortex Type 5 some air bubbles are entrained from water surface and are transported down to the intake. In the strongest Type 6 vortex, a stable air core is formed in the centre of the vortex and air is entrained into the power tunnel steadily [1]. To prevent formation of an air core vortex, a minimum operating depth, called critical submerged depth ‘hcr’ is recommended for the intakes. Submerged depth is defined as the distance between water surface and the axis of the intake (Figure 2). Many researchers have tried to find a relationship for hcr based on flow parameters such as the intake Froude number defined as:

gD

VFr where V is intake velocity, g is gravitational acceleration and D is

the power tunnel diameter (Berge 1966, Gordon 1970, Reddy and Pickford 1972,

Coherent surface swirl

Surface dimple coherent swirl at surface

Dye core to intake coherent swirl throughout water column

Vortex pulling floating trash but not air

Vortex pulling air bubbles to intake

Full air core to intake



Amphlett 1976, Chang 1977) [2–4, 11, 12]. However, since many factors such as the reservoir geometry affect the strength of vortices these relationships are not very accurate and are valid within their data limits and experimental setups. Therefore in design of power intakes, especially for large dams, physical model study is required.

Figure 2: Formation of an air core vortex at the intake.

Critical submerged depth can be determined in a physical model. It should be noticed that since the operating level of the power intake should not be reduced below the critical submerged depth, the volume of water in this region can not be used for power generation. Therefore increasing the submerged depth of the intake for prevention of vortex formation may not always be an economical solution. On the other hand, construction of deeper intakes may be more expensive. A strategy for preventing vortex formation or in another word decreasing the critical submerged depth is to employ anti-vortex devices. Considering the factors which affect vortex strength, it can be concluded that vortex formation can be prevented or its strength can be reduced if there is a disturbance and an increase in flow resistance along the vortex path. Knauss [1] introduced various anti-vortex devices for intakes. These devices include: i) Vertical walls on top of the intake which induce friction on vortices and reduce their strength, ii) half cylinder walls in front of the intakes which cuts the vortex path and cause additional resistance to flow rotation iii) floating plates at reservoir water surface, which prevents vortex rotation due to friction. Since this device is floating on water surface it can adjust itself above the power intake at any water surface elevation, and iv) horizontal plates installed on top of the intakes This device though is fixed above the intake, cuts the vortex path and prevents its rotation. (Figure 3). In the present work formation of vortices was studied on power intakes of Godar-e-Lander Dam. To reduce the minimum operating water level of the reservoir, performance of anti-vortex walls were examined on the intakes.

S



Vertical Wall

Half Cylender wall

Floating Perforated Plate Horizontal perforated Plate

Figure 3: Few types of anti-vortex devices used at horizontal intakes (courtesy of Water Research Institute, Tehran, Iran).

In additional experiments the effect of trash rack and spillway operation simultaneously with the power intakes on type of vortices was studied at different reservoir levels.


Godar-e-Lander is rock fill dam 170 m high. Four power tunnels with 10 m diameter were constructed to supply the 8 power units (Figure 4). Design discharge of each intake was 375 m3/s. Elevation of intake axis was 340.98 m above the sea level (masl). Maximum and minimum reservoir operation level was designed at 372 masl and 363 masl respectively. As a part of design, a 1:66.67 scale model of the dam was constructed in Water Research Institute Laboratory in Tehran. The hydraulic model of Godar-e-Lander Dam included: dam body, the reservoir, spillways, power intakes and a part of their downstream tunnels (Figure 5). To include the effect of reservoir geometry on inlet flow pattern, an extension of 933 m of the reservoir upstream of the dam body and width of 667 m along the dam axis in prototype scale was constructed in the model. To be able to see the flow pattern at power intakes the reservoir walls and power intakes and tunnels were made from clear Perspex. Standard orifice plates installed downstream of each power tunnel in the model were used for discharge measurement. A point gauge with 0.1 mm

Intake



accurate was also used to measure elevation of water surface in the reservoir. To distinguish the vortex type in each test, light colorful objects were released in the flow. This model study was used to collect hydraulic design data for the anti-vortex wall. The hydraulic information obtained from the model study included: type of vortices, vortex stability in various alternatives of anti-vortex walls.

Figure 4: Plan view of power intakes and spillways.

Figure 5: 1:66.67 scale model of the Godar-e-Lander Dam.



3 Model results

Model studies showed that with operation of the intakes, at reservoir level of 367.5 masl that is 4.5 m above the designed minimum operation water level, stable vortex Type 4 formed. These tests were repeated with installation of trash rack on intakes and different combinations of intakes operation with similar results (Table 1). It was also observed that operation of the spillways simultaneously with the power intakes reduces the vortices strength by inducing turbulence in the reservoir. Additional tests showed that, since the vortex location was closer to the intake head wall at elevations below 365.5 masl, the vortex type reduced one level (Table 1 and Figure 6) [5,6].

Figure 6: Side and front view of dam, power intake location and anti-vortex wall.

Owing to viscous effects in small scale models, strengths of vortices may be underestimated and therefore stronger air core vortices may form in the prototype. Considering negative consequences of vortex formation and air entrainment at power intakes, design of an anti-vortex wall was proposed to enable the dam operators to use the power units until the designed minimum water level of 363 masl. In hydropower tunnels air should not be drawn into the intakes. This corresponds to vortex Type 5 and 6. To evaluate the performance of the anti-vortex wall and considering the scale effects it was decided not to have vortices stronger than Type 2 in the model. This is especially acceptable since the cost of constructing anti-vortex walls is not high. A wall was installed in the model at elevation 365.5 masl above each intake on top of the intake head wall and with the same slope (Figure 6). The top elevation of this anti-vortex wall was selected 368.5 masl which is the height above it no vortex was observed (see Table 1). Different combinations for operation of intakes were tested at 3 reservoir surface elevations of 363 masl, 367.5 masl and 372 masl and 2 discharges for spillway operation and formation of vortices was monitored. The results of these tests showed that the strongest vortex with the presence of the anti-vortex wall was Type 2 at elevation 367.5 masl. The vortices with the presence of wall were also very unstable (Table 1). These results showed the successful performance of the anti-vortex wall.



Table 1: Summary of results on vortex formation and the performance of an anti-vortex wall at different conditions.

Reservoir Level (masl)

Discharge of

Spillway (m3/s)

Trash rack Number of Active Intakes

Anti-Vortex Wall

The more Strength Type of Vortices

Stability

372 Closed No Trash rack 1,2 No wall No Vortex No Vortex

367.5 Closed No Trash rack 1,2 No wall 4 Stable

363 Closed No Trash rack 1,2 No wall 3 Stable

367.5 2500 No Trash rack 1 No wall 2 Stable

367.5 2500 No Trash rack 1,2 No wall 2 Stable

367.5 Closed No Trash rack 1,2,3,4 No wall 3 Stable

367.5 2500 No Trash rack 1,2,3,4 No wall 2 Unstable

367.5 5000 No Trash rack 1,2,3,4 No wall No Vortex No Vortex

372 Closed Trash rack 1,2 No wall No Vortex No Vortex

367.5 Closed Trash rack 1,2 No wall 4 Stable

363 Closed Trash rack 1,2 No wall 3 Stable

367.5 2500 Trash rack 1 No wall 2 Stable

367.5 2500 Trash rack 1,2 No wall 2 Stable

367.5 Closed Trash rack 1,2,3,4 No wall 3 Stable

367.5 2500 Trash rack 1,2,3,4 No wall 2 Unstable

367.5 5000 Trash rack 1,2,3,4 No wall No Vortex No Vortex

372 Closed Trash rack 1,2 yes No Vortex No Vortex



Table 1: Continued.

Reservoir Level (masl)

Discharge of

Spillway (m3/s)

Trash rack Number of Active Intakes

Anti-Vortex Wall

The more Strength Type of Vortices

Stability

367.5 Closed Trash rack 1,2 yes 2 Unstable

363 Closed Trash rack 1,2 yes 1 Unstable

367.5 2500 Trash rack 1 yes No Vortex No Vortex

367.5 2500 Trash rack 1,2 yes No Vortex No Vortex

367.5 Closed Trash rack 1,2,3,4 yes 2 Unstable

367.5 2500 Trash rack 1,2,3,4 yes 1 Unstable

367.5 5000 Trash rack 1,2,3,4 yes No Vortex No Vortex

4 Summary and conclusions

Formation of vortices at power intakes is an undesirable phenomenon, which cause number of problems. Stronger vortices have more negative effects on performance of a hydropower plant. Godar-e-Lander Dam and Hydropower Plant is placed in west of Iran. It is a rock fill dam with 170 m height from the foundation and a hydropower plant with the capacity of 2000 MW. In physical model study it was found that vortices of Type 4 were formed at the intakes at depths more than the minimum designed reservoir water level. Owing to scale effects stronger air core vortices may form in the prototype. Experiments also showed that operation of spillways has a great effect on reducing or eliminating vortices. It was also observed that trash rack effect on reducing type of vortices was negligible. To reduce the strength of vortices anti vortex walls were installed on top of each intake head wall and with the same slope. In fact the intake head wall was extended an extra 3 m. Results of experiments showed that type of vortices reduced from Type 4 to Types 2 and 1 when anti-vortex walls were installed. Moreover vortices became very unstable when anti-vortex walls were present. The anti-vortex wall was then recommended as a cheap method to prevent vortex formation in the prototype.

Acknowledgement

The authors would like to thank Water Research Institute, Tehran, Iran for their permission in using the model data.



References

[1] Knauss, J, “Swirling Flow Problems at intakes”, Balkema, 1987. [2] Gordon, J.L, “Vortices at Intakes”, Water Power, 1970. [3] Reddy, Y.R. & Pickford, J.A., “Vortices at Intake in Conventional Sumps”,

Water Power, No.3, 1972. [4] Amphlet, M.B, ”Air Entraining Vortices at Horizontal Intake”, HRC

Wallingford Rep., No .OD/7,1976 [5] Sarkardeh, H, Safavi, K & Karaminejad, R “Experimental study of vertical

anti-vortex wall at power intakes of Karun III dam”, 2nd IJREW on Hydraulic Structures, Italy, 2008.

[6] Sarkardeh, H, Zarrati, A.R & Roshan, R “Effect of intake head wall and trash rack on type and strength of vortices”, Journal of Hydraulic Research, Submitted for publication, 2008.

[7] Lugt, H.J, “Vortex Flow in Nature and Technology”, John Wiley & Sons, 1983.

[8] Hecker, G.E, “Model-Prototype Comparison of Free Surface Vortices”, Journal of Hydraulic Engineering, Vol.107, No.10, pp 1243-1259, 1981.

[9] Anwar, H.O., Weller, J.A & Amphlett M.B, “Similarity of Free Vortex at Horizontal Intake”, Journal of Hydraulic Research,No.2,pp 95-105, 1978

[10] Jain, A.K, Raju, K.G.R & Grade, R.J, “Vortex Formation at Vertical Pipe Intake”, Journal of Hydraulic Engineering, Vol.100, No.10, pp 1427-1445, 1987.

[11] Berge, J.P. (1966). A Study of Vortex Formation and Other Abnormal Flow in a Tank with and without a Free Surface. La Houille Blanche, No.1.

[12] Chang, E. (1977). Review of Literature on Drain Vortices in Cylindrical Tanks. BHRA Report, TN.1342, March.



Minimum fluidization velocity, bubble behaviour and pressure drop in fluidized beds with a range of particle sizes

B. M. Halvorsen1,2 & B. Arvoh1

1Institute of Process, Energy and Environmental Technology, Telemark University College, Norway 2Telemark Technological R&D Centre (Tel-Tek), Norway

Abstract

Fluidized beds are used in the production of pure silicon for solar cells. The particles are fully consumed during the reaction and the particles in the reactor have a large range of diameters. When the range of particle sizes is wide, the particles have a tendency to segregate. A series of experiments are performed to study the particle segregation and the influence of particle segregation on the bubble formation and flow behaviour. Experiments are performed in a two dimensional bed. The minimum fluidization velocity and the pressure drop have been measured. Spherical glass particles with different ranges of particle sizes are used in the experiments. Superficial gas velocities well above the minimum fluidization velocities are used in the study of segregation and bubble formation. Corresponding simulations are performed by using the commercial CFD code Fluent 6.3. The computational results are compared to the experimental data and the discrepancies are discussed. Keywords: fluidized bed, minimum fluidization velocity, particle size, pressure drop, particle segregation, fluent.

1 Introduction

Fluidized beds are widely used in industrial operations due to their large contact area between phases, which enhances chemical reactions, heat transfer and mass transfer. The efficiency of a fluidized bed reactor is highly dependent on the flow conditions which also control the mixing of the bed. Particle sizes, range of



particle sizes, particle size distribution and superficial gas velocity are influencing on the flow behaviour and mixing. Fluidized bed reactors are used in production of silicon for solar cells. The gas-particle reaction is a continuous process where the particles are fully consumed during the reaction and the particles in the reactor may therefore have particle diameters that range from 0 to about 500 µm. In a reactor like this, the temperature becomes very high and it is therefore extremely important to keep the particles fluidized and well mixed. The aim of this work is to study how the range of particle sizes and the particle size distribution influence on the flow behaviour in fluidized bed. When the powders have a large range of particle sizes, the chance of segregation is significant. Segregation and low degree of mixing may give very high temperatures in parts of the bed, and the consequence may be melting of the reactor. Knowledge about mixing and segregation are therefore essential.

2 Fluidized bed dynamics

2.1 Characterization of particles

Computational studies have been performed on a two dimensional fluidized bed. Spherical particles with a mean diameter of 154 µm, 488 µm and 960 µm and a density of 2485 kg/m3 are used. The behaviour of particles in fluidized beds depends on a combination of the particle size and density. Geldart fluidization diagram [1], shown in Figure 1, is used to identify characteristics associated with fluidization of powders. The dots represent the powders used in this study. The powder with mean diameter of 154 µm is close to group A particles, whereas the powder with mean diameter of 960 µm is close to group D particles. The fluidization properties for these groups of particles differ significantly from each other.

Figure 1: Geldart classification of particles [1].



Particles characterized in group A are easily fluidized and the bed expands considerably before bubbles appear. This is due to inter-particle forces that are present in group A powders [2]. For group B particles the inter-particle forces are negligible and bubbles are formed as the gas velocity reaches the minimum fluidization velocity. The bed expansion is small compared to group A particles. Geldart group D describes large and/or dense particle powders. These powders need a large amount of gas to get fluidized, and bubbles may occur if the gas velocity is kept close to the minimum fluidization velocity. Group D powders give low degree of solid mixing and gas back-mixing compared to group A and B powders [3].

2.2 Model

Computational studies have been performed on a two dimensional fluidized bed. The computational work is performed by using the commercial CFD code Fluent 6.3. The model is based on an Eulerian description of the gas and the particle phases. The combinations of models that are used in this work are presented in Table 1:

Table 1: Models used in Fluent.

Property Model Drag Syamlal & O’Brien

Granular viscosity Syamlal & O’Brien Frictional viscosity Schaeffer Frictional pressure Based-ktgf

Solid pressure Ma-ahmadi In a bubbling fluidized bed the concentration of particles varies from very low to very high. In dilute regions, the kinetic of the particles will dominate the solids viscosity, and the solid pressure will be close to zero. In regions with higher concentration of particles, the collisions between particles will dominate the solids viscosity, and the solid pressure will increase. At very high concentration of particles, the frictional stresses dominate the solid viscosity. The drag describes the momentum exchange between phases and is expressed by the drag coefficient in the momentum equation. The Syamlal & O’Brien drag model is expressed by [4]:

sr

sgggsDsg dv

UUC 24

3 −=Φ

ρεε (1)

where εg and εs are the gas and solid fractions, ρg is the gas density, Ug and Us are the gas and solid velocities and ds is the particle diameter. The terminal velocity correlation for the solid phase, vr, is a function of void fraction and Reynolds number. An empirical equation for vr is developed by Garside and Al Dibuouni [5]. The drag factor is proposed by Dalla Valle [6] and is expressed by:



2

/Re8.463.0

+=

rsD v

C (2)

The minimum fluidization velocity can be developed from the buoyant-equals-drag balance:

( )( ) )(1 sgg

sggsg uug −

Φ=−−

ερρε (3)

where the drag coefficient is developed by Syamlal & O’Brien. The equation for minimum fluidization velocity is [7]:

( )( )sg

gsmfmfmf

gU

Φ

−−=

ρρεε 12 (4)

Multiple particle phases are used in the simulations to account for the particle size distribution. The model, Syamlal-O’Brien-symmetric, is used to express the particle-particle momentum exchange [8]. The radial distribution function included in the Syamlal-O’Brien-symmetric equation is expressed by Ma and Ahmadi [9].

3 Results

Flow behaviour in a two dimensional bed with a uniform air distribution is studied experimentally and computationally. The purpose of this work is to study minimum fluidization velocity, particle segregation, bubble activity and pressure drop on a simple well-defined model. The experimental results are used to validate the CFD model that is used in the simulations.

3.1 Experimental set-up

A lab-scale fluidized bed with a uniform air distribution is constructed. The width, depth and height of the bed are 0.20 m, 0.025 m and 0.80 m respectively. The pressure is measured at five positions in the bed. A digital camera is used to record the bubble behaviour and particle segregation. The experimental set-up is shown in Figure 2. Spherical glass particles with density 2485 kg/m3 and different particle diameters are used in the experiments. The mean particle size and the range of particle sizes are presented in Table 2.

Table 2: Particle diameters.

Experiment no. Particle range [µm] Mean diameter [µm] 1 100-200 153 2 400-600 488 3 750-1000 960 4 (100-200)+( 400-600) 320 5 (100-200)+( 750-1000) 556 6 ( 400-600) +( 750-1000) 724



For all the powders and mixture of powders, the minimum fluidization velocities are observed. Experimental studies of bubble behaviour and particle segregation are performed with velocities well above the minimum fluidization velocity. The pressure is measured as a function of time. Photos are taken initially and after a few minutes. The experiments have been run for 2 to 10 minutes.

Figure 2: Experimental set-up.

3.2 Computational set-up

The simulations are performed with particles with diameters equal to the mean diameters of the glass powders used in the experiments. The data are given in Table 3. Two particle sizes are used to simulate the mixtures of two powders with different mean particle size.

Table 3: Particle diameters.

1 particle phase 2 particle phase Simulation no. Mean diameter [µm] Mean diameter [µm]

1 153 2 488 3 960 4 153(50%)+488(50%) 5 153(50%)+960(50%)

The simulations are run with the same velocities and initial bed heights as in the experiments. Two-dimensional Cartesian co-ordinate system is used to



describe the geometry. The grid is uniform in both horizontal and vertical direction, and the grid size is 0.5x0.5 mm. The models used are given in Table 1. The simulations have been run for 10 to 30 s.

3.3 Results

The measured and the theoretical minimum fluidization velocities are presented in Figure 3. The theoretical fluidization velocities are calculated from eqn (7) using the Syamlal and O’Brien drag model. The minimum fluidization velocity is influenced of the void fraction in the bed. The theoretical minimum fluidization velocities are calculated for a range of void fractions, from 0.37 to 0.44. The deviation between the theoretical and the experimental fluidization velocities increases with increasing void fraction. The experimental minimum fluidization velocity is about equal to the theoretical for the small particles (100-200 µm). For the larger particles the experimental minimum fluidization velocities are significantly lower than the theoretical. In the mixtures, the experimental minimum fluidization velocities are about equal to the minimum fluidization velocity for the smallest particles in the mixture. The experiments with small particles, mixture of 50% small and 50% medium particles and mixture of 50% small and 50% large particles give minimum fluidization velocities of 0.033, 0.033 and 0.04 m/s respectively. This indicates that the theoretical minimum fluidization velocity cannot be based only on the mean particle sizes. In a bed with a large range of particle sizes, the smaller particles will fill the void between the larger particles, and the bulk density can be rather high compared to a bed with one sized particles. This will highly influence on the fluidization condition, bubble formation and bubble behaviour in the bed.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 200 400 600 800 1000 1200

Mean particle diameter [µm]

Min

imum

flui

diza

tion

velo

city

[m/s

]

Ex. Small+mix smallEx. Medium + mix mediumEx. LargeCalc., void=0.37Calc., void=0.38Calc., void=0.40Calc., void=0.42Calc., void=0.44

Figure 3: Experimental and calculated minimum fluidization velocities.

Experiments are performed with superficial gas velocities well above the minimum fluidization velocities for the different powders and mixture of



powders. Figure 4 and 5 show the results from experiment and simulation with the smallest particles. The mean particle diameter is 154 µm and the superficial velocity is 0.1 m/s. The superficial velocity is well above the minimum fluidization velocity. The simulations are performed with one particle phase. The computational results agree well with the experimental results, both with respect to bubble behaviour and pressure drop. The pressure drop over the bed is about 50 mbar.

t=2 s t=10 s Initially t=2 min

Figure 4: Bubble behaviour. Small particles. Superficial velocity 0.1 m/s.

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6

Height above gas distributor [m]

Pres

sure

gau

ge [m

bar]

SimulationExperiment

Figure 5: Pressure as a function of height above the gas distributor, small particles, superficial gas velocity 0.1 m/s.

The mixture of 50% small and 50% medium particles gives a mean diameter of 320 µm. The theoretical minimum fluidization velocity is 0.10-0.17, depending on the void fraction. The measured minimum fluidization velocity is 0.033. Superficial velocity 0.1 m/s is used in experiment and simulation. The simulation is performed with two particle phases. The computational and experimental flow behaviour is presented in Figure 6. The experimental results



show very clearly the segregation of particles. The bubble formation occurs above the layer of large particles in the lower part of the bed. The tendency of particle segregation is also observed in the simulations, where high concentration of large particles is located close to the gas distributor. Figure 7 shows the comparison of the experimental and computational pressure drop for the mixture. The experiment gives higher pressure than the simulations in the upper part of the bed. This may be due to a higher bed expansion in the experiment. In the experiments, the pressure is not measured at positions below 0.185 m above the distributor.


Figure 6: Bubble behaviour. Mixture of small and medium particles. Superficial velocity 0.1 m/s.

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6

Height above the gas distributor [m]

Pres

sure

gau

ge [m

bar]


Figure 7: Pressure as a function of height, 50% small and 50% medium particles.

Figure 8 shows results from experiments and simulations performed with a mixture of 50% small and 50% large particles. The mean particle diameter of the mixture is 556 µm and the superficial velocity used is 0.1 m/s. The theoretical



minimum fluidization velocities range from 0.21 to 0.34 m/s for void fraction 0.37 to 0.44 respectively. The minimum fluidization velocity observed in the experiments is 0.04 m/s. The simulations are performed with two particle phases with diameter 153 µm and 960 µm. Bubbles are formed in the simulations although the superficial gas velocity is well below the theoretical minimum fluidization velocity for the mixture. Segregation of particles is very clear both in the simulation and experiment. The pressure as a function of bed height is presented in Figure 9. The computational pressure agrees well with the experimental pressure measurement.


Figure 8: Bubble behaviour. Small and large particles. Superficial velocity 0.1 m/s.

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6Height above gas distributor [m]

Pres

sure

gau

ge [m

bar]


Figure 9: Pressure as a function of height, 50% small and 50% large particles.

Experiments and simulations have been performed with medium and large particles. The superficial velocity is 0.2 m/s for medium particles, 0.33 m/s for



large particles. The velocities are well above the experimental minimum fluidization velocities, but below the theoretical minimum fluidization velocities. The experiments give bubbles, whereas the simulations give flow through a fixed bed. The simulations are performed with one particle phase which means that all the particles in the simulation have the same diameter. This gives a rather high void fraction and therefore a higher gas flow is required to get the particles fluidized. For these cases the fluidization does not occur at current velocities in the simulations. Figure 10 and 11 show the flow behaviour in the bed with medium and large particles respectively.

t=10 s Initially t=2 min

Figure 10: Bubble behaviour. Medium particles, superficial gas velocity 0.2 m/s.

t=10 s Initially t=2 min

Figure 11: Flow behaviour. Large particles, superficial gas velocity 0.33 m/s.



4 Conclusion

Flow behaviour in a two dimensional bed with a uniform air distribution is studied experimentally and computationally. The experiments are performed with spherical glass particles with different particle sizes and range of particle sizes. The minimum fluidization velocity is measured for all the different powders. The pressure is measured at different heights of the bed. The bubble behaviour is observed by using a digital camera. Corresponding simulations are performed by using the commercial CFD code Fluent 6.3. The model is based on a multi-fluid Eulerian description of the phases. The measured minimum fluidization velocities are compared to theoretical minimum fluidization velocities. The theoretical minimum fluidization velocities are developed from the buoyant-equals-drag balance by using Syamlal & O’Brien drag coefficient. The theoretical minimum fluidization velocity is significantly higher than the observed experimental velocity except for the smallest particles. This may be due to the particle size distribution and range of particle sizes in the experimental powder which influence on the void fraction and fluidization properties. The theoretical minimum fluidization velocity is based on a mean particle diameter. Powders with mono-sized particles have a significantly higher void fraction than powders with a particle size distribution. 50%-50% mixtures of two powders has a about the same minimum fluidization velocity as the smallest particles in the mixture. Comparison of computational and experimental bubble behaviour is performed, and the results show good agreement according to bubble formation, pressure drop and particle segregation for the small and mixtures of small and larger particles. Simulations with medium and large particles give no fluidization, whereas fluidization and bubbles are obtained in the corresponding experiments. This indicates that more than one particle phase is needed in the simulation to account for the particle size distribution in the powder.

References

[1] Geldart, D., Types of Gas Fluidization, Powder Technology, 7, pp. 285-295, 1973.

[2] Geldart, D., Gas Fluidization Technology, John Wiley & Sons Ltd., 1986. [3] Kunii, D., Levenspiel, O., Fluidization Engineering, Second Edition,

Butterworth-Heinemann, Newton, US, 1991. [4] Syamlal, M., O’Brien, T.J., A Generalized Drag Correlation for Multi-

particle Systems, Morgantown Energy Technology Center, 1987. [5] Garside, J. & Al Dibuouni, M.R., Velocity-Voidage Relationships for

Fluidization and Sedimentation, I&EC Process Des. Dev., 16, pp. 206-214, 1977

[6] Dalla Valle, J.M., Micromeritics, Pitman, London, 1948. [7] Gidaspow, D., Multiphase Flow and Fluidization, Academic Press, Boston,

1994.



[8] Syamlal, M., The Particle-Particle Drag Term in a Multiparticle Model of Fluidization., National Technical Information Service, Springfield, VA, 1987.

[9] Ma, D., Ahmadi, G., Thermodynamical Formulation for Dispersed Multi-phase Turbulent Flows, Int. J. Multiphase Flow, 16, pp. 323-351, 1990.



Section 3 Turbulent flow

Mathematical modelling on particle diffusionin fluidised beds and dense turbulenttwo-phase flows

R. GrollCenter of Applied Space Technology and Microgravity,Germany

Abstract

Volume-fraction weighted and Reynolds averaged momentum transport equationsare solved in an Euler/Euler approach to numerically simulate the turbulent,dispersed two-phase flow in a two-dimensional channel and a three-dimensionalconic diffuser flow. Particular attention is given to the modelling of turbulentdiffusion and particle wall interaction, assuming local equilibrium, but introducingindividual terms for particle/fluid drag interaction, particle collisions and trajectorycrossings. These influences have been quantified in terms of partial viscosities, arestitution power and a turbulence structure parameter. Boussinesq approximationshave been used for each phase and the formulation of their interaction wasprovided in the framework of the eddy-viscosity modelling concept.Keywords: two-phase flow, particle diffusion, particle collision, fluidised bed.

1 Introduction

The momentum transport equation includes a turbulent diffusion term, whichcharacterises motions that are not resolved by the convective term. This turbulentdiffusion depends on the turbulent kinetic energy and the turbulence characterisingEulerian time scale. Based on ”Csanadys Approximation” [3], the time scale ofthe dispersed phase is coupled with the turbulent time scale of the continuousphase. This time scale quantifies the diffusion intensity [5] and is influenced bythe drag interaction of the particles with the viscous gas phase and inter-particlecollisions [4, 9]. The fluctuation of a filtered variable φ is written:

φk = φ− < φ >k . (1)



The index k stands for C (continuous phase) or D (dispersed phase). Thefollowing negation notation is used here, i.e. C = D, D = C. The drag forcerelaxation time scale τ kp depends on the particle Reynolds number, based on thevelocity difference between the continuous and the dispersed phase [12], see e.g.Crowe et al. (1998) [2]. The equation terms are described with the help of anaveraging operator< . >k [11]:

< uki >l= αluki

αl; uki =< uki >

l +uki l . (2)

Summarising the different diffusion generating components to a combinedparticle-turbulence shear stress model the diffusion of dispersed particles ispredicted very well, especially inside turbulent shear layers.

2 Transport equations

Modelling the diffusion character of particle flows transported in a viscous carrierphase the diffusion is not dominated by the eddy dissipation as it is in turbulentshear flows in a continuous gas phase. This dispersed phase diffusion is charac-terised by unsteady drag influences and crossing trajectory characteristics definedby the correlation of velocities of the continuous phase and the dispersed phaseitself. The Reynolds averaged, volume-fraction weighted momentum transportequations for the continuous and the dispersed phase read:

∂

∂t(ρkαk < uki >

k)+ ∂

∂xj(ρkαk < uki >

k< ukj >k)

= −α ∂pC

∂xi+ α

∂σ kij

∂xj+ ρk

αk

τ kp(< uki >

k − < uki >k)

+ ρkαkgi − ∂

∂xj(ρkαk < uki kukj k >k) (3)

Collision and crossing trajectory terms were approximated by Grad [4] andCsanady [3]. The pressure gradient of the continuous phase is equivalent to thelift force. And the sum would be zero in a hydrostatic case. σkij defines the viscousstresses inside the phase k. Because of the solid character of the dispersed phaseσDij is zero.

The underlying turbulence model for both phases is based on the Boussinesqanalogy, employing eddy viscosity as the model quantity, whose formulation wasprovided in the framework of the standard k− ε modelling concept. kC defines theturbulent kinetic energy of the continuous phase. The corresponding variable ofthe dispersed phase kD describes the particle velocity variance at a given position.



∂

∂t(ρkαkkk)+ ∂

∂xj(ρkαkkk < ukj >

k)

= ∂

∂xj

(αk

(μk + ρk

νkt

σ kk

)∂kk

∂xj

)− ρkαk < uki kukj k >k

∂ < uki >k

∂xj

with σCk = 1, 0 and σDk = 2.5 (4)

The transport equations for the turbulent kinetic energy kk and its dissipation rateεkt differ from those for a single-phase flow by several additional production termsand the modified dissipation. This total turbulent kinetic energy loss εkα in thepresent model arises from relative drag and particle collision processes.

∂

∂t

(ρkαkεkt

)+ ∂

∂xj

(ρkαkεkt < u

kj >

k)

= ∂

∂xj

(αk(μk + ρk

νkt

σ kε

)∂εkt

∂xj

)− C1

εkt

kkρkαk < uki kukj k >k

∂ < uki >k

∂xj

− C2ρkαk

εkt εkα

kk+ C3

εkt

kkρkαk

τ kp

(qk − 2kk

)

with σCε = 1.3, C1 = 1.44, C2 = 1.92, C3 = 1.2 (5)

Because of the different phase velocities inside the drag relation term a generalformulation is needed for the complimentary index of k. The velocity covarianceqk =< uki kuki k >k of He and Simonin (1993) [5] represents the trace of thevelocity vector correlation tensor of both phases:

∂

∂t

(ρkαkqk

)+ ∂

∂xj

(ρkαkqk < ukj >

k)

= ∂

∂xj

(ρkαkνkα

∂qk

∂xj

)+ ρk

αk

τ kp

(2Zkkk + 2kk −

(1 + Zk

)qk)

− ρkαk < uki kuj k >k∂ < uk >k

∂xj− ρkαk < uki kukj k >k

∂ < uk >k

∂xj

with Zk = ρkαk

ρk αk, (6)

which completes the present three-equation model for each of both phases. Veloc-ity correlations, representing the turbulent momentum diffusion, are modelled bythe following Boussinesq approximations:

− < uki kukj k >k

= −2

3kkδij + νkt

(∂ < uki >

k

∂xj+ ∂ < ukj >

k

∂xi− 2

3

∂ < ukl >k

∂xlδij

)(7)



− < uki kukj k >k

= −1

3qkδij + νkα

(∂ < uki >

k

∂xj+ ∂ < ukj >

k

∂xi− 2

3

∂ < ukl >k

∂xlδij

). (8)

Modelling the viscous continuous phase the eddy viscosity is modelled based oneddy dissipation relating components. Because of the phase interaction influenceon the turbulent diffusion of both phases an additional transport equation for thevelocity covariance has to be defined. The diffusion coefficients of momentum andturbulent transport equations are given by the characteristic diffusion time scale τ kα :

νkt = τ kα · 2

3kk and νkα = τ kα · 2

3qk . (9)

To close the present formulation of particle and carrier gas phase motion this timescale has to be modelled. Based on this new kind of particle diffusion modellingalso equilibrium-turbulence boundary conditions are modified by the influences ofinter-particle collisions and phase-interactions of dispersed and continuous phase.

3 Diffusion modelling

Based on this model of momentum diffusion, which depends on the velocitygradients of the diffusing phase, the characteristic diffusion time scales haveto be defined by the velocity correlation and its associated loss rate εkα . Localequilibrium describes the equivalence of production and loss of turbulent kineticenergy. Assuming ∂/∂x1 ≈ 0 yields an expression (i = 1, j = 2) for the non-diagonal element of the Reynolds stress tensor.

εkα = − < uk1kuk2k >k∂ < uk1 >

k

∂x2(10)

⇒ − < uk1kuk2k >k =(< uk1kuk2k >k

)2εkα

∂ < uk1 >k

∂x2

With the definition of the turbulence structure parameter Ckα of the phase k

Ckα = −< uk1kuk2k >k23kk

(11)

⇒ νkt∂ < uk1 >

k

∂x2= − < uk1kuk2k >k=

(23C

kαkk)2

εkα

∂ < uk1 >k

∂x2

the turbulent viscosity is calculated using the turbulent kinetic energy kk , its lossrate and the turbulence structure parameter. Based on the diffusion definition (eq.9)



Table 1: Restitution power and turbulence loss components.

γ model πCγ εCγ πDγ εDγ

β Jones et al. 23βε

Cβ εCt 0. 0.

d Csanady 0. 0. 23kD

τCα

√1 + CCβ (ξ

Cr )

2 0.

p Wallis 23

2kC−qCτCp

2kC−qCτCp

23

2kD−qDτDp

2kD−qDτDp

c Jenkins et al. 0. 0.σDcτDp

23kk kk

τDc

(1 − (eDc )

2)

the turbulent time scale is also defined by these values:

νkt = −2

3τ kαk

k ⇒ τ kα = νkt23kk

= 2

3

(Ckα

)2 kk

εkα(12)

With the definition of the restitution power πkα , the turbulent viscosity is deter-mined by the turbulent kinetic energy and the restitution power.

πkα = 2

3

kk

τ kα= εkα(

Ckα)2 ⇒ − < uk1kuk2k >k=

(23kk)2

πkα

∂ < uk1 >k

∂x2(13)

The power πkα describes the restitution of turbulent shear forces based on thedissipation and structure of turbulence and reduces the turbulent diffusion.

This restitution power consists of the partial powers defined by four differenteffects (see Table 1), which are described in the following subsections with theindices β, p, c and d . The total turbulence loss rate εkα is given by the sum ofindividual loss rates (εkγ , see Table 1):

εkα =∑γ

εkγ = εkβ + εkd + εkp + εkc (14)

The different diffusion rates and turbulence loss rates are induced by the viscousturbulent shear stress (πkβ, ε

kβ : Jones et al. [7]), crossing trajectory effects (πkd , ε

kd :

Csanady [3]), drag forces (πkp, εkp : Schiller and Naumann [12]) and collision terms

(πkc , εkc : Jenkins and Richman [9]). Adding together these influences, the new

restitution power term of the turbulent diffusion is modelled.

πkα =∑γ

πkγ = πkβ + πkd + πkp + πkc (15)



0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

y/h

UD/UC0

Eul./Eul. [2] UD

Eul./Eul. [1] UD

Eul./Lag. UD

Exp. UD

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25

y/h

uDmag/UC

0

Eul./Eul. [2] uDmag

Eul./Eul. [1] uDmag

Eul./Lag. uDmag

Exp. uDmag

Figure 1: Normalised stream-wise particle mean velocity UD and normalisedstandard deviation uDmag of the particle velocity magnitude in the fully-developed, particle-laden channel flow.

Simulating the general restitution power πkα and energy loss εkα the resultingturbulence structure parameter Ckα is computed in the following way:

πkα = 2

3

kk

τ kα= εkα(

Ckα)2 ⇒ Ckα =

√εkα

πkα(16)

The turbulent time scale τ kα depends on the sum of all diffusion rates:

τ kα =23kk

πkβ + πkd + πkp + πkc(17)

This way of calculation yields a deterministic method to compute the turbulenttime scale, which is needed for the calculation of the general turbulent viscosity ofboth phases (eq.12).

4 Computational results and discussion

This model was validated using experimental data of 70 μm copper particlesin a fully-developed channel flow ([10]; experimental results). The results usingthe present model (Eul./Eul. [2]) were also compared with the results obtainedby an Euler/Lagrange scheme (Eul./Eul.) [6, 8] and a well-known Euler/Euleriandiffusion approach (Eul./Eul. [1]) [5, 11].

Gravity acts in the positive x-axis direction. The channel flow Reynolds number,based on channel height (2 h = 40 mm) and single phase channel centrelinevelocity (UC0 = 10.5 m/s) is Re2h = 27600. The flow is regarded as fullydeveloped after 125 channel heights and at this position it is assumed that theparticle velocity and particle turbulence has reached an asymptotic state. Thecopper particles have a density of ρD = 8800 kg/m3 and a diameter of Dp =70 μm. The inlet mass loading of particles is ZD0 = 10% and the parameter of



0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000 6000

y/h

Restitution Power [m2/s3]

DC

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

y/h

Turbulence Structure Parameter

DC

Figure 2: Turbulent kinetic energy loss rate εkα and turbulence restitution power πkαof the dispersed phase (D) and the continuous phase (C).

elasticity is eDc = 0.90. The results obtained using the present method were alsocompared with the computational results obtained by an Euler/Lagrange scheme[8].

Comparing the present model (Eul./Eul. [2]) with a standard particle diffusionmodel (Eul./Eul. [1]), the velocity and the standard deviation of the velocitymagnitude of the present model agree better with the Euler/Lagrangian results.Because of the assumed isotropy of the present model, the standard deviation ofthe stream-wise particle velocity does not agree as well with the predicted standarddeviation of the particle velocity magnitude.

The characteristic model values describing diffusion: the restitution power πkαand the turbulence structure parameter Ckα are shown in Fig. 2. The dissipationloss of the continuous phase and the restitution power of the dispersed phasedominate over the respective values of the other phase. As expected the turbulencestructure parameter of the dispersed phase CD decreases near the wall, becauseof the decreasing ratio

√εDα /π

Dα against the nearly constant turbulence structure

parameter of the continuous phaseCCα . Comparing the positions of these restitutionmaxima of the momentum diffusion with local minima of the turbulence structureparameter CDα (Fig. 2, right) near the wall the difference between the differentkinds of diffusion, with and without viscous turbulence dissipation, in dispersedand continuous phase are shown.

Validating this new particle diffusion model a test case of Bohnet and Triesch [1]has been chosen. Simulating a fluidised bed with a particle loading Z = 1 insidea rising vertical diffuser flow. Glass particles are used approaching the dispersedphase. The half-cone angle of the conic diffuser is 6 and the outlet/inlet diameterratio is D2/D1 = 1.45. The particle laden flow enters the diffuser part of thechannel after a distance, which is long enough to develop a fully turbulent flow.

Analysing the velocity profiles (Fig. 3) local loading, gas and particle velocitiesare shown at entrance and exit of the diffuser. The gas phase inlet-boundarycondition is a velocity block profile of U0 = 26.0 m/s. The symmetry line isat x = 0. Inside the diffuser the gas velocity decreases and is overtaken by theparticles. The particle velocity decreases during the relaxation process upside the



Figure 3: Normalised radial profiles of axial gas velocity, particle velocity andloading in comparison with experimental results.

diffuser. The particle loading has a similar maximum at the exit of the diffuser.Also this maximum decrease during the homogenising diffusion process upsidethe diffuser. The numerical data are compared with experimental data of Bohnetand Triesch [1].

5 Conclusions

Using this kind of diffusion blending, simulations are able to give better results forturbulent wall layers of the dispersed phase, including their turbulence production.The prediction of turbulent particle diffusion is limited by the quality of modellingof the momentum diffusion and the turbulence production of the dispersed phase.

Compared to classical equilibrium models, which solve an additional differ-ential equation for the energy loss, coming up to the dissipation rate in viscoussystems, the energy loss of the dispersed phase is given by algebraic equations. Theturbulence structure parameter remains nearly constant in the dissipative systemsexamined here. So the ratio of restitution and dissipation power of the involvedsub-models keeps also nearly constant.

For the not viscous, dispersed phase the turbulence structure parameterdecreases corresponding to the high restitution and the locally low momentumdiffusion inside the wall layer. This characteristic behaviour is based on the addi-tional restitution power without the corresponding loss rate induced by crossingtrajectories effects. Basically the added modelling of the turbulence structureparameter influences the production of particle velocity variance near the wall inthat kind that the gradients of the stream-wise particle velocity agree with themeasurement data.

References

[1] Bohnet A. and Triesch O., Influence of particles on fluid turbulence, Chem.Eng. Techn. 26:1254ff., 2003.



[2] Crowe C.T., Sommerfeld M., and Tsuji Y., Multiphase flows with dropletsand particles. CRC Press LLC, Florida, 1998.

[3] Csanady G.T., Turbulent diffusion of heavy particles in the atmosphere. J.Atm. Sc. 20:201ff., 1963.

[4] Grad H., On the kinetic theory on rarefied gases. Communications on Pureand Applied Mathematics 2(4):331ff., 1949.

[5] He J., and Simonin O., Non-equilibrium prediction of the particle-phasestress tensor in vertical pneumatic conveying. ASME-FED : Gas-Solid Flows166:253, 1993.

[6] Huber N., and Sommerfeld M., Characterization of the cross-particle con-centration distribution in pneumatic conveying systems. Powder Techn.79:191ff., 1994.

[7] Jones W. P., and B. E. Launder, The prediction of laminarization with a two-equation model of turbulence. Int. J. Heat and Mass Transf. 15:301ff., 1972.

[8] Kohnen G., ber den Einfluss der Phasenwelchselwirkungen bei turbu-lenten Zweiphasenstrmungen und deren numerische Erfassung in derEuler/Lagrange Betrachtungsweise (tranl. On the influence of phase inter-actions of turbulent two-phase flows and their numerical description in theEuler/Lagrangian approach), Shaker Verlag, Aachen, Germany, 1997.

[9] Jenkins J. T., M. W. Richman, Grads 13-moment-system for a dense gas ofinelastic spheres. Arch. Ration. Mech. Anal. 87:355ff., 1985.

[10] Kulick J. D., Fessler J.R., Eaton J.K., Particle response and turbulencemodification in fully developed channel flow. J. Fluid Mech. 277:109ff.,1994.

[11] Politis S. 1989. Prediction of two-phase solid-liquid turbulent flow in stirredvessels, PhD Thesis, Imperial College London, 1989.

[12] Schiller L., A. Naumann, Uber die grundlegenden Berechnungen bei derSchwerkraftaufbereitung (transl.: On the basic calculations in gravity con-ditioning), VDI-Journal 77:318ff., 1933.



A numerical study of the scale effects affecting the evolution and sediment entrainment capacity of a gravity current, propagating over a loose bed containing large-scale roughness elements

T. Tokyay & G. Constantinescu Department of Civil and Environmental Engineering, The University of Iowa, USA

Abstract

When compositional gravity currents propagate over a loose bed, they can entrain, carry, and deposit large quantities of sediment over considerable distances from the entrainment location. The capacity of compositional gravity currents to entrain sediment of various sizes from a loose bed containing large-scale roughness elements in the form of two-dimensional (2D) ribs is investigated using high-resolution Large Eddy Simulation (LES). The compositional gravity current is produced by the instantaneous release of a large volume of heavier lock fluid in a straight open channel. The present study focuses on the influence of the Reynolds number and the presence of 2D ribs on the evolution of the lock-exchange gravity current and on the distributions of the bed friction velocity. At the higher Reynolds number considered in the present investigation, the structure of the gravity current is closer to the inviscid state often assumed in simplified models. Keywords: gravity currents, sediment entrainment, Large Eddy Simulation, large-scale roughness.

1 Introduction

Gravity currents are mainly horizontal flows moving under the influence of gravity and generated by predominantly horizontal density differences within a



fluid or between two fluids. Predicting and understanding the evolution of gravity currents is of considerable interest to many applications in geophysics, in particular due to their impact on the environment. In the case in which they propagate over a loose bed (e.g., in rivers, lakes or oceans), compositional gravity currents can entrain substantial amounts of sediment and induce the formation of a turbidity gravity current. The turbulent motions within the body of the current detach particles from the bed and entrain them into the channel to form a turbidity current. Additionally, in most cases the loose bed over which the gravity current propagates is not smooth and flat. The presence of large-scale roughness elements can significantly modify the capacity of the compositional gravity current to entrain sediment with respect to the case in which the bed is flat. For example, erosion by gravity currents is one of the main causes for the formation of submarine canyons. Detailed measurements of the velocity and density fields within the gravity current are seldom available from experimental studies. In many applications involving turbidity currents or compositional currents propagating over loose beds, information on the spatial and temporal distributions of the bed friction velocity are essential to determine the amount of sediment entrained and carried by the current. High-resolution three-dimensional numerical simulations using eddy-resolving techniques like Direct Numerical Simulation (e.g., see Hartel et al., [3]) and Large Eddy Simulation (e.g., Ooi et al., [6]) have the advantage that can predict the spatial and temporal evolution of these variables. In particular, LES allows performing simulations at Reynolds numbers that are closer to the range typically encountered in practical applications. In many applications in river and ocean engineering, the Reynolds numbers are very high and the structure of the gravity current approaches the inviscid limit. Laboratory studies are typically conducted at much lower Reynolds numbers.

Figure 1: Sketch of a lock exchange flow in a channel after the gate was removed with associated front velocities of the heavier (Uf) and lighter (Ufl) currents.

In this work we investigate the effect of the Reynolds number on the evolution and capacity to entrain sediment of a lock-exchange compositional Boussinesq gravity current propagating over a flat bed containing an array of 2D ribs. We are also studying how the capacity of the gravity current to entrain sediment changes with respect to the case when the gravity current propagates over a flat bed. As a result of the density differences between the lock fluid and the surrounding fluid and of the hydrostatic pressure differences, the heavy fluid starts spreading underneath the lighter fluid, as shown in Figure 1. In the set up



considered in this study, the gravity current containing lock fluid propagates in a rectangular horizontal plane channel. We consider the case in which the initial volume of the release (heavier lock fluid) is high (x0>>H) and occupies the whole depth of the channel (full-release case). The channel is long enough to avoid interactions of the gravity current with the end walls during the simulation time. Thus, this paper considers only the evolution of a gravity current during the slumping phase. The wavelength of the square ribs is 3H and their height is 0.15H (see Figure 2), where H is the channel depth. The changes in the structure of the gravity current and its capacity to entrain sediment are investigated between Re=48,000 (Re=ubH/ν, where ν is the molecular viscosity,

Hgub '= is the buoyancy velocity and g’ is the reduced gravity), in the range

at which most laboratory experiments are conducted, and Re=106, closer to field conditions.

Figure 2: Sketch showing dimensions and spacing of the 2D ribs on the bottom surface.

2 Numerical model

A finite-volume LES code is used to solve the governing equations on non-uniform Cartesian meshes. A semi-implicit iterative method that employs a staggered conservative space-time discretization is used to advance the equations in time while ensuring second order accuracy in both space and time. A Poisson equation is solved for the pressure using multigrid. The algorithm discretely conserves energy, which allows obtaining solutions at high Reynolds numbers without artificial damping. A dynamic Smagorinsky model is used to estimate the subgrid-scale viscosity and diffusivity. All operators are discretized using central discretizations, except the convective term in the advection-diffusion equation solved for the concentration for which the QUICK scheme is used. Detailed validation of the code for 3-D LES simulations of cavity flows with or without an incoming turbulent flow is described in Chang et al. [1]. The same code was successfully used by Chang et al. [2] to predict ejection of buoyant pollutants from bottom channel cavities and by Ooi et al. [6] to predict the evolution of intrusive gravity currents. The density difference between the lock fluid and the ambient fluid is small enough to use the Boussinesq approximation. The Navier-Stokes equations and the advection-diffusion equation for the concentration are made dimensionless using the channel depth, H, and the buoyancy velocity, ub. The time scale used in the discussion of the results is t0=H/ub. The non-dimensional concentration is defined as )/()( minmaxmin CCCCC −−= , where maxC , minC represent the



maximum (lock fluid) and minimum (ambient fluid) concentrations in the domain and C is the dimensional concentration. The lock gate is positioned in the middle of the computational domain (x/H=0.0). The top and bottom surfaces are simulated as no-slip smooth walls. The flow is assumed to be periodic in the spanwise direction (z). A zero normal gradient boundary condition is assumed for the concentration at the top, bottom and at the two end boundaries. All simulations discussed in this paper were conducted with a value of the viscous Schmidt number of 600 corresponding to saline water. No assumptions are needed on the value of the turbulent Schmidt number as the dynamic procedure (Chang et al. [1]) directly estimates the value of the subgrid-scale diffusivity based on the resolved velocity and concentration fields. The flow field was initialized with the fluid at rest. The length of the computational domain in the streamwise and spanwise directions was 40H and H, respectively. The mesh contained over 40 million cells and the mesh spacing in the wall normal direction was sufficiently small to resolve the viscous boundary layer (no wall functions were used) in all the simulations. The time step was 0.001t0. To maintain the anti-symmetry of the forward and backward propagating currents containing lock and ambient fluid, respectively, the roughness elements were placed on the bottom surface in the region with x/H>0 and on the top surface in the region with x/H<0. This allows analyzing only the evolution of the forward propagating current containing heavier lock fluid. In the case in which 2D ribs were present the evolution of the gravity current is analyzed for t<50t0. During this period the interaction of the gravity currents containing lock and ambient fluid with the end walls is negligible. The two simulations conducted with a flat bed are denoted Flat_LR (Re=48,000) and Flat_HR (Re=106), respectively. The simulations conducted with an array of equally-spaced 2D ribs are denoted Ribs_LR (Re=48,000) and Ribs_HR (Re=106), respectively. At both Reynolds numbers, the gravity current and the flow in the region behind the front is highly turbulent, similar to gravity currents of practical relevance in engineering and geophysics. We consider the gravity current is highly turbulent if velocity spectra contain a clear inertial range as the upstream part of the gravity current is convected over streamwise locations situated at a sufficient distance from the lock gate ( ≅ 4H in our simulations).


The presence of the ribs induces significant changes in the structure of the gravity current. For example, Figure 3 compares the distributions of the spanwise-averaged velocity magnitude inside the forward propagating current between the high Reynolds number simulations (Re=106) at t=32t0. In the flat bed case (Flat_HR) the velocity magnitude in the tail of the gravity current increases monotonically with the distance from the lock-gate position (x/H=0.0) until the start of the dissipative wake region (x/H~14). In contrast to this, in the case in which 2D ribs are present (Ribs_HR), the velocity magnitude is strongly



amplified as the lock fluid inside the gravity current is convected over the top of the rib. The amplification is the highest in the jet-like flow that forms as the heavier gravity current fluid separates from the top of the rib. The jet-like flow reattaches some distance downstream of each rib (~0.5H), at which point the high-speed jet-like flow becomes parallel to the bed. The velocity magnitude continues to be high in the near-bed region until the presence of the next rib is felt by the gravity current. As this happens, the flow inside the gravity current starts decelerating again. Observe also that the front of the gravity current in the Ribs_HR simulation is situated about 2.5H behind the front position in the Flat_HR simulation. This is expected, as the presence of large-scale roughness elements at the bed introduces an additional drag force that slows down the current with respect to the case when it propagates over a flat bed.

Figure 3: Distributions of the spanwise-averaged velocity magnitude at t=30t0 in the Flat_HR (top) and Ribs_HR (bottom) simulations.

Shin et al. [7] performed full-depth lock-release experiments with high Reynolds number gravity currents propagating over a flat bed. They found that the front velocity during the slumping phase was consistent with Benjamin’s half-depth solution (Uf/ub=0.5). Several other experiments (e.g., Keulegan [5]) have shown that the non-dimensional front velocity of full-depth lock-release Boussinesq currents during the slumping phase increases with the Reynolds number and eventually approaches 0.5. Our 3D LES simulations of gravity currents propagating over a flat bed confirm this trend (Uf/ub=0.49 for the Flat_HR simulation at Re=106). A very good agreement between the predicted (Uf/ub=0.45 for Flat_LR simulation) and measured (Uf/ub~0.45) non-dimensional front velocities is observed at Re=48,000. As expected, and consistent with the results in Figure 3, the presence of the ribs slows down the propagation of the front of the current. Figure 4 compares the trajectories of the front, xf/H, in the Ribs_HR and Flat_HR simulations. The trajectories are very close until the front of the current approaches the first bottom-attached rib in the simulation containing the ribs. Then, the two trajectories start diverging. As expected, the slope of the front trajectory in the Flat_HR simulation can be considered constant starting at the end of the acceleration phase (t~3t0). As the front was situated at more than 2H from the end-wall during the time interval the simulation was run, the gravity current remained in the slumping phase (Uf/ub=0.49). The slope of the front trajectory in the Ribs_HR simulation shows a larger variation in time after the front approaches the first rib (t~10t0). This is mainly



because the front is decelerating as it approaches the upstream face of each rib due to the adverse pressure gradients induced by the rib obstacle. Moreover, the velocity in the head region has to further decrease as the head rises above the top of the rib. During this time, the potential energy increases at the expense of the head losing some of its kinetic energy. Then, the front velocity increases as the head is projected downwards toward the bed. Still, the results in Figure 4 clearly show the mean slope of the trajectory can be considered in a good approximation constant for t>15t0. This means that during the slumping phase the front velocity of a gravity current propagating over a surface containing 2D ribs is approximately constant. The value inferred from the Ribs_HR simulation is Uf/ub=0.36, which corresponds to a 27% reduction with respect to the flat-bed case. The mean front velocity predicted during the slumping phase is 0.34 for the Ribs_LR simulation and 0.45 for the Flat_LR simulation. Thus, in the lower Reynolds number simulations the reduction in the front velocity is close to 24%.

Figure 4: Time history of the front-position xf/H for the Flat_HR (dashed line) and Ribs_HR (solid line) simulations.

Figure 5: Time history of the potential energy, Ep, kinetic energy, Ek, and integral of the total dissipation, Ed, in the Ribs_LH (dashed line) and Ribs_HR (solid line) simulations.

Analysis of the global energy balance for the Ribs_LR and Ribs_HR simulations (Figure 5) allows comparing scale effects in the temporal variations of the kinetic energy, Ek, potential energy, Ep, and of the integral in time of the



total dissipation rate integrated over the whole domain (ε), ∫=t

d dtE0

)()( ττε .

In the case of compositional gravity currents (e.g., see Ooi et al. [6]) the sum of these three terms should be constant in time ( dpk EEE ++ =constant). Scale effects are negligible for the variation of the potential energy. The effect of the increase in the Reynolds number is to increase the front speed and the kinetic energy. Most of the increase in the kinetic energy is compensated by a decrease in the dissipation term Ed. This is because as the Reynolds number increases, the local dissipation rate that is proportional to the molecular viscosity (1/Re) is expected to decrease despite the fact that the velocity gradients within the turbulent gravity current are larger. The regions of relative minima observed in the temporal variation of Ek correspond to the time intervals when the front starts interacting with the upstream face of the rib and the head is convected over the top of the rib. The effect of the increase of the Reynolds number on the temporal evolutions of Ek, Ep and Ed in the flat-bed simulations is qualitatively similar. The only difference is the fact that the relative increase in Ek and the corresponding decrease in Ed are larger compared to the case when the bed surface contains ribs. Thus, the importance of scale effects decreases in the case in which the bed contains large-scale roughness elements. The bed friction velocity distributions are needed to estimate the amount of sediment entrained from the bed by a compositional gravity current propagating over a loose bed. This is because most sediment entrainment formulae predict the entrainment rate function of the bed-friction velocity. For example, in the case of Van Rijn’s [8] model the local pick up rate, P, is proportional to the difference between the actual bed friction velocity and the critical bed friction velocity obtained from Shields’ diagram for a given size of the sediment particles. Figure 6 compares the distributions of the non-dimensional bed-friction velocity, uτ/ub, at t=32to for the simulations conducted at Re=106. One obvious feature of the distributions of uτ/ub is the streaky structure of the region behind the front. The bed-friction-velocity streaks are due to the formation of streaks of high and low streamwise velocity at a small distance from the bed in the region where the flow inside the gravity current is strongly turbulent. The streaks disappear only in the regions situated upstream and downstream of each rib. After the jet-like flow past the top of each rib reattaches to the bed, the streaks need a certain distance to form again beneath the gravity current flow. Then the streaks disappear again as the near-bed flow decelerates close to the upstream face of the next rib. Thus, one expects the formation of the streaks will be suppressed if the spacing of the ribs is sufficiently decreased. Another relevant feature of the distribution of uτ/ub in the Flat_HR simulation (Figure 6) is the fact that, with the exception of the head region, there are no regions of strong amplification of uτ/ub with respect to the mean value (~0.013). This means that in the later stages of the slumping phase (t>20to) the interfacial billows shed in the dissipative wake region are not strong enough to amplify significantly the bed friction velocity values beneath them. This is because the



cores of the Kelvin Helmholtz billows are strongly disturbed in the formation region, in part due to their interactions with the lobes and the clefts present at the front of the current. This explains why the distribution of the bed friction velocity is relatively uniform behind the front region during the later stages of the slumping phase for a gravity current propagating over a smooth surface. The distributions of uτ/ub are qualitatively similar for the Flat_LR simulation but, as expected, the mean value of uτ/ub is larger (~0.028).

Figure 6: Distribution of the non-dimensional friction velocity, uτc /ub, on the bottom wall (x/H>0) at t=32t0 in the Flat_HR (top) and Ribs_HR (bottom) simulations. Also shown are the corresponding concentration contours (aspect ratio is 1:2) in an x-y section.

In the Ribs_HR simulation results, the distribution of uτ/ub is strongly modulated by the presence of the roughness elements. The bed-friction velocity is strongly amplified downstream of each of the ribs, in the region where the flow inside the gravity current plunges downwards in the form of a jet-like flow and reaches the flat part of the bed surface. After the front passes several ribs, the largest amplification does not always occur downstream of the rib situated the closest to the front, as was the case in the initial stages of the slumping phase. This is because as a result of the interaction between the front and the ribs, the mixing in the head region is very high (see concentration contours for Ribs_HR simulation in Figure 6) and the mean concentration in the head region is smaller than the one in the upstream part of the gravity current (e.g., around the second rib behind the front in Figure 6). The lowest values of the bed-friction velocity are observed in the region situated upstream of each rib, where the heavier gravity current fluid decelerates and increases its potential energy such that it can be convected over the top of the rib. Figures 7 and 8 allow a direct comparison of the spanwise averaged bed friction velocity among the four simulations in the later stages of the slumping phase (t=32to). The streamwise distributions of the bed friction velocity are qualitatively similar in the low and high Reynolds number simulations.



However, important differences are observed in the distribution of this variable between the simulations with a flat bed and the simulations with a bed containing ribs. Consistent with the results in Figure 6, the presence of the ribs induces large-scale variations in the streamwise distribution of the spanwise-averaged bed friction velocity. The peak values present in the region where the current reattaches to the flat part of the bed in the simulation containing the ribs are larger than the values observed in the same region in the simulation with a flat bed conducted at the same Reynolds number. For example, the nondimensional peak values of the bed friction velocity behind the head region are around 0.02 in the Ribs_HR simulation, while the corresponding values in the Flat_HR simulation are around 0.015 (see Figure 7). The differences are even larger in the simulations conducted at Re=48,000. The peak values in the Ribs_LR simulation are close to 0.045 while the corresponding values in the Flat_LR simulation are around 0.03 (see Figure 8).

Figure 7: Streamwise variation of the spanwise-averaged friction velocity on the bottom wall (x/H>0) at t=32t0 in the Flat_HR (dashed line) and Ribs_HR (solid line) simulations.

Figure 8: Streamwise variation of the spanwise-averaged friction velocity on the bottom wall (x/H>0) at t=32t0 in the Flat_LR (dashed line) and Ribs_LR (solid line) simulations.

Based on the results in Figure 7, the mean value of the spanwise-averaged bed friction velocity in between x/H=0.0 and the front is 0.011 in the Ribs_HR simulation and 0.013 in the Flat_HR simulation. This means that a gravity current propagating over a flat bed in a high Reynolds number (~106) lock-exchange flow will have a slightly higher capacity to entrain fine sediment compared to one propagating over a bed containing large-scale roughness elements in the form of 2D ribs. Same trend is observed in the simulations conducted at Re=48,000 where the relative difference between the streamwise-averaged values in between x/H=0.0 and the front is similar (0.026 in the Flat_LR simulation and 0.021 in the Ribs_LR simulation). Of course, at both Reynolds numbers the capacity of the gravity current to entrain coarser sediment is expected to be higher in the case in which the gravity current propagates over an array of 2D ribs mainly due to the strong amplification of the bed friction



velocity in the region where the jet-like flow of heavier fluid reattaches to the bed. While in the flat bed case one expects the capacity of the current to entrain sediment will peak behind the front which is propagating with constant velocity, in the case in which ribs are present the capacity of the current to entrain sediment will be the highest downstream of the first 3-4 ribs behind the front. To quantitatively compare the capacity of the gravity current to entrain sediment particles of a certain size in the four simulations, the flux of the sediment entrained from the bed per unit time and unit width is calculated for a certain sediment size and Shields critical shear velocity, uτc, using:

∫=A

dAPW

)t(F 1 (1)

where W (=H) is the width of the channel, A is the bed area and P is the local pick-up rate for the sediment. In the present work the expression proposed by van Rijn (1984) was used to estimate P when uτ>uτc.

20

80606051

2

22 1000330.

....

c

c dg)s(

u

uu.P

ντ

ττ −

−= (2)

The pick-up rate P is expressed in units of volumetric flux per unit area per unit time. If at a certain location uτ<uτc, then the local sediment entrainment is equal to zero (P=0). In Equation (2), d is the diameter of the sediment, g is the gravitational acceleration, ν is the molecular viscosity and s is the ratio between the density of the sediment and that of the fluid.

Figure 9: Time history of the flux of sediment entrained at the bed (x/H>0) for particles with a threshold value for entrainment of uτc/ub=0.03 (top) and uτc/ub=0.01 (bottom) in the Flat_LR (circles) and Ribs_LR (squares) simulations.

Assuming a length scale h=1.7m corresponding to ∆ρ/ρ0=0.02, the buoyancy velocity in the Re=106 simulations is ub=0.58m/s and t0~3s, if the ambient fluid



is water. The flux F(t) was plotted in Figure 9 for the simulations conducted at Re=106 for four sediment sizes (d=14, 30, 50 and 100 µm). Using Shields diagram, the critical bed friction velocity uτc for sediment entrainment is 0.006, 0.008, 0.01, 0.012 m/s for the four sediment sizes with d=14, 30, 50 and 100 µm. Past the initial stages of the slumping phase (t>40s during which the front of the gravity current passes the first rib in the Ribs_HR simulation) the capacity of the gravity current propagating over ribs to entrain particles with d<15 microns (Figure 9a) is similar to that of the gravity current propagating over a flat bed. For particles with d=30 µm (Figure 9b), the flux of sediment entrained at the bed for t>40s is three to five times larger for the gravity current propagating over ribs. For d>50 µm (Figures 9c and 9d) the gravity current propagating over a flat bed cannot entrain sediment past the initial stages of the slumping phase as uτ<uτc. In the case of the gravity current propagating over a bed containing ribs uτ remains larger than uτc in some of the regions where the jet-like flow re-attaches downstream of the first couple of ribs behind the front of the current. The flux of coarser sediment is modulated in time by the formation of a region of strong bed friction velocity amplification each time the gravity current overtakes a new rib.

4 Conclusions

The erosional properties of high Reynolds number gravity currents with a large volume of release propagating over a flat bed and over a flat bed containing an array of equally-spaced 2D ribs was investigated for 48,000<Re<106. Present results show that a gravity current with a high volume of release propagating over an array of 2D ribs reaches a regime in which the front velocity is nearly constant (slumping phase). The presence of an array of equally spaced (3H) square 2D ribs of size 0.15H*0.15H reduced the velocity of the front by about 24% at Re=48,000 and by about 27% at Re=106 compared to the flat-bed case. In the flat-bed simulation at Re=106 the predicted non-dimensional front velocity during the slumping phase (0.49) was very close to the value obtained from theory for inviscid currents (0.5). In the case the ribs were present, the largest bed friction velocity values were observed downstream of the reattachment line of the plunging jet-like flow that forms as the gravity current is convected over the top of the ribs. The largest values did not necessarily occur at all times downstream of the first rib behind the front, especially after the gravity current overtook the first couple of ribs in the series. This is because the heavier lock fluid in the head region mixed with the lighter surrounding fluid. At both Reynolds numbers, the peak bed friction velocity values in the simulations containing 2D ribs were larger than the ones recorded at the same nondimensional time for the gravity current propagating over a flat bed. This has important consequences in terms of the capacity of the gravity current to entrain coarser material for which the critical bed shear stress value for entrainment is around or above the maximum bed shear stress value predicted in the flat-bed case.



At both Reynolds numbers, past the initial stages of the slumping phase (t>15t0) the mean value of the spanwise-averaged bed friction velocity in between x/H=0.0 and the front was about 20% lower in the case the bed contained ribs. However, the presence of the ribs induced the formation of bands of high bed-friction velocity. In these regions larger particles can be entrained by the gravity current. Indeed, analysis of the flux of sediment entrained at the bed in the Re=106 simulation (h=1.7m, ∆ρ/ρ0=0.02, ub=0.58m/s) past the initial stages of the slumping phase, confirmed that the capacity of the gravity current propagating over a flat bed to entrain fine sediments (d<15 microns) is at least equal to that of the same gravity current propagating over a bed containing ribs. However, for larger particles the capacity to entrain sediment of the compositional gravity current propagating over a flat bed was much smaller than the one estimated for the same current propagating over a bed containing ribs.

References

[1] Chang, K.S., Constantinescu, G., Park, S-O., Analysis of the flow and mass transfer processes for the incompressible flow past an open cavity with a laminar and a fully turbulent incoming boundary layer. J. Fluid Mech. 561, pp. 113–145, 2006.

[2] Chang, K.S., Constantinescu, G., Park, S-O., The purging of a neutrally buoyant or a dense miscible contaminant from a rectangular cavity. Part II: The case of an incoming fully turbulent overflow. J. Hydraulic Engineering, 133(4), pp. 373–385, 2007.

[3] Härtel, C., Meiburg, E., and Necker, F., Analysis and direct numerical simulation of the flow at a gravity-current head. Part 1: Flow topology and front speed for slip and no-slip boundaries. J. Fluid Mechanics, 418, pp. 189–212, 2000.

[4] Hatcher, L, Hogg, A. and Woods, A., The effect of drag on turbulent gravity currents. J. Fluid Mechanics, 416, pp. 297–314, 2000.

[5] Keulegan, G.H., An experimental study of the motion of saline water from locks into fresh water channels. U.S. Natl. Bur. Stand. Rep. 5168, 1957.

[6] Ooi, S.K., Constantinescu, S.G. and Weber, L., A numerical study of intrusive compositional gravity currents. Physics of Fluids, 19, 076602, 2007.

[7] Shin, J., Dalziel, S., and Linden, P.F., Gravity currents produced by lock exchange. J. Fluid Mech., 521, pp. 1–34, 2004.

[8] Van Rijn, L.C., Sediment pick-up functions. J. Hydraulics Engineering, 110(10), pp. 1494–1503, 1984.



Effect of relative motion between bubbles and surrounding liquid on the Reynolds stress as a mechanism controlling the radial gas holdup distribution

K. Ueyama Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Japan

Abstract

Relations are derived between time-averaged quantities of bubble turbulence, liquid velocity, static pressure and gravitational force, based on a careful treatment of the Navier-Stokes equations with certain approximations. The liquid phase is modeled as a combination of a bulk phase and a cloud phase. Time-integration of the substantial derivative term in the Navier-Stokes equation for the cloud phase yields the difference between the total convective transport of liquid momentum from the cloud phase to the bulk phase and that from the bulk phase to the cloud phase, throughout the time of integration. The difference can be interpreted as bubble turbulence entering into the bulk phase, which should be convectively transported in the bulk phase. Finally, relations between time-averages of the bubble turbulence, liquid velocity, static pressure and gravitational force are obtained through time-integration of the Navier-Stokes equation for the cloud phase. The resulting equation can be used to deduce, analytically or numerically, the macroscopic properties of gas-liquid multiphase flow in combination with a two-fluid model equation, for instance. These equations are applied to recirculating turbulent flow in bubble columns operating at high gas feed rates, and together with a simple model for the bubble turbulence we derive the well-known parabolic distribution of gas holdup. Keywords: gas-liquid multiphase flow, macroscopic property, time-average, Navier-Stokes equation, cloud phase, gas holdup distribution, two-fluid model, recirculating turbulent flow.



1 Introduction

Gas-liquid multiphase flow is widely used in the chemical, pharmaceutical and petrochemical industries to realize effective mass transfer, heat transfer and chemical reactions between gas and liquid phases. Gas-liquid multiphase flow in industrial apparatus usually involves a high gas volume fraction, to maintain high contact surface area between the two phases. Numerous studies have been made to determine and control the macroscopic properties of gas-liquid multiphase flow. Previous studies can be classified into three categories: numerical studies based on a two fluids model equation; approaches that build up from an understanding of the interaction between individual bubbles and the surrounding flow field; and studies of macroscopic properties of gas-liquid multiphase flow. Numerical studies have become common as a result of the development of computer hardware and software. These works are based on a two-fluid model equation in which the interfacial force term comprises forces acting on a single sphere [7,16]. However, a volume-mean, for instance, of the interfacial force acting on bubble surfaces in a reference volume should reflect contributions of the forces acting on a bubble and the distribution of gas holdup. This argument is valid for any two-fluid model equation regardless of the details of the averaging. The interfacial force term in the two-fluid model equation therefore should represent the combined effect of forces acting on a bubble and the spatial distribution of gas holdup. The approach that proceeds from an understanding of the interaction between a single bubble and the surrounding flow field has already been helpful in studying dynamics of a single bubble in various flow fields [4,5,8]. This approach needs adapting to deal with gas-liquid multiphase flow with relatively high gas hold up and the effects of surrounding bubbles. Studies based on macroscopic properties of gas-liquid multiphase flow proceed from known macroscopic properties to provide an understanding of basic phenomena such as interaction between a single bubble and the surrounding flow field. Sato et al. [6] introduced the notion of bubble turbulence, and showed that the measured radial distribution of the mean liquid velocity accurately coincides with model prediction based on the measured radial distribution of gas holdup together with the eddy diffusivity for fully developed turbulent flow and the calculated Reynolds stress due to bubble turbulence. The success of this model implies that bubble turbulence is important in determining macroscopic properties of gas-liquid multiphase flow. Zhang and Ahmadi [16] showed that a modification of εκ − model taking bubble turbulence into account could accurately reproduce the time-averaged velocity profile in bubbly flow based on a two-fluid model equation. In the present work, a phenomenological definition of bubble turbulence is given by time-averaging the Navier-Stokes equations as proposed by Ueyama and Miyauchi [9], for a cloud phase surrounding the bubbles. A general relation between bubble turbulence and turbulent flow quantities is derived which can be used to deduce macroscopic properties of gas-liquid multiphase flow. The



assumptions used to obtain this relation are analyzed and shown to be acceptable. Finally, the well-known parabolic distribution of gas holdup for recirculating turbulent flow in bubble columns operated at a high gas feed rate [2,3,10–12,15] is obtained as an analytical solution of our relation.

2 Definition and notation for time-averaging

We can time-integrate the Navier-Stokes equations, at a fixed point in space, for the bubble phase, liquid phase or for any fixed quantity of fluid we define mathematically by its bounding surface. The time-averaged Navier-Stokes equations for the bubble or liquid phase give relations between time-averaged physical quantities for those phases. Turbulence induced by bubble motion relative to the surrounding liquid, henceforth referred to as bubble turbulence, plays an important role in a mechanism controlling the gas holdup distribution [6,16]. Let us consider the liquid phase as comprising a cloud phase and a bulk phase in discussing the effect of bubble turbulence on the mechanism of gas-liquid multiphase flow. We suppose that each bubble is surrounded by a cloud, in which bubble turbulence is generated by relative motion between the bubble and liquid. At this point we are not concerned whether the cloud includes multiple bubbles and the physical definition of the cloud is not necessary for the time-averaging procedure. Bellow, the notations involved in the time-averaging procedure are set out, and the definition of the cloud phase is stated. Time-integration is performed at a fixed point in space over a sufficiently long intervalΛ to obtain reliable time-averaged values. The gradient of a physical quantity q can be time-averaged as in eqn (1).

( ) ( ) ( ) ( )

∑ −+∑ −+

∇=

∫ ∫∇=∇=∇≡∇

===

===

+

′′

M

j tt

N

i ttcc

t

t

cc

qqqqq

qdtqqqdt

TTTT lj

aj

ai

li

c

11

1

1 1

ξξξξΛΛ

Λ

ΛΛΛ

Λ

Λ Λ

Λ (1)

Here, the suffix c refers to the cloud phase and the superscript Λ denotes the

time-averaged value over a time intervalΛ . A notation Λc denotes the sum of individual time lengths during which the cloud phase is continuously observed.

A superscript c

refers to the value averaged over a time interval Λc , and ∫Λc

dt

refers to integration over the time during which the cloud phase is continuously observed. We denote by T a

i and Tli the arriving and leaving time of the i-th

bubble. Notations T aj′ and T l

j′ respectively denote the arriving and leaving

time of the j-th cloud. The values of T ai , Tl

i , T aj′ and T l

j′ depend on position. The symbols M and N denote the total numbers of clouds and bubbles,



respectively. The vector ξ is the gradient vector of the surface Tt ai= , Tt l

i= ,

Tt aj′= or Tt l

j′= .

T∇=ξ (2) If us is a moving velocity vector of bubble surface, it follows that:

1=⋅ξus (3) If there is no mass transfer across the surface, the following eqn (4) is obtained, because ξ is normal to the surface.

( ) 0=⋅− ξuu s (4) In the absence of mass transfer across the surface, it then follows that:

1=⋅ξu (5)

3 Time-integrations of terms in the Navier-Stokes equations

3.1 Substantial derivative term

Upon applying eqns (3), (4) and (5), a substantial derivative term can be time-averaged, as follows:

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ∑

⋅−−⋅−−

⋅∇=

∑ ⋅−⋅+∑ −−

∑ ⋅−⋅+∑ −−

⋅∇=

∫

⋅∇+∂∂

=

==

==

′′′′

′′′′′′

M

j

c

M

j

M

j

N

i

N

i

c

Ts TTs T

TTTTTT

TTTTTT

aj

aj

lj

lj

lj

lj

aj

aj

lj

aj

ai

ai

li

li

ai

li

c

dtt

1

c

11

11

c

1

11

11

1

ξuuuξuuuuu

ξuuξuuuu

ξuuξuuuuuu

uuu

ρρΛ

ρΛΛ

ρρΛ

ρρΛ

ρρΛ

ρρΛ

ρΛΛ

ρρΛ Λ

(6)

Here ρ and u respectively denote the density and velocity vector of the liquid. A vector ξT a

j′ can be rewritten using the normal unit vector n at the

surface, as:

( ) nunξ⋅

=′

′s T

Taj

aj

(7)

Let us choose the direction of n toward the movement of the cloud surface, in order to interpret the second term on the right hand side of the final term in eqn (6). The term ( ) ξuuu Ts T

aj

aj ′′⋅−− ρ can be rewritten as:

( ) ( ) ( )

nu

nuuuξuuu

⋅

⋅−−=⋅−−

′

′′′ s T

s TTs T

aj

aj

aj

aj

ρρ (8)

Since liquid is flowing into the cloud phase from the bulk phase, and the cloud surface is moving towards the bulk phase at Tt a

j′= , the numerator on the



right hand side of eqn (8) is the flux of momentum transported convectively from the bulk phase to the cloud phase. The denominator is the volume swept by the surface per unit time. The right hand side of eqn (8) can be understood as the total impulses, divided by the total time, per unit volume added at the instance when the cloud arrives at the time-averaging point, that is due to convective transport of momentum from the bulk phase to the cloud phase.

t=T’ia

t=Tia

t=Til

t=T’il

i-th bubble

cloudcloud phase

bulk phaset=T’ia

t=Tia

t=Til

t=T’il

i-th bubble

cloudcloud phase

bulk phase

Figure 1: Schematic diagram of the bubble, cloud and bulk phases.

We thus conclude that the second term on the right hand side of the final term in eqn (6) is the difference between liquid momentum transported convectively from the cloud phase to the bulk phase and that from the bulk phase to the cloud phase; that is, the convective transport term of the bubble turbulence momentum,

u′bρ , into the bulk phase; this represents an input of bubble turbulence momentum to the bulk phase. The input of bubble turbulence momentum is immediately transported by liquid flow in the bulk phase, and it should therefore be balanced with the time-averaged convective transport of bubble turbulence, uu′⋅∇− b

bb ρΛ , for the bulk

phase. Here, Λb is a sum of individual time durations during which the bulk

phase is continuously observed. The superscript b signifies a time-averaged

value over the time intervalΛb . The balance between input and convective transport terms for bubble turbulence is expressed as:

( ) ( ) uuξuuuξuuu ′⋅∇−∑

⋅−−⋅−−=

=′′′′ bTs TTs T

bb

M

jaj

aj

lj

lj

ρΛρρ1

0 (9)

From eqns (6) and (9), time-integration of the substantial derivative term leads to eqn (10).



( ) ( )

′′⋅∇+

′′⋅∇+

′′⋅∇+

⋅∇=

′⋅∇+

⋅∇=

∫

⋅∇+∂∂

uuuuuuuu

uuuu

uuu

bbbbLL

b

bbccccllc

bbcc

dttc

ρΛΛρ

ΛΛρ

ΛΛρ

ΛΛ

ρΛΛρ

ΛΛ

ρρΛ Λ

1

(10)

Here, u′L is turbulence other than the bubble turbulence u′b , and is assumed

to be independent of the bubble turbulence. The superscript l refers as usual to a time-averaged value over the time interval ΛΛΛ cbl += , that is the entire duration during which the liquid phase is observed.

uuuuuu ′+′+=′+= Lbll , (11)

In eqn. (11), u′ is a fluctuating component of the liquid velocity. Eqn (12) holds in the cloud phase, because the bubble turbulence is newly generated there:

uuuu ′′>>′′ LLbbcc (12)

By applying the condition (12) in eqn (10), we find that:.

( ) ( )

′′⋅∇+

⋅∇≈∫

⋅∇+∂∂

uuuuuuu bbllllcdt

tc

ρΛΛρ

ΛΛρρ

Λ Λ

1 (13)

The second term on the right hand side of eqn. (13) is obtained from the general property of time-averaging. The right hand side is given by the physical components time-averaged over the entire duration for which the liquid phase is observed. This will enable us to construct a simple physical model for the bubble turbulence.

3.2 Static pressure term

We have:

( ) ( ) ∑

−+∑ −+

−∇=∫ ∇−

==′′′′

M

j

N

i

ccTTTTTTTT a

jaj

lj

lj

li

li

ai

ai

c

PPPPPdtP11

111 ξξξξΛΛΛ

ΛΛ Λ

(14)

The second and third terms on the right hand side are surface terms representing the effect of static pressure at the bubble surface and cloud surface, respectively. In the conventional two-fluid model equation, the surface term for the static pressure acting on unit volume of a particular phase is given as the product of the mean static pressure and the gradient of a local fraction of the phase;

( )

∇=

∑

−+∑ −

==′′′′ Λ

ΛΛ

ccM

j

N

iPPPPP TTTTTTTT a

jaj

lj

lj

li

li

ai

ai 11

1 ξξξξ (15)

From eqns.(14) and (15), it follows that:

( ) ( )PPPdtP cccccc

c

∇−=

∇+

−∇=∫ ∇−

ΛΛ

ΛΛ

ΛΛ

Λ Λ

1 (16)



3.3 Stress tensor term

We have:

( )

( ) ∑

⋅−⋅+∑ ⋅−⋅+

⋅−∇=

∫ ⋅∇−

==′′′′

M

j

N

i

ccTTTTTTTT

aj

aj

lj

lj

li

li

ai

ai

c

dt

11

11

1

ξτξτξτξττ

τ

ΛΛΛΛ

Λ Λ (17)

The second term on the right hand side is a surface term due to the viscous force at the bubble surface. This can be calculated by time-averaging the Navier-Stokes equations for the gas phase in gas-liquid multiphase flow, neglecting the density and viscosity terms[9,13,14]:

( ) ( )PllN

i TTTT li

li

ai

ai

∇−

−≈∑ ⋅−⋅= Λ

ΛΛΛ 1

1 ξτξτ (18)

The third term on the right hand side of eqn (17) is a surface term due to the shear stress at the cloud surface. It should cancel with the effect of shear stress at the bubble surface when the cloud thickness is negligible. The third term vanishes when the cloud phase entirely occupies the liquid phase. Hence, the third term can be approximated as:

( )Pl

l

lclM

j TTTTaj

aj

lj

lj

∇−−

≈∑

⋅−⋅

=′′′′ Λ

ΛΛΛΛΛ

Λ 1

1 ξτξτ (19)

The right hand side goes to ( )Pll ∇

−ΛΛΛ when Λc vanishes, and is zero

when ΛΛ lc = . By substituting eqns. (18) and (19) into eqn (17), we have:

( ) ( ) ( )Pdt l

l

lccc

c

∇−

−

⋅−∇≈∫ ⋅∇−

ΛΛΛΛΛ

ΛΛ

Λ Λττ1 (20)

3.4 Gravitation term

We have:

( ) gg ρΛΛρ

Λ Λ

c

c

dt =∫1 (21)

4 Time-integrations of the Navier-Stokes equations for the cloud phase

We now have, from eqns.(13), (16), (20) and (21):

( ) gτuuuu ρΛΛ

ΛΛ

ΛΛρ

ΛΛρ

ΛΛ cccl

l

cllllcPbb +

⋅∇−∇−=

′′⋅∇+

⋅∇ (22)

The absolute value of the second term on the right hand side is usually very small compared to that of the second term on the left hand side, the first term on the right hand side, or the third term on the right hand side, in which case:



+∇−≈

′′⋅∇+

⋅∇ guuuu ρ

ΛΛ

ΛΛρ

ΛΛρ

ΛΛ ll

l

cllllcPbb

(23)

Ueyama has derived the following equation from the Navier-Stokes equations time-averaged for the liquid phase in gas-liquid multiphase flow [9,13,14]:

gτuu ρΛΛ

ΛΛρ

ΛΛ llllll

P +

⋅∇−

⋅−∇=∇ (24)

By substituting eqn (24) into eqn (23), and neglecting the second terms on the right hand sides of eqn(24) in comparison with the Reynolds stress term in the first term on the right hand side of eqn (24), we have:

⋅∇≈

′′⋅∇+

⋅∇ uuuuuu

ll

l

cllllcbb ρ

ΛΛ

ΛΛρ

ΛΛρ

ΛΛ (25)

The gravitation term cancels in deriving eqn (25). The physical quantities in eqns (23) and (25) are all time-averaged values over the time interval Λl , which is the total duration during which the liquid phase is observed. Eqn (23) gives a relation between bubble turbulence, static pressure and gravitational force, and eqn (25) is a relation between the tensors uu ΛΛ ll , uu ′′ bb

lΛ and uuΛl . In combination with the two-fluid model equation, eqns (23) and (25) allow us to deduce the macroscopic properties of gas-liquid multiphase flow.

5 Gas holdup distribution for recirculating turbulent flow

In this section, eqn (23) is applied to recirculating turbulent flow in large scale bubble columns operating at high gas feed rates. The radial and angular component of the time-averaged velocity are both zero, the axial and angular gradient of time-averaged quantities are zero, and no stationary swirl flow is observed in such a flow field [2,3,10–12,15]. The radial distribution of gas holdup for the recirculating turbulent flow can be expressed as:

( )φεε n−= 10 (26)

Table 1: Values of n for the gas holdup distribution.

D T [m] U G [m/s] gas distributor n

single nozzle

single nozzle

single nozzle

perforated plate

perforated plate

Here, ε is the gas holdup defined as the ratio

ΛΛl , and ε 0 is the gas holdup at

the center of the column. We define a dimensionless radial coordinate R

r=φ ,



and n is a fitting parameter. Ueyama [10, 11] obtained numerical values of n which fit the measured distribution of gas holdup, as shown in Table 1. The values of n in Table 1 are scattered around n = 2. Eqn. (27) was obtained from the Navier-Stokes equations, time-averaged for gas-liquid multiphase flow [13].

( ) ( )∫

−−−=′′−

φφτρεεφ

φρε λ

0

21 dR

gRuu wcmc zr

c (27)

Here, ε m is a cross sectional mean of the gas holdup, εW is the gas holdup at the column wall, and τW denotes the shear stress. Ueyama and Saitoh [14] recently showed that the radial distributions of the Reynolds stress measured by Degaleesan [1] agree very well with eqn. (27) for the parabolic distribution of gas holdup, as shown in Figure 2.

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

φ [-]

τrz [N/m

2]

UG=0.096m/s

DT=0.14m

UG=0.12m/s

DT=0.14m

UG=0.10m/s

DT=0.44m

0.102

0.125

0.0775 (εm-εw)|fitting

Eq.(27)

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

φ [-]

τrz [N/m

2]

UG=0.096m/s

DT=0.14m

UG=0.12m/s

DT=0.14m

UG=0.10m/s

DT=0.44m

UG=0.096m/s

DT=0.14m

UG=0.12m/s

DT=0.14m

UG=0.10m/s

DT=0.44m

0.102

0.125

0.0775 (εm-εw)|fitting

Eq.(27)

Figure 2: Prediction based on a parabolic gas holdup distribution and

Degaleesan’s data [1] for the radial distribution of Reynolds stress.

It can be concluded that the radial gas holdup distribution is parabolic for the recirculating turbulent flow in bubble columns. Below, we shall derive the parabolic distribution from eqn. (23). Eqn. (28) is obtained from Navier-Stokes equation time-averaged for gas-liquid multiphase flow [9,13,14].

( ) gRz

PmW

lρετ −−−=

∂∂ 12 (28)

The axial component of eqn (23) becomes, upon exploiting eqn. (28):



( ) ( )

−+≈′′−

∂∂ g

Ruurrr m

lWbzbr ρεεταρε 211 (29)

Here, Λ

Λαl

c= . The term ( ) uu bzbrl

′′− ε1 on the left hand side of eqn (29)

is defined as follows:

( ) ∫ ′′=′′−ΛΛ

εl

dtuuuu bzbrl

bzbr11 (30)

The value of this expression depends strongly on the motion of bubbles surrounding the spatial point at which time-averaging is performed. We now construct a simple model to deduce the value of the left hand side, using a Cartesian coordinate system for gas-liquid multiphase flow with gas holdup distributed in the x-direction and homogeneous in the y- and z-direction. When bubbles are rising, in the z-direction, the value of ∫ ′′

Λl

dtuu bzbx consists of a

contribution due to bubbles which have passed through the x+ region, which is a region with x-value greater than that of the time-averaging point, and a contribution from bubbles in the x− region. Represent the contribution of the

x+ bubbles as ( ) λε +xF , where ( )λε +x is the gas holdup at λ+x . The

contribution of the x− bubbles can be expressed as ( ) λε −− xF , since the sign of the bubble turbulence velocity, u bx′ , to be coupled with u bz′ is opposite

to that of the contribution of the x+ bubbles for the same value of u bz′ . Hence, the value of ∫ ′′

ΛΛ l

dtuu bzbx1 is given as:

( ) ( )

( ) ( )( )εεελ

λελεΛ

εε

Λ

fx

xFx

xFxFdtuu

xxxxx

bzbrl

=== ∂

∂=

∂∂

≈

+−+=∫ ′′

2

1

(31)

Here, ( ) ( )εε

λξ Fddf 2≡ . Eqn (31) is different from the model proposed by

Sato et al. [6], which took the value to be proportional to the gas holdup ε : however, the present model may fit the phenomenon better. By substituting eqn (31) into eqn (29), we have:

( ) ( )

−+≈

Rgfdd

dd

mW εερταφε

φεφ

φ2 (32)

Upon taking a polynomial expression for ε and ( )εf , the following solutions are obtained in the case where the parameter α is constant.

( )φεεεε 200 −+= W (33)



( ) ( ) ( )

−+−

= εερτ

εεαε W

W

W RgRgf 4

8 0

(34)

Eqn. (33) gives the parabolic radial distribution of gas holdup, which has been known empirically for forty years. The most dubious assumption involved in deriving the parabolic distribution is that the value of

ΛΛα

lc= is constant

regardless of the gas holdup value. However, the assumption of constant α is acceptable because, even if the value of α depends on the gas holdup, there should be a minimum positive value of α which we can adopt in deriving eqn. (29).

6 Conclusion

The Navier-Stokes equations have been time-integrated for the cloud phase which surrounding each bubble. A phenomenological definition of bubble turbulence is clearly introduced as the liquid momentum generated in the cloud phase. It is also shown that the bubble turbulence generated in the cloud phase acts as an input of liquid momentum in the bulk phase, and is in balance with convective transport in the bulk phase. Eqs. (23) and (25) above show the relation between the bubble turbulence and the time-averaged velocity field of gas-liquid multiphase flow. By applying eqn. (23) to the recirculating turbulent flow in bubble columns, the well-known parabolic radial distribution of gas hold up is obtained as an analytical solution. Eqns. (23) and (25) can be applied to any kind of gas-liquid multiphase flow, so as to deduce macroscopic properties of the multiphase flow.

Symbols

DT : column diameter [ ]m g : gravitational acceleration [ ]sm 2−⋅ g : vector of gravitational acceleration vector [ ]sm 2−⋅ M : total number of clouds n : fitting parameter in eqn. (27) [ ]− N : total number of bubbles P : static pressure [ ]Pa q : physical quantity r : radial coordinate [ ]m R : column radius [ ]m t : time [ ]s

T ai : arrival time of the i-th bubble [ ]s



Tli : time of leaving of the i-th bubble [ ]s

T aj′ : arrival time of j-th cloud [ ]s

T lj′ : time of leaving of the j-th cloud [ ]s

u : velocity [ ]sm 1−⋅ u : velocity vector [ ]sm 1−⋅ u′ : fluctuating velocity vector [ ]sm 1−⋅

u′b velocity vector of bubble turbulence [ ]sm 1−⋅

u′L vector of turbulence independent of bubble turbulence [ ]sm 1−⋅

U G : superficial gas velocity [ ]sm 1−⋅ x : Cartesian coordinate [ ]m y : Cartesian coordinate [ ]m z : axial coordinate [ ]m α : parameter defined by

ΛΛα

lc= [ ]−

ε : gas hold up defined by ( )Λ

Λε l−= 1 [ ]−

ε 0 : gas holdup at column center [ ]−

εW : gas holdup at column wall [ ]−

εm : cross sectional average of gas holdup [ ]− Λ : duration for time averaging [ ]s

Λb : total time during which bulk phase is observed [ ]s

Λc : total time during which cloud phase is observed [ ]s

Λl : total time during which liquid phase is observed [ ]s ξ : gradient vector at surface [ ]ms 1−⋅ τ : tensor of shear stress [ ]mPa 1−⋅ φ : dimensionless radial coordinate defined by

Rr=φ [ ]−

Suffixes and other notations b : bulk phase or bubble turbulence c : cloud phase l : liquid phase

T ai : value at surface Tt a

i=

Tli : value at surface Tt l

i=

T aj′ : value at surface Tt a

j′=



T lj′ : value at surface Tt l

j′= b

: time-averaged value for duration Λb c

: time-averaged value for duration Λc l

: time-averaged value for duration Λl Λ

: time-averaged value for duration Λ

∫Λc

dt : sum of the time integration for individual time integral during which the

cloud phase is continuously observed

References

[1] Degaleesan, S., Fluid dynamic measurements and modelling of liquid mixing in bubble column, PhD Thesis, Washington University, St. Louis, USA, 1997

[2] Hills, J.H., Radial non-uniformity of velocity and voidage in a bubble column, Trans. Inst. Chem. Engrs., 52, 1-10, 1974.

[3] Katoh, Y., M. Nishinaka and S. Morooka, Gas holdup distribution in bubble columns, Kagakukogaku Ronbunshu, 1,530-535,1975.

[4] Kurose, R., R. Misumi and S. Komori, Drag and lift forces acting on a spherical bubble in a linear shear flow, Int. J. Multiphase Flow, 27, 1247-1258, 2001.

[5] Legendre, D. and J. Magnaudet, The lft force on a spherical bubble in a viscous linear flow, J. Fluid Mech., 368, 81-126, 1998.

[6] Sato, Y., M. Sadatomi and K. Sekoguchi, Momentum and heat transfer in two-phase bubble flow, Int. J. Multiphase Flow, 7, 167-177, 1981.

[7] Tabib, M. V., S. A. Roy and J. B. Joshi, CFD simulation of bubble column –an analysis of interphase force and turbulence model, Chem. Eng. Journal, 139, 589-614(2008).

[8] Tomiyama, A., Y. Nakahara, Y. Adachi and S. Hosokawa, Shapes and rising velocity of single bubbles rising through an inner subchannel, J. Nuclear Science and Technology, 40, 136-142, 2003

[9] Ueyama, K. and T. Miyauchi, Time-averaged Navier-Stokes equations as Basic Equations for multiphase flow, Kagakukogaku Ronbunshu, 2, 595-601, 1976.

[10] Ueyama, K. and T. Miyauchi, Behavior of bubbles and liquid in a bubble column, Kagakukogaku Ronbunshu, 3, 19-23, 1977.

[11] Ueyama, K., Study on gas-liquid multiphase flow in bubble columns, Doctoral thesis, University of Tokyo, Tokyo, Japan, 1978.

[12] Ueyama, K. and T. Miyauchi, Properties of recirculating turbulent two phase flow in gas bubble columns, AIChEJ., 25, 258-266, 1979.



[13] Ueyama, K., Force balance controlling radial gas holdup distribution for the recirculating turbulent flow in bubble columns, J. Chem. Eng. Japan, 39, 1-6, 2006.

[14] Ueyama, K and M. Saitoh, Overview of multiphase flow phenomena in moving time-averaged space, Chem. Eng. Technol., 29, 1102-1106, 2006.

[15] Yamagoshi, T., Bachelor Thesis, Study on bubble columns, University of Tokyo, Tokyo, Japan, 1969.

[16] Zhang, X. and G. Ahmadi, Eulerian-Lagrangian simulations of liquid-gas-solid flows in three-phase slurry reactors, Chem. Eng. Sci., 60.5089-5104, 2005.



Velocity and turbulence measurements of oil-water flow in horizontal and slightly inclined pipes using PIV

W. A. S. Kumara1, B. M. Halvorsen1,2 & M. C. Melaaen1,2 1Telemark University College, Norway 2Telemark R & D Centre (Tel-Tek), Norway

Abstract

Oil-water flows in horizontal and slightly inclined pipes are investigated using Particle Image Velocimetry (PIV). PIV offers a powerful non-invasive tool to study such flow fields. The experiments are conducted in a 15 m long, 56 mm diameter, inclinable steel pipe using Exxsol D60 oil (viscosity 1.64 mPa s, density 790 kg/m3) and water (viscosity 1.0 mPa s, density 996 kg/m3) as test fluids. The test pipe inclination is changed in the range from 5° upward to 5° downward. The experiments are performed at mixture velocity 0.25 m/s and inlet water volume fraction 0.25. The instantaneous local velocities are measured using PIV, and based on the instantaneous local velocities mean velocities and turbulence profiles (U-rms, V-rms and Reynolds stresses) are calculated. The time averaged cross sectional distributions of oil and water phases are measured with a traversable gamma densitometer. The flow regimes are determined based on visual observations. The measured flow regimes, water hold-up, slip ratio and velocity and turbulence profiles show a strong dependency with pipe inclination. Keywords: oil-water flow, horizontal flow, inclined flow, particle image velocimetry, gamma densitometry, hold-up, slip ratio, turbulence measurements.

1 Introduction

The flow of two immiscible liquids is often encountered in many industrial applications, especially in the petroleum industry. However, despite their abundance, liquid-liquid flow systems have not drawn so much attention as gas-liquid flows. Only recently, they are attracting more and more interest due to



continuous improvements required by technological applications in offshore oil industry. The current trend in offshore oil production is characterized by deep-waters, smaller oil fields with thin oil layers, increased water production and development of horizontal and branched wells to easily penetrate into the large areas of the reservoir. Offshore deep-water production would involve transportation of oil-water mixtures from the wellhead to a central gathering station or a platform as shown schematically in fig. 1. The terrain is not flat and flows through inclined pipes are often encountered. Oil-water mixtures may sometimes be transported through hilly terrain as well. The pipe inclination may lead to more mixing of the oil-water phases, and flow patterns, phase distributions and pressure drop will be highly affected. The knowledge about the hydrodynamic properties associated with these flows is extremely important to ensure safe design and efficient operation of offshore transportation pipelines. Nevertheless, limited number of studies on oil-water flow in inclined pipes has been performed in the past [1–4]. The existing literature covers mainly the flow patterns, pressure drop and hold-up measurements of oil-water flow in inclined pipes. The data on mean velocities, velocity fluctuations, Reynolds stresses and other turbulence properties have not been measured with sufficient details and accuracy. The present paper reports the mean axial velocity and turbulence measurements of oil-water flow in pipes at different pipe inclinations (-5º, -1º, 0º, +1º and +5º) using Particle Image Velocimetry (PIV). The time averaged cross-sectional distribution of oil and water is measured using a gamma densitometer. The flow regimes are determined by visual observations.

Well Head

Gas

Oil

Water

Horizontal Well

Riser

Processing Platform

Transportation Pipeline

Sea

Transportation Pipeline

Figure 1: Schematic of a subsea processing facility.

2 Experimental set-up

The experiments were carried out in the multiphase flow facility at Telemark University College, Porsgrunn, Norway.



2.1 Multiphase flow loop

A simplified flow sheet of the experimental rig is shown in fig. 2.

SEPARATORR100

P100

OIL TANK T100

V103V104

V105

V111

V116

103LT

WATER TANKT101

V109 V177

177FC

108FC

P101

V114 V168

113FC

168FC

114AFT V115A

V115B

114BFT

115FT

110FT

127LZ

126LZ

125LZ

104LT

GAMMA

V112

PLEXIGLASS

120PDT

121PDT

109AFT V110A

V110B

109BFT

TEST SECTION

Figure 2: Simplified flow sheet for the test rig.

The experiments were performed using water (density 996 kg/m3, viscosity 1 mPa s) and Exxsol D60 oil (density 790 kg/m3, viscosity 1.64 mPa s) as test fluids. Oil and water are stored in separate tanks (T100 and T101 for oil and water, respectively) and circulated using volumetric pumps P100 and P101. The mass flow, density and temperature are measured for each phase before entering the test section using Coriolis flow meters (FT109B, FT110, FT114B and FT115). A controller based on LabView® allowed to set the input oil and water flow rates and to select the appropriate pumps and flow meters. The test section is a 15 m long steel pipe with inner diameter equal to 56 mm. The test pipe inclination is changed in the range from 5° upward to 5° downward. Towards the end of the test section, there is a short transparent section for visual observations and PIV measurements.

2.2 Particle Image Velocimetry (PIV)

The fundamental methodology employed in the PIV system is relatively straightforward and actually rather simple in principle. The PIV measurements include illuminating a cross section of the seeded flow field, typically by pulsing light sheet, recording multiple images of the seeding particles in the flow using a camera located perpendicular to the light sheet, and analyzing the images for displacement information. The velocity and turbulence distributions are estimated based on particle displacements. The PIV system used for this study consists of a pulsed laser, CCD video camera, synchronizing system and a personal computer all linked to each other. The illumination beam is produced by dual cavity Nd:YAG laser with peak emission of 532 nm waves with the output



energy of 50 mJ/pulse. The laser beam is directed through an optical system to produce a thin planer sheet of high intensity laser light. This sheet of laser light is aligned to illuminate the vertical plane across the pipe center. The oil and water phases are seeded for PIV measurements. In general, these particles should be small enough to be good flow tracers and large enough to scatter sufficient light for imaging. Polyamide seeding particles (PSP) having mean particle diameter of 20 µm are used in both oil and water phases. The flow images are recorded with a X-StreamTMXS-3 cross correlation CCD camera of 1260×1024 pixel resolution. The camera is focussed normal to the illuminated flow field. A Dantech FlowMap processor is used to synchronize the laser and camera and to process the resulting image pairs. The PIV system allows the acquisition of flow images with the sample frequency of 50 Hz. The first and higher order turbulence statistics are estimated based on 4084 flow images. The flow images as shown in fig. 3(a) are divided into rectangular regions called interrogation areas. In the present experiments, the interrogation area is 32×32 pixels, which yields to a spatial resolution of 1.92×1.92 mm. For each of these interrogation areas the images from the first and second pulse of the light sheet are cross-correlated to produce average particle displacement vectors. The instantaneous velocity vectors are estimated based on average displacement vectors and time delay between laser pulses. From a series of instantaneous velocity measurements, the mean velocity and turbulence properties can be estimated. The number of tracer particles in the flow is very important in obtaining a good signal peak in the cross correlations. As a rule of thumb, 10 to 25 particle images should be seen in each interrogation area. Fig. 3(b) shows distribution of tracer particles in four interrogation areas.

(a) (b)

Figure 3: PIV flow images: (a) Particle image, (b) Tracer particles in four interrogation areas.

2.3 Gamma densitometry

The gamma densitometry offers non-intrusive measurements of local phase distributions in oil-water flow. It exploits the fact that electromagnetic radiation is attenuated as it passes through matter owing to the interaction of its photons with the matter. The gamma densitometer is equipped with a 45 mCi Americium-241 source and a detector [sodium iodide (NaI) scintillation crystal



doped with thallium]. The source and the detector are located diametrically opposite to each other on a pipe section with collimator structure used to ensure the production of a narrow gamma beam. The degree of attenuation experienced by a narrow beam of gamma radiation is a function of the gamma beam photons’ energy and the density of the absorbing matter. In two-phase flow, this can be calibrated to measure the local water volume fraction in the volume covered by the gamma beam. By traversing the gamma densitometer, the cross-sectional distribution of oil and water phases can be measured. In the experiments, the vertical interface position is measured by traversing horizontal gamma beams.

2.4 Investigated flow conditions

All the experiments are performed at mixture velocity 0.25 m/s and inlet water cut 0.25. The mixture velocity for oil-water flow is defined as:

AQQ

U wom

+= (1)

where Qo and Qw are the inlet volumetric flow rates of oil and water, respectively and A is the pipe cross-sectional area. The inlet water cut (λw) is defined as:

wo

ww QQ

Q+

=λ (2)

The experiments are performed at five different pipe inclinations (-5º, -1º, 0º, +1º and +5º).


The effect of pipe inclinations on the flow patterns, hold-up, slip ratio, velocity and turbulence in oil-water flow is investigated in this section.

3.1 Water hold-up and slip ratio measurements

The water hold-up (ηw) for oil-water flow is defined as:

AAw

w =η (3)

where Aw is the cross sectional area occupied by water phase. The vertical distance from the bottom of the pipe to the point, where the local water volume fraction is equal to 0.50 is considered as the interface height and the interface is treated as a flat surface in order to estimate the flow areas for different phases. It is possible to measure the liquid fractions as function of time at a given cross section of the pipe: the so-called in-situ water hold-up. In oil-water flows, the in-situ hold-up will be time-dependent due to, for example, wave motion and mixing at the interface. An averaged hold-up value can be obtained by taking some average of the time dependent signal. The water hold-up values reported in this paper is estimated using time averaged local water volume fraction measurements from gamma densitometry.



The ratio between the averaged in-situ velocities of the two phases is often given as slip ratio, S, and it can be calculated as follows:

sw

so

o

w

w

o

UU

AA

UU

S == (4)

where Ao is the flow area occupied by the oil phase. Uso and Usw are superficial velocities for oil and water phases, respectively. The superficial velocities for oil and water phases are defined as follows:

A

QU o

so = and A

QU w

sw = (5)

The slip ratio is dependent on physical properties combined with flow rates, flow pattern and pipe geometry. S>1 means that the oil travels faster than water in the pipe while S<1 indicates that water is the faster phase.

0

0.1

0.2

0.3

0.4

0.5

0.6

-6 -4 -2 0 2 4 6

Pipe inclination [0]

Wat

er h

old-

up [-

]

Water hold-up

Inlet watervolume fraction

0

1

2

3

4

-6 -4 -2 0 2 4 6

Pipe inclination [0]

Slip

ratio

[-]

(a) (b)

Figure 4: The water hold-up and slip-ratio as a function of the pipe inclination: (a) Water hold-up, (b) Slip ratio.

Fig. 4 (a) shows the variation of water hold-up of oil-water flow as a function of pipe inclination at mixture velocity 0.25 m/s and inlet water cut 0.25. The constant inlet water volume fraction used in all the experiments is also indicated. It is important to note that the hold-up which is measured, is in almost all instances significantly different from the input water volume fraction. In upward flow there is an expected trend of in-situ accumulation of the denser phase, in this case water. Hence, the measured water hold-up reaches higher values compared to horizontal flow as can be seen in the fig. 4(a). The measured hold-up value for horizontal (0º) flow is 0.34 and it increases up to 0.52 and 0.58 for upward flow at +1º and +5º. The water hold-up increase from 0º to +1º appears greater than from +1º to +5º. In addition, it is very dependent on the pipe inclination when the pipe is nearly horizontal. The physical reason for this is as follows. There is a shearing force at the interface separating the oil and the water due to the velocity difference between these phases. At inlet water cut 0.25, the oil moves faster than the water so that the water is dragged along by the oil



phase. When the pipe has an upward inclination, gravity will act to pull the more dense water phase downwards. Thus, the total force dragging the water up the inclined pipe is now reduced due to the action of the gravitational force. As a result, the water moves slower. With the constant volume flow of fluids and conservation of mass, a slower water velocity implies that the cross section of the pipe occupied by the water phase must increase. In other words, the water hold-up increases. When the pipe is inclined downwards, the opposite will happen: gravity works in the same direction as the shearing force and water hold-up is low compared to the horizontal flow. The measured water hold-up values are 0.21 and 0.14 for pipe inclination angles -1º and -5º, respectively. The deviation of measured water hold-up values from input water volume fraction is higher for upward flow compared to the downward flow as shown in fig. 4(a). The slip ratio of oil-water flow in horizontal and slightly inclined pipes is presented in fig. 4(b) at mixture velocity 0.25 and inlet water cut 0.25. It shows some similarities with the hold-up distribution presented earlier, giving an indication of the strong link between the water hold-up and the slip ratio of oil-water flow in pipes. The measured slip ratio for horizontal flow is 1.54. In this case, oil flows faster than water phase due to difference in wetted perimeter. The slip ratio varies widely with pipe inclination for near horizontal flows. The measured slip ratios are 3.25 and 4.08 for pipe inclinations +1º and +5º, respectively. At low inlet water fraction (λw=0.25), the oil, as the less dense phase, travels at a considerably higher velocity than the water, so that S is always above one for upwardly inclined flows. Previous studies indicate that at low upward inclination generally results in higher water hold-up than in horizontal flow giving higher slip ratios [5–7]. The relevant systems are described by Scott [5] at +15º and +30º, Lum et al [6] at +5º and Abduvayt et al [7] at +0.5º and +3º. The reported S values are in general above one, for upwardly inclined oil-water flows and show a good agreement with present observations. In addition, the measurements show that the slip ratio increases largely when the pipe is inclined from 0º to +1º compared to from +1º to +5º. This could be due to the increased mixing at +5º, higher water hold-up is still favoured but is tempered slightly by the increased level of mixing and interfacial waves. The slip ratios of downwardly inclined pipes show fewer deviations from horizontal flow compared to the upwardly inclined pipes. The measured slip ratios for pipe inclinations -1º and -5º are 0.79 and 0.47. In downwardly inclined oil-water flows at these inclination angles, the gravity would favour faster water flow and the slip ratio is below one. The available data on slip ratio measurements in downward oil-water flow are by Cox [8] at -15º and -30º and Abduvayt et al [7] at -0.5º and -3º. Abduvayt et al [7] showed that the slip ratio is always less or equal to one for downwardly inclined oil-water flows. Nevertheless, Cox [8] reported generally low slip ratios, although not always below one. The present slip ratio profile shows that the slip ratio can be above one for downwardly inclined oil-water flow at very small pipe inclinations. Hence, the present measurements show a good accordance with data presented by Cox [8].



3.2 Local water volume fraction, velocity and turbulence measurements

The PIV measurements are performed on two-phase oil-water flow at mixture velocity 0.25 m/s and inlet water cut 0.25 for five different pipe inclinations. The measurements of mean axial velocity, root mean squared velocities and Reynolds shear stresses are presented. In every plot, the local water volume fraction data from gamma densitometry is included to indicate the position of the interface. The still flow images are used as the background of the graphical representation of measured data for better visualization of the flow.

3.2.1 Horizontal flow Fig. 5 presents local water volume fraction, mean axial velocity and turbulence profiles for oil-water flow in horizontal pipe. The direction of the flow is from left to right with the oil phase in the upper part of the pipe. The flow regime in this case is smooth stratified flow without dispersion. The flow is gravity dominated and the phases are segregated having a smooth interface.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.2 0 0.2 0.4 0.6 0.8 1

Water vol.frac [-], U-mean [m/s], Reynolds stress [kg/sm2]

Pos

ition

[-]

-0.04 0 0.04 0.08 0.12 0.16 0.2

U-rms[m/s], V-rms[m/s]

Water vol.fracU-meanReynolds stressU-rmsV-rms

Figure 5: Mean axial velocity and turbulence measurements for horizontal flow.

The local water volume fraction is expected to be zero in the oil phase and one in the water phase. At the interface it will be somewhere in between. Good agreement between gamma densitometry measurements and visual observations can be seen. The mean axial velocity profile shows that mean axial velocity is highest in the oil phase, as expected. In this case, the Reynolds numbers for oil and water phases are 5685 and 8344, respectively. The turbulence statistics of the flow field are calculated based on instantaneous velocity measurements. The root mean-squared velocity components of streamwise (U-rms) and wall normal



(V-rms) directions are presented together with Reynolds stress (ρuv ). Here, V is the mean velocity component in wall normal direction, u and v represent the fluctuating velocity components in streamwise and wall normal directions and ρ is the fluid density. The root mean squared velocity components represent time averaged values of fluctuating velocities. The largest values of U-rms, V-rms and Reynolds stress are observed in the regions where the mean axial velocity gradient is largest highlighting the intimate connection between turbulence production and sheared mean flow. The streamwise intensity, U-rms, is produced by mean shear and two peaks are observed very close to the wall in oil and water phases. The wall normal stress component, V-rms, is fed by redistribution of intensity from U-rms, hence fall below U-rms. The difference between U-rms and V-rms implies an anisotropic structure of the turbulence. The maximum mean axial velocity is observed at normalized radial position 0.33 in the oil phase. At this point, the mean axial velocity gradient is zero and hence no turbulence is produced. Nevertheless, the values of U-rms and V-rms do not decrease very much because vigorous eddy mixing transports turbulent fluid from nearby regions of high turbulence production towards this region. A slight increase in U-rms is observed close to the interface. The impenetrability of the pipe wall suppresses the normal component of the turbulence. Therefore the maximum values of U-rms are observed very close to the wall whereas the maximum values of Reynolds stress are located at a certain distance away from the pipe wall. The Reynolds stress is proportional to the shear rate and it can be expressed as:

yUuv t ∂∂

−= µρ (6)

where µt is the turbulent eddy viscosity. The expression is often known as Boussinesq eddy viscosity hypothesis. As shown in fig. 5, the mean velocity gradient is positive in the water phase and Reynolds stress is negative. This follows the above argument based on Boussinesq eddy viscosity hypothesis. The Reynolds stress is zero at the point where the maximum mean axial velocity is observed. This is due to the change of the sign of the mean velocity gradient. The Reynolds stress has a special behaviour close to the oil-water interface. It decreases towards zero close to the interface. The oil-water interface acts as a moving wall and the stable density stratification close to the interface suppresses turbulence fluctuations normal to the interface. This may be the reason for the observed smaller values of Reynolds stress close to the interface.

3.2.2 Upward flow The effect of pipe inclination on flow pattern, local water volume fraction, mean axial velocity and turbulence profiles of upwardly inclined oil-water flows are presented in fig. 6. The measurements are performed at two different pipe inclinations, +1º and +5º. The introduction of a small inclination in the pipeline, as well as the size of the inclination, has a significant impact on oil-water flow in pipes. The observed flow patterns at +1º and +5º are very different from the smooth stratified flow observed in the horizontal pipe for the same flow



conditions. In deviated flows, the gravitational force has components both normal and parallel to the pipe axis. The normal component promotes segregation of the phases as in the horizontal flow, while the parallel one can act either in the direction of the flow (downward flow) or in the opposite direction (upward flow). Therefore, a declination in the pipe causes higher in-situ water velocity than in the corresponding horizontal case; conversely, the in-situ water velocity in an inclined pipe is lower than in a horizontal pipe. As shown in fig. 6(a) stratified flow with some mixing at the interface is observed at the pipe inclination +1º. The parallel gravity component introduces significantly higher slip between oil and water phases at +1º compared to the horizontal flow as shown in fig. 4(b). The increased relative movement between the phases results in the development of vortex motion at the boundary of the two liquids. Mutual penetrations of vortices take place in each of the phases. This interface disturbances lead to the formation of liquid drops. The dynamic and buoyant forces are acting simultaneously on the drops. The former, which tend to spread the drops throughout the pipe section are not large enough to overcome the settling tendency of the counteracting gravity force and both kinds of drops remain close to the interface. Some interfacial waves are also observed. As shown in fig. 6(b) for inclination +5º the smooth interface of horizontal flow is completely replaced by large amplitude waves together with some mixing at the interface. The oil-water interface is noticeably wavier and more irregular at +5º than at +1º. In comparing the current observations with those of previous investigators there is a good consensus on the flow behaviour at this range of flow conditions. At low mixture velocities, where the transition from stratified to dual continuous flow occurs, a general agreement exists that in upward flow dispersion appears at lower velocities than in the horizontal case [5, 9 and 10]. The experimental data of Oddie et al [9] do indicate the enhancement of mixing with increased pipe inclination. Lum et al [2] have investigated oil-water flow patterns at mixture velocity 0.7 m/s and inlet water fraction 0.32 for upwardly inclined pipes. They have reported stratified flow with smooth interface for horizontal flow whereas stratified wavy flow pattern has been observed in upwardly inclined pipes for similar conditions. The degree of waviness correspondingly increased with pipe inclination so that the flow at +10º is wavier than at +5º. Therefore, the present observations closely adhere to the data presented by previous investigators on flow patterns of oil-water flow in upwardly inclined pipes. Fig. 6(a) shows local water volume fraction, mean axial velocity and turbulence measurements at pipe inclination +1°. The local water volume fraction measurements agree well with the visual observations. The oil layer at the top moves faster than the water layer and there is a sharp transition in velocity across the interface. The observed higher slip ratio (S=3.25) causes an increase in the level of mixing at the oil-water interface. The maximum mean velocity is observed again in the oil phase and it is about 30% higher than the corresponding maximum velocity for horizontal flow. The estimated Reynolds numbers for oil and water phases are 10316 and 4208, respectively. The turbulence level in the oil phase has increased whereas it has decreased in the



-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.2 0 0.2 0.4 0.6 0.8 1


Pos

ition

[-]

-0.05 0 0.05 0.1 0.15 0.2 0.25

U-rms [m/s], V-rms [m/s]

Water vol. frac.U-meanReynolds stressU-rmsV-rms

(a)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.5 -1 -0.5 0 0.5 1


Pos

ition

[-]

-1.2 -0.8 -0.4 0 0.4 0.8



(b)

Figure 6: Mean axial velocity and turbulence measurements for upward flow: (a) +1º, (b) +5º.

water phase compared to the results for horizontal flow. The maximum U-rms value is located close to the wall in oil phase and it is about 30% of the mixture velocity and about 60% higher than the observed value for horizontal flow. The U-rms values show fewer variations over a wide range in the water phase due to



relatively constant mean shear rate. The minimum value of U-rms in the oil phase is observed at the normalized radial position 0.34 where the mean axial velocity is largest. Some fluctuations of U-rms are observed at the interface. The measured V-rms values in the oil phase are slightly larger than the values in the water phase due to higher turbulence level. The V-rms profile is relatively flat in the water phase. The maximum Reynolds stress value is located close to the pipe wall in the oil phase and it is about 140% of the corresponding maximum vale observed for horizontal flow. The measured Reynolds stress profile changes its sign at the normalized radial position 0.34 and smaller values are observed close to the interface due to stable density stratification. The local water volume fraction, mean axial velocity and turbulence measurements are presented in fig 6(b) for pipe inclination +5º. The measured interface region by gamma densitometry extends from normalized position -0.29 to 0.52. The sharp interface observed in horizontal flow and at +1º is absent and the interface is associated with transient effects due to waves and mixing. The gamma densitometry produces time-averaged measurements and it cannot capture the transient effects of the flow. The image used in fig. 6(b) shows the instantaneous flow behavior. Therefore, time averaged interface position measurements deviates from the visual observations. At pipe inclination +5º the water phase travels much slower than the oil phase. Most of water moves concurrently with the oil as it flows up the pipe. However, very close to the bottom of the pipe, gravity and frictional effects overcomes the flowing momentum of the water, resulting in the fall back of the water phase at the base of the pipe. This is the reason for the observed negative axial velocities very close to the wall in the water phase. The slip ratio further increases up to 4.08. The maximum mean axial velocity is located in the oil phase at normalized radial position 0.48 and it is about 63% higher than the corresponding value observed for horizontal flow. The Reynolds numbers for oil and water phases are 10944 and 4080, respectively. The measured U-rms and V-rms values are higher in the oil phase compared to the water phase. The measured U-rms value closest to the pipe wall in the oil phase is approximately 40% of the mixture velocity. The U-rms and V-rms values slightly increase towards the interface both in oil and water phases. The measured Reynolds stress profile changes its sign at the normalized radial position 0.48 due to zero mean axial velocity gradient. Surprisingly, large Reynolds stress values are observed around the wavy oil-water interface in contrast with the results for horizontal and upwardly inclined flow at +1º. As shown in fig. 5 and 6(a) smaller Reynolds stress values are observed close to the interface because of the damping effect of wall normal component of the turbulence fluctuations due to stable density stratification. As inclination is increased, the effect of gravitational forces tended to act strongly upon the fluid parallel to the pipe, but against flow, thus increasing the rolling motion of the phases around the interface generating interfacial waves. The rolling motion of the fluid around the interface enhances the turbulence fluctuations close to the interface. Therefore, U-rms, V-rms and Reynolds stress values increase towards the oil-water interface.



3.2.3 Downward flow Fig. 7 shows the measurements for downwardly inclined pipes. The experiments are performed at two different pipe inclinations, -1º and -5º. At pipe inclination, -1º stratified flow with clear interface is observed as shown in fig. 7(a). No sign of dispersion at the interface was observed in contrast with the results at +1º. At -5º, the mixing between the two phases is enhanced, and the flow regime can be identified as stratified flow with mixing at the interface. A clear thin water layer is observed at the bottom of the pipe. In contrast to the current work, stratified wavy flow pattern has been observed in downward flow by other investigators [7–9]. This could be due to the different ranges of velocities, pipe diameters and liquid properties used. The present measurements show an extensive level of increase in mixing between oil and water phases at -5º compared to +5º and it has bean observed previously by Lum et al [6]. Fig. 7(a) shows local water volume fraction, mean axial velocity and turbulence measurements at pipe inclination -1º. A good consensus is observed between gamma measurements and visual observations. The PIV measurements of mean axial velocity profile show the maximum velocity in the water phase as expected. The maximum mean axial velocity is located at normalized radial position -0.58 and it is very close to the oil-water interface. The fast moving water phase drags the oil layer and mean axial oil velocity increases towards the interface. The estimated Reynolds numbers for oil and water phases are 5387 and 10199, respectively. The turbulence intensity has been increased in the water phase whereas it has been decreased in the oil phase compared to the horizontal flow. The highest U-rms value is now observed in the water phase and it is about 20% of the mixture velocity. The maximum values of V-rms in oil and water phases are approximately equal. The highest magnitude of the Reynolds stress is located in the water phase and is about 70% higher than the corresponding value in the oil phase. The measured axial velocity profile is having smaller gradients around normalized radial position 0.10. It may be the reason for the observed smaller Reynolds stress values in that region. In addition, the Reynolds stresses are damped around the interface due to stable gravity stratification. Fig. 7(b) presents measurements at pipe inclination -5º. A thick interface region is visually observed due to increased mixing at the interface and it is validated by gamma measurements. The measured local water volume fraction values close to the bottom pipe wall is one and it indicates the existence of the thin continuous water layer. The thin water layer moves considerably faster than the oil phase. The measured maximum mean axial velocity in the water phase is about 25% higher than the corresponding value at -1º. The mean axial velocity in the oil phase increases towards the interface. The calculated oil and water phase Reynolds numbers are 5253 and 12056, respectively. The U-rms value measured closest to the wall in the water phase is about 26% of the mixture velocity. PIV gives smooth continuous turbulence measurements in the oil phase whereas the measurements in the thin oil layer and close to the interface show some fluctuations. The measured values of both U-rms and V-rms slightly increase towards the interface. Higher Reynolds stress values are observed close to the



-1

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tion

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Water vol. fracU-meanReynolds stressU-rmsV-rms

(a)

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(b)

Figure 7: Mean axial velocity and turbulence measurements for downward flow: (a) -1º, (b) -5º.

interface compared to the results at -1º. This may be due to the slightly wavy structure of the interface.



4 Conclusions

The effect of upward and downward inclination during oil-water pipe flow on flow regimes, water hold-up, slip ratio, velocity and turbulence profiles were investigated experimentally. The experiments were performed in a 15 m long, 56 mm diameter, inclinable steel pipe using Exxsol D60 oil and water as test fluids. The test pipe inclination was changed in the range from 5° upward to 5° downward. All experiments were performed at mixture velocity 0.25 m/s and inlet water cut 0.25. The mean axial velocity and turbulence profiles were measured using PIV, and gamma densitometry is used to measure the cross sectional distribution of oil and water phases. The flow regimes were determined by visual observations. The results were also compared with published inclined oil-water flow studies. The stratified flow regime was observed with smooth interface for horizontal flow. However, at +1° stratified flow with mixing at the interface was observed due to increased slip between oil and water phases. The interface was found to become wavier as the degree of inclination increased from the horizontal. The oil-water interface is noticeably wavier and more irregular at +5º. Stratified flow with clear interface was observed in downward flow at -1º. No sign of mixing at the interface is observed in contrast with the results at +1º. Nevertheless, significant mixing effect at the interface was observed at -5º probably because of increased water velocity and tendency of water to disperse the oil. In general, the present observations on flow regimes show a good consensus with previous investigations. The measured water hold-up in almost all instances are significantly different from the input water volume fraction. As expected, the slip ratio and water hold-up increased with pipe inclination, from -5º to +5º. In general, the increases are more prominent in the upward inclinations, while in the downward inclination the increase was more moderate. In addition, the water hold-up and slip ratio are very dependent on the pipe inclination when the pipe is nearly horizontal (-1º and +1º). The maximum mean axial velocity is observed in the oil phase for horizontal and upwardly inclined flows whereas it is located in the water phase for downwardly inclined flows. At +5º, the gravity effects overcome the flowing momentum of the water phase resulting in backflow of thin water layer very close to the bottom pipe wall. In general, higher U-rms, V-rms and Reynolds stress values are observed close to the wall due to large mean axial velocity gradients. A damping effect of Reynolds stresses is observed close to the interface due to stable density stratification for horizontal and near horizontal flows. This may be due to the suppression of turbulence fluctuations normal to the interface by gravitational forces. Surprisingly, larger Reynolds stress values are observed towards the interface at pipe inclination +5º. In this case, the rolling motion of the phases around the interface generates interfacial waves enhancing turbulence fluctuations close to the oil-water interface. Slightly higher Reynolds stress values are observed close to the interface at pipe inclination -5º.



References

[1] Rodrigues, O.M.H. & Oliemans, R.V.A., Experimental study on oil-water flow in horizontal and slightly inclined pipes, Int. J. Multiphase Flow, 32, pp. 323-343, 2006.

[2] Lum, J.Y.L., Al-Wahaibi, T. & Angeli, P., Upward and downward inclination oil-water flows, Int. J. Multiphase Flow, 32, pp. 413-435, 2006.

[3] Hasan, N.M. & Azzopardi, B.J., Imaging stratifying liquid-liquid flow by capacitance tomograpy, Flow measurement and instrumentation, 18, pp. 241-246, 2007.

[4] Fairuzov, Y.V., Transient gravity-driven countercurrent two-phase liquid-liquid flow in horizontal and inclined pipes, Int. J. Multiphase Flow, 29, pp. 1759-1769, 2003.

[5] Scott, G.M., A study of two phase liquid-liquid flow at variable inclinations, M.S. Thesis, University of Texas at Austin, USA, 1985.

[6] Lum, J.Y.L., Lovick, J. & Angeli, P., Low inclination oil-water flows, Canad. J. Chem. Eng., 82, pp. 303-315, 2004.

[7] Abduvayt, P., Manabe, R., Watanabe, T. & Arihara, N., Analysis of oil-water flow tests in horizontal, hilly-terrain and vertical pipes, In: Proc. Annual SPE Tech. Conf., Houston, Texas (SPE 90096), in CD ROM, 2004.

[8] Cox, A.L., A study of horizontal and downhill two-phase oil-water flow, M.S. Thesis, University of Texas at Austin, USA, 1985.

[9] Oddie, G., Shi, H., Durlfosky, L.J., Aziz, K., Pfeffer, B. & Holmes, J.A., Experimental study of two and three phase flow in large diameter inclined pipes, Int. J. Multiphase flow, 29, pp. 527-558, 2003.

[10] Lum, J.Y.L., Lovick, J. & Angeli, P., Low inclination oil-water flows, Canad. J. Chem. Eng., 82, pp. 303-315, 2004.



Section 4 Environmental multiphase flow

Meandering of a particle-laden rivulet

P. Vorobieff1, A. Mammoli1, J. Coonrod2, V. Putkaradze1,3

& K. Mertens4

1Department of Mechanical Engineering, The University of New Mexico,USA2Department of Civil Engineering, The University of New Mexico, USA3Department of Mathematics, Colorado State University, USA4Carolina Center for Interdisciplinary Applied Mathematics,The University of North Carolina, USA

Abstract

The behavior of a rivulet flowing down an incline is a fundamental problem inhydrodynamics, with many important applications in water resources engineering,largely because of its connection with natural meandering flows (rivers andstreams). Recent advances in the understanding of laboratory rivulet flows revealseveral important features of the rivulet behavior that are directly relevant to thisconnection with natural flows. Rivulet meandering is triggered by irregularities inthe flow rate at the origin, and amplified by re-absorption of droplets left on theinclined surface by the previous meanderings. This leads to a statistically non-trivial behavior, with the spectrum of an ensemble of rivulet deviations from itscentreline obeying a power law. Some of the statistics of the laboratory rivulets(e.g., the area swept by the rivulet) closely resemble those of real rivers (Hack’slaw). However, there are many important physical differences between rivuletsand real rivers. Among them is the fact that the flow in real rivers and streamsis often multiphase (sediment-laden). Here we present the results of laboratoryexperiments with rivulets, where the flow down an inclined, partially wetting planecarries a well-characterized particulate to model sediment flow. The most notablechange produced by the addition of the particles is the formation of a stationarymeandering pattern, which does not occur under the same conditions for the flowwithout particles.Keywords: meandering, free-surface flow, flow with particles, rivulet.



1 Introduction

Rivers and streams meander, with their paths gradually changing, leaving behindcrescent lakes, eroding the shores [1], transporting sediment, and in the processchanging and defining the shape of Earth’s surface (Fig. 1a). Some of the mostfamous natural monuments (such as the Grand Canyon) owe their existence toriver meandering [2]. Flow of ancient rivers is likely responsible for some veryprominent features on the surface of Mars ([3], also see Fig. 1b). As recentlyrevealed by the Cassini/Huygens probe, rivers also flow and meander on Titan [4],the satellite of Saturn, although it is liquid methane, not water, that flows therebetween the shores of ice at a chilly 93 Kelvin (Fig. 1c).

Meandering is affected by many complex factors, including turbulence in thewater, erosion of the soil on the bottom, unevenness of the soil properties, andvariations in flow rates due to the seasons. However, there are some simplestatistical properties that apply to all rivers and streams. The most notable of themis Hack’s law, which defines a relationship between the length of a stream or riverL and the size of its drainage basin A (the area of land where water from rainand snow melt drains downhill into the river). This law was discovered by anAmerican geophysicist J.T. Hack in 1957 [5]. Hack studied the streams of theShenandoah Valley and adjacent mountains in Virginia, and obtained a power-lawformulaL = 1.4Ah, where h (Hack’s exponent) is a constant value which turns outto be the same or almost the same for Shenandoah Valley and for several thousandsof other river basins for which measurements were made. The most common valueof h is about 0.57.

It is notoriously hard to answer exactly which of the multitude of the possiblyimportant and highly intertwined geological and meteorological factors contributesmost to the river meandering. As in many other cases where there is an interestingnatural phenomenon which is both too complicated and too big to fit in a labon a modest budget, physicists came up with an experimental model of gravity-driven meandering flow – a rivulet on an inclined plane. This model can be easilyreproduced in a kitchen sink by placing a tilted cookie sheet under a slightlyopened faucet. In this kitchen-sink experiment, the tiny rivulet will meander,producing a curious pattern not unlike a miniature river, although in a properlymaintained kitchen, there should be no precipitation, erosion, or sedimentationinterfering with the flow. Similar observations on somewhat larger sheets inlaboratory led to the conclusion that, since the rivulet always meanders, there mustbe some inherent instability that causes it to behave this way. It was also commonlythought that meandering of a rivulet has some kind of a most frequently occurringspatial scale (called characteristic wavelength λ) associated with it.

However, the basic premise of many rivulet studies – that the rivulet will alwaysmeander – was recently demonstrated to be incorrect, with perfectly straight, stablerivulets (Fig. 1d) obtained experimentally [6,7]. The key element in stabilizing therivulets is the flow rate. It has to be maintained at a constant value, free of anyfluctuations. A discharge through a small hole in the bottom of a very tall reservoirwould have such fluctuation-free quality; a discharge from a kitchen faucet would



Figure 1: Natural (a-c) and laboratory (d-e) flows, view from above, flow directionis from top to bottom where known. a, Flow of Machadinho river, Brazil.The image extent is about 12 km. Source: Google Earth/TerraMetrics.b, Mars Express image of Nanedi Valles on Mars captured on October3, 2004. Source: European Space Agency (ESA)/DLR/FU Berlin (G.Neukum). The image extent is about 100 km. c, Cassini/Huygens imageof a river channel on Titan. The image extent is about 4 km. Source:ESA/NASA/JPL/University of Arizona. d, Non-meandering flow (con-stant flow rate) of a 50%-50% water-glycerine mix on acrylic substrate.e, Meandering flow under the same conditions, with flow rate fluc-tuations induced by a pulsed electromagnetically operated valve. Theoverlay shows the centerline (gray dashed line), the coordinate system,and the velocity components. The image extent in d and e is 2.4 m.

not. In most natural and laboratory flows, the flow rate will fluctuate, and thus thestream will meander. Remove the fluctuations though, and you get a stable rivulet,sometimes forming a pretty braiding pattern explained in our earlier work witha simple theoretical model [7]. This study, however, did not attempt to provide asimilar explanation for the meandering flow regime that would emerge in the casethe flow rate was not constant.



2 Meandering rivulet studies

In our subsequent experiments with a specially built inclined plane apparatus, wetried several ways of perturbing the flow. We found that the best way to producemeandering was to add an electronically controlled solenoid valve to the dischargetube feeding fluid from the tall reservoir to the incline. With this valve squeezingthe tube every few seconds, the flow would start to meander (Fig. 1e). With thevalve turned off, it would gradually straighten out.

We took several hundred pictures of the meandering rivulet and started ana-lyzing their statistics. We assembled a plot showing how often each meanderingfrequency f , and corresponding length scale l = 1/f , show up in the picturedatabase. This plot (a power spectrum) revealed that for a wavenumber k = 2π/l,the corresponding value of the power spectrum S(k) was proportional to k−5/2.This relationship (a power law) holds in a range from the characteristic streamwidth (about 5 mm) to the largest scale we can measure (2.4 m, the extent of ourinclined plane). Now, if a characteristic wavelength λ existed, it would correspondto a characteristic wavenumber kmax = 2π/λ, where the spectrum would peak.But the spectra we measure show no such peaks, in contradiction with what earlierworks had suggested.

Power-law statistics are Nature’s way of telling us that something simple andprofound is happening. Thus we tried to explain the behavior of the stream in ourexperiment with a model that only takes into consideration the most basic featuresof our flow. We averaged out velocity variations in the stream cross-section. Wereplaced the actual (variable) value of the contact angle between the fluid, the air,and the surface, with another average [8]. Finally, we radically simplified one veryimportant feature of the stream. A real meandering stream on a partially wettingplane sheds droplets as it changes its course, like a meandering river leavingcrescent lakes in its old path (of course, in the latter case, the process is much morecomplex). These droplets may get reabsorbed into the rivulet during subsequentmeanderings. The physics of each such event are rather complex, but, insteadof trying to model it exactly, we account for such stream-droplet interactions byadding a random forcing term to our equations. This term has to be statisticallyfaithful to what we see in experiment, but it is much simpler to deal with. Wederived two equations (for the two coordinates in the plane) describing the fluidmoving due to a combination of surface tension, gravity, shear in the flow nearthe surface (lubrication approximation), viscous dissipation inside the fluid, andrandom forcing due to droplets. We added the law of conservation of mass (inthe form of a continuity equation). Thus we obtained a simple system that couldbe quickly solved numerically using an ordinary personal computer. This modelsystem can produce rivulet shapes in the plane that are very similar to those wesaw in experiment, and with the same statistical properties.

We repeated our experiments on several surfaces made of different materials,with different static contact angles. The higher the contact angle, the easier it isfor a droplet to roll off the surface. Fewer droplets mean weaker random forcing,and that reflects in a slower growth of the meandering amplitude, again in nice



agreement with our theory [9]. However, on each surface we studied, the scaling ofthe power spectrum was the same: S(k) ∼ k−5/2. Working with different surfaces,we found another curious thing. A real river has a drainage basin. Our rivuletalso has a drainage basin of a sort – it can encounter droplets left by the previousmeanderings, and these droplets are typically confined between the rivulet path andthe rivulet’s initial course before we began perturbing the flow (the straight line).What turned out is that the area of this “drainage basin” obeys the same scaling lawas the drainage basin of a real river – Hack’s law! Moreover, our theory predictshow the value of Hack’s law exponent is related to the scaling of the meanderingpower spectrum. If that scaling is -5/2, then Hack’s law exponent should be 4/7 [8].Surprisingly, 4/7 approximately equals 0.57, the same value we see in laboratoryrivulets and in real rivers.

3 Observations of stationary meandering

One of the notable differences between our work and some of the earlier experi-ments lay in the rarity of observed stream pinning events. Pinning (or stationarymeandering) occurs when a meandering flow pattern ceases to change with time.While it was reported to occur fairly commonly by other researchers [10], in ourexperiments pinning was rare and could always be attributed to the inclined planebeing contaminated with dust or dried-up residue from earlier experiments. Withthe incline wiped clean and the experiment restarted, pinning did not re-occur.Meandering would persist for as long as the flow rate was perturbed by the solenoidvalve. If the flow rate perturbation ceased, the flow would gradually transition to astraight rivulet stationary state.

Unlike our laboratory rivulet, natural flows are always multi-phase to someextent, carrying sediments, organic matter (e.g., dead skunks), and debris. As theflow rate changes in an alluvial system, scour or deposition will occur, furtheraffecting meandering patterns [11]. Sediment transport was believed to be nec-essary for flow meanders [12], however, experiments without sediment transportalso manifest meandering [8, 10, 13]. While the notion of meandering as aninherent global flow instability that arose in the 1980s [13] has been experimentallydisproven [6, 7], transverse shear stress distribution along the wetted perimeterof the meandering stream [14] likely still plays a role in meandering. It is alsonotable that, in the case of natural flows, their course can also persist for a longduration. This persistence can be aided by erosion of the riverbed. On the otherhand, strong (catastrophic) changes in the natural flow rate can lead to changes inthe established course of the stream. Thus we attempted to bring our experimentone step closer to these natural flows by adding a particulate phase to the fluid.In the experiments described here, the fluid was Albuquerque tap water (withsmall naturally occurring salinization), and we added a particulate phase to it. Thelatter was comprised of small, nearly monodisperse (500 ± 75 μm), and almostneutrally buoyant (specific gravity ∼ 1.05) polystyrene microspheres, in fractionsthat were varied from 0.5 to 5% of the total volume of the flow. The water-particlemixture was driven into the tall reservoir by a peristaltic pump from a reservoir



that collected the runoff from the bottom of the inclined plane. On the time scalecharacterizing the experiment, settling of the particles in the top reservoir wasnegligible.

The resulting behavior of the multiphase meandering flow (Fig. 2) turned out tobe quite interesting. There was no qualitative difference between the pure waterflow and the flow of the water with particles when the flow rate was periodicallydisrupted by the solenoid valve (flow reduction by 25% for 0.1 s every 2 s) – inboth cases, the flow continued to meander. The difference emerged, however, whenthe valve was turned off. The pure water flow in that case eventually reverted toa straight rivulet, with relaxation times between 40 minutes and 2.5 hours. Underthe same conditions, the flow with particles simply retained its course for as longas the experiment was run (up to 4 hours).

4 Discussion and future work

Consider a stationary bend in the rivulet. It doesn’t have to be stationary in thestrict sense, just stable enough to persist on the time scale of interest (severalminutes). For this bend, the hydrodynamic centrifugal force is balanced by thepressure gradient. At least, this should be the case for the flow of a pure fluid.However, if a heavy (non-neutrally buoyant) particle is present in the flow, it willbe subject to the centrifugal force as it passes the bend.

This particle can cause pinning if it gets trapped near the edge of the rivulet asshown in Fig. 3, A-A. The unbalanced force acting upon a particle of radius a is

F = ρa3U2

R,

whereρ is the density difference between the particle and the fluid,U is the flowvelocity and R is the radius of curvature of the bend. Drift velocity towards theedge can be estimated as

v = F

6πρaν= ρ

ρ

a2

6πν

U2

R,

where the denominator represents Stokes drag on the particle. Sedimentation nearthe edge happens if vτ ≤ d , where τ = L/U is the time it takes for theflow to travel the length of the bend, and d is the characteristic cross-streamdistance. Assuming L πR, which holds exactly if the bend is a semi-circle,the probability p of sedimentation for a single particle traversing the bend is

p = ρ

ρ

a2U

6νd. (1)

For our flow, the characteristic values are as follows. U ∼ 1 m/s, d ∼ 0.01m,ρ/ρ 0.05, ν ∼ 10−6 m2/s, a 5 × 10−4 m. Thus p ∼ 2 × 10−3. Once aparticle passes through the bend without being stuck it will move to next bend and



Figure 2: Snapshots of meandering multiphase flow. The flow direction is fromtop to bottom, the physical extent of the imaged area in this directionis 2.4 m. The images are separated with an 8-second interval. In thefirst three images (top row, a-c), the flow is subjected to a periodicperturbation in the flow rate, causing continuous meandering. Betweenthe third (c) and fourth (d) image, the flow rate perturbations cease,producing a stationary meandering pattern (bottom row, d-f). In theimages shown, the flow is a mixture of 99% water with 1% 500-micron polystyrene microspheres by volume, and the flow rate is about100 cm3/s.



Figure 3: Left – view of a rivulet flow carrying particles through a bend. L andR are the characteristic length and radius of the bend. A-A (bottomcenter) shows a cross-section view of the rivulet, with d being thecharacteristic cross-sectional size. Top center – photo close-up of astationary meandering rivulet (note that the contact angles on both sidesare the same). Top right – photo close-up of a transient (non-stationary)meandering rivulet with different advancing and receding contact angles.

so forth. An interesting feature of equation (1) is that the probability of being stuckat the bend does not depend on the bend’s radius.

Let us now consider a more realistic situation when many particles are movingthrough the bend at the same time. Suppose the density of particles per unit volumeis α. Since the cross-section of the stream is of the order d2, there particles aremoving through the fluid bend with the rate

n αd2U .

Thus, after the time t ,N = αd2Ut particles will have passed through the bend. Theprobability of a single particle passing through the bend is 1−p. The probability ofN independent particles passing through (assuming that the particles are spread farenough for each sticking event to be statistically independent) is (1−p)N e−Np.Thus the probability of at least one ofN particles sticking ps as they go through is

ps = 1 − e−Np = 1 − e−αd2Utp = 1 − et/Ts . (2)

Again, interestingly enough, the probability of pinning is characterized by theflow parameters, but is not dependent on the bend’s characteristics. We can thusdefine the time scale Ts for particle pinning in a single bend as

Tp ∼ 1

αd2Up. (3)

After the time t Tp, the meandering trajectory of the rivulet will be frozen intime with probability being almost 1. More precisely, if we define Tp as the time



after which the probability of meandering being frozen is, say, 90%, then we get amore precise version of formula (3):

Tp = log 10

αd2Up. (4)

Consider the typical particle concentration producing pinning in our experi-ments – 5% by volume. This means that 1 cm3 contains about α = 95 particleswith the radius of 500 microns. Assuming a characteristic cross-sectional streamarea d2 ∼ 0.1 cm2, the volume of the flow traversing this area in one second is10 cm3, about a thousand particles will be carried through each cross-section persecond. Then, (4) gives Ts ∼ 1.2 sec, which explains a near-instantaneous pinningof the stream. While equation (4) is of course just a rough estimate, it is usefulin evaluating the order of magnitude for the pinning time, which is clearly small.More detailed evaluation of the pinning time can be obtained by constructing amore detailed model of particle motion very close to the contact line, improvingon a rather simple-minded version of the Stokes drift we have utilized here.

Note that pinning does not occur when the flow rate is subjected to fluctuationsas described in the previous part. In the immediate future, we will study thesensitivity of a pinned (stationary) rivulet to flow-rate perturbations varied induration, frequency, and intensity. While our simple model deliberately doesnot include many aspects of real streams (such as an erodible riverbed), therelationship between flow-rate fluctuations and the transition between stationaryand non-stationary meandering can provide some insights into one importantaspect of the behavior of natural rivers. Most of these rivers change their coursesomewhat infrequently, although their flow rates do fluctuate quite a bit. However,an unusually strong fluctuation (for example, due to a particularly intense rainfall)still can destabilize the course of a real river. With global climate change affectingthe precipitation patterns throughout the world, any quantitative insights into whatamount of fluctuation in the flow rate could cause a catastrophic change in thecourse of a natural stream would be very helpful.

Acknowledgements

This research is supported by the National Science Foundation New MexicoEPSCoR (Experimental Program to Stimulate Competitive Research) award EPS-0447691. This work was also partially supported by the US Department of Energy(DOE). The financial support does not constitute an endorsement by the DOE ofthe views expressed in this paper

References

[1] Langbein, W. & Leopold, L., River meanders – theory of minimum variance.U.S. Geological Survey Professional Paper 422-H, 1966.



[2] Stevens, L., Schmidt, J., Ayers, T. & Brown, B., Flow regulation, geomor-phology, and Colorado River marsh development in the Grand Canyon,Arizona. Ecological Applications, 5(4), pp. 1025–1039, 1995.

[3] Malin, M. & Edgett, K., Evidence for persistent flow and aqueous sedimen-tation on early Mars. Science, 302, pp. 1931–1934, 2003.

[4] C.C. Porco et al., Imaging of Titan from the Cassini spacecraft. Nature, 434,pp. 159–168, 2005.

[5] Hack, J., Studies of longitudinal stream profiles in Virginia and Maryland.U.S. Geological Survey Professional Paper 294-B, 1957.

[6] Mertens, K., Putkaradze, V. & Vorobieff, P., Braiding patterns on an inclinedplane. Nature, 430, p. 165, 2004.

[7] Mertens, K., Putkaradze, V. & Vorobieff, P., Morphology of a stream flowingdown an inclined plane. Part 1. Braiding. Journal of Fluid Mechanics, 531,pp. 49–58, 2005.

[8] Birinr, B., Mertens, K., Putkaradze, V. & Vorobieff, P., Morphology of astream flowing down an inclined plane. Part 2. Meandering. Journal of FluidMechanics, 607, pp. 401–411, 2008.

[9] Birinr, B., Mertens, K., Putkaradze, V. & Vorobieff, P., Meandering fluidstreams in the presence of flow-rate fluctuations. Physical Review Letters,101, p. 114501, 2008.

[10] Le Grand-Piteira, N., Daerr, A. & Limat, L., Meandering rivulets on a plane:A simple balance between inertia and capillarity. Physical Review Letters,96, p. 254503, 2006.

[11] Engelund, F. & Skovgaard, O., On the origin of meandering and braiding inalluvual streams. Journal of Fluid Mechanics, 57, pp. 289–302, 1973.

[12] Parker, G., On the cause and characteristic scales of meandering and braidingin rivers. Journal of Fluid Mechanics, 76, pp. 457–480, 1975.

[13] Nakagawa, T. & Hotsuta, M., Note on boundary effects on stream meander-ing. Sedimentology, 31, pp. 119–122, 1984.

[14] Nakagawa, T., Boundary effects on stream meandering and river morphology.Sedimentology, 30, pp. 117–127, 1983.



Experimental study on the rheological behaviour of debris flow material in the Campania region

A. Scotto di Santolo, A. M. Pellegrino & A. Evangelista Department of Hydraulic, Geotechnical and Environmental Engineering, University of Naples “Federico II”, Italy

Abstract

The rheological behaviour of the natural material collected in the source area of three debris flows occurring in the mountainsides of the North-western Campania region (southern Italy) has been investigated. The tests have been carried out at different solid volumetric concentration Cv with a standard rheometer equipped with two geometric configurations to avoid disturbing effects. All materials behave like a Non-Newtonian fluid with a threshold shear stress τy (yield stress) that increases with solid volumetric concentration. The experimental data have been fitted with standard model generally used for fluids. A simple relation between Cv and τy has been obtained. Keywords: rheology, debris flow, pyroclastic soils, laboratory activity, fluids model, yield stress.

1 Introduction

Debris flows represent serious hazards in slopes of the North-western Campania region (southern Italy). Therefore, the evaluation of constitutive laws for the material involved represents a key requirement for the hazard mitigation. Traditionally, debris flows have been regarded as homogeneous fluids and flow behaviour has been considered to be controlled by the properties of the ‘matrix’ (a mixture of fine sediment and water in which coarse particles are dispersed) [1, 2]. Existing physical theories to describe flow and depositional process of debris flow are mainly divided into theories which are based on the treatment of material as one phase (rheological approaches) [3–6] or as two or more phases



(Coulomb mixture approaches) [7, 8]. The Coulomb mixture approach specifies distinct constitutive equation for the solid phase, the liquid phase and the phase interaction force. Conversely, using a rheological approach, the bulk mixture behaviour can be characterized by a limited number of parameters, relating shear stress and viscosity to shear rate. The present study focuses the results from rheological point of view.

2 Debris flow rheology

Natural debris flows are often classified on the relative concentration of fine and coarse sediment that is used to characterize the main flow regime behaviour [9, 10]. Above a critical solid concentration Cv, a particle-friction-collision regime dominates the flow process. Models based on the work of Bagnold [11] are used to describe the flow behaviour of these mixtures. If the solid concentration is less than the critical one, the flow behaves like Non-Newtonian fluids. They are also called time independent fluids and are subdivided into several groups (fig. 1). Generally, the debris flows mixtures behave like viscoplastic fluids, as indicated in curve (4), fig. 1. It is considered that, for such a fluid, exists an abrupt change in debris flows behaviour around a given shear stress value, the yield stress, which needs to be overcome before flow take place. The viscoplastic character of debris flow mixtures has often been reported in literature [4–6, 12–17]. Phenomenological laws like the Bingham generalized model are usually used to describe the rheological behaviour of these mixture [6, 10, 18], which is written as:

ny kγττ += (1)

In eqn. (1), τy is the yield stress, γ is the shear rate, k is the consistent coefficient and n the pseudoplastic index. When the index n is equal to the unity, the eqn. (1) becomes the Bingham model and the coefficient k becomes the Bingham viscosity ηB (curve (5) in fig. 1).

(1) Newtonian fluids behaviour;(2) Shear thinning behaviour (Pseudoplastic fluids);(3) Shear thickening behaviour (Dilatant fluids);(4) Shear thinning yield stress fluids (viscoplastic fluids);(5) Bingham plastic fluids.

a) b)

(1) Newtonian fluids behaviour;(2) Shear thinning behaviour (Pseudoplastic fluids);(3) Shear thickening behaviour (Dilatant fluids);(4) Shear thinning yield stress fluids (viscoplastic fluids);(5) Bingham plastic fluids.

a) b)

Figure 1: a) Flow curves; b) Viscosity curves.



ΩΩ

ΩΩ

Figure 2: The rotational rheometer AR 2000ex (TA Instruments) and the two rheometrical system used: parallel plates and vane rotor system.

3 Experimental setup and procedures

3.1 Rheometer apparatus

In order to ensure the validity of rheometrical measurements and to reduce the risk of misinterpretation, the rotational rheometer AR 2000ex (TA Instruments) equipped with two different geometry systems (parallel plates – PP and vane rotor – VR) has been utilized (fig. 2). The parallel plates are composed of a lower stationary steel plate and an upper rotational one with a diameter of 40 mm. The distance between two plates (the gap H) is larger than ten times maximum particles diameter, dmax [19]. For a gap equal to 1 mm, dmax must be smaller than 0.1 mm. The shear rate and the shear stress are evaluated with following eqns:

( )H

RRR⋅

==Ωγγ

(2)

( )R

3R

3R ddT

R2R2T3

γπγ

πγτ ⋅

⋅⋅+

⋅⋅⋅

= (3)

where r is the distance from the central axis, R is the upper plates radius, H is the gap, Ω is the rotation velocity (rad/s) and T is the torque. The vane rotor geometry consists of four thin blades arranged at equal angles around a small cylindrical shaft: the blades radius is 14 mm and the blades height is 42 mm. It is immersed in the sample (approximately equal to 27 ml) contained in a cylindrical cup with 15 mm in radius. The rotor is rotated around its axis at a given rotational speed Ω and the torque T is measured by the transducer of the rheometer. During the test the material is trapped in the blades and the shear is achieved around a fictitious cylinder within the mixture [10]. The shear stress and shear rate are:

( )12

1

RRR−⋅

=Ω

γ (4)

LR2T

21 ⋅⋅⋅

=π

τ (5)



where R1 and R2 are, respectively, the blades radius and the cup radius, L is the material depth. The eqns. (4) and (5) are usually applied when the ratio R1 / R2 is close to the unity.

3.2 Materials

The materials tested have been collected from the source area of three debris flows occurred in Campania region (southern Italy), fig. 3a). Material A has been sampled in Nocera, Salerno (March 2005). Material B in Monteforte Irpino, Avellino (May 1998). Material C in Astroni, Naples (December 2005). The soil type ,involved in thickness of about a meter, regards the most recent pyroclastic deposits deriving from the volcanic activity of mount Somma/Vesuvius for materials A and B and from the volcanic activity of the Phlegrean Fields for material C. The grain size distributions of the samples are reported in fig. 3b). Soil A and soil B are sandy silt with a small clay fraction and soil C is gravely silty sand. These materials are well documented in the literature [20–22]. The substratum underlying the soil is of the same volcanic nature for material C and of a carbonatic nature for materials A and B. Mean physical properties are reported in table 2 (GS is the specific gravity of soil particles, γd and γ are the dry and total weight of soil per unit volume respectively, n is the porosity and Sr is the degree of saturation).

Table 1: Main physical properties of the tested debris flow materials.

Debris flow Substratum Material GS γd

(Kn/m3) γ

(Kn/m3) n Sr

Nocera Carbonatic A 2.62 9.08 11..35 0.66 0.35 Monteforte Irpino Carbonatic B 2.57 7.11 12.11 0.71 0.71

Astroni Pyroclastic C 2.54 8.99 9.84 0.67 0.24

3.3 Laboratory procedures

The analyzed debris flow mixtures have been tested in rate-controlled mode at constant temperature (20 ± 0.5°C). The flow curves have been determined during one experiment by applying successive shear rate level, ranging from 0.014 to 1400 s-1. In order to certify the reproducibility of the test, each one has been repeated at least three times and the averaged values of the experimental results have been considered. Due to the geometry dimension of the used rheometer, only the flow curves of material samples with grain sizes smaller than 0.1 mm have been derived. The experiments have been carried out with mixtures of different water content. The solid volumetric concentration Cv, i.e. the ratio of the amount of solids to the total mixture, has been considered. The total solid volumetric concentration Cv is defined as:

ws

sv VV

VC

+= (6)

where Vw and Vs are, respectively, the volume of water and solid in the sample. In order to consider a significant range of the sediment concentration for the material tested, mixtures with Cv changing from 0.20 to 0.40 have been prepared.



Figure 3: a) Location of the studied debris flows and distribution of the main pyroclastic deposits; b) Grain size distribution of the tested debris flow materials.

The experimental program performed is shown in Table 2.

Table 2: Experimental programme.

# Material Geometry system Cv (%) Test 1 0.20 A – 0.20 – PP 2 0.30 A – 0.30 – PP 3

PP 0.40 A – 0.40 – PP

4

A

VR 0.20 A – 0.20 – VR 5 0.20 B – 0.20 – PP 6 PP 0.30 B – 0.30 – PP 7 0.20 B – 0.20 – VR 8

B VR 0.30 B – 0.30 – VR

9 0.20 C– 0.20 – VR 10 0.30 C– 0.30 – VR 11

C VR 0.40 C– 0.40 – VR

4 Experimental results

4.1 Preliminary evaluations

Disturbing effects due to material properties and rheometrical geometry features could leading to erroneous data interpretation [6, 10, 13, 17]. Depending on the geometry apparatus, shear conditions (related to samples volume) and disturbing effects (related to geometry features and fluid types) change in a different way.

Volcanic Systems

Calcareous bedrock covered by pyroclastic materialsBedrock covered by ashes pyroclastic materials

Bedrock covered by ashes and pumices pyroclastic materials

Bedrock on flysch

Somma - Vesuvio

Island of Ischia

Phlegrean Fields

Roccamonfina

Piedmont Plans

Plans on pyroclastic materials

Plans on IgnimbriteVolcanic Systems



Bedrock on flysch

Somma - Vesuvio

Island of Ischia

Phlegrean Fields

Roccamonfina

Piedmont Plans





Bedrock on flysch

Somma - Vesuvio

Island of Ischia

Phlegrean Fields

Roccamonfina

Piedmont Plans





Bedrock on flysch

Somma - Vesuvio

Island of Ischia

Phlegrean Fields

Roccamonfina

Piedmont Plans


Plans on Ignimbrite

Gulf of Naples

Gulf of Salerno

Volcanic Systems

Calcareous bedrock covered by pyroclastic materials

Bedrock covered by ashes pyroclastic materials Bedrock covered by ashes and pumices pyroclastic materials

Bedrock on flysch

Somma - Vesuvio Island of Ischia Phlegrean Fields Roccamonfina

Piedmont Plans


Plans on Ignimbrite

B

A C

Gulf of Naples

Gulf of Salerno

Volcanic Systems



Bedrock on flysch


Piedmont Plans


Plans on Ignimbrite

B

A C

Gulf of Naples

Gulf of Salerno

Volcanic Systems



Bedrock on flysch


Piedmont Plans


Plans on Ignimbrite

Gulf of Naples

Gulf of Salerno

Volcanic Systems



Bedrock on flysch


Piedmont Plans


Plans on Ignimbrite

B

A

C

0

10

20

30

40

50

60

70

80

90

100

0,000 0,001 0,010 0,100 1,000 10,000 100,000

Particle diameter, d (mm)

Perc

ent f

iner

(%)

Material AMaterial BMaterial C

Clay Silt Sand Gravel

a)

b)



Data obtained by different facilities should be in agreement only if the material tested has been homogeneously sheared, in each geometry, as predicted by theory [6, 10]. In order to understand how the geometry system influences the experimental results, preliminary tests on a typical Newtonian fluid, liquid paraffin on sale, have been carried out with facilities, parallel plates and vane rotor. In order to certify the reproducibility of the test, each test has been repeated three times and the averaged values of the experimental results have been considered. Fig. 4 reports the flow curves of the liquid paraffin analyzed obtained using the two mentioned geometries. A quantitatively difference between the experimental results obtained using the parallel plates system and the experimental data obtained using the vane rotor system has always been noted. Comparing the theoretical viscosity (0.01 Pa·s) and the measured viscosity values, an overestimate of the rheological parameters has been observed when the parallel plates geometry has been used.

Figure 4: Liquid paraffin on sale. Comparison of parallel plates results and vane rotor results.

Figure 5: Material A, Cv = 0.20. Comparison between the experimental results obtained with the parallel plates and the vane rotor system.

0,001

0,01

0,1

1

10

0 20 40 60 80 100Shear rate (1/s)

She

ar s

tress

(Pa)

ParallePlatesVane Rotor

0,001

0,01

0,1

1

0 20 40 60 80 100Shear rate (1/s)

Vis

cosi

ty (P

a s)

ParallePlatesVane Rotor

0,001

0,01

0,1

1

10

100

0 200 400 600 800

Shea

r stre

ss (P

a)

Shear rate (1/s)

Vane rotorParallel plates

Critical shear rate

Yield stress

0,01

0,1

1

10

0 20 40 60 80 100

She

ar s

tress

(Pa)

Shear rate (1/s)

Vane rotorParallel plates

Minimum



Considering the Newtonian behaviour of the liquid paraffin ( γητ ⋅= with η =cost), a scale factor to go from the results obtained with the parallel plates system to the results obtained with the vane rotor system has been evaluated. For a Non-Newtonian fluid like the mixtures tested, quantitative and qualitative differences have been observed. In fig. 5 for material A with Cv equal to 0.20, the flow curves obtained with the two geometric facilities have been reported. At small shear rate, a minimum in the flow curves attained from the tests carried out using the vane rotor system have been noted: the shear stress decrease with shear rate, followed by a subsequent increase at larger shear rate values. Such a minimum for concentrated suspensions has been reported by several authors and only the increasing part of the flow curves has been considered [6, 10, 15, 17, 23–25]. The vane rotor response is higher than the parallel plates response after a shear rate value equal to 50 s-1, which called the critical shear rate (fig. 5). The critical shear rate is about constant for the materials tested and the solid volumetric concentrations considered. After the critical shear rate, the mixtures analyzed behave like a Newtonian fluid according to fig. 4. A scale factor similar to that used for the experimental data of the liquid paraffin has been considered and the flow curves obtained with the parallel plates system have been scaled by this factor. Probably, some disturbing effects have been occurred during the tests like changing of the material free surface, edge/crack effects, heterogeneities in particle distribution (particle settling and migration due to particle inertia and secondary flow) and the phenomenon of wall slip. The occurrence of some disturbing effects has been evaluated. Sedimentation certainly occurred because the difference ∆ρ between the densities of the disperse phase ρd and the continuous phase ρc is greater than 103 [26], as shown following:

33

33

cd mkg10

mkg106.110002600 >⋅=−=−= ρρρ∆ (7)

Particle inertia occurred because the particle Reynolds number ReP is greater than 10−1 [27], as shown following:

113

2

c

2c

p 10105.1210

05.0501000aRe −−

−>⋅=

⋅⋅=

⋅⋅=

ηγρ

(8)

It is possible to estimate the influence of settling through the calculation of the experimental time texp required for a single sphere to migrate over a length l (l is equal to H, the gap height) as shown following:

sec115.005.081.91600

11029

agl

29t 2

3

2c

exp =⋅⋅

⋅⋅=

⋅⋅

⋅⋅=

−

ρ∆η

(9)

where ∆ρ = |ρd−ρc|, g is the acceleration of gravity and a is the radius of the particle with maximum size in the mixtures [27]. In the following the flow curves for shear rate more than the critical shear rate have been shown.



4.2 Experimental results and model fitting

In fig. 6 the experimental data and the theoretical flow curves in a semi logarithmic scale diagram for each material analyzed, at different solid volumetric concentration Cv, have been reported: the solid line represents the theoretical model function and the points indicate the experimental data. The best fitting model of the experimental data is the theoretical model of Bingham. The model fitting parameters have been reported in Table 3. First of all, it is noted that, after the critical shear rate, all the debris flow mixtures investigated behave like a shear-thinning yield stress fluids: shear stress increases with the increase of shear rate and the viscosity decreases. The influence of the solid volumetric concentration Cv on the rheological parameter of debris flow material mixtures tested has been evaluated. Proportionally higher values of shear stress and viscosity with increasing of the solid volumetric concentration have been noted. Moreover, on equal solid volumetric concentration, the yield stress and the viscosity of material B are higher than the yield stress and the viscosity of materials A and C. The rheological parameters of material C are the lowest according to the volcanic particle nature.

Table 3: Bingham model parameters.

Material Cv (%)

τy (Pa)

ηB (Pa · s) R2

0.20 0.732 0.0037 0.998 0.30 0.927 0.0059 0.999 A 0.40 1.337 0.0233 0.999 0.20 3.792 0.0145 0.991 B 0.30 16.88 0.0333 0.969 0.20 0.047 0.0033 0.999 0.30 0.181 0.0042 0.998 C 0.40 0.607 0.0074 0.999

In fig. 7 the yield stress τy (obtained by the results of the vane rotor) versus the solid volumetric concentration Cv has been reported: the yield stress τy exponentially increase with the increase of solid volumetric concentration Cv. According to some previous study [4–6, 10, 12, 13, 17], the following relation could be used:

vCy e ⋅⋅= βατ (10)

where α and β are fitting parameters. Their values have been reported in table 4.

Table 4: Fitting parameters α and β.

Material α β A 0,2464 0,1215 B 0,1333 0,1909 C 0,0089 0,1066



Figure 6: Experimental data and theoretical flow curves at different

volumetric concentration Cv : a) material A; b) material B; c) material C.

0,1

1

10

100

50 200 350 500 650 800Shear rate (1/s)

She

ar s

tress

(Pa)

Cv 0.20Cv 0.30Cv 0.40

0,1

1

10

100

50 200 350 500 650 800Shear rate (1/s)

She

ar s

tress

(Pa)

Cv 0.20Cv 0.30

0,1

1

10

100

50 200 350 500 650 800

Shear rate (1/s)

Shea

r stre

ss (P

a)

Cv 0.20Cv 0.30Cv 0.40

a)

b)

c)



Figure 7: Yield stress τy versus solid volumetric concentration Cv.

5 Conclusion

In order to evaluate the rheological behaviour of natural debris flow material, laboratory tests involving soils taken from the source area of three debris flows occurred in Campania region (southern Italy) have been carried out. Mixtures with varying concentration of fine sediment with maximum diameter less than 0.1 mm and distilled water have been prepared. The debris flow mixtures have been investigated in a standard rheometer with two different geometries, the parallel plates system and the vane rotor system. The vane geometry seems to be an appropriate rheometrical tool for quantitative evaluation of the rheological behaviour of debris flow materials (but using the parallel plates system is also possible to give some simple qualitative ideas about debris flow mixtures). The comparison between two geometry configurations allows checking the range of shear rate where there are not disturbing effects and misleading evaluations. In this range of shear rate all the debris flow mixtures tested behave like a shear thinning yield stress fluid: shear stress increases with the increase of shear rate and viscosity decreases when increasing shear rate. These experimental data have been fitted with the theoretical Bingham model. The increase in solid volumetric concentration produces an increase of shear stress and viscosity. The yield stress dependent exponentially on the solid volumetric concentration. The experimental results presented have to be preliminary, because they have been carried out only on particles less than 0,1 mm. For determining the rheological behaviour of gravel-sand mixtures, new apparatus have been set up.

References

[1] Costa, J.E. & Williams, G.P., Debris flow dynamics (videotape), US Geological Survey Open file 84 – 606, 22min, 1984. www. pubs.usgs.gov/of/1984/ofr84-606/

0,01

0,1

1

10

100

10 100Solid volumetric concentration (%)

Yiel

d st

ress

(Pa)

Material AMaterial BMaterial C

20 30 40



[2] Johnson, A.M., Debris flow. Topics in Slope Instability, eds. D. Brunsden and D.B. Prior, Wiley, New York, pp. 257-361, 1984.

[3] Coussot, P. & Piau, J.M., On the behaviour of fine mud suspensions. Rheologica Acta, 33, pp. 175-184, 1994.

[4] O’Brien, J.S. & Julien, P.Y., Physical properties and mechanics of hyperconcentrated sediment flow. Proc. of the Specialty Conference on Delineation of landslide, flash flood and debris flow hazard in Utah, Utah Water Research Laboratory, General Series, UWRL/G-85/03, pp. 260-278., 1984.

[5] Phillips, C.J. & Davies, T.R.H., Determining rheological parameters of debris flow material. Geomorphology, 4, pp. 573-587, 1991.

[6] Major, J.J. & Pierson, T.C., Debris flow rheology: experimental analysis of fine – grained slurries. Water Resources Research, 28 (3), pp. 841-857, 1992.

[7] Savage, S.B. & Hutter, K., The motion of a finite mass of granular material down a rough incline. Journal of Fluid Mechanics, 199, pp. 177-215, 1989.

[8] Iverson, R.M., The physic of debris flow. Reviews of Geophysics, 35, pp. 245-296, 1997.

[9] Takahashi, T., Debris flow Mechanics, Prediction and Countermeasures, Taylor and Francis Group: London, pp. 35-38, 2007.

[10] Coussot, P., Mudflow Rheology and Dynamics, IAHR Monograph Series, A.A. Balkema: Rotterdam, pp. 252, 1997.

[11] Bagnold, R.A., Experiment on a gravity-free dispersion of large solid sphere in a Newtonian fluid under shear. Proc. of The Royal Society London, Series A, pp. 49-63, 1954.

[12] O’Brien, J.S. & Julien, P.Y., Laboratory analysis on mudflow properties. Journal of Hydraulic Engineering, 144, pp. 877-887, 1988.

[13] Coussot, P. & Piau, J.M., A large-scale field cylinder rheometer for the study of the rheology of natural coarse suspensions. Journal of Rheology, 39 (1), pp. 105-123, 1995.

[14] Contreras, S.M. & Davies, T.H.R., Coarse-Grained Debris Flows, Hysteresis and Time-Dependent Rheology. Journal of Hydraulic Engineering, 126, pp. 938-941, 2000.

[15] Ancey, C. & Jorrot, H., Yield stress for particle suspensions within a clay dispersion. Journal of Rheology, 45 (2), pp. 297-319, 2001.

[16] Schatzmann, M., Rheometry of large particle fluids and debris flows, PhD Dissertation No 16093, ETH, Zürich, Switzerland, pp. 192, 2005.

[17] Kaitna, R., Rickenmann, D. & Schatzmann, M., Experimental study on rheological behaviour of debris flow material, Acta Geotechnica, 2, pp. 71-85, 2007.

[18] Nguyen, Q.D. & Boger, D.V., Measuring the flow properties of yield stress fluids, Annual Review of Fluid Mechanics, 24, pp. 47-88, 1992.

[19] Chhabra, R.P. & Richardson, J.F., Non-Newtonian Flow in the Process Industries. Butterworth-Heinemann. Oxford, pp 436, 1999.



[20] Scotto di Santolo, A., Analisi geotecnica dei fenomeni franosi nelle coltri piroclastiche della provincia di Napoli, PhD thesis, University of Naples “Federico II” and Rome “La Sapienza”, 2000.

[21] Ruopolo, S., Analisi dei fenomeni franosi nella coltre piroclastica non satura del cratere degli Astroni, Graduate thesis, Department of Geotechnical Engineering, University of Naples “Federico II”, 2006.

[22] Papa, R., Indagine sperimentale di una copertura piroclastica di un versante della Campania, PhD thesis, University of Naples “Federico II”, 2007.

[23] Alderman, N.J., Meeten, G.H. & Sherwood, J.D., Vane rheometry of bentonite gels, Journal of Non-Newtonian Fluids Mechanics, 39, pp. 291-310, 1991.

[24] Nguyen, Q.D. & Boger, D.V., Direct yield stress measurement with the vane method, Journal of Rheology, 29, pp. 335-347, 1985.

[25] Pignon, F., Magnin, A. & Piau, J.M., Thixotropic colloidal suspension and flow curve with a minimum: identification of flow regimes and rheometric consequence, Journal of rheology, 40, pp. 573-587, 1996

[26] Larson, R.G., The Structure and Rheology of Complex Fluids. Oxford Univ. Press, New York, 1999.

[27] Macosko, C.W., Rheology. Principles, Measurements and Applications. Wiley-VCH, Inc. pp 550.1994



Experimental and numerical investigation of mixed flow in a gallery

S. Erpicum1, F. Kerger1,2, P. Archambeau1, B. J. Dewals1,2 & M. Pirotton1 1Research Unit of Hydrology, Applied Hydrodynamics and Hydraulic Constructions – HACH, ArGEnCo Department, Liege University, Belgium 2Belgian Fund for Scientific Research – FRS-FNRS, Belgium

Abstract

Experimental investigations on a physical model of a gallery performed in the Laboratory of Structures Hydraulics at the University of Liege are presented. The study focuses on the influence of the gallery aeration rate on the mixed flow pattern and the pressure distribution. In particular, the effect of air vents on the flow patterns and the release capacity of the gallery are assessed. An unusual mechanism leading to a two-phase instability is also pointed out when the aeration rate is not sufficient. Experimentations are completed with numerical computations performed with an original 1D model developed by the authors to give new insight into the mechanisms involved. Keywords: hydraulics, mixed flows, civil engineering, aeration.

1 Introduction

Mixed flows, characterized by the simultaneous occurrence of free-surface and pressurized flows, are frequently encountered in rivers networks, sewer systems, storm-water storage pipes, flushing galleries… As a matter of fact, some hydraulic structures are designed to combine free-surface and pressurized sections (e.g. water intakes). Dynamic pipe filling bores may occur in hydraulic structures designed only to convey free-surface flow. During such a transition, highly transient phenomena appears and may cause structural damages to the systems [1], generate geysers through vertical shafts [2] and engender flooding. What is more, air/water interactions may arise in such structure (particularly at the transition bore [3]) and thoroughly alter the flow regime. Sometimes the



interactions produce beneficial effects. However, more often than not, the effects are not beneficial and the remedial actions are expensive. For instance, the presence of air in pipelines can severally affect the water carrying capacity of the line, increase head losses, produce undesirable pressure rise, and even induce water hammer or blowbacks by air evacuation [4]. The effect of air in such situations depends on the location and amount of the un-dissolved air as well as the configuration of the hydraulic structure. In addition, the prediction of such flows by means of a numerical scheme remains challenging by the lack of suited mathematical model and closure relations [5]. Traditionally, the common design of hydraulic structures was conducted assuming either stratified or pressurized single phase flow. The need for additional flexibility in the hydraulic electricity production on the one hand and more rigorous restriction due to environmental and political demands on the other hand lead to an increasing demand for further research on air/water interaction for civil engineering applications. As pointed out in [6], multiphase flow in hydraulic structures is a topic of recent interest. In fact, the only comprehensive description of the relevant flow patterns could be found in approaches originating from chemical and process engineering. As the typical internal diameter of the conduits in chemical engineering is at least one order of magnitude smaller than in hydraulic engineering, the application of these concepts to hydraulic works remains challenging. In this paper, the results of the experimental study of stationary mixed flows taking place in a gallery are presented in detail. Experimental investigations have been carried out on a model of a gallery in the Laboratory of Structures Hydraulics (HACH) of the Liege University. The model includes a Plexiglas pipe linking two tanks. The topography of the upstream and downstream tanks has been built regarding realistic in-situ conditions. The aim of the experimental study was to: 1. Determine the expected flow discharge and the influence of the aeration rate

on the flow discharge. 2. Identify hydrodynamic characteristics of the flows appearing in the gallery

and appearance conditions for each pattern. In particular, the experimental apparatus is clearly described in this paper as well as the measurement system and the investigation method. Finally, numerical simulations have been performed with an original 1D model for mixed flows, developed by Kerger et al. [5]. The model is implemented in the 1D module of the software package WOLF. WOLF is a finite volume flow simulation modelling system developed within the Laboratory of Hydrology, Applied Hydrodynamics and Hydraulic Constructions (HACH) at the University of Liege. Stationary numerical results give new insight into the mechanisms regulating flows in the gallery.

2 Experimental set up

2.1 Physical model

The experimental facilities are made of two tanks, an upstream and a downstream one, linked by a circular gallery 5 m long with a .14 m diameter.



The natural topography of a mountain river bed is represented in both tanks, as the gallery bypasses a river meander. The gallery inlet and outlet are located in the right bank of the river, at the level of the river bottom, and the constant gallery slope is 6.96%. The inlet structures are profiled to decrease the head losses at the gallery entry and a .17 m long square to circle transition links the inlet to the gallery. The gallery is not straight but counts for an upstream bend of 27.68 degrees with a curvature radius of 1.81 m and a sharp angle of 2.5 degrees at its middle (figure 1). The outlet is rectangular with a radial gate .12 m high and .12 m wide to control the flows through the gallery. A .17 m long circle to square transition links the gallery to the outlet. The tanks are made of steel; the tanks topography has been build with concrete blocks and mortar painted with latex. The gallery is in transparent Plexiglas and the inlet and outlet are made of aluminium and PVC. The roughness height of the gallery has been estimated to be 2.10-5 m. In a second part of the study, three air vents with a .02 m diameter have been added upstream of the gallery, on the top of the circular cross section, at the beginning of the constant slope. The purpose of these vents is to feed the gallery with air to prevent the formation of low pressures within the flowing water. Air vents have been designed according to the recommendations in [7].

Figure 1: Sketch of the experimental device.

2.2 Water alimentation and boundary conditions

The water feeding system is a closed circuit with a pump taking water from a 400 m³ underground reservoir to inject it through under pressure galleries in an alimentation basin in the upstream tank. The alimentation basin is separated from

Air vents

Circular pipe

Inlet Radial gate

Natural river bed

Natural river bed

Upstream tank

Downstream tank

Alimentation basin

Outlet

Natural river bed

Natural river bed

Outlet

Circular pipe

Inlet

Air vents

Upstream tank

Alimentation basin

Downstream tank



the upstream tank by a permeable screen to make uniform the velocity fields entering the physical model. Downstream of the model, the water is collected in a free-surface channel to go back to the underground reservoir. The discharge in the upstream tank is the upstream boundary condition. The head level upstream of the gallery regulates naturally regarding the gate opening rate and the system release capacity. Downstream of the physical model, the natural topography is very steep so no specific boundary condition is needed (supercritical flow).

Figure 2: Details of the physical model.

2.3 Measurement system

The model is equipped with the following measurement system: The upstream discharge is measured with an electromagnetic dischargemeter

(accuracy of l l/s) on the pumping system; The water level in the upstream tank is measured using a limnimeter

(accuracy of 0.1 mm) and a Pitot tube (accuracy of 0.1 mm); 9 Pitot tubes are regularly distributed along the gallery to measure the

pressure head in the gallery (accuracy of 0.1 mm); 14 graduated scales are fixed on the gallery perimeter to measure the water

level for stratified flows.

3 Numerical model

In the last decades, large literature [8, 9] has been dedicated to a new approach in experimental hydraulics consisting in coupling experimental investigations with CFD computations. Comparison of the experimental data with numerical results

a. General view of the physical model

b. Downstreamtank

c. Upstreamtank

d. Gallery

e. Inlet structure in the upstream tank



from a modelling system is shown to give new insight into the flow conditions and to significantly reduce the cost of the physical models. The numerical model used in this paper is an original 1D model for mixed flows developed by Kerger et al. [5]. It is implemented in the one-dimensional module of the modelling system WOLF. WOLF is finite volume scheme developed within the Laboratory of Hydrology, Applied Hydrodynamics and Hydraulic Constructions (HACH) of the University of Liege [10]. WOLF 1D is based on a set of hyperbolic Partial Differential Equations (PDE), usually called Saint-Venant equations [11], describing one-dimensional unsteady open channel flow. The Saint-Venant equations are derived from cross section integration of the Navier-Stokes equation:

2

f

0QAZQ gA SQt x

A x

(1)

where A[m²] is the cross section, Q[m³/s] is the flow discharge, g[m²/s] is the gravity, Z[m] is the free surface elevation, Sf[-] is the friction slope resulting from the resistance law. Friction slope SF may be computed using the Darcy-Weisbach relation and the Colebrook relation for the friction factor:

2 1 2.51 with 2 log

2 3.7 Re

F

h h

fu kS

D f D f (2)

with Dh[m] the hydraulic diameter of the cross-section, k[m] the roughness height, u[m] the water velocity and Re[-] the Reynolds Number. Pressurized flows are commonly described through the Allievi equations [12]. According to the Preissmann slot model [13], pressurized flow can equally be calculated through the Saint-Venant equations by adding a conceptual slot on the top of a closed pipe [5]. When the water level is above the cross section maximum level, it provides a conceptual free surface flow, for which the gravity

wave speed is fc g T (Tf is the slot width). Physically, the slot accounts

naturally for the water compressibility and the section dilatation under a variation of pressure. In order to simulate pressurized flows with a piezometric head below the top of the pipe section, an original concept, called negative Preissmann slot [5], has been developed.

4 Results

Investigations focused mainly on stationary flows and aimed at determining the flow discharge as a function of the upstream pressure head and downstream gate opening. In this case, strong air/water interactions alter the flow behaviour. In particular, the flow discharge through the gallery is strongly influenced by air/water interaction and depends of the aeration rate. In this section, the influence of both the upstream pressure head and the aeration conditions on the release capacity of the gallery is analyzed.



4.1 Without air vents

Figure 4 shows the experimental relation between the flow discharge through the gallery and the upstream pressure head (zero level is set at the upstream reservoir bottom level) without air vents. Various two-phase flow patterns are observed according to the flow discharge. 5 areas corresponding to the 5 flow patterns that are usually mentioned in literature [14] can be defined. Details of each flow pattern are provided on figure 3: 1. Pure water fully free surface flow (or smooth stratified flow) is observed for

pressure head below 30 cm (figure 3.a). It is characterized by both phases, air and water, flowing separately by gravity. A smooth interface between phases appears only if both phases flow with almost the same velocity.

2. Wavy stratified flow is observed for pressure head between 30 cm and 40 cm (figure 3.b). This flow does not differ much from smooth stratified flow in terms of hydro-mechanic characteristics. Surface ripples and waves are building up and create a rather rough interface. The volume of airflow, entering by inlet as dissolved air in water and through a vertical vortex appearing at the gallery intake, is equal to the air volume insufflated into the flow by self-aeration plus the air volume flowing above the water surface as a result of the air-water shear forces.

3. Intermittent flow is observed for pressure head between 40 cm and 55 cm. This latter category includes slug flow (figure 3.c), where waves touch the top of the tube and form a liquid slug which passes rapidly along the gallery; as well as plug flow (figure 3.d), in which there are large bubbles flowing near the top of the tube. As the transition from plug to slug flow is gradual and not very sharp, slug and plug flow together are often simply referred to as intermittent flow patterns with no further specification. This flow pattern is not strictly speaking a steady flow as it is characterized by a time periodic oscillation between a free surface flow and slug flow, according to the aeration rate and the amplitude of the waves. This leads to significant fluctuations in the measured data.

4. Bubbly flow, characterized by the entrainment of small bubbles dispersed in the liquid continuum, is observed for pressure head between 55 cm and 70 cm (figure 3.e). The larger bubbles (but smaller than the resulting pockets during plug flow) propagate below the conduit ceiling due to buoyancy. Smaller bubbles, primarily transported by liquid turbulence, may be detected dispersedly over the whole cross-section. Bubbles appear where the pressure of the liquid falls below the atmospheric pressure.

5. A pure water pressurized flow is observed for pressure head above 70 cm (figure 3.f).

Two curves computed with WOLF are represented on the graph of figure 4. The dotted line is computed assuming that a free surface appears if the water height is below the cross section top (air phase above the free surface is at atmospheric pressure – high aeration rate). The continuous line is computed by activating the negative Preissmann slot (sub-atmospheric pressurized flow – low aeration rate).



Figure 3: Flow patterns visualization.

Numerical results are in good accordance with experimental data for smooth stratified flows and fully pressurized flows. Bubbly and intermittent flows show a similar behaviour to the sub-atmospheric pressurized flows. This point underlines the aeration rate in the gallery is too small to create a free surface flow.

W a te r F low

Wa t e r F l o w

Wa t e r F l o w

Wa t e r F l o w

Wa t e r F l o w

Wa t e r F l o w

a. Smooth Stratified flow

b. Wavy Stratified flow

c. Intermittent – Slug flow

d. Intermittent – Plug flow

e.Bubbly flow

f. Pressurized flow



Figure 4: Experimental relation (upstream pressure head-flow discharge) and flow patterns observed.

4.2 Periodic instabilities

A periodic instability between two unstable steady flow regimes occurs in the area denoted by wavy stratified flows (figure 4). The instability induces large period (10 to 60 seconds) oscillations of the water level in the upstream reservoir. The amplitude of the oscillations can reach 2 cm. In addition, pressure oscillations, whose amplitude can reach 4 cm, are observed all along the gallery. The inception of the instability is intimately linked with the aeration rate of the gallery, and in particular the amount of air entrained through the vertical vortex appearing at the water intake. Indeed, the minimum of the oscillation corresponds to a highly aerated stratified flow below an air phase at sub-atmospheric pressure. The distribution of pressure along the gallery in this case is given in figure 5.b. As pointed in figure 4, this flow pattern gives the minimum water carrying capacity for a given upstream head. If the upstream reservoir is supplied with a constant flow discharge, the water level in the reservoir increases. As the water level arises in the reservoir, the amount of air entrained through the vortex decreases [15]. Decrease of the air void fraction in the flow generates the formation of Kelvin-Helmholtz instabilities characteristic of intermittent flows (pressurized flow pattern). The maximum of the oscillations corresponds to this poorly aerated intermittent flow. The distribution of pressure along the gallery in this case is given in figure 5.a. As pointed in figure 4, this flow pattern gives the maximum water carrying capacity for a given upstream head. Then, the water level in the reservoir decreases because the flow discharge in the gallery is higher than the flow discharge supplying the reservoir. The cycle of the instability is then explained.

0

10

20

30

40

50

60

70

80

90

5 15 25 35 45 55

Ups

trea

m p

ress

ure

head

[cm

]

Flow discharge [l/s]

Experimental

Numerical (Atmospheric pressure)

Numerical (Sub‐Atmospheric pressure)

Smooth stratified flow

Bubbly flow

Pure water fully pressurized flow

Intermittent flow: Plug/Slug flow

Wavy stratified flow

Upstream reservoir bottom level

Oscillations



Figure 5: Computed total head and pressure head distribution for an intermittent flow sub-atmospheric pressurized flow and free-surface flow computation.

Figure 6: Effects of the addition of air vents on the experimental relation head/discharge and the flow pattern observed.

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

Alt

itud

e (c

m)

Abscissa(m)

Topography

Pipe crown

Numerical total head

Numerical Pressure head/Free surface level

Experimental total head

Experimental Pressure head/Free surfacelLevel

a. Pressure profile for a discharge of 38.4 l/s : Sub-atmospheric pressurized flow

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,6

Alt

itud

e (c

m)

Abscissa(m)

Topography

Pipe crown





b. Pressure profile for a discharge of 38.4 l/s : Free surface flow

0

10

20

30

40

50

60

70

80

90

5 15 25 35 45 55

Ups

trea

m p

ress

ure

head

[cm

]


ExperimentalExperimental -Air vents addedNumerical (Atmospheric pressure)Numerical (Sub-Atmospheric pressure)


Bubbly flow







4.3 Air vents effect

In this section, the effect of the 3 air vents added upstream of the gallery, on the top of the circular cross section, at the beginning of the constant slope is assessed. The purpose of these vents is to enable air to enter the gallery to prevent the formation of low pressures within the flowing water (figure 5.a). The experimental relations between the flow discharge through the gallery and the upstream pressure head is shown on figure 6. The graph compares the new relation with the relation without air vent. Addition of air vents does not affect pure water flow, as expected. On the opposite, a diminution of the water carrying capacity of the gallery is observed for multiphase flow patterns. Area of observation of each two-phase flow pattern is affected as well. The ranges of flow discharge corresponding to bubbly flows and corresponding to intermittent flows are narrowed to the benefit of the wavy stratified flow area.

5 Conclusions

Experimental investigations on a physical model of a gallery have been presented. They were completed with CFD computations to give more insight into the mechanisms involved. Flow patterns description and mathematical models originating from chemical/process engineering served as a referential basis. The key issue of the present work is to show concepts develop for small size pipe can be adapted for conduit sizes significantly larger. Flow pattern identification has been performed for various aeration rates. In particular, the effect of the aeration rate and the flow pattern over the water carrying capacity has been assessed. Particular insight has been given on the nature of the transition between free surface and pressurized flows. Finally, a two-phase instability has been outlined. The fundamental concepts introduced pave the way for further research. Experimental research is required to derive air entrainment predictive relation and flow pattern maps more specific to hydraulic structures.

Acknowledgement

A part of the experimental results is the property of EDF-CIH. The HACH gratefully acknowledge EDF-CIH for the authorization to publish those results.

References

[1] Zhou, F., F.E. Hicks, and P.M. Steffler, Transient Flow in a Rapidly Filling Horizontal Pipe Containing Trapped Air, Journal of Hydraulic Engineering, 128(6): p. 625-634, 2002

[2] Guo, Q. and C. Song, Dropshaft Hydrodynamics under Transient Conditions, Journal of hydraulic Engineering, 117(8): p. 1042-1055, 1991



[3] Vasconcelos, J. and S. Wright, Experimental Investigation of Surges in a Stormwater Storage Tunnel, Journal of hydraulic Engineering, 131(10): p. 853-861, 2005

[4] Estrada, O.P., Investigation on the Effects of Entrained Air in Pipelines, in Eigenverlag des Instituts für Wasserbau der Universität Stuttgart. 2007, Universität Stuttgart. p. 200.

[5] Kerger, F., S. Erpicum, P. Archambeau, B.J. Dewals, and M. Pirotton. Numerical Simulation of 1D Mixed Flow with Air/Water Interaction in Multiphase Flow New Forest, 2008

[6] Keller, U., in Versuchanstalt für Wasserbau, Hydrologie und Glaziologie der Eidgenössichen. 2006, ETH Zürich: Zürich. p. 250.

[7] Falvey, H.T., Air-Water Flow in Hydraulic Structures. Engineering Monogaph. Vol. 41: United States Department of the Interior. 1980

[8] Dewals, B.J., S. Andre, M. Pirotton, and A. Schleiss. Quasi 2D-numerical model of aerated flow over stepped chutes. in 30th IAHR Congress, Greece, 2003

[9] Erpicum, S., P. Archambeau, B.J. Dewals, S. Detrembleur, A. Lejeune, and M. Pirotton. Interactions between Numerical and Physical Modelling for the design and Optimization of Hydraulic Structures - Example of a Large Hydroelectric Complex. in International Symposium on Hydraulic Structures, XXII Congresso Latinoamericano de Hidrahulica, Ciudd Guayana, Venezuela, 2006

[10] Dewals, B.J., S. Erpicum, P. Archambeau, S. Detrembleur, and M. Pirotton, Depth-Integrated Flow Modelling Taking into Account Bottom Curvature, Journal of Hydraulic Research, 44(6): p. 787-795, 2006

[11] Cunge, J.A., F.M. Holly, and A. Verwey, Practical Aspects of Computational River Hydraulics. [Monographs and surveys in water resources engineering], 3. Boston: Pitman Advanced Pub. Program. 1980

[12] Wylie, E.B. and V.L. Streeter, Fluid transients. Première ed, ed. M.-H. Inc., 385, 1978

[13] Preismann, A. Propagation des intumescences dans les canaux et rivieres. in First Congress of the French Association for Computation, Grenoble, France, 1961

[14] Wallis, G.B., One-dimensional Two-phase Flow, ed. M.-H.B. Company. 410, 1969

[15] Quick, C.M., Efficiency of Air-Entraining Vortex Formation at Water Intake, Journal of the Hydraulics Division, Proceedings of the American Society of Civil Engineers, 96(7): p. 1403-1416, 1970



Sediment transport via dam-break flowsover sloping erodible beds

M. Emmett & T. B. MoodieUniversity of Alberta, Canada

Abstract

When a semi-infinite body of homogeneous fluid initially at rest behind a verticalretaining wall is suddenly released by the removal of the barrier, the resultingflow over a horizontal or sloping bed is referred to as a dam-break flow. Whenbed resistance is neglected the exact solution, in the case of a stable horizontalbed, may be obtained on the basis of shallow-water theory via the methodof characteristics and the results are well known. Discrepancies between theseshallow-water based solutions and experiments have been partially accounted forby the introduction of flow resistance in the form of basal friction. This addedfriction significantly modifies the wave speed and flow profile near the head ofthe wave so that the simple exact solutions no longer apply. Various asymptoticor numerical approaches must be implemented to solve these frictionally modifieddepth-averaged shallow-water equations. When the bed is no longer stable so thatsolid particles may be exchanged between the bed and the fluid, the dynamics ofthe flow become highly complex as the buoyancy forces vary in space and timeaccording to the competing rates of erosion and deposition. It is our intentionhere to study dam-break flows over erodible sloping beds as agents of sedimenttransport, taking into account basal friction as well as the effects of particleconcentrations on flow dynamics including both erosion and deposition. We shallconsider shallow flows over initially dry beds and investigate the effects of changesin the depositional and erosional models employed, in the nature of the drag actingon the flow, and in the slope of the bed. These models include effects hithertoneglected in previous studies and offer insights into the transport of sediment inthe worst case scenario of the complete and instantaneous collapse of a dam.Keywords: gravity current, dam-break flow, sediment transport, dilute sediment.



1 Introduction

Dam-break flows, which are represented by the sudden release of fluid containedin a semi-infinite reservoir behind a vertical barrier, are of both practical andfundamental importance in fluid mechanics, engineering, and geomorphology.They have played a crucial role in underpinning simple models for a numberof natural and catastrophic events, such as break-out floods from the failure ofend moraine dams and various sheet flow events, as well as the formative stagesof lahars or debris flows [1–3]. Although in practice the release of water uponcollapse of the retaining barrier will often be more gradual than that in the idealisedmathematical models, one can view these models as providing the worst casescenario for these events [3, 4].

The earliest work on dam-break flows considered single phase, low aspect ratio,frictionless flows in rectangular geometry taking the shallow-water equations asthe governing model equations. With the bed below the dam assumed horizontaland dry, the solution for the flow is a centred simple wave that was first developedby Ritter [5]. If the initial depth of water behind the vertical dam is h0, the front ofthe flow advances as a wave over the dry bed with constant speed 2

√gh0, while

the reduction of depth spreads back from the initial position of the dam with speed√gh0, where g is the acceleration due to gravity. In the disturbed region between

the two extremes of depth, the velocity u and the depth of the flow h are given by

u = 2

3

(xt

+√gh0

), and (1)

√gh = 1

3

(2√gh0 − x

t

), (2)

where x measures distance downstream of the original position of the dam andt measures the time elapsed since its collapse. Although these solutions doprovide a reasonably good match to the experimental observations when thetransients associated with the initial release have died down there are still importantproperties of the flow that are not captured by the classical shallow-water modelused in the construction of the solutions displayed in (1) and (2). It has beenobserved in particular [3, 6] that for the dam-break flow experiments the waternear the tip piles up and the front speed is appreciably less than that predicted bythe simple theory.

In order to account for this blunting of the tip and the slowing down of thefront several authors [4, 7, 8] have postulated that near the tip, where the depthof flow drops to zero, frictional resistance and the resulting turbulence dominatethe flow. To account for this basal friction a Chezy resistance term is added tothe momentum equation [4, 8]. Various asymptotic procedures were employed[4, 7, 8] to determine the influence of this resistance and it was found that itsinclusion brought theory and experiment into closer accord. Since we are interestedin developing and exploiting models for sediment transport that employ dam-break flows on down-sloping topography as paradigms for certain geological andengineering processes we shall extend the model beyond what is discussed in this



paragraph while appreciating the significant gains achieved through the additionof the resistance term. In fact, basal friction with realistic parameterisations forgeological applications appears to have a much greater influence on flow dynamicsthan does the presence of particles in suspension [9].

Recent studies [1, 2] have employed dam-break flows as agents of sedimenttransport. In [2] the authors explore dam-break flows over beds that consist offine sediment that can be entrained into the water column and transported insuspension. The sediment transport was passive in that the suspended particles didnot influence the flow dynamics which could then be totally specified employingthe simple exact solution of the shallow-water equations for both a dry bed [5] anda bed with ‘tail water’. Recent studies by the current authors [9, 10] have shownthat under the assumption of dilute suspensions employed in [2] the suspendedparticles will play a relatively minor role in modifying the flow dynamics so thatpassivity of sediment transport does not produce large errors. The omissions ofbasal friction and bed topography however, as was the case in [2], were shown tohave a more profound negative influence on the accuracy of the results [10].

In the present work we shall employ dam-break flows over erodible beds asagents of sediment transport. The inclusion of a velocity dependent basal frictionas well as bottom topography and non-passive particle transport adds severalimportant mechanisms that were absent from previous studies. We shall assumethat our flows are shallow so that the pressure remains hydrostatic throughout theflow regime [2, 8–13]. We shall also assume that the particle concentration in theflow remains sufficiently low so that we may treat the particles as being isolatedand employ a Boussinesq approximation whereby the particles appear in themomentum equations only in the buoyancy terms. These assumptions put definiteconstraints on the range of particle volume fractions φ(x, t) in our well-mixedsuspensions. When erosion exceeds deposition so that particle concentrations areincreasing we shall assume that our model calculations are valid up until φ ≈ 0.05[9, 10]. Although the bottom boundary shear stress could be calculated from thefull governing equations we shall adopt the common and much simpler approachof introducing a Chezy drag coefficient CD which when viscous effects are small(large Reynolds number flows) gives the boundary shear stress as τb = CDρf u

2,where ρf is the density of the fluid and u a depth averaged horizontal velocity[4,8–10]. The Chezy drag coefficient is dimensionless and usually falls in the range0.01-0.001 for most environmental flows [8]. With all of our additions to the modelof Pritchard and Hogg [2] (basal drag, bottom topography, and particle modifiedflow dynamics) the simple shallow-water based solutions [5] will no longer applyand the approach adopted in [2] is not available so that an alternative approach willhave to be adopted.

2 Model development

We consider the two-dimensional flow resulting from the sudden release of waterinitially held at rest behind a plane vertical retaining wall of height h0. The bedbelow the dam which is initially located at x = 0, is gently varying and specified



b(x, t)

x

z

u(x, t)h(x, t)

ρ(x, t) = φ(x, t)ρp + (1 − φ(x, t))ρf

reservoir

Figure 1: Schematic for the dam-break flow on a dry sloping and erodible bed.

by the topography z = b(x, t). It is assumed that initially there is no waterbelow the dam and that the bed is comprised of fine or cohesive material which,once a threshold shear stress is exceeded, is entrained into the water columnand transported in suspension to possibly be deposited downstream. With ourassumption of dilute suspensions the erosion or deposition depth is small relativeto the depth of the current so that the position of the bottom boundary may be keptfixed in most of our calculations. Although in the early stages of such dam-breakflows the stream-wise and vertical scales of the motion will be comparable, theresoon comes a time when stream-wise scales dominate the vertical ones and theflow may be considered to be a shallow flow with negligible vertical accelerationsand a hydrostatic pressure distribution can be adopted [2, 8].

We shall assume that once particles have been suspended into the water columnthey are always vertically well-mixed by the turbulence of the flow [2,11,12,14] sothat the volume fraction of particles in suspension φ is a function of the horizontalcoordinate x and the time t elapsed since collapse of the dam. We shall alwaysassume that the water initially behind the dam is particle free. The bulk density ρof the suspension is then given by

ρ(φ) = ρpφ + (1 − φ)ρf , (3)

where ρp (> ρf ) is the particle density. The setting under which the model isdeveloped is summarised by the schematic presented in Figure 1.

The depth-averaged continuity and momentum equations for the system underthe Boussinesq approximation (φ 1) and the hydrostatic assumption [10] are

∂h

∂t+ ∂

∂x(hu) = 0, and (4)

∂

∂t(hu)+ ∂

∂x

(βhu2 + 1

2

g

ρfρh2

)= −ρgh

ρf

∂b

∂x− CDu

2, (5)

where β is the Boussinesq coefficient (shape factor) [8]. The magnitude of β ≥ 1corresponds to the amount of shear present in the horizontal velocity field and maydepend on such factors as the Reynolds number or the boundary roughness [8]. The



particle mass conservation and bed evolution equations [9, 10, 12] are

∂

∂t

(ρpφh

)+ ∂

∂x

(ρpφhu

) = qe − qd, and ρp∂b

∂t= −qe + qd. (6)

where qe is the mass erosion flux and qd is the mass deposition flux.Observations have shown that qe and qd are functions of both the fluid velocity

u and the volume fraction of particles in suspension φ [15]. We adopt the usualexpression for mass deposition rate, that is, qd = ρpφvs , where vs is the Stokessettling velocity [2,9–14]. With no particles in suspension behind the dam we needonly consider deposition when u > uc, where uc is some critical velocity belowwhich particles are not entrained into the fluid column. We adopt the expression

qe(u) =ρpve

(u2

u2c

− 1)n

for |u| ≥ uc,

0 for |u| < uc

(7)

for the erosion rate qe, where ve is a sediment entrainment rate. This model isused to describe the erosion of sediment from a cohesive bed or from a bed of finecohesionless material where some critical shear stress must be exceeded in orderto entrain particles from the bed into the water column [2, 9, 10, 15–17].

We make all equations non-dimensional using the non-dimensionalisation andscaling scheme presented in [10]. Of particular importance is the typical volumefraction scale φ0 and velocity scale U . The velocity scale U is the familiar√gh0 wherein g is a ‘modified gravity’ which is given by (γ φ0 + 1)g where

γ = (ρp−ρf )/ρf . Rendering all equations non-dimensional gives

∂

∂th+ ∂

∂x(hu) = 0, (8)

∂

∂t(hu)+ ∂

∂x

[βhu2 + 1

2h2(1 − + φ

)] = −h(1 + φ)bx − CDu

2, (9)

∂

∂t(φh)+ ∂

∂x(φhu) = qe − qd, (10)

∂b

∂t= φ0

(qd − qe

), (11)

qd = udφ, and qe(u) =

⎧⎪⎨⎪⎩ue

(u2

u2c

− 1

)nfor |u| ≥ uc,

0 for |u| < uc

(12)

where we have introduced the non-dimensional parameter = 1 − 1/(γ φ0 + 1).

3 Analysis of the model

In this section we highlight various aspects of dam-break flows over erodible bedsbased upon our model equations. The parameter values used in the simulations aretypical for flows of a geological scale with h0 = 20m [11].



0

0.5

1

h(x,t

)(a) h(x, t) vs x

−300 −200 −100 0 100 200 300 400x

0

1

2

u(x,t

)

(b) u(x, t) vs x

−300 −200 −100 0 100 200 300 400x

Figure 2: Solutions of (a) height and (b) velocity of the particle free modified dam-break flow over a flat bottom with basal drag at t = 300. Solid lines shownumerical solutions with β = 1.0, 1.1, and 1.2 (from left to right, at thefront). Dashed line shows the Ritter solution. Parameter value used isCD = 0.001.

We begin by presenting an exploration of the effects of those mechanisms(velocity shear, bed slope, and drag) that do not necessarily involve suspendedparticles. To do so we examine solutions of the initial value problem correspondingto the dam-break flow for the non-dimensional pair of coupled equations (8) and(9) with ≡ 0. Subsequently, we present an overview of the effects of suspendedparticles by allowing the flows to entrain particles through bed erosion but fixingthe bed so that b(x, t) is in fact independent of t . A more detailed study ofthese scenarios is presented in [10]. Finally, we present a brief note regarding theevolution of the bed topography when the bed is eroded according to (11).

3.1 Modified dam-break flows with drag

In this subsection we present solutions for modified dam-break flows with basaldrag in the absence of bed topography or particles. That is, following the approachemployed in [2, 6–9, 18] we retain the basal drag term in order to bring our modelcalculations into closer accord with the experimental results of [7] wherein dam-break flows over a horizontal bed were examined.

In Figure 2 we have plotted both the height and velocity profiles, respectively,for both the classic solution [5] and the numerical solution including basal drag.



We note immediately that the presence of drag has significantly altered the shapeof the depth profile in the immediate vicinity of the leading edge as well as thevelocity structure of the flow in this region. We also note that, in agreement with theexperimental results [7] and the theoretical approach adopted in [4], the velocity isnearly uniform in the blunt snout. That is, in the deformed tip basal drag retards theflow so that the velocity profile is approximately horizontal there. The assumptionthat the velocity in the tip depends only on time was crucial to the theoreticaldevelopment in [4] and appears to be confirmed by our numerical work.

Furthermore, we note that the effect of vertical shear (β > 1) in the horizontalvelocity profile is most dramatic in the immediate vicinity of the leading edgewhere the depth, and hence momentum, of the flow is small allowing the effect ofvertical shear in the horizontal velocity to be accentuated.

3.2 Modified dam-break flows with drag over a linear slope

In this subsection we will examine modified dam-break flows over sloping bedsin order to isolate the effects of the interplay between the bottom slope and basaldrag. We shall take the bottom topography to be specified by

b(x, t) = −sx(x) (13)

where s is a small non-dimensional parameter and is the Heaviside stepfunction. Employing this simple linear form for the bottom topography allows usto appeal to our intuition while interpreting both theoretical and numerical results.Furthermore, the stream-wise gradient of a linear slope is constant, affording usthe opportunity to perform an asymptotic expansion over the bed slope s, whichwas presented in [10].

In Figure 3 we have plotted the numerical solutions for both the height andvelocity profiles of a particle free dam-break flow over a sloping bottom with dragfor β = 1. We note that both the height and velocity profiles are nearly horizontalin the bulk of the flow over the linearly sloping bed. In the presence of a slopingbottom the blunt snout in the height profile has become more abrupt and fallssteeply to zero at the front. As demonstrated in previous sections, the effect of dragis to retard the front and create a blunt snout, while the effect of a sloping bottomis to draw out the fluid, reducing its height in the bulk of the flow over the slopingbed, and slightly increasing its height directly behind the front. Furthermore, incontrast to the drag-free case, the presence of a sloping bottom has a significanteffect on the front position of the flow. As one may expect, the front position isgreater for flows over steeper beds.

3.3 Modified dam-break flows with drag and sediment

In this subsection we will examine modified dam-break flows as agents of sedimenttransport over flat erodible beds while allowing the particle volume fraction tochange through the mechanisms of particle advection, deposition, and entrainmentthrough bed erosion. In our previous work [9] we demonstrated the role played



0

0.5

1

h(x,t

)(a) h(x, t) vs x

−100 0 100 200x

0

1

2

u(x,t

)

(b) u(x, t) vs x

−100 0 100 200x

Figure 3: Solutions of (a) height and (b) horizontal velocity for particle free dam-break flows over linear slopes with drag. Solid lines show numericalsolutions at t = 100 with slope s = 0, 0.001, and 0.01 (from left toright, at the front). Dashed lines show the Ritter solutions. Parametervalues used are CD = 0.01 and β = 1.

by the critical bed velocity uc in the long term competition between erosion anddeposition and displayed the strong effect of basal drag on the ultimate outcome ofthis competition. Furthermore, we observed that the inclusion of particles did nothave a significant effect on the height or velocity profiles of the flows. Generally,flows with particles were slightly faster compared to analogous particle free flows,but only by a few percent at most.

The maximum rate of sediment entrainment occurred at the front and was nearlyuniform within the snout. Peaks in the volume fraction of sediment profiles wereobserved directly behind the front where the height of the fluid decreased sharplyto zero. These peaks in the volume fraction of sediment profiles where highest forshort post-release times since the velocity was also highest for short post-releasetimes, and decreased with time.

In Figure 4 we have plotted a typical numerical solution of the full modelequations with drag, particle deposition, and bed erosion over a flat bed for aparameter configuration in which the drag coefficient CD and critical bed velocityuc were small enough so that particles were entrained by the snout for the durationof the flow.

The interplay between basal drag and the critical bed velocity was furtherdemonstrated in our recent work [9] by examining the particle flux at a fixed



0

0.5

1

φ

φ(x, t) vs x

0 100 200 300 400 500x

t = 60

t = 600

Figure 4: Solutions (volume fraction) of the dam-break flow over a flat bed withbasal drag, particle deposition, and particle entrainment at various timest = 60, 120, . . . , 540, 600. Parameter values used are CD = 0.001,φ0 = 0.01, γ = 2.5, ud = 0.005, ue = 0.0015, n = 1.2, uc = 0.5,and β = 1.

station. Flows with high basal drag did not continue to erode the bed and entrainparticles once the front had passed a given station. Flows with a critical bedvelocity above 2/3 were also dominated by deposition since the long term Rittervelocity is less than 2/3 regardless of the presence of basal drag. Flows withlow drag and a critical bed velocity below 2/3 continued to erode the bed andadvect entrained particles downstream. As time progressed, the horizontal velocityat the station approached its Ritter solution and the particle flux due to erosionapproached a steady value while the particle flux due to deposition increased withincreasing volume fraction until an equilibrium between erosion and depositionwas reached.

3.4 Modified dam-break flows with drag and sediment over a linear slope

In this subsection we will examine modified dam-break flows as agents of sedimenttransport over sloping erodible beds.

In our previous work [10] we observed that, as in the previous case with flatbeds, the inclusion of particles did not have a significant effect on the height orvelocity profiles of the flows. Particles entrained into the flow maintained theirrelative position within the flow which resulted in a nearly linear volume fractionprofile that increased in the downstream direction. Furthermore, the maximumattained by the volume fraction continued to increase for all post-release times,in contrast to the flat bed case. The peak in the volume fraction occurred directlybehind the front, and was primarily due to advection coupled with the nearlyhorizontal velocity profile over the sloping bed.

In Figure 5 we have plotted the horizontal velocity u and volume fraction ofsediment φ for a typical numerical solution to the full model equations with basaldrag, particle deposition, bed erosion, and bed slope. These plots elucidate therelationship between the velocity and volume fraction of sediment entrained bythe flow.



0

1

2

u

(a) u(x, t) vs x

−100 0 100x

t = 20t = 100

−0.1

0

0.1

0.2

0.3

0.4

0.5

φ

(b) φ(x, t) vs x

−100 0 100x

t = 20

t = 100

Figure 5: Solutions (a) horizontal velocity and (b) volume fraction of the dam-break flow with basal drag, particle deposition, and bed erosion at varioustimes t = 20, 40, . . . , 80, 100. Parameter values used are CD = 0.001,φ0 = 0.01, γ = 2.5, ud = 0.005, ue = 0.0015, n = 1.2, uc = 0.5,β = 1, and s = 0.001.

The linear nature of the volume fraction profile shown in Figure 5 is in contrastto the flat case (see Figure 4) in which the volume fraction was highest in the snoutand decayed in a non-linear fashion in the upstream direction [9]. In the slopingcase, the peak in the volume fraction is primarily due to advection coupled withthe nearly horizontal velocity profile, and the peak increases for all post-releasetimes.

3.5 Modified dam-break flows with drag and sediment over a variable bed

In this subsection we will examine modified dam-break flows as agents of sedimenttransport over initially flat erodible beds. We allow the bed topography to changewith time through the mechanism of erosion according to (11).

In Figure 6 we have plotted the bed topography b(x, t) and horizontal velocityu of the numerical solution to the full model equations with basal drag, particledeposition, and bed erosion. These plots show the scour pit resulting from bederosion. We note that the pit is deepest slightly downstream from the originalposition of the dam. There is peak in the horizontal velocity directly above thescour pit since the fluid gains momentum as it falls into the pit.



−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01b(x,t

)(a) b(x, t) vs x

−100 0 100 200x

0

1

2

u(x,t

)

(b) u(x, t) vs x

−100 0 100 200x

Figure 6: Solutions of (a) bed height and (b) horizontal velocity for particle ladendam-break flows over variable beds with drag at time t = 120. Solidlines show numerical solutions with CD = 0.001, φ0 = 0.01, γ = 2.5,ud = 0.005, ue = 0.0015, n = 1.2, uc = 0.5, β = 1, and s = 0.001.Dashed lines show the Ritter solutions.

Further exploration of the complex interplay between the fluid, sediment, andbed dynamics will be carried out in subsequent research.

4 Discussion

We have developed a model to describe the transport of dilute sediment underdam-break flows over sloping beds with basal drag and presented numeri-cal solutions to this model in order to investigate the influence of variousmodel parameters. The model developed is an extension to previous models,includes a velocity dependent basal drag force, incorporates the effects ofa spatially dependent bed topography, and allows the variable concentrationof suspended particles, through the mechanisms of deposition and erosion,to influence the flow dynamics. The model is unique from existing modelswhich do not couple the flow and sediment dynamics and do not includebasal drag [2] or bed topography [9]. The numerical results show that thiscoupling is especially important for the sediment dynamics and should notbe ignored when the understanding of sediment processes is vital to a givenstudy.



References

[1] Capart, H. & Young, D., Formation of a jump by the dam-break wave over agranular bed. J Fluid Mech, 372, pp. 165–187, 1998.

[2] Pritchard, D. & Hogg, A., On sediment transport under dam-break flow. J.Fluid Mech, 473, pp. 265–274, 2002.

[3] Stansby, P., Chegini, A. & Barnes, T., The initial stages of dam-break flow. J.Fluid Mech, 374, pp. 407–424, 1998.

[4] Whitham, G., The effects of hydraulic resistance in the dambreak problem.Proc R Soc London, Ser A, 227, pp. 399–407, 1955.

[5] Ritter, A., Die fortpflanzung der wasserwellen. Z Verein Deutsch Ing, 36, pp.947–954, 1892.

[6] Dressler, R., Hydraulic resistance effect upon the dam-break functions. J. ResNat Bur Stand, 49, pp. 217–225, 1952.

[7] Dressler, R., Comparison of theories and experiments for the hydraulic dam-break wave. Proceedings of the Commission des Eaux de Surface at theAssemble Gnrale de Rome 1954, International Association of HydrologicalSciences, volume 38, pp. 319–328, 1954.

[8] Hogg, A. & Pritchard, D., The effects of hydraulic resistance on dam-breakand other shallow inertial flows. J Fluid Mech, 501, pp. 179–212, 2004.

[9] Emmett, M. & Moodie, T., Dam-break flows with resistance as agents ofsediment transport. Phys Fluids, 20(8), p. 086603, 2008.

[10] Emmett, M. & Moodie, T., Sediment transport via dam-break flows oversloping erodible beds. Stud Appl Math, 2009. Submitted.

[11] Bonnecaze, R., Huppert, H. & Lister, J., Particle-driven gravity currents. J.Fluid Mech, 250, pp. 339–369, 1993.

[12] Moodie, T., Pascal, J. & Swaters, G., Sediment transport and deposition froma two-layer fluid model of gravity currents on sloping bottoms. Stud ApplMath, 100, pp. 215–244, 1998.

[13] Moodie, T. & Pascal, J., Axisymmetric particle-bearing gravity flows onsloping bottoms. Can Appl Math Q, 7, pp. 17–47, 1999.

[14] Moodie, T. & Pascal, J., Non-hydraulic effects in particle-driven gravitycurrents in deep surroundings. Stud Appl Math, 107, pp. 217–251, 2001.

[15] Teisson, C., Ockenden, M., Hir, P.L., Kranenburg, C. & Hamm, L., Cohesivesediment transport processes. Coastal Eng, 21, pp. 129–162, 1993.

[16] Kerswell, R.R., Dam break with Coulomb friction: a model for granularslumping? Phys Fluids, 17(5), pp. 057101, 16, 2005.

[17] Blanchette, F., Strauss, M., Meiburg, E., Kneller, B. & Glinsky, M., High-resolution numerical simulations of resuspending gravity currents: Condi-tions for self-sustainment. J Geophys Res, 110, p. C12022, 2005.

[18] Whitham, G., Linear and Nonlinear Waves. Wiley, New York, 1974.



Section 5 Bubble and drop dynamics

Hydrodynamic drag and velocity of micro-bubbles in dilute paper machine suspensions

A. Haapala1, M. Honkanen2, H. Liimatainen1, T. Stoor1 & J. Niinimäki1 1University of Oulu, Fibre and Particle Engineering Laboratory, Finland 2Tampere University of Technology, Energy and Process Engineering, Finland

Abstract

This paper studies hydrodynamic drag forces acting on freely rising micro-bubbles in dilute paper machine suspensions under turbulent flow conditions. Dissolved, colloidal and numerous solid materials i.e. process chemicals, wood extractives, fillers and wood fibre fractions present in these suspensions disturb the rise of micro-bubbles increasing their drag. The aim of this study is to characterise the terminal velocities and drag coefficients of the bubbles as a function of their Reynolds number in several paper machine circulation waters, i.e. white waters, and in some model suspensions. Characterisation is performed experimentally with a high-speed CMOS camera and a submersed back-light illumination in a pressurised bubble column. Image sequences of bubbly flow are analysed with automatic image processing algorithms that measure not only the bubble size and velocity, but also the velocity of the fluid surrounding bubbles, revealing the initial slip velocity of each bubble. Bubbles are tracked in time to provide time series data for every bubble that passes the focal plane of the imaging system. Results show how some suspension properties – concentration, apparent viscosity and surface tension – affect the motion of micro-bubbles. Results also show the changes in micro-bubble formation with pressure drop and differences of bubble size distributions in a variety of suspensions and solutions. Finally, a mathematical model describing the bubble rise velocities and drag coefficients with respect to the bubble Reynolds number is developed for the investigated white waters. Keywords: bubble motion, drag coefficient, drag force, white water, papermaking, high-speed imaging, high-speed imaging, bubble sizing.



1 Introduction

Bubbly gasses and their interaction with other substances in pulp suspensions may cause substantial problems in papermaking processes. Most of them relate to stock filtration, dewatering, sheet formation or pumping but also process measurements. For example, consistency can be affected as recently presented by Stoor [1] and Helle et al [2]. Deaeration in papermaking usually takes place in a pressurised tank or a special gas removing pump, while passive methods rely solely on bubble rise and evacuation from suspension. Today, different solutions for gas removal are developed using mathematical models and computational fluid dynamics (CFD) tools that describe the phenomenon of bubble motion. These models in turn require experimental knowledge on the kinetics and hydrodynamics behind the free gas bubble rise in process suspensions. The rise of free gas bubbles in papermaking suspensions is hindered by apparent viscosity and solids, while surface tension of the liquid phase affects the size distribution of formed bubbles [3–8]. This is also typical for fibre-free filtration waters drained from sheet former, commonly referred as white waters. Suspended and dissolved solids are the main components contributing to bubble drag in white waters i.e. fibre fines fractions and inorganic particles such as clay fillers [7, 8]. Their composition and overall chemical state varies from one paper machine to another, whilst process waters slowly accumulate dissolved and colloidal materials from raw materials, process chemicals and fresh water. While the effect of surfactants on the drag coefficient of rising bubbles was shown previously in the work of Okazaki [6] in 1962, so far research on experimental values for bubble drag and rise velocity in complex papermaking white waters has not been published. Image-based measurements have become a powerful tool to determine the gas bubble size, velocity and the velocity of suspension surrounding the gas bubble [8, 9]. Robust image analysis algorithms [10, 11] can automatically recognise individual, in-focus bubbles from image sequences of complex multiphase suspensions, where bubbles commonly overlap. Optical measurements are, however, limited to dilute multiphase flows, because they need undisturbed optical access to the measurement volume. Dispersed phase particles and bubbles in these suspensions scatter the incident light and damp the light intensity, which restricts the penetration of light through the suspension. In addition, they scatter light on the optical path between the illumination source and camera causing image distortions. Opaque multiphase suspensions, such as white waters, are often testing the limits of digital imaging methods. However, an underwater measurement probe [13] or illumination also provides a way to visualise these opaque multiphase suspensions inside large vessels. In this study, the bubbly multiphase flows are visualised close to the column wall with submerged back-light illumination optics. The aim of the study is to investigate the effect of white water properties on size distribution and motion of micro-bubbles in white waters in order to produce consistent experimental data for the validation of CFD models that describe the deaeration phenomenon of these suspensions. Micro-bubbles are generated in



characterised white waters and model waters of varying consistencies, apparent viscosities and surface tension by dissolving air under pressure and causing a sudden pressure drop. Generated micro-bubbles are then visualised using a high-speed CMOS camera and submerged diode laser back-light. Bubble size distributions, rise velocities of bubbles and bubble drag coefficients (CD) are defined for each white water as a function of bubble Reynolds number.

2 Materials and methods

2.1 Studied suspensions

The examined white waters (WWs) originate from 8 paper machines that utilise various raw materials and produce a variety of paper grades: mechanical pulp to produce newsprint and magazine grades while eucalyptus, pine or similar kraft pulp to produce different fine paper grades. Thus fines and fillers content in white waters varied accordingly for each machine. Analyses on waters were made according to the following standards and methods; pH (SFS 3021), conductivity (SFS-EN 27888) and dry matter content (SFS-EN 20638). Surface tension was measured with a Krüss K8600 (du Noüy’s ring method) and apparent viscosities with a Haake 501 viscometer. In addition, apparent mean particle size of white water solids were determined using a multi-wavelength particle size analyser (Beckman Coulter LS 13 320). Physico-chemical properties of tested white waters are summarised in table 1.

Table 1: Properties of white waters.

WW 1 WW 2 WW 3 WW 4 WW 5 WW 6 WW 7 WW 8pH (20°C) 7.9 4.0 6.9 7.7 7.7 7.7 7.7 7.6

Density (40°C) [g/L] 988 989 988 987 991 991 995 989Conductivity

(20°C) [µS/cm] 1.5 2.0 1.0 0.9 1.0 0.9 1.1 1.4Dry Matter Content [%] 0.58 0.49 0.50 0.25 0.22 0.22 0.25 0.21

Surface tension (20°C) [mN/m] 53 51 48 59 58 53 66 61

Apparent viscosity (40°C) [mPas] 1.7 1.8 1.8 1.4 1.3 1.2 1.4 1.5

Mean particle size, (from <90%) [µm] 7.9 (20.7) 10.0 (26.3) 6.5 (20.8) 7.3 (23.6) 7.4 (20.0) 8.3 (26.3) 10.4 (33.3) 7.1 (19.4)

In addition to process waters some measurements were made on model waters. Carboxymethyl cellulose (CMC) and n-butanol were added into deionised water in the absence and presence of dry refined pine kraft pulp fibres (average fibre length 1.52 mm). CMC was used to increase apparent viscosity, n-butanol to lower suspension surface tension and wood fibres to promote micro-bubble formation through nucleation and also to increase bubble drag through bubble-solid interactions in model waters. Micro-bubble motion in a variety of these suspensions was similarly measured in order to characterise the effects of fibre and fines content, suspension viscosity and surface tension to micro-bubble movement and size distribution. Properties of these model waters are presented in table 2.



Table 2: Properties of model suspensions.

Water CMC CMC + fibre n-but n-but + fibrepH (20°C) 8.6 - - - -Dry Matter Content [%] 0.0 0.0 0.15 0.0 0.15

Surface tension (20°C) [mN/m] 72 68 70 54 54

Apparent viscosity (20°C) [mPas] 1.0 5.6 5.8 1.5 2.2

2.2 Measurement setup

Experiments were carried out in a batch process, where 20 litres of suspension was pressurised inside a 1200 mm long cylindrical bubble column with a diameter of 167 mm. A steady internal overpressure of 3 bars was maintained by feeding 25°C pressurised air through sintered porous media at 4 bars, allowing continuous air flow-through in column. The amount of dissolved oxygen in water increased until the suspension saturation level was reached. This was monitored with a dissolved oxygen analyser (Hach Orbisphere 3600). After saturating the suspension, a pressure drop was created by opening a solenoid valve on top of the column. Images of generated micro-bubbles were gathered over a period of 30 seconds after the first micro-bubble appeared in the measurement volume. Illustration of the experimental setup is presented in fig. 1. The column had a 300 mm high midsection of transparent polycarbonate piping through which imaging took place. The mill waters were tested at 40°C to achieve similar physical suspension characteristics that occur in real process environments, while tests on model suspension took place at 20°C.

Figure 1: The test setup consisting of pressurised bubble column and camera.



2.3 Measurement technique

A high-speed digital imaging setup was used to visualise the rise of micro-bubbles near the bubble column wall. Imaging was limited only to near wall regions, as contaminants blocked the view deeper within the suspension. The flow was illuminated with a submerged light diffuser connected to a pulsed diode laser (Cavilux Smart, 400 W, 690 nm) with an optical fibre. Cavilux control unit synchronised the laser and the high-speed CMOS camera (PCO 1200hs) while the image acquisition was user controlled with a laptop computer. As shown in fig. 1, a camera was placed outside the column opposite to the submerged light diffuser to provide shadow images of micro-bubbles in the flow between the light diffuser and the column wall. To eliminate the image distortions due to the curved column wall, an external cubical basin filled with water was placed around the bubble column. Camera image was geometrically calibrated with a measuring rod and a scaling of 18.6 µm per image pixel was obtained. Micro-bubbles and their motion were then detected as shown in fig. 2.

Figure 2: Original snapshot image of micro-bubbles in white water on left, the measured velocity vector field in the middle and a multi-frame image of detected micro-bubbles on right.

High-speed image sequences were analysed automatically with image analysis algorithms while a velocity vector field was provided by cross-correlation of two consecutive images utilising the 32x32 pixel interrogation areas. Fast Fourier transform (FFT) and spatial vector validation were made with DaVis 7.2 software that is commonly utilised in particle image velocimetry (PIV) experiments. Visual detection of micro-bubbles in the image was difficult even for a human eye. Micro-bubble recognition (human and automated) relies on the following assumptions: micro-bubbles produce dark, circular shadow images with a bright spot at the centre of the shadow and a sharp outline, whose curvature is nearly constant. We also assumed a size range from 50 µm to 1 mm for micro-bubbles and that they are rising upwards, thus eliminating bubbles stuck on backlight or column walls, one of which is present up in the rightmost picture in fig. 2.



2.4 Image analysis

Automatic micro-bubble recognition algorithm included several phases that are briefly described in the following 5 step list: 1. Image equalisation: Image background was computed for each image with a low-frequency filter. Spatial image equalisation was obtained by dividing the image with the computed background image. 2. Detection of micro-bubble outlines: Local (5x5) standard deviation and a local kurtosis (4th moment) of greyscales emphasised the image areas of high greyscale variance. As local kurtosis on micro-bubble outline was greatly lower than that of curly fibre. Due to the presence of a smooth continuous change in the curvature of a bubble perimeter, the subtraction of a scaled local kurtosis image from the local standard deviation image returned an image emphasising only the outlines of micro-bubbles. Finally, a search for modified local maximum was carried out to recognise the outlines of sharp micro-bubble images. 3. Fitting circles on micro-bubble outlines: Micro-bubble was recognised by fitting a circle on its recognised outline. A fast and robust 3-point circle fit (modification of Kamgar-Parsi and Netanyahu [14]) was utilised. 4. Validation of circle fit: Fitted circles were validated based on size range, ratio of circle perimeter to recognised outline length and a ratio of circle area to area that overlaps with other circles, based on a criteria that the centre of the circle had to have higher original greyscale value than the inner side of the circle perimeter. 5. Lagrangian tracking of micro-bubbles in image sequence: After the recognition of micro-bubbles, three consecutive image frames were analysed to cluster images that belonged to the same micro-bubble. Thus each analysed micro-bubble had to be detected in at least three consecutive images. Only the rising micro-bubbles were of interest and thus the allowed velocity range was limited upwards. A constant rise velocity and size for each micro-bubble were assumed allowing a maximum of 20% variation in pseudo-distance s; defined as:

( ) ( ) ( )212

,2

,, )()()()()()( jdkdjykyjxkxs iiieiieijk −+−+−= − (1) where the bubble coordinates and size were predicted from the first two frames for the third image frame, and estimates (xi,e,yi,e and di-1) were compared to potential pair image values (xi,yi and di). The tracking method corresponded to a best-estimate method presented by Ouellette et al [15].

2.5 Bubble size and velocity discretisation

Digital imaging technique provided plenty of information on objects under study: object size, shape and concentration in the image. The relation of an object image size and real object size was straight-forward when the objects were spherical and the geometrical calibration between the image plane and the object plane was provided, as in the case of this study. The relation between object concentration in image and in suspension can be obtained in dilute concentrations [16] and the size and shape of non-spherical objects can be statistically measured from a large set of images [12]. The gas



bubbles in this application were spherical and their size range was narrow, so the computation of size distributions was provided directly from image analysis results. Size distributions are discretised into bubble size classes with equal width of 0.05 mm and size range from 0.1 to 1 mm.

2.6 Computation of the bubble drag co-efficient

Bubble size, shape, rise velocity and the velocity of the surrounding fluid were measured with the imaging technique. Assuming a steady flow where only drag and buoyancy forces affect bubble motion, the bubble drag co-efficient was estimated. Suspensions were so dilute that the interactions of micro-bubbles could be neglected, but interactions between suspension contaminants and micro-bubbles clearly took place. These interactions were considered in the drag co-efficient (CD) of a bubble. Thus, we obtained a simplified momentum equation that covers micro-bubble motion:

( ) ( ) 021

=⋅⋅−+⋅−⋅−⋅⋅=∑ gVAUUUUCF BLBBLBLBLD ρρρ , (2)

where AB, VB and UB are bubble’s surface area, volume and rise velocity, ρ is fluid density and UL is liquid velocity, obtained as the instantaneous mean liquid velocity in the whole measurement volume. In the case of spherical gas bubbles, momentum eq. (4) reduces to present CD in the following form:

( )( ) ( )22 3

434

LB

B

LB

B

L

LBD

UU

gd

UU

gdC

−⋅≈

−⋅⋅

−−=

ρρρ . (3)

The drag co-efficient of a gas bubble rising in a stagnant pure liquid can be computed as a function of the bubble’s Reynolds number. Depending on the bubble Reynolds number range used, ReB can be from either Hadamard-Rybzynski (for ReB ≤ 11) [17] or Moore’s solution (for 11 < ReB ≤ 500) [18]. However, Mei et al [19] obtained an empirical correlation that matches both correlations, valid for micro-bubble Reynolds numbers presented in this study:

( )

+++=

−− 12/1Re315.3121

Re81

Re16

BBB

DC . (4)

To further quantify measured rise velocities, a comparative analysis was performed where the measured data was fitted to Mei’s eqn (4), describing drag coefficients required to model the ascent of spherical bubbles in suspensions.


Bubble dynamics in suspension flow differ from dynamics of a pure air/water suspension. During their rise, bubbles have to push solids away from their path. In addition, fibres in model suspensions dampen the flow velocity fluctuations and resist the flow from becoming turbulent, as reported in [3, 4, 12]. In process waters, bubbles were seen to align in swarms more strongly than in pure gas-liquid suspension. The first bubble of a bubble swarm rises slower than the bubbles in its wake increasing the chances of bubble coalescence. On the other hand, surface active contaminants present in these suspensions induce bubble



surface stabilisation, prevent bubble coalescence and lower bubble surface tension [6]. High-speed image sequences on model suspensions show that particles often attach on micro-bubbles and decrease their rise velocity, similar to phenomena seen in pulp suspensions [4]. Commonly, the buoyancy of a micro-bubble is not sufficient to detach the bubble from flocculated fines and thus some bubbles tend to remain trapped within suspension, making them naturally tedious to deaerate completely without any special process stages [2, 4, 20].

3.1 Micro-bubble size distribution in tested suspensions

Distributions of micro-bubbles (db < 1.0 mm) after depressurising air-saturated suspensions from 3 bars to normal pressure are presented in fig. 3. The bubbles generated in pure liquids are larger than bubbles in fibre suspensions. The presence of contaminants in every suspension shifts the distribution towards smaller bubble size. Fibre content appears to have even higher impact on mean bubble size than changes in surface tension and viscosity. Although increased fibre consistency is also noted to promote large bubble formation [3, 4], that trend is not visible here. Actually fewer micro-bubbles are generated in fibre suspensions than in pure liquids.

Figure 3: Micro-bubble size distribution on pressure drop for model

suspensions and white waters.

Differences in bubble size distributions between white waters are small despite the differences in water properties. All white waters have sufficient solid contents to act as nucleation sites in the event of depressurisation and bubble formation, thus increasing the probability of small bubble formation. The differences in oxygen saturation levels on white waters are measured in few ppms and thus it seems that the chemical and physical properties of these white waters have little effect on the formed bubble size distribution. White waters produce micro-bubbles with similar size distribution to 0.125M n-butanol solution that has similar surface tension and apparent viscosity. However, the generated micro-bubble concentrations differ significantly.

3.2 Bubble behaviour in model suspensions

Fig. 4 shows sequences of three consecutive images overlaid on top of each other. The circles on top of the images present the detected in-focus bubbles and



the blue arrows correspond to their velocities. Fig. 4 demonstrates how already the experimental images reveal the differences of suspensions. For example, CMC solution produces large bubbles that have higher tendency to attach on solid surfaces. The stagnant bubbles remain undetected and are not analysed.

Figure 4: Micro-bubble analysis images on some measured cases: left,

micro-bubbles in deionised water, 0.2% CMC solution, solution of 0.2% CMC and 0.15% fibres, and on right bubbles in 0.125 M n-butanol.

Figure 5: Bubble drag coefficients and mean bubble rise velocities according

to micro-bubble Reynolds number and size for model suspensions.

Fig. 5 presents the measured drag coefficients (CD) with respect to bubble Reynolds number (ReB) and the measured rise velocities with respect to bubble diameter. As seen in tables 1 and 2, the model suspensions had lower solid contents than any process water. Suspensions with CMC had high viscosities and those with n-butanol were at equal level to mill waters both in terms of apparent viscosity and surface tension. As can be seen in Fig. 5 micro-bubble drag is elevated by suspension viscosity and lowered by surface tension – presence of fibres don’t seem to have significant impact on bubble drag, likely due to channelling of rising bubbles to areas with lowest consistencies.



The bubble rise velocity in water and liquid-solid suspensions with low solids consistencies can be reasonably estimated using the Mei’s model. Measured CD differs somewhat from eqn (4) for bubbles with low ReB, seemingly experiencing an exponential decline as ReB grows. Deviations from Mei’s model seem to arise in suspensions of higher apparent viscosity, which was expected given that the model is based upon uncontaminated liquid medium. Presence of fibres appears less significant suggesting that apparent viscosity is the property most hindering micro-bubble rise in suspensions. Similarly, bubbles in the most viscous suspensions had by far the slowest rise velocities within all size categories, as seen in fig. 5. While all model waters had reduced rise velocity in relation to pure water, viscous (over 5.5 mPas) CMC-solutions affected bubble motion most. Bubble rise in surfactant treated suspensions with viscosity around 2 mPas was faster, but still far from reference waters. Fibres are supposed to hinder bubble rise mainly through collisions and attachment, even though these have a minimal affect bubbles can readily avoid entanglement and no web formation occurs in waters this diluted. Also fibres present in CMC-solutions appear insignificant in relation to viscous drag of the liquid phase.

3.3 Bubble behaviour in white waters

Small changes in physico-chemical properties of white waters made detailed analysis on drag or terminal velocity of bubbles problematic. In relation to deionised water, the drag of the smallest bubbles in papermaking suspensions was, however, clearly elevated, as seen in fig. 6 left. As in model waters, the most remarkable deviations from Mei’s theory [19] occur for bubbles with the smallest ReB values. This is mainly attributed to surface contamination of bubbles that reduces surface tension and to elevate viscosity of the white water’s liquid phase. The uniformity of drag coefficient development on suspensions allows generalisations to be made in modelling work for most typical mill white waters as bubble rise media. Drag coefficient of micro-bubbles is higher in white waters than in pure water when ReB < 10 and lower when ReB > 10.

Figure 6: Bubble drag coefficients and mean bubble velocities according to

the micro-bubble Reynolds number and size for white waters.



The non-linearity of certain drag and micro-bubble velocity measurements can be attributed to a small population of detected bubbles in each size class and to the attachment of some, especially larger micro-bubbles on to pulp fragments. This was mostly seen on white waters 2, 3 and 4 in fig. 6. Based on experiences with different suspensions, we can conclude that the differences in solid contents and deviations in apparent viscosity or suspended particle sizes are so minor that they can be approximated with a single model. However, the rise velocity of micro-bubbles appears to be the weakest in suspensions WW1-WW3 and WW5 that mostly contain materials originating from mechanical pulp, e.g. wood based extractives, colloidal pitch, etc. Their appearance is not seen in present analyses of water properties and would thus make ideal continuation for this line of study.

4 Conclusions

A digital imaging method was used to present the degree on which apparent viscosity, surface tension and solids consistency affect micro-bubble formation, drag and rise in dilute suspensions. Similarity of micro-bubble formation and uniformity of drag coefficient development in white waters was shown with data needed to devise a novel model describing drag forces affecting micro-bubble rise in white waters. Effects of contaminants in process water on dynamics of micro-bubbles were shown to greatly affect the rate of bubble evacuation and thus the kinetics of suspension deaeration process. Results revealed that contamination decreases bubble size and reduces the micro-bubble concentration. The results can be explained with the changes in surface tension and viscosity of the suspension. In white waters, the drag co-efficient of micro-bubbles is higher than in pure water when ReB < 10 and lower when ReB > 10.

Acknowledgements

Authors would like to thank Tekes (The Finnish Funding Agency for Technology and Innovation), PaPSaT graduate school and Academy of Finland for financially supporting this research.

References

[1] Stoor, T. Air in pulp and papermaking processes, Oulu Univ. press, pp. 66, 2006.

[2] Helle, T-M. Qualitative and quantitative effects of gas content on papermaking. Paper & Timber, 82(7), pp. 457-463, 2000.

[3] Heindel, T.J. Bubble size in cocurrent fiber slurry. Industrial and Engineering Chemistry Research, 41, pp. 632-641, 2002.

[4] Reese, J., Jiang, P. & Fan, L-S. Bubble characteristics in three-phase systems used for pulp and paper processing. Chemical Engineering Science, 51(10), pp. 2501-2510, 1996.



[5] Margaritis, A., te Bokkel, D.W. & Karamanev, D.G. Bubble rise velocities and drag coefficients in non-Newtonian polysaccharide solutions. Biotechnology and Bioengineering, 64(3), pp. 257-266, 1999.

[6] Okazaki, S. The Velocity of Air Bubble Ascending in Aqueous Solution of Surface Active Substance and Inorganic Electrolyte. Colloid & Polymer Science, 185, pp. 154-157, 1962.

[7] Hubbe, M.A. Water and papermaking 2, white water components. Paper Technology, 48(2), pp. 31-40, 2007.

[8] Garver, T.M., Xie, T.B. & Kenneth H. Variation of white water composition in a TMP and DIP newsprint paper machine. Tappi Journal, 80(8), pp.163-173, 1997.

[9] Lindken, R. & Merzkirch, W. A novel PIV technique for measurements in multiphase flows and its application to two-phase bubbly flows. Experiments in Fluids, 33, pp. 814-825, 2002.

[10] Honkanen, M., Saarenrinne, P., Stoor, T. & Niinimäki, J. Recognition of highly overlapping ellipse-like bubble images. Measurement Science and Technology, 16, pp. 1760-1770, 2005.

[11] Honkanen, M. & Marjanen, K. Analysis of the overlapping images of irregularly-shaped particles, bubbles and droplets. Proc. of Int. Conf. on Multiphase Flow, Leibzig, Germany, paper 559, 2007.

[12] Honkanen, M. Direct optical measurement of fluid dynamics and dispersed phase morphology in multiphase flows. Univ. print, Tampere, pp. 80, 2006.

[13] Honkanen, M., Eloranta, H. & Saarenrinne, P. Submersible, planar shadow image velocimetry system for online, in-situ analysis of multiphase flows in industrial processes. Proc. of the 13th Int. Symposium on Flow Visualization, Nice, France, paper 222, 2008.

[14] Kamgar-Parsi, B. & Netanyahu, N.S. A nonparametric method for fitting a straight line to a noisy image, IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(9), pp. 998-1001, 1989.

[15] Ouellette, N.T., Xu, H. & Bodenschatz, E. A quantitative study of three-dimensional Lagrangian particle tracking algorithms. Experiments in Fluids, 40, pp. 301-313, 2006.

[16] Frank, X., Li, H. & Funfschilling, D. An analytical approach to the rise velocity of periodic bubble trains in non-Newtonian fluids. European Physics Journal E, 16, pp. 29-35, 2005.

[17] Hadamard, J. Mouvement permanent lent d’une sphère liquide et visqueuse dans un liquide visqueux. Comptes Rendus Mathématique. Académie des Sciences, Paris, 152, pp. 1735-1738, 1911.

[18] Moore, D.W. The boundary layer on a spherical gas bubble. Journal of Fluid Mechanics, 16, pp. 161-176, 1963.

[19] Mei, R., Klaussner, J.F. & Lawrence, C.J. A note on the history force on a spherical bubble at finite Reynolds number. Physics of Fluids, 6, pp. 418-420, 1994.

[20] Haapala, A., Stoor, T., Liimatainen, H., Nelo, M. & Niinimäki, J. Passive white water deaeration efficiency in open channel flow. 62nd Appita Ann. Conf., Rotorua, NZ, 2008.



Effects of physical properties on the behaviour of Taylor bubbles

V. Hernández-Pérez, L. A. Abdulkareem & B. J. Azzopardi Department of Chemical and Environmental Engineering, Faculty of Engineering, University of Nottingham, UK

Abstract

Gas-liquid flow in vertical pipes, which is important in oil/gas wells and the risers from sea bed completions to FPSOs, was investigated to determine the effects of physical properties on the characteristics of the mixture such as void fraction, structure frequency and velocity as well as the shape of 3D structures. These are difficult to visualise using conventional optical techniques because even if the pipe wall is transparent, near-wall bubbles would mask the flow deep in the pipe. Therefore, more sophisticated methods are required. Two advanced wire mesh sensors (WMS) were used and the two-phase mixtures employed were air-water and air-silicone oil. The effect of fluid properties is accounted for in terms of the Morton number. It was found that the flow pattern is affected by the fluid properties as the results revealed that contrary to what is commonly assumed when modelling pipe flow, the flow is not symmetric, with a lot of distortion, which is even higher for the air-silicone oil mixture. Keywords: gas/liquid, vertical, wire mesh sensor, slug flow, void fraction, structure velocity.

1 Introduction

One of the most common structures found in gas-liquid flows is the Taylor bubble; it is related directly to the slug flow pattern in upward vertical flow, for instance in oil/gas applications. This bubble occupies the greater part of the pipe cross section. Liquid between the Taylor bubble and the pipe wall flows around this bubble as a thin film. Traditionally the Taylor bubble has been stereotyped as a bullet-shape bubble and most hydrodynamic models of pressure drop for



slug flow take on this shape as suitable, but in reality the behaviour of Taylor bubbles is rather complex and they can adopt different shapes affected by different parameters, such as pipe diameter and fluid properties. Several authors have reported in the literature a lack of slug flow in relatively large pipe diameters. The behaviour of bubbles in turn can affect parameters such as pressure drop and heat transfer coefficients. Of particular importance can be the role that fluid properties play in oil/gas production, where they can vary from one well to another. The simple change of fluid from one mixture to another can result in the variation of several important properties such as viscosity, density and surface tension. In fact, the viscosities measured for different heavy oils can vary by orders of magnitude. However in the literature, there is a lack of information regarding the effect of fluid properties on the behaviour of Taylor bubbles, particularly at high Reynolds numbers. Most of the work using different gas-liquid mixtures has been directed towards the measurement of average liquid holdup and pressure drop and flow pattern transition, for instance Weisman et al. [1] as well as Nädler and Mewes [2], have studied effect of fluid properties on flow patterns in horizontal two-phase flow. For vertical flow, not surprisingly, most of the extensive work reported has been regarding the motion of single Taylor bubbles using air-water mixtures, as it is the basis for the study of more complicated flows. One of the first investigations on these large bubbles was carried out by Davies and Taylor [3], who carried out viscous potential flow analysis of cap bubbles and found that the terminal rise velocity was simply related to the curvature radius of the cap bubble. However due to disturbances induced by the bubbles in the liquid, it is evident that the Taylor bubbles in a train of bubbles will behave different from a single bubble in static liquid. In general, the theoretical approach has been limited to the low Reynolds number regime and several studies have been published over the years, for instance Moore [4] carried out a study of a gas bubble in a viscous liquid and more recently Tomiyama et al. [5] studied the terminal velocity of a single bubble rising through an infinite stagnant liquid in a surface tension force dominant regime theoretically and experimentally. In many cases investigators have followed the experimental approach in order to tackle the behaviour of Taylor bubbles in more attention demanding problems. Nicklin et al. [6] established a correlation for translational velocity of Taylor bubbles in moving liquid while White and Beardmore [7] used dimensionless groups to account for the combined effect of several variables, including fluid properties. Later developments in instrumentation have allowed researchers to look further at the two-phase flow; Van Hout et al. [8] measured the translational velocity of elongated bubbles in continuous slug flow, Hassan et al. [9] studied two-phase flow field and 3D structures in bubbly flow using particle image velocimetry (PIV). However, no effect of physical properties has been reported. Not until recently have a few studies been presented on numerical simulation of Taylor motion, usually they are limited by the sort of assumptions that need to be made, such as flow symmetry and stagnant liquid. For example Clarke and Issa [10] modelled the motion of a periodic train of Taylor bubbles in vertical flow by



imposing cyclic conditions at the inlet and outlet of the slug unit based on the assumption that the flow pattern repeats itself over consecutive slug units. Taha and Cui [11] among others have highlighted the use of dimensionless groups in the study of Taylor bubbles along with the numerical approach, as correlations of experimental data are generally developed in terms of dimensionless groups rather than in terms of the separate dimensional variables in the interests of compactness and in the hope of greater generality. The shape of the bubbles can change with the local hydrodynamic conditions, adding new degrees of freedom to an already complex problem. Knowledge of the Taylor bubble behaviour in moving liquid is fundamental to our understanding of multiphase flow, particularly when fluid properties are varied. In this work, a study of the effect of liquid properties on Taylor bubble behaviour is performed based on experimental data obtained at high Reynolds numbers. Particular emphasis is put in the extraction of 3D structures, with the use of advanced instrumentation known as wire mesh sensor, looking forward to use this information for validation of computational models for determination of gas-liquid interface.

2 Experimental facility

Two advanced wire mesh sensors (WMS), developed at Forschungszentrum Rossendorf-Dresden, have been used in a two-phase flow facility at Nottingham. The basic part of the facility has been frequently described elsewhere, for example Hernández-Pérez et al. [12] and Azzopardi et al. [13], and for the sake of simplicity details are omitted here. The sensors are described in detail by Prasser at al. [14] and Da Silva et al. [15] respectively. Both of them have a grid of 24 × 24 measurement points evenly distributed across the pipe cross-section given by a 24 × 24 wire configuration in two planes. The first is based on conductivity measurements and is suitable for water; the second is based on capacitance measurements and works with non-conductive materials such as oil. Figure 1 shows a picture of the wire mesh sensor. The two-phase mixtures employed were air-water and air-silicone oil. The latter liquid has a surface tension about one third that of water and a viscosity of ~5x water. The physical properties of the fluids used are given in Table 1.

Table 1: Properties of the fluids.

Fluid Density (kg/m3)

Viscosity (kg/ms)

Surface tension (N/m)

Air 1.224 0.000018 0.072

Water 1000 0.001

Silicone oil 900 0.005 0.02



Figure 1: Wire-mesh sensor (2 × 24 electrode wires).

The data were gathered in two campaigns: In the first campaign the air-water mixture was used with the conductivity wire mesh sensor and a pair of capacitance probes in order to obtain structure velocity, as reported by Hernández-Pérez et al. [12]. In the second campaign air-silicone oil was employed with the capacitance wire mesh sensor as well as Electrical capacitance tomography (ECT), as described by Abdulkareem et al. [16]. The test pipe is 6 m long and 67 mm diameter. In both campaigns, the wire mesh sensor was located at 4.92 m from the mixing or inlet section and we focus on vertical flow. Conditions studied were superficial velocities: for air ranging from 0.05 to 4.7 m/s and for liquid from 0.0 m/s to 0.7 m/s for both water and silicone oil. The data were taken at a data acquisition frequency of 1000 Hz over an interval of 40 s for the wire mesh sensors and 200 Hz for both capacitance probes and ECT. In addition, high speed video system was used in order to obtain real images of the flow regimes under different conditions.


The behaviour of the Taylor bubble is described by means of several parameters such as time series of cross-sectional area averaged void fraction. Further analysis of these time series will allow flow patterns and structure frequencies to be extracted and compared. Finally the full 3D structures will be presented. Applying dimensional analysis, the effect of the properties can be put in terms of dimensionless numbers. For gas bubbles rising in liquids, the viscosity ratio and density ratio tend to be very small and therefore it is generally sufficient to consider only three dimensionless groups, namely the Morton number (M), the Eotvos Number (Eo) and the Froude number (Fr) as identified by White and Beardmore [7]. M is also called the properties number and together with the Eötvös number, is useful to characterize the shape of bubbles. These numbers are defined respectively as:



324 )( LGLL ρρgM (1)

/)( 2DgEo GL (2)

and

LGLTB gDUFr /)(/ (3)

Here g is the gravity acceleration, µ is the viscosity, ρ is the density, σ is the surface tension, D is the pipe diameter and UTB is the velocity of a Taylor bubble rising in motionless liquid. The subscripts L and G refer to liquid and gas respectively. Therefore for air-water M = 2.64 10-11 and Eo=610.9 whereas for air–silicone oil M = 1.36 10-9 and Eo=1970. According to Bhaga and Weber [17], since M for both of the two-phase mixtures employed in this work are lower than 4 10-3, the bubble behaviour is expected to be a function of both Morton and Reynolds numbers. It can be observe from eqn (1) that variations in M are mainly due to the factor, µ4, since ρ and σ do not vary much from water to silicone oil. Water is usually considered as a low M number fluid.

3.1 Time series of void fraction

The time series of cross sectional average void fraction show in a simple way the occurrence of structures as the gas-liquid mixture flows along the pipe. It also constitutes the raw data for application of statistical analysis to judge the flow behaviour. Figure 2 shows a typical run in which slug flow is present, the appearance of Taylor bubbles can be identified as high void fraction intervals whereas the low void fraction intervals correspond to the liquid slugs. The irregular variation of the void fraction reveals the transient nature of the Taylor bubbles. In addition, the liquid film variation can be observed to have an irregular shape.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.6

0.8

1

time (s)

void

fra

ctio

n

Figure 2: Typical time series of cross-section average void fraction.

3.2 Flow patterns and structure frequency

In considering Taylor bubbles, it is important to know the flow conditions under which these particular structures occur. Statistical analysis of the time series data can provide an insight into flow pattern identification. In this sense, a widely used and accepted method is Probability Density Function (PDF). It is also important to characterise the structures in terms of their occurrence frequency,



which was found from the time series utilizing the Power Spectral Density (PSD) technique. Figure 3 shows the time series, the PDF and the PSD for both two-phase mixtures at a particular flow condition. In this figure a time interval of 10 s is used to plot the time series however for processing the PDF and the PSD, the data gathered in 40 s have been considered. It is apparent from both the time series and PDF plots that the fluctuation amplitude of void fraction is bigger for the case of air-water mixture. Also, the double peak shape of the PDF plot, characteristic of slug flow is not well defined; indeed the flow is greatly distorted as illustrated by the 3D structures in section 3.3.

0 5 100

0.2

0.4

0.6

0.8

1

Time (s)

Voi

d fr

actio

n

0 0.5 10

1

2

3

4

Void Fraction

PD

F (

%)

0 1 2 3 40

1

2

3

4x 10

5

Frequency (Hz)

PS

D

0 5 100

0.2

0.4

0.6

0.8

1

Time (s)

Voi

d fr

actio

n

0 0.5 10

1

2

3

4

Void Fraction

PD

F (

%)

0 1 2 3 40

0.5

1

1.5

2x 10

5

Frequency (Hz)

PS

D

Air-water air-silicone oil

Figure 3: Effect of fluid properties on flow pattern transition, liquid superficial velocity 0.2 m/s and gas superficial velocity 0.94 m/s.

Due to the random distortion and fluctuations of Taylor bubbles, in practice, it is usually more convenient to study its behaviour in terms of average parameters such as the frequency. Looking at Figure 3, it can be observed that the frequency of Taylor bubbles is also affected by the change in physical properties of the liquid phase, for a particular condition the structure frequency increases when the viscosity is increased as the dominant frequency predicted with PSD is higher for the case of silicone oil liquid. Also, the liquid film in the Taylor bubble is in



general thinner for the case of the air-water mixture. This is in agreement with Goldsmith and Mason [18] who found a similar behaviour. Figure 4 shows that, the change of liquid from water to silicone oil, which means an increase in the Morton number, keeping the liquid superficial velocity constant produces a displacement of the bubbly-cap bubble boundary towards a higher gas superficial velocity. This behaviour was observed to be evident at higher liquid superficial velocities (0.7 m/s). This is similar to the findings of Weisman et al. [1] in terms of the flow pattern boundaries being affected by the physical properties of the fluids and can be thought of as an indication of a longer development distance required by a more viscous flow. Most of the PDFs in Figure 4 show a shape that corresponds to the cap bubble flow pattern.

0

2

4

6

8

10

Bubbly flow

Usg=0.05 m/s

Bubbly flow

0.15 m/s

Cap bubble

0.28 m/s

Cap bubble

0.47 m/s

Cap bubble

0.94 m/s

Churn flow

1.4 m/s

0 0.5 10

5

10

Cap bubble

PD

F (

%)

0 0.5 1

Cap bubble

0 0.5 1

Cap bubble

Void Fraction0 0.5 1

0

5

10

Cap bubble

0 0.5 1

Taylor bubble

0 0.5 1

Taylor bubble

Figure 4: Probability density function of the time series of the cross-section average void fraction at liquid superficial velocity 0.7 m/s. Lower row air-water, upper row air-silicone oil.

3.3 Shape of 3D structures

The detection of the sharp gas-liquid interfaces plays an important role in identification of flow patterns. The wire mesh sensors employed in this study are capable of providing the Taylor bubbles shapes by creating a 3D reconstruction of the flow as shown in Figure 5, where the Taylor bubbles have been brought out for two cases. A time interval of 2 s has been used in both figures 5 and 6. The main features observed in Figure 5 are asymmetry and distortion of the Taylor bubbles, the distortion of bubbles appears to be higher in the air-silicone oil mixture, which can be due to the lower surface tension of the silicone oil, as the surface tension acts as the force that restrains deformation, despite its higher viscosity, as the main effect of viscosity on the displacement of a bubble is the production of a drag force that tends to slow down its relative motion in the surrounding fluid. From the 3D representation in Figure 5 it is clear that the physical properties greatly influence the shape of the structures. For the conditions used in this work the typical Taylor bubble shape is rarely observed for the case of silicone oil.



air-water air-silicone oil

Figure 5: Comparison of Taylor bubble shapes in 3D. Superficial velocities: liquid 0.7 and gas 1.4 m/s.

Both the increase of viscosity and the decrease of surface tension in silicone oil with respect to water contribute to the increase of the Morton number. Therefore the more deformation observed for the air-silicone oil is in agreement with Duineveld [19], who observed that for water, low Morton number, the bubbles rising have relatively low deformation. The deformation of a Taylor bubble is also related to the stresses generated from the translation movement; as a result, as the mixture velocity increases, it can be observed in Figure 6 that the Taylor bubble is totally broken when the gas superficial velocity reaches 0.94 m/s. This phase interaction mechanism might be the reason why, as reported by Azzopardi et al. [13], liquid structures inside the gas core of the Taylor bubble (Wisps) have been found to exist in the churn flow regime. The deformation due to the high mixture velocity is also related to the high level of turbulence intensity in the flow. The flow conditions employed in the present work, involve large Reynolds numbers. Indeed the classical bullet shape Taylor bubble is rarely observed at the conditions of liquid superficial velocity 0.7 m/s for both water and silicone oil. It is generally accepted that turbulence enhances bubble breakup, however currently turbulence remains an



unsolved problem although different turbulence modelling approaches have emerged. Bubble distortion increases not only with the Morton number but also with Eotvos number, which involves the pipe diameter. This is congruent with the fact that Taylor bubble distortion is proportional to pipe diameter, as it has been reported in the literature.

Usg= 0.15 0.28 0.47 0.94 m/s

Figure 6: Effect of mixture velocity on Taylor bubble shape, air-water at liquid superficial velocity 0.7 m/s and different gas superficial velocities.

3.4 Translational velocity

The translational velocity, which is an essential parameter to characterise Taylor bubbles, has been obtained by cross correlating a pair of signals delivered by the capacitance probes for the case of air-water and ECT for air-silicone oil. Some of these results are plotted in Figure 7. They show a higher translational velocity for the case of the silicone oil. The higher translational velocity is related to the higher frequency shown in the PSD plot in Figure 3 and the higher overall liquid holdup in the pipe for the air-silicone oil. This finding is in agreement with Van Hout et al. [8], who found that the drift velocity for continuous slug flow is enhanced by the dispersed bubbles in the liquid slug body. Similarly, Hills and Darton [20] found considerable enhancement in the velocity of large bubbles in bubble swarms. Based on observations of Hills [21] who found that when a cap bubble rises in a swarm of small bubbles, the small bubbles never coalesce with the nose of the cap bubble, they suggested that the enhancement is due to the shape change in the cap bubble. However, Hills’ arrangement consisted of a two-dimensional bubble column of width so much larger than the cap bubble. Therefore the small bubbles in most cases manage to escape from the cap bubble driven by the liquid velocity field.



Figure 7: Effect of physical properties on translational velocity. Liquid superficial velocity 0.2 m/s.

4 Conclusion

A study of the effect of the change of fluid properties on Taylor bubble behaviour has been carried out with the use of advanced instrumentation, and a clear effect has been observed. For a higher Morton number, which corresponds to the silicone oil, there is more distortion. Other flow features such as the liquid film and structure frequency are also affected. This comparison shows a remarkable effect of the physical properties on the flow pattern. These results can be used to validate the numerical modelling of Taylor bubbles, which is increasingly gaining popularity, as computational fluid dynamics codes are becoming more widespread.

Acknowledgements

This work has been undertaken within the Joint Project on Transient Multiphase Flows and Flow Assurance. The Authors wish to acknowledge the contributions made to this project by the UK Engineering and Physical Sciences Research Council (EPSRC) and the following: - Advantica; BP Exploration; CD-adapco; Chevron; ConocoPhillips; ENI; ExxonMobil; FEESA; IFP; Institutt for Energiteknikk; PDVSA (INTERVEP); Petrobras; PETRONAS; Scandpower PT; Shell; SINTEF; StatoilHydro and TOTAL. The Authors wish to express their sincere gratitude for this support. V. Hernández-Pérez was supported by Mexican Council for Science and Technology (CONACyT) with a PhD scholarship. L. Abdulkareem would like to thank The Kurdish Government for supporting his PhD study.



References

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[2] Nädler, M. & Mewes, D., Effects of the liquid viscosity on the phase distributions in horizontal gas-liquid slug flow, Int. J. Multiphase Flow, Vol. 21, pp. 253-266, 1995.

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[5] Tomiyama, A., Celata, G.P., Hosokawa, S. & Yoshida S., Terminal velocity of single bubbles in surface tension force dominant regime, Int. J. Multiphase Flow Vol. 28, pp. 1497-1519, 2002.

[6] Nicklin, O.J., Wilkes, J.O., & Davison, J. F., Two-phase flow in vertical tubes, Trans. I. Chem. Engrs. Vol. 40, pp. 61-68, 1962.

[7] White, E.R. & Beardmore, R.H., The velocity of rise of single cylindrical air bubbles through liquids in vertical tubes, Chem. Eng. Sci. Vol. 17, pp. 351-361, 1962.

[8] Van Hout, R., Barnea, D., & Shemer, L., Translational velocities of elongated bubbles in continuous slug flow, Int. J. Multiphase Flow Vol. 28, pp. 1333-1350, 2002.

[9] Hassan, Y.A., Schmidl, W. & Ortiz-Villafuerte, J., Investigation of three-dimensional two-phase flow structure in a bubbly pipe flow, Meas. Sci. Technol. Vol. 9, pp. 309–326, 1998.

[10] Clarke, A. & Issa, R. I., A numerical model of slug flow in vertical tubes”, Computers & Fluids, Vol. 26, pp. 395-415, 1997.

[11] Taha, T. & Cui, Z.F., CFD modelling of slug flow in vertical tubes, Chem. Eng. Sci. Vol. 61, pp. 676-687, 2006.

[12] Hernández-Pérez, V., Azzopardi, B.J. & Morvan, H., Slug flow in inclined pipes. 6th Int. Conf. Multiphase Flow, Leipzig, Germany 5 to 9 July 2007.

[13] Azzopardi, B.J., Hernández-Pérez, V., Kaji, R., Da Silva M.J., Beyer, M., & Hampel, U., Wire mesh sensor studies in a vertical pipe, HEAT 2008, Fifth International Conference on Transport Phenomena in Multiphase Systems, Bialystok, Poland, 2008.

[14] Prasser, H.-M., Bottger, A., & Zschau J., A new electrode-mesh tomograph for gas-liquid flows, Flow Meas. Instr. Vol. 9, pp. 111-119, 1998.

[15] Da Silva, M.J., Schleicher E. & Hampel, U., Capacitance wire-mesh sensor for fast measurement of phase fraction distributions. Meas. Sci. Tech. Vol. 18, pp. 2245-2251, 2007.

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[18] Goldsmith, H. L. & Mason, S. G., The movement of single large bubbles in closed vertical tubes, J. Fluid Mech. Vol. 14, pp 42-58, 1962.,

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[21] Hills, J.H., The rise of a large bubble through a swarm of smaller ones, Trans. I. Chem. Engrs. Vol. 53, pp. 224-233, 1975.



Numerical simulation of one-dimensional mixed flow with air/water interaction

F. Kerger1,2, S. Erpicum1, P. Archambeau1, B. J. Dewals1,2 & M. Pirotton1 1Laboratory of Hydrology, Applied Hydrodynamics and Hydraulic Constructions (HACH), Liège University, Belgium 2Belgian Fund for Scientific Research F.R.S-FNRS, Belgium

Abstract

An original one-dimensional unified numerical model dealing with aerated mixed flow, characterized by a simultaneous occurrence of free-surface and pressurized flow, is derived and applied to the case of a gallery. The mathematical model is based on a new integration of the Homogeneous Equilibrium Model (HEM) over the cross section of a free-surface flow and consists of a simple set of equations analogous to the Saint-Venant equations. In addition, both free-surface and pressurized flows are mathematically modeled by means of the free-surface set of equations (Preissmann slot model). The original concept of the negative Preissmann slot is proposed to simulate sub-atmospheric pressure. This model is shown to be particularly well suited for the simulation of bubbly and intermittent flows. Keywords: hydraulics, Preissmann slot, homogeneous equilibrium model.

1 Introduction

Mixed flows, characterized by the simultaneous occurrence of free-surface and pressurized flows, are frequently encountered in river networks, sewer systems, storm-water storage pipes, flushing galleries, bottom outlets,… As a matter of fact, some hydraulic structures are designed to combine free-surface and pressurized sections (e.g. water intakes). In addition, dynamic pipe filling bores may occur in hydraulic structures designed only for conveying free-surface flow under an extreme water inflow or upon starting a pump. During such a transition, highly transient phenomena appear and may cause structural damages to the



system [1], generate geysers through vertical shafts [2], engender flooding,… What is more, air/water interactions may arise, particularly at the transition bore [3], and thoroughly alter the flow regime and its characteristics. On account of the range of applications affected by mixed flows, a good prediction of mixed flow features is an industrial necessity. Numerical simulation of mixed flow remains, however, challenging for two main reasons. Dissimilarity in the pressure term arises between the classical sets of equations describing free-surface and pressurized flows. Air/water interaction has to be taken into account through a two-phase flow model. Different mathematical approaches to describe mixed flows have been developed to date. Firstly, the so-called shock-tracking approach consists of solving separately free-surface and pressurized flows through different sets of equations [4, 5]. Secondly, the Rigid Water Column Approach [6] treats each phase (air/water) separately on the basis of a specific set of equations. The latter approach succeeds in simulating complex configurations of the transition. However, using this method for practical application is not possible because of the complexity and specificity of the algorithm. Thirdly, the so-called shock-capturing approach is a family of methods that computes pressurized and free-surface flows by using a single set of equations [7–9]. In this paper, a shock-capturing approach is used, based on the widespread model of the Preissmann slot [7]. Free-surface flow and pressurized flow are in this way equally solved through a free surface set of equations. An original concept developed by the authors, the negative Preissmann slot, extends the Preissmann slot model to simulate sub-atmospheric pressurized flows. Computing air-water interaction requires using a two-phase flow model. On the one hand, to the authors’ knowledge, no mixed flow model takes into account the effect of entrained air in the water flow. Only the air phase pressurization is usually modeled, as in the Rigid Water Column [6] and in the shock-capturing model of Vasconcelos et al. [8]. On the other hand, the usual multiphase flow investigations focus mainly on fully pressurized flow in small diameter pipes for chemical and mechanical engineering applications. There have been only a few attempts, often based on a transport equation [10], to simulate air entrainment in large hydraulic structures. Consequently, the current research aims at applying the classical model for multiphase flow to civil engineering applications. In this paper, an Homogeneous Equilibrium Model (HEM) coupled with the Preissmann slot model is derived by using the time and area averaging methods proposed by Ishii and Hibiki [11]. These developments have been carefully implemented in the one-dimensional module of the software package WOLF. WOLF is finite volume flow simulation modeling system developed within the Laboratory of Hydrology, Applied Hydrodynamics and Hydraulic Constructions (HACH) of the University of Liege. Application to this new model to the case of flows in a gallery is presented in this paper. Experimental results from a physical model build in the Laboratory of Structures Hydraulics of the University of Liege are used for comparison with numerical results.



2 Mathematical model

2.1 Homogeneous equilibrium model for free-surface flow

Homogenous flow theory provides the simplest technique for analyzing multiphase flows. Using suitable averaged properties, the fluid is treated as a pseudo-fluid that obeys the usual equations of a single-component flow. Under the assumption of equality between air and water properties, one obtains the Homogeneous Equilibrium Model (HEM) as demonstrated by Ishii and Hibiki [11]. This assumption is particularly suited for dispersed bubbly flow. The model is commonly used for the simulation of heat exchangers [12, 13], two-phase flow in ducts [14],…

2.1.1 3D Time-averaged governing equations HEM may be considered as a simplification of the Drift Flux Model [11] if the drift or diffusion of mass is negligible regarding the continuity equation for the gas phase. Thus, HEM 3D model is derived through the time averaging of the Local Instant Formulation for multiphase flow, the introduction of suitable mixture variables and the assumption of equilibrium between phases. For further details, we refer the interested reader to the classical book of Ishii and Hibiki [11]. The resulting equations write:

mm m

gg m g

Tm mm m m m m m m

0 continuity equationt

diffusion equationt

pt

momentum equation

v

v

vv v g M

(1)

where ρm[kg/m³] is the mixture density, vm[ms-1] is the mixture velocity vector (under the assumption of velocity equilibrium, vm= vwater= vair), g[-] is the air void fraction, g[s

-1] the phase change volume generation, pm[Nm-2] is the mixture pressure, m[Nm-2] and T[Nm-2] are the viscous and turbulent stress tensors, g[ms-2] is the gravity and Mm[kgs-2m-2] is the interfacial momentum source. Closure of eqn (1) requires the definition of the mixture variables and a constitutive equation. Air and water are supposed to be incompressible Newtonian fluid, and the mixture properties write then:

m g g g w g w

T

m g g g w

1 1

1 . .

v v (2)

At this point, no assumption is needed for the constitutive equations of the turbulent stress T, the phase change volume generation g, the pressure distribution pm and the mixture momentum source Mm.



2.1.2 Area average of three-dimensional homogeneous equilibrium model 1D HEM equations are obtained by area-averaging eqns (1) and (2) over the cross section of the flow. The originality of the following development is to consider a free-surface flow. It is indeed shown in section 3 how we use the free-surface set of equations to simulate pressurized flow as well. The whole process of integration is beyond the scope of this paper. It is performed by analogy to the integration of the Saint-Venant equation for pure water flow, as exposed in [15]. The flow is assumed to flow in the x-direction. Successive integration over the flow width (y-abscissa) and the flow depth (z-abscissa) results in a set of conservative hyperbolic partial differential equations that describe the flow below a free-surface. A non-dimensional analysis of eqn (1) leads to the following expression for the mechanical constitutive equation:

b

z

m w b

h

p(x, z, t) g cos dz 1 g z h

(3)

where hb is the distance between the z-abscissa and the bottom of the flow area. Eqn (3) simply states that the pressure distribution over the flow area is hydrostatic. 1D HEM for free-surface flow writes at the end of the day:

mg

m

2m

m F w

u

1 1 u 0t x

Z1 u g 1 S gp1 ux x

(4)

s

b

h

w s

h

where p h z l z dz

, [m²] is the flow cross-section area, um is the

flow velocity, Z[m] is the free surface elevation, SF[-] is the friction slope (resulting from the integration of the viscous, turbulent shear stress and the interfacial momentum source). The area-average of a given function f is defined as:

1

, , , ,

f g

b d

h l

h l

f x t f x y z t dydz

(5)

As pointed in [13], two-phase friction slope SF may be computed using the Darcy-Weisbach relation. The Colebrook relation for the friction factor still holds if a suitable “virtual viscosity” for the mixture is computed:

2 1 2.51 with 2 log

2 3.7 Re

1Re with 1

mF

h h

w m hm w

m

fu kS

D f D f

u D

(6)

with Dh[m] the hydraulic diameter of the cross-section, k[m] is the roughness height, Re[-] the Reynolds Number and µw[kgm-1s-1] the dynamic viscosity.



To close the partial differential system given by eqn (4), we still need to give an expression for the phase change volume generation g. Literature is abundant for empirical relations. To keep the generality of the model, a very fundamental relation given in [10] for air entrainment in free-surface flow is used:

g eqm

(7)

where and g are constants calibrated with experimental results. The onset of air entrainment is controlled by the parameter m=1 or m=0. Discretization of eqns (4)-(7) is performed by means of a finite volume scheme with an original flux vector splitting [16]. Time discretization is based on an explicit Runge-Kutta scheme to enhance the convergence in steady state applications.

2.2 Preissmann slot model

Pressurized flows are commonly described through the Allievi equations [17]. According to the Preissmann slot model [7], pressurized flow can be equally calculated through the free-surface equations by adding a conceptual slot at the top of a closed pipe (Figure 1b). When the water elevation is above the pipe crown, it provides a conceptual free-surface flow, of which the gravity

wavespeed is given by sc g T (Ts is the slot width). Strictly speaking, the

pressure wave celerity of a flow in a full pipe, referred by a[m/s], depends on the properties of the fluid, the pipe, and its means of support. In first approximation, its value is not dependant of the pressure value and may be computed on the basis of solid mechanics relations [17]. It is then easy to choose a slot width Ts which equalizes the gravity wavespeed c to the water hammer wavespeed a:

s

g dpT with a² A

a² d A

(8)

From a hydraulic point of view, all the relevant information is summarized in the relation linking the water height and the flow area (H-A). A specific relation corresponds to each geometry of the cross section (Figure 1a). Adding the Preissmann slot leads to linearly extend the relation beyond the pipe crown head. In order to simulate pressurized flows with a piezometric head below the pipe crown, the authors propose a new concept, called negative Preissmann slot. It consists in extending the Preissmann straight line for water height below the pipe crown (Figure 1c). Two values of the flow area correspond to each water level below the pipe crown: one for the free surface flow and one for the pressurized flow. The choice between the two relations is done according to the local aeration conditions. For steady flow applications, the choice of the slot width may be arbitrary. On the one hand, the wave celerity does not affect the steady state of a flow. On the other hand, explicit numerical schemes are characterized by a time step t that is limited by a CFL condition of the form

m

xNbC 1 with NbC max u c *

t

(9)



Figure 1: The Preissmann slot method under different flow conditions.

Figure 2: Description of the experimental setup.

It seems then reasonable to impose a wider slot than the width calculated with eqn (4) in order to decrease the computation time.

3 Steady flow application

This section outlines the application of the one-dimensional HEM solver for simulating stationary mixed flows taking place in a gallery. Numerical results are compared with experimental results provided by experimental investigations carried out in the Laboratory of Structures Hydraulics (HACH) of the University of Liege. The model (figure 2) includes a Plexiglas circular pipe linking two tanks. Topography of the upstream and downstream tanks has been built

a. Free surface flow b. Pressurized flow c. Sub-atmospheric pressurized flow

Pre

issm

ann

slot

mod

el

(inc

l. ne

gati

ve s

lot)

Water level

Slot accounts for the radial flow increase and the watercompressibility

Slot accounts for the radial flow decrease and the waterdilatation

Rel

atio

nfl

ow a

rea/

wat

er h

eigh

t

2sgAT

a

0.00

0.25

0.50

0.75

1.00

1.25

0.00 0.50 1.00 1.50 2.00

Non

-dim

ensi

onal

flo

w a

rea

(-)

Non-dimensional water height (-)

Free-surfaceflow

pipe

cro

wn

heig

ht

0.00

0.25

0.50

0.75

1.00

1.25

0.00 0.50 1.00 1.50 2.00

Non

-dim

ensi

onal

flo

w a

rea

(-)


Free-surfaceflow

Pressurized flow

pipe

cro

wn

heig

ht

0.00

0.25

0.50

0.75

1.00

1.25

0.00 0.50 1.00 1.50 2.00

Non

-dim

ensi

onal

flo

w a

rea

(-)


Sub-atmosphericpressurized flow

Pressurized flow

Choice based on aerationconditions

pipe

cro

wn

heig

ht

2sgAT

a

Circular pipe

Inlet Radial gate

Natural river bed

Natural river bed

Upstream tank

Downstream tank

Alimentation basin

Outlet

Natural river bed

Natural river bed

Outlet

Circular pipe

Inlet

Upstream tank

Alimentation basin

Downstream tank

a. Sketch of the experimental model b. View of the physical model



regarding realistic in-situ natural conditions. The gallery inlet and outlet structures are also represented. Experimental apparatus, measurement systems and results are described in details in [18].

3.1 Experimental investigations

Investigations focus mainly on stationary flows and aims at determining the flow discharge through the gallery as a function of the upstream pressure head. Strong air/water interactions may alter the flow behavior. In particular, the flow discharge through the gallery is strongly influenced by air/water interaction, and consequently depends of the aeration rate as well. Various two-phase flow patterns are observed according to the flow discharge through the gallery. Figure 4 shows the experimental relation between the flow discharge and the upstream pressure head (zero level is set at the upstream reservoir bottom level). The curve defines 5 areas corresponding to the 5 flow patterns (Figure 3) traditionally mentioned in the literature [13]: 1. A pure water fully free surface flow or smooth stratified flow. 2. A wavy stratified flow. 3. An intermittent flow that includes slug flow as well as plug flow. 4. A bubbly flow. 5. A pure water pressurized flow.

Figure 3: Experimental relation (upstream pressure head-flow discharge).

3.2 Pure water simulation

Simulations are performed under the assumption of a pure water flow (void fraction is equal to zero). They use a discretization step x=3.33cm, a CFL number limited to 0.5 and a roughness height k = 2.10-5m. The flow discharge varies between 5l/s and 55l/s. Results provide new insight into the flow behavior. A first head/discharge relation (dotted line in figure 4) is computed with the HEM model and assuming a free surface in each mesh if the water height is below the pipe crown (air phase above the free surface is at atmospheric pressure). The second head/discharge relation (continuous line) is computed by activating the negative Preissmann slot (sub-atmospheric pressurized flow).

0

10

20

30

40

50

60

70

80

90

5 15 25 35 45 55

Ups

trea

m p

ress

ure

head

[cm

]



Bubbly flow





Water Flow

Water Flow

Water Flow

Water Flow

Water Flow

Water Flow

Smooth Stratified flow

Wavy Stratified flow

Slug flowIntermittent

Plug flowIntermittent

Bubbly flow

Pressurized flow



Figure 4: Computed flow discharge relation for pure water simulations.

Figure 5: Computed total head and pressure head distribution for a smooth stratified flow and a pressurized flow.

Numerical results are in good accordance with experimental data for smooth stratified flows and fully pressurized flows. Bubbly and intermittent flows show a similar behavior two the sub-atmospheric pressurized flows. A periodic instability between to unstable steady flow regimes occurs in the area of wavy stratified flows. The instability induces large period (10s to 60s) oscillation of the water level in the upstream reservoir. The amplitude of the oscillation reaches 2cm. Consequently, very large pressure oscillations (up to 4cm) are observed in the gallery. For further details over this regime, we refer the interested reader to the paper of Erpicum et al. [18]. Experimental and numerical data for the distribution of the total head and the pressure head (water level for free surface flow) along the gallery length are given in figure 5 for a smooth stratified flow (discharge of 9.5l/s) and a fully

0

10

20

30

40

50

60

70

80

90

5 15 25 35 45 55

Ups

trea

m p

ress

ure

head

[cm

]


Experimental


Numerical (Sub‐Atmospheric pressure)


Bubbly flow





-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

Alt

itud

e (c

m)

Abscissa (m)

Topography

Pipe crown




Experimental Pressure head/Free surface level

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

Alt

itud

e (c

m)

Abscissa(m)

Topography

Pipe crown





a. Pressure profile for a discharge of 9.5 l/s

b. Pressure profile for a discharge of 48.4 l/s



pressurized flow (discharge of 48.4l/s). In the latter case, results are in full agreement. In the former case, a slight discrepancy is observed in the total head curve. It results from the effect of the air phase flowing above the free surface that is not taken into account in the computation.

Figure 6: Computed total head and pressure head distribution for an intermittent flow sub-atmospheric pressurized flow and free-surface flow computation.

A comparison of the results given by the computation for an intermittent flow of 38.4l/s discharge is shown in figure 6. Pressure distribution along the gallery is computed in figure 6b under the assumption of a free surface flow. Large discrepancies of the results are observed. The upstream pressure head is overestimated. In figure 6a, activation of the negative Preissmann slot gives the curve corresponding to a pressurized flow. We consequently identify a large area of sub-atmospheric pressure in the upstream part of the pipe. Results are now in better accordance and it has been concluded that the aeration rate of the pipe is not sufficient to induce the apparition of a free surface flow. However, some differences still remain.

3.3 Air-water mixture simulation

Application of the Homogeneous Equilibrium Model enables to overcome the results discrepancy observed in section 3.2 for bubbly and intermittent flows (figure 4 and 6). The effect of the entrained air on the water flow is accurately computed by using the eqn (7) for the phase change volume generation g. The parameter is set at 25 and g is calibrated according to the flow pattern observed. For bubbly flows, as bubbles arise from the air dissolved in water,

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5A

ltit

ude

(cm

)

Abscissa(m)

Topography

Pipe crown





a. Pressure profile for a discharge of 38.4 l/s : Sub-atmospheric pressurized flow

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,6

Alt

itud

e (c

m)

Abscissa(m)

Topography

Pipe crown





b. Pressure profile for a discharge of 38.4 l/s : Free surface flow



equilibrium void fraction is chosen between 0.5% and 2%. For intermittent flows, an additional air supply is provided through a vertical vortex appearing at the water intake. Equilibrium void fraction is then chosen between 2% and 4.5%. Figure 7a shows a comparison between experimental and numerical data for the relation between the flow discharge through the gallery and the upstream pressure head. Taking into account air/water interaction in the computation obviously gives more accurate results for bubbly and intermittent flows. The void fraction relation corresponding to this new relation is given in Figure 7b. A comparison between experimental data and numerical results computed with the HEM scheme is drawn on figure 8. Computation is performed with a flow discharge of 38.4l/s and a void fraction of 4.5%. Results are in full agreement.

Figure 7: Results of air-water mixture simulation.

Figure 8: Computed total head and pressure head distribution for a bubbly (flow discharge of 38.4 l/s and void fraction of 4.5%).

4 Conclusion

The original mathematical model derived in this paper is a first step towards a completely unified model for the simulation of highly transient mixed flow in multi-scale hydraulic structures. Thanks to the Preissmann slot method, both free-surface and pressurized flow are calculated through the free-surface set of equation by adding a narrow slot at the top of the pressurized sections. In

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Alt

itud

e (c

m)

Abscissa(m)

Topography

Pipe crown





0

10

20

30

40

50

60

70

80

90

5 15 25 35 45 55

Ups

trea

m p

ress

ure

head

[cm

]


Experimental


Numerical (Sub‐Atmospheric pressure and void fraction variable)


Bubbly flow




Upstream reservoir bottom level0

1

2

3

4

5

6

7

5 15 25 35 45 55

Air

Voi

d F

ract

ion

[%]


Pure w

ater fully pressurized flow

Bubbly flow

Intermittent flow



b. Void fraction distribution in bubbly and intermittent flowsa. Head/Discharge relation computed with variable void fraction



addition, an original negative Preissmann slot has been added to simulate sub-atmospheric pressure. Area-integration of the Homogeneous Equilibrium Model (HEM) over the cross section give a simple set of equations, analogous to the Saint-Venant equations, for analyzing air-water flows. This assumption has been shown to be particularly well-suited for the simulation of bubbly and intermittent flows. The fundamental concepts introduced in the previous pages pave the way for further research. Experimental research is required to develop appropriate source terms as phase change volume generation and interfacial momentum source. Development of a stratified air/water model would give us insight into wavy stratified flows. All results should be then easily extended to multidimensional problems.

References

[1] Zhou, F., F.E. Hicks, and P.M. Steffler, Transient Flow in a Rapidly Filling Horizontal Pipe Containing Trapped Air, Journal of Hydraulic Engineering, 128(6): p. 625-634, 2002

[2] Guo, Q. and C. Song, Dropshaft Hydrodynamics under Transient Conditions, Journal of hydraulic Engineering, 117(8): p. 1042-1055, 1991

[3] Vasconcelos, J. and S. Wright, Experimental Investigation of Surges in a Stormwater Storage Tunnel, Journal of hydraulic Engineering, 131(10): p. 853-861, 2005

[4] Cardle, J. and C. Song, Mathematical Modeling of Unsteady Flow in Storm Sewers, International Journal of Engineering Fluid Mechanics, 1(4): p. 495-518, 1988

[5] Politano, M., A.J. Odgaard, and W. Klecan, Numerical Evaluation of Hydraulic Transients in a Combined Sewer Overflow Tunnel System, Journal of Hydraulic Research, 133(10): p. 1103-1110, 2007

[6] Li, J. and A. McCorquodale, Modeling Mixed Flow in Storm Sewers, Journal of hydraulic Engineering, 125(11): p. 1170-1180, 1999

[7] Preissmann, A. Propagation des intumescences dans les canaux et rivieres. in First Congress of the French Association for Computation, Grenoble, France, 1961

[8] Vasconcelos, J., S. Wright, and P.L. Roe, Improved Simulation of Flow Regime Transition in Sewers : The Two-Component Pressure Approach, Journal of Hydraulic Engineering, 132(6): p. 553-562, 2006

[9] Bourdarias, C. and S. Gerbi, A Finite Volume Scheme for a Model Coupling Free Surface and Pressurized Flows in Pipes, Journal of Computational and Applied Mathematics, 209: p. 109-131, 2007

[10] Dewals, B.J., S. Andre, M. Pirotton, and A. Schleiss. Quasi 2D-numerical model of aerated flow over stepped chutes. in 30th IAHR Congress, Greece, 2003

[11] Ishii, M. and T. Hibiki, Thermo-fluid dynamics of two-phase flow. First ed, ed. U. Springer Science: Springer Science, USA. 430, 2006



[12] Clerc, S., Numerical Simulation of the Homogeneous Equilibrium Model for Two-phase Flows, Journal of Computational Physics, 161: p. 354-375, 2000

[13] Wallis, G.B., One-dimensional Two-phase Flow, ed. M.-H.B. Company. 410, 1969

[14] Guinot, V., Godunov-type Schemes: An introduction for engineers, ed. E. Science. Amsterdam. 480, 2003

[15] Cunge, J.A., F.M. Holly, and A. Verwey, Practical Aspects of Computational River Hydraulics. [Monographs and surveys in water resources engineering], 3. Boston: Pitman Advanced Pub. Program. 1980

[16] Dewals, B.J., S. Erpicum, P. Archambeau, S. Detrembleur, and M. Pirotton, Depth-Integrated Flow Modeling Taking into Account Bottom Curvature, Journal of Hydraulic Research, 44(6): p. 787-795, 2006

[17] Wylie, E.B. and V.L. Streeter, Fluid transients. Première ed, ed. M.-H. Inc., 385, 1978

[18] Erpicum, S., F. Kerger, P. Archambeau, B.J. Dewals, and M. Pirotton. Experimental and numerical investigation of mixed flow in the flushing gallery of a dam. in Multiphase Flow New Forest, 2008



Geometry effects on the interaction of two equal-sized drops in simple shear flow at finite Reynolds numbers

S. Mortazavi & M. Bayareh

Department of Mechanical Engineering, Isfahan University of Technology, Iran

Abstract

The effect of geometry on the interaction of two equal-sized drops in shear flow is presented. The full Navier-Stokes equations are solved by a finite difference/front tracking method. The interaction of drops was studied at finite Reynolds numbers for viscosity ratio (λ) of one. The distance between drop centres along the velocity gradient direction (z) was measured as a function of time. The interaction of two drops contains approach, collision, and separation. Based on experimental data, we simulated different geometries by changing the offset and size of drops. It was found that ∆z increases after collision and reaches

∆z, during three stages of interaction, increases with the increasing initial offset. To investigate the drop shape evolution, we calculated the deformation and the orientation angle formed by the drop major axis and horizontal direction. The deformation of the drops is maximum when the drops are pressed against each other and minimum when they are drawn a part. Our results show that the time of approaching of drops at low initial offset is greater than the other ones, but the maximum deformation is the same for equal drop sizes. The deformation decreases with the decreasing size of drops. As the initial offset increases, the drops rotate more quickly and the available contact time for film drainage decreases. We found that the trajectories of drops in the approaching stage are different owing to the different initial offsets. However, after the drops come into contact, it can be seen that they follow the same trajectories, similar to experimental results. Keywords: two-phase flow, front tracking, Reynolds number, Weber number, capillary number, offset, drainage time.


to a new steady-state value after separation. The values of

doi:10.2495/MPF090321


1 Introduction

Immiscible polymer blending plays a fundamental role in determining material properties of industrial interest. It is important to understand and control the size and size distribution of the dispersed drops because the properties of the blend depend on them. The final size distribution is determined by a balance between flow-induced break up and coalescence. While break-up involves a single drop and is not very affected by the presence of neighbouring drops in a blend (Leowenberg and Hinch [1], coalescence is the result of many-particle interaction process. The majority of numerical simulations are based on the interaction of two deformable drops in a shear flow, the drainage of the thin film between two colliding drops and the problems of coalescence of two deformable drops. Wang et al. [2] computed the coalescence of two undeformed spherical drops. Leowenberg and Hinch [3] presented the numerical simulations of the interaction between deformable drops based on boundary integral calculations. They showed that if capillary number is much smaller than one, the tendency for coalescence is greatest when drops are pressed against each other by the shear flow. Viscosity ratio effects on film drainage between interacting drops were studied by Bazhlekov et al. [4]. Cristini et al. [5] simulated the drop break-up and coalescence by an adaptive mesh algorithm. Effects of inertia on the rheology of a dilute emulsion of drops in shear flow are investigated by Zhao [6] using direct numerical simulation. The drop shape and flow are computed by solving the Navier-Stokes equations in two phases using front tracking method. On the other hand, most of experimental works are based on blending studies that analyses the drop size distribution of a blend or a concentrated emulsion. The collision of two equal-sized drops immersed in an immiscible liquid phase undergoing a shear flow in a parallel apparatus was investigated by Guido and Simeone [7] over a range of capillary numbers. Trajectories of a pair drops and their deformations were presented. The coalescence efficiency of two drops in a simple shear flow was also investigated by Mousa et al. [8]. The effect of viscosity ratio on the flow-induced coalescence of two equal-sized drops with clean interfaces was investigated by Yoon et al. [9]. Their studies showed that when the viscosity ratio is greater than O(0.1), the critical capillary number decreases with increasing offset only for the smallest offset. Zhao [10] investigated the drop break up in dilute Newtonian emulsions in simple shear flow by using high-speed microscopy over a wide range of viscosity ratio, focusing on high capillary number. He showed the final drop size distribution intimately links to the drop break up mechanism, which depends on viscosity ratio and capillary number. In this article, we present numerical simulation data describing the motion of a pair of drops under simple shear flow at finite Reynolds numbers. We consider the special case of drops with the same viscosity as the continues-phase fluid. Formulation and numerical method are described in §2, results are presented in §3, and concluding remarks are given in §4.



2 Formulation and numerical method

2.1 Formulation

The governing equations for the motion of unsteady, viscous, incompressible, immiscible two-fluid systems are the Navier-Stokes equations in conservative form:

.)()(.. ST dXxnuuPuu

tu

−+∇+∇∇+−∇=∇+∂∂

∫ βδκσµρρ (1)

Here u is the fluid velocity, p is the pressure, ρ is the fluid density, µ is the fluid viscosity, σ is the surface tension coefficient, g is the acceleration due to gravity.

βδ is a two- or three-dimensional delta function (for β=2 and β=3) respectively. κ is the curvature for two-dimensional flows and twice the mean curvature for three-dimensional flows. n is a unit vector normal to the drop surface pointing outside of the drop. x is the position in Eulerian coordinate and X is the position of front in Lagrangian coordinate. Both of immiscible fluids are taken to be incompressible, so the divergence of velocity field is zero:

.0. =∇ u (2) Equations of state for the density and the viscosity are:

,0=DtD ρ

.0=DtD µ (3)

Continuity of stresses at the fluid boundary shows that the normal stresses are balanced by surface tension. The force due to surface tension is

.knF σ=∆ (4) Three governing non-dimensional numbers of the flow are the Reynolds number (bulk and particle Reynolds numbers), the Weber number and the capillary number. Only two of these non-dimensional numbers are independent (one Reynolds number and the Weber or capillary number):

,Re0

20

µρ GH

b =

,Re0

20

µρ GR

P =

,32

0

σρ RG

We =

.0

σµ GR

Ca = (5)

Here is the density of ambient fluid, is the viscosity of the ambient fluid, R is the initial radius of the drop, H is the width of the channel and G is the shear rate. The shear rate is

.H

uuG bt −= (6)

where ut and ub are the velocity of top and bottom walls, respectively. It is usual to define a scalar measure of the drop deformation (the Taylor deformation) by:

.blblD

+−

= (7)

where l, b are the major and minor semi-axes of the drop (defined by the largest and smallest distances of the surface from the centre). In addition, the collision or film drainage time is the time between the points where the centre-to-centre distance is equal to one undeformed drop diameter to the instant of coalescence.

0ρ 0µ



2.2 Numerical method

Various methods have been used to simulate the two-phase flows. These methods include the Marker-And-Cell (MAC) method, the Volume-Of-Fluid (VOF) method, and the level set method. In general, the interface representation can be explicit (moving mesh) or implicit (fixed mesh) or a combination of both. The front-tracking method is combination of fixed and moving mesh method. Although an interface grid tracks the interface, the flow is solved on a fixed grid. The interface conditions are satisfied by smoothing the interface discontinuities and interpolating interface forces from the interface grid to the fixed grid. In this method, the governing equations are solved separately for each fluid. Front capturing has two difficulties. The first is a sharp boundary between the fluids and the second is accurate computation of surface tension. Different attempts have been made in overcoming these problems. For the simulations presented here, the method developed by Unverdi and Tryggvason [11] is used. They simulated the motion of buoyant bubbles in a periodic domain. Eqns (1), (2), and (3) are solved in a rectangular three-dimensional domain with a finite difference method. The spatial differentiation is calculated by a second order finite difference scheme on a staggered Eulerian grid. We use an explicit second-order time integration method. Combining the incompressibility condition and momentum equations results in a non-separable elliptic equation for the pressure. Due to the similarity in density between the drop and the ambient fluid, a quick poisson solver solves the pressure equation. The force due to surface tension on each element of front is

.∫∆

=S

SndF σκδ σ (8)

In three-dimensional flow, the average surface curvature is .)( nnn ×∇×=κ (9)

Then, the force on each element surface is

.)( ∫∫∫ ×=×∇×==S

SA

AA

A ndtndnndF σσκσδσσ

σ (10)

The integration is over the boundary of each element representing the front. t and n are the tangent and the normal vector to each element, respectively.

3 Results

The reference system to describe the results is shown in fig. 1. According to experiments of Yoon et al. [9], initial offset is defined as the shortest distance from the centre of the drop to the inflow axis (∆) divided by drop radius R. The centre-to-centre distance between drops is 4R as shown in fig. 1. The coordinate axes are oriented as follows: the x-axis is parallel to flow direction, the y-axis is parallel to the vorticity direction, and the z-axis is parallel to the velocity gradient. The relative trajectory of the two drops will be expressed in terms of the differences

12 zzz −=∆ and12 xxx −=∆ , where

ix and iz are the centre-of-



mass coordinates of the ith drop. The difference ∆y between the y-coordinates of the two drop centres is zero. In all plots, ∆x and ∆z will be made dimensionless by the radius R of the undeformed drops as the characteristic length. We will compare our results with experimental results of Guido and Simeone [7] and Yoon et al. [9] and numerical results of Loewenberg and Hinch [3]. Sequences (1-4) show the interactions between two drops in simple shear flow (fig. 2). Initially, each drop has the steady shape under same flow conditions.

z

y

x

Rd 4

Figure 1: Schematic of the relative trajectory between a pair of deformable interacting drops in shear flow (Offset = ∆ / R).

1 2 3 4

Figure 2: Sequences (1-4) showing the interaction between two drops in simple shear flow with Ca = 0.13, Offset = 0.512, λ = 1.

In fig. 3, ∆z is plotted as a function of ∆x during approach, collision and separation between two drops. The data correspond to sequence depicted in fig. 2. It can be seen that ∆z starts increasing after the drops come into apparent contact (∆x ~ -2R), reaches a maximum value, and, after separation, reaches a new steady-state value. The final value of ∆z (which is 1.4 for offset = 0. 2, 1.52 for offset = 0.512, and 1.72 for offset = 0.8) is greater than the value before collision. In the other words, if the drops were made to collide again by reversing the flow direction, ∆z increased further. So, the effect was irreversible, and repeated collisions lead to increasing values of ∆z until drop interaction became negligible (Guido and



Simeone [7]). This is also an agreement with the numerical simulations of Loewenberg and Hinch [3]. Experimental results of Guido and Simeone [7] was based on λ = 1.4 and numerical results of Loewenberg and Hinch [3] were presented for viscosity ratio of one.

Figure 3: Cross-flow separation (velocity gradient direction) versus ∆x / R between interacting drops with Ca = 0.13, λ = 1 (present work) and λ = 1.4 (experiment), and different offsets.

The deformation parameter is shown in fig. 4 as a function of dimensionless time with viscosity ratio of one and Ca = 0.3. Based on experimental observation of Guido and Simeone [7], deformation of two drops is the same. Deformation slightly increases, and then reaches a maximum, a minimum, a second maximum, and eventually reaches a steady state value from before the collision. Numerical simulations of Loewenberg and Hinch [3] show no difference before and after collision at low-Reynolds numbers. Comparison between the results shows that the time of approaching of drops at low initial offset is greater (for current simulations). Following Allan and Mason [12] and Guido and Simeone [7], the minimum value of deformation is lower than the steady-state value before (or after) collision. This can be explained as the result of two processes: (i) relaxation of drop shape once they leave the compressional axis and (ii) action of the surrounding fluid on the drops. At fixed initial offsets, as the size of drops increases, the deformation increases as shown in fig. 5. We see that the approach-collision-separation times of drops are different owing to the different initial sizes. The film drainage time increases with increasing size of drops. The dimensionless drainage time is 5.21 for D / H = 0.3 and 5.86 for D / H = 0.36. Fig. 6 shows the trajectories of the drops for different initial offsets with the same capillary number for a viscosity ratio of 1. As the initial offset increases,

0 5 10X / R

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Z/R

Offset = 0.2

Offset = 0.512

Offset = 0.8

Expriment (Guido & Simeone)

∆



the drops rotate more quickly and the available time for film drainage decreases as shown in fig. 6. Therefore, we should expect that the critical capillary number for coalescence will decrease with increasing offset. This was found in studies of Yang et al. [13], where the viscosity ratio was 0.096.

Figure 4: The deformation parameter as a function of dimensionless time between interacting drops with Ca = 0. 3, λ = 1 and different offsets.

Figure 5: The deformation parameter as a function of dimensionless time between interacting drops with Ca = 0. 075, λ = 1 and different size of drops.

In fig. 7 we see that the approaching parts of the trajectories are different from each other owing to different initial offsets. However, after the drops come into

0 10 20t*

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Def

orm

atio

n

Offset = 0.12Offset = 0.2Numerical simulations (Loewenberg & Hinch)

0 5 10 15t*

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Def

orm

atio

n

D / H = 0.3D / H = 0.36



contact, it can be seen that they follow the same trajectories of separation distance versus orientation angle (α) formed by the drop major axis and horizontal direction. This is an agreement with the experimental results of Yoon et al. (2005). They said that they have no explanation for this result.

Figure 6: Trajectories of drops for different initial offsets, separation distance versus dimensionless time, with Ca = 0.00481, λ = 1 (present work), and λ = 1.2 (experiment).

0 25 50 75 100

Angle (deg.)

0

0.5

1

1.5

2

2.5

Dis

tance

,d/2

R

Offset = 0.08Offset = 0.12Expriment (Yosang Yoon et al.), Offset = 0.08Expriment (Yosang Yoon et al.), Offset = 0.12

Figure 7: Trajectories of drops for different initial offsets, separation distance versus orientation angle (α), with Ca = 0.00481, λ = 1 (present work), and λ = 1.2 (experiment).

0 5 10 15 20

t*

0

0.5

1

1.5

2

2.5

Dis

tance

,d/2

R

Offset = 0.08Offset = 0.12Expriment (Yosang Yoon et al.), Offset = 0.08Expriment (Yosang Yoon et al.), Offset = 0.12



4 Conclusion remarks

The effects of geometry include initial offset and size of drops on the interaction of two equal-sized drops in simple shear flow has been presented using finite difference/front tracking method. Simulations were studied at finite Reynolds numbers for viscosity ratio of one. The deformation, relative trajectories, and film drainage time were examined by changing the initial offset and size of drops. We changed offset and size of drops, Based on experimental data. It was found that ∆z increases after collision and reaches to a new steady-state value after separation. The values of ∆z, during the interaction, increases with the increasing initial offset. Our results showed that the time of approaching of drops at low initial offset is greater than the other ones, but the maximum of deformation is the same for equal drop sizes. The deformation decreases with the decreasing size of drops. As the initial offset increases, that time for film drainage decreases. Also, the approaching parts of the trajectories are different from each other owing to the different initial offsets. However, after the drops come into contact, it can be seen that they follow the same trajectories of separation distance versus orientation angle formed by the drop major axis and horizontal direction, similar to experimental results.

References

[1] Loewenberg, M. & Hinch, E.J., Numerical simulation of a concentrated emulsion in shear flow. J. Fluid Mech., 321, pp. 395-419, 1996.

[2] Wang, H., Zinchenko, A.Z. & Davis, R.H., The collision rate of small drops in linear flow-fields. J. Fluid Mech., 265, pp. 161-188, 1994.

[3] Loewenberg, M. & Hinch, E.J., Collision of two deformable drops in shear flow. J. Fluid Mech., 338, pp. 299-315, 1997.

[4] Bazhiekov, I.B., Chesters, A.K. & Van De Vosse, F.N., The effect of the dispersed to continuous-phase viscosity ratio on film drainage between interacting drops. Intl J. Multiphase Flow, 26, pp. 445-466, 2000.

[5] Cristini, V., Blawzdziewicz, J. & Loewenberg, M., An adaptive mesh algorithm for evolving surface: simulation of drop break-up and coalescence. J. Comput. Phys. 168, pp. 445-463, 2001.

[6] Zhao, X., Effects of inertia on the rheology of a dilute emulsion of drops in shear flow. J. Rheology, 49, pp. 1377-1394, 2005.

[7] Guido, S. & Simeone, M., Binary collision of drops in simple shear flow by computer-assisted video optical microscopy. J. Fluid Mech., 357, pp. 1-20, 1998.

[8] Moua, H., Agterof, W. & Mellema, J., Experimental investigation of the orthogenetic coalescence efficiency of droplets in simple shear flow. J. Colloid Interface Sci., 240, pp. 340-348, 2001.

[9] Yoon, Y., Borrell, M., Park, C.C. & Leal L.G., Viscosity ratio effects on the coalescence of two equal-sized drops in a two-dimensional linear flow. J. Fluid Mech., 525, pp. 355-379, 2005.



[10] Zhao, X., Drop break up in dilute Newtonian emulsions in simple shear flow: new drop break up mechanism. J. Rheology, 51, pp. 367-192, 2007.

[11] Unverdi, S.O. & Tryggvason, G., Computations of multi-fluid flows. Phys. Fluids, D60, pp. 70-83, 1992.

[12] Allan, R.S. & Mason, S.G., Particle motion in sheared suspensions. XIV. Coalescence of liquid drops in electric and shear fields. J. Colloid Interface Sci., 17, pp. 383-408, 1962.

[13] Yang, H., Park, C.C, Hu, Y.T. & Leal, L.G., The coalescence of two equal-sized drops in a two-dimensional linear flow. Phys. Fluids, 13, pp. 1087-1106, 2001.



Section 6 Flow in porous media

Modelling the tide effects in groundwater

J. MlsFaculty of Science, Charles University, Czech Republic

Abstract

It is well known that in aquifers in contact with the ocean, sinusoidal fluctuationsof groundwater level occur in response to tides. Similar semidiurnal fluctuationshave been observed in the piezometric head of confined aquifers without anycontact with the ocean. These fluctuations are of interest to hydrogeologists andgeophysicists as they indicate that the observed wells are sensitive indicators ofthe surrounding material deformations either by changes in the surface loading orby applied strain.

The aim of the contribution is to model the pressure-head fluctuations in aleaky confined aquifer as a result of periodic changes of the load originating inthe tidal changes of gravity acceleration. The Biot approach was used to derivethe governing equations. The aquifer characteristics and the imposed boundaryconditions were determined according to the known geology of the particularsite where the water level fluctuations were observed. The problem was solvednumerically and the obtained results were compared with the measured values.Keywords: groundwater tides, water level fluctuations, tide potential, tide acceler-ation, aquifer, solid phase compressibility, Biot’s approach.

1 Introduction

According to Rojstaczer and Agnew [5], high porosity aquifers show high sensitiv-ity to atmospheric loading while high sensitivity to the applied areal strains occursfor low porosity aquifers, and both increase with decreasing compressibility of thesolid matrix.

Involving an inertia term, Bodvarsson [2] generalized the Darcy law and gavea theory of strain-induced pressure fluctuations in a well-aquifer system. In hispaper, he emphasized the necessity of the wells being connected with a sufficientlyconfined aquifer.



Response of the water level in a well to Earth tides and atmospheric loadingunder unconfined conditions was studied by Rojstaczer and Riley [6]. The authorsdefine physical properties of aquifers to exhibit a significant effect of water levelfluctuation: high specific storage, high hydraulic conductivity, and low specificyield.

Boreholes where measurable values of tide oscillation have been found indicatelocations where the solid-phase tension of the geological layer does not match thetide forces. At such places, the pressure of the liquid phase takes over a part of thesolid-phase tension.

During hydrogeological investigation of the Police basin, situated in northeast-ern Bohemia, oscillations of water table were found in several boreholes (Krasnyet al. [3]). Subsequent more detailed research checked this phenomenon with theeffects of atmospheric pressure and groundwater storage due to recharge/dischargeprocesses. The relation was confirmed in all but one of the cases. Water table ofthe borehole V-34 oscillated in its own way regardless of atmospheric pressureand the pumping process in its neighbourhood. According to the frequency of thewater table changes, Krasny et al. [3] came to the conclusion that the observedphenomenon were tidal oscillations of groundwater hydraulic head.

The knowledge of the geology of the investigated region and the knowledge ofhydraulic characteristics of the aquifer and surrounding layers make it possible toformulate a simple model of the two phase system and numerically simulate thesupposed process. To make this, the Biot [1] approach has been applied connectingthe solid-phase elastic deformation with the pressure and flow of the liquid phasewithin the pore system.

Figure 1: Water level fluctuations indicated in boreholes VS-20 and V-34.



Figure 2: Position of boreholes VS-20 and V-34 and of the faults.

2 Geological conditions

Figure 1 shows a water level fluctuation as it was recorded in two boreholes, VS-20and V-34, in the Police basin. Initially, it was explained by pumping from a watersupply borehole lying about 1,5 km north of the measured boreholes. It was laterrealised that the borehole V-34 was separated from the pumping site by the Skalskfault. Subsequent test confirmed the assumption that the fault built an imperviousobstacle which did not allow for any propagation of hydraulic head changes.

The borehole V-34 is situated in a narrow strip bounded by two faults, Skalskfault and a shorter parallel one, see Figure 2. The borehole is open to a confinedaquifer, the part of which, situated between the planes of the faults, is affected bytension originated in tide forces.

Hydrogeology of the region, the Police basin, and hydraulic characteristics ofparticular layers are known from the study published by Krsn et al. [3]. This makesit possible to define the conceptual model of the investigated process: the domain,its parameters and the boundary conditions.

The studied domain is the aquifer between the faults. The problem is supposed tobe one dimensional defined in horizontal direction normal to the plane of the faults.The coordinate is oriented in the northwest direction and denoted x. The domain



Figure 3: North part of the sectional view AA′.

is then = (0, L), L being the distance of the faults. In Figure 3, A2 denotesthe aquifer, A1/A2 denotes the underlying layer and A/C denotes the overlyinglayer. The underlying layer is supposed to be an aquiclude and the overlying layeris supposed to be an aquitard. Hydraulic conductivities of these layers, specificstorativity of the aquifer and pressure head in the aquifer lying above the layer A/Care known from (Krsn et al. [3]). The region north of the Skalsk fault is elevatedabout 100 m above the southern part of the basin. Hence, it is supposed that thepart of the massif situated south of the domain , having smaller load, is easierdeformable than the northern one. Consequently, it is supposed that the northernpart of the basin is rigid and the modelled part of the aquifer is pushed against theplane of the fault.

3 Governing equations

Part of the tension arising within geological layers originates in tide accelerationacting upon the mass of the layers. In most cases, the solid phase, due to itstoughness, holds most of the tension. There are, however, places where the tideforces affect the liquid phase within the pore space in such a way that thecorresponding pressure-head fluctuation is measurable.

Biot [1] published his theory of consolidation which he considered as a processof squeezing water out of a deformable solid-phase body. The Biot theory connectslaws of elastic deformation with laws of Darcian flow of fluid in porous media. Inthis way, it makes it possible to relate the solid-phase tension with liquid-phasepressure and to formulate governing equations of the investigate process.

The initial Biot’s assumptions are

∂ui

∂xj+ ∂uj

∂xi= 1

μτi,j − σ

μ(1 + σ)ϑ δi,j + 2p

3Hδi,j , i, j = 1, 2, 3, (1)

and

θ = θ0 + ϑ

3H+ p

R. (2)



Hence, taking into account the equilibrium conditions of a stress field

∂τi,j

∂xj= 0, i = 1, 2, 3, (3)

the resulting equations of the Biot theory are

μui + μ

1 − 2 σ

∂ε

∂xi− α

∂p

∂xi= 0, i = 1, 2, 3, (4)

and

α∂ε

∂t+(

1

R− α

H

)∂p

∂t= k

ηp, (5)

where x1, x2, x3 are the space coordinates, t is time, u = (u1, u2, u3) is the solid-phase displacement vector, τi,j is the tensor of the solid-phase stress, p is theliquid-phase pressure, θ is the water content, δi,j is the Kronecker tensor, μ isthe shear modulus of the solid phase, σ is the Poisson ratio, H and R are Biot’scoefficients: 1/H is a measure of compressibility of the solid phase for a change inwater pressure and 1/R is a measure of the change in water content for a change inwater pressure, k is the permeability of the aquifer, η is the liquid-phase dynamicviscosity,

ϑ = τj,j , (6)

ε = ∂uj

∂xj, (7)

and

α = 2(1 + σ)μ

3(1 − 2 σ)H. (8)

4 The solved problem

According to the assumptions of our model, Equations (4) and (5) are rewritten inthe one-dimensional form as

2μ(1 − σ)

1 − 2 σu− α

∂p

∂x= 0, (9)

and

α∂2u

∂t∂x+(

1

R− α

H

)∂p

∂t= k

ηp. (10)

One-dimensional form of Equation (1) is

τ = 2μ(1 − σ)

1 − 2 σ

∂u

∂x− α p. (11)



Now, every triplet of functions u(x, t), p(x, t), f (t) satisfying equation

2μ(1 − σ)

1 − 2 σ

∂u

∂x− α p = f (12)

satisfies also Equation (9). From Equations (11) and (12), it follows that thefunction f represents solid-phase stress and that it can be determined providedfunction τ is known as a function of time at a single value of coordinate x.Consequently, denoting T (t) the tension at a boundary point, it follows from (12)

∂2u

∂t∂x= α(1 − 2 σ)

2μ(1 − σ)

∂p

∂t+ 1 − 2 σ

2μ(1 − σ)T ′(t). (13)

The last equation makes it possible to exclude the unknown function u fromEquation (10) and to solve the resulting second order parabolic equation with theliquid-phase pressure as the unknown function. According to the aquifer and itsneighbouring layers, the solved equation is either linear or nonlinear. Solving theproblem for a leaky confined aquifer and horizontally oriented domain , it isnecessary to introduce a source term and to deal with the continuity equation ofthe liquid phase in the form

∂θ

∂t+ div v = q, (14)

where v is vector of the liquid-phase flux density and q is the source term, i.e.the volume of water arising in unit volume of the aquifer per unit time. As thehydraulic head of the overlying aquifer as well as the height and the permeabilityof the aquitard are known, the source term can be determined as a function of theliquid-phase pressure:

q = k

η

(ρw g + p − p

Z

), (15)

where ρw is the liquid-phase density, g is the Earth’s gravity acceleration, p is theliquid-phase pressure at the top of the aquitard and Z is height of the aquitard.

Biot [1] discussed consolidation of saturated soils as a special case of his generalequations. He comes to the conclusion that following equation holds for saturatedsoils:

α = 1 andH R

H − α R→ ∞. (16)

Hence, the diffusivity coefficient of the liquid-phase pressure is

c = 2μk (1 − σ)

η (1 − 2 σ). (17)

Concerning the boundary conditions, it is supposed that the top of the aquifer isthe only part of its boundary that allows for discharge or recharge. Consequently,the boundary ∂ is impervious and the Neumann boundary condition is imposedin the form

∂p

∂x(x, t) = 0, t > 0, x ∈ ∂. (18)



Figure 4: Comparison of observed water level fluctuations with results of a simu-lation.


The tension T is not known in our case. On the other hand, the amplitude of thedetected oscillations can be estimated. Hence, function T can be determined bymeans of the inverse formulation of the problem.

Assuming that the tension T (t) originates in tide acceleration, it can beexpressed as

T (t) = MN γ (t), (19)

where MN is a constant and γ is the projection of the tide acceleration vector intoaxis x (the normal to the Skalsky fault). Choosing a value of MN , the forwardproblem can be solved as described above. Figure 4 presents results of such asimulation.

In this study, the tide acceleration is supposed to be determined by Sun andMoon, the effect of other planets being neglected. Function γ was calculated fromknown coordinates of tide acceleration vector. The code SPZ SM 01 (Ondovcin[4]), returning values of tide potential and tide-acceleration vector to given timeand position on Earth, was used for this purpose.

It can be seen from Figure 4 that in this case, the value ofMN was overestimated.On the other hand, the figure shows the most important result of the simulation:there is clear correspondence between the simulated tide effects and the observeddata. Consequently, the results confirm the previous conclusion made by Krasnyet al. [3].

The oscillations of the computed curve are determined by function γ and thevalue of MN affects just the amplitude. Hence, it is a simple problem to iterate forits proper value. The value MN = 2 × 108 kg/m2 was found as the best match ofthe observed amplitude.



Acknowledgements

This paper is based upon work supported by the Grant Agency of the CzechRepublic under grant No. 205/07/1311 and by the Ministry of Education of theCzech Republic under grant No. MSM 0021620855. The author is also indebtedto the Journal of Geological Sciences for the permission to use Figures 1, 2, and 3.

References

[1] Biot, M. A., 1941, General theory of three-dimensional consolidation, Journalof Applied Physics, 12, 155–164

[2] Bodvarsson, G., 1970, Confined fluids as strain meters, Journal of GeophysicalResearch, 75, 14, 2711–2718

[3] Krsn, J., Buchtele, J., ech, S., Hrkal, Z., Jake, P., Kobr, M., Mls, J., Łantrek, J.,Łilar, J., Valeka, J., 2002, Hydrogeology of the Police basin: Optimisation ofgroundwater development and protection, Journal of Geological Sciences, 22,5–100 (in Czech)

[4] Ondovcin, T., 2007, Description of the code SPZ SM 01, UK, Prague, (unpub-lished)

[5] Rojstaczer, S. and Agnew, D. C., 1989, The influence of formation materialproperties on the response of water levels in Wells to Earth tides and atmo-spheric loading, Journal of Geophysical Research, 94, B9, 12, 403–12, 411

[6] Rojstaczer, S. and Riley, F. S., 1990, Response of the water level in a wellto Earth tides and atmospheric loading under unconfined conditions, WaterResources Research, 26, 1803–1817

[7] van der Kamp, G., and Gale, J. E., 1983, Theory of Erth tide and baromet-ric effects in porous formations with compressible grains, Water ResourcesResearch, 19, 538–544



Modelling of diffusion in porous structures

E. du Plessis & S. WoudbergApplied Mathematics, Department of Mathematical Sciences,Stellenbosch University, South Africa

Abstract

An existing pore-scale model is used to predict the effective diffusivity ofstaggered two-dimensional rectangular unconsolidated arrays through the use ofa Representative Unit Cell concept. A tri-diagonal matrix algorithm is used tosolve the diffusive flux field and to compute the effective diffusion coefficient forconcentration gradients of staggered arrays. The numerical results and analyticalmodel are compared critically with theoretical and numerical studies, as well asexperimental data reported in literature. The good correlations obtained for theeffective diffusivity provide confidence in both the computational and analyticalwork.Keywords: diffusion, porous media, pore-scale, effective diffusion coefficient,modelling, microstructure, fibres.

1 Introduction

The study of molecular diffusion forms an important cornerstone in the analysis ofmore involved multiphase processes, such as dispersion and combustion in porousmedia, which form part of many processes in the coal industry, e.g. the recovery ofmethane from coal beds (Kim et al. [1]) and similar industries. Natural substances,such as coal, vary considerably in structural morphology. The microstructure isseldom simple and most often present a mixture of different types of porous envi-ronments. The flow processes thus often consist of a mixture of convective flow inmacropores and molecular diffusion in micropores. Following up on considerablesuccess with a pore-scale model on the modelling and subsequent predictionof drag during convection in porous media, our next goal is the modelling ofdiffusion making use of the same geometrical and modelling practices. To this end,descriptions of diffusion in two-dimensional rectangular arrays of non-staggered



and of fully staggered solid rectangles are needed to formalize the theory and tocompare it with published experimental, computational and theoretical models.

2 Diffusion equation

Fick’s second law of diffusion or simply the diffusion equation is given by

∂ρA

∂t+ ∇·D∇ρA = 0, (1)

whereD is the mass diffusivity or diffusion coefficient for component A diffusingthrough component B and ρA is the mass concentration of species A. The diffusiv-ity D is a property of a specific system, dependent upon the system’s temperature,pressure and composition (Welty et al. [2]). Eqn. (1) applies to a stationary,incompressible fluid without chemical production. For a steady incompressiblefluid, constant diffusion coefficient, no chemical production and no fluid motionthe Laplace equation for mass transport is obtained (Welty et al. [2]), i.e.

∇2ρA = 0. (2)

3 Effective diffusion coefficients of porous media

Equation (2) is a point equation which governs the transport of mass of thechemical species considered. When regarding diffusion in porous media, eqn(2) needs to be solved at every point within the porous medium which leads toenhanced mathematical complexity. The characterization of diffusion in porousmedia therefore relies on the use of macroscopic dependent variables (Saez et al.[3]). Eqn (2) thus needs to be volume averaged over a representative portion ofthe porous domain according to the volume averaging theory (e.g. Whitaker [4]).Boundary conditions also need to be imposed together with conditions at the fluid-solid interfaces (Saez et al. [3]). For the present work no mass transfer betweenthe phases will be considered. Volume averaging of eqn (1) leads to the followingmacroscopic diffusion equation:

∂〈ρA〉f∂t

= Deff : ∇∇〈ρA〉f , (3)

where Deff is the effective diffusion coefficient which is only a function of thegeometric structure of the porous medium at the pore level (Saez et al. [3]) andmay thus be expressed as a function of the porosity alone, i.e.

Deff

D= f (ε). (4)

In the next section a geometric pore-scale model will be introduced which willbe used to predict the effective diffusivity analytically.



n

dds

UsUf

Uo

L

Figure 1: RUC model for fibre beds.

4 Geometric pore-scale model for fibre beds

The geometric pore-scale model for fibre beds was originally introduced by DuPlessis [5]. It is based on a rectangular Representative Unit Cell, abbreviated RUC,which is defined as the smallest rectangular control volume, Uo, into which theaverage geometrical properties of the porous medium may be embedded. The RUCmodel for fibre beds is shown in Fig. 1.

Only cross-flow – flow perpendicular to the prism axis – will be considered, asindicated by the streamwise direction n in Fig. 1. Cross-flow through prismaticporous media can therefore be approximated as two-dimensional flow. Two arraysare considered, namely a fully staggered array in which maximum staggeringoccurs in the streamwise direction and a non-staggered array in which no stag-gering occurs in the streamwise direction. The fluid filled volume within the RUCis denoted by Uf and Us denotes the volume of the solid phase. The RUC modelis assumed to be homogeneous and isotropic with respect to the average geometricproperties of the fibre bed. The porosity ε of the RUC model is defined as

ε = Uf

Uo. (5)

4.1 Volume partitioning

The total fluid filled volume Uf within the RUC may be expressed as

Uf = U‖ + U⊥ + Ug + Ut, (6)

where the streamwise volume U‖ is the total fluid volume parallel to the stream-wise direction, the transverse volumeU⊥ is the total fluid volume perpendicular tothe streamwise direction, in the stagnant volume, Ug, the fluid remains stationaryand in the transverse fluid volume, Ut , no wall friction occurs due to the absenceof adjacent solid surfaces. From eqn (5) and quantification of the respective fluidvolumes, the following relationship between the linear dimensions of the RUC



model for fibre beds is obtained in terms of the porosity (Du Plessis [5]):

ds = d√

1 − ε. (7)

Diedericks and Du Plessis [6] defined the geometrical tortuosity as the ratio ofthe streamwise displacement d to the total path length de of the fluid traversingthrough the constant cross-sectional area Ap‖ in the RUC. The tortuosity may alsobe expressed as a volumetric ratio, but with stagnant volumes excluded, yielding[7]

χ = de

d= U‖ + Ut + ξU⊥

Ap‖d= U‖ + Ut + ξU⊥

U‖ + Ut, (8)

where the coefficient ξ was introduced to account for the reduction in the tortuositydue to the splitting of the streamtube into two equal but directionally oppositetransverse parts in a fully staggered array. It thus follows that ξ = 1/2 for thefully staggered array and ξ = 0 for the non-staggered array. Quantification of thetortuosity leads to the following expression for the fully staggered array

χ = 1 + 1

2

√1 − e, (9)

and for the non-staggered array χ = 1. In order to account for the effect of stagnantregions a geometric factor ψ was introduced and defined as (Lloyd et al. [7])

ψ ≡ Uf

U‖ + Ut= Uf

Ap‖d= U‖ + Ut + U⊥ + Ug

U‖ + Ut. (10)

The geometric factor expressed as a function of the porosity alone yields

ψ = ε

1 + √1 − e

. (11)

4.2 Effective diffusivity predicted by the RUC model

According to Kim et al. [1] it is common for isotropic processes to express theeffective diffusivityDeff over the diffusivityD as

Deff

D= ε

χ. (12)

Making use of eqn (9) for the tortuosity obtained by the RUC model, yields

Deff

D= ε

1 + 12

√1 − e

, (13)

for the fully staggered array. An alternative expression for the effective diffusioncoefficient can be obtained by using the geometric factor supplied by the RUC



model, which takes into account stagnant regions, instead of the tortuosity, i.e.

Deff

D= 1 + √

1 − e. (14)

Equation (14) applies to both the staggered and non-staggered arrays.Weissberg [8] made use of a variational approach which they applied to a bed

of spheres to obtain the following expression for the effective diffusivity:

Deff

D= ε

(1 − 1

2ln ε

)−1

. (15)

Maxwell (Kim et al. [1]) analyzed a dilute suspension of spheres analyticallyand obtained the following expression for the effective diffusivity:

Deff

D= ε

[1 + 1

2(1 − ε)

]−1

. (16)

The micropore-macropore model of Wakao and Smith [9] is given by

Deff

D= ε2. (17)

Based on their experimental data Kim et al. [1] proposed the following empiricalequation for the effective diffusivity:

Deff

D= ε1.4. (18)

5 Series-parallel formulae

A popular concept to model diffusion in arbitrary composites is to mix twoformulae that applies to composites made up of laminates of different diffusivities,namely the series and the parallel formula (Crank [10]). The assumptions for thesemodels are that each laminate has uniform diffusive properties and the diffusivetransport is uni-directional. It will be assumed that the solids are impenetrable, i.e.Ds = 0 (Crank [10]).

5.1 Series-parallel (SP) model

In the SP model (Crank [10]) the composite is split into thin cross-stream strips,after which an effective diffusion coefficient is calculated for each strip using theparallel formula and, finally, by summing the strips in series an effective diffusioncoefficient is obtained for the composite as a whole. Thus, in the SP-model theparallel formula is first applied, followed by the series formula.



Diffusion

Figure 2: Overlapping fully streamwisely staggered array.

5.2 Parallel-series (PS) model

In the PS model (Crank [10]) the composite is split into thin strips parallel to thestreamwise direction, after which an effective diffusion coefficient is calculated foreach strip using the series formula and, finally, by summing the strips in parallelan effective diffusion coefficient is obtained for the composite as a whole. Thus, inthe PS-model the series formula is first applied, followed by the parallel formula.

6 Effective diffusion coefficients of ordered arrays

For the present work only the overlapping fully streamwisely staggered array willbe discussed. A regular array, a non-overlapping fully staggered array and anoverlapping fully transversally staggered array have also been studied but will notbe included in this work.

In a regular array no staggering and no overlap of the solids occur in anyof the two principle directions. In the non-overlapping fully staggered array thesolid rectangles are staggered in both principle directions but with no overlap ofthe solids. In the overlapping fully streamwisely staggered array staggering andoverlapping of the solids occur only in the streamwise direction. A schematicrepresentation of an overlapping fully streamwisely staggered array is shown inFig. 2. The unit cell is indicated by the dashed lines and also shown in Fig. 3.

Both the SP and the PS models were applied to the unit cell. The results of theoverlapping and non-overlapping streamwisely staggered arrays are presented inTable 1.



Figure 3: Unit cell of streamwise fully staggered array with overlaps.

Table 1: Diffusion coefficients based on the SP- and PS-models.

Non-overlapping fully

staggered array: SP model:DSP

D=[

1 + ds‖ds⊥d‖(2d⊥ − ds⊥)

]−1

PS model:DPS

D= d⊥ − ds⊥

d⊥

Overlapping fully streamwisely

staggered array: SP model:DSP

D=[

1 + ds‖ds⊥d‖(2d⊥ − ds⊥)

]−1

PS model:DPS

D= 0

7 Weighted average of the SP and PS models

The following weighted average is suggested by Bell and Crank [11] to predict theeffective diffusivity:

Deff = θDSP + (1 − θ)DPS, (19)

where DSP and DPS are the estimates produced by the SP and PS models and

θ = 0.56 − 0.5

(1

2ds‖

)+ 0.4

(1

2(d⊥ − ds⊥)

). (20)



0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

100h=100×(d⊥−1/2ds

⊥)

Def

f/ D

DSP

and DPS

models

Weighted averageNumerical data

PS curve

SP curve

Figure 4: Effective diffusion coefficient as a function of the transverse pore widthh for σ = 0.2.

For the present analysis let the streamwise solid dimension be denoted by σ =12ds‖ and the transverse pore width be denoted by h = 1

2 (d⊥ − ds⊥).

8 Numerical computations

A tri-diagonal matrix algorithm was used to solve the diffusive flux field and tocompute the concentration gradients from the discritized form of eqn (2). Theconcentration values were used to calculate the effective diffusion coefficient. Theratio of Deff /D was calculated as the ratio of the sum of the fluxes of all the cellswithin the porous matrix over the ratio of the total flux when no solids are present.

8.1 Fully staggered array of rectangles

Fig. 4 shows the effective diffusion coefficient as a function of the transverse porewidth h for σ = 0.2. In RUC notation it follows that for this specific case d‖ = d⊥and 1

2ds‖ = 0.2, whilst 12 (d⊥ − ds⊥) is allowed to vary. For 0 ≤ h ≤ 0.5 the SP

and PS models for the overlapping fully streamwisely staggered array were usedand for 0.5 ≤ h ≤ 1 the SP and PS models for the non-overlapping fully staggeredarray were used. The predictions by both the SP and PS models are shown in Fig.4 together with the weighted average of the two models. The agreement betweenthe weighted average model and the numerical data is satisfactory. Fig. 4 is similarto Fig. 9(a) of Bell and Crank [11].



0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Porosity,∈

Def

f/ DRUC model (eqn. (14))RUC model (eqn. (13))Weissberg [8]Maxwell (Kim et al [1])Wakao and Smith [9]Emp eqn. (Kim et al [1])Exp. data (Currie [12])Exp. data (Kim et al [1])Exp. data (Hoogschagen [13])Num. data (present work)Num. data (Kim et al [1])

Figure 5: Effective diffusion coefficient as a function of porosity for a fullystaggered array of squares.

8.2 Fully staggered array of squares

The effective diffusion coefficient for the fully staggered isotropic array wascomputed by setting 2 d‖ = d⊥, because if a square cell with solid squares is used,then the effective diffusion coefficient cannot be computed for porosities less than0.5 because the solid squares will begin to overlap. In the way Bell and Crank [11]formulated their cells it is most probable that they encountered this restriction forfurther analysis. The effective diffusion coefficients predicted by the SP and PSmodels for square arrays can be obtained from table 1 as a function of the porosityby setting d‖ = d⊥ = d and ds‖ = ds⊥ = ds .

Fig. 5 compares the numerical data to the predictions for the effective diffusivityobtained by the RUC model and with experimental and numerical data as well asseveral analytical models from literature.

The model of Wakao and Smith [9] underpredicts the diffusivities obtained fromthe experimental data whilst both RUC models are in good agreement with theexperimental data as well as all the other analytical models. The experimentaldata of Currie [12] and Hoogschagen [13] were obtained from beds of spheres.The RUC model involving the tortuosity proves to be more accurate at porositiesgreater than 0.8 than the RUC model based on the geometric factor. Nevertheless,both RUC models as well as the numerical results obtained within this work are ingood agreement with the experimental and numerical data from literature as wellas with the other analytical models from literature. Kim et al. [1] states that simpletwo-dimensional models can be used to predict the transport properties of isotropicsystems which is confirmed by the present work.



9 Conclusions

A pore-scale model of simple rectangular geometry is used to predict the effectivediffusion coefficient for isotropic systems. The present results show that theeffective diffusivity can be accurately predicted in terms of only the porosity andthe particle geometry. Numerical computations were performed to compute theeffective diffusion coefficient for staggered arrays of solid rectangles. Both theanalytical RUC model and the numerical computations are in good agreementwith data and predictive models from literature. The successful interim resultspave the way for a more advanced study towards analysis of coupled convective-diffusive processes in double porosity media which is important in the study ofcoal processing.

References

[1] Kim, J., Ochoa, J.A. & Whitaker, S., Diffusion in Anisotropic Porous Media.Transport in Porous Media, 2, pp. 327–356, 1987.

[2] Welty, J.R., Wicks, C.E. & Wilson, R.E., Fundamentals of Momentum, Heatand Mass Transfer. John Wiley and Sons, 1969.

[3] Saez, A.E., Perfetti, J.C. & Rusinek, I., Prediction of Effective Diffusivities inPorous Media using Spatially Periodic Models. Transport in Porous Media,6, pp. 143–157, 1991.

[4] Whitaker, S., The Method of Volume Averaging. Kluwer Academic Publish-ers, 1999.

[5] Du Plessis, J.P., Saturated crossflow through a two-dimensional porousmedium. Advanced Water Resources, 14(3), pp. 131–137, 1991.

[6] Diedericks, G.P.J. & Du Plessis, J.P., On tortuosity and areosity tensors forporous media. Transport in Porous Media, 20, pp. 265–279, 1995.

[7] Lloyd, C.A., Du Plessis, J.P. & Halvorsen, B.M., On Closure Modelling ofVolume Averaged Equations for Flow Through Two-Dimensional Arrays ofSquares. Proceedings of the Fifth International Conference on Advances inFluid Mechanics, March 2004, Lisbon, Portugal, pp. 85–93, 2004.

[8] Weissberg, H.L., Effective Diffusion Coefficient in Porous Media. Journal ofApplied Physics, 34(9), pp. 2636–2639, 1963.

[9] Wakao, N. & Smith, J.M., Diffusion in catalyst pellets. Chemical Engineer-ing Science, 17, pp. 825–834, 1962.

[10] Crank, J., The Mathematics of Diffusion. Clarendon Press, Oxford, 1975.[11] Bell, G.E. & Crank, J., Influence of imbedded particles on steady-state

diffusion. J Chem Soc, Farabay Trans 2, 70, pp. 1259–1273, 1974.[12] Currie, J.A., Gaseous diffusion in porous media part 1. - a non-steady state

method. British Journal of Applied Physics, 11, pp. 314–324, 1960.[13] Hoogschagen, J., Diffusion in Porous Catalysts and Adsorbents. Industrial

and Engineering Chemistry, 47, pp. 906–913, 1955.



Measurement and prediction for air flow dragin different packing materials

C. Rautenbach1, B. M. Halvorsen2, 3, E. du Plessis1, S. Woudberg1

& J. P. du Plessis1

1Department of Mathematical Sciences, Applied Mathematics Division,Stellenbosch University, South Africa2Institute for Process, Energy and Environmental Technology,Telemark University College, Norway3Telemark Technological R&D Centre (Tel-Tek), Norway

Abstract

Packed bed reactors are widely used in industry to improve the total contact areabetween two substances in a multiphase process. In some cases, like for the pack-ing elements of some CO2 absorption towers, the packing material can be of suchgeometric nature that during discharge through them different flow conditions canbe present in different parts of the packing. This renders prediction of pressuredrops quite difficult. This paper concerns experimental and modelling activitiesto improve predictive equations for pressure drops over a packed bed of Raschigrings. It is shown that the application of some corrective measures can dramaticallyimprove the correlation between theory and experiment, but that more research isneeded in this field, regarding both carefully controlled experiments and mathe-matical modelling.Keywords: pressure drop, porous media, packing materials, Raschig rings, dragmodels.

1 Introduction

In the chemical engineering industry packed bed reactors are widely used to improvethe total contact area between two substances in a multiphase process. The processtypically involves forced convection of liquid or gas through either structured ordumped solid packings. Applications of such multiphase processes include mass



(a) (b)

Figure 1: Typical examples of random packings. (a) Metallic Raschig rings and(b) glass Raschig rings.

transfer to catalyst particles forming the packed bed and the adsorption of gases orliquids on the solid packing.

For reactor design the drag laws are needed to predict the pressure drops oversuch reactors. In many cases the packing material can be of such geometric naturethat during discharge through them different flow conditions can be present in dif-ferent parts of the packing. This renders prediction of pressure drops quite difficult.This paper concerns experimental and modelling activities to improve predictiveequations for pressure drops over a packed bed of Raschig rings.

An experimental study on the determination of air flow pressure drops over dif-ferent packing materials was carried out at the Telemark University College inPorsgrunn, Norway. The packed bed consisted of a cylindrical column of diameter0.072m and height 1.5m, filled with different packing materials. Air was pumpedvertically upwards through a porous distributor to allow for a uniform inlet pres-sure. Resulting pressure values were measured at regular height intervals withinthe bed. Due to the geometric nature of a Raschig ring packing the wall effects,namely the combined effects of extra wall shear stress due to the column sur-face and channelling due to packing alignment adjacent to a solid column surface,were assumed to be negligible. Several mathematical drag models exist for packedbeds of granular particles and an important question arises as to whether they canbe generalized in a scientific manner to enhance the accuracy of predicting thedrag for different kinds of packing materials. Problems with the frequently usedErgun equation, which is based on a tubular model for flow between granules andthen being empirically adjusted, will be discussed. Some theoretical models thatimprove on the Ergun equation and their correlation with experimental work willbe discussed. It is shown that a particular pore-scale model, that allows for differ-ent geometries and porosities, is superior to the Ergun equation in its predictions.Also important in the advanced models is the fact that it could take into accountanomalies such as dead zones where no fluid transport is present and surfaces thatdo neither contribute to shear stress nor to interstitial form drag. The overall con-clusion is that proper modelling of the dynamical situation present in the packing



0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

500

1000

1500

2000

2500

3000

glass Raschig ring data

Experimentalerror

Figure 2: Experimental results for glass Raschig rings.

can provide drag models that can be used with confidence in a variety of packedbed applications.

There is a wide range of different packing materials available. The packingmaterial used varies from application to application. Factors that need to be con-sidered include the pressure drop produced by the packed bed, chemical stabilityof the packing and size of the packing, to name but a few. Porous media createdby using the packing materials illustrated in Figure 1 are called random dumpedpackings, as they are randomly placed into the container. Raschig rings and smallglass spheres were provided by the TUC in Porsgrunn Norway and were used toproduce the experimental results presented in this study.

An example of data retrieved during the experiments conducted at the TUC isgiven in Figure 2. As indicated in Figure 2, there is a data point that does not followthe trend of the rest of the data. This is assumed to be caused by experimental errorand is ignored for the remainder of the analysis.

2 Existing models used to predict flow behavior through porouspacked beds

In this section some existing available models are discussed. For convenience ofcomparison all models will be rewritten in terms of a F , defined by

∆pL

= µqF, (1)



with ∆p the measured pressure drop, corrected for gravity, and L the bed height.The Reynolds number,Rep, is defined as:

ReDp ≡ ρqDp

µ, (2)

for use in the present study.

2.1 Ergun equation

The Ergun equation [1], empirically based on results obtained from experimentalpacked beds of identical spheres, is given in the present notation, with Dp theparticle diameter, as:

FD2p =

150(1 − ε)2

ε3+

1.75(1 − ε)ε3

· ReDp . (3)

The empirically based constants compensate for the assumptions made in the cap-illary model. One assumption is that the porous medium is statistically uniform sothat there is no channelling. Of course this is a crude assumption as channellingis common place in practical applications. Another more practical assumption isthat the column diameter is large in comparison to the particle dimensions. TheErgun equation [1] also assumes a uniform particle size. As more data on irregularparticles becomes available, the modelling can be improved to represent a widerspectrum of packed bed geometries.

2.2 RUC model

An RUC drag model, [2], is used in this work to give a possible prediction ofsingle phase flow through the packing elements of a CO2 absorption tower. Thisuniversal model can be applied to different types of porous media and for this workour interest is in the granular and foam versions.

2.2.1 Granular RUC modelThe granular RUC model is model aims to approximate porous media such assand, consisting of small granular particles. The expression for the drag factor forgranular porous media is expressed as:

FD2p =

25.4(1 − ε)4/3

(1 − (1 − ε)1/3)(1 − (1 − ε)2/3

)2 +cd(1 − ε)

2ε(1 − (1 − ε)2/3

)2ReDp . (4)

The form drag coefficient, cd, should typically be determined either numericallyor empirically and is frequently assigned the value 1.9. The pressure drop can thenbe determined via the drag factor F from equation (4).



2.2.2 Foam modelThe RUC foam model was developed to accurately predict flow behavior throughfoamlike porous media [3]. A typical example is a spongelike metallic foams. Thetwo variations that exist in the foam model are the doubly staggered model andthe singly staggered model, the latter yielding a smaller pressure drop for the samedischarge. The drag factor is given as:

Fd2 =24ψ2 (ψ − 1)

ε2+

cd ψ2(ψ − 1)

2 ε2(3 − ψ)· ρqdµ, (5)

in the case of the doubly staggered model and as:

Fd2 =36ψ2 (ψ − 1)

ε2+

cd ψ2(ψ − 1)

ε3(3 − ψ)· ρqdµ, (6)

for the singly staggered model. An expression for the geometric factor, ψ, as givenin equations (5) and (6), is given as:

ψ = 2 + 2cos[4 π3

+13cos−1(2ε− 1)

], (7)

for foamlike media. The micro-scale parameter d is given by the length of a cubethat would produce N cubes in the total packing volume. The total number ofparticles in a fixed bed is represented by N .

2.3 The Sonntag correction

A small change in the porosity has a large impact on the pressure drop and thusthis effect can have a large influence on the pressure drop, predicted by the models.Sonntag [4] introduced the influence of the fraction m of the inner volume Vi

of a Raschig ring that is stagnant and does not contribute to the shear stress norto the interstitial form drag. After experimental correlations he stated that onlyapproximately 20% of the inner volume of the ring is available for flow, i.e. m =0.2. The effect of the decrease in the volume available for flow is a decrease in theeffective porosity.

2.4 Nemec’s equation

Nemec [4] applied the Sonntag correction to the Ergun equation. In Nemec’s workthe experimentally determined values for the tortuosity, χ, and the friction fac-tor, f , are kept the same as stated by Ergun. The reasoning is that when Sonntagderived his 20% criterion he used the original Ergun equation. Thus if adapted val-ues for the tortuosity, χ, and the friction factor, f , are used, say for a bed consisting



of equivalent solid cylinders, the value of Sonntag’s correction would change. Theequation put forward by Nemec can be written as:

Fd2e =

150(1 − ε)2

ε3

[ε3

(1 − (1 − ε) (Vfc −mVi) /Vp)3

]×

[de (Sfc +mSi)

6Vp

]

+1.75deρq(1 − ε)

µε3

[ε3

(1 − (1 − ε) (Vfc −mVi) /Vp)3

]

×[de (Sfc +mSi)

6Vp

]2

, (8)

with Vi the volume of the inner void cylinder, Vp the particle volume and Vfc thevolume of a hypothetical full cylinder with the same outer dimensions. In equation(8) the surfaces are indicated by an S and the subscripts have the same meaning aspreviously mentioned volumes. The fraction of the inner void of each ring availablefor flow is denoted by m and can be taken as 20% according to Sonntag [4]. Theequivalent particle diameter is defined as 6Vp/Sp and is denoted by de in equation(8).

2.5 Mackowiac’s equation

Using experimental results from Raschig ring packings, Mackowiak [5] arrived atthe following drag equation for perforated Raschig ring packings when he investi-gated the influence of the fluid-solid interface on the drag:

F =1µq

(725.6Rev

+ 3.203)

(1 − ϕ)(

1 − ε

ε3

) (F 2

v

dpK

). (9)

Here the form factor ϕ becomes zero for non-perforated packings, like Raschigrings.

According to Mackowiak the value for the resistance coefficient, ψ0, has beendetermined experimentally and has been found to be given by:

ψ0 =(

725.6Rev

+ 3.203), (10)

with Rev the modified Reynolds number and is defined as:

Rev =qdp

(1 − ε)µK. (11)

3 Comparative results

The comparisons of the models to the experimental data for glass Raschig ringsare given in Figure 3. The models were compared to the data and to each other,for metallic Raschig rings in Figure 4 (a). The metallic rings produced a higher



0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

500

1000

1500

2000

2500

3000

q [m/s]

∆ p/

L [P

a/m

]

DataNemec eqn. (5.1.37)Doubly staggered foam eqn. (2.2.12)Singly staggered foam eqn. (2.2.13)Granular model eqn. (2.2.9)Ergun eqn. (2.3.4)Mackowiak eqn. (5.1.48)

Figure 3: Comparison of model with the data for glass Raschig rings (refer toFigure 1(a)).

porosity than the glass ring bed. The result is that the effect of the wall on the overall pressure drop is lower with the metallic rings. This is also slightly evident fromFigures 3 and 4 (a). It is suspected that wall effects are the cause of the differencein curvature of the models and the actual data. Confirmation of this suspicion isthat when pressure drop measurements through small spherical particles were col-lected, there were no curvature discrepancy. In Figure 4 (b) the comparison of theGranular RUC model to the data acquired with flow through non-uniform sphericalparticle is given. Thus the difference in curvature could not be caused by incorrectdata processing or incorrect experimental methods.

4 Adaptations

4.1 Shape-factor

In the case of irregular shaped packings, shape factors can be used to determine theequivalent diameter of a sphere with the same volume as the element or particle(nominal diameter). The sphericity, φs, of an element is the ratio of the surfaceof the equivalent sphere to the actual surface area of the element. In Figure 5 theeffect of a shape factor (sphericity) is investigated. It is evident that the sphericitymarks a significant improvement on the RUC granular model and Ergun equation’saccuracy.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

200

400

600

800

1000

1200

q [m/s]

∆ p/

L [P

a/m

]

DataNemec eqn. (8)Doubly staggered foam eqn. (5)Singly staggered foam eqn. (6)Granular model eqn. (4)Ergun eqn. (3)Mackowiak eqn. (9)

(a)

0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.0220

1

2

3

4

5

6x 10

4

q [m/s]

∆ p/

L [P

a/m

]

Granular model eqn. (2.15), with a particle diameter of 0.0001mGranular model eqn. (2.15), with a particle diameter of 0.0002mGranular model eqn. (2.15), with a particle diameter of 0.0003mdata

(b)

Figure 4: (a) Comparison of model with the data for metallic Raschig rings Figure1 (b). (b) Comparison of the RUC granular model [2] with experimentaldata given a range of particle sizes. 100 − 200µm powder was used toacquire the data.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

500

1000

1500

2000

2500

3000

q [m/s]

∆ p/

L [P

a/m

]


Figure 5: Investigation of the effect of the sphericity on the Ergun equation and theRUC granular model.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1000

2000

3000

4000

5000

6000

q [m/s]

∆ p/

L [P

a/m

]


Figure 6: (a) Comparison of models with the Sonntag correction equal to 20%(with glass Raschig rings). (b) Pressure gradients of the different modelsand the data.



4.2 The Sonntag correction

This stagnant region within the rings can also be the reason for the under predictionof the Ergun equation. In Figure 6 the effect of the Sonntag correction is given.Thus using the sphericity and the Sonntag correction the best correlation to theexperimental data is obtained.

5 Conclusion

The major contribution of this work was the generation of a set of data by meansof experiments and analyses of possible predictive models. They produced satis-factory correlations to data and thus provide confidence in the capability of math-ematical models to predict experimental trends for various fixed bed reactors.

The initial aim of using the spherical particle powders in the experimental partof this study was to check how experimental results would compare with the wellknown Ergun equation. As the Ergun equation was adapted empirically, based onthe data obtained for flow through uniform spheres, it is expected to describe theflow through non-uniform spheres to some extent. The reason for such an assess-ment was because of the non-linearity of the data acquired with Raschig rings. Thecurvature differed from what the models predicted and the suspicion arose that thedata may have been processed incorrectly. Thus, after the powders produced datathat had the same behavior as the models, especially the Ergun equation, it can beconcluded that the problem with the Raschig rings was not the result of incorrectdata processing. The main cause is expected to be wall effects, due to the columnto ring size ratio being small, only about 10. If this ratio is much larger, i.e. ifdc/dp >> 10, the effect of the wall can be ignored [5].

With the incorporation of the Sonntag correction all the models perform better.On the modelling side the Sonntag correction was applied with great success toexisting empirical and pore-scale models. Using any of the models discussed in thiswork without Sonntag’s correction will result in a under-prediction of the pressuredrop for Raschig ring experiments.

The deviations in the models from the data could be attributed to a variety ofdifferent effects. Almost none of these effects could be pin-pointed satisfactorilyin this work due to time constraints. It is thus advisable to verify which effectspredominated in both the Raschig ring and powder beds. The next logical stepwould then also be to find ways in which to combat the effects or to adapt themodels to take these effects into account.

References

[1] Ergun, S., Fluid flow through packed columns. Chemical EngineeringProgress, 48, pp. 89–94, 1952.

[2] du Plessis, J.P. & Woudberg, S., Pore-scale derivation of the ergun equation toenhance its adaptability and generalization. Chem Eng Sci, 63, pp. 2676–2586,2008.



[3] Crosnier, S., du Plessis, J.P., Riva, R. & Legrand, J., Modeling of gas flowthrough isotropic metallic foams. Journal of Porous Media, 9, pp. 35–54, 2006.

[4] Nemec, D. & Levec, J., Flow through packed bed reactors: 1. single-phaseflow. Chemical Engineering Science, 60, pp. 6947–6957, 2005.

[5] Mackowiak, J., Extended channel model for prediction of the pressure dropin single-phase flow in packed columns. Chemical Engineering Research andDesign, 2008.

[6] Mcdonald, I.F., El Sayed, M.S., Mow, K. & Dullien, F.A.L., Flow throughporous media - the ergun equation revisited. Ind Eng Chem Fundam, 18,pp. 199–208, 1979.



CFD simulation with multiphase flows in porous media and open mineral storage pile

S. Torno, J. Toraño, I. Diego, M. Menéndez, M. Gent & J. Velasco

School of Mines, University of Oviedo, Spain

Abstract

In open storage piles in bulk solids port terminals, power stations and cement factories, not only the solid and porous barrier behaviour in front of the pile is important, but also the effect that porous and solid barriers produce when they are behind the pile. Considering the dust propagation behind the pile, the solid barriers are more effective than the porous ones. But, the effect of the porous barriers on the wind velocity distribution, mainly in zones between the barriers and the piles should be taken into account, regarding the total dust emission to the atmosphere and its propagation. In these studies, wind and dust concentration measurements in laboratory and field were carried out. A hot-wire anemometer (Velocicalc Plus (TSI)) was used in the wind study and two dust collectors (E-Sampler, Met One Instruments, Inc., Oregon, USA) were used to measure the dust concentration (Total Particle like PM10); a meteorological station (E-Sampler, Met One Instruments, Inc., Oregon, USA) attached to a PM10 collector was also employed. All these measurements were used to adjust the 3D CFD computational model (Ansys CFX 10.0): In the wind case through a k-epsilon turbulence model and the dust case by Lagrangian method. These adjusted models allow us to carry out several simulations combining the effect of solid and porous fences in front of and behind the pile, as well as pile shape modifications and behaviour analysis according to the dust emission from several special configurations and its relation to the wind gust preferential directions. Keywords: particle emission, Computational Fluid Dynamics (CFD), open storage pile, air pollution.



1 Introduction

The movement of minerals in open storage piles produces particle emissions to the atmosphere, which were studied to minimize their effect in the area more or less near the pile. In previous research [1], the use of solid barriers in storage pile protection which reduces dust emission to 66% in the worst environmental and industrial conditions, was shown. Nevertheless, the use of this solid barrier produces a high velocity vortex between the barrier and the pile, causing a greater dust emission to the windward side of the pile, [2–4]. The authors, based on their own experience and other research like [5] and [6], changed a solid barrier for a 30% porosity barrier, proving that the dust emission is reduced by 78%. The 30% porosity is obtained from research carried out by [7], who have determined from the “Particle Tracking Velocimitry” that a porosity of 30% is the more effective. Once the barrier effect (solid and porous ) in front of the open storage pile is studied, the need of studying the barrier effect behind the pile arises, thus the pile is protected against the wind with a barrier in front of it and the emitted dust is collected with a barrier behind the pile. In this paper, the barrier effectiveness, solid and porous, in each of the studied position, is shown. The 3D Computational Fluid Dynamics or CFD code Ansys CFX 10.0 is used to carry out the simulations, through the K-epsilon turbulence model [8], which was adjusted by the measurement campaigns carried out in the Mine of Carbonar S.A. situated in Asturias (North of Spain).

2 Experimental description

2.1 Introduction

The scale tests were carried out in the installations of Carbonar, which are shown in fig. 1. The equipment consisted of a metalic cone of 52cm in height (h) and 69cm in diameter, a metalic ventilation pipe of 3m in length and 300mm in diameter, which is connected to a 12kw fan, which generates the wind flow. Two barriers are added to these elements, which are placed in front of and behind the cone.

2.2 Barrier in front of the cone

In this case, the barrier has 52cm in height and 1.78m in length such as it was explained in [1] for the solid barrier, and it is placed at 25cm from both the pipe and the cone. In fig. 2 distribution of the elements used in the scale tests for the porous barrier, is shown.



Figure 1: Zone where the equipment was installed.

2.3 Barriers behind the cone.

In this case the barrier is placed at 3.5m behind the cone and it has 1m in height and 2.04m in length. In fig. 3 the distribution of the elements for measurements in field with the dust collectors at 2.5m behind the barrier, is shown.

2.4 Velocity and concentration measurements.

Flow velocity measurements were carried out with a hot-wire anemometer (Velocicalc Plus (TSI)), which was also used by other authors like [9] and [10], in 73 measurement points distributed in the cone zone and the barrier. Two dust collector (E-Sampler, Met One Instruments, Inc) measuring total particles and PM10 (<10 micron) were used to measure dust concentration. Meteorological conditions, relative humidity of atmosphere, temperature, wind velocity and the wind direction were measured by a meteorological station (E-Sampler, Met One Instruments, Inc) attached to a PM10 collector. These field measurements were used to validate CFD simulation.

3 Modelling

The mathematical model is based on three equations: continuity, momentum and energy. These expressions form a partial differential equation system, coupled in linear form and they were used to achieve the solution. The porosity input in the model is carried out from the research developed by [11] and [12] in which the absolute porosity value is calculated by two coefficients, the Linear Resistance Coefficient (CR1) and the Quadratic Resistance Coefficient (CR2): Determination of these coefficients is included in the discretisation of the Navier-Stokes equations. This discretisation produces a set of equations which



describe the fluid properties, that is, momentum equation (1) and continuity equation (2):

( ) ( ) 0zw

xu

t=

∂ρ∂

+∂ρ∂

+∂ρ∂

, (1)

Mijj

i2

iij

ij

i Sxx

uxx

Pxu

ut

u+

∂∂∂

µ∂∂

+∂∂

−=

∂∂

+∂∂

ρ . (2)

I II III IV V

Figure 2: Distribution of the elements used for the porous barriers.

Figure 3: Distribution of the elements with the dust collectors.



where, I is the Acceleration term, II is the Advection term, III is the Pressure term, IV is the Diffusion term and V is the Source term (3):

i2Ri1RMi uUCuCS −−= (3) where, ui is the velocity dimensional component along the 3 directions (x,y,z), U

is the velocity vector, U is the speed, P is pressure, ρ is the density of incompressible air, µ is the dynamic viscosity and SMi are the source terms in the three directions; it indicates the porous barriers in the equation. CR1 is set to 0 (Kg /m3 s1) assuming negligible viscous losses in the ambient air flow (µ=1.79x10-5 N s/m2). CR2 is set to 18.45 (kg/m4), in a 30% porosity, [11]. These coefficients are related to absolute porosity via Kloss (empirical loss coefficient) [CR2·ε2/ρ], which is included in Darcy´s Law (4):

ilossii

uUkukx

Pρ+

µ=

∂∂

− (4)

In fig. 4, the geometry carried out by using SolidWorks for the barrier placed in front of the cone, is shown. The domain of the two models has the same dimensions (8x5x25).

Figure 4: Geometry of the model with the barrier in front of the cone.

In fig. 5, the meshing carried out by using IcemCFD with the barrier placed behind the cone, is shown. This model was meshed with a total of 1,011,030 elements and the other model (barrier placed in front of the cone) was meshed with a total of 1,107,611 elements. In both cases it can be seen a finer meshing zone, where the dust collectors are situated, in which dust concentration measurements are taken for the simulations (fig. 5). The fluid domain is a 25º air affected by the gravity laws, not taking into account the heat transference and considering k-epsilon as the turbulence model.



The boundary conditions have been an Inlet that corresponds to a 12 m/s wind velocity coming out of a metallic pipe; the ground surface, cone, pipe and barrier as Wall and considering the rest as Opening.

Figure 5: 3D meshing of the model with the barrier behind the cone.

4 Wind results

4.1 Barriers in front of the cone

Taking into account the results obtained in the model with a solid barrier (when porosity is 0%), [1], it was decided to change the barrier porosity to 30%. This porous barrier makes the high velocity vortex that is produced behind the solid barrier, disappear, avoiding the dust set in suspension to windward zone of the cone. In fig. 6 two planes of mean wind velocity for both cases, solid and porous barrier, are shown. Besides, how the porous barrier eliminates the wind recirculation between the barrier and the cone, which is responsible for the dust emission on windward surface of the pile, is shown too. These velocity values obtained from our simulations have been validated through velocity measurements in field for the same points. In the solid barrier case, the equation that relates field measurements to those of the model is y=0.9495x-0.1191 with a correlation coefficient R2= 0.8859. In the 30% porosity barrier case, the equation that relates field measurements to those of the model is y=1.5216x-1.2837 with a correlation coefficient R2= 0.8493. The high values of the correlation coefficients indicate the suitability of the models.



(a)

(b)

Figure 6: Planes of mean wind velocity for solid barrier (a) and porous barrier (b).

4.2 Barriers behind the cone

When the barrier is behind the cone, the solid barrier is the most effective, since it traps more quantity of dust emitted by the pile than the porous barrier. In fig. 7 two planes of mean wind velocity perpendicular to the cone for solid and porous barrier, are shown. It can be seen in the porous barrier case, how the air reaches the dust collectors with a higher velocity than in the solid barrier case. This is due to the recirculation zone of low wind velocity that is produced behind the solid barrier,



As in the previous section, the velocity values obtained from our simulations have been validated through velocity measurements in field for the same points. In the solid barrier case, the equation that relates field measurements to those of the model is y=0.9999x-0.2841 with a correlation coefficient R2= 0.8756. In the porous barrier case, the equation that relates field measurements to those of the model is y=2.5135x-3.9428 with a correlation coefficient R2= 0.8463.

(a)

(b)

Figure 7: Planes of mean wind velocity for solid barrier (a) and porous barrier (b).



5 Concentration results

Models are validated when field measurements coincide with simulations carried out by CFX. The concentration value in software Ansys CFX is achieved through `Particle Tracking`, which is Lagrangian type solver that simulates particle trajectories from calculated velocity fields. This method starts from the continuous phase calculation (air) and they are used as input date for the dispersed phase (particles). The millions of particles which are set in suspension in reality are simulated from the thousands of representative particles with a certain mass quantity; this is known as “Particle Number Rate” [8]. The concentration calculation is obtained from de calculation of the area under the Concentration-Time curve obtained by the PM10 dust collector. This calculation is necessary as a permanent phenomenon (concentration by CFX) and a transitory phenomenon (concentration in field) are compared. In table 1 the results of dust concentration obtained by modelling and those obtained in field for the two types of barriers, in front and behind the cone, are shown. The porous barrier placed in front of the cone presents less particle emission to the atmosphere. The lower concentration value behind the cone corresponds to a solid barrier, as this barrier traps the most particles avoiding they spread farther. As it can be seen, in table 1, the effect of porous barriers is to trap some particles and decrease the velocity of those crossing the barrier.

Table 1: Comparison between concentration measurements by CFX and in field.

CONCENTRATION (mg/m3)

In front of the cone Behind the cone BARRIER TYPE

Experimental Simulation Experimental Simulation

Solid 12.27 12.02 30.54 29.53

Porous 7.86 8.124 34.05 34.97

6 Conclusion

Ansys CFX software is a good tool to simulate the problems of dust set in suspension and its spreading in the surrounding area of an open storage pile. When emissions of particulated material to the atmosphere are important it is necessary to use protection systems in front of the pile and behind it. These protection systems are solid and porous barriers and the effect they produce on dust emission from the pile depend on the position of this pile.



The use of a porous barrier in front of the pile decreases the effect of the high velocity vortex which is produced behind the solid barrier, avoiding in this way the particle emission on windward surface of the pile. Behind the pile, a solid barrier traps more particles than a porous one.

References

[1] Diego, I., Toraño, J., Torno S. & Garcia B., 2008. Experimental tests and Computational Fluid Dynamics (CFD) simulations of barriers installed around open storage piles of raw materials. Advances in Fluid Mechanics, 7, pp. 101-109.

[2] Lee, S.J. & Lim, H.CH., 2001. A numerical study on ow around a triangular prism located behind a porous fence. Fluid Dynamics Research, 28, pp. 209-221.

[3] Toraño, J., Rodríguez, R., Diego, I., Rivas, J.M. & Pelegry, A., 2007. Influence of the pile shape on wind erosion CFD emission simulation. Applied Mathematical Modelling, 31, pp. 2487–2502.

[4] Diego, I., Toraño, J., Torno, S. & García, B., 2008. Experimental tests and Computational Fluid Dynamics (CFD) simulations of barriers installed around open storage piles of raw materials. Seventh International Conference on Advances in Fluid Mechanics (AFM 2008), The New Forest, UK.

[5] Lee, S.J. & Kim, H.B., 1999. Laboratory measurements of velocity and turbulence field behind porous fences, Journal of Wind Engineering and Industrial Aerodynamics, pp. 311-326.

[6] Kim, H.B. & Lee, S.J., 2001. Hole diameter effect on flow characteristics of wake behind porous fences having the same porosity. Fluid Dynamics Research, 28, pp. 449–464.

[7] Lee, S.J., Park, K.C. & Park, C.W., 2002. Wind tunnel observations about the shelter effect of porous fences on the sand particle movements. Atmospheric Environment, 36, pp. 1453–1463.

[8] Diego, I., Pelegry, A., Torno, S., Toraño, J. & Menendez M., 2009. Simultaneous CFD Evaluation of Wind Flow and Dust Emission in Open Storage Piles. Applied Mathematical Modelling. Article in press.

[9] Coleman, H.W., Steele, W.G, 1989. Experimentation and Uncertainty Analysis for Engineers. John Wiley & Sons., Inc., New York.

[10] Kim, H.G., Lee, Ch.M., Lim, H.C. & Kyong, N.H., 1997. An experimental and numerical study on the flow over two-dimensional hills. Journal of Wind Engineering and Industrial Aerodynamics, 66, pp. 17-33.

[11] Tiwary, A., Morvan, H.P. & Colls, J.J., 2005. Modelling the size-dependent collection efficiency of hedgerows for ambient aerosols. Journal of Aerosol Science, 37, pp. 990-1015.

[12] Wang, H. & Takle, E.S., 1995. A numerical simulation of boundary-layer flows near shelterbelts. Boundary-Layer Meteorology, 75, pp. 141-173.

[13] ANSYS CFX-Solver, Release 10.0: Theory, 2008. Flow in porous media, pp. 65-67.



Powered addition applied to the fluidisationof a packed bed

P. D. de Wet1, B. M. Halvorsen2, 3 & J. P. du Plessis1

1Department of Mathematical Sciences, Applied Mathematics Division,Stellenbosch University, South Africa2Institute for Process, Energy and Environmental Technology,Telemark University College, Norway3Telemark Technological R&D Centre (Tel-Tek), Norway

Abstract

Upon analysis of a set of collected data of Newtonian flow through a porousmedium, it is evident that asymptotes exist for some variable dependencies in thetransition from packed to fluidised bed. The transition between such asymptotesis governed, amongst others, by parameters such as particle size, particle sizedistribution, superficial gas velocity and bed height. The powered addition to apower, s, of such asymptotes f0 and f∞, leads to a single correlating equationthat is applicable over the whole range of flow rates, namely f s = f s0 + f s∞.This procedure circumvents the introduction of ad hoc curve fitting measuresin the cross-over regions between the asymptotes and subsequent, unwantedjumps in piecewise fitted correlative equations for the dependent variable(s). Theaforementioned method of powered addition is applied to the experimental dataand the outcomes are discussed.

1 Introduction

A fluidised bed is formed when a fluid, usually a gas, is passed upwards througha bed of particles. The packed bed of particles, supported on some kind ofdistributor, is converted into an expanded, suspended bed and in the process takeson many liquid-like properties: the bed has zero angle of repose, pressure increaseslinearly with distance below the surface, wave motion is observed, denser objectssink and lighter objects may be floated on its surface, their movement almostunhindered [1].



A fluidised system has a number of highly useful properties that may be utilisedin industrial applications. Although the mechanism may be both physical andchemical in nature, the dominating attribute utilised in a specific industrial process,will determine its application. A broad classification of fluid bed applications isgiven in [2], from which it is evident that fluidisation is an interdisciplinary fieldof inquiry. To optimally reap the benefits of this promising process there thus existsa particular need for the development of predictive models; holding challenges forboth fundamental and applied research. The underlying motivation for this projectis the aspiration towards establishing a sound modelling framework for analyticaland computational predictive measures.

2 Powered addition as curve-fitting technique

It is common practice to represent the general trend in a set of collected data bydrawing a line through the individual datum points on the plot. The better thepredictive line on the graphical presentation corresponds to the physical reality,especially in the limits of the independent variable, the greater the trustworthinessof obtained results. The basic procedure of asymptotic matching by straightfor-ward addition of the expressions for the asymptotic conditions is a method thathas been in use for some time, especially in engineering practice. However, thearticle by Churchill and Usagi [3], which appeared in 1972, for the first time reallyformalised the use and accentuated the wide application possibilities of the methodand variations thereof.

In many continuum processes the value of a sought after parameter, thedependent variable, is expressible as a function of certain known parameter(s),the independent variable(s), at low and high values. The latter may be regardedas asymptotic conditions of the dependent variable. Let a certain parameter, f , ofsuch a process be described at low values of x by

f0 ≡ f x → 0, (1)

and the dependence at the upper extremal value of x by

f∞ ≡ f x → ∞. (2)

The simple addition of two asymptotic solutions or approximations is ofteneffected to obtain a single solution that holds over the entire range of theindependent variable, i.e.

f = f0 + f∞. (3)

Equation (3) may now be considered as a matching curve connecting the twodependencies as it satisfies the asymptotic conditions and also provides values forf at intermediate values of the independent variable, x. Frequently the values ofthe dependent variable at the transition between the asymptotic extremities do notlie exactly on this matching solution. Churchill and Usagi [3,4] and Churchill [5,6]demonstrated that the use of powered addition, as shown in equation (4), may lead



to dramatic improvement in correlation with experimental data

f s = f s0 + f s∞. (4)

By adjusting the value of the shifting exponent, s, the level of the solution maybe modified so as to more closely trace the expected or empirical values, yieldingbetter correspondence between predictive equation and experimental results.

3 Experimental procedure

Experimental data was obtained from measurements performed on laboratory-scale fluidized beds at Telemark University College, Porsgrunn, Norway. The bedwas contained by a cylindrical perspex tube with an inner diameter of 72mm. Sincethe project was only concerned with the pressure drop across a specified sectionof the bed (i.e. between two pressure sensors), the porosity of the plate on whichthe bed was supported was irrelevant. It was only required not to allow particles todrop through into the antechamber and to function as a uniform gas distributor.

The beds were filled with glass powders consisting of spherical glass particles,with a density of 2485 kg/m3 and available in three different diameter-ranges:100 μm–200 μm; 400 μm–600 μm; and 750 μm–1000 μm. These particles allfall into Group B and D according to the Geldart powder classification [7]. Sincethe particles are manufactured they were assumed, for the sake of simplicity, tobe perfectly spherical in shape and thus have a Waddell sphericity factor, ψ = 1[8]. In all of the experiments performed the fluid used to fluidise the bed wasair at ambient conditions, with a density of 1, 2 kg/m3 and viscosity of 1.78 ×10−5 N·s/m2.

By using different gas velocities and only one particle diameter-range or amixture of the particle diameter-ranges, the parameters of the experiment couldbe varied.

A minimum of three runs were performed for each of the three differentdiameter-ranges mentioned above. A run consisted of a gradual increase untiland beyond fluidisation, followed by a gradual and controlled decrease to zerofluid flow. It was noted that hysteresis only became apparent if the bed wasallowed a period of rest between consecutive fluidisations. Consequently, runs fora particular diameter-range were conducted in succession allowing the obtainedpressure-values to be averaged. It is to this averaged data that the curve-fittingtechnique discussed in this paper was applied.

4 Curve fitting

Three regimes are to be identified, namely the two regimes corresponding respec-tively to the physical conditions related to the two asymptotes f0 and f∞ and thechange-over regime connecting the two. This latter regime surrounds the criticalpoint, xc, where the asymptotes meet and is of particular interest in the presentstudy.



4.1 The lower asymptote

Before the onset of fluidisation, the bed may be regarded as a packed bed orporous medium consisting of spherical particles. To describe the pressure dropof Newtonian flow through such a structure, the Ergun equation has proven tobe satisfactory in most applications as is evident from its extensive utilisation inchemical engineering. In their paper, Du Plessis and Woudberg [9] compare theRUC (representative unit cell) model to the Ergun equation for the description ofNewtonian flow through a packed bed of uniformly sized spherical granules andfind the agreement to be satisfactory. The choice, in this paper, of the RUC modelto describe the lower asymptote is due to the fact that it is adaptable to differentphysical situations, whereas the Ergun equation is empirically based and will thusvary according to the situation to which it is applied. Furthermore it allows theusage of the average bed porosity and is applicable over both the entire porosityand laminar Reynolds number ranges.

In the original Ergun equation, which is already a special case of poweredaddition with shifter, s = 1,

p

H= A

(1 − ε)2

ε3

μq

D2h

+ B1 − ε

ε3

ρf q2

Dh, (5)

the values of coefficients A and B were acquired experimentally and are givenas 150 and 1.75 respectively. Here p denotes the finite pressure difference(measured in the stream-wise direction of fluid flow), H the bed height, ε theporosity or bed voidage,μ the fluid viscosity, ρf the fluid density, q the superficialvelocity of the traversing fluid and Dh the hydraulic diameter (which is equal tothe diameter of the spherical particles).

The work of Du Plessis & Woudberg allows one to purge equation (5) of itsempirical elements. Pore-scale analysis of interstitial flow conditions lead to thefollowing expression of coefficients

A = 25.4ε3

(1 − ε)2/3(1 − (1 − ε)1/3)(1 − (1 − ε)2/3)2, (6)

and

B = ε2cd

2(1 − (1 − ε)2/3)2. (7)

They thus succeed in rewriting (5) such that it is independent of, and not limitedby, the range of porosities used. Here the particle Reynolds number, Rep, is definedas

Rep ≡ ρf qDh

μ, (8)

and as in [9] the value of the form drag coefficient, cd in (7), was taken to be 1.9,presenting the most empirical aspect of the procedure.



4.2 The upper asymptote

As noted by Geldart [8], the pressure drop across a fluidised bed, given by

p = m0g

Ac= ρ0U0g

Ac= ρ0gH, (9)

is the only parameter that can be predicted with accuracy, since at all times duringfluidisation the downward force, i.e. the weight of the bed,m0g, is balanced by theupward force,pAc. Division of equation (9) by the bed height, yields

p

H= m0g

AcH= ρ0U0g

AcH= ρ0g, (10)

which forms the upper limiting asymptote. Here m0 denotes the bulk mass, ρ0 thebulk density,U0 the bulk volume, g acceleration due to gravity (taken as 9.81 m/s2)and Ac the cross-sectional area of the bed.

4.3 Powered addition of the asymptotes

The original Ergun equation, (5), was obtained through simple addition of theBlake-Kozeny and Burke-Plummer equations, the former being a Darcy-typeequation predominating in the regime where Rep → 0 and the latter dominatingin the Forchheimer regime. If powered addition, as discussed in Section 2, is usedto match the asymptotic conditions – i.e. equations (5) and (10) is combined – asingle correlative measure,

p

H=[(A(1 − ε)2

ε3

μq

D2h

+ B1 − ε

ε3

ρf q2

Dh

)s+ (ρ0g)

s

]1/s

, (11)

is obtained for the pressure drop over the bed. Here coefficients A and B are asexpressed in equations (6) and (7).

4.4 Critical point and shifting-exponent

The central or critical point, xc, of the matching curve is the value of theindependent variable at which the asymptotes meet. Since the asymptotes intersecthere, the numerical value of their respective functional expressions must be equal,that is

f0x = f∞x. (12)

As both functions, f0 and f∞, contribute equally to the added solution at this point,the resultant curve is most sensitive to variations in the value of the sifter, s, in thevicinity of xc.



Hence, to determine the value of the central point of the fluidised bed, we set

A(1 − ε)2

ε3

μq

D2h

+ B1 − ε

ε3

ρf q2

Dh= ρ0g, (13)

which yields a quadratic equation in q . Let qc be the value at which the asymptotesmeet, i.e. the critical point. Solving q = qc in equation (13) yields

qc = A

B

μ

2ρfD(1 − ε)

⎡⎣−1 ±

√(1 + B

A2

ε3

(1 − ε)3

4ρ0ρf gD3

μ2

)⎤⎦ . (14)

Since B

A2ε3

(1−ε)34ρ0ρf gD

3

μ2 ≥ 0 in equation (14) and qc ≥ 0, it follows that wemay disregard the negative root. Substitution of this qc value into equation (11)will yield the function value at the intersection of the asymptotes. Determining thevalue of the shifting-exponent, s, we use the same argument as above in equation(12). Thus,

f sc = f s0 + f s∞ = 2f s0 = 2f s∞, (15)

whence it follows that (fc

f∞

)s=(fc

f0

)s= 2. (16)

The value of s may now be determined straightforwardly from equation (16) as

s = ln 2

ln fc − ln f∞= ln 2

ln fc − ln f0. (17)

In performing an experiment, it is therefore advantageous to arrange the physicalconditions in such a manner that the independent variable (here q) is in closevicinity of xc (here qc). Whenever the experimental value of fc = f xc is known,we proceed to determine the value of the shifter by equation (17).

Alternatively, visual inspection by trial and error adjustment of the correlationbetween the predictive curve and data points may lead to an assignment of a valueto s. As noted by [3] the matched curve is relatively insensitive to variations ins; the required acuity being determined by considerations such as the processinvolved, tunability of other parameters and allowable error-margin.

5 Correlation of experimental results

The pressure drop was plotted against the superficial velocity for each of thediameter ranges. Figures 1 and 2 serve to illustrate the influence of the use ofthe minimum (dmin), average (dmean) and maximum (dmax) particle diameters onthe orientation of the lower asymptote. A similar plot may be drawn for particlediameter-range 400 μm–600 μm. It is important to note that the asymptotes neednot be straight lines; this is apparent in Figure 2 where the quadratic nature of theRUC model – equation (5) with coefficientsA and B as expressed in equations (6)



0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

2000

4000

6000

8000

10000

12000

14000

16000

Particle diameter: 100µm − 200µm

Superficial velocity, q [m/s]

Δ p/

H [

Pa/

m]

experimentf∞f0, d

min

f0, d

mean

f0, d

max

fcritical

Figure 1: Particle diameter-range 100 μm–200 μm. Influence of chosen particlediameter on the positioning of the lower asymptote, f0, and the positionof the critical point, fc. The RUC model is used to describe the lowerasymptote.

0 0.1 0.2 0.3 0.4 0.50

2000

4000

6000

8000

10000

12000

14000

16000Particle diameter: 750µm − 1000µm


Δ p/

H [

Pa/

m]

experimentf∞f0, d

min

f0, d

mean

f0, d

max

fcritical

Figure 2: Particle diameter-range 750 μm–1000 μm. The choice of particle diam-eter determines the orientation of the lower asymptote, f0, and criticalpoint, fc. Note that the asymptotes need not be straight lines – slightcurvature of f0 due to the quadratic nature of equation (5) is evident.



100 − 200μm 400 − 600μm 750 − 1000μm

dmin -0.593 -0.825 -1.878

dmean -2.248 -3.155 -8.193

dmax -14.280 -9.567 -11.786

Table 1: Calculated values of shifter-exponent, s, for the different particle diameterpossibilities.

and (7) – starts to dominate due to an increase in the superficial velocity before theonset of fluidisation.

From these graphical results the particle-diameter yielding the best correlationwith the experimental data was chosen to be used in the fitting of a predictive curve.In the case where the data points were not noticeably favouring a specific particlediameter, the experimental qc-value closest matching the theoretically predictedvalue of the critical point, as expressed by equation (14), was used to determinean s-value. The corresponding curves were plotted and the best match was chosenby visual inspection. From equation (14) it is clear that the diameter of the particleimpacts on the value of qc, and thus on fc = f qc. The latter in turn has a directinfluence on the value of s, as calculated by equation (17). Calculated values of sare shown in Table 1.

For diameter-range 100 μm–200 μm, shown in Figure 1, the experimentalvalues lie between the average, dmean, and maximum, dmax, particle diameters.Curves for the matched solution were plotted for a particle diameter of both153 μm and 200 μm – the former yielding results that closer match the trendof the data; illustrated by Figure 3.

In the case of diameter-range 400 μm–600 μm, the data points were distributedaround the asymptote predicted by the upper limit of the range, dmax. Use of aparticle diameter of 600 μm yields the best graphical results, as shown in Figure4.

Examining diameter-range 750 μm–1000 μm, it was once again difficult todiscern the particle diameter to be used by merely regarding data point distributionabout the asymptotes. Figure 2 shows experimental points crossing the asymptotesfor both dmax and dmean. Curves were plotted for both solutions; the averagediameter of 960μm more closely followed the trend exhibited by the experimentaldata. The graphical result of a plot with this diameter is shown in Figure 5.

6 Conclusion

Although powered addition with a shifter, s, has not yet been proven to accuratelydescribe the relative behaviour of the different parameters during a transferprocess, it may be argued to be more appropriate, since the rate of change betweenthe two asymptotes can be adjusted to fit experimental readings. For instance, the



0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

2000

4000

6000

8000

10000

12000

14000

16000

Particle diameter: 100µm − 200µms = −2.2476


Δ p/

H [

Pa/

m]

experimentfcritical

dmean

= 153μm

Figure 3: Curve-fitting by powered addition to experimental readings for particlesin diameter-range 100 μm–200 μm. The functional relation is given byequation (11). An s-value of -2.248 was used.

0 0.05 0.1 0.15 0.2 0.250

2000

4000

6000

8000

10000

12000

14000

16000



Δ p/

H [

Pa/

m]

experimentfcritical

dmax

= 600μm

Figure 4: Powered addition-curve fitted to data for particles in diameter-range400μm–600μm. The functional relation is once again given by equation(11); s-value of -9.567 was used.

higher the value of the shifter-exponent, the more abrupt the changeover betweenthe two asymptotic processes. Furthermore, irrespective of the rate at which thechangeover occurs, it will be a smooth transition as is expected for the crossoverfrom one continuum process to another. Thus, although primarily a curve-fitting



0 0.1 0.2 0.3 0.4 0.50

2000

4000

6000

8000

10000

12000

14000

16000



Δ p/

H [

Pa/

m]

experimentfcritical

dmean

= 960μm

Figure 5: Curve-fitting by the method of powered addition of asymptotic solutionsas applied to particles in diameter-range 750 μm–1000 μm. Equation(11) represents the functional relation with an s-value of −8.193 havingbeen used.

exercise, this procedure leads to better physical modelling since the only ’tuneable’parameter is the shifter-exponent, s. Adjusting its value does not change the valueof the asymptotic conditions and leaves the double-asymptote character of thetransfer process intact. The inherent simplicity of the method suggests that in sucha curve-fitting exercise the greater deal of effort should be exerted in determinationof the asymptotes, the value of the exponent, s, and possible relation of the latterto some quantifiable parameter.

References

[1] Davidson, J.F. & Harrison, D., Fluidised Particles. Cambridge UniversityPress: London, 1963.

[2] Geldart, D., Introduction (Chapter 1). Gas Fluidization Technology, ed.D. Geldart, John Wiley & Sons, Inc.: Chichester, pp. 1 – 10, 1986.

[3] Churchill, S.W. & Usagi, R., A general expression for the correlation of ratesof transfer and other phenomena. American Institute of Chemical EngineersJournal, 18(6), pp. 1121 – 1128, 1972.

[4] Churchill, S.W. & Usagi, R., A standardized procedure for the productionof correlations in the form of a common empirical equation. Industrial andEngineering Chemistry, Fundamentals, 13(1), pp. 39 – 44, 1974.

[5] Churchill, S.W., The Interpretation and Use of Rate Data: The Rate Concept.McGraw-Hill: New York, 1974.

[6] Churchill, S.W., Derivation, selection, evaluation and use of asymptotes.Chemical Engineering and Technology, 11, pp. 63 – 72, 1988.



[7] Geldart, D., Characterization of fluidized powders (Chapter 3). Gas Fluidiza-tion Technology, ed. D. Geldart, John Wiley & Sons, Inc.: Chichester, pp. 33– 54, 1986.

[8] Geldart, D., Single particles, fixed and quiescent beds (Chapter 2). GasFluidization Technology, ed. D. Geldart, John Wiley & Sons, Inc.: Chichester,pp. 11 – 32, 1986.

[9] du Plessis, J.P. & Woudberg, S., Pore-scale derivation of the Ergun equationto enhance its adaptability and generalization. Chemical Engineering Science,63(1), pp. 2576 – 2586, 2008.



Section 7 Heat transfer

Tube bundle’s cooling by aqueous foam J. Gylys1, S. Sinkunas2, T. Zdankus1, M. Gylys1 & R. Maladauskas2 1Energy Technology Institute, Kaunas University of Technology, Lithuania 2Department of Thermal and Nuclear Energy, Kaunas University of Technology, Lithuania

Abstract

Single-phase coolants, such as water, oil or air, are mostly used in industrial apparatus. But usage of two-phase coolants, such as aqueous foam, can significantly reduce material and energy expenditures, simultaneously sustaining proper heat transfer intensity on heated surfaces. Comparing with single-phase coolant, the two-phase foam coolant has additional possibility to change the intensity of heat transfer by changing volumetric void fraction of foam. This enables wider range of regulation of heat transfer intensity. An experimental investigation of heat transfer between in-line tube bundle and aqueous foam flow was performed. One type of aqueous foam – statically stable foam – was used as a coolant. Vertically downward moving foam flow crossed the in-line tube bundle. Spacing between the centres of the tubes across the in-line tube bundle was 0.03 m and spacing along the bundle was 0.03 m. During an experimental investigation it was determined dependence of heat transfer intensity on flow parameters: flow velocity, volumetric void fraction and liquid drainage from foam. Apart of this, influence of tube position in the bundle on heat transfer was investigated. Experimental results were summarized by criterion equations, which are suitable for the design and calculation of foam apparatus. Mentioned experiments are the continuation of our previous investigation with foam flow moving downward after 180-degree turning. Keywords: heat transfer, aqueous foam flow, in-line tube bundle, void fraction of foam.

1 Introduction

Heat transfer between different heated surfaces and aqueous foam flow is the area of our researches. It was noticed that usage of two-phase coolants, such as



aqueous foam, could significantly reduce material and energy expenditures, simultaneously sustaining proper heat transfer intensity on heated surfaces. Comparing with single-phase coolant, the two-phase foam coolant has additional possibility to change the intensity of heat transfer by changing volumetric void fraction of foam. Small density and mentioned properties of foam type coolant enables to create compact, light and economic heat exchanger with simple and safe operation using two-phase foam flow. Characteristics of one type of aqueous foam, namely statically stable foam, demonstrate its perfect availability for heat transfer process [1]. It appears that investigated by us statically stable foam keeps its initial structure and bubbles' dimensions within broad limits of time intervals, from several minutes to days, even after termination of the foam generation. Thus, this type of coolant was used as heat transfer working fluid in our investigation. The statically stable foam has all specific peculiarities of aqueous solution foams: drainage of liquid from foam [2, 3], diffusive transfer of gas between bubbles [4], division and collapse of foam bubbles [1, 4]. During foam flow contact with heated surfaces some foam bubbles are destroyed and additional liquid flow appears. All those mentioned above phenomena are closely linked with each other and make extremely complicated an application of analytic methods for their study. Thus experimental method of investigation was selected in our work. In our previous researches heat transfer of alone cylindrical surface–tube and then of tube line to upward statically stable foam flow was investigated [1]. Next experimental series with staggered tube bundle [5, 6] and in-line tube bundles of different geometry [7–9] in upward and downward after 180-degree turning foam flow followed. Presently the experimental set-up design was changed and foam flow moved downward without any turning. During an experimental investigation it was determined dependence of heat transfer intensity on flow parameters: flow velocity, volumetric void fraction and liquid drainage from foam. Apart of this, influence of tube position in the bundle on heat transfer was investigated. The results of investigation were compared with the results of our previous researches and influence of foam flow turning on heat transfer intensity of different tubes of the bundle was determinate. Results of investigation were generalized using relationship between Nusselt and Reynolds numbers and volumetric void fraction of foam. The obtained generalized equation can be used for the designing of foam heat exchangers and calculating of heat transfer intensity of the in-line tube bundle.

2 Experimental set-up

Experimental set-up (fig. 1) consisted of the following main parts: vertical experimental channel, in-line tube bundle, gas and liquid control valves, gas and liquid flow meters, liquid storage reservoir, liquid level control reservoir, air fan, electric current transformer and stabilizer. Cross section of the experimental channel had dimensions 0.14 x 0.14 m2; height of it was 1.8 m. Walls of the



Figure 1: Experimental set-up scheme: 1 – liquid reservoir; 2 – liquid level control reservoir; 3 – liquid receiver; 4 – gas and liquid control valve; 5 – flow meter; 6 – foam generation plate; 7 – experimental channel; 8 – tube bundle; 9 – thermocouples; 10 – transformer; 11 – stabilizer; 12 – valve.

channel were made from the transparent material in order to observe foam flow visually. Statically stable foam – one type of aqueous foam – was used as coolant for our experiments. Statically stable foam flow was generated from the detergents water solution. Concentration of the detergents was kept constant at 0.5 % in all experiments. Foam-able liquid was supplied from the reservoir onto the special perforated plate. Foam flow was generated during gas and liquid contact. Foam flow parameters control was fulfilled using gas and liquid valves.

Liqu

id

11

10

~

Foam

V

A

9

mV

6

8

1

2

Gas

3

12

4

4 5

5

7



Perforated plate for foam generation was installed at the upside of the experimental channel and was made from stainless steel plate with a thickness of 2 mm; orifices were located in a staggered order; their diameter equal 1 mm; spacing between the centres of the holes equal 5 mm. In-line tube bundle was used during experimental investigation. This bundle of the tubes consisted of six rows with five tubes in each. Spacing between centres of the tubes across and along the tube bundle was equal to 0.03 m (s1=s2=0.03 m). All tubes had an external diameter of 0.02 m. Schematic view of the experimental section with tube bundle is presented in fig. 2. One tube-calorimeter was heated electrically. This tube was made of copper and had an external diameter of 0.02 m also. The ends of the heated tube was sealed and insulated to prevent heat loss through them. During the experiments calorimeter was placed instead of one of the bundle’s tube. An electric current value of heated tube was measured by an ammeter and voltage by a voltmeter. Temperature of the calorimeter surface was measured by eight calibrated thermocouples: six of them were placed around the central part of the tube and two of them were placed in both sides of the tube at a distance of 50 mm from the central part. Temperature of the foam flow was measured by two calibrated thermocouples: one in front of the bundle and one behind it.

Figure 2: In-line tube bundle in downward and downward after turning foam flow.

Measurement accuracies for flows, temperatures and heat fluxes were of range correspondingly 1.5%, 0.15–0.20% and 0.6–6.0%. During the experimental investigation a relationship was obtained between an average heat transfer coefficient h from one side and foam flow volumetric void fraction β and gas flow Reynolds number Reg from the other side:

s 2

A6

A5

A4

A3

A2

A1

B6

B5

B4

B3

B2

B1

C6

C5

C4

C3

C2

C1

Foam flow

s1

d

s 2

D6

D5

D4

D3

D2

D1

E6

E5

E4

E3

E2

E1

F6

F5

F4

F3

F2

F1

s1

d

Foam flow



( )gf Re,fNu β= . (1) Nusselt number was computed by formula

f

fhd

Nuλ

= . (2)

Here λf is the thermal conductivity of the statically stable foam flow, W/(m·K), computed by the equation ( ) lgf λββλλ −+= 1 . (3) An average heat transfer coefficient we calculated as

T

qh w

∆= . (4)

Gas Reynolds number of foam flow we computed by formula

g

gg A

dGRe

ν= . (5)

Foam flow volumetric void fraction we expressed by the equation

lg

g

GG

G

+=β . (6)

Experiments we performed within limits of Reynolds number diapason for gas (Reg): 190–410 (laminar flow regime) and foam volumetric void fraction (β): 0.996–0.998. Gas velocity for foam flow was changed from 0.14 to 0.30 m/s.

3 Results

The process of heat transfer between tubes of in-line tube bundle and vertical downward laminar foam flow was the object of our experimental investigation. Comparison of heat transfer intensity (Nuf) of the third line tubes A3, B3 and C3 to the downward foam flow is shown in the fig. 3. The heat transfer intensity of tubes located at different places across and along the experimental channel is under the influence of distribution of local flow velocity and local foam void fraction across and along the channel. The maximum value of the foam flow local velocity is in the centre of the channel cross-section and decreases following to the walls of the channel. It is different with local void fraction of the foam. The foam is dryer in the centre of the channel cross-section and it is wetter near the channels walls. The distribution of the foam local void fraction decreases by foam flow passing the tube bundle. Side tubes A and C were located at the same distance from the vertical axis of the experimental channel, therefore foam local void fraction and foam flow local velocity had correspondingly the same values near the side tubes A and C and the heat transfer intensity of those tubes was identical. Some points of results data of the tubes A and C covered each other. Therefore an average heat transfer intensity of side tubes (AC) was calculated for the better experimental results analysis.



0

200

400

600

800

1000

150 200 250 300 350 400 Re g

Nu fB3 (0.996)B3 (0.997)B3 (0.998)AC3 (0.996)AC3 (0.997)AC3 (0.998)

Figure 3: Heat transfer intensity of the tubes A3, B3 and C3 to downward foam flow, β=0.996, 0.997 and 0.998.

By increasing of foam flow gas Reynolds number (Reg) from 190 to 410, heat transfer intensity (Nuf) of the tube B3 from middle-column to downward foam flow increases by 1.6 times (from 567 to 880) for foam with volumetric void fraction β=0.996; by 1.6 times (from 476 to 785) for β=0.997, and by 1.7 times (from 285 to 498) for β=0.998. The heat transfer intensity of the tube B3 is on average by 1.9 times higher to the wettest foam flow (β=0.996) in comparison with the driest foam flow (β=0.998). The Nuf of the side-column third tubes A3 and C3 grows by 1.6 times (from 637 to 1047) for β=0.996; by 1.8 times (from 491 to 871) for β=0.997, and by 1.7 times (from 340 to 584) for β=0.998 (Reg =190–410). The heat transfer intensity of the side-tubes (A3 and C3) is on average by 1.8 times greater to the wettest foam flow in comparison with the driest foam flow. An average heat transfer intensity of the third side-tubes A3 and C3 (AC3) is higher than that of the third middle-tube (B3) by 15% for β=0.996; by 12% for β=0.997, and by 13% for β=0.998 (Reg =190–410). Situation is different when foam flow initially moved vertically upward then after the 180-degree turning moved vertically downward crossing the tube bundle [8]. Distribution of foam flow local void fraction across the channel changes during flow turning. This transformation depends mainly on liquid drainage from the foam; therefore drainage must be taken into account during analysis. Gravity forces act along the upward and downward foam flow, but in the foam flow turning those forces act across foam flow also. Therefore liquid drains down from the foam near upper channel wall and local void fraction increases (foam becomes drier) here as well. After the turn, local void fraction of foam is lower (foam is wetter) on the internal left side of the channel cross-section (near tube column D), and local void fraction of foam is higher (foam is



0

200

400

600

800

1000

1200

150 200 250 300 350 400 Re g

Nu fD3 (0.996)E3 (0.996)F3 (0.996)D3 (0.998)E3 (0.998)F3 (0.998)

Figure 4: Heat transfer intensity of the tubes D3, E3 and F3 to downward after 180-degree turning foam flow, β=0.996, 0.997 and 0.998.

drier) on the external right side of the cross-section (near tube column F). Flow velocity distribution in cross section of the channel transforms after turn also. Comparison of heat transfer intensity of the third line tubes D3, E3 and F3 to the downward after 180-degree turning foam flow for the volumetric void fraction β=0.996 and β=0.998 is shown in the fig. 4. Within the interval of foam flow gas Reynolds number (Reg) from 190 to 410, heat transfer intensity (Nuf) of the tube D3 is higher on average 31% than heat transfer intensity of the tube E3 and heat transfer of the tube E3 is higher on average 73% than that of the tube F3, for β=0.996. Heat transfer of the tube D3 is higher on average 7% than heat transfer intensity of the tube E3, and heat transfer of the tube E3 is higher on average 41% than that of the tube F3, for β=0.998. Heat transfer of the tube D3 to the wettest foam flow (β=0.996) is by 2.3 times higher than that to the driest foam flow (β=0.998). An average heat transfer rate of middle-column tubes (B and E) was calculated in order to analyse the influence of foam flow turning on the heat transfer intensity of the tubes in the middle-column of the bundle. An average heat transfer intensity of the middle-column tubes (B) to downward foam flow and an average heat transfer intensity of the middle-column tubes (E) to downward after 180-degree and R=0.17 m radius foam flow turning is presented in the fig. 5. Foam is wetter and local velocity of foam is higher in the centre of the channel cross-section in the case of foam flow without the turning. Therefore an average heat transfer intensity of the middle-column tubes B is higher than that of the middle-column tubes E (downward after 180-degree turning foam flow)



0

200

400

600

800

1000

150 200 250 300 350 400 Re g

Nu fB (0.996)B (0.997)B (0.998)E (0.996)E (0.997)E (0.998)

Figure 5: An average heat transfer intensity of the tubes B to downward foam flow and an average heat transfer intensity of the tubes E to downward after 180-degree turning foam flow, β=0.996, 0.997 and 0.998.

on average by 22% for β=0.996; by 28% for β=0.997, and by 27% for β=0.998 (Reg=190–410). Experimental results of investigation of heat transfer from the in-line tube bundle to downward and downward after 180-degree turning foam flow were generalized by criterion equation using dependence between Nusselt number Nuf and gas Reynolds Reg number. This dependence within the interval 190 < Reg < 410 for the in-line tube bundle in downward and downward after 180-degree turning foam flow with the volumetric void fraction β=0.996, 0.997, and 0.998 can be expressed as follows: m

gn

f RecNu β= . (7) For the entire middle-column (B) in the downward foam flow on average c=3.4, n=–500, m=41.4(β–1.018). For the entire middle-column (E) in the downward after 180-degree turning foam flow on average c=16.1, n=518, m=140.7(1.003–β).

4 Conclusions

Heat transfer of in-line tube bundle to vertical downward laminar foam flow was investigated experimentally. Heat transfer intensity of the third side-tubes (A3 and C3) is on average by 13–15% higher than that of the third middle-tube (B3) for β=0.996–0.998 and Reg =190–410.



Foam is wetter and local velocity of foam is higher in the centre of the cross-section of the channel in the case of downward foam flow without turning. Therefore heat transfer intensity of the middle-column tubes to downward foam flow is on average by 22–28% higher than that of the middle-column tubes to downward after 180-degree turning foam flow. Criterion equation (7) may be applied for calculation and design of the statically stable foam heat exchangers with in-line tube bundles.

Nomenclature

A – cross section area of experimental channel, m2; c, m, n – coefficients; d – outside diameter of tube, m; G – volumetric flow rate, m3/s; h – average coefficient of heat transfer, W/(m2⋅K); Nu– Nusselt number; q – heat flux density, W/m2; Re – Reynolds number; T – average temperature, K; β – volumetric void fraction; λ – thermal conductivity, W/(m⋅K); ν – kinematic viscosity, m2/s.

Indexes

f – foam; g – gas; l – liquid; w – wall of heated tube.

References

[1] Gylys, J., Hydrodynamics and Heat Transfer Under the Cellular Foam Systems, Technologija: Kaunas, pp. 50–314, 1998.

[2] Nguyen A. V., Liquid Drainage in Single Plateau Borders of Foam. Journal of Colloid and Interface Science, 249(1), pp. 194–199, 2002.

[3] Fournel B., Lemonnier H., Pouvreau J., Foam Drainage Characterization by Using Impedance Methods. Proc. of the 3rd Int. Symp. on Two-Phase Flow Modelling and Experimentation, p. [1–7], 2004.

[4] Tichomirov V., Foams. Theory and Practice of Foam Generation and Destruction, Chimija: Moscow, pp. 11–106, 1983.

[5] Gylys, J., Zdankus, T., Miliauskas, G.; Sinkunas, S., Influence of vertical foam flow liquid drainage on tube bundle heat transfer intensity. Experimental Heat Transfer, 20(2), pp. 159–169, 2007.

[6] Gylys, J., Sinkunas, S., Zdankus, T., Analysis of staggered tube bundle heat transfer to vertical foam flow. International Journal of Heat and Mass Transfer, 51(1–2), pp. 253–262, 2008.

[7] Gylys, J.; Sinkunas, S.; Zdankus, T.; Giedraitis, V., Balcius, A., Foam flow turn influence on the in-line tube bundle heat transfer intensity. Proc. of the Computational Methods and Experimental Measurements XIII, Prague, Czech Republic, pp. 457–464, 2007.



[8] Gylys, J., Sinkunas, S., Zdankus, T., Influence of the foam flow turning on the staggered and in-line tube bundles heat transfer intensity. J. WSEAS Transactions on Heat and Mass Transfer, 1(3), pp. 268–273, 2006.

[9] Gylys, J.; Sinkunas, S.; Zdankus, T.; Giedraitis, V., Different type tube bundle heat transfer to vertical foam flow. Proc. of the 5th International Conference on Nanochannels, Microchannels and Minichannels, ICNMM 2007, Pueble, Mexico, pp. 449–455, 2007.



Desulfurization of heavy crude oil by microwave irradiation

A. Miadonye1, S. Snow1, D. J. G. Irwin1, M. Rashid Khan2 & A. J. Britten1 1School of Science & Technology, Cape Breton University, Canada 2King Abdullah University of Science & Technology, Saudi Arabia

Abstract

Heavy crude oils normally have a high sulfur content and are usually very viscous. To improve the quality of the refinery fractions and subsequent consumers’ products, it is imperative to remove the impurities and contaminants and, where possible, upgrade the heavy crude oil. In this project, the desulphurization process of Arabian heavy sour crude oil was studied by a novel method of microwave irradiation. The heat transfer characteristics of various mineral additives were studied for use as microwave sensitizers. Crude oil samples containing various combinations of hydrogen donor additives, catalysts, and microwave sensitizers were studied. The samples were exposed to different irradiation periods at different power levels in a modified domestic microwave oven. The results indicate that crude oil microwave absorption characteristics can be improved fourfold with charcoal and doubled with polar solvents, but they showed negligible change with serpentine, due to poor hear transfer properties. The sulfur content of the original crude oil was reduced by 2.3% with H2 at 20 atmosphere pressure and 5 minutes irradiation period; and by 33.8% with ethanolamine as the hydrogen donor and 25 minutes irradiation period. For the crude oil fractions, the sulfur reductions were up to 48% and 10% for lighter and heavier fractions respectively. Analysis with GC-FID showed strong evidence of fragmentation and recombination reactions in samples irradiated for 20 and 25 minutes with a temperature of 300°C and above. Keywords: crude oil desulfurization, microwave irradiation, petroleum upgrading, hydro-desulfurization, microwave sensitizers.



1 Introduction

The application of radiation chemistry in the oil industry gained prominence in the early 1960s when only light hydrocarbon substances were used as models in radiation processing experiments [1, 2]. Radiation processing was rather expensive then and it was not until the 1990s that the concept of the ‘hydrocarbon enhancement electron-beam technology’ (HEET) was developed. More recently, microwave irradiation has been used in the petroleum industry for inspecting coiled tubing and line pipes, measuring multiphase flow, and the mobilization of asphaltic crude oil [2–5]. Gunal and Islam [3] observed the permanent alteration of asphaltene in the colloidal structures of the crude oil molecules and an increase in viscosity when exposed to microwave irradiation, due to the re-orientation of molecular structures rather than thermal breakdown. They noted that when exposed to electromagnetic irradiation, the presence of asphaltene caused permanent changes in crude oil rheology due to the polar nature of asphaltene molecules. Zaykin et al. [6, 7] reported the evidence of much branching and breaking of the paraffin chain during irradiation of paraffinic oil. Thus, microwave heating has been identified to offer numerous advantages, such as short start up time, rapid heating, energy efficiency, and precise process control. Microwave energy can be delivered directly to the reacting or processing species by using their dielectric properties or by adding absorbing material, which converts electromagnetic energy into heat. Thus, microwave energy has the ability to crack hydrocarbons and create a method of desulphurization. The sulfur content in heavy crude oil varies from 0.1% to 15%, and most is heavy molecular organic sulfur compounds – any dissolved elemental sulfur and/or hydrogen sulfide represent only a small part of the total sulfur. The sulfur containing compounds in crude oil have been identified to include the following compounds: sulfides, disulfides, mercaptans (thiophenes), benzothiophenes, dibenzothiophenes, benzonaphthothiophenes, and dinaphthothiophenes [8]. Desulphurization of crude oil is an important preliminary step to improve the quality and yield of gasoline products. Currently, the method of desulphurization in the chemical industry has fundamental limitations, such as costly energy and material consumption, extreme processing conditions and expensive catalysts. New products and processing routes are continually being sought including current methods in microwave irradiation. Through the use of microwave power, along with additives, hydrocarbons high in sulfur content and/or composed of primarily heavy hydrocarbons can be made into useful commercial products that can be burned cleanly and efficiently as a fuel oil, as demonstrated in several patents for the use of microwave irradiation [9].

2 Experimental methods

The experiments involved the reconstruction of a domestic microwave to accommodate the devices required for monitoring the irradiation process, and the analysis of the reaction products using appropriate analytical instrumentation.



2.1 Microwave set-up and irradiation process

The desulphurization process was carried out in a domestic microwave oven which was modified to allow for the accommodation of high temperature and moderate pressure reactors, mixing device, and a device for reconstitution of volatile fractions. Also included in the modification was a provision for monitoring the temperature and pressure of the process. In a typical experiment, Saudi Arabia heavy crude was mixed with one or more of hydrogen, light hydrocarbon liquid, polar additives, hydrotreating catalysts, microwave sensitizers, and exposed to various dosages of microwave radiation at low pressure. The selection of microwave sensitizers was based on their dielectric constant obtained from literature [10]. The power level and irradiation intensity was at level high (recorded in this microwave oven as Power Level 10), and the maximum irradiation period was 25 minutes. Table 1 illustrates the different samples used in the process. Polar additives used were ethanolamines, to examine their influence on desulphurization and, on the microwave radiation characteristics of the Saudi Arabia heavy crude oil.

Table 1: List of materials.

Arabian Heavy Crude Oil °API (27.31); Sulfur content (3.066%)

Pure Hydrogen Gas Activated Charcoal Sensitizer

Palladium Oxide Catalyst

Serpentine Sensitizer Ethanolamines Polar Additives

Mongstad, Norway Crude Oil

Model Compound

2.2 Analytical methods

The products formed after irradiation and the control samples were analyzed using Gas Chromatography with Flame Ionization Detector (GC-FID), GC with Mass Spectrometry (GC-MS), and Fourier Transform Infrared (FT-IR) spectroscopy among other analytical methods. The GC instrument is a 5890 Series II Plus Gas Chromatograph coupled with a FID and a 5872 Mass Selective Detector (MS) fitted with a fused silica capillary column. The column contained the non-polar stationary phase used for simulated distillation analyses (ICB-1, 30mx0.25mm ID, 0.25µm film thickness; J & K Scientific, Milton, Ontario, Canada). The initial column temperature 70°C, held for 1 minute, then increased at a rate of 30°C/min to 310°C and held for 1 minute. The injection volume was 0.5µl. The carrier gas was ultra-high purity hydrogen (from Air Liquide Canada, Sydney, Nova Scotia, Canada) at an initial pressure of 4.1 psi (at 70°C) and flow rate of 0.50ml/min. The FID was held at 350°C with H2 at 35ml/min; air, 350



ml/min, make-up N2, 25ml/min. The MS interface temperature was 320°C. It was operated in the scan mode (50-550 amu) with 1-minute solvent delay period. Each analysis was 10 minutes. Carbon standards were analyzed and retention times of n-C5, 10,15,20,25 and 30 were determined.


The physical properties of the pure (original) heavy crude oil were determined. The ºAPI was obtained by hydrometer to be 27.31, with sulfur content of 3.066 per cent and viscosity of 34.84 cSt at 25.2°C. The distillation fractions of the pure heavy crude oil obtained between 154°C and 452°C for 50 gram sample and their mass per cent of sulfur contents before and after microwave irradiation are compared in Table 2. The sulfur contents of the light distillates were reduced to 39% and 48%, while those of heavy distillates were reduced to 0.9% and 10%. The results showed approximately 50% desulphurization can be achieved in the lighter fractions.

3.1 Effects of sensitizers and additives in hydro-desulfurization

The main purpose of hydro-desulphurization (HDS) is to improve the quality of the heavy crude oil and thus, meeting the required specifications for its particular

Table 2: Sulfur content analysis for the fractions.

Distillation

Fractions

(50 g)

Temp

(°C)

Irradiation

Time

(mins)

Mass % sulfur

Non-

irradiated

Irradiated for 10

mins with

catalyst

1

154.5 to

250.0 10 1.859 0.9624 (48.3%)

2

260.0 to

306.2 10 0.3110 0.1902 (38.8%)

3

318.2 to

380.1 13 0.9030 0.8128

(10%)

4

396.4 to

452.2 25 2.528 2.506

(0.89%)



use. Depending on the process conditions the HDS process can be classified as “destructive” or “non-destructive”. The destructive HDS process is characterized by molecular fragmentation and hydrogenation saturation of the fragments to produce lower boiling fractions, and the non-destructive HDS process requires milder conditions (referred to as hydrotreating) and provides a means of removing simple souring compounds [8]. In the microwave irradiation process it is difficult to meet the requirements of the HDS destructive process in the absence of sensitizers. The prevailing conditions in microwave process generally favor non-destructive HDS due to the low temperature conditions obtainable with microwave irradiation. Since crude oil absorbs little microwave radiation, sensitizers and other polar solvents have been used to improve its absorption. As indicated in Figure 1, the presence of sensitizers and additives improved absorption by the oil of microwave radiation, identified by the increase in the temperature of the oil.

Figure 1: Microwave absorption characteristics of heavy crude oil with and without additives.

To ascertain the amount of power absorbed by the charcoal and serpentine used as sensitizers, the energy absorbed at Power Level 10 by 60 ml water core immersed in 650 ml crude oil jacket was measured. Each sensitizer was incrementally added into the crude oil jacket and irradiated for a prescribed period. Table 3 illustrates the microwave absorption characteristics of the sensitizers as given by the average power absorbed by the oil in the jacket. It is evident from the results that serpentine is a poor microwave sensitizer with 73 Watts (perhaps this is unique with the type of serpentine used for this experiment



Table 3: Microwave absorption characteristics of the sensitizers.

Samples Composition Final

Temperature (ºC)

Irradiation Time(s)

Power (Watt)

AH50 (original crude) only

650 ml crude oil

32.2 90 100

AH50 + 60g charcoal

650 ml oil and charcoal

91.8 90 975

AH50 + 40g charcoal + 60ml water

water tube in 650 ml oil

oil: 33.2 water: 102.0

45

oil: 383 water: 461

AH50 + 20g charcoal + 60ml water

water tube in 650 ml oil

oil: 28.3 water: 101.0

43

oil: 241 water: 476

AH50+ 3.25g serpentine

650 ml oil + serpentine

30.8 90 73

as their composition varies depending on location). The activated charcoal however, improved the microwave absorption characteristics of the crude oil from 100 to 975 Watts and thus, one of the effective sensitizers for crude oil. The non-irradiated sample was also pressurized with pure hydrogen gas at 20 and 30 atmosphere pressure and then heated in a high pressure steel reactor and maintained at 84.5ºC and 100ºC respectively for 30 minutes over a palladium-silica based catalyst. The results of sulfur analysis are given in Table 4 for the heavy oil and Table 2 for the fractions. The change in the sulfur content for the samples that were subjected to high pressure hydro-desulphurization reaction in the autoclave heating was negligible, between 1.8% and 2.3%, compared to 16% and 33% for irradiated samples as shown in Table 4. In agreement with the required process conditions for hydro-desulphurization discussed above, it is obvious that the temperatures for the autoclave process were too low to initiate the reaction for effective reduction in sulfur content of the heavy oil. On the other hand, for irradiated samples desulphurization was not much affected by high temperature as it was with irradiation periods, with the optimum period being 25 minutes for 33.8% desulphurization. The results showed that ethanolamine has potential as a desulphurization agent for sour crude oil.

3.2 Effects of irradiation on heavy oil composition

The samples exposed to five different microwave irradiation periods; 0, 5, 10, 15, 20, and 25 minutes were analyzed using GC-FID and GC-MS. The results



from the analysis showed no change in molecular structure for majority of the samples after being subjected to microwave irradiation. At irradiation temperatures of up to 300°C, corresponding to approximately 590KJ/kg, there was no noticeable change in molecular structure for the different samples as shown with the GC-FID analysis (Figure 2). The desired enthalpy to achieve breaking of the hydrocarbon bonds may not have been obtained at these temperatures.

Table 4: Sulfur content of the heavy crude after irradiation.

Samples Temp.

(ºC)

Irradiation

Time(min)

Sulfur

Content (%)

% Sulfur

Reduction

Heavy oil (AH50) only - - 3.066 -

AH50-

PM(10%)H2=30atm 84.5

autoclave

heating 3.011 1.8

AH50-

PM(10%)H2=20atm 100

autoclave

heating 3.012 1.8

AH50-

PM(5%)H2=20atm 84.5

autoclave

heating 2.997 2.3

AH50+1.5g

Palladium cat 228.8 25 3.06

No

change

AH50+10% charcoal

on cat 243.7 25

No

observation -

AH50+10%DEA+15%

charcoal on cat 381.6 25 2.574 16.1

AH50-(10%DEA) 193.6 25 2.062 32.8

AH50-(5%DEA) 196.1 25 2.031 33.8

Bond energy or enthalpy is essentially the average enthalpy change for a gas reaction to break all the similar bonds. Looking at the enthalpy attained in each reaction media (samples 1 to 6 in Table 5), it is apparent that only a few reaction medium attained the energy necessary to bring about bond cleavage. Samples number 5 and 6 have enthalpies of 601.3 KJ/kg and 762.5 KJ/kg respectively. The highest temperature achieved when using charcoal as the sensitizer was 381.6°C (sample number 6). The GC-FID chromatogram of this sample showed a noticeable change in molecular structure (Figure 3) compared to the chromatogram for the pure crude (sample #1). There was an evident shift in the peaks of the chromatogram of this sample compared to that of the original heavy



Figure 2: GC-FID of microwave irradiated samples.

Table 5: Energies of the reaction medium after 25 minutes microwave irradiation.

Sample number

Sample composition

Final Temperature

(°C)

ΔH (KJ/kg)

1 AH50 Pure Crude

100g 53.80 -

2 AH50 Pure Crude 100g + 10%

DEA 193.6 362.1

3 AH50 pure crude 100g + 1.5g

palladium catalyst 228.8 437.1

4 AH50 pure crude 100g + 10%

DEA/1g palladium catalyst 298.8 586.2


DEA/ 10% charcoal/ 1g palladium catalyst

305.9 601.3


DEA/ 15% charcoal/ 1g palladium catalyst

381.6 762.5

Sample #1

Sample #2

Sample #4



Figure 3: Comparison of GC-FID of pure heavy crude oil and crude oil sample containing additives irradiated for 25 minutes.

crude and other samples with lower reaction temperatures. These results suggest that the molecular structure of the sample has changed and higher molecular weight hydrocarbon chains were formed through chemical bonding. With 15 per cent charcoal on palladium catalyst (sample #6), the highest temperature of 381.6°C was obtained approximately 10 minutes into the set irradiation time of 25 minutes. Once this temperature was achieved, it decreased to 346.1ºC over the remaining 15 minutes. The extra irradiation time allowed the reaction to proceed further causing the hydrocarbons to bond and create a higher molecular weight material. It appears that the optimum microwave irradiation time for this sample is 10 minutes which is when the highest temperature was reached. At this time, there might have been some breaking of the heavier hydrocarbons. Similar trend was obtained with sample #5, attaining the temperature of 305.9°C in approximately 17 minutes before reducing to slightly lower temperature. Viscosity tests were conducted at 30°C for each microwave sample to verify and compare the changes in molecular nature of the crude at different microwave irradiation time. The results are shown in Figure 4 for three samples. Viscosity results do not explicitly confirm the presence of light fractions, but illustrate the strong possibility of abundance of high molecular weight hydrocarbons. This is an indication of the domination of recombination

Sample #1

Sample #6



Figure 4: Comparison of viscosity of different samples exposed to different irradiation times.

reactions over fragmentation reactions in microwave process. This observation is in agreement with the GC-FID chromatogram obtained for irradiated and pure samples given in Figure 3.

4 Conclusion

The application of microwave irradiation process for desulphurization and upgrading of heavy crude oil is illustrated. The results show that sensitizers improve crude oil absorption of microwave radiation and that with appropriate composition of polar additive, catalyst and sensitizer up to 40% desulphurization is achievable. Erratic change in the viscosity of the heavy crude oil indicates the occurrence of fragmentation and recombination reactions at different irradiation time. Overall, the viscosity of irradiated samples is slightly higher than that of non-irradiated sample which means that recombination reaction is prevalent. This study shows that microwave radiation promotes both desulphurization and upgrading of heavy oil at low temperatures where similar reactions are not feasible by the thermal process. The optimum condition for desulphurization and upgrading of heavy crude oil by microwave irradiation depends on the composition and type of catalyst, microwave sensitizer, polar additive and

30

40

50

60

70

80

90

0 5 10 15 20 25 30

Irradiation time (mins.)

Vis

cosi

ty (

cSt.)

Pure heavy crude oil (sample #1)Crude oil with additives (sample #5)Crude oil with additives (sample #6)



irradiation period. For concurrent desulphurization and upgrading process 25 minutes irradiation time, 381°C, 10% ethanolamine, and 15% activated charcoal on palladium catalyst were identified as the optimum.

Acknowledgements

The authors thank Jamie Tunnicliff for assistance with the experiment, and gratefully acknowledge financial support from the Saudi Aramco Ltd, Cape Breton University Research Policy (RP) grant and the Natural Science and Engineering Research Council of Canada (NSERC).

References

[1] Panchenkov, G.M., Erchenkov, V.V., Radiation-Chemical Processes. Chemistry and Technology of Fuels and Oils, 16(7-8), pp. 433-436, 1980.

[2] Ashton, S.L., Cutmore, N.G, Rooch, G.J., Watt, J.S., Zastawny, H.W., McEwan, A.J., Development and Trial of Microwave Techniques for Measurement of Multiphase Flow of Oil, Water and Gas. SPE Asia Pacific Oil and Gas Conference, Melbourne, Australia. SPE paper N0.28814, 1994.

[3] Gunal, O.G., Islam. M.R., Alteration of Asphaltic Crude Rheology with Electromagnetic and Ultrasound Irradiation. Journal of Petroleum Science and Engineering, 26, pp. 263-272, 2000.

[4] Stanley, R.K., Methods and Results of Inspecting Coiled Tubing and Line Pipe. SPE/IcoTA Coiled Tubing Roundtable, Houston, Texas, SPE paper No. 68423, 2001.

[5] Zaykina, R.F., Zaykin, Yu-A., Radiation Technologies for Production and Regeneration of Motor Fuel and Lubricants. Radiation Physics and Chemistry, 65, pp. 169-172, 2002.

[6] Zaykin, Yu.A., Zaykina, R.F. & Silverman, J., Radiation Thermal Conversion of Paraffinic Oil. Radiation Physics and Chemistry, 69, pp. 229-238, 2004.

[7] Zaykina, R.F., Zaykin, Yu-A., Mirkin, G., Nadirov, N.K. Prospects for Irradiation Processing in the Petroleum Industry. Radiation Physics and Chemistry, 63, pp. 617-620, 2002.

[8] Speight, J.G. The Chemistry and Technology of Petroleum, 3rd edition. Mercel Dekker Inc., New York.

[9] U.S. Pat. # 4,148,614, April 10, 1979; U.S. Pat. # 4,749,470, June 7, 1988; U.S. Pat. # 6,824,746; U.S. Pat. # 4,279,722, Nov. 15, 1994; Belgian Pat. # 481,341; Hungarian Pat. # 19,498.

[10] asiinstr.com/technical/Dielectric constants.htm



Section 8 Image processing

Reconstruction of a three-dimensional bubble surface from high-speed orthogonal imaging of dilute bubbly flow

M. Honkanen Tampere University of Technology, Energy and Process Engineering, Finland

Abstract

This paper presents image analysis algorithms to reconstruct a three-dimensional gas bubble surface from the time series of two orthogonal projections and to also recognize bubble outlines when the bubbles are partly overlapping in the image. Overlapping bubble images raise a large problem for three-dimensional, tomographic imaging of bubbly flows. Here, a novel approach to recognize the outlines of irregularly-shaped, overlapping bubbles based on the overlapping object recognition algorithm is presented. The main objective is to investigate the accuracy of a back-light imaging technique to measure bubble properties from a) a single projection or b) from two, orthogonal projections. The time series data of the bubble surface area, volume and orientation, location, velocity and trajectory direction is obtained. Temporal fluctuations in the measured bubble volume reveal the measurement accuracy in the case of a bubble swarm rising from a single orifice. Keywords: image analysis, three-dimensional imaging, high-speed imaging, bubble sizing, overlapping object recognition, optical tomography.

1 Introduction

The direct imaging technique for particle and bubble tracking and sizing (e.g., [1, 2]) is not yet widespread; it has only become feasible with high-resolution CCD cameras and fast image processing possibilities [3]. However, direct imaging potentially offers size estimates of irregularly-shaped objects. Difficulties arise with the definition of the size of the observation volume and with optical access,



and a compromise is made between observation area and size resolution. Therefore, the direct imaging technique is more applicable for sizing larger particles in dilute systems [3]. In the experimental images of bubbly flows, image segments are often created by groups of bubbles and not by individual bubbles, i.e. objects. If a grey-scale thresholding method is applied and two or more objects overlap in the image, they are detected as one big object and only one size and velocity is measured for the whole group. This causes errors in the measured size and velocity distributions of the studied objects. Therefore, overlapping object recognition algorithms [4–10] are necessary to increase measurement accuracy and sample rate by recognizing individual, in-focus objects. Utilization of the specific pattern recognition algorithms is discussed in [10]. The properties of irregularly-shaped bubbles are difficult to measure. Image analysis retrieves the properties of one or several two-dimensional projections of the bubble. The relation between the three-dimensional properties (bubble volume, characteristic size, surface area, orientation) and the measured image properties (projected area diameter, perimeter, major and minor dimensions) must be determined. Bubble volume can be estimated as a volume of an equivalent sphere with a diameter equal to the projected area diameter of the bubble image. A more precise estimation is obtained assuming that the bubble has a symmetrical shape along its vertical axis, e.g. ellipsoidal shape, and that the third main axis of the ellipsoid is predicted with the axes visual in the image [4]. Typically, image-based bubble sizing techniques rely on single projection. Here, two orthogonal views of bubble swarm rising from a single orifice are obtained with a high-speed camera, two mirrors and a LED back-light. Two orthogonal views (direct and mirror images) are geometrically mapped to the same scale and a three-dimensional Lagrangian bubble tracking algorithm is utilized to obtain time series data of bubble surface area, volume and orientation, location, velocity and trajectory direction. Temporal fluctuations in measured bubble volume reveal the measurement accuracy. For comparison, bubble volume is measured from a single view based on common assumptions on bubble shape.

2 Measurement setup

High-speed digital imaging setup is used to visualize the rise of air bubbles in stagnant tap water, in a bubble column. The flow is illuminated with a continuous, white LED array. High-speed CMOS camera (Photron SA1) acquires megapixel images at the speed of 1000 frames per second. Two mirrors are placed at 45 degrees angles on both sides of the bubble column to provide also the side view of the column with the same camera. Photograph of the measurement setup without mirrors is shown in Fig. 1. Camera image is geometrically calibrated with a measuring rod placed in the middle of the column. Field of view is chosen to cover the column cross-section in the front view and in the side view through the side mirror. Image scaling of 0.135 mm per image pixel is obtained for the front view and 0.149 mm/pixel for



the side view. Figure 2 shows images of measuring rod and bubbles in the measurement volume. The aperture of the objective (Sigma 105mm macro) is closed to f#16 to view all bubbles in-focus in both views. The front and side views are cropped from the original image and the front-view image is down-sampled to match the size of the side-view image. That provides us front-view and side-view images whose vertical coordinates are equal. The vertical location and dimension of each bubble image can be approximated equal in both views, because bubbles rise up in the middle of the column.

Figure 1: Measurement setup of bubble column, HS-camera and LED light.

x [mm]

y [m

m]

0 20 40 60 80 100 120

0

20

40

60

80

100

120

x [mm]

y [m

m]

0 20 40 60 80 100 120

0

20

40

60

80

100

120

Figure 2: Image of measuring rod on the left and an experimental image on the right.



3 Image analysis

High-speed image sequences are analyzed automatically with the following image analysis algorithms. Algorithms can be divided into 4 groups: image segmentation, overlapping object recognition, three-dimensional Lagrangian particle tracking, reconstruction of three-dimensional bubble surface and computation of bubble properties. Image processing and analysis algorithms are implemented in MATLAB computing software.

3.1 Image segmentation

Firstly, bubble shadows have to be distinguished from image background. An instantaneous image background image Ibg is computed individually for each image utilizing a sliding (40x40 pixel) greyscale maximum filter. The background greyscale level is estimated as 90% of the sliding maximum value. Niblack [11] has presented an image segmentation method that scales the greyscale threshold value with a local (15x15 pixel) standard deviation of the image greyscales Istd.

stdbgthr III 12.0 , (1)

The image pixels with greyscale value lower than the value in the greyscale threshold image Ithr, belong to bubble segments. Bubble segments are filled with morphological filter to include also the bright holes of bubble shadows in the bubble segments. Bubble segments are labelled and their properties are computed individually from both views. Bubble image properties include centre point coordinates, bounding box dimensions, projected area, perimeter, major and minor axis, aspect ratio and orientation. The major and minor axes (i.e. principal components) of the bubble image are obtained with direct least square ellipse fitting method [12].

3.2 Analysis of overlapping bubble images

Bubbles may overlap in one or both of the projections as can be seen in the example image in Fig. 2. The cases of bubble image overlap are recognized via solidity criterion: A bubble segment is bounded with a convex line by minimizing the length of the line. If the solidity (i.e. the ratio of the segment area and the area inside the convex line) is less than the solidity-threshold value (~0.9), the segment must include more than one bubble. Image segments of overlapping bubbles are analyzed with a watershed algorithm [8] or with a contour curvature-based OOR algorithm [9] that fits ellipses on contour arcs or produces breaklines between the connecting points of overlapping bubbles. The accuracy of contour curvature computation is further improved utilizing the adaptive curvature functions presented by Urdiales et al. [13]. Figure 3 shows the overlapping object recognition results of the three algorithms for the example image (Fig. 2). Watershed segmentation is sensitive to the changes on the bubble outline resulting in recognition of too many bubbles in case of the side view in Fig. 3a. The contour-curvature based OOR algorithms provide correct results. The advantage of breakline detection method in



comparison to ellipse fitting method is clear in this case of non-ellipsoidal bubble images. However, watershed and breakline detection methods cut off overlapping segment parts, whereas the ellipse fitting method can estimate the full sizes of partly overlapping bubbles.

a)

b)

c)

Figure 3: Analyzed segment of overlapping bubbles by a) Watershed algorithm or by Contour-curvature analysis with b) ellipse fitting and c) with breakline detection.

3.3 Three-dimensional Lagrangian particle tracking

There are several ways to implement three-dimensional Lagrangian particle tracking for a high-speed image sequence [14]. Guezennez et al. [15] proposed to carry out particle tracking in time prior to the projective matching especially in case of orthogonal imaging. They concluded that Lagrangian particle tracking in two-dimensions followed with the matching of particle trajectories between two projections is easier than individual particle matching followed by three-dimensional Lagrangian particle tracking. In this study, bubble concentration is so low that bubble matching and tracking tasks are trivial. Both tasks are realized as least squares error minimization functions. Projective matching of recognized bubble images relies on the similarity of the vertical location and dimension of each bubble image in both views. Note that in the case of perfect overlap of bubbles in one of the views, the projective matching can return several matches for single bubble projection. Three-dimensional Lagrangian particle tracking



algorithm assumes constant rise velocity and size of each bubble allowing maximum of 20% variation in the computed pseudo-distance s:

21

2,

2,

2,, )()()()()()()()( jdkdjzkzjykyjxkxs iiieiieiieijk

(2)

where bubble trajectory k is compared to the bubble j in image i. The bubble coordinate estimates (xi,e,yi,e ,zi,e) for image i are obtained from bubble trajectory assuming constant rise velocity of the bubble. di-1 is the previously measured diameter of bubble k. All possible bubble trajectories are compared to the bubble values (xi, yi ,zi and di) in the next image frame and the match is found minimizing the pseudo-distance, eqn. (2). Bubble properties along the bubble trajectory are stored only in frames, in which the algorithm has returned a consistent match. Bubble tracking is not stopped if the correct match is not found in the next image frame, but in that case the search is continued to the subsequent frames and the particle location estimate is updated.

Figure 4: Four views of an analyzed bubble image. Bubbles are colour-coded so as to recognize the same bubbles in four views.

3.4 Reconstruction of a three-dimensional bubble surface

Following steps are taken to reconstruct a three-dimensional bubble surface from two projections at each time step:

a) A three-dimensional bounding box is generated for each bubble assuming that the scaled, vertical dimensions equal in both projections.

b) The bounding box is scanned with a horizontal plane that is moved along the vertical axis from top to bottom. Three-dimensional bubble image is reconstructed based on ellipses fitted on the horizontal planes, where the ellipse location and main axes correspond to the bubble image centre locations and widths in both projections, at the height of the horizontal plane. The ellipse orientation remains unknown, because the horizontal plane is not visible in neither of the two projections.



c) The 3d bubble image is smoothed with a 3x3x3 smoothing filter to provide continuous bubble surface.

d) The 3d bounding box is placed in the coordinate system of the measurement volume and an iso-surface of the 3d bubble image is formed in that location. The colour of the surface is defined individually for each bubble trajectory, so every bubble is colour-coded making it easier to follow it in image sequence. Finally, lightings are added to the 3d image.

Figure 4 shows four views of a single 3d bubble image in different angles. Horizontal planes are clearly visible on the reconstructed bubble surface because of the low spatial resolution of the measurement. However, a larger field of view allows inspection of bubbles over 13 cm vertical distance.

3.5 Computation of bubble properties

The bubble properties are measured both from single projections and from two, orthogonal projections. Volume-equivalent diameter is chosen to represent the characteristic size of a bubble. It can be computed from a single projection a) as a projected area diameter dprojArea of the bubble image, eqn. (3), or b) assuming an ellipsoidal shape or more precisely oblate spheroid, setting the second horizontal axis of the ellipsoid equal to the visible horizontal axis [4], eqn. (4).

proj

projArea

Ad 2 (3) 312 bad projPCA (4)

31cbadest (5)

316

V

dnum (6)

Computation of volume-equivalent diameter from two orthogonal projections can rely on ellipsoidal shape-assumption estimating the diameter with the main axes of the ellipsoid by eqn. (5), or it can be computed directly from the 3d bubble image counting the bubble volume V in voxel by voxel-basis and following eqn. (6). Measurement of bubble surface area is more uncertain than the volume, because it can vary in time and the bubble surface is curved in all dimensions. Bubble surface area can be estimated from surface areas of volume-equivalent sphere and ellipsoid, and it can be computed directly from the reconstructed 3d bubble image. Bubble orientation is provided in two projections, but not in the horizontal (x-y) plane. For that, a third view would be necessary.


Three-dimensional, time-resolved measurement results are difficult to present in a two-dimensional, stationary form. An attempt is made in Figure 5 to present the three-dimensional motion of bubbles. Colour-coded bubble trajectories are shown in front and side views on left and the 3d bubble image with the same colour-coding is shown on right, where column’s front-right corner is in front. The 4-mm bubbles rise up at about 0.3 m/s. Every bubble is sampled about 400 times during its 13 cm rise from the orifice.



Figure 5: Two views of bubbles with measured outlines and colour-coded trajectories and the reconstructed 3d image of bubbles.

Time series measurement results enable statistical analysis of measurement accuracy. The volume of an air bubble can be assumed constant during the 13 cm bubble rise in water. The volume-equivalent bubble diameter is measured in six different ways. Time series results of bubble velocity, surface area and volume- equivalent diameter for two example bubbles are shown in Figure 6. Average and standard deviation values of six measures of volume-equivalent bubble diameter are presented in Table 1 for 25 bubble trajectories. Measures from single projection have standard deviation of about 7% and the measure based on ellipsoidal-shape assumption performs slightly better than a projected area diameter. The same confidence interval is found by Rodriguez-Rodriguez et al. [15], who investigated errors produced by processing a two-dimensional image of the three-dimensional ellipsoidal bubble. They concluded that typically the sizing error is bounded by 0.85 < d / dp < 1.08. In our case, the projected area diameter clearly underestimates the bubble size, whereas ellipsoid-assumption overestimates it. The measures from two projections are clearly more accurate than the estimates from single projections. Direct 3d bubble volume computation



a)

b)

Figure 6: Time series of bubble velocity, surface area and volume-equivalent diameter for example bubbles (a) and (b).



provides standard deviation of 0.1 mm, which is only 2/3 image pixels. The principal axes of 3d ellipsoid return a size measure with STD of 0.168 pixels, which is a good result. This shows that direct least squares ellipse fit returns reasonable size estimates also for irregularly-shaped bubbles with an ellipse-like shape. Surface area of a bubble is measured a) as the area of the fitted ellipsoid, 3D-PCAest, b) as the area of the sphere with a diameter equal to the mean of two projected area diameters, ProjArea, and c) directly from the 3d bubble image. The 3d bubble image is reconstructed from two projections and therefore, the projected areas relate to the 3d bubble image providing nearly the same surface area. As expected, the surface area of a fitted ellipsoid is clearly smaller than the true surface area of a bubble.

Table 1: Statistics of measured volume-equivalent bubble diameter.

Size measure Average [mm] Standard deviation [mm]

Measures from two projections:

Computed from 3d image, dnum 3.959 0.104

3d ellipsoid-assumption, dest 4.259 0.168

Measures from single projection:

Projected area diameter dprojArea 1 3.683 0.304

Projected area diameter dprojArea 2 3.882 0.273

Projected principal axes dprojPCA 1 4.127 0.289

Projected principal axes dprojPCA 2 4.402 0.281

Bubble formation and detachment from the orifice is interesting. Figure 7 shows time series data of a bubble during the first 100 ms of its trajectory. The bubble is first recognized when its size is 2 mm in diameter and it is still attached to the orifice. Its size increases linearly 35 ms until it detaches from the orifice and starts rising up to the water surface. The detachment causes a peak in the rise velocity. The surface area of a bubble has its maximum value at the time of the detachment. After that the surface area decreases and starts oscillating as seen in Fig. 6. All volume-equalized bubble diameter measures, except directly computed value (3Dnum), overestimate bubble size, when the bubble is attached to the orifice.

5 Conclusions

The presented high-speed orthogonal imaging approach is simple and effective method to measure gas bubble properties in three-dimensions. The three-dimensional bubble surface is reconstructed from two projections and the



bubbles are tracked 13 cm long path from the orifice. The volume-equivalent bubble diameter is measured with a sub-pixel accuracy having relative error of only 2.5%. Also two-dimensional image-based measurements provide accurate estimates of bubble size with a root mean square error of about 7%. However, third view would be necessary to measure the 3d orientation of a bubble. The presented overlapping object recognition algorithms enable the analysis of three-dimensional bubble trajectories also in more complex bubbly flows. Future work is to apply the presented imaging techniques in different applications.

Figure 7: Time series of bubble velocity, surface area and volume-equivalent diameter during the bubble generation and ejection from the orifice.

Acknowledgement

The financial support of Academy of Finland is gratefully acknowledged.

References

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[6] Pla, F. Recognition of Partial Circular Shapes from Segmented Contours. Computer Vision and Image Understanding vol. 63, pp. 334-343, 1996.

[7] Shen, L., Song, X., Iguchi, M., Yamamoto, F. A method for recognizing particles in overlapped particle images. Pattern Recognition Letters 21, pp. 21-30, 2000.

[8] Meyer, Fernand, Topographic distance and watershed lines. Signal Processing, 38, pp. 113-125, 1994.

[9] Honkanen, M., Saarenrinne, P., Stoor, T. & Niinimäki, J. Recognition of highly overlapping ellipse-like bubble images. Meas Sci Technol, 16, pp. 1760-1770, 2005.

[10] Honkanen, M. Direct optical measurement of fluid dynamics and dispersed phase morphology in multiphase flows. Univ. print, Tampere, 2006, http://dspace.cc.tut.fi/dpub/handle/123456789/78 .

[11] Niblack, W. An Introduction to Digital Image Processing, pp. 115-116, Prentice Hall, 1986.

[12] Fitzgibbon, A., Pilu, M., Fisher, R. B. Direct least square fitting of ellipses. Pattern Analysis and Machine Intelligence, IEEE Transactions 21, pp. 476 -480, 1999.

[13] Urdiales, C., Bandera, A., Sandoval, F. Non-parametric planar shape representation based on adaptive curvature functions. Pattern Recognition 35, pp. 43-53, 2002.

[14] Ouellette, N.T., Xu, H. & Bodenschatz, E. A quantitative study of three-dimensional Lagrangian particle tracking algorithms. Exp Fluids, 40, pp. 301-313, 2006.

[15] Guezennec, Y. G., Brodley, R. S., Trigui, N., Kent, J. C., Algorithms for fully automated three-dimensional particle tracking velocimetry. Exp Fluids 17, pp. 209-219, 1994.

[16] Rodriguez-Rodriguez, J., Martinez-Bazan, C., Montanes, J. L. A novel particle tracking and break-up detection algorithm: application to the turbulent break-up of bubbles. Meas Sci Technol 14 1328-1340, 2003.



Experimental investigation on air entrainment below impinging jets by means of video observations and image processing

D. V. Danciu, M. J. da Silva, M. Schmidtke, D. Lucas & U. Hampel Institute of Safety Research, Forschungszentrum Dresden-Rossendorf e.V., Germany

Abstract

Air entrainment as a result of a water jet plunging into a pool of water was studied by means of visualization techniques. Image processing algorithms were used to analyze the recorded sequences of the phenomenon. Data regarding the penetration depth and bubble size distribution was gathered for different jet impact velocities and jet lengths. The accumulated results are further used to validate the models implemented in computational fluid dynamics (CFD) codes. Keywords: air entrainment, plunging jet, penetration depth, image processing, bubbles size distribution.

1 Introduction

The phenomenon of air entrainment as a result of an impinging jet plunging into a pool of water occurs in many industrial processes (steel industry, wastewater treatment, food industry), as well as in nature (cascades, breaking waves). A water jet impinging on a free surface of a pool causes air entrainment as soon as the jet velocity is high enough (inception velocity). A swarm of bubbles appears as result of the impingement. Impinging jets are also important in different scenarios considered in nuclear reactor safety analyses. One example is the emergency core cooling (ECC) injection into a cold leg partially filled with steam and water. Such a situation is accounted in some scenarios of loss of coolant accidents. In this case, the injected cold water impinges as a jet on the surface of the hot water. Depending on the velocity of the jet, steam bubbles may



be entrained below the surface by the impinging jet. These bubbles contribute to heat exchange and mixing of the fluids. Heat transfer between cold and hot water and mixing in the cold leg play an important role since the mixed water enters the reactor pressure vessel and may cause high temperature gradients at the wall of the vessel (pressurized thermal shock). The mechanism of air entrainment has been studied over the past years by different researchers. Recent works in this area are those of Chirichella et al. [5] and Cummings and Chanson [7]. The experiments described in [5] studied air entrainment produced by a translating axisymmetric laminar water jet and distinguished between four entrainment regimes. In [6] air entrainment was studied by means of visualization experiments. The inception conditions of air-water flows were found to depend critically upon the jet turbulence. Air entrainment was also observed to be caused by the presence of foam bubbles at the jet-plunge pool intersection. In the experimental study described in [1], the entrainment regimes are studied and a proper nozzle configuration is being searched for, in order to minimize the turbulence effects of the jet. Oguz [11] experimentally and theoretically studied the role of the surface disturbances in the entrainment of bubbles. He showed that a key aspect of the process is the jet roughness. The results gathered from the study show the importance of the jet surface properties, in contrast to previous investigations. The average roughness of the jet was discovered to be controlling the air entrainment rate and the average wavelength of the disturbances to be correlated to the bubble size distribution. In a theoretical and experimental study on air entrainment [11] is showed that an undisturbed jet does not entrap air even at relatively high Reynolds numbers. Some studies have been made regarding air entrainment caused by means of impinging jets. We found though there is a certain lack of systematic data in the literature with regard to the penetration depth of the bubbles and the gas void fraction. We developed our experimental setup to study the air entrainment phenomenon under different velocity and jet length conditions. Main goal is to gain experimental data to validate the models implemented in the computational fluid dynamics (CFD) codes.


Figure 1 shows a schematic of the experimental setup. The experiments were carried out in a 0.3m x 0.3m x 0.5m water tank constructed with transparent acrylic walls for visualization purposes. The water level in the tank was kept constant at 0.28 m throughout the experiments. Water was pumped out of the tank and re-injected through a smooth 6 mm diameter, 50 mm length steel pipe used as nozzle to produce a vertical falling round jet. A rotameter was used for the measurement of the flow rate. Images of the impact between the jet and the water pool were captured by a high-speed camera (DRS Technologies). For each experimental condition, a sequence of images of the region below the surface was taken. The camera was operated with a frame rate of 200 frames per second.



Backlighting with high luminosity LED panels was used during the experiments in order to have a proper exposure at the required filming rate. Experiments were realized for different nozzle heights (Lj in Fig. 1) and volumetric flow rates. The experiments consisted in a vertical liquid jet of different lengths and velocities impinging on the calm surface of the water pool at an angle of 90°. The velocity of the jet ranged between 0.9 m/s and 2.5 m/s at the nozzle exit and the jet length was varied between 0.01 m and 0.2 m. The test matrix is shown in Table 1. The velocity of the jet at the plunge point is calculated as:

jj gLvv 220 (1)

where v0 is the velocity of the jet at the nozzle exit, Lj is the jet length and g the earth acceleration. In (1) free falling is assumed for the water after leaving the nozzle.

3 Preliminary observations

In some cases it is difficult to determine the conditions for the inception of air entrainment. Air entrainment can also occur for a very short period of time immediately after the impact, regardless of the nozzle height and the velocity of the jet. Similar to [5], where it was stated that the incipient air entrainment takes place with the appearance of at least one bubble, in the absence of other bubbles in the water pool, which lasts longer than 2s beneath the surface of the pool, we also observed three different regimes during our experiments.

Figure 1: Schematic of the experimental setup.



Table 1: Test matrix.

Lj [m] 0.01

0 0.02

5 0.05

0 0.07

5 0.10

0 0.12

5 0.15

0 0.17

5 0.20

0 v0

[m/s] Rej Impact velocity vj [m / s]

0.88 530

5 0.99 1.13 1.33 1.50 1.66 1.80 1.93 2.05 2.17

0.98 589

5 1.08 1.21 1.40 1.56 1.71 1.85 1.98 2.10 2.21

1.08 648

4 1.17 1.29 1.47 1.62 1.77 1.90 2.03 2.15 2.26

1.18 707

4 1.26 1.37 1.54 1.69 1.83 1.96 2.08 2.20 2.31

1.28 766

3 1.35 1.46 1.62 1.76 1.90 2.02 2.14 2.25 2.36

1.38 825

2 1.44 1.54 1.69 1.83 1.96 2.08 2.20 2.31 2.41

1.47 884

2 1.54 1.63 1.78 1.91 2.03 2.15 2.26 2.37 2.47

1.57 943

1 1.63 1.72 1.86 1.99 2.11 2.22 2.33 2.43 2.53

1.67 10021 1.73 1.81 1.94 2.06 2.18 2.29 2.39 2.49 2.59

1.77 10610 1.82 1.90 2.03 2.14 2.26 2.36 2.46 2.56 2.66

1.87 11200 1.92 1.99 2.11 2.23 2.33 2.44 2.54 2.63 2.72

1.96 11789 2.01 2.09 2.20 2.31 2.41 2.51 2.61 2.70 2.79

2.06 12379 2.11 2.18 2.29 2.39 2.49 2.59 2.68 2.77 2.86

2.16 12968 2.21 2.27 2.38 2.48 2.58 2.67 2.76 2.85 2.93

The first entrainment regime is the one where during the impingement

no bubbles appear; although for all our experimental conditions we had a fully turbulent flow developed in the nozzle (Re>2300).

The second one is the medium entrainment regime, where either only one or two bubbles appear after a period of time, remain trapped under the surface for a short time and disappear after a few seconds (incipient entrainment), or a swarm of bubbles is being entrained from time to time (intermittent entrainment).

The last entrainment regime is that of continuous air entrainment, when large air pockets and bubble swarms are continuously being entrained.



a. b. c. d.

e. f.

Figure 2: Development of the air meniscus and its breakage at a jet impact velocity of 2.3m/s. The times are, from a. to f., 15ms, 55ms, 85ms, 115ms, 145ms, 160ms; 0ms being considered at the moment when the jet makes first contact with the water pool.

Figure 2 shows the occurrence of air entrainment. At the moment of the impact between the jet and the pool, an air meniscus forms around the jet at the plunging point and the pool surface curves downward in the form of an inverted meniscus. As the meniscus moves forward, toward the bottom of the tank, it stops growing (c) and the jet water begins to break through its’ bottom (d). As the jet water moves downwards, the void retracts towards the free surface. According to [11], the water from the jet is found around the wall of the void and during the penetration it does not mix with the water from the pool; instead it forms a layer around the void and separates it from the surrounding water. This observation was possible due to the fact that dyed water was used for the jet. Figure 3 illustrates the distribution of the entrainment regimes dependent on impact velocity and jet length for the experimental points of this work.



0.00 0.05 0.10 0.15 0.20 0.25

1.0

1.5

2.0

2.5

3.0

3.5

v j [m

/s]

Lj [m]

no entrainment incipient/intermittent entrainment continuous entrainment

1.2

2.2

0.9

v0(m/s)

Figure 3: Entrainment regimes as a function of vj, the jet impact velocity and the jet length, Lj.

Figure 4: Air entrainment at v0 = 1.18m/s, for Lj = 0.025m; 0.05m; 0.175m; 0.2m (from left to right).

In the case of turbulent vertical liquid jets, the surface tension forces no longer dominate the process of keeping the surface of the jet smooth. Due to the friction between the liquid and the gas, the surface of the jet becomes rough and bumpy. The length of the jet along with the gravitational acceleration plays an important role in the inception of air entrainment. For different flow rates, air entrainment does not occur. We observed that as soon as the length of the jet is being increased even with just a few millimeters, air entrainment occurs.



A distinctive and interesting situation takes place for v0=1.2 m/s, where all four entrainment regimes are present. They vary with the jet length, starting with no entrainment, then jumping directly to the continuous entrainment regime, switching to incipient/intermittent entrainment followed by no entrainment for several jet lengths, returning to intermittent entrainment and continuous entrainment. Figure 4 presents the pictures corresponding to the jet lengths for which air entrainment takes place.

4 Methodology and analysis

The objective of our experiments was to gather information about the bubble size distribution, the penetration depth of the plume, the width of the plume and their variation in time. In each experiment a sequence of 2000 frames was recorded for different flow rates and lengths of the jet. Image processing algorithms were applied to extract the sought information.

4.1 Penetration depth

During the experiments, we observed that the length of the jet plays an important role in the entrainment process. We determined the penetration depths’ variation in time for different experimental conditions. Due to the fact that the plume fluctuates, one cannot refer to the penetration depth at a distinct moment. Therefore, to find the penetration depth for a set of experimental conditions, the best way is to average it over the entire sequence. Images were subdued to background subtraction, removal of the small air bubbles at the walls and averaging. Our algorithm averages over the entire 2000 frames sequence. It is noticeable in the averaged imagine (Fig. 5, bottom) that, at the tip, the plume is rarified, meaning that only a small amount of bubbles travels that deep. Therefore, we have chosen to consider as penetration depth the first grey level value from the bottom of the plume which represents only ten percent of the maximum grey level value of the averaged plume, found usually in the core of the plume. Figure 5 shows an example of the penetration depth variation for two different flow rates, but identical impact velocity, vj = 2.5 m/s. The penetration depth of the air bubbles decreases with the increase of the impact velocity, most of the time. The plume becomes denser with the velocity and, due to the fact that the buoyancy forces surpass the viscous drag exerted by the downward moving jet, the plume retracts towards the surface of the water. According to Bin [1], the penetration depth can be calculated as:

67.00

775.01.2 dvH jp (2)

which means that for constant impact velocity, the penetration depth should be constant. The penetration depth as a result of eq. 2 is found between 12 cm and 16 cm for impact velocities between 2 m/s and 3 m/s. Figure 6 shows the variation of the penetration depth for three different nozzle exit velocities and different jet lengths, as a result of our image processing algorithms. We observe



that the behavior of the penetration depth is not monotonous with the impact velocity, opposite to eq. 2. However, the values of the penetration depth are found in the range given by Bin’s formula, although we get different penetration depths for the same impact velocity and nozzle diameter, but different nozzle velocities and jet lengths. In our experiments we vary the impact velocity by modifying the length of the jet and the nozzle exit velocity. We can observe that for v0 = 2 m/s and v0 = 2.2 m/s, the penetration depth has a downward trend (see Figure 5). It first decreases with the jet length and then fluctuates slightly between 13 and 14 cm. By increasing the jet length for a constant nozzle exit velocity, we increase the impact velocity and the momentum transport of the jet into the pool. This should lead to a higher penetration depth, as predicted by eq. 2. In our case, the penetration depth does not increase with the impact velocity.

v0=1.8m/s, Lj=15cm v0=2.2m/s, Lj=7.5cm

Figure 5: Example of the penetration depth variation for constant impact velocity vj=2.5m/s, for the two different nozzle exit velocities and jet lengths: snapshots (top) and processed sequences (bottom).

This might be caused by higher gas void fractions in the plume which have been observed for higher jet lengths. High gas void fractions inside the plume counteract the jet momentum by buoyancy driven friction. Hence, the two opposite effects are responsible for the behavior of the penetration depth. However, in the case of v0 = 1.8 m/s the penetration depth behaves non-monotonously, similar to the entrainment behavior in Figure 4. In Figure 4 (v0 = 1.2 m/s) the amount of entrained gas varies not monotonous with the jet length. This indicates that the jet surface instabilities which trigger gas



entrainment do not increase monotonous with the jet length. Maybe different kinds of surface instabilities play a role here: surface instabilities triggered by the nozzle edge and surface instabilities which grow with the jet length, such as Rayleigh instabilities.

12.00

13.00

14.00

15.00

16.00

17.00

1.8 2 2.2 2.4 2.6 2.8 3

vj [m/s]

hP [c

m]

v0 = 1.8 m/s v0 = 2 m/s v0 = 2.2 m/s Bins' Formula

Figure 6: Processed penetration depth variation for three different nozzle exit velocities and jet length ranging between 0.025m and 0.2m.

4.2 Bubbles size distribution/estimation

Two kinds of air bubbles are found in the plume: very small bubbles, which travel deeper into the tank, and bigger bubbles, which travel together towards the surface of the tank, in the form of a cone. During their movement, the bubbles often change shape. Only the very small bubbles maintain their initial spherical form. The larger bubbles take an elliptical form which is also subdued to distortion. Even for higher jet velocities there are bubbles which escape from the cone and travel alone towards the surface. This happens frequently at the boundaries of the cone, where the buoyancy forces overcome the jet momentum. Coalescence and break-up phenomena were also observed. Bubble sizes were estimated from single images by means of image processing through subsequent background subtraction, cell segmentation, bubble detection and bubble size calculation by means of a Hough transform based algorithm. It was found that the latter algorithm could only be used for experiments with a small number of bubbles in the plume. Otherwise, bubbles sizes could only be estimated at the edge of the plume where the bubbles do not overlap. Figure 7 shows an example of detected bubbles in a thin plume.



Figure 7: Image with bubbles detected (centre positions and radii marked).

The Sauter mean diameter in the region of the rising bubbles was about 3 - 4mm, in agreement with previous data from the literature [1]. The smallest measurable bubbles were found to range between 0.3mm and 1.5mm. Some of the largest bubbles had diameters around the value of 7mm.

5 Conclusions

To study the occurrence of air entrainment by means of impinging circular jets, we used a video observation technique. Experiments have been carried out in a square tank for various jet exit velocities and jet lengths. The experimental data was processed and will be used to validate the models implemented in CFD. Some key parameters in air entrainment occurrence are: the velocity of the jet, the physical properties of the liquid, the length of the jet (falling height) and the turbulence of the jet. The penetration depth of the plume was measured and compared with an empirical formula by Bin. It has been found that in addition to the impact velocity, the amount of entrained gas has a significant effect on the penetration depth. Further experiments are planned at FZD for a better observation of the liquid field velocities by PIV methods and quantitative void fraction measurements with x-ray radiography.

References

[1] Bin A. K., Gas entrainment by plunging liquid jets. Chem. Eng. Sci., Vol. 48, No. 21, pp. 3585-3630, 1993.

[2] Bonetto F. & Lahey Jr. R.T., An experimental study on air carryunder due to a plunging liquid jet. Int. J. Multiphase Flow, Vol. 19, No. 2, pp. 281-294, 1993.



[3] Chanson H., Aoki S. & Hoque A., Physical modelling and similitude of air bubble entrainment at vertical circular plunging jets. Chem. Eng. Sci., Vol. 59, pp. 747-758, 2004.

[4] Chanson H. & Brattberg T., Experimental investigations of air bubble entrainment in developing shear layers. Univ. of Queensland, Dep. of Civil Eng., Report CH48/97, 1997.

[5] Chanson H. & Manasseh R., Air entrainment processes in a circular plunging jet: void-fraction and acoustic measurements. J. of Fluids Eng., Vol. 125, pp. 910-921, 2003.

[6] Chirichella D., Ledesma R. Gomez, Kiger K. T. & Duncan J. H., Incipient air entrainment in a translating axisymmetric plunging laminar jet. Physics of Fluids, Vol. 14, pp. 781-790, 2002.

[7] Cummings P. D. & Chanson H., An experimental study of individual air bubble entrainment at a planar plunging jet. Trans. I. Ch. E., Vol. 76, pp. 159-164, 1999.

[8] Davoust L., Achard J. L. & El Hammoumi M., Air entrainment by a plunging jet: the dynamical roughness concept and its estimation by a light absorption technique. Int. Journal of Multiphase Flow, Vol. 28, pp. 1541-1564, 2002.

[9] El Hammoumi M., Achard J. L. & Davoust L., Measurements of air entrainment by vertical plunging liquid jets. Experiments in Fluids, Vol. 32, pp. 624-638, 2002.

[10] Lin T. J. & Donnelly H. G., Gas bubble entrainment by plunging laminar liquid jets. A. I. Ch. E. Journal, Vol.12, pp. 563-571, 1966.

[11] Oguz H. N., The role of surface disturbances in the entrainment of bubbles by a liquid jet. J. Fluid Mech., Vol.372, pp. 189-212, 1998.

[12] Soh W. K., Khoo B. C. & Yuen W. Y. D., The entrainment of air by water jet impinging on a free surface. Exp. in Fluids, Vol. 39, pp. 496-504, 2005.

[13] Zhu Y., Oguz H. N. & Prosperetti A., On the mechanism of air entrainment by liquid jets at a free surface. J. Fluid Mech., Vol.404, pp. 151-177, 2000.



Section 9 Interfacial behaviour

LBM simulation of interfacial behaviour of bubbles flow at low Reynolds number in a square microchannel

Y. Y. Yan & Y. Q. Zu School of the Built Environment, University of Nottingham, UK

Abstract

Bubbles flow in a microchannel is an interfacial and surface tension dominated problem. The paper reports results of numerical modelling and simulation of interfacial behaviour of bubbles flow and coalescence in a square microchannel. The lattice Boltzmann method (LBM) is developed and applied to the simulation in which a simple linear function is applied to the order parameter to approximate the density within the interface of gas-liquid. The evolution of two isothermal air bubbles flowing through a water-filled microchannel at low Reynolds number and the interactions between the flow fields and the interface of gas-liquid are simulated and investigated numerically. Keywords: bubbles, coalescence, microchannel, LBM, interfacial behaviour.

1 Introduction

To understand bubbles interfacial behaviour in two-phase flow has long been an important topic of research in physical science and engineering; it becomes particular important for studying the flow in a confined system such as microchannel. Over the past few years, the study of gas-liquid two-phase flow in microchannels has drawn much attention from scientists and engineers due to increasing demands for developing micro-fluidic devices, micro-heat-exchangers, micro-reactors, micro-actuators, etc. On experimental study, in the last decades, many results on adiabatic gas-liquid two-phase flow patterns in micro/mini-channels were reported [1–3], but they poorly agree with the previous transition models and correlations for macro-channels. Chen et al. [4] tested nitrogen-water two-phase flow in circular mini/micro channels to report flow patterns including bubbly, slug, bubble-train



slug, churn and annular flows. Other experimental studies of gas-water flow in microchannels can also be identified [5–9]. On numerical study, in recent years, along with extensive applications of CFD to the study of two-phase flow [10–13], the lattice Boltzmann method (LBM) has become an established numerical scheme for simulating multiphase fluid flows. The key idea behind the LBM is to recover correct macroscopic motion of fluids by incorporating the complicated physics of problems into simplified microscopic models or mesoscopic kinetic equations. In LBM, kinetic equations of particle velocity distribution functions are first solved; macroscopic quantities are then obtained by evaluating hydrodynamic moments of the distribution function. This intrinsic feature enables the LBM to model phase segregation and interfacial dynamics of multiphase flow. Therefore, the LBM has a potential and broad applicability as well as many computational advantages such as parallel of algorithm and simplicity of programming [14–15]. Since the last 20 years, different LBM models for simulating multiphase flow have been developed. Gunstensen et al. [16] proposed a multi-component model based on the two-component lattice gas method. Shan and Chen [17] presented a two-phase/component flow model of mean-field interactions. Later, Swift et al. [18, 19] proposed a free energy model; and He et al. [20] developed the model using index function to track the interface of multi-phase flow with large density ratios. In 2004, Inamuro et al. [21] developed a LBM model based on the projection method to predict the behaviours of incompressible bubbles/particles in bulk liquid. The method calculates two distribution functions of particle velocity to track the interface and to predict velocities; the corrected velocity field satisfying the continuity equation is obtained by solving the Poisson equation. Recently, a further work of the LBM for incompressible two-phase flow on a partial wetting surface with large density ratio was presented by the author [22]. In the present paper, two isothermal air bubbles motion and coalescence in a water-filled rectangular microchannel are studied to test the suitability of the LBM for simulating the interfacial behaviour of the bubbles flow in a micro channel.

2 Methodology

2.1 The lattice Boltzmann method

A new scheme of the lattice Boltzmann method for simulating two-phase fluid of large density ratio is proposed and described below. In a 3-dimensional 15-velocity (D3Q15) LBM model, as shown in Fig. 1, the particle velocity,

)14 ..., ,1 ,0( =ααe , is given by

−−−−−−−−−−−−−−−

=111111111001000111111110100100111111110010010

],,,,,,,,,,,,,,[ 14131211109876543210 eeeeeeeeeeeeeee

. (1)



Figure 1: Discrete velocity set of 3-D 15-velocity (D3Q15) model.

To simulate two-phase fluid flow, two distribution functions of fluid particle velocity, αf and αg , are introduced. Function αf is used to calculate the order parameter φ , which distinguishes the two phases. Function αg is used to calculate the predicted velocity, *u , of the two-phase fluids. The evolution of the particle distribution functions ),( tf xα and ),( tg xα with particle velocity αe at point x and time t is calculated by the following equations:

),(),( )( tftf eqtt xex ααα δδ =++ ; (2)

),(),( )( tgtg eqtt xex ααα δδ =++ ; (3)

where, 1=tδ is the time step in which the particles travel the lattice spacing; )(eqfα and )(eqgα are the corresponding equilibrium states of αf and αg , given by

( )

ααα

ααααα

φω

φωφφφφ

eGe

uex

⋅⋅′+

⋅′+

∇−∇−+=

)(

36

),(

220

)(

k

kkpFHtf eq

; (4)

( ) ( )

αααααα

αααααα

µρ

ωφρ

φρ

ω

ω

euueGe

euueuueuex

⋅∇+∇⋅∇+∇−⋅⋅′+

⋅∇+∇⋅′+−⋅′+⋅′+=

)]([1332)(

])(43

23

2931[ ),(

2

22)(

kFk

tg eq

; (5)

where,

41 ,... ,76 ,... ,1

0 ,72/1

,9/1,9/2

===

=

ααα

ωα , 41 ,... ,7

6 ,... ,10

,24/1

,3/1,3/7

===

−=

ααα

αF , 41 ,... ,1

0 ,0

,1

=

=

=

α

α

αH ;

and

IG 2

23))((

29)( φφφφ ∇−∇∇= (6)

In the above equations, k is a constant parameter for determining the width of interface and the strength of surface tension; I is a unit tensor of second-order. Given that )(φψ is the bulk free-energy density, then

11

8

x

zy

15

3

4

2

6

9 14

7

12

13 10

0



ψφψφ −∂∂

=0p . (7)

The macroscopic quantities, *u , φ , ρ , µ can be evaluated as

∑=α

αφ f , ∑=α

αα geu* , (8)

>

≤≤

<

+−−−

=

L

LG

G

L

GGLGL

G

G

φφ

φφφ

φφ

ρ

ρρρφφφφ

ρ

ρ

,

,)(

,

, (9)

GGLGL

G µµµρρρρµ +−−−

= )( , (10)

where Lφ and Gφ are respectively the maximum and minimum order parameter for marking bulk liquid and gas; Lρ and Gρ are respectively the density of liquid and gas phases; Lµ and Gµ are respectively the dynamic viscosity of liquid and gas phases. In Eq. (10), a simple linear function is applied to approximate the density within the interface; this enables the present method to obtain ),()( tf eq xα and ),()( tg eq xα in a simple form and thereby improve computation efficiency. For example, the calculations of second-order tensor

)(ρG , first partial derivative of ρ , etc., can be avoided in the present model, but they have to be calculated in the models such as [21]. To enable the method to treat two-phase fluids interacting with confined solid surfaces with wetting boundary potentials, for the current isothermal system, a simple form of representation of the free energy density )(φψ , as suggested in [23], rather than the van der Waals free energy used in the traditional model, is applied in the present simulation, namely,

bbLG p−+−−= φµφφφφβφψ 22 )()()( ; (11) where β is a constant relating to interfacial thickness; bµ and bp are the bulk chemical potential and bulk pressure, respectively. By substitution of Eq. (11), Eq. (7) becomes

bGLGLGL pp +−−−−−= )3)()(( 20 φφφφφφφφφφφβ . (12)

In a plane interface under an equilibrium condition, the density profile across the interface is on equilibrium and can be represented as [22]

−

++

=D

GLGL ξφφφφξφ 2tanh22

)( ; (13)

where ξ is the coordinate normal to the interface; the interface thickness D is given by

βφφ 24 kD

GL −= . (14)



The fluid-fluid (liquid-gas) surface tension force σ is expressed as [24]

βφφσ kGL 26

)( 3−= . (15)

2.2 Correction for pressure

It should be pointed out that the predicted velocity *u is not divergence free. To obtain the velocity field which satisfies the continuity equation ( 0=⋅∇ u ), *u is corrected by following equations:

ρp∇

−=− *uu , (16)

∇⋅∇=⋅∇

ρp*u ; (17)

where p is the pressure of the two-phase fluid. Eq. (17) can be approximated by the LBM framework equation:

*13

)],(),([1),()1,( uxxxex ⋅∇−−−=++ρ

ωω

τα

ααααα npnhnhnh ; (18)

where, n is the number of iterations and ρτ /15.0 += is the relaxation time. The pressure at step 1+n is given by

∑ +=+α

α )1,()1,( nhnp xx . (19)

The convergent pressure p is determined when ε<+−+∈∀ |)1,()1,(| , npnpV xxx ; (20)

where V denotes the whole computational domain. Substituting the newly obtained pressure p into and solving Eq. (16) gives the corrected u , the velocity field.

2.3 Boundary treatment

Applying the present LBM model, no-slip boundary conditions can be implemented by simply specifying the zero velocity on the solid boundaries, i.e. the boundary velocities u and *u in Eqs. (4-5) and (16-18) are given by

0=wu , 0* =wu . (21) As there is always a thin liquid layer in the vicinity of the solid boundary surface due to the intermolecular forces between the liquid and solid substrate [25, 26], it is assumed that a thin liquid occupies one layer of the lattice spacing, the order parameter on the boundary used in Eqs. (4-5) is then determined by

Lw φφ = . (22) In the present simulation, the finite-difference of the order parameter on the boundary are given by

∂∂

−∂∂

+∂∂

−≈∂∂

==== 21002

2

4321

ζζζζζφ

ζφ

ζφ

ζφ ; (23)



where, ζ is the direction perpendicular to the wall. In this scheme, the first term on the right hand side of Eq. (23) is determined by a right-handed finite-difference; the second term is calculated by a standard centred finite-difference formula. Finally, it is found empirically that the best choice for the third term is a left-handed finite-difference formula taken back into the wall, namely,

( )012

2

4321

====

+−≈∂∂

ζζζφφφ

ζφ

z

. (24)


The motion of air bubbles surrounded by water flow in a three dimensional rectangular microchannel is considered. The gravitational force is taken into account by adding the term gGz )/1(3 ρρω αα −− e to the right hand side of Eq. (3), where g is the dimensionless gravitational acceleration. Fig. 3 shows the computational domain and initial and boundary conditions of the modelling. Initially, two air bubbles with same diameter md µ200= are

placed mµ300 apart in water inside the channel of the length mLx µ1200~= , the

width and the height mLL zy µ300~~== . The channel has an inlet boundary on

the left hand side of the channel and a free outflow boundary on the right hand side of the channel. The other four sides of the channel are no-slip solid walls. Naturally, the densities of two fluids are set at 3/1000~ mkgL =ρ and

3/0.1~ mkgG =ρ (making the density ratio to be 1000); meanwhile the dynamic viscosities of them are mskgL /101~ 3−×=µ , mskgG /101.2~ 5−×=µ , respectively. The initial surface tension between water and air is of

23 /101~ skg−×=σ and the gravitational acceleration is at 2/8.9~ smg = . To relate the physical parameters with simulation parameters, a length scale of

mL 50 101 −×= , time scale of sT 7

0 101 −×= and mass scale of kgM 140 101 −×=

are applied; these lead to the dimensionless parameters: 100=Lρ ; 1.0=Gρ ;

1.0=Lµ ; 3101.2 −×=Gµ ; 4.0=Lφ ; 1.0=Gφ ; 05.0=k ; and 9108.9 −×=g , respectively. Unless otherwise specified, the following simulations are within a computational domain occupied by 3030120 ×× cubic lattices; ε in Eq. (20) is set at 6101 −×=ε . The velocity distribution at the inlet boundary is specified as,

=

=

−−=

;0),,0(

;0),,0(

;)/())((16),,0( 2

zyu

zyu

LLyzzLyLUzyu

z

y

zyzyx

(25)



Figure 2: Computational domain and initial/boundary conditions.

where, U is the maximum value of ),,0( zyux . Thus, the Reynolds number is defined as

L

zLULµ

ρ=Re . (26)

The bubbles flow and behaviour in the microchannel at Re=100 is first simulated. The evolution with time of bubbles shapes and the behaviour of interactions are shown in Fig. 4. It can be seen clearly that the bubbles move in x-direction by the thrust force of surrounding water flow and meanwhile go up in y-direction due to the effect of buoyancy force; and with time marching, the two bubbles coalesce into a lager one. To focus only on the shape evolution of the left bubble at the early stage, it is found that the lower part of the bubble moves more quickly in x-direction than the upper part, which is caused mainly by the effects of velocity boundary layer near the solid wall of the channel.

0.5ms

1ms

1.5ms

2ms

x

yz

Figure 3: Evolution with time of bubble shapes and behaviour at Re=100.

The velocity fields are obtained through the numerical modelling. For example, at st 3102~ −×= , the velocity distribution at different cross sections of



y-z plane in Fig. 4 such as 2/xLx = , 3/2 xL and 4/3 xL , respectively, is shown in Fig. 5; where the solid line, the constant density line, indicates the interface between the two phases. As both pressure and velocity distributions across the interface are normally excellent indicators of numerical stability for the LBM calculations [27], Figs. 5 and 6 have actually shown that the present LBM can be used to obtain reasonable and stable velocity fields. Indeed, similar to the conventional CFD, the numerical instabilities of the LBM for two-phase flow of large density ratios are mainly caused by spurious velocities and/or the large oscillation of the pressure distribution across the phase interface. However, in the present method, the velocity and pressure are both corrected by solving an additional Poisson equation after each collision-stream step. Such corrections are able to ensure the velocity to satisfy the continuity equation and smooth pressure distributions even across the interface, so that to ensure the numerical stability.

Figure 4: Velocity field at different cross section, mst 2~ = , Re=100: (A: 2/xLx = , B: 3/2 xLx = , C: 4/3 xLx = ).

Figs. 6(a) and (b) show the velocity vector and the vorticity contours, respectively, at st 3102~ −×= , and at 2/yLy = on x-z plane. It can be seen that the local distribution in coherent structures is evident; the shape of the coalescent bubbles is a result of the interaction between the fields of velocity and density concentration, and this is mainly affected by the effects of buoyancy force of the bubbles [28]. Fig. 7 shows the evolution of the bubble shapes at a low Reynolds number (Re=50). The results show an interesting evolution of the bubbles flow and

2ms

x A

B C

0 Buoyancy forces drive the coalescent bubble upward

High pressure region

A B C



2ms

y

x

z

vortices

centres

(a)

(b)

Figure 5: Velocity vector and vorticity contours of coalescent bubble at 2/yLy = on x-z plane for mst 2~ = , Re = 100.

0.5ms 1ms 1.5ms

2ms 2.3ms 2.4ms

2.5ms 2.6ms 2.7ms

Figure 6: Time evolution of bubble shapes at Re=50.

coalescence. At the early stage of the flow, the evolution of the bubbles is quite similar to that at Re=100, two bubbles coalesce at 2~5.1 .0 ms; however, at the later stage, the newly coalescent bubble re-breaks up into two bubbles; this shows probably a typical phenomenon of the flow at low Reynolds number.



Obviously, such separation is caused by the strong effect of shear boundary layer near the upstream boundary. As at low Reynolds number, the buoyancy effect is more evident; which forces the bubbles migrating to the upper boundary of the channel and meanwhile results in the displacements in x-direction smaller and forms a relative returning flow in the vicinity of the bubbles interface as the sheer stress tending to different directions.

4 Conclusions

In this paper, a newly modified LBM model is developed to simulate bubbles flow in a rectangular microchannel. In the current model, a simple linear function of order parameter is applied to approximate the density within the interface of two fluids; meanwhile, a new form of the free energy density (rather than the van der Waals free energy) is used to enable the model treat confined surface and wetting boundary conditions. Based on the developed LBM model, the evolution of two isothermal air bubbles move through a water-filled microchannel are investigated numerically. Both the bubble shapes and the velocity files are imported to analyze the bubble-water and bubble-bubble interactions. The effect of Reynolds number on the flow is also examined. It is found that two bubbles can finally coalesce into one larger bubble at the relatively high Reynolds number; however, under the lower Reynolds number (e.g. Re=50), the newly coalesced bubble can be separated again by the stronger shear flow upstream.

Acknowledgement

This work is supported by the EPSRC under grant EP/D500125/1.

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Experimental investigation of a 2d impinging jet on a liquid surface

R. Berger1,2, S. Depardon1, P. Rambaud2 & J. M. Buchlin2 1Division of Research and Automotive Innovation, PSA Peugeot Citroën, France 2Environmental and Applied Fluid Dynamics Department, von Karman Institute, Belgium

Abstract

An experimental investigation of impinging 2d planar air jet on a water surface is performed by means of Particle Image Velocimetry (PIV). The difficulty encountered to measure the two dimensional velocity and turbulence fields close to the interface requires a special treatment of the PIV data obtained in the air and water sides. The resulting technique enables the localisation of the moving interface, the use of the inter-correlation and the calculation of statistics in both phases. The results show that the proposed PIV interface detection technique is in good agreement with the LeDaR detection technique developed at the von Karman Institute. The analysis of the air and water velocity field close to the interface highlights the topological differences between dimpling and incipient splashing configurations. The results obtained in this study will serve to model later turbulence transfer in impinging jet configuration. Keywords: turbulence, 2d impinging jet, interface, PIV – LeDaR.

1 Introduction

Gas jets impinging a liquid surface are encountered in a broad swathe of engineering processes related to automotive, metallurgical, chemical industries and safety systems. The gas jet impingement causes a depression at the liquid surface. The development of the lateral gas flow from the impact point induces water motion and wave formation. However, the physical interaction between a gas (air) and a liquid (water) involves a number of complex phenomena occurring in



both phases at the vicinity of the interface. The momentum transfer from the air to the water produces waves on the surface of the liquid and involves underwater flow and turbulence. The study of the flow structure above and below the interface is a very challenging task, especially in the presence of waves. The airflow structure above water surface has been studied for 50 years but the contribution of the near surface turbulence to momentum exchange between the two fluids is not well quantified and its modelling often in default. This lack of knowledge comes mainly from the difficulties to measure with accuracy physical quantities in the near interface region. In carrying a study of wind-wave flumes in oceanography, Shaikh and Siddiqui [1] show that PIV is a reliable optical non-intrusive measurement technique for investigating interface problem. However, it is also known that PIV measurements are very sensitive to the quality of the image [2], so that the moving interface needs to be accurately located. Another limitation in the previous near interface PIV studies is the lack of simultaneous measurements. Indeed, lots of measurements and analysis are done either in the liquid phase [3, 4] or in the gas phase [1], but very none are addressed to simultaneous measurements in both phases. The present study is focused on the development of a PIV technique to be applied both in air and water above and below an interface in the case of a planar 2d impinging jet. It is a pioneer study, which reports an analysis of the structure of the flow in both phases in the case of an impinging jet configuration and which yields a better understanding of the related mechanisms.


The experiments are conducted on a dedicated setup designed and built at the von Karman Institute for fluid dynamics in Rhode-Saint-Genèse, Belgium (Figure 1). The facility is composed of a 1m long, 30cm wide and 15cm high transparent Plexiglas tank that allows optical measurements. In this tank, a removable double floor is used to set the water depth around 15mm and the distance with the nozzle outlet close to 30 mm. The 2d planar jet is created by a nozzle with an outlet slot of 5mm. The nozzle is composed of a “stagnation” chamber and a convergent section. The nozzle is fed a 10 bars compressed air system through a pressure regulator that sets the flow rate, which is determined from the measurement of pressure inside the stagnation chamber. The turbulence in the jet potential core is quantified by means of a hot wire and evaluated at 4%. The jet Reynolds number, Re, is based on the nozzle slit. In the present study the Re value of 2000 and 2700 are considered. They correspond to dimpling jet and incipient splashing jet, respectively, as defined by Molloy [5] (see Figure 2). PIV [2], with parameters summarised in Figure 3, is used to measure 2d instantaneous velocity fields in a plane perpendicular to the water surface along the centre line of the tank. The measurement chain is made of a 250 mJ two cavities pulsed Nd-Yag laser and a PCO 1280 x 864 pixels CCD camera with a 35mm objective and their related acquisition system. The camera and the laser are synchronised at 3 Hz by means of a timing hub. The laser sheet is created and



Figure 1: Schematic of the setup.

Figure 2: Impinging jet configurations – Molloy [5].

focused in the measurement plane with the use of a spherical and a cylindrical lens, which guide it to the measurement region by the side of the tank for the water PIV and by the bottom of the tank for the air PIV as shown in Figure 4. The instantaneous character of the PIV measurements allows the computation of detailed statistics of the flow field in terms of mean and fluctuating quantities.

Dimpling Splashing Penetrating



Air phase PIV Water Phase PIV seeding particle oil vapour Vestosint

Liquid Water 10mg/L Water – fluoreseine mixture

Particle diameter ~1 µm ~5 µm field of view size ~35 mm x 25 mm separation time ~0.04 ms ~4ms

Processing software WIDIM [6] initial window size 96 x 96 refinement step 2

Overlapping 75% final resolution 1 vector / ~0.17 mm

Figure 3: PIV parameters.

Cylindrical lens Spherical lens

Prism

Water phase PIV

Air phase PIV

Figure 4: Laser arrangement overview.

In addition to PIV measurements, the LeDaR interface tracking technique developed at the VKI [7] is also applied to validate by comparison the interface detection from the PIV images. The LeDaR database is composed by 3000 images sampled at 2.5 kHz rate by means of an 800 x 600 pixels high speed camera. The measurements allow also the detailed statistics of the interface deformation (mean and fluctuations), the dynamic analysis of the data.

3 Interface detection

In order to use WIDIM PIV algorithm developed at the VKI, a special treatment is used for each phase. The objective of these treatments is to provide images only in the medium (air or water) of interest. For both phases, the PIV images are



used to detect the position of the interface. Once the interface is detected, a simple filtering is applied to blank the part of image that is not in the investigated phase. However, the procedure to determine the interface position is different depending on whether one uses air or water PIV data.

3.1 LeDaR interface detection method

Level Detection and Recording (LeDaR) algorithm is a reference measurement method for interface detection developed at the VKI. The principle and the performance of this technique are described by Planquart [8] and Bouchez [9]. The algorithm used is an improved version developed by Toth [10] for grey level sensor camera. The enhanced version used is based on the Maximum Forward Step Filter method that has already been shown [10] as the most robust detection method among the others available in LeDaR. An example of LeDaR interface detection is shown in Figure 5. Based on the accuracy of the algorithm and the spatial resolution, the resulting uncertainty of the interface location is estimated to be within ±70 µm.

Figure 5: Example of LeDaR application: a – raw image, b – enhanced images with resulting interface (white line).

3.2 Water phase interface detection

From the water PIV raw images, the technique is relatively complex because the interface doesn’t appear clearly due to the “Mirage effect” (Dias [11, Nogueira [12]). Therefore, an image texture separation has been developed and used (Figure 6). The different steps of the detection procedure are successively:

• Grey level and Sobel filtering of the raw image to highlight the particle in the images. The use of the fluorescein is justified at this step for a better contrast and an efficient filtering of the image background and the “Mirage effect”

• Texture segmentation and closure. This step appears as the most critical because highly depend on the seeding density. It is required to be homogeneous between all the images for an automated process. The result is a binary image where the value “0” corresponds to the air phase and the value “1” to the water phase.

• Localisation of the binary jump for each pixels column, defined as the interface location

a

b



• Spatial smoothing of the detected interface. • Blanking of the above interface part in order to obtain the final

images used for the cross-correlation process. The process is applied to a set of 1000 images. The results show that small deviations of the interface appears due to strong light reflections on the interface by floating particles or low seeding density zones within less than 10% of the images. The difference between the computed interface and the real interface is manually checked on a set of 20 images randomly selected and for each one the maximal deviation recorded. The resulting uncertainty of the interface location is estimated to be within ±180 µm. This ambiguity is mainly due to the closure and segmentation parameters, to the seeding density and the spatial resolution of the images.

Figure 6: Water side PIV interface detection method illustration.

3.3 Air phase interface detection

The detection scheme from the air phase PIV is completely different. Indeed to avoid light saturation and to provide useful pictures for the interface detection, the setting respects the schematic drawn in Figure 7. The way of detecting the interface in the air phase PIV image relies on the assumption that the deformed surface assumption has a 2d character. Instead of detecting the interface in its real position (A in Figure 7), the interface is recorded as the image of the laser reflection footprint (B in Figure 7) on the wall of the double floor through the deformed interface (B’ in Figure 7). This point is recorded as point b on the camera picture. In addition we assume that the line of

Raw images

Sobel filter +

Grey level filter

Segmentation +

closure

Spatial Smoothing

Interface location + blanking



high intensity on the picture is the real interface deformation with a small shift that is very easy to determine manually. So applying the LeDaR algorithm [7] on the raw image and using a shift parameter we are able to accurately determine the position of the real interface (Figure 8). Applying the same accuracy evaluation than for the water phase, the interface is found to be localised within a tolerance of ±120 µm. The determination from the air side is more accurate than from the water side thanks to the accuracy of the LeDaR algorithm.

Figure 7: Detail of Air side PIV image.

Figure 8: Air phase PIV based interface detection illustration: a – raw images, b – raw images with highlighted interface, c – blanked image, d – image with velocity vectors.

a

b

c

d

Real Interface

LeDaR Detected




4.1 Interface tracking scheme evaluation

The PIV based interface localisation gives statistical information on the position of the interface that can be compared to the data obtained with a specific interface tracking measurement technique. The previous data extracted from the PIV are confronted with data obtained by the LeDaR technique. Figures 9 and 10 are typical examples. Figure 9 shows that good agreement is obtained between the interface tracking methods for the dimpling configuration of the impinging jet. The trough (0<X/b<3) of the water surface described by Molloy [5] is retrieved by all the methods (“b” being the jet slot width).

Figure 9: Interface location comparison for the dimpling configuration.

Figure 10: Interface location comparison for splashing configuration.

In the incipient splashing configuration (Figure 10), both techniques agree on the mean interface position in the depression region (0 < X/b < 2.5) and the

Shaded area in waterphase PIV



“interface flow” region (X/b > 4.5). However, the water PIV based interface position is underestimated in the region of the “lip” (2.5 < X/b < 4.5). This inconsistency can be explained by lighting and seeding artefact. To obtain images respecting the needs of our interface detection scheme, the laser sheet is carried by the side of the tank. The consequence is the missing part upside the one detected is present in a shadow and low seeded region. The resolution of this problem is still under investigation. The consequence is a loss of PIV data in this region even though the technique gives satisfactory results in overall.

4.2 PIV qualitative analysis

PIV provides a series of snapshots of the velocity field at fixed time interval. In a fixed Eulerian coordinate system, a time series of the velocity data at any spatial location between the top and the trough of the waves cannot be extracted. This limitation is mainly due to the fact that a given spatial location can either lie in the water or in the air. Hsu and Hsu [13] already show that time-averaged quantities do not exist in a useful form at a fixed position in the crest-trough region of a wavy interface. Two ways of computing the quantities are given:

• Conditional averaging. This method uses fixed spatial location and takes in account only the data that are in the investigated phase.

• Wave-following Eulerian averaging: in order to obtain time-averaged quantities especially for high amplitudes waves, Hsu and Hsu [13] as Shaikh and Siddiqui [1] use another system they show as more accurate and meaningful. They transformed the fixed Eulerian coordinate system into a wave-following Eulerian system. The transformation imposes that the new vertical origin of the coordinate system is located on the interface for each instantaneous field. The quantities are then arranged with reference to the water surface using a new vertical coordinate system ζ, such that ζ=0 is the location of the interface in the each field and the positive ζ-axis is pointing upwards.

Since the first method could be sufficiently accurate for a smooth interface as observed in the dimpling condition, the issue could be more chaotic in the incipient splashing configuration where the surface becomes strongly wavy. That is why all the present study statistics are computed using the wave following coordinate system. Figure 11 shows the averaged velocity field on both phases for the two jet

configurations. We can clearly distinguish the typical behaviour of each configuration with the lip presence (X/b~3). For both flow configurations the bulk rotating motion in the water phase, created by the water surface dragging [14] is captured. Compared to the dimpling configuration, the centre of the rotating motion in the incipient splashing configuration is located outside the field of view, because of higher velocity and wider water depression at the impingement. For both cases it is worthwhile focussing on the region close to the interface.

The measurement resolution is not sufficient to check the velocity continuity



across the interface due to the one two order of magnitude difference between the velocities in both phases. Indeed, to be able to confirm continuity, we need to access the viscous sub-layer velocity in each phase as shown by Davis [15], who argues that the viscosity has a significant influence on the momentum transfer across the interface. That very small layer has not been resolved in the present study.

Figure 11: Water and air sides averaged velocity magnitude fields and streamlines – X/b>0: splashing configuration – X/b<0: dimpling configuration.

However, Figure 11 shows several regions (I) close to the interface where the streamlines cross the interface. The possible reasons of this wrong behaviour of streamlines are the lack of resolution close to the interface and the computation of the statistics (effect of the wave-following co-ordinate system, convergence). The resolution of these problems and the improvements of the results are under investigation.

4.3 Momentum interaction between water and air phases

The air and water phenomena are clearly related as already mentioned by Komori et al. [16]. The air velocity magnitude reaches its maximum at the lip crest, at the same position that the maximum velocity in the water. The air velocity decrease in the “interface flow” region goes along with a water velocity decrease. It is obvious that the air flow is responsible for the water motion but the description of the fluid-to-fluid interaction at the interface remains incomplete. From the PIV averaged fields, two main zones are

I



found be important for analysis of turbulence; the part at the crest of the lip, and the part at the trough of the lip named here as the “interface flow” region. As PIV measurements provide instantaneous velocity fields, they contain

both mean and turbulent component of the velocity. As summarised by Shaikh and Siddiqui [1], the instantaneous velocity over a wavy surface is the sum of mean, turbulent and wave induced velocities. The conventional fluctuating velocity component u’ and v’ are computed using the Reynolds decomposition: U (t) = Um + u’ (t)

V (t) = Vm + v’ (t)

where Um and Vm are the time averaged mean velocity components, U (t) and V (t) the instantaneous one. The Reynolds stresses are then computed as ''.vu− . As mentioned by Kato and Sano [18], this computed term represents the

total Reynolds stress as u’ and v’ taken into account both wave-induced and turbulent velocities. Shaikh and Siddiqui [1], Friebel [19] make the observation that the positive Reynolds stress in the air phase just above an interface indicates that the transport of the turbulence is done towards the water surface. Using the same analogy, Shaikh and Siddiqui [1] observes that negative Reynolds stress in the air at the interface corresponds to the momentum flux transported from the fluctuating liquid surface to the gas phase. A similar discussion holds for water PIV analysis. In the present configuration, the Reynolds stress distribution is the same in the vicinity of the water surface for both dimpling and incipient splashing jet configurations. Furthermore, positive Reynolds stresses in the water come face to face with positive Reynolds stresses in the air, meaning that the turbulence in the air is transported through the surface to the water and vice versa for negative ones. Indeed, three important Reynolds stresses regions are denoted in the close vicinity of the surface.

As observed in Figure 13, upstream the lip crest (A), the Reynolds stresses remain positive, so the turbulence is transported toward the water. Then, at the lip crest (B), the Reynolds stresses becomes negative. It means that in this region, the water surface brings some turbulence to the air flow. And finally, downstream the crest (C), the Reynolds stress becomes positive, so the exchange of the turbulence is once again from the air to the water. The mechanism of the turbulence transport in the 2d impinging jet can now be described. The dynamic pressure of the jet imposes at the impingement a depression on the water surface. In this depression cavity, the turbulence of air flow associated with the pressure gradients impose to the liquid a motion. Moreover, the unsteadiness of the jet drives, by flapping on the surface, the interface into motion at the lip region. The continuous motion of the lip imposes the air flow to be continuously deviated at the crest, bringing to the air flow a new quantity of turbulence (wave-induced turbulence) highlighted by negative Reynolds stresses. Once the crest is reached, the generated turbulence is transported towards the liquid. The waves, because of their limited size, do not



Figure 12: Reynolds stresses distribution along and in the vicinity of the interface in the air phase (splashing configuration).

Figure 13: Reynolds stresses distribution along and in the vicinity of the interface in the water phase (splashing configuration).

act on the turbulence of the air flow and the Reynolds stresses remain positive. So it would be possible to correlate the unsteady phenomenon in air with unsteady motions of the interface. This issue is beyond the scope of the present study and will be a part of a forthcoming paper where a deeper analysis of the turbulence transport phenomena will be addressed. These results demonstrate that the non-intrusive PIV technique can measure the velocity field above and below a water-air interface and properties of the field on both sides are closely linked by the turbulence properties of the field in the impinging jet configuration.

5 Conclusion

An experimental investigation of the flow field in the vicinity of an interface resulting from the impingement on a water layer of a 2d dimpling and incipient

AB C

A


B C


splashing air jet is performed by the means of PIV. An innovative PIV technique is introduced to measure two dimensional velocity fields in both water and air phases. For each phase, a method is developed in order to detect the interface using only the raw PIV images. These 2 methods are confronted to a well-tried interface tracking method in order to be validated. The results show that the interface is determined with a good accuracy for each phase in both jet configurations. These methods give access to the turbulence properties of the flow of each phase in the very close vicinity of the interface. The analysis of the velocity and Reynolds stress fields highlights the turbulence transport mechanisms from the air jet towards the water flow. The interaction between the interface motion and the flow fields is also emphasized. The present study demonstrates that PIV is an efficient technique to measure with a good accuracy the phasic velocity fields in the near interface region of a gas jet impinging on a water surface. The perspectives of this study are twofold; a deeper analysis of the turbulence transfer phenomena that take place in the interface region and subsequently the use of the database to validate CFD simulation such as VOF-LES.

References

[1] Shaikh N. and Siddiqui K., Air velocity measurements over the wind-sheared water surface using Particle Image Velocimetry, Ocean dynamics 58, 2008, pp65-79.

[2] Riethmuller M.L, Particle Image Velocimetry and associated techniques, 2000, VKI lecture series 2000 – 01.series 2nd revised edition 2007 – 01.

[3] Pierson W.L., Measurement of surface velocities and shears at a wavy air-water interface using PIV, Exp in Fluids 23, 1997, pp427-437.

[4] Misra K.S., Thomas M., Kambhametu J.T., Estimation of complex air water interfaces from PIV images, Exp. in Fluids 40, 2006, pp764-775

[5] Molloy N.A., Impinging jet flow in a two-phase system: The basic flow pattern, J. Iron Steel Industry, October 1970, pp943-950.

[6] Scarano F., Particle Image Velocimetry, development and application, PhD thesis, universita di Napoli Frederico Secundo, Italy, 2000

[7] Toth B., Anthoine J., Riethmuller M.L., Mesure dynamiques de la deformée d’une surface libre, Congres francophone de techniques laser, 2006

[8] Planquart Ph., Real time optical detection and characterisation of water model free surface. IN proceedings of the 4th European Continuous Casting Conference, 2002, Birmingham, UK

[9] Bouchez D., Zimmer L., Riethmuller M.L, Optical detection and characterization of interfaces, in Proceedings of the 9th Symp. of flow visualizations, August 2000.

[10] Toth B., Two phase flow investigation in a cold gas solid rocket motor model through the study of the slag accumulation process, PhD thesis, 2008, von Karman Institute ULB.



[11] Dias Pereira M., Bubble formation at a multiple orifice plate submerged in quiescent liquid, PhD Thesis, 1999, von Karman Institute - ULB

[12] Nogueira S., Flow around a single Taylor bubble rising through stagnant and co-current flowing Newtonian liquids

[13] Hsu C.T, Hsu E.Y., On the structure of turbulent flow over a progressive water wave: theory and experiments in a transformed, water-followed co-ordinate system. JFM 105, 1983, pp87-117

[14] Forrester S.E., Evans G.M., Computational modelling study of a plane gas jet impinging onto a liquid pool, Inter. Conf. in Mineral & Metal Processing and Power Generation, 1997, pp313-320

[15] Davis R.E., on turbulent flow over a wavy boundary, JFM 42, 1970, pp721-731

[16] Komori S., Nagaosa R., Murakami Y., Turbulence structure and mass transfer across a sheared air-water interface in wind-driven turbulence, JFM 249, 1993, pp161-183

[17] Hassain AKMF. Reynolds W.C., The mechanics of organized wave in turbulent shear flow, JFM 41, 1970, pp241-258.

[18] Kato H, Sano K, An experimental study of turbulent structure of wind over water waves. Rep Port Harb Res Inst 10, 1971, pp3-42

[19] Friebel HC., Air momentum flux measurements in the surf zone over braking waves, 2005, PhD thesis SIT Hoboken, New Jersey, USA



Author Index

Abdulkareem L. A. .................. 355 Akbarzadeh K. ........................... 85 Alke A...................................... 157 Andrade T. H. F. ...................... 193 Archambeau P.................. 317, 367 Arvoh B. .................................. 227 Azzopardi B. J. ........................ 355 Bayareh M. .............................. 379 Bédat B. ................................... 147 Berger R................................... 507 Bothe D.................................... 157 Britten A. J............................... 455 Buchlin J. M. ..................... 27, 507 Castro A. V. ............................. 169 Chabane S. ................................. 27 Constantinescu G. .................... 251 Coonrod J................................. 295 Crivelaro K. C. O..................... 193 da Silva M. J. ........................... 481 Damacena Y. T. ....................... 193 Danciu D. V. ............................ 481 De Leebeeck A. ......................... 99 de Wet P. D.............................. 431 Debaste F. .................................. 15 Decker R. K. .............................. 45 Depardon S. ............................. 507 Dewals B. J. ..................... 317, 367 Diego I. .................................... 421 du Plessis E. ..................... 399, 409 du Plessis J. P................... 409, 431 Dutta A. ................................... 181 Emmett M. ............................... 329 Erpicum S. ....................... 317, 367 Eskin D. ..................................... 85 Esmaili E...................................... 3 Evangelista A........................... 305

Farias Neto S. R....................... 193 Gent M..................................... 421 Geraldelli W. O........................ 169 Golak S. ..................................... 67 Groll R. .................................... 241 Gylys J. .................................... 445 Gylys M. .................................. 445 Haapala A. ............................... 343 Halloin V. .................................. 15 Halvorsen B. M............... 227, 277,

409, 481 Hampel U................................. 481 Haut B........................................ 15 Herman C................................... 15 Hernandez-Perez V.................. 355 Heynderickx G. J. .................... 181 Höhne T. .................................. 123 Honkanen M. ................... 343, 469 Hua J........................................ 111 Irwin D. J. G. ........................... 455 Karube K. .................................. 77 Kerger F........................... 317, 367 Kourakos V. G........................... 27 Kroeger M. .............................. 157 Kumara W. A. S. ..................... 277 Leyssens T. ................................ 15 Liimatainen H. ......................... 343 Lima A G. B. ........................... 193 Lindvig T. .................................. 85 Line A........................................ 15 Lo S. .......................................... 77 Lucas D.................................... 481 Maekawa M. .............................. 77 Mahinpey N. ................................ 3


Maladauskas R......................... 445 Mammoli A.............................. 295 Marin G. B. .............................. 181 Meier H. F.................................. 45 Melaaen M. C. ......................... 277 Menéndez M. ........................... 421 Mertens K. ............................... 295 Miadonye A. ............................ 455 Mimura K. ................................. 77 Mls J. ....................................... 391 Moodie T. B............................. 329 Mori M............................... 45, 169 Mortazavi S.............................. 379 Niinimäki J. ............................. 343 Noriler D.................................... 45 Nossen J. .................................. 111 Nydal O. J. ................................. 99 Pellegrino A. M. ...................... 305 Pierrat D..................................... 27 Pirotton M........................ 317, 367 Pržulj V.................................... 135 Przyłucki R. ............................... 67 Putkaradze V............................ 295 Quan S. .................................... 111 Raeckelboom J......................... 181 Rajaie E.................................... 205 Rambaud P......................... 27, 507 Rashid Khan M. ....................... 455 Ratulowski J. ............................. 85 Rautenbach C........................... 409

Ropelato K............................... 169 Roshan R. ................................ 217 Sarkardeh H. ............................ 217 Schäfer M. ................................. 55 Schmidtke M. .......................... 481 Scotto di Santolo A.................. 305 Shala M.................................... 135 Sinkunas S. .............................. 445 Snow S..................................... 455 Sohrabi M. ............................... 205 Sternel D.................................... 55 Stoor T. .................................... 343 Tokyay T. ................................ 251 Toraño J. .................................. 421 Torno S. ................................... 421 Ueyama K................................ 263 Vallée C. .................................. 123 Velasco J.................................. 421 Vorobieff P. ............................. 295 Wacławczyk T. .......................... 55 Warnecke H.-J. ........................ 157 Woudberg S. .................... 399, 409 Yan Y. Y.................................. 495 Zarrati A. R.............................. 217 Zdankus T................................ 445 Zeren Z. ................................... 147 Zu Y. Q.................................... 495


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