washington university sever institute of ...crelonweb.eec.wustl.edu/theses/jiang/jiang thesis...

WASHINGTON UNIVERSITY SEVER INSTITUTE OF TECHNOLOGY

DEPARTMENT OF CHEMICAL ENGINEERING ___________________________________________________________________

FLOW DISTRIBUTION AND ITS IMPACT ON PERFORMANCE

OF PACKED-BED REACTORS

by

Yi Jiang

Prepared under the direction of

Prof. M. H. Al-Dahhan & Prof. M. P. Dudukovic

___________________________________________________________________

A dissertation presented to the Sever Institute of

Washington University in partial fulfillment

of the requirements for the degree of

DOCTOR OF SCIENCE

December, 2000

Saint Louis, Missouri, USA

WASHINGTON UNIVERSITY SEVER INSTITUTE OF TECHNOLOGY

DEPARTMENT OF CHEMICAL ENGINEERING ___________________________________________________________________

FLOW DISTRIBUTION AND ITS IMPACT ON PERFORMANCE

OF PACKED-BED REACTORS

by

Yi Jiang

Prepared under the direction of

Prof. M. H. Al-Dahhan & Prof. M. P. Dudukovic

___________________________________________________________________

December, 2000

Saint Louis, Missouri, USA

___________________________________________________________________

Packed-bed reactors are used in numerous industrial applications. Extensive

efforts in academic and industrial research have been made to improve the understanding

of the hydrodynamics and modeling the effect of particle-scale and bed-scale phenomena

on reactor performance. The studies conducted so far have been limited to

phenomenological approaches focusing on the global and mean quantities and utilizing

only the mean properties in model development. Many of these models are homogeneous

and pseudo-homogenous in nature, whereas few models consider the heterogeneity of the

bed structure. This research shows that the performance of packed-bed reactors could be

better modeled by properly accounting for the heterogeneity of the bed structure and of

the flow.

The first part of this study is focused on modeling of flow distribution in packed

beds. Two modeling approaches are developed for simulating single-phase and

ii

multiphase flow distribution in packed beds. One is the discrete cell model (DCM)

approach, which essentially rests on the minimization of the total mechanical energy

dissipation rate proposed by Holub (1990). Another approach is the k-fluid computational

fluid dynamics (CFD) approach, which solves the ensemble-averaged Naiver-Stokes

equations for stationary solid phase and flowing phase(s) with appropriate closures. The

predictions of flow distribution by these two different approaches (DCM and CFD) are

comparable for single- and two-phase flow systems. An agreement between the predicted

flow velocities and the experimental data in the literature is also achieved. The k-fluid

CFD modeling approach is superior in computational efficiency particularly for a packed

bed of large size. It is capable of simulating the system with different inlet flow

distributions (e.g., uniform and nonuniform; steady state and unsteady state). A statistical

implementation of bed porosity distribution into the model has been developed and

adopted in both DCM and CFD flow simulations. This shows significant promise in our

ability to predict the flow structure in the bed.

The second part of this study is devoted to utilizing the flow distribution

information in industrial practice, and to quantifying the impact of flow distribution on

reactor performance. From an engineering point of view, the developed flow distribution

models have proven useful in the following applications:

• Quantification of the relationship between bed structure, flow distribution and

operating conditions.

• Obtaining the multiphase flow structure in a bench-scale packed bed by

performing flow simulation, interpretation of the irregular or scattered bench-

scale experimental data, and exploration of scale-down issues.

• Development of a combinational modeling scheme for flow and reaction in

packed-bed reactors based on the ‘mixing-cell network’ concept, in which the

simulated flow results can be used as input data for the cell network model. Such

a modeling strategy is definitely useful for the diagnostic analysis of the operating

units because of its capability of providing the mapping information on the bulk

flows and the species concentration for a given kinetics.

iii

Contents Page

Tables.............................................................................................................................. ix

Figures ............................................................................................................................ x

Acknowledgements........................................................................................................ xxiv

Nomenclature................................................................................................................. xxvii

1. Introduction to Flow Distribution in Packed Beds ............................................... 1

1.1 Research Motivation ..................................................................................... 2

1.1.1 Discrete Cell Model (DCM) Revisited........................................... 2

1.1.2 k-Fluid CFD Model Development and Applications ..................... 4

1.1.3 Impact of Flow Maldistribution on Packed-Bed Performance....... 4

1.2 Research Objectives ...................................................................................... 6

1.2.1DCM Revisited................................................................................ 6

1.2.2 k-Fluid CFD Model for Packed Beds............................................. 6

1.2.3 Applications of k-Fluid CFD Model .............................................. 7

1.2.4 Impact of Flow Maldistribution on Reactor Performance ............. 7

1.3 Thesis Structure............................................................................................. 7

2. Experimental Observations: Liquid Flow Distribution in Trickle Beds.............. 10

2.1 Introduction ................................................................................................... 10

2.2 Experiment-I: 2-D Liquid Flow Imaging ...................................................... 12

2.2.1 2-D Packed Bed and CCD Setup.................................................... 12

2.2.2 Imaging and Processing ................................................................. 13

iv

2.2.3 Liquid Flow Imaging...................................................................... 14

2.2.4 Experimental Results and Discussion ............................................ 14

2.2.4.1 Non-prewetted bed (dry bed) .......................................... 14

2.2.4.2 Prewetted bed (wet bed).................................................. 15

2.2.4.3 Comparison of liquid flow in non-prewetted bed

and prewetted bed........................................................................ 15

2.3 Experiment-II: Exit Flow Measurements in 3-D Bed ................................... 24

2.3.1 Experimental Objectives ................................................................ 24

2.3.2 3-D Column and Exit Flow Measurement ..................................... 25

2.3.3 Experimental Results and Discussion ............................................ 26

2.4 Conclusions ................................................................................................... 32

3. Discrete Cell Model Approach Revisited: I. Single Phase Flow Modeling .......... 33

3.1 Introduction ................................................................................................... 33

3.2 Non-Parallel Gas Flow Models ..................................................................... 34

3.2.1 Vectorized Ergun Equation Model................................................. 34

3.2.2 Equation of Motion Model............................................................. 35

3.2.3 Discrete Cell Model (DCM)........................................................... 35

3.3 Discrete Cell Model (DCM).......................................................................... 38

3.4 CFDLIB Formulation.................................................................................... 42

3.5 Modeling Results and Discussion ................................................................. 43

3.5.1 Model Packed Bed ......................................................................... 43

3.5.2 Analysis of the Energy Dissipation Equation ................................ 45

3.5.3 Comparison of DCM and CFDLIB................................................ 52

3.5.4 Comparison of DCM/CFDLIB and Exprimental Data .................. 58

3.5.5 Case Studies by DCM .................................................................... 63

v

3.6 Concluding Remarks ..................................................................................... 69

4. Discrete Cell Model Approach Revisited: II. Two Phase Flow Modeling............ 71

4.1 Introduction ................................................................................................... 71

4.1.1 Spatial Sacles in Trickle Beds........................................................ 73

4.1.2 Governing Principles for Flow Distribution................................... 73

4.2 Extended Discrete Cell Model ...................................................................... 75

4.3 Modeling Results and Discussion ................................................................. 79

4.3.1 Comparison of DCM and CFD Simulations .................................. 81

4.3.2 Effect of Liquid Distributor............................................................ 86

4.3.3 Effect of Particle Prewetting .......................................................... 90

4.4 Conclusions and Final Remarks.................................................................... 94

5. Computational Fluid Dynamics (CFD): I. Modeling Issues ................................. 96

5.1 Introduction and Background........................................................................ 96

5.1.1 CFD Applied to Multiphase Reactors ............................................ 96

5.1.2 CFD and Other Modeling Approaches to Multiphase Flow in

Packed Beds ............................................................................................ 100

5.2 Spatial and Temporal Characteristics of Flow in Packed Beds .................... 102

5.3 Structure Implementation.............................................................................. 104

5.4 k-Fluid Approach and CFDLIB Code ........................................................... 108

5.4.1 Eulerian k-Fluid Model .................................................................. 108

5.4.2 k-Fluid Model in CFDLIB ............................................................. 109

5.5 CFD Modeling Issues.................................................................................... 113

5.5.1 Significance of Terms in the Momentum Balance......................... 113

5.5.2 Closures for Multiphase Flow Equations ....................................... 115

5.5.3 Interfacial Tension Effect, Wetting Correction.............................. 120

vi

5.5.4 Effect of Mesh Size on Computated Results.................................. 123

5.5.5 Boundary Conditions ..................................................................... 129

5.6 Conclusions and Remarks ............................................................................. 130

6. Computational Fluid Dynamics (CFD): II. Numerical Results &

Compariosn with Experimental Data ......................................................................... 131

6.1 Introduction ................................................................................................... 131

6.2 Comparison of CFD Simulation and Experimental Results.......................... 133

6.2.1 Liquid Upflow in Packed Beds ...................................................... 134

6.2.2 Gas and Liquid Cocurrent Downflow in Trickle Beds .................. 142

6.3 Simulation of Feed Distribution Effects........................................................ 149

6.4 Conclusions ................................................................................................... 153

7. CFD Applications in Scale-Down and Scale-Up of Packed-Bed Reactors........... 154

7.1 Introduction ................................................................................................... 154

7.2 Model Bench-Scale Packed Beds.................................................................. 157

7.3 Modeling Results of Bench-Scale Packed Beds ........................................... 159

7.4 Statistical Nature of the Bed Structure and Flow.......................................... 171

7.4.1 Bed Structure.................................................................................. 171

7.4.2 Multiscales of Flow and Role of Various Forces. .......................... 172

7.4.3 Link of Macroscale and Cell-Scale Hydrodynamics...................... 173

7.4.4 Statistical Quantities....................................................................... 173

7.5Modeling Result and Correlation Development............................................. 174

7.5.1 Model Packed Beds........................................................................ 174

7.5.2. Capillary Force Effect ................................................................... 176

7.5.3. Porosity Distribution Effect .......................................................... 180

vii

7.5.4. Correlation Development .............................................................. 181

7.5.5. Superficial Velocities at the Inlet .................................................. 184

7.6 Conclusions and Remarks ............................................................................. 187

8. A Combined k-Fluid CFD Model and the Mixing-Cell Network Model ............. 188

8.1 Introduction ................................................................................................... 188

8.2 k-Fluid CFD Model for Flow Simulation .................................................... 192

8.3 Mixing-cell Network Model.......................................................................... 197

8.4 Concluding Remarks ..................................................................................... 203

9. Thesis Accomplishments and Future Work ........................................................... 205

9.1 Summary of Thesis Accomplishments.......................................................... 205


9.1.2 k-Fluid CFD Model........................................................................ 206

9.1.3 Mixing-Cell Network Model.......................................................... 207

9.2 Recommendations for Future Research ........................................................ 208


9.2.2 k-Fluid CFD Model........................................................................ 208

9.2.3 Mixing-Cell Network Model.......................................................... 210

Appendix Comparison between Trickle Bed and Packed Bubble Column

Reactor Performance for the Hydrogenation of Biphenyl ........................................ 211

A1 Introduction ................................................................................................... 211

A2 Reactor Models.............................................................................................. 213

A2.1 Kinetic Model................................................................................. 213

A2.2 Key Assumptions ........................................................................... 214

A2.3 Cocurrent Trickle Bed and Packed Bubble Flow Bed Model ........ 214

viii

A3 Results and Discussion.................................................................................. 216

A3.1 Flow Characteristics and Flow Regimes ........................................ 216

A3.2 Trickle-bed Reactor Performance .................................................. 217

A3.3 Packed Bubble Flow Reactor Performance.................................... 218

A3.4 Sensitivity of Model Parameters .................................................... 219

A4 Conclusions ................................................................................................... 220

References ...................................................................................................................... 228

Vita.................................................................................................................................. 244

ix

Tables Table Page

3-1. Dimensions of the model bed and physical properties of the fluids in the simulations

............................................................................................................................... 45

4-1. Summary of operating conditions used in flow simulations...................................... 87

5-1. Current Status of CFD Modeling in Multiphase Reactors......................................... 99

5-2. Typical Ranges of Force Rations in Two-Phase Flow in Packed Granular packing

(adapted from Melli et al., 1990)......................................................................... 114

5-3. Models for Drag Coefficients .................................................................................. 119

6-1. Statistical Description of Porosities and CFD Simulated Velocities ....................... 136

6-2. Parameters Used in the Discretization of the Radial Porosity Profile, and in the

Generation of 2D Porosity Distribution .............................................................. 149

6-3. Feed Velocities and Holdups at to Ten Sections from the Center to the Wall ........ 151

7-1. Statistical description of the porosity distribution ................................................... 176

A-1. Kinetic parameters for hydrogenation of biphenyl (Sapre and Gates, 1981).......... 221

A-2. Summary of various correlations used in this study ............................................... 221

A-3. Properties of the catalyst particles .......................................................................... 221

A-4. List of gas and liquid feed velocity......................................................................... 222

x

Figures Figure Page

2-1a. Schematic diagram of 2-D rectangular packed bed ................................................. 14

2-1b. Experimental setup using CCD video camera imaging technique........................... 14

2-2. Effect of liquid superficial mass velocity on liquid rivulet flow at single-point inlet in

a non-prewetted packed bed: the radius of the liquid rivulet increases with the liquid

superficial mass velocity .......................................................................................... 18

2-3. Effect of liquid irrigation rates on the local radial spreading of the liquid rivulet at a

point source inlet in the non-prewetted bed (ROI Size: 3 cm × 2 cm)..................... 19

2-4. Cross-sectional liquid distribution in a 3D-rectanglar non-prewetted bed of glass

beads (dp = 1.6 mm), [From Ravindra et al (1997)]. Upper left- at the top layer with

single-point liquid inlet; lower left- at the layer 12 cm far from the top with single-

point liquid inlet; upper right- at the top layer with single-line liquid inlet; lower

right- at the layer 12 cm far from the top with single-line liquid inlet..................... 19

2-5. Local liquid distribution at (a) the top region and (b) the bottom region at a mass

superficial velocity of 7.04 kg/m2/s in a non-prewetted bed.................................... 20

2-6. The steady state liquid distribution in a prewetted bed at different liquid superficial

mass velocities.......................................................................................................... 20

xi

2-7. The development of finger-type liquid flow in a prewetted bed at a superficial mass

velocity of 0.74 kg/m2/s ( t: starting time, second) ................................................. 21

2-8. Liquid flow distribution in (a) non-prewetted bed and (b) prewetted bed with single-

point liquid inlet without gas flow. Part (c) shows the image intensity profiles at

specific vertical position (z = 6 cm from the top) in cases (a) and (b)..................... 22

2-9. Transient behavior of reaction rates in non-prewetted and prewetted beds for

oxidation of SO2 with the active carbon particles as catalyst. [Data are extracted

from Ravindra et al. (1997) at T = 25 C, P = 1 atm.]............................................... 23

2-10. Dependence of the global reaction rate on liquid velocity in non-prewetted and

prewetted beds (uniform liquid inlet) for oxidation of SO2 with active carbon

catalyst. [Data are extracted from Ravindra et al (1997) at T = 25 °C; P = 1 atm.]. 23

2-11. Packing image taken from the front of the 2-D rectangular packed bed ................. 24

2-12. Schematic diagram of the experimental setup for a 3D column with exit flow

measurement and periodic liquid feed controller..................................................... 27

2-13. Liquid collector with 25 individual tubes located at the bottom of the packed bed 28

2-14. Liquid flow measurements in the non-prewetted bed: dimensionless liquid flow

velocity data from 25 individual tubes at different liquid superficial mass velocities

(H = 6 ft, G = 0.0 m/s, uniform liquid inlet) ............................................................ 28

2-15. Individual points measurements in the prewetted bed: dimensionless liquid flow


(H = 6 ft, G = 0.049 m/s, uniform liquid inlet) ........................................................ 29

2-16. Effect of time split in On/Off periodic operation mode on liquid radial profiles with

uniform liquid inlet (H=2.0 ft, G=0.049 m/s, L = 1.5 kg/m2/s)............................... 30

2-17. Effect of time split in On/Off periodic operation mode on liquid radial profiles with

point source liquid inlet (H=2.0 ft, G=0.049 m/s, L = 1.0 kg/m2/s) ........................ 31

xii

3-1. Model packed bed ('2D' rectangular as example) and velocity at each interface of cell

j. (Note that Sx,j equals to Sx+∆x ,j in the '2D' rectangular packed bed) ...................... 38

3-2. Porosity distribution of model bed (32 cells x 8 cells). ............................................. 44

3-3. Contribution of each energy dissipation rate term at each cell to the total energy

dissipation rate. V0 = 0.5 m/s (gas flow without internal obstacles); Re'= 28.51 ..45

3-4a. Contribution of each energy dissipation rate term at each cell to the total energy

dissipation rate. V0 = 0.5 m/s (gas flow with two internal obstacles at Z/dp = 30,

66); Re' =28.5 .......................................................................................................... 48

3-4b. Contribution of each energy dissipation rate term at each cell to the total energy

dissipation rate. V0 = 0.5 m/s (gas flow with two internal obstacles at Z/dp = 30,

66); Re' =28.5 (zoom-in) .......................................................................................... 48

3-5. Contribution of each energy dissipation rate term at each cell to the total energy

dissipation rate. V0 = 0.1 m/s (liquid flow without internal obstacles); Re' = 47.5 . 50

3-6. Comparison of CFD simulations and DCM predictions at a superficial velocity of 0.1

m/s at different axial positions (Z/dp) (Re' = 5.7)..................................................... 54


m/s at different axial positions (Z/dp) (Re' = 28.5)................................................... 55


m/s at different axial positions (Z/dp) (Re' = 170.1)................................................. 56

3-9a. Comparison of superficial velocity (Vj) between CFD and DCM predictions for gas

flow in the Reynolds number (Re') range of 5 to 171. ............................................. 57

3-9b. Comparison of relative interstitial velocity (Uj/U0) between CFD and DCM

predictions for gas flow in the Reynolds number (Re') range of 5 to 171. (U0 =

V0/εB) ....................................................................................................................... 58

xiii

3-10a. Comparison of predicted interstitial velocity component in the Z direction (Uz) by

two methods in liquid up-flow system: liquid superficial velocity V0 = 0.1 m/s (Re'

= 47.5). ..................................................................................................................... 59

3-10b. Comparison of predicted interstitial velocity component in X direction (Ux) by

two methods in liquid up-flow system: inlet liquid superficial velocity V0 = 0.1 m/s

.................................................................................................................................. 60

3-11a. Influence of gas feed superficial velocity on DCM predicted cell interstitial

velocity profiles........................................................................................................ 61

3-11b. Effect of particle Reynolds number (Rep) on the calculated relative cell superficial

velocity profile inside a bed using DCM ................................................................. 61

3-12. Comparison of experimental data of Stephenson and Stewart (1986) and CFDLIB

simulated results for relative velocity in a packed bed with D/dv = 10.7 and dv =

0.7035 cm (cylindrical particles). Physical properties of liquid: Liquid -B for

condition at a Rep of 5, ρ = 1.125 g/cm3; µ = 0.474 g cm/s. Liquid -C for condition

at a Rep of 80, ρ = 1.027 g/cm3; µ = 0.114 g cm/s ................................................... 62

3-13a. Interstitial velocity field in a packed bed with two internal obstacles and gas

uniform feed from the top at a superficial velocity of 0.5m/s (Re' = 28.5) (U0 =

120.5 cm/s); (velocity vector plotting). .................................................................... 65

3-13b. Interstitial velocity field in a packed bed with side gas feed (top-left) and internal

obstacles. Inlet gas mean superficial velocity: 0.5m/s (Re' = 28.5) (U0=120.5 cm/s)

(point source inlet from left side, inlet point superficial velocity is of 4.0 m/s)

(velocity vector plotting).......................................................................................... 66

3-14a. Pressure field in a packed bed with two internal obstacles and gas uniform feed

from the top at a superficial velocity of 0.5m/s (Re' = 28.5) (The relative values of

pressure with respect to the inlet operating pressure are plotted). Two obstacle

xiv

plates are placed in this bed, one is located at Z/dp of 66 (at the left side), another

is at Z/dp of 30 (at the right side). The width of the obstacle plate (i.e. the length that

it protrudes into the bed) is half of the width of bed (4 cells). ................................. 67

3-14b. Dimensionless pressure drop in a packed bed with two internal obstacles and a gas

point feed from top-left side at an equivalent feed superficial velocity of 0.5m/s (Re'

= 28.5) (Dimensionless pressure drop, ψρG

G gPZ

=

1 ∆∆

is plotted). Two obstacle

plates are placed in this bed, one is located at Z/dp of 66 (at the left side), another is

at Z/dp of 30 (at the right side). The width of the obstacle plate (i.e. the length that it

protrudes into the bed) is half of the width of bed (4 cells) ..................................... 68

4-1. Tne coordinate system and velocity conventions for α phase in the cell .................. 76

4-2a. Local porosity distribution in model bed; Random internal porosity (0.36 ~ 0.44).

Darker color corresponds to higher porosity............................................................ 80

4-2b. Average porosity profiles in X and Z directions in model bed................................ 80

4-3a, 3b, 3c and 3d. Comparison of the predicted gas interstitial velocities (relative) at the

specific axial level by DCM and CFD. Ul = 0.00148 m/s (UF); Ug = 0.05 m/s (UF);

Completely prewetted packed bed. (The relative interstitial velocity is defined as

the local interstitial velocity (Vi) divided by the overall interstitial velocity (V0).

The value of V0 in this case is equal to 0.1205 m/s). ............................................... 83

4-3e. Comparison of the predicted gas interstitial velocities (relative) for all the cells by

DCM and CFD. Inlet superficial velocities (uniform): Ul = 0.00148; Ug = 0.05m/s;

Completely prewetted packed bed ........................................................................... 84

4-4. Comparison of predicted liquid holdup at specific levels by DCM and CFD. Single

point source liquid inlet: Ul = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05m/s;

Non-prewetted packing ............................................................................................ 85

xv

4-5a. Liquid holdup distribution with single liquid point source inlet (located at No. 5

cell from left) by DCM. Ul = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05m/s;

Non-prewetted packing ............................................................................................ 87

4-5b. Liquid holdup distribution with two liquid points source inlet (located at No. 3 cell

and No. 6 cell from left) by DCM. Ul = 0.00148 m/s (Ul (PS2)=0.00592 m/s); Ug =

0.05m/s; Non-prewetted packing ............................................................................. 88

4-5c. Liquid holdup distribution in whole domain of the non-prewetted packed bed with

uniform liquid distributor by DCM. Ul = 0.00148 m/s; Ug = 0.05m/s..................... 89

4-5d. Comparison of liquid flow maldistribution calculated by DCM along the bed for

different liquid distributors. Ul = 0.00148 m/s; Ug = 0.05m/s. ................................ 90

4-6a. Liquid holdup distribution in the whole domain of the completely prewetted packed

bed (f = 1). Ul = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05 m/s; Point liquid

distributor (PS1)........................................................................................................ 92

4-6b. Liquid holdup distribution at specific levels (Z/dp) in the completely prewetted

packed bed (f = 1), Ul = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05 m/s; Point

liquid distributor (PS1); Overall liquid holdup = 0.0758.......................................... 93

4-6c. Liquid holdup distribution at specific levels (Z/dp) in the completely non-prewetted

packed bed (f = 0), Ul = 0.00148 m/s (Ul (PS1) = 0.01184 m/s); Ug = 0.05 m/s; Point

liquid distributor (PS1); Overall liquid holdup = 0.0716.......................................... 93

5-1. Generated pseudo-Gaussian distribution of porosity under three constraints: (1) ε0 =

0.36; (2) Longitudinally averaged radial porosity profile (white filled circles)

reported by Stephenson and Stewart (1986). (Dr = 7.6 cm, dp = 0.703 cm, Section

size = 0.05R = 0.19 cm). (a)-contour plot; (b)-radial profiles; (c)-histogram

(standard deviation of porosity, σB = 12% ε0). ...................................................... 108

xvi

5-2. Block, sections and cells in CFDLIB for packed bed modeling: (a) physical block,

(b) logical block consists of a number of sections, (c) a section consists of a cell or a

number of cells. ...................................................................................................... 112

5-3. Comparison of Xkl values from different models [Ug = 6 cm/s]: A- Two-fluid

interaction model (Attou et al., 1999a); H- Single slit model (Holub et al., 1992);

SC- Relative permeability model (Saez and Carbonell, 1985). ............................. 117

5-4. Effect of liquid superficial mass velocity on liquid holdup (hp) and particle external

wetting efficiency (wt) at a gas superficial velocity of 6 cm/s. Holub model (Single

slit model, see Holub et al., 1992); S & C model (Relative permeability model, see

Saez and Carbonell, 1985). Particle external wetting efficiency values (wt) were

calculated by the correlation of Al-Dahhan and Dukovic (1995). wt-S & C model

means the pressure-drop value used in calculating wt value was from S & C model;

wt-Holub model means the pressure-drop value used in calculating the wt value was

from Holub model. ................................................................................................. 118

5-5. Comparison of the calculated capillary pressure values from two different

expressions, Eq 5-17a and 5-17b for air-water system (dp = 0.003m; θs =0.63). .. 120

5-6. Simulated liquid upflow velocity component, Vx contour (a) and profiles (b) using

mesh1 (10 × 20) and mesh2 (20× 40) at a superficial velocity of 10 cm/s ............ 125

5-7. Simulated liquid upflow velocity component, Vz contour (a) and profiles (b) using

mesh1 (10 × 20) and mesh2 (20× 40) at a superficial velocity of 10 cm/s ............ 126

5-8. Initial solid volume fraction distribution at 10 ×15 section-discretization (section size

= 1.0 cm) for gas-liquid cocurrent downflow simulation (zoom: x = 0 ~ 4; z = 4 ~

8)............................................................................................................................. 127

xvii

5-9. Gas phase holdup contours and gas interstitial velocity vectors in the 4 × 4 cm zone

marked in Figure 5-9 (a) cell size =1.0 cm; (b) cell size = 0.5 cm; (c) cell size =

0.25 cm (zoom: x = 0 ~ 4; z = 4 ~ 8)..................................................................... 128

5-10. Effect of the mesh sizes (a, b, c) on the cell-scale gas holdup values ................... 129

6-1a. Generated sectional porosities (RN1) plotted in the radial direction and longitudinal

averaged radial porosity profile of Stephenson & Stewart (1986). Statistics of the

RN1 distribution are given in Table 6.1. ................................................................ 136

6-1b. Generated sectional porosities (RN2) plotted in the radial direction and

longitudinally averaged radial porosity profile of Stephenson & Stewart (1986).

Statistics of the RN1 distribution are given in Table 6.1. ...................................... 137

6-2a. Comparison of longitudinally averaged radial velocity profiles at different Reynolds

numbers and experimental data of Stephenson & Stewart (1986). ........................ 138

6-2b. Comparison of longitudinally averaged radial velocity profiles at different

Reynolds numbers and experimental data of Stephenson & Stewart (1986).

Statistics of the RN2 bed are available in Table 6.1; PA bed: sectional porosities are

only varying in the radial direction). ...................................................................... 139

6-3a. Frequency distribution of axial interstitial velocity (Re = 5): RN1-CFD simulation

based on random porosity set 1; RN2-CFD simulation based on random porosity set

2; Exp. –Experimental data reported by Stephenson and Stewart (1986) ............. 140

6-3b. Frequency distribution of axial interstitial velocity (Re = 280): RN1-CFD

simulation based on random porosity 1 (ε: std/µ = 0.0916/0.3534; Vx: std/µ

=1.879/0.2034; Vz: std/µ = 3.864/7.0915); RN2-CFD simulation based on random

porosity 2 (ε: std/µ = 0.0916/0.3534; Vx: std/µ =1.879/0.2034; Vz: std/µ =

3.864/7.0915). Exp. –Experimental data reported by Stephenson and Stewart (1986)

................................................................................................................................ 141

xviii

6-4a. Discretization of the radial porosity profile into sectional porosity values (dp =

3mm): From the wall to the center: sectional mean = 0.411, 0.363, 0.363, 0.365,

0.362, 0.362, 0.363,.364, 0.362, 0.366; sectional std/mean = 20%, 15%, 10%, 10%,

10%, 10%, 10%, 10%, 10%, 10%..................................................................... 143

6-4b. Solid volume-fraction distribution generated based on the data in Table 2 in a pilot

scale packed bed..................................................................................................... 145

65. Simulated phase volume-fraction distribution at liquid superficial velocity of 0.45

cm/s and gas superficial velocity of 22 cm/s in a pilot-scale packed bed. (a) liquid;

(b) gas..................................................................................................................... 146

6-6. Comparison of CFD k-fluid model and other phenomenological models prediction of

liquid saturation with the experimental data of Szady and Sundaresan (1991) (gas

superficial velocity is 22 cm/s). The f values used in CFD modeling are evaluated

by the particle external wetting efficiency correlation by Al-Dahhan and Dudukovic

(1995). exp- use measured static liquid holdup (0.022) in Saez & Carbonell model;

cal- use the correlation-estimated value (0.05) in Saez & Carbonell model.......... 147

6-7. Comparison of CFD k-fluid model and phenomenological models prediction of

pressure gradient with the experimental data of Szady and Sundaresan (1991) (gas

superficial velocity is 22 cm/s) The f values used in CFD modeling are evaluated by

the particle external wetting efficiency correlation by Al-Dahhan and Dudukovic

(1995). exp- use measured static liquid holdup (0.022) in Saez & Carbonell model;

cal- use the correlation-estimated static liquid holdup (0.05) in Saez & Carbonell

model...................................................................................................................... 148

6-8a. Comparison of liquid holdup distribution under nonuniform (left) and uniform

(right) feed conditions at Ul0 = 0.295 cm/s, Ug0 = 22 cm/s .................................... 151

xix

6-8b. Comparison of gas holdup contour and gas interstial velocity vector plot under

nonuniform (left) and uniform (right) feed conditions at Ul0 = 0.295 cm/s, Ug0 = 22

cm/s ........................................................................................................................ 152

7-1. Bench-scale cylindrical packed-bed and its porosity description (a) computer

generated 2D axisymmetric solid volume fraction distribution; (b) radial porosity

profile, ε (r); (c) axial porosity profile, ε (z). ......................................................... 159

7-2. Contours of packed bed structure and corresponding hydrodynamic parameters: (a)

solid holdup-THE1; (b) liquid holdup-THE2; (c) gas holdup-THE3; (d) axial liquid

interstitial velocity-V2; (e) axial gas interstitial velocity -V3; (f) pressure at Ug0 = 6

cm/s and Ul0 = 0.3 m/s at steady state operation. ................................................... 162

7-3a.Relative interstitial velocity profiles of the gas and liquid phase, profiles for

porosity, gas and liquid volume fraction in the radial direction at low flow rates (Ul0

= 0.05 cm/s; Ug0 = 6.0 cm/s). ................................................................................. 163

7-3b.Relative interstitial velocity profiles of the gas and liquid phase, profile for porosity,

gas and liquid volume fraction in the radial direction at low flow rates (Ul0 = 1.0

cm/s; Ug0 = 12.0 cm/s). .......................................................................................... 163

7-4a. Relative interstitial velocity profiles of the gas and liquid phase, porosity profile in

the radial direction at low flow rates (Ul0 = 0.05 cm/s; Ug0 = 6.0 cm/s). ............... 164

7-4b. Relative interstitial velocity profiles of the gas and liquid phase, porosity profiles in

the radial direction at low flow rates (Ul0 = 1.0 cm/s; Ug0 = 12.0 cm/s). ............... 164

7-5. Histogram of the relative interstitial velocities of the gas and liquid phase at low flow

rates (1: Ul0 = 0.05 cm/s; Ug0 = 6.0 cm/s) and high flow rates (2: Ul0 = 1.0 cm/s; Ug0

= 12.0 cm/s)............................................................................................................ 165

xx

7-6. Liquid holdup distribution in a periodic liquid inflow model (15s-on and 45s-off)

(left) and steady sate model (right) in 1-inch cylindrical packed bed at Ug0 =6 cm/s

and Ul0 =0.3 cm/s. .................................................................................................. 167

7-7. (a) Solid volume-fraction (THE1 = 1.0 - Bed Porosity) distribution in the model 2D

rectangular bed; (b) liquid holdup (THE2) contour at steady state liquid feed;

snapshot of liquid holdup (THE2) contours at (c) t =15s; (d) t = 25s; (e) t = 40s; (f) t

= 55s from start of the liquid ON cycle (left) in comparison with steady state holdup

contours (right)... .................................................................................................... 169

7-8. Comparison of cross sectional liquid holdup profiles at different axial locations

under steady (filled squares) and unsteady state operation ((a), Z= 1.8 cm; (b), Z=

18.9 cm; (c), Z= 26.1 cm); (c), Z= 28.8 cm) (!-15s; !-25s; "-40s; "-55s).. .. 170

7-9. Trickle bed and model bed with 500 cells ............................................................... 175

7-10. Transverse averaged profiles of porosity (hard line), liquid holdup (square) and

liquid saturation (least line) vs. longitudinal position (z) at different wetting states

(a) f = 0.0; (b) f = 0.5; (c) f = 1.0 at Ul = 0.3 cm/s, Ug = 6.0 cm/s ......................... 178

7-11. Distribution of gas and liquid interstitial velocity components in non-prewetted bed

(f = 0) (up-2 rows plots) and in prewetted bed (f = 1) (low-2 rows plots) at Ul = 0.3

cm/s, Ug = 6.0 cm/s (G-gas, L-liquid) ................................................................... 179

7-12. Contours of solid volume fraction (=1.0-porosity) distribution of model beds (II,

III, IV) for CFDLIB ............................................................................................... 181

7-13. Contours of CFDLIB simulated liquid volume fraction (holdup) distribution in

model beds (II, III, IV) ........................................................................................... 182

7-14. Standard deviation (S.D.) of the liquid holdup distribution from CFD simulations

and from Eq (7-9) calculations vs. bed wetting factor (f) in model Bed-II, III, IV183

xxi

7-15. Standard deviation (S.D.) of the liquid holdup distribution vs. standard deviation

of the bed porosity at two wetting limits at Ul = 0.3 cm/s, and Ug = 6.0 cm/s....... 184

7-16. Liquid holdup (filled squares) (Holub et al., 1992) and particle external wetting

efficiency from correlation (empty circle) (Al-Dahhan & Dudukovic, 1995)....... 185

7-17. Liquid holdup values from CFDLIB simulations (mean +/- S.D.) and Holub

correlation (1992): εB = 0.399; dp = 3 mm; Ug = 0.03 m/s (hL: liquid holdup) ..... 186

8-1a. Two-dimensional interconnected cell network ...................................................... 191

8-1b. Fluid superficial velocities and concentrations of species i at interface of the cell j,

where C3 i,j = C4 i,j due to the well-mixing ............................................................... 191

8-2a. Porosity distribution at spatial resolution of 1 cm (dp = 3 mm, L = 50 cm, D = 10

cm).......................................................................................................................... 193

8-2b. Histogram of porosity distribution (Gaussian distribution) used in the k-fluid CFD

model...................................................................................................................... 194

8-3a. Simulated liquid superficial velocity component (Ux, m/s): Ul0 = 0.003 m/s; Ug0 =

0.06 m/s .................................................................................................................. 195

8-3b. Simulated liquid superficial velocity component (Uz, m/s): Ul0 = 0.003 m/s; Ug0 =

0.06 m/s .................................................................................................................. 196

8-4. Computed cell scale mass transfer coefficient (kls, cm/s): Ul0 = 0.003 m/s; Ug0 = 0.06

m/s .......................................................................................................................... 197

8-5. Cells with different inflow and outflow configurations........................................... 200

8-5. Concentration contour of species B in the liquid phase (m = 0.0; n = 1.0; r = 1.0),

CBl, 0 = 5.4 kmol/m3, Ul0 = 0.003 m/s; Ug0 = 0.06 m/s............................................ 201

xxii

8-7. Longitudinally averaged concentration profile of species B and liquid velocity

component (Uz) profiles in the X direction. (CBl - filled circle; Uz – blank square;

CBl, 0 = 5.4 kmol/m3; Ul0 = 0.003 m/s; Ug0 = 0.06 m/s).......................................... 202

8-8. Calculated concentration profiles of species B at k = 1.E-04 m3/kg.s, n = 1.0, m =

0.0 by (i) plug flow; (ii) ADM (De = 2.53E-04 m2/s calculated from Sater and

Levenspiel, 1966, Pe = 5.92); (iii) Mixing-cell network model; (iv) ADM (adjusted

De = 1.5E-04 m2/s, Pe = 10). CBl, 0 = 5.4 kmol/m3; Ul0 = 0.003 m/s; Ug0 = 0.06 m/s

................................................................................................................................ 203

A-1. Contacting pattern in the trickle flow regime and bubble flow regime .................. 222

A-2. Effect of pressure on the flow regime in downflow packed bed using the flow chart

of Larachi et al. (1993)........................................................................................... 223

A-3. Species concentration profiles along the reactor. D = 11.3 m; Ul = 0.0075 m/s; Ug =

0.06 m/s; P = 70 atm; T = 603K; CaL : hydrogen; CbL : biphenyl; CcL : product .... 223

A-4. Effect of operating pressure on the exit biphenyl conversion and global

hydogenation rate (Re). H= 4 m, D = 11.3 m ........................................................ 224

A-5. Effect of liquid superficial velocity on the exit biphenyl conversion and global

hydrogenation rate (Rg). H = 4 m; D = 11.3 m...................................................... 224

A-6. Effect of liquid superficial velocity on the exit biphenyl conversion in the upflow

packed bed.............................................................................................................. 225

A-7. Comparison of biphenyl conversion profile and hydrogen concentration profile in

liquid phase in the up-flow and down-flow packed beds. H = 4 m; D = 4 m; Ul =

0.0075 m/s; Ug = 0.12 m/s; P = 70 atm; T = 648K; mass transfer coefficients used:

for the up-flow mode, 0.1× (ka)gl = 0.0144 s-1, 0.1× (ka)ls = 0.129 s-1; for the down-

flow mode, 0.1×(ka)gl = 0.00344 s-1, 0.1× (ka)ls = 0.0338 s-1. (For the directly

xxiii

calculated values of mass transfer coefficients, the corresponding exit biphenyl

conversion for up-flow is 72.35%, and for down-flow, 71.56%) .......................... 226

A-8. Sensitivities of the model with respect to gas-liquid, liquid-solid mass transfer

coefficients in trickle-bed reactor at 605 K, 70 atm. with flow conditions: Ug = 0.06

m/s; H=32 m; D=4 m ............................................................................................. 227

xxiv

Acknowledgments

This thesis is a result of over three-years of research carried out in the framework

of packed-bed studies at the Chemical Reaction Engineering Laboratory (CREL),

Washington University in St. Louis. The visions, ideas, models and results contained in

this thesis could not have been achieved without the supports of many people inside and

outside CREL.

I wish to express my deep gratitude to two advisors Prof. M. H. Al-Dahhan and

Prof. M. P. Dudukovic for their guidance and freedom while working towards this thesis.

Their encouragements of independent thoughts made the final thesis possible. Special

thanks to my co-advisor, Prof. M. P. Dudukovic for his constant patience and support,

and for challenging me to improve my writing and critical reviewing skills, which made

my publications possible.

It has been my pleasure to work at CREL, the place has provided me great

opportunities to experience the developments of various multiphase flow reactors and to

learn extensively from great colleagues inside and outside CREL. My first assignment at

CREL was truly memorable. The assignment was sponsored by Monsanto Company to

investigate the feasibility of using high-pressure trickle-bed reactors to substantially

improve the productivity of complex reaction networks. The project brimmed with

challenges, which made my first-year life in the United States full of excitant. Many

enjoyable hours were spent with my co-worker, M. R. Khadilkar, from whom I have

learned so much in professional attitude, which made the project successful and later on

was helpful in several aspects of my thesis work. Thanks to Dr. R. Kahney, Dr. Sh. Chou,

and Dr. G. Ahmed at Monsanto Company for our invaluable discussions, and for their

helpful suggestions.

xxv

My special thanks is to Dr. R. A. Holub who initialized the flow distribution

study at CREL as a part of his doctoral research, and provided me the original program of

the discrete cell model (DCM), which allows me to revisit and to extend this approach. I

greatly appreciate his contribution to the development of the original DCM, and his

kindness in agreeing to sit on my thesis committee.

I would like to thank Prof. P. A. Ramachandran for our invaluable discussions, for

his helpful suggestions and for acting as my thesis committee member. Thanks also to

Prof. R. A. Gardner from the Department of Mechanical Engineering for being on my

thesis committee, for taking interest in my work and providing useful comments and

suggestions.

I wish to thank Dr. P. L. Mills at DuPont Company for his invaluable advice

regarding many interesting issues in trickle-bed research.

Particularly important for the progress of this work were the numerous

discussions I had with my past and present colleagues at CREL. My sincere gratitude

goes to Dr. M. R. Khadilkar, Dr. S. B. Kumar, Dr. J Chen and Dr. S. Roy, whose help

and encouragement were invaluable throughout my stay at Washington University.

I sincerely acknowledge the help and assistance offered by all the past and present

members of CREL including Dr. A. Kemoun, Dr. K. Balakrishnan, G. Bhatia, J. A.

Castro, P. Chen, Dr. S. Degaleesan, Dr. P. Gupta, J. Mettes, Dr. J. Lee, K. Ng, B-Ch.

Ong, Dr. Y. Pan, N. Rados, A. Ramohan, Dr. Y. Wu, Dr. Z. Xu, J. Xue and many others.

I also wish to thank the entire Chemical Engineering Department, particularly the

secretaries for their help with numerous formalities.

I am also glad that I was able to enjoy the three-month internship at DuPont

Company sponsored by Conoco Inc. in 1999. Special acknowledgements go to Dr. Tiby

Leib of DuPont Company and Dr. Harold Wright of Conoco Inc. for this great

opportunity and for their professional guidance, which allowed me to gain invaluable

experiences in multiphase reactors other than packed-bed reactors.

I wish to express my deep gratitude to Prof. G. Gao in China, who has given me

constant support in the professional development since I worked as an assistant professor

at Jiangsu Institute of Petrochemical Technology (JIPT) in 1989. Thank you for

xxvi

introducing me to CREL, and thank you for numerous help you offered during the last

ten years.

Last, but not least, I would like to thank my wife, Feixia, who supported my

decision to embark in graduate studies and fulfill my career goals, despite the significant

changes it involved in our lives. She has also endured many long hours waiting for me to

come home from the lab, and has provided stability to our family by taking charge of our

home and our daughter’s education. I thank my daughter, Sheri, for her understanding

and encouragement to finish this thesis. My deep thanks to my parents, brothers and

sisters in China for their prayerful supports both in my decision to go on to graduate

studies and to start my new career at Conoco Inc.

Yi Jiang

Washington University, St. Louis

December, 2000

xxvii

Nomenclature In Chapter 2 dsp the radius of the liquid channel (i.e., filament)

G gas superficial mass velocity, kg/m2/s

H length of paced bed, m

L liquid superficial mass velocity, kg/m2/s

P pressure, atm

R radius of packed column, m

r radial distance from the center, m

T temperature, °C

Vav cross-section averaged liquid volumetric flow rate, m3/s

V measured liquid volumetric flow rate, m3/s

In Chapters 3 & 4 a constant in Leverett’s function (= 0.48 for air-water system)

aj, working variable (E1(1-εj)2µα/(ραε3jdp

2))

b constant in Leverett’s function (= 0.036 for air-water system)

bj working variable (E2(1-εj)/(ε3jdp))

D width of model bed, m (= 0.072 m), 8 cells

dp particle diameter, m (= 0.003 m)

dv equivalent diameter of particle (m)

E1, E2 Ergun constants

Ev,j mechanical energy dissipation in the cell jth, J/s (based on Vj)

Ev,j,α energy dissipation rate of phase α in the jth cell, J/s (based on Vj,α)

xxviii

Ev, bed total mechanical energy dissipation rate in the bed, J/s

f particle wetting factor

f1,, f1,j resistance factor, 150(1-εj)2µ/(ρ εj3

φ2dp2 )

f2, f2,j resistance factor, 1.75(1-εj) ρ/(εj3 φdp)

Gaα, j Cell Galileo number of the α phase, gdP3εj

3/(µα2(1-εj)3)

gi, i=x,z gravitational acceleration in the i direction, (gx = 0; gz = 9.8 m/s2)

H height of model bed, m (= 0.288 m), 32 cells

N total number of the cells (8 × 32 = 264)

Nc number of cells in each row

Pc pressure at the center of the cell, N/m3

P0 pressure, dyn/cm2

Pz pressure in the z direction, N/m2

kp0 pressure ( pp k −0 non-equilibrium pressure)

∆P/∆Z pressure drop per unit cell length, N/m3

r number of cells in each row (X direction)

R Radius of packed beds, m

Re’ Reynolds number, V0dpρ/6/(µ (1-εB))

Rep particle Reynolds number, V0dPρ/µ

Reα, j cell Reynolds number of the α phase, VαdP/(µα(1-εj))

si cell face area at a given coordinate direction i ,m2

Sw, j liquid saturation in cell j (εL,j /εj)

Ti energy dissipation rate due to the inertial term, J/s

Tk energy dissipation rate due to the kinetic term, J/s

Tv energy dissipation rate due to the viscous term, J/s

Uj local interstitial velocity, m/s (=Vj / εj)

u0 material velocity, (cm/s)

ku material k interstitial velocity, cm/s (ρkuk ≡ < αkρ0u0 >)

ku' fluctuating part of material k interstitial velocity, cm/s

xxix

U0 input interstitial velocity, m/s (=V0 / εB)

V velocity vector

Vj, α superficial velocity of phase α in the jth cell, m/s

(Vol)j volume of cell j , m3 (=∆Z×∆X× ∆Y for rectangular bed)

Vc, j volume of the cell j, m3 (=S z,j×∆Z)

Vj superficial velocity in the jth cell, m/s

V0 input superficial velocity, m/s

∆Z,∆X,∆Y size of the cell, m (in this work, 3dp= 0.009 m)

Greek Letters

kα material indicator (=1 if material k is present; =0 otherwise)

kα! material derivative

εB bed porosity (= 0.415)

εj porosity in the jth cell

εj, α holdup of phase α in the jth cell

σ liquid surface tension, N/m (0.072 for water)

γi, i=x,z the angle of each axis with horizontal plane

"Φ the gravitational potential,

φ particle shape factor, (φ = 1 for Spherical particle)

µ viscosity of fluid, Pa s ( gas:1.8×10-5 Pa s; liquid:1.0×10-3 Pa s)

µα viscosity of phase α in the bed, Pa s (µL=1.0e-3 Pa s; µG=1.8e-5 Pa s)

θ k material k volume fraction (θ k = < α k > )

τ0 deviatiric stress

ρ density of fluid, kg/m3 ( gas: 1.2 kg/m3; liquid: 1000 kg/m3)

ρk density of material k, g/cm3 (≡ < αkρ0 >)

ρα density of phase α in the bed, kg/m3 (ρL=1000 kg/m3; ρG=1.2 kg/m3)

ψG local gas flow dimensionless pressure drop, ψρG

G gPZ

=

1 ∆∆

xxx

ψα,j dimensionless pressure-drop for phase α, (= ∆ ∆P Lg

α

αρ/

+1)

Subscripts

X x coordinate for the rectangular cell or bed

Z axial coordinate along the length of bed

< > ensemble average (note: cross-sectionally average in Eq.5-15)

In Chapters 5 & 6 Agl, Ags, Aks parameters defined in Table 5-3

Bgl, Bgs, Bks parameters defined in Table 5-3

Bo Bond number

Ca Capillary number

C1,j, C2,j inflow concentration of species i in j-cell

C3,j, outflow concentration of species i in j-cell

dp particle diameter, m

de equivalent diameter of particle

dmin minimum equivalent diameter of the area between three spheres in contact

( pdd5.0

min 5.03

−=

π)

Dr diameter of packed bed, m

E1, E2 Ergun constants (E1 = 180; E2 = 1.8)

f particle wetting factor

F pressure factor

FD(k-l) Drag between phases k and l

g gravity, 9.81m/s2

H height of model bed, m

J J-function

kr relative permeability parameter in Eq (5-17)

ls size of section, m

xxxi

n time step number

p, P pressure, N/m3

Pc capillary pressure, N/m3

PG pressure in gas phase, N/m3

PL pressure in gas phase, N/m3


kp0 pressure (=k

k pθ

α >< 0 )

r radial position in cylindrical coordinate, m

R Radius of packed beds, m

Rep Reynolds number, V0dpρ /µ

Sk saturation of phase k

S.D. standard deviation

t time, s

THE1, 2, 3 solid, liquid and gas volume fraction


ku material k interstitial velocity, cm/s (ρkuk ≡ < αkρ0u0 >)

ku' fluctuating part of material k interstitial velocity, cm/s

U0 input superficial velocity, m/s (=V0 × εB)

U1,j, U2,jinflow velocity of j-cell

U3,j, outflow velocity of j-cell

Ug gas superficial velocity, m/s

Ug0 gas feed superficial velocity, m/s

Ul liquid superficial velocity, m/s

Ul0 liquid feed superficial velocity, m/s

V0 input interstitial velocity, m/s (=U0 / εB)

Vg gas interstitial velocity, m/s

Vg0 gas feed interstitial velocity (= Ul0 / εB), m/s

Vl liquid interstitial velocity, m/s

xxxii

Vl0 liquid feed interstitial velocity (= Ug0 / εB), m/s

Vx interstitial velocity component in horizontal or radial direction, cm/s

Vz interstitial velocity component in axial direction, cm/s

V velocity vector

Z axial position, cm or m

Xkl momentum exchange coefficient between phases k and l

Greek Letters

kα material indicator (=1 if material k is present; =0 otherwise)


εB mean porosity of packed bed

εg gas holdup

εl liquid holdup

εl0 static liquid holdup

φ particle shape factor, (φ = 1 for Spherical particle)

µ viscosity of fluid, Pa s

kθ material k volume fraction (θ k = < α k > ), k = G, L, S

0Lθ static liquid holdup


ρ density of fluid, kg/m3


ρ0 material density, kg/m3

ρg density of gas phase, kg/m3

ρl density of liquid phase, kg/m3

σs surface tension

σB standard deviation of porosity distribution

< > ensemble average

∇⋅ divergence

xxxiii

ϕ angular coordinate

In Chapters 7 dp particle diameter, m

E1, E2 Ergun constants (E1 = 180; E2 = 1.8)

f fractional wetting value

f (xj) probability density function

FD Drag force

g gravity, cm/s2


S.D. standard deviation

t time, s

V2 interstitial liquid velocity components, cm/s

V3 interstitial gas velocity components, cm/s

Vr superficial relative velocity based on gas flow, as defined in Eq (7d), cm/s

Vx, Vz interstitial velocity components, cm/s


|ukl| slip interstitial velocity between phase k and phase l, cm/s

ku material k interstitial velocity vector, cm/s

ku' fluctuating part of k interstitial velocity vector, cm/s

U0 input superficial velocity, cm/s

Ul0 liquid superficial velocity, cm/s

Ug0 gas superficial velocity, cm/s

x horizontal position in x-z coordinate

Xkl momentum exchange coefficient between phase k & l

xj variable of system

z axial position in x-z coordinate

xxxiv

Greek Letters

α1, α2, α3 parameters

kα material indicator (=1 if k is present; =0 otherwise)


εB bed porosity

ε section porosity

ε(r) longitudinally averaged radial porosity profile

ε(z) cross-section averaged porosity profile

θ k material k volume fraction (θ k = < α k > )


ρ density of fluid, kg/m3 ( gas: 1.2; liquid: 1000)


µ mean value

µα viscosity of phase α

σs surface tension

σB standard deviation of porosity distribution

σl standard deviation of liquid holdup

γ1 Skewness of statistical data

γ2 Kurtosis of statistical data

< > ensemble averaged

In Chapter 8 a0 a basis for cell cross-section area, m2

ak k interface area of cell, m2

aL gas-liquid mass transfer area per unit cell volume, m2/m3

CAg,0 concentration of A in the feed gas, kmol/m3

CAg,k concentration of A in the gas phase enter the cell, kmol/m3

CAg,out concentration of A in the gas phase leaving the cell, kmol/m3

xxxv

CAl,k concentration of A in the liquid phase enter the cell, kmol/m3

CAl,out concentration of A in the liquid phase leaving the cell, kmol/m3

CAs concentration of A at particle surface, kmol/m3

CBl,0 concentration of B in the feed gas, kmol/m3

CBl,k concentration of B in the liquid phase enter the cell, kmol/m3

CBl,out concentration of B in the liquid phase leaving the cell, kmol/m3

CBs concentration of B at particle surface, kmol/m3

De effective intraparticle diffusivity of the species, m2/s

HA Henry's law solubility coefficient of A, Ag/Al

kr reaction-rate constant [m3/kg.s][m3/kmol] m+n-1(i.e., 1.e-4)

k* dimensionless rate constant

kl liquid-film mass transfer coefficient, m/s

kg gas-film mass transfer coefficient, m/s

ks solid-film mass transfer coefficient, m/s

KL overall gas-liquid mass transfer coefficient, m/s, defined as

( ) ( ) ( )ALgAALlALL akHakaK

111 +=

Sp external surface area of a catalyst particle, m2

t time, s

Ug superficial gas velocity, m/s

Ul superficial liquid velocity, m/s

Vc volume of cell, m3

Vp volume of a catalyst particle, m3

xAl dimensionless concentration of A in liquid phase

xBl dimensionless concentration of B in liquid phase

xBs dimensionless concentration of B at particle surface

yAg dimensionless concentration of A in gas phase

yAs dimensionless concentration of A at particle surface

xxxvi

Greek Letters

αA, αB dimensionless gas-liquid mass transfer coefficients defined by Eq (6)

βA parameter defined as Eq (6)

ε void fraction in the jth cell

η catalyst effectiveness factor, defined as ηφ φ φ

= −

1 13

13tanh

ηCE catalyst particle wetting efficiency

γ stoichiometric coefficient of A in the reaction

γA, γB parameters defined as Eq (6)

ρ density of the catalyst particle, kg/m3 (=ρp = 2500 kg/m3)

φ Thiele modulus, defined as ( )φ ρ=

∫−

VS

kA B D r c dcP

PP j I

mj In

e c

Cj I

, , ,

.,

20

0 5

Ω Reaction rate per unit volume of catalyst particle, kmol/m3.s

In Appendix Ci,L, concentration of species i in liquid phase

ci,L, dimensionless concentration of species i in liquid

DEL,i diffusivity of species i in liquid phase

G gas mass superficial velocity, kg/m2/s

Ug gas superficial velocity, m/s

uSL, Ul liquid superficial velocity, m/s

ki rate constant

(ka)gl gas-liquid mass transfer coefficient. 1/s

(ka)ls liquid-solid mass transfer coefficient. 1/s

L liquid superficial velocity, kg/m2/s

T temperature, K

P operating pressure, atm

Pe Péclet number

Qg gas volumetric flow rate, m3/s

xxxvii

Ql liquid volumetric flow rate, m3/s

ri reaction rate based on the species I

Rv global hydrogenation rate, mol/s/m3

X conversion of biphenyl, %

Greek Letters

αG,L dimensionless parameter

αL,S dimensionless parameter

β1,,β1, dimensionless parameter

ξ dimensionless position

Subscripts

A hydrogen

B biphenyl

C cyclohexylbenzene

D H2S

e equilibrium

i input

1

Chapter 1

Introduction to Flow in Packed Beds

Fluid flowing through packed grain-like material constitutes a large part of our

natural environment as well as a substantial fraction of man-made processes. The

heterogeneity of the packed media and its impact on global transport properties have been

important subjects of study in various science and engineering disciplines including

hydrology, oil recover, chemical engineering, composite material processing, biology,

and medicine (Bideau and Hansen, 1993; Stanek, 1994; Keller, 1996; Helmig, 1997;

Ingham and Pop, 1998; many others). Although there is a broad spectrum of length scales

involved in such a multidisciplinary research field, a certain similarity does exist in

different disciplines in both geometrical aspects and the flow transport phenomena. Thus,

the research result derived from one discipline makes a certain contribution in other

related disciplines. The work described in this thesis has been carried out in the

framework of chemical reaction engineering, particularly for catalytic packed-bed

reactors, in which the uniformity of the flow field is important in assessing reactor

performance. The understanding and prediction of the flow structure (i.e. pattern) are

crucial for improving the yield of chemical reaction. Moreover, the significance of this

research is due to the major applications of packed beds in petroleum, petrochemical and

biochemical processes in terms of number, capacity and annual value of products (Sie

and Krishna, 1998).

To systematically study the bed structure and flow phenomena in packed beds, it

is necessary to clarify several important issues. For example, due to various interaction

forces that exist in the system and contribute to the flow-pattern formation in packed

2

beds, it is essential to assess the relative importance of the interaction forces before

making any approximation in the flow equations. Because of the statistical nature in both

voidage structure and flow distribution in packed beds, it is reasonable to come up with a

statistical methodology for describing the spatial distributions of voidage space and fluid

flow. How to effectively represent the real 3-D structure in 2-D coordinates is one of the

basic issues to be resolved before preaching the virtues of 2-D flow simulation. In the

scale-up and scale-down of packed-bed reactors, the scale-dependency of the structure-

flow relation is the main concern when one uses this relationship in design practice.

The major goal of this study is to conduct systematic theoretical modeling of fluid

flow in packed beds, and to assess the impact of flow distribution on the reactor

performance. In addition, some experimental work has been performed to validate the

model development. The first part focuses on developing the flow distribution model

using either engineering approach or a computational fluid dynamics method. The second

part focuses on the applications of the developed flow distribution models in the scale-up

and scale-down of packed beds as well as in modeling of reactor performance.

1.1 Research Motivation 1.1.1 Discrete Cell Model (DCM) Revisited Modeling the complex fluid dynamics in packed bed reactors is important in

design and scale-up. Previous studies have developed various models to predict the single

phase or two phase flow distribution in packed beds. Although they have provided the

insights into fluid flow distribution at a certain level, most of them still could not capture

a number of experimental observations. Furthermore, some of available models are too

complicated for applications, others are too simple to reflect the actual flow textures

inside packed beds. It is desirable to develop a model that can capture most of the

important experimental observations and predict the results within engineering accuracy.

Based on the assumption that the flow is governed by the minimum rate of total

energy dissipation in the bed, Holub proposed a discrete cell model (DCM) for single

phase and two phase flow distribution (Holub, 1990). In DCM, the packed bed is treated

3

as a number of interconnected cells, and for each cell, one can write the mechanical

energy dissipation rate for all phases. The model structure is so clear that it provides

potential for further upgrading of the model. For example, it has been noticed that the

original DCM results were based on a small size two-dimensional packed bed due to the

limitation of computational power. Moreover, the model could not distinguish the liquid

flow distribution in prewetted and non-prewetted beds because of the lack of

consideration of the capillary force in the model equations. Our current research

overcomes those obstacles and extends the utility of DCM.

The main assumption of DCM is that the flow is governed by the minimum rate

of total energy dissipation in the bed. The theoretical justification for this assumption has

been provided only for linear systems, in which the fluxes and driving forces have a

linear relationship, and rests on the principle of minimization of entropy production rate

(Jaynes, 1980). For non-linear systems, examples can be constructed for which the

'principle of energy minimization' does not hold and, hence, that demonstrates that it is

not a general 'principle' at all (Jaynes, 1980). Nevertheless, this energy minimization

approach was reported to be valid for some classes of nonlinear systems such as particle

flow in circulating fluidized beds (Ishii et al., 1989; Li et al., 1988, 1990). Hence, for any

specific nonlinear system one needs to conduct a detailed verification study before

considering 'energy minimization' as the governing principle for flow distribution (Hyre

and Glicksman, 1997). Regarding the flow distribution in packed beds, it is necessary to

revisit DCM by examining how well can this 'principle' be used to describe the flow. This

can be done by comparing the results of the DCM either to accepted solutions of the

ensemble-averaged momentum and mass conservation equations or to reliable

experimental data. Unfortunately, there is very few experimental data for single phase

velocity profiles inside packed beds available in the literature due to the limitations on the

non-intrusive velocity measuring techniques (McGreavy et al., 1986; Stephenson and

Steward, 1986; Peurrung et al., 1995). Fortunately, recent advances in understanding of

multiphase flow and development of robust computation codes make the extension of this

work as well as the verification of DCM predictions feasible. Such comparison study

should generate a better appreciation of what the concept of minimization of the total

4

energy dissipation rate can and cannot do. The intent of this part of our study is not to

replace the fluid dynamic simulations by the minimization of total energy dissipation

rate, but to examine whether an alternative of engineering accuracy to a CFD model

exists and can be used.

1.1.2 k-fluid CFD Model Development and Applications

The superiority of the k-fluid CFD model in computational efficiency,

particularly for large-scale packed beds, has motivated us to undertake such model

development. Although there have been many studies in utilizing the CFD approach to

simulate the flow pattern in fluidized beds and bubble column reactors (Kuipers and van

Swaaij, 1998), there is no detailed study of CFD in the multiphase packed beds because

of the difficulty in incorporating the complex geometry (e.g. tortuous interstices) into the

flow equations, and the difficulty in accounting for the fluid-fluid (gas-liquid)

interactions in presence of complex fluid-particle (e.g., partial wetting) contacting. The

intent of this part of study is to find an efficient way to solve the above problems.

To be successful in scaling up multiphase packed beds, it is important to quantify

the flow structures in bench-, pilot- and commercial-scale reactors by either flow

measurements or reliable numerical flow simulations. Once we ensure that the k-fluid

CFD model can predict the macroscopic flow structure with engineering accuracy, then

we can conduct extensive numerical flow modeling in the beds of different sizes. Those

flow simulation can help us to understand how the flow distribution varies with the

reactor size; how the relationship of bed structure and flow pattern varies with the

operating conditions.

1.1.3 Impact of Flow Distribution on Packed-Bed Performance Many phenomenological reactor models for multiphase packed beds have been

developed and utilized for several decades by assuming simple flow patterns without

solving the momentum balances (El-Hisnawi et al., 1982). To account for the non-ideal

flow patterns in reactor modeling, efforts made in the literatures include a two-region cell

model (Sims et al., 1994), a cross-flow model (Tsamatsoulis and Papaynnakos, 1995),

5

and other models based on liquid flow maldistribution (Funk et al., 1990), the stagnant

liquid zones (Rajashekharam et al., 1998), and the one-dimensional variations of gas and

liquid velocities along the reactor (Khadilkar, 1998) etc. The ways used to incorporate the

multiphase flow pattern, however, do not make these models suitable as a diagnostic tool

for operating units, which are normally operated under conditions not amenable to the

model assumptions.

In principle, the performance of multiphase reactors can be predicted by solving

the conservation equations for mass, momentum and (thermal) energy in combination

with the constitutive equations for species transport, chemical reaction and phase

transition. However, because of the incomplete understanding of the physics, plus the

nature of the equations- highly coupled and nonlinear, it is difficult to obtain the

complete solutions unless one has reliable physical models, advanced numerical

algorithms and sufficient computational power. Although the full probability density

function (PDF) method has some promises in solving the single-phase reactive flow (Fox,

1996), for most multiphase reactive flows, the challenge exists in both numerical

technique and physical understanding. The use of direct numerical simulation (DNS) on

single particle and single void scale in micro-flow modeling requires complete

characterization of solids boundaries and voids configuration, which is obviously

undoable for a massive packed bed. To focus on the macroscale flow distribution, a

statistical method for implementing the porosity distribution has potential for success in

multiphase flow modeling using ensemble-averaged equations of motion (i.e., k-fluid

computational fluid dynamics, CFD, model) because both porosity and flow structures

are statistical in nature. The intent of this part of our study is to utilize the simulated flow

distribution results by the k-fluid CFD or DCM to assess the impact of flow pattern on

the reactor performance for a given kinetics.

For the systems in which the flow patterns are not substantially affected by

reaction, the sequential modeling of flow and reaction(s) is a good alternative for quick

evaluation of the reactor performance based on flow considerations. Such sequential

modeling scheme of flow and reaction allows one to deal with the packed beds with a

complex flow pattern and complicated chemical kinetics. The modeling results provide

6

the map of both flow and species concentration distribution in packed beds, which are

particularly valuable for the diagnostic analysis of the operating commercial reactors.

1.2 Research Objectives The overall objectives of this study are outlined below. Details of the

implementation of each are discussed in Chapter 2 through Chapter 9.

1.2.1 DCM Revisited The objectives of this study are:

• To analyze the contribution of each energy dissipation term in DCM equations,

and to compare its predictions of single phase flow with the k-fluid CFD

simulation results and available experimental data, and to fully assess the

applicability of this engineering approach for single phase flow modeling in

packed beds.

• To extend the DCM for predictions of gas and liquid two phase flow distributions

in trickle beds. The developed model has the ability to take into account the state

of particle external wetting and the distributor effects. It is desire to compare the

predictions of the extended DCM with the k-fluid CFD simulation and other

independent modeling results, and to reach same conclusion regarding the

application of this model.

1.2.2 k-Fluid CFD Model for Packed Beds The objectives of this study are:

• To analyze the importance of each basic force in ensemble-averaged conservation

equations of mass and momentum, such as inertial force term, Reynolds stress

term, gravity term and capillary force term.

• To develop a statistical method to implement the porosity distribution in the k-

fluid model simulation.

7

• To establish the way to compute the momentum exchange coefficients used in the

k-fluid CFD model.

• To compare the predictions of the k-fluid CFD model with the experimental data

available in the literature.

1.2.3 Applications of k-Fluid CFD Model The objectives of this study are:

• To perform a series of numerical flow simulations using the k-fluid CFD model to

quantify the relationship between the bed structure, flow distribution and particle

external wetting at different operating conditions.

• To perform the flow simulation in bench-scale packed beds in order to provide a

basis for interpretation of irregular experimental data.

1.2.4 Impact of Flow Distribution on Reactor Performance The objective

of this study is to develop a methodology for modeling flow and reaction in multiphase

packed-bed reactors. The model can provide the mapping information on both flow and

species concentration in the entire reactor domain. Based on the concept of the mixing-

cell network, a combinational modeling scheme is to be developed in which the k-fluid

CFD model or the DCM model can provide the detail flow distribution information, then

the mixing-cell network model can provide the distribution information of species

concentration for a given kinetics.

1.3 Thesis Structure To be consistent with the thesis format requirement, and also for convenience of the

reader, the thesis has been organized in the following manner: each Chapter is written as

a full manuscript which consists of (i) introduction of the topics, (ii) results and

discussion, and (iii) conclusions. In the course of flow distribution study, there have been

many opportunities to work on other aspects of packed-bed reactors, which are relevant

to the flow pattern directly or indirectly. One of such typical research accomplishments

8

is documented as Appendix. The comparison study of the commercial scale trickle-bed

reactor performance under upflow and downflow conditions is performed, in which the

effect of large-scale flow pattern on the reactor performance is discussed.

Although the Chapters are independent as topics, they are structured in a certain

logical sequence. The main body of this thesis consists of Chapter 2 to Chapter 8, in

which the experimental and modeling results are presented and discussed in detail.

Chapter 1 is a general introduction in which the general motivation and the overall

objectives of this study are given. In Chapter 9, the thesis accomplishments are

summarized; the issues that deserve future research efforts are highlighted.

To obtain a better physical understanding of particle external wetting and its impact on

the formation of liquid structures in packed beds, in Chapter 2, we present some

experimental observations of liquid distribution in pseudo two-dimensional and real

three-dimensional packed beds, which provide a physical background for developing the

flow distribution model.

In Chapters 3 and 4, an engineering approach, discrete cell mode (DCM) (Holub, 1990)

has been revisited and extended. The main assumption used in DCM, that the flow

distribution is governed by the minimum total energy dissipation rate in packed beds, has

been examined in detail. In Chapter 3 we focus on the single-phase flow system, whereas

in Chapter 4 we deal with the two-phase flow system such as that in trickle-bed reactors.

The flow distribution results obtained by the DCM approach have been compared with

the solutions of ensemble-averaged Naiver-Stokes equations (i.e., k-fluid CFD model),

and with the experimental results. A reasonable agreement is achieved for engineering

applications.

In Chapters 5 and 6, we focus on the development of the k-fluid model in the

framework of computational fluid dynamics (CFD) for the prediction of macroscopic

flow pattern in packed beds. A statistical method is developed for implementing the

complex bed structure in the k-fluid CFD model. Several important issues in using the k-

fluid model for packed beds are discussed in Chapter 5. The numerical results of the k-

fluid CFD model at steady state and unsteady state feed conditions are presented in

9

Chapter 6. The comparison of the model predictions with experimental data is also

provided.

In Chapter 7, two case studies demonstrate the applications of the k-fluid CFD

simulations in scale-down and scale-up of multiphase flow packed beds. The first case

study presents the multiphase flow simulation in bench-scale packed beds for the first

time. The simulation results provide valuable insights on the distributions of velocity,

pressure, and phase holdup, which are useful in interpreting scattered experimental data.

In the second case study, the quantitative relationships among bed structure, operating

condition, particle external wetting, and the resultant flow distribution are developed in a

statistical manner through a series of the k-fluid CFD simulations. This work revealed

that the contribution of capillary forces to liquid maldistribution is significant in the case

of partial particle wetting; however, the effect of porosity non-uniformity in packed beds

can be reduced if the particles are prewetted well.

In Chapter 8, we focus on the impact of flow distribution on packed-bed

performance. A combinational modeling strategy of flow and reaction in packed-bed

reactors has been developed based on the concept of the mixing-cell network. Such a

methodology provides an efficient way to utilize the flow information obtained by DCM

or CFD simulation in the prediction of reactor performance. The spatial mapping

information on the bulk flow and the species concentration are valuable in the diagnostic

analysis of the operating commercial units.

10

Chapter 2

Experimental Observations: Liquid Flow Distribution in Trickle Beds 2.1 Introduction

Trickle-bed reactors with cocurrent gas and liquid downflow have found various

applications in petroleum, petrochemical and biochemical industries. Gas and liquid

distribution play an important role in determining the reactor performance. To develop an

advanced model for the design of new units and for the diagnostic analysis of operating

units, the bed structure and flow distribution need to be incorporated into the reactor

model. In fact, several researchers have shown that the prediction of packed bed

performance can be improved if the nonuniformities of the bed structure are properly

accounted for (Lerou and Froment, 1977; Delmas and Froment, 1988; Daszkowski and

Eigenberger, 1992). Because of the complex structure of the interstitial space between

particles plus the complicated interactions between particles and fluids, reliable flow

distribution modeling in trickle beds has been the challenging subject for several decades.

In the literature, certain approximations have been made in solving the flow equations,

particularly for flow in a commercial scale packed bed. For example, the k-fluid model,

based on the volume-averaged or ensemble-averaged Navier-Stokes equations, has shown

promise in dealing with the flow in packed beds, because it can avoid solving for the

tortuous particle boundaries, and just treats the gas, liquid and even the solid as

continuous but penetrating phases. In fact, such a model has been developed not only for

11

one-dimensional (1-D) trickle beds to predict the global hydrodynamics and flow regime

transition (Attou et al., 1999; Attou and Ferschneider, 2000), but also for simulating the

flow in 2-D beds (Anderson and Sapre, 1988). It has been realized that the progress in

using the k-fluid model in packed beds relies on better closures for momentum exchange

coefficients and efficient ways for implementing the porosity distribution information

into the model.

To establish reliable formulae for computing the various momentum exchange

coefficients and for describing the porosity distribution, well-designed fine-scale

experimental studies using advanced techniques are essential. The non-invasive

monitoring of macroscopic flow pattern provides the physical mirror for validation of

large-scale flow modeling. Moreover, in the beginning of model development,

experiments even using conventional techniques can still offer useful evidence for

justifying the importance of each term in the model equations. The motivation for the

experimental study presented in this Chapter is to obtain some experimental evidence of

the effects of particle wetting and inflow-operating mode on the liquid distribution in

packed beds.

The indirect flow visualization techniques such as radioactive computer-

tomography (CT), magnetic resonance imaging (MRI) and electric capacity tomography

(ECT) have shown the capabilities of obtaining the spatial distribution of multiphase

flows at certain resolution (Lutran et al., 1991; Kantzas, 1994; Toye et al., 1997; Chaouki

et al., 1997; Sederman et al., 1997; Reinecke et al., 1998). Nevertheless, the direct flow

visualization, such as digital imaging technique, at some cases, are valuable for studying

the parameter dependence and monitoring the course of flow development. By zooming-

in and –out the region of interest (ROI), one can obtain the information at different scale.

In this Chapter, we present some experimental observations of liquid distribution

in a bench-scale pseudo two-dimensional (2-D) rectangular packed-bed and in a pilot-

scale three-dimensional (3-D) cylindrical packed-column. The flow modeling results

based on the same dimensions of these two packed beds are given in Chapters 4 and 6.

A Charge-Couple-Device (CCD) video camera was used to visualize the liquid

texture in the 2-D transparent packed bed at both bed scale and particle scale by simply

12

zooming-in and –out the ROI. To track the development of the finger-type liquid texture

after introducing the liquid into the bed, both the bed scale and particle scale images were

recorded during the most of liquid flow development. The particle prewetting effects

were confirmed in both 2-D and 3-D packed beds. The following issues have been

targeted:

In the 2-D bench-scale rectangular packed-bed, we focus on

• Particle prewetting effect at particle and bed scales on liquid flow distribution

• Liquid texture development at trickling flow condition

• Causes of liquid filament formation

In the 3-D pilot-scale cylindrical packed-bed, we focus on

• Particle prewetting effect at particle and bed scales on liquid flow distribution

• Liquid distributor effect on the bed scale liquid distribution

• Unsteady state liquid feed on the bed scale liquid distribution

2.2 Experiment I: 2-D Liquid Flow Imaging

2.2.1 2-D Packed Bed and CCD Setup A 2-D rectangular packed-bed was made of Plexiglas with a height of 30 cm, a

width of 7.2 cm and a thickness of 1.25 cm as shown in Figure 2-1a. The schematic

diagram of the experimental setup for visualizing the liquid flow using a computer-based

CCD image technique is depicted as Figure 2-1b. The packing consists of glass beads of

3mm in diameter. The packing height of the bed is about 27 cm. This setup can be

operated with gas and liquid co-current flows and with different liquid distributors (e.g.,

single-point liquid inlet and multi-point liquid inlets). The gas feed is uniform during all

experimental runs. Working fluids are air and colored water (in black) at room

temperature (~25 °C). Pressure drops with and without including the collector plate, are

13

measured by manometers. The video imaging was taken during the experiment running

from the front side and the rear side of the bed, and then was processed after that.

2.2.2 Imaging and Processing In general, the computer-based CCD video imaging system consists of a computer

with a plug-in image acquisition board (e.g., DT-3851 from Data Translation Co.; IMAQ

PCI-1408 from National Instruments Co.) and a CCD video camera. At times, different

lighting apparatus may be necessary to condition the image for easier image processing

or to illuminate the scene under low-level light conditions. Technically, one may think of

lighting as analogous to signal conditioning. If the scene is properly lighted (conditioned)

then the image is easier to process. The image acquisition board uses a high-speed analog

to digital converter to digitize the incoming video signal. With the emergence of

multimedia, image acquisition hardware has become less expensive and more powerful.

Application software, which may be a graphical or text-based language, controls the

image board as well as processes and displays the incoming video.

The system applied in this study includes a Sony CCD TRP-279 video camera

connected to the DT-3851 image acquisition hardware, an adjustable lighting

background, and a high-speed Pentium-II computer installed with Global Lab image

processing software. This setup allows showing the acquisition of full live liquid flow

video during single experiment run.

14

2D Rectangular Bed I2D Rectangular Bed I

72 mm

Liquid

Gas

Liquid out

Gas vent

Pres

sure

dro

p I

Pres

sure

dro

p I

Pres

sure

dro

p II

Pres

sure

dro

p II

LiquidLiquidPoint SourcePoint Source

Pack

ing

with

3 m

m sp

here

sPa

ckin

g w

ith 3

mm

sphe

res

280

- 300

mm

280

- 300

mm

Gas - LiquidGas - LiquidSeparatorSeparator

Figure 1. 2D packed-bed I

LiquidLiquidUniform InletUniform Inlet

Liquid Gas

2.2.3 Liquid Flow Imaging A set of experiments was conducted at different liquid superficial mass velocities,

in the range of 0.5 to 11.0 kg/m2/s. To confirm the particle prewetting effect on liquid

texture, the experiments were performed in nonprewetted bed and prewetted bed at the

same flow conditions. The particle scale flow images were obtained by zooming in the

region of interest (ROI). This allows us to see how the particle scale liquid flow pattern

varies with time and with the superficial liquid feed velocity.

2.2.4 Experimental Results and Discussion 2.2.4.1 Non-prewetted bed (dry bed)

The experiments were run in non-prewetted beds at the liquid mass velocity range

from 0.74 to 11.12 kg/m2/s. Figure 2-2 shows the effect of liquid superficial velocity on

liquid channel ling (rivulet) flow as single-point liquid inlet was used. Clearly, the liquid

2D Bed with two phase flow inlet and outlet

CCD Camera

Monito Compute

Background

Figure 2-1a Schematic diagram of 2-D rectangular packed bed.

Figure 2-1b Experimental setup using CCD video camera imaging technique.

15

from the distributor follows the previously established flow path without making any new

rivulet while the superficial liquid velocity increases. The radius of the liquid channel or

rivulet (i.e., filament), dsp, increases in the increase of liquid superficial mass velocity, L.

The relationship between L and dsp, however, does not follow the ‘square’ rule (i.e., L ∝

dsp2). When liquid superficial velocity increases from 0.74 to 3.52 kg/m2/s as shown in

Figure 2-2, the liquid saturation of the channel gradually increases. When the liquid

superficial velocity is doubled to 7.04 kg/m2/s, an apparent spreading of the liquid

filament takes place since the center of the filament has already saturated the voidage

space. If one looks at the specific region by zooming on ROI in Figure 2-2, as one can see

from Figure 2-3, the liquid easily occupies the interstitial voidage without radially

spreading in the non-prewetted bed. A similar liquid flow pattern was reported in

Ravindra et al. (1997a) for non-prewetted beds as given in Figure 2-4, except for one

difference. In Figure 2-4 the radius of the liquid rivulet increases along the liquid path

through the packing from the top to the bottom. However, in our experiments, the radius

of the liquid rivulet decreases along the liquid path downwards in the vertical direction.

Such a difference in the rivulet paths could be caused by the different particle sizes [1.6

mm in Ravindra et al (1997a); 3 mm in this work], or perhaps by the different surface

tension of the liquid due to the different color additives used [organic material in

Ravindra et al. (1997a); the black inorganic color material in this work], and by the

different diameters of the liquid inlet tubes. By setting the ROI at the top layer and the

bottom layer, it has been observed that the relatively large radius rivulet at the top of the

bed is mainly caused by the liquid inlet jet, as shown in Figure 2-5(a). At high mean

irrigation rates the packing immediately below the point inlet is not only completely

filled by the liquid, but, part of the liquid actually can not penetrate into the void

available below the mouth of the nozzle and spreads radially over the top surface of the

bed. The narrow jet thus effectively transforms into a disc whose radius depends on the

mean irrigation rate and the geometry of the packing. A similar experimental observation

has also been reported and analyzed by Stanek (1977) in which an attempt has been made

to calculate the radius of the disc distributor obtained from the central jet by using the

modified Ergun equation. Once this disc type of initial liquid distribution is formed, the

16

liquid channel keeps flowing through the bed, which is not exactly straight except at very

high liquid irrigation rate. The liquid saturation at bottom of the bed (see Figure 2-5b) is

obviously higher than that at the top of the bed (see Figure 2-5a) due to the surface

tension effect. The radial liquid spread decreases along the flow path from the top to the

bottom and the liquid droplet or channel becomes more filament type. Different trends in

liquid rivulet path observed in Ravindra et al (1997a) as shown in Figure 2-4 could be

due to the small particle size (1.6 mm). The effect of liquid superficial mass velocity on

the radius of the liquid rivulet is also clear: the higher the liquid irrigation rate is, the

larger the rivulet radius is as seen in Figure 2-2.

2.2.4.2 Prewetted bed (wet bed)

Before these experiments were started, the bed was prewetted by flooding it with

clear (non-colored) water. The bed was then allowed to drain until no liquid dropped out.

Figure 2-6 shows the steady state liquid distribution in the 2-D prewetted bed at different

liquid superficial mass velocities with single liquid inlet. Liquid textures become

complicated, and more liquid spreading and more particle wetting is observed. Figure 2-7

shows how the liquid texture is developed after starting the liquid irrigation. Apparently,

while the liquid follows the established paths, the new liquid paths are also formed, and

eventually, a tree-type steady state flow textures are generated after a certain time period.

2.2.4.3 Comparison of liquid flow in non-prewetted and prewetted beds

The significant difference in liquid flow texture is clearly shown in Figure 2-8.

One can further examine the intensity profiles of the two images at a specific axial

position, 6 cm away from the top (see Figures 2-9 and 2-10). Clearly, more liquid

spreading occurs in the prewetted bed whereas liquid rivulet flow is the dominant flow

pattern. The impact of such a difference in liquid textures on the performance of trickle-

bed reactor has been experimentally demonstrated by Ravindra et al (1997b) through the

oxidation of sulfur dioxide with the active carbon particles as catalyst. It was found that

the reaction took a long time to reach the steady state in the non-prewetted bed, and the

17

global reaction rate was lower than that in the prewetted bed at the same liquid superficial

mass velocity.

So far, the presented experimental work is limited to the bench-scale 2-D

rectangular bed at steady state flow condition. It is believed that the liquid flow

distribution in a 3-D cylindrical column is different from those obtained in a 2-D

rectangular bed. As one can see from an image of particle packing in the 2-D bed (see

Figure 2-11), by counting the particles, it was found that there is no significant porosity

difference between the central region and the wall regions because most of the particle

confinement arises due to the front and rear walls of the bed. In other words, such 2-D

packed bed can be a representation of the packing zone with relatively uniform porosity

at a scale of two or three particle diameters. Hence, the liquid flow distribution observed

in such a 2D bed presents the flow situation inside large scale packed beds. To examine

the similar parameter effects and unsteady state operation in a 3-D column, we conducted

flow experiments using exit flow measurement in a pilot scale cylindrical column packed

with the same size of glass beads.

18

L = 0.74 kg/m2/s L = 1.48 kg/m2/s L = 3.52 kg/m2/s L = 7.04 kg/m2/s

(ROI Size: 10cm × 6 cm)

Figures 2-2 Effect of liquid superficial mass velocity on liquid rivulet flow from single-point inlet in a non-prewetted packed

bed: the radius of the liquid rivulet increases with the liquid superficial mass velocity.

18

19

L = 0.74 kg/m2/s L = 3.52 kg/m2/s L = 7.04 kg/m2/s

(a) (b) (c)

Figure 2-3. Effect of liquid irrigation rates on the local radial spreading of the liquid

rivulet at a point source inlet in the non-prewetted bed (ROI Size: 3 cm × 2 cm).

H = 0H = 0 H = 0H = 0

H = 12H = 12 H = 12H = 12

L = 1 kg/m2.s; G = 0.05 kg/m2.s

Figure 2-4. Cross-sectional liquid distribution in a 3D-rectanglar non-prewetted bed of

glass beads (dp = 1.6 mm). [From Ravindra et al (1997a)]. Upper left- at the top layer

with single-point liquid inlet; lower left- at the layer 12 cm far from the top with single-

point liquid inlet; upper right- at the top layer with single-line liquid inlet; lower right- at

the layer 12 cm far from the top with single-line liquid inlet.

20

a. Top region b. Bottom region

Figure 2-5. Local liquid distribution at (a) The top region and (b) the bottom region at a

mass superficial velocity of 7.04 kg/m2/s in a non-prewetted bed.

L= 1.48 3.52 7.04 (kg/m2/s)

Figure 2-6. The steady state liquid distribution in a prewetted bed at different liquid superficial mass velocities.

21

t = 5 s t = 11 s t = 25 s

Figures 2-7. The development of finger-type liquid flow in a prewetted bed at a

superficial mass velocity of 0.74 kg/m2/s (t: starting time, second).

22

(c)

Figure 2-8. Liquid flow distribution in (a) non-prewetted bed and (b) prewetted bed with

single-point liquid inlet without gas flow. Part (c) shows the image intensity profiles at

specific vertical position (z = 6 cm from the top) in cases (a) and (b).

0

50

100

150

200

250

0 3 6 9 12 15 18 21 24X (dp)

Inte

nsity

of i

mag

e

non-prewetted bedprewetted bed

Axial position: 6 cm down from topdp = 0.3 cm

a. Non-prewetted bed b. prewetted bed

23

0.00.51.01.52.02.53.03.54.0

0.0 2.0 4.0 6.0 8.0Time (hr)

Ra

x 10

8 (gm

ol/c

m3 .s

)

prewetted bed

nonprewettedb d

Figure 2-9. Transient behavior of reaction rates in non-prewetted and prewetted beds for

oxidation of SO2 with the active carbon particles as catalyst. [Data are extracted from

Ravindra et al. (1997b) at T = 25 °C; P = 1 atm].

1.0

1.5

2.0

2.5

3.0

3.5

0.0 2.0 4.0 6.0 8.0

L (kg/m2.s)

Ra

x 10

8 (gm

ol/s

.cm

3 ) prewetted bed

nonprewetted bed

Figure 2-10. Dependence of the global reaction rate on liquid velocity in non-prewetted

and prewetted beds (uniform liquid inlet) for oxidation of SO2 with the active carbon

particles as catalyst. [Data are extracted from Ravindra et al. (1997b) at T 25 °C; P = 1

atm].

24

wall core

Figure 2-11 Packing image taken from the front of the 2-D rectangular packed bed.

2.3 Experiment II: Exit Flow Measurement in 3D Bed

2.3.1 Experiment Objectives The particle scale and bed scale liquid textures have been visualized and

presented in Section 2.2 in which the particle prewetting effect on liquid distribution has

been clearly demonstrated. These liquid flow observations, however, have been limited to

two dimensional bench scale packed beds under steady state liquid feed condition and

with single point liquid inlet because of the technique limitation (e.g. using colored

liquid). Obviously, we need to confirm those liquid distribution phenomena such as

particle wetting effect in a pilot scale cylindrical column. Moreover, we would like to

gain some information on liquid distribution under unsteady state liquid feed such as

periodic liquid feeding. Thus, the objectives for the experiments based on exit flow

measurement are as follows:

(i) Verify the particle prewetting effect on liquid distribution in a pilot scale cylindrical

column

(ii) Explore the possibility of using exit flow measurement to detect the difference in

liquid distribution at steady state liquid feed and at periodic liquid feed

25

(iii) In periodic liquid feed, examine the effect of liquid cycle split ratio (i.e., the ratio of

liquid ON time and liquid OFF time) on the radial liquid velocity profiles

2.3.2 3D Column Setup and Exit Flow Measurement The schematic diagram of the 3D column setup is shown in Figure 2-12. The pilot

scale column was made of Plexiglas with an inner diameter of 5 5/8 inch (14.3 cm) and a

height of 6ft (2 m). The same size of glass beads as used in the 2D bed (i.e., 3 mm) was

employed as packing. The total height of the packing was varied in the range of 2 ft to 6

ft. The fluid media used were air and water at room temperature (~25 °C). Two types of

liquid distributors were used: uniform and a point source. There are 182 holes with a

mean diameter of 0.6 cm on the uniform distributor (about 33 % of the area of the

distributor is open). Gas enters the reactor through the separate tubes located high than

the level of the liquid inlet. In addition, attention has been paid to the proper design of

gas distributors to avoid maldistribution problem. A uniform liquid inlet distribution was

obtained when the liquid superficial velocity is beyond 1.0 kg/m2/s in the presence of gas

flow. The performance of the liquid distributor was checked using the same procedure

reported in Kouri and Sohlo (1987, 1996): by locating the distributor just above the liquid

collector and measuring the liquid distribution of the distributor by the liquid collecting

annulii. At a low liquid inlet irrigation rate such as 0.5 kg/m2/s, it was found that the

uniform liquid initial distribution could not be obtained due to the wettability of Plexglass

materials. Higher liquid irrigation rate and/or gas flow can eliminate the above problem

to get a uniform initial liquid inlet.

For investigation of periodic operation, a solenoid valve is fixed in close

proximity to the reactor inlet and is connected to the flexible timer that regulates the

on/off operation of the valve within an adjusted cycle. The timer handles a wide range of

on/off cycles, which can range from 0.1 to 2000 seconds. For the off-time period we used

a bypass to let the water go through the pump back to the feed tank. In the present study,

three types of time split of the ON/OFF setting are tested such as ON/OFF (defined as

SR) = 20 sec/ 40 sec.= 0.5; 30 sec./30sec.= 1.0; 40 sec./20 sec. = 2.0.

26

At the bottom of the column is a collector connected to 25 flexible tubes. The

collector has five rings to separate the water in the radial direction and also walls to

separate the liquid in these rings to get the liquid distribution in the azimuthal direction

(see Figure 2-13), therefore, the liquid distribution at the bottom of the column was

measured at each set of conditions by collecting the liquid flow in 25 different sections of

the cross-sectional area. Most of the radial liquid distribution data is based on the six

annular sections. As recommended by Kouri et al (1996), the liquid radial distribution

data reported in this Chapter are expressed as dimensionless liquid velocity, V/Vav,

against the square of the dimensionless radius, (r/R)2, as the square of the dimensionless

radius is proportional to the area of the sampling section under consideration. The axial

pressure drop data along the column were measured for a couple of experiments using a

water manometer as shown in Figure 2-12.

2.3.3 Experimental Results and Discussion Since there were significant differences in liquid textures in the prewetted and

non-prewetted 2D packed beds with single point liquid inlet were observed. We

performed similar experiments in the 3D column with non-prewetted particles and

prewetted particles, and then determined the liquid distribution from 25 individual tubes

located at the bottom of the cylindrical column. As shown in Figures 2-14 and 2-15, the

same conclusion about the particle prewetting effect in 3D packed column can be drawn

as established for 2D column. The liquid paths in non-prewetted bed are relatively stable,

even when increasing the liquid superficial mass velocity from 0.5 kg/m2/s to 10

kg/m2/s, as shown in Figure 2-14. More uniform liquid distribution is found in the

prewetted bed, which seems independent of liquid flow rate (see Figure 2-15).

Figures 2-16 and 2-17 show the measured radial liquid velocity profiles at

different cycle split ratios (SR = 0.5, 1.0 and 2.0) with a uniform liquid inlet (Fig. 2-16)

and with a point liquid inlet (Fig 2-17). In the uniform liquid inlet case, the small cycle

split ratio (SR = 20-on/40-off = 0.5) yields more uniform radial liquid flow distribution in

the time-averaged sense. For a given time-averaged liquid superficial velocity, the

27

smaller SR value means higher superficial velocity during the liquid ON period. More

liquid is driven by the higher momentum of liquid flow to the radial direction, and causes

the maximum velocity position to move in the direction of the wall (see Figure 2-16). For

the point source liquid inlet, there is no conclusive effect of SR value on liquid radial

spreading as shown in Figure 2-17.

Figure 2-12 Schematic diagram of the experimental setup for a 3D column with exit flow

measurement and periodic liquid feed controller.

air Tape

1

2

3

4

5

6 Manomet

Timer

Collector and Measuring

Tank

A B

Pum

Packing: 3 mm Glass

28

Figure 2-13 Liquid collector with 25 individual tubes located at the bottom of the packed

bed.

0.000.501.001.502.002.503.003.504.00

1 3 5 7 9 11 13 15 17 19 21 23 25

Tube #

V / V

av

L=0.5 kg/m2.sL=1.0 kg/m2.sL=5.0 kg/m2.sL=10 kg/m2.s

Figure 2-14 Liquid flow measurements in the non-prewetted bed: dimensionless liquid

flow velocity data from 25 individual tubes at different liquid superficial mass velocities

(H = 6 ft, G = 0.0 m/s, uniform liquid inlet).

1

2

4

3 5

6 7

89

10

11

12

13 14

15 16

17 18 19

20

21

22 23

24

25

29

0.00

1.00

2.00

3.00

4.00

1 3 5 7 9 11 13 15 17 19 21 23 25

Tube #

V / V

av

L=3.0 kg/m2.sL=5.0 kg/m2.sL=7.0 kg/m2.s

Figure 2-15 Liquid flow measurements in the prewetted bed: dimensionless liquid flow


(H = 6 ft, G = 0.049 m/s, uniform liquid inlet).

30

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1(r/R)2

V / V

av

40 (ON)/20(OFF)

30/30

20/40

Figure 2-16. Effect of time split in On/Off periodic mode on liquid flow radial profiles

with uniform liquid inlet (H=2.0 ft, G=0.049 m/s, L = 1.5 kg/m2/s) with uniform liquid

inlet.

31

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1(r/R)2

V / V

av

40(ON)/20(OFF)

30/30

20/40

Figure 2-17. Effect of time split in On/Off periodic mode on liquid radial profiles with

point source liquid inlet (H = 2.0 ft, G = 0.049 m/s, L = 1.0 kg/m2/s).

The comparison of the radial liquid flow exit profiles at steady state and periodic

operations, reveals no significant difference of the model of operation. However, such

exit flow measurement cannot give the flow distribution information inside the column

and can only provide the time-averaged radial flow profile at exit. Moreover, many

factors may contribute to such measurement, in particular, the arrangement of collecting

tubes, because unequal flow resistances in individual tube can results in redistribution of

32

liquid flow, and change the liquid flow profiles completely. Hence, the experimental data

obtained by exit flow measurement shown in this Section are inconclusive except for the

effect of particle prewetting.

2.4 Conclusions The first major goal of this study is to model multiphase flow distribution in

packed bed reactors. The liquid flow observations presented in this Chapter indeed

provide helpful information for the formulation of flow equations, which will be utilized

in Chapters 3, 4 and 5. This information are summarized as follows:

(i) The flow distribution experiments by direct liquid flow visualization in 2-D bed and

by indirect exit-flow measurement in 3-D column have demonstrated the significant

particle external wetting effect on the formation of liquid texture. The particle scale

liquid textures indicate the apparent influence of capillary pressure on the liquid

spreading. Proper implementation of the capillary force in the flow model equations

is important.

(ii) The liquid flow textures in the prewetted bed and non-prewetted bed are apparently

different. Filament flow is dominant in the non-prewetted bed whereas the film liquid

texture exists in the well-prewetted packed bed.

(iii)In general, inflow distributors play an important role in flow distribution. The proper

design of the liquid jets is essential in determining the liquid distribution at the top

layer of the packing.

(iv) Exit flow measurement is not a recommended way to compare the liquid flow

distribution at steady-state liquid feed and in periodic flow mode. For the liquid

ON/OFF mode with uniform liquid inlet, the cycle split ration (SR) does have an

impact on the time-averaged radial liquid flow profiles. There is no effect of the SR

values on the time-averaged radial liquid flow profiles when the point source liquid

inlet is used. The numerical flow simulation definitely can contribute to the

understanding of dynamic flow pattern at quantitative way.

33

Chapter 3

Discrete Cell Model Approach

Revisited: I. Single Phase Flow Modeling 3.1 Introduction

Gas flow through packed beds is commonly encountered in industrial applications

involving mass or/and heat transfer both with and without chemical reaction. Complete

understanding of the gas flow distribution in packed beds is of considerable practical

importance due to its significant effect on transport and reaction rates. It was shown that

reaction and radial heat transfer can only be modeled correctly if the radial

nonuniformities of the bed structure are properly accounted for (Lerou and Froment,

1977; Delmas and Froment, 1988; Daszkowski and Eigenberger, 1992). Therefore, over

the years, a number of studies investigated the radial variation of the axial gas velocity in

packed beds. This included axial velocity measurement at various radial positions,

measurement of radial porosity profiles (Morales et al., 1951; Schwartz and Smith, 1953;

Benenati and Brosilow, 1962; Lerou and Froment, 1977; McGreavy et al., 1986;

Stephenson and Stewart, 1986; Volkov et al., 1986; Peurrung et al., 1995; Bey and

Eigenberger, 1997), and modeling of the radial variation of axial velocity (Schwartz and

Smith, 1953; Stanek and Szekely, 1972; Cohen and Metzner, 1981; Johnson and Kapner,

1990; Ziolkowska and Ziolkowski, 1993; Cheng and Yuan, 1997; Bey and Eigenberger,

1997; Subagyo et al., 1998). It was noted, however, that in industrial packed beds, some

34

nonuniformities either due to the presence of internal structures (Bernier and Vortmeyer,

1987a, 1987b), or due to irregular gas inlet design (Szekely and Poveromo, 1975) could

cause the flow not to be one dimensional and the gas velocity to vary in both radial and

axial direction. Such two dimensional flow is called “non-parallel” flow in the literature

(Stanek, 1994). Hence, for industrial applications of packed beds, it is certainly important

to be able to effectively model the non-parallel gas flow. In general, three types of

mathematical models have been developed for the treatment of non-parallel gas flow in

packed beds. They are briefly summarized below.

It should be noted that our goal here is simulation and prediction of single phase

flow on a bed scale, i.e. the capture of the gas velocity profile on a scale of a couple of

particles, not on the scale of the individual tortuous passages in the bed. We are not

attempting to model the flow on a particle scale but to find the means for effectively

computing the bed scale flow distribution provided the voidage distribution is known.

3.2 Non-Parallel Gas Flow Models 3.2.1 Vectorized Ergun Equation Model

This model is based on the assumption that a packed bed can be treated as a

continuum. Therefore, it is assumed that the Ergun equation can be used in the

differential, vector form as shown by Equation (3-1).

( )VffVP 21 +=∇− (3-1)

The intent is to utilize the empirical Ergun equation, which is shown to hold well for

overall pressure drop in macroscopic beds with unidirectional flow, for an infinitesimal

length of the bed and apply it in the direction of flow. For an incompressible fluid,

applying the curl operator (∇× ) to Equation (3-1) yields Equation (3-2), which is a vector

equation containing the velocity vector V as the only dependent variable. The

components of the velocity vector also have to satisfy the continuity Equation (3-3).

( )[ ] 0VfflnVV 21 =+∇×−×∇− (3-2)

0=⋅∇ V (3-3)

35

The solution for the velocity components can be obtained by solving Equations (3-2) and

(3-3). A number of investigators (Stanek and Szekely, 1972, 1973, 1974; Szekely and

Poverromo, 1975; Beminger and Vortmeyer, 1987a) utilized this method to model two-

and three-dimensional flow in packed beds.

3.2.2 Equations of Motion Model

In principle, the mass conservation (continuity equation) and momentum balance

(Navier-Stokes equations) can be solved for the flowing phase provided the solid

boundaries are precisely specified. Such direct numerical simulation (DNS), however, is

beyond reach at present for large industrial scale packed beds (Joseph, 1998). By

employing the effective viscosity as an adjusting factor, Ziolkowska and Ziolkowski

(1993) and Bey and Eigenberger (1997) tried to develop a mathematical model for the

interstitial velocity distribution based on the Navier-Stokes equations, but porosity was

only considered as a function of radial position in such models. To take into account the

complex fluid-particle interactions and the multi-dimensional variation of bed voidage in

packed beds, a k-fluid (interpenetrating fluid) model provides a viable alternative

(Johnson et al., 1997). By ensemble averaging, the continuity and momentum equations

for the flowing phase are formulated in a multi-dimensional form and the interphase

interaction is described via an appropriate drag correlation. The resulting equations can

be solved via packaged computational fluid dynamics codes such as CFDLIB (Kashiwa

et al., 1994).

3.2.3 Discrete Cell Model (DCM)

This two-dimensional model is based on the concept that the bed may be

represented by a number of interconnected discrete cells (Holub, 1990), with the bed

porosity allowed to vary in two directions from cell to cell. The fluid flow is assumed to

be governed by the minimum rate of total energy dissipation in the packed bed (i.e. flow

follows the path of the least resistance). Ergun equation is assumed to be applicable to

36

each cell. Therefore, the solution for velocity at each cell interface can be achieved by

solving the non-linear multi-variable minimization problem.

Although the vectorized Ergun equation model (Stanek and Szekely, 1972) has

provided a good description for non-parallel gas flow (1D axial flow), it is still difficult to

capture the nonuniformity of flow at the cell scale (few particles). It is also cumbersome

to model the flow in beds with an internal random porosity profile because of the

difficulties in assigning discrete porosity values to points in a continuum. Another

difficulty of this model is the inability to set a no-slip boundary conditions at the walls.

The validity of the vectorized form of the Ergun equation was demonstrated only by

comparison of the predicted exit velocity profile with experimental measurements. This

kind of comparison is only reasonable for the parallel flow system that exhibits no effect

of the packing support plate on the flow. Because of the above considerations, the

discrete cell model was formulated as an alternative that may offer advantages in solving

these problems. For example, the cell model is capable of capturing the non-parallel flow

on a cell scale (few particles) due to the character of the cell model. The appropriate

voidage can be assigned easily for each cell and the no-slip wall condition can be

simulated by the extra cell method (the detail discussion will be given later). It is

assumed that the Ergun equation is applicable at the cell scale. This assumption is

reasonable because the original Ergun equation was derived from the experimental

measurements in small laboratory-scale packed beds (Ergun, 1952). The cell size has to

be small compared to the bed scale (i. e., bed diameter), to obtain the desired resolution

of the bed properties and flow distribution, but large compared to the particle scale (i.e.,

particle diameter) in order to apply the Ergun equation (1952) to each discrete cell. The

appropriate cell dimensions that satisfy these criteria were discussed by Vortmeyer and

Winter (1984), who concluded that homogeneous models of packed bed heat transfer

failed in beds with a tube to particle diameter ratio less than three. While this conclusion

was not reached for the exact situation considered here, a minimum linear dimension of

about three particle diameters for each cell can be considered appropriate (Holub, 1990).

37

The second assumption of DCM is that the flow is governed by the minimum rate

of total energy dissipation in the bed. The theoretical justification for this assumption has

been provided only for linear systems, in which the fluxes and driving forces have a

linear relationship, and rests on the principle of minimization of entropy production rate

(Jaynes, 1980). For non-linear systems, examples can be constructed for which the

'principle of energy minimization' does not hold and, hence, that demonstrates that it is

not a general 'principle' at all (Jaynes, 1980). Nevertheless, this energy minimization

approach was reported to be valid for some classes of nonlinear systems such as particle

flow in circulating fluidized beds (Ishii et al., 1989; Li et al., 1988, 1990). Hence, for any

specific nonlinear system one needs to conduct a detailed verification study before

considering 'energy minimization' as the governing principle for flow distribution (Hyre

and Glicksman, 1997). Regarding single phase flow distribution in packed beds, it is

necessary to revisit DCM by examining how well can this 'principle' be used to describe

the flow. This can be done by comparing the results of the DCM to either accepted

solutions of the ensemble-averaged momentum and mass conservation equations or to

reliable experimental data. Unfortunately, there is very few experimental data for the

velocity profiles inside packed beds available in the literature due to the limitations on the

non-intrusive velocity measuring techniques (McGreavy et al., 1986; Stephenson and

Steward, 1986; Peurrung et al., 1995). Thus, the objectives of this study are (i) to perform

a series of numerical comparison studies of DCM predictions and CFD two-fluid model

simulations, (ii) to compare the numerical results of DCM/CFD with the limited

experimental data available in the literature, and finally, (iii) to reach a conclusion

regarding the applicability of the minimization of energy dissipation concept in modeling

single phase flow distribution in packed beds.

Another motivation for this study is the fact that the concept of minimization of

the rate of energy dissipation was never tested against the solution of the full set of

equations of motion for a non-parallel flow system. Now, we provide such a test for flow

distribution in packed beds. The results should generate a better appreciation of what the

concept of minimization of the total energy dissipation rate can and cannot do.

38

3.3 Discrete Cell Model (DCM) The discrete cell model based on the minimization of energy dissipation rate

presented and discussed here is adapted from the concept originally proposed by Holub

(1990). Although a 2D-model bed is considered here, its extension to 3D axi-symmetric

cases is readily accomplished. The 2D rectangular model bed shown in Figure 3-1 is

divided into a number of cells, each of which is assumed to have uniform porosity within

itself and have two fluid velocity components (Vz and Vx) at each cell-interface. The

porosity can vary from cell to cell. The rate of energy dissipation for each cell can then be

derived from the macroscopic mechanical energy balance and results in Equation (3-4) in

X-Z coordinates for either two dimensional rectangular (2D) or three dimensional axi-

symmetric cylindrical (3D) situation. The differences in Equation (3-4) for 2D

rectangular and 3D axial symmetric cylindrical bed are the expressions for the interface

areas (Si) and the cell volumes (Vc,j).

Figure 3-1. Model packed bed ('2D' rectangular as example) and velocity at each

interface of cell j. (Note that Sx,j equals to Sx+∆x ,j in the '2D' rectangular packed bed).

X

Z

VZ, j

VZ+∆Z, j

VX, j VX+∆X, j

39

The detail derivation of DCM Equations was given in Holub (1990), and rests

on the macroscopic mechanical energy balance. Here we give the main steps of these

derivations. For the jth cell, the rate of energy dissipation in X-Z coordinates can be

expressed by (Eq. 3-4).

−

Φ+

+

⋅

∑

=inj

iiii s

sVs

sVPs

sV

sV

,

4

1

2

21 ρρ

0,

4

1,

2

=

−

Φ+

+

⋅

∑

= jVjoutj

iiiEsVsVPsVVεε

ρεεε

ρ (3-4)

where the superficial velocity (Vj) and the corresponding energy dissipation rate for the

cell (EV,j) are used. Rearrangement of Equation (3-4), by substituting the expression for

the area for each cell interface, yields Equation (3-5).

−+−=

∆+∆+∆+∆+

jSVSVSVSVjVE ZZZZZZ

jj

XXXXXX

j

33

2

33

211

2, εε

ρ

( ) ( ) ZZjZZZZZjZZXXjXXXXXjXX SVPSVPSVPSVP ∆+∆+∆+∆+∆+∆+ −+−+ ,,,,

( ) ( ) ZZjZZZZZjZZXXjXXXXXjXX SVSVSVSV ∆+∆+∆+∆+∆+∆+ Φ−Φ+Φ−Φ+ ,,,,ˆˆˆˆ ρρρρ (3-5)

The difference in potential energy terms ( ∆EP , Eq. 3-6) (shown as the last two terms in

Eq.3-5) can be considered negligible ( ∆E P≈ 0) for gas flow at normal or low pressure

since the gravitational force on the gas is very small.

( ) ( ) ZZjZZZZZjZZXXjXXXXXjXXP SVSVSVSVE ∆+∆+∆+∆+∆+∆+ Φ−Φ+Φ−Φ=∆ ,,,,ˆˆˆˆ ρρρρ ≅ 0 (3-6)

The pressure terms at each cell interface (e.g. PZ and PZ+∆Z), however, can be considered

to be equal to the pressure at the cell center plus the pressure gradient between the center

and the interface multiplied by the appropriate distance. For the Z direction, as an

example, the desired relationships can be written as follows.

2Z

ZP

czPZPZZ

∆

∆∆−+=

∆+

(3-7)

40

2Z

ZP

czPZZPZZ

∆

∆∆−−=∆+

∆+

(3-8)

Rearranging Equation (3-5), by substituting Equations (3-6), (3-7) and (3-8), gives:

−+−=

∆+∆+∆+∆+ jSVSVjSVSVE ZZZZZZ

jXXXXXX

jjV

332

332,

112 εερ

( ) ( )ZZjZZZjZCZXXjXXXjXCX SVSVPSVSVP ∆+∆+∆+∆+ −+−+ ,,,,

∆∆−+

∆∆−+

∆∆−+

∆∆−+ ∆+

∆+∆+

∆+jZZ

ZZjZ

ZjXX

XXjX

X

j VZPV

ZPV

XPV

XPVol

,,,,2 (3-9)

For each cell, we can write Equation (3-10) based on the mass balance as follows

( ) ( ) 0,,,, =−+− ∆+∆+∆+∆+ ZZjZZZjZXXjXXXjX SVSVSVSV (3-10)

Since the magnitudes of PCZ and PCX have to be the same at the central point of the cell j,

the substitution of the mass balance Equation (3-10) into Equation (3-9) eliminates the

central pressure term. To completely eliminate the pressure terms from Equation (3-9),

the body force terms, represented by the pressure gradient, can be replaced by an

appropriate drag force model which relates pressure drop to the local superficial velocity.

In this work, a specially abbreviated form of the Ergun equation (Ergun, 1952) for each

coordinate direction (X and Z) will be used to simplify the equations. For example, for

the Z direction, we have

j,Zj,Zj,2j,Zj,1Z

VVfVfZP1 +=

∆∆

ρ− (3-11)

where the pressure loss per unit cell is caused by simultaneous viscous and kinetic energy

losses. The resulting expression for calculating the energy dissipation rate per unit cell

can be obtained, as shown by Equation (3-12), and the total energy dissipation rate for the

entire bed is then obtained by the summation of Equation (3-12) over all the cells.

−ερ+−

ερ=

∆+∆+∆+∆+ jSVSVjSVSV21

j,VE ZZ3

ZZZ3Z2

jXX

3XXX

3X2

j

( ) jCjXXXXjjXXjjXjXjjXj VVVfVfVVfVf ,,2

,22

,,1,2

,,22

,,1 ∆+∆+∆+ ++++

41

( ) j,Cj,ZZj,ZZj,j,ZZj,j,Zj,Zj,j,Zj, VVVfVfVVfVf ∆+∆+∆+ ++++ 22

21

22

21 (3-12)

In Equation (3-12) f 1,j and f 2,j are Ergun coefficients (Ergun, 1952) defined as follows

( )( ) 32

2

1

1150

jP

jj, d

fεφε−µ

= (3-13)

( )( ) 32

1751

jP

jj, d

.f

εφε−ρ

= (3-14)

In this study, we use the 'universal values' (E1=150, E2=1.75) to calculate f1,f and

f2,f as done by most other investigators (Vortmeyer and Schuster, 1983; Stanek, 1994;

Bey and Eigenberger, 1997, etc.). Although E1 and E2 values can vary from macroscopic

bed to bed due different structures of the packing in the bed (MacDonald, et al., 1979),

this effect can be accounted for by the assignment of a non-uniform porosity distribution

instead of using the average porosity value for the bed.

The complete model for determining the gas flow distribution in the bed requires

the minimization of the rate of total energy dissipated with the cell velocities as variables.

It is a nonlinear, multivariable minimization problem (Eq.3-15) subject to mass balance

constraint for each cell (Eq.3-16, based on constant fluid density assumption), and

constraints for bed boundaries. The setting of cell boundary conditions reflect the internal

structural nonuniformities and operating conditions. In other words, this model can

predict the gas flow distributions in packed beds with various operating conditions (i.e.

side gas feed) and with different internal structural nonuniformities.

[ ] [ ]∑=

=N

1jj,Vbed,V EMinEMin (3-15)

( ) ( ) 0=−+− ∆+∆+∆+∆+ jZZZZZZjXXXXXX SVSVSVSV (3-16)

The subroutine DN0ONF from the International Mathematical Statistics Library (IMSL)

was used to solve this constrained nonlinear minimization problem and obtain the fluid

velocity components Vx and Vz for each cell in the bed.

42

3.4 CFDLIB Formulation CFDLIB, a library of multiphase flow codes developed by Los Alamos National

Laboratory (Kashiwa et al., 1994), has been used to obtain the results for comparison

with the DCM predictions. The solution algorithm is a cell-centered finite-volume

method applied to the time-dependent conservation equations (Kashiwa et al., 1994). The

governing equations that serve as the basis for the CFDLIB codes are:

Equation of continuity:

>=<∇+ kkkkk ut

αρρ∂

∂ρ!. (3-17)

The terms on the left hand side of Equation (3-17) constitute the rate of change in mass of

phase k at a given point, and the term on the right hand side is the source term due to

conversion of mass from one phase to the other. In present study this term is equal to zero

since no phase change, reaction or mass transfer is considered in this cold flow modeling.

Equation of momentum:

=⋅∇+ kkkkk uu

tu

ρ∂

∂ρ (rate of change in k th phase momentum)

><+ ku αρ !00 (net mass exchange source of k)

><∇⋅− kkk uu ''0ρα (multiphase Reynolds stress)

pk ∇−θ (accln. by the equilibration pressure)

><∇⋅+ 0τα k (accln. due to average material stress)

)( 0 pp kk −∇− θ ( accln. by nonequilibrium pressure)

gkρ+ (accln. by body force)

>∇⋅−−<+ kIpp ατ ])[( 00 (momentum exchange terms) (3-18)

43

This set of equations is exact with no approximations other than the ensemble

averaging used in the two fluid model approaches (Ishii, 1975). The special case of one

fixed phase (the catalyst bed) has been incorporated in the code for single phase flow

simulation (Kumar, 1995). In Equations (3-17) and (3-18), the mass source term is

considered as zero due to absence of reaction or interphase transport. The important term

is the interphase momentum exchange term, which is modeled by the choice of the

appropriate drag closure. Contribution of Reynolds stress can be ignored for most cases

for flow through packed beds. The detail discussions of this term will be given later. One

of the advantages of CFDLIB is that there are options for specifying user defined drag

forms based on each combination of the phases under consideration. In this study, the

same drag force formulation as used in the Ergun equation is employed for both CFDLIB

(Exchange term in Eq.3-18) and DCM simulations. This is a realistic drag correlation at

the cell scale as mentioned earlier, and it has been used by many other investigators

point-wise in packed beds (Vortmeyer and Schuster, 1983; Stanek, 1994; Song et al.,

1998; etc.). CFDLIB code also allows the choice of velocity and pressure boundary

conditions for inflow, outflow and free slip or no slip at the wall boundaries. To keep the

consistency with the discrete cell approach used in DCM, the spatial discretization of the

model bed is the same in both methods as the cell scale (few particles). Regarding the

dependency of the flow simulation result on the grid size, one will see in Chapter 5, that

the macro-scale velocities simulated by the k-fluid CFD model are grid independent.

The comprehensive discussions of CFD modeling are given in Chapter 5 in

which the detail implementations of bed structure and interaction forces are presented.

3.5 Modeling Results and Discussion

3.5.1 Model Packed Bed

The model bed used for this numerical study is a two dimensional packed bed

with a predetermined pseudo-random porosity distribution as shown in Figure 3-2. The

average porosity of this bed is 0.415, and was obtained experimentally in an identical '2D'

44

rectangular bed with spherical particles of 3 mm diameter (see Chapter 2). The porosity

profiles in the internal region of the bed were generated by a computer program under

certain constraints (Range: 0.360 ~ 0.440; mean: 0.406), which is fairly close to that

obtained by dumping spheres into beds (Tory et al., 1973). A relatively higher porosity of

0.44 was assigned to the wall and the support plate regions based on the typical porosity

profiles reported in the literature (Benenati and Brosilow, 1962; Haughey and Beveridge,

1969). The dimensions of the model bed and of the cells as well as physical properties of

the fluid (gas) are given in Table 3-1. The bed walls are considered to be impermeable in

the normal direction (X direction) and allow free-slip in the parallel direction (Z

direction). In order to implement the no-slip boundary conditions in DCM, the ‘ghost

cell’ method can be used in which an extra column of cells outside the bed can be set and

assigned an extremely low porosity (i.e. less than 0.01). Thus, the effect of bed wall and

no slip boundary condition on gas flow could in principle be considered in this way. It

should be noted that the use of DCM is not limited to spherical particles. It can be applied

to any shape of particles by taking into account the particle shape factor, φ, in Equations

(3-13) and (3-14).

45

Figure 3-2. Porosity distribution of model bed (32 cells x 8 cells): Total average porosity:

0.415; internal region: 0.36~0.44 (random distribution); wall region: 0.44; Two limits

(0.36 and 0.44) correspond to the dense packing and loose packing porosity. When two

obstacle plates are placed in this system, one is located at Z/dp of 66 (at the left side),

another is at Z/dp of 30 (at the right side) as marked in the above figure. The width of the

obstacle plate (i.e. the length that it protrudes into the bed) is half of the width of bed

(4 cells).

Table 3-1 Dimensions of the model bed and physical properties of the fluids in the

simulations.

Dimension Properties

D = 0.072 m; dp = 0.003 m ρG = 1.2 kg/m2; ρL = 1000 kg/m2;

H = 0.288 m µG=1.8 x 10-5 Pa.s; µL=1.0 x 10-3 Pa.s;

3.5.2 Analysis of the Energy Dissipation Equation

As shown in Equation (3-12), there are three terms contributing to the total energy

dissipation rate per unit cell: inertial loss (Ti), viscous loss (Tv), and kinetic energy loss

46

(Tk). The contribution of the gravitational potential term has been ignored for gas flow

due to the low density of the fluid (this term is accounted for when liquid flow is

considered, see Eq 3-5). The expressions for these three terms in cell j are given below.

jSVSVjSVSVT ZZZZZZj

XXXXXXj

j,i

∆+∆+∆+∆+ −

ερ+−

ερ= 33

233

2 22 (3-19)

( ) j,Cj,ZZZZj,j,Zj,Zj,j,XXXXj,j,xj,Xj,j,k VVVfVVfVVfVVfT ∆+∆+∆+∆+ +++= 22

22

22

22 (3-20)

( ) j,Cj,ZZj,j,Zj,j,XXj,j,Xj,j,V VVfVfVfVfT 21

21

21

21 ∆+∆+ +++= (3-21)

-1.00E-040.00E+001.00E-042.00E-043.00E-044.00E-045.00E-046.00E-047.00E-048.00E-049.00E-04

0 48 96 144 192 240 288cell number

Ener

gy d

issi

patio

n ra

te, J

/s

TiTkTv

Figure 3-3. Contribution of each energy dissipation rate term at each cell to the total

energy dissipation rate. V0 = 0.5 m/s (gas flow without internal obstacles); Re’ = 28.5;

sJTn

i /1085.6 5256

1

−

=×−=∑ ; sJT

nk /1037.1 1

256

1

−

=×=∑ ; sJT

nv /1087.6 2

256

1

−

=×=∑ ;

( ) sJTTTn

vki /10056.2 1256

1

−

=×=++∑ (the cell number is counted from the top left of the

bed in the X direction)

47

As derived earlier, the pressure drop term is substituted by Tk,j and Tv,j to

eliminate the pressure term (see Equations 3-9 and 3-11). This is rigorously true only

when inertial terms are zero and no source terms due to interphase transport are present in

the continuity equation. Hence, we still consider the inertial terms in Equation (3-12) so

as to account for flow with abruptly changing direction. The significance of this term is

examined for a low density gas flow (where it is expected to be negligible), a high

density liquid flow, and gas flow with internal obstacles (where it can approach in

magnitudes the other terms). For a non-parallel gas flow test case (Reynolds number, Re’

of 28.5), Figure 3-3 shows the contribution of each energy dissipation rate term to the

total energy dissipation rate. One should note that the Reynolds number (Re') in this

paper is defined on the basis of the input superficial velocity V0 and the inverse of the

specific surface of particles as the length scale (see Notation) which is the same as that in

Stanek (1994). It can be converted to the particle Reynolds number (Rep) used in some

studies by multiplying it with a factor of ( )6 1−εB (~3.51 in this study). It was found that

when no internal obstacles are present and the flow is nearly parallel, the inertial term

(Ti) is negligible compared to the other two terms (Tk and Tv). The viscous term (Tv) is

about one third of the total energy dissipation rate, and the kinetic term (Tk) is two thirds

of the total energy dissipation rate. However, when two obstacle plates are placed in the

above packed bed to create significantly nonparallel flow (see Figure 3-2), their effect on

the total energy dissipation rate per unit cell is significant as shown in Figure 3-4a. The

total energy dissipation rate is almost 50% higher compared to the one without the

internal obstacles. The inertial term (Ti) is still negligible compared to the other two

terms (Tk and Tv) except in the very proximity of the obstacles as shown in Figure 3-4b.

The values of Tk and Tv are scattered, but of the same order. Higher values of Tk are

observed at the obstacle regions as shown in Figure 3-4b. It is clear that internal obstacles

make the gas flow more non-uniform. The possibility of a dominant inertial term was

examined for a case of high density fluid by simulating a saturated liquid flow case. Here,

the kinetic term (Tk) is seen to be dominant in the energy dissipation rate per unit cell at

liquid superficial velocity, U0 of 0.1 m/s (Re' = 47.5). The inertial term (Ti) is not

48

significant even in this case as shown in Figure 3-5. It can be concluded that the inertial

term is not important except in the obstacle region which is in agreement with the

simulations reported in the literature (Choudhary et al., 1976). This also justifies the

substitution of the pressure drop by the Ergun equation terms (Tk and Tv) and elimination

of the pressure term from the equation completely. In general, however, it is still

advisable to include the inertial term in the formulation of the total energy dissipation per

unit cell to account for those highly non-uniform flow situations in which the inertial

terms could be important in affecting the nature of flow (Choudhary et al., 1976).

49

-1.00E-03

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

7.00E-03

0 48 96 144 192 240 288

Cell number

Ener

gy d

issi

patio

n ra

te, J

/sTiTkTv

Figure 3-4a. Contribution of each energy dissipation rate term at each cell to the total energy dissipation rate. V0 = 0.5 m/s (gas flow with two internal obstacles at Z/dp = 30,

66); Re' =28.5. sJTn

i /1022.1 4256

1

−

=×−=∑ ; sJT

nk /1015.2 1

256

1

−

=×=∑ ;

sJTn

v /10905.0 1256

1

−

=×=∑ ; ( ) sJTTT

nvki /1006.3 1

256

1

−

=×=++∑ (The dashed line region will

be re-illustrated in Figure 4b).

-1.00E-040.00E+001.00E-042.00E-043.00E-044.00E-045.00E-046.00E-047.00E-048.00E-049.00E-04

0 48 96 144 192 240 288Cell number

Ener

gy d

issi

patio

n ra

te, J

/s

TiTkTv

Figure 3-4b. Contribution of each energy dissipation rate term at each cell to the total

energy dissipation rate. V0 = 0.5 m/s (gas flow with two internal obstacles at Z/dp = 30,

66); Re’ =28.5; Ebed = ( ) sJTTTn

vki /1006.3 1256

1

−

=×=++∑ .

50

-1.00E-03

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

0 48 96 144 192 240 288

Cell number

Ener

gy d

issi

patio

n ra

te, J

/sTiTkTv

Figure 3-5. Contribution of each energy dissipation rate term at each cell to the total

energy dissipation rate. V0 = 0.1 m/s (liquid flow without internal obstacles); Re' = 47.5;

sJTn

i /1079.5 4256

1

−

=×−=∑ ; sJT

nk /101.9 1

256

1

−

=×=∑ ; sJT

nv /1053.1 1

256

1

−

=×=∑ ;

( ) sJTTTn

vki /10624.10 1256

1

−

=×=++∑ .

Regarding the contribution of the Reynolds stress term to the cell-scale velocity

distribution in packed beds, we performed CFDLIB simulations of liquid up-flow at a

high particle Reynolds number (Rep) of 600 (Vl = 0.2 m/s) with and without turning on a

simple turbulence model based on the mixing-length concept (using particle diameter as a

sample of mixing-length). The relative differences in simulated liquid velocity profiles in

the two cases are negligible (less than 0.1 %). This implies that the contribution of the

Reynolds stress term to the cell-scale (i.e. 0.9 cm = 3 particles) flow distribution in

packed beds is negligible. However, such term may become important if one attempts to

model the local particle scale (less than one particle diameter) flow field. As a matter of

fact, the transition between laminar and turbulent flow regime occurs at a certain particle

Reynolds number range (Rep), which may vary with particle diameter. For instance, the

critical Reynolds number range of 150 to 300 was reported by Jolls and Hanratty (1969)

51

for particles of 1.27cm in diameter while Latifi et al. (1989) reported the range of

110~370 for 0.5 cm diameter glass beads. This data was locally measured by using

micro-electrodes with a diameter of 25 µm (Latifi et al., 1989) and reflects the flow

behavior in the interstitial space in packed beds. The recent fine-mesh CFD simulation by

Nijemeisland et al. (1998) did find the stronger turbulent eddies in the gaps in between

the spheres at higher Reynolds number flow conditions.

In the development of the DCM, we made use of the fact that pressure for the

orthogonal directions X and Z, Pcx and Pcz, has the same magnitude at the center of the

cell. We could then eliminate the central pressure term from the expression for the energy

dissipation rate per unit cell (Eq 3-9) by using the mass balance for each cell. In order to

ensure that this formulation is consistent, we have back calculated the central pressure

(Pcx and Pcz) based on the two dimensional velocity solution (Vz and Vx) and verified that

they do have the same values at the center of each cell as required, although the pressure

drop in the X and Z directions may have different values.

Due to the nonlinearity of the equations (cubic in velocity), another important

consideration is the uniqueness of the velocity obtained by the solution of the

minimization problem solved in DCM. To examine this, different starting guess values

varying over two orders of magnitude were used for a test case (input superficial velocity,

V0 = 0.1 m/s). For this case, starting guessed values anywhere between –1.0 m/s to +1.0

m/s converged to a unique solution for velocity based on the minimum total energy

dissipation rate.

With regard to the computational efficiency of DCM, for the cases considered in

this study (total cell number: 264 = 256packing zone + 8supporting plate; the corresponding

number of variables in the optimization is 569), the computation time is comparable with

that required to execute the CFDLIB code with identical discretization. It is noted,

however, that simulation of a case with a larger number of cells would require a more

effective non-linear multi-variable optimization algorithm to get better computational

efficiency.

52

3.5.3 Comparison of DCM and CFDLIB

The verification of DCM predictions can be obtained by comparing them with the

fluid dynamic model solutions (CFDLIB) under identical physical and operating

conditions. For the simplified case of 'parallel flow' Stanek (1994) argued that the

solutions for velocity obtained by the two methods, Differential Vectorial Ergun

Equation Model (based on momentum equation) and minimum rate of energy dissipation

method are identical in both limiting ranges of the Reynolds number (fast flow, i.e. Re' ≥

150, and slow flows, i.e. Re' ≤ 1.5). This conclusion was reached by comparing the

analytical solutions of the two methods. In the transition region (1.5 < Re' < 150),

however, the minimum rate of energy dissipation method yielded smaller velocities

(Stanek and Szekely et al., 1974; Stanek, 1994) than the vectorized Ergun Equation. As

mentioned earlier, the rate of energy dissipation term due to inertia was ignored in the

differential vectorized Ergun equation model (Kitaev et al., 1975). For the case of two

dimensional ''non-parallel flow'', which is of interest in this study, the conclusions

regarding the applicability of the minimum rate of energy dissipation concept in

providing a comparable solution for the gas velocity at cell scale need to be reconsidered.

However, analytical solutions of the fluid dynamic equations for "non-parallel flow" are

unavailable; therefore, the numerical results from computational fluid dynamic solution

(CFDLIB) are used for verification of the DCM simulation results. In order to compare

them effectively, the same operating conditions and the same structure of the bed are

used in the simulations. To cover a wide range of Reynolds numbers, three sets of

superficial gas velocity of 0.1, 0.5, 3.0 m/s are chosen. The corresponding Reynolds

numbers (Re') are 5.7, 28.5, and 170.9 respectively. Three sets of results at different

Reynolds number are shown in Figures 3-6, 3-7 and 3-8 at different axial positions (Z/dp

= 4.5, 19.5, 34.5, 49.5, 64.5, 79.5 from the top of bed).

Following the work of Stephenson and Stewart (1986), and Cheng and Yuan

(1997), we use the (relative) local superficial velocity (dimensionless superficial velocity)

defined as:

53

(Relative) local superficial velocity 0V

VUU j

jj

jj =><

=εε

(3-22)

i.e, the local interstitial velocity times the local porosity (for single phase flow) divided

by the cross-sectionally averaged superficial velocity as given in Equation (3-22). It is

found that the simulated local (i.e. cell scale) dimensionless gas superficial velocity

profiles by both DCM and CFD at each given axial position track the porosity profile

very well. Higher local porosity corresponds to higher local velocity. The difference in

prediction between DCM and CFD simulation was found to be less than 10 % over the

whole range of Reynolds numbers (Re’ = 5 ~ 171) as shown in Figures 9a and 9b. It is

also shown that the dimensionless local superficial velocities vary in the range of 0.8 to

1.2 for the given system with a porosity variation of a cell scale of 0.9cm. All this

indicates that the velocity profiles from DCM and CFDLIB compare well at three

different Reynolds numbers. Reasonable comparisons of the two modeling approaches

are achieved even at high Re' number (Re' = 170.9 at V0 = 3.0 m/s). This implies that

DCM based on the minimum rate of energy dissipated can provide us with gas velocity

predictions comparable to those obtained by CFD, which rests on ensemble-averaged

mass and momentum conservation equations.

54

Z/dp =4.5 (from top)

0.10

0.30

0.50

0.70

0.90

1.10

1.30

0 3 6 9 12 15 18 21 24

X/dp

rela

tive

velo

city

0.100.150.200.250.300.350.400.450.50

CFDDCMporosity

Z/dp =19.5

0.100.300.500.700.901.101.30

0 3 6 9 12 15 18 21 24X/dp

Rel

ativ

e ve

loci

ty

0.10

0.20

0.30

0.40

0.50

z/dp=34.5

0.100.300.500.700.901.101.30

0 3 6 9 12 15 18 21 24X/dp

rela

tive

velo

city

0.100.150.200.250.300.350.400.450.50

Z/dp=49.5

0.1

0.3

0.5

0.7

0.9

1.1

1.3

0 3 6 9 12 15 18 21 24

X/dp

Rel

ativ

e ve

loci

ty

0.100.150.200.250.300.350.400.450.50

Z/dp=64.5

0.10

0.30

0.50

0.70

0.90

1.10

1.30

0 3 6 9 12 15 18 21 24

X/dp

Rel

ativ

e ve

loci

ty

0.100.150.200.250.300.350.400.450.50

Z/dp=79.5

0.100.300.500.700.901.101.30

0 3 6 9 12 15 18 21 24

X/dp

Rel

ativ

e ve

loci

ty

0.10

0.20

0.30

0.40

0.50

Figure 3-6. Comparison of CFD simulations and DCM predictions at a gas superficial velocity of 0.1 m/s at different axial

positions (Z/dp) (Re' = 5.7). Left axis is relative cell superficial velocity; Right axis is cell porosity. 54

55


0.10

0.30

0.50

0.70

0.90

1.10

1.30

0 3 6 9 12 15 18 21 24

X/dp

rela

tive

velo

city

0.100.150.200.250.300.350.400.450.50

DCMCFDporosity

Z/dp =19.5

0.100.300.500.700.901.101.30

0 3 6 9 12 15 18 21 24X/dp

Rel

ativ

e ve

loci

ty

0.10

0.20

0.30

0.40

0.50

z/dp=34.5

0.100.300.500.700.901.101.30

0 3 6 9 12 15 18 21 24X/dp

rela

tive

velo

city

0.100.150.200.250.300.350.400.450.50

Z/dp=49.5

0.1

0.3

0.5

0.7

0.9

1.1

1.3

0 3 6 9 12 15 18 21 24

X/dp

Rel

ativ

e ve

loci

ty

0.100.150.200.250.300.350.400.450.50

Z/dp=64.5

0.10

0.30

0.50

0.70

0.90

1.10

1.30

0 3 6 9 12 15 18 21 24

X/dp

Rel

ativ

e ve

loci

ty

0.100.150.200.250.300.350.400.450.50

Z/dp=79.5

0.100.300.500.700.901.101.30

0 3 6 9 12 15 18 21 24

X/dp

Rel

ativ

e ve

loci

ty

0.10

0.20

0.30

0.40

0.50


positions (Z/dp) (Re' = 28.5).

55

56


0.100.300.500.700.901.101.30

0 3 6 9 12 15 18 21 24

X/dp

rela

tive

velo

city

0.10

0.20

0.30

0.40

0.50

DCMCFDporosity

Z/dp =19.5

0.100.300.500.700.901.101.30

0 3 6 9 12 15 18 21 24X/dp

Rel

ativ

e ve

loci

ty

0.10

0.20

0.30

0.40

0.50

z/dp=34.5

0.100.300.500.700.901.101.30

0 3 6 9 12 15 18 21 24X/dp

rela

tive

velo

city

0.10

0.20

0.30

0.40

0.50

Z/dp=49.5

0.1

0.3

0.5

0.7

0.9

1.1

1.3

0 3 6 9 12 15 18 21 24X/dp

Rel

ativ

e ve

loci

ty

0.100.150.200.250.300.350.400.450.50

Z/dp=79.5

0.100.300.500.700.901.101.30

0 3 6 9 12 15 18 21 24X/dp

Rel

ativ

e ve

loci

ty

0.10

0.20

0.30

0.40

0.50

Z/dp=64.5

0.100.300.50

0.700.901.101.30

0 3 6 9 12 15 18 21 24

X/dp

Rel

ativ

e ve

loci

ty

0.100.150.200.250.300.350.400.450.50


positions (Z/dp) (Re' = 171).

56

57

To examine the effect of fluid density and gravity, the calculations by both

methods were repeated for liquid flow through a liquid-saturated bed. In practice, this

would be the case of liquid up-flow through a packed bed. It should be noted that the

gravity term has now to be accounted for (see Eq. 3-5) because Equation 3-6 is not

satisfied for liquid flow. The difference in prediction of Vz (velocity component in the Z

direction) between DCM and CFD simulation was found to be less than 10 % for the

liquid superficial velocity of 0.1 m/s (Re' = 47.5). DCM yields a 1~2% lower prediction

of Vz than CFD as shown in Figure 10a. Correspondingly, a lower prediction of Vx

(velocity component in X direction) was found in CFD as show in Figure 3-10b. This

implies that in a liquid-solid system the prediction by DCM is a little bit more sensitive to

the bed structure such as porosity distribution than CFD. From a practical point of view,

this feature of DCM prediction for liquid flow does not diminish its appeal as the method

of providing a reasonable solution. This also implies that DCM is applicable for modeling

of the high pressure gas-solid systems in which the density of the gas is high.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

V from CFD (m/s)

V fro

m D

CM

(m

/s)

-10%

+10%

Figure 3-9a. Comparison of superficial velocity between CFD and DCM predictions for

gas flow in the Reynolds number (Re') range of 5 to 171.

58

0.5

0.7

0.9

1.1

1.3

1.5

0.5 0.7 0.9 1.1 1.3 1.5U/U0 from CFD

U/U

0 fro

m D

CM

-10%

+10%

(b)

Figure 3-9b. Comparison of the relative interstitial velocity (Uj/U0) between CFD and

DCM predictions for gas flow in the Reynolds number (Re') range of 5 to 171. (U0 =

V0/εB ).

3.5.4 Comparison of DCM/CFDLIB and Experiment Data

As discussed earlier, most experimental studies in the literature reported the

velocity profiles at the bed exit (Morales et al., 1951; Szekely and Poveromo, 1975; Bey

and Eigenberger, 1997 etc.). They provide the data only for validating the model

prediction for the velocity profile downstream of the bed (see Bey and Eigenberger,

1997; Subagyo et al., 1998). For non-parallel flow system of interest in this study, the

exit velocity profile cannot represent the flow behavior inside the bed (Lerou and

Froment, 1977; McGreavy et al., 1986). Hence, experimental data inside packed beds is

needed to perform the proper comparison of DCM/CFDLIB predictions and experimental

results. The liquid velocity profile inside the bed of Stephenson and Stewart (1986) is

useful for such a comparison because both porosity and velocity data were reported in

59

their paper. However, the data is still inadequate for a very rigorous comparison of the

numerical simulation and experiments since only one set of data was reported, and this

was an ensemble-averaged result based on a large number of 'cell' measurements.

Nevertheless, for lack of better data, this information has been used by others for model

validation (Cheng and Yuan, 1997; Subagyo et al., 1998). The single phase flow

distribution data of McGreavy et al (1986) inside the packed bed is only good for a

qualitative test of numerical simulations because the corresponding porosity data was not

reported (see Figure 7 in McGreavy et al., 1986).

20.0

22.5

25.0

27.5

30.0

20.0 22.5 25.0 27.5 30.0Uz(DCM), cm/s

Uz(

CFD

), cm

/s

+10%

-10%

Figure 3-10a. Comparison of predicted interstitial velocity component in the Z direction

(Uz) by two methods in liquid up-flow system: liquid superficial velocity V0 = 0.1 m/s

(Re' = 47.5).

60

-4.0

-2.0

0.0

2.0

4.0

-4.0 -2.0 0.0 2.0 4.0

Ux (DCM), cm/s

Ux

(CFD

), cm

/s

Figure 3-10b. Comparison of predicted interstitial velocity component in X direction (Ux)

by two methods in liquid up-flow system: inlet liquid superficial velocity V0 = 0.1 m/s.

Figures 3-11a and 3-11b display the DCM results indicating the effect of fluid

superficial inlet velocity (or particle Reynolds number, Rep) on the velocity profile inside

the bed, which are qualitatively comparable with the experimental data of McGreavy et al

(1986) (see Figures 7 and 8 in that paper). The high velocity zones match the high

voidage regions, as would be expected, and as the flow rate increases the magnitude of

these peaks become more pronounced. Figure 3-12b is also comparable with the recent

independent modeling result of Subagyo et al (1998) (See Figure 9 in their paper). The

same conclusions are evident as reported by Subagyo et al (1998) that for Rep less than

500, the velocity profile is dependent on the particle Reynolds number. On the other

hand, the effect of the Reynolds number on the velocity profile is no longer significant

for Rep higher than 500.

61

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 3 6 9 12 15 18 21 24

X/dp

uj,

m/s

0.15

0.25

0.35

0.45

0.55

0.65

poro

sity

0.1m/s (Rep=20) 0.5m/s (Rep=100)porosity

Figure 3-11a. Influence of gas feed superficial velocity on DCM predicted cell interstitial

velocity profiles.

0.7

0.8

0.9

1.0

1.1

1.2

0 3 6 9 12 15 18 21 24X/dp

Vj/V

0

0.15

0.25

0.35

0.45

0.55

0.65

poro

sity

0.1m/s (Rep=20) 0.5m/s (Rep=100)3.0m/s (Rep=600) porosity

Figure 3-11b. Effect of particle Reynolds number (Rep) on the calculated relative cell

superficial velocity profile inside a bed using DCM.

62

0.3

0.6

0.9

1.2

1.5

1.8

0 4 8 12 16 20cell # (interval #)

Rel

ativ

e ce

ll su

perf

icia

l ve

loci

ty

0.200.250.300.350.400.450.500.550.600.650.70

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5r (cm)

poro

sity

Vz/V0 (Exp.) Vz/V0 ( Rep = 5) (Cal.)Vz/V0 (Rep = 80) (Cal.) porosity (Exp.)

Figure 3-12 Comparison of experimental data of Stephenson and Stewart (1986) and

CFDLIB simulated results for relative velocity in a packed bed with D/dv = 10.7 and dv =

0.7035 cm (cylindrical particles). Physical properties of liquid: Liquid -B for condition at

a Rep of 5, ρ = 1.125 g/cm3; µ = 0.474 g cm/s. Liquid -C for condition at a Rep of 80, ρ =

1.027 g/cm3; µ = 0.114 g cm/s.

The quantitative comparison of our CFD numerical simulations (CFDLIB) has

been carried out with the data of Stephenson and Stewart (1986) in which the velocity

and voidage data were obtained by using optical measurements for Reynolds numbers of

5 and 80 in beds with D/dp ratio of 10.7. Velocity was measured inside a bed of

cylindrical particles (dv = 0.7035 cm) with liquid flows of very different physical

properties ( ρ µL L; ). To simulate the experimental bed, a 2D axi-symmetric bed in

cylindrical coordinates (r-Z) is chosen in CFDLIB simulation. The radial spatial

discretization (Nc = 20) is the same as that used as the viewing zone for collecting each

63

experimental data point (i.e a space interval of ∆ = 0 05. R ). Regarding the dependency of

flow simulation result on the grid size, one will see in Chapter 5, that the simulated

macro-scale velocities by k-fluid CFD mode are grid independent.

In addition, no-slip wall boundary and liquid gravity effect are accounted for in

the simulations. The experimentally reported radial porosity profile is used in CFDLIB

simulation. Figure 3-12 shows the comparison of CFDLIB simulated relative superficial

velocity profile (Vz/V0) and the measured data at Rep numbers of 5 and 80. Good

agreement is achieved. This implies that even for a cell size less than a particle diameter,

the CFDLIB code can still provide a reasonable prediction of the velocity profile. The

same agreement between DCM and the experimental data of Stephenson and Stewart

(1986) is expected since DCM and CFDLIB have always provided results with 10% of

each other as discussed in Section 3.5.3. One should note that the above comparison of

CFDLIB and the experimental data still rests on the one-dimensional porosity variation in

the radial direction. Because of the lack of the two-dimensional measured porosity

distribution and velocity distribution data reported in the paper by Stephenson and

Stewart (1986), it is impossible to conduct the full comparison of simulated two-

dimensional velocity field by DCM/CFDLIB with two-dimensional experimental data of

flow distribution at the cell scale.

3.5.5 Case Studies by DCM

Since the validity and accuracy of DCM are established in the previous sections,

DCM can be used in engineering applications as demonstrated in the case studies

considered here. Because of the discrete nature of DCM, boundary conditions can be

easily set. The local variation of porosity can also be accommodated readily. It is possible

to use DCM to model two- or three-dimensional non-parallel flow fields. Two cases are

considered to demonstrate these claims: (i) A bed with pseudo-random porosity

distribution and internal obstacles was considered; (ii) Two types of gas flow inlets (top

and side gas inlets) are examined using the DCM method. Velocity vector plots and

pressure and dimensionless pressure drop contour plots are shown in Figures 3-13a and

3-13b, 3-14a and 3-14b, respectively. Figures 3-13a and 3-13b illustrate the dependency

64

of the gas velocity field on the internal structure nonuniformities inside the beds (i.e. two

internal obstacle plates) and the effect of irregular gas feed (i.e. side gas input) as well as

of the pseudo-random porosity distribution. No vortices appear in the vicinity of the

obstacle plates or at least they are not larger than the cell size. The predicted results for

velocity are almost symmetric with respect to the obstacle plate, which is in good

agreement with entrance region, but also in downflow regions as shown in Figure 3-13b

at given operating conditions, although no effect could be detected at the exit. Therefore,

it is difficult to draw the proper conclusion about the effect of side gas feeding on the

flow field based only on the exit velocity measurements (Szekely and Poveromo, 1975).

It is expected, however, that the effect of side feed will depend on the magnitude of the

side feed gas velocity. The full pressure field in the packed bed with two internal

obstacles is shown in Figure 3-14a. Higher local pressure drop occurs at the regions

around internal obstacles. This is not surprising due to the higher velocity and higher

flow resistance in these regions. Figure 3-14b shows the dimensionless local pressure

drop (ψρG

G gPZ

=

1 ∆∆

) in the case of the side gas feed. Higher values of ψG are evident

in the entry and obstacle regions. In contrast, lower values of ψG are apparent in the

corner regions.

65

Figure 3-13a. Interstitial velocity field in a packed bed with two internal obstacles and

gas uniform feed from the top at a superficial velocity of 0.5m/s (Re' = 28.5) (U0=120.5

cm/s); (velocity vector plotting).

66

Figure 3-13b. Interstitial velocity field in a packed bed with side gas feed (top-left) and

internal obstacles. Inlet gas mean superficial velocity: 0.5m/s (Re' = 28.5) (U0=120.5

cm/s) (point source inlet from left side, inlet point superficial velocity is of 4.0 m/s)

(velocity vector plotting).

67

Figure 3-14a. Pressure field in a packed bed with two internal obstacles and gas uniform

feed from the top at a superficial velocity of 0.5m/s (Re' = 28.5) (The relative values of

pressure with respect to the inlet operating pressure are plotted). Two obstacle plates are

placed in this bed, one is located at Z/dp of 66 (at the left side), another is at Z/dp of 30 (at

the right side). The width of the obstacle plate (i.e. the length that it protrudes into the

bed) is half of the width of bed (4 cells).

68

Figure 3-14b. Dimensionless pressure drop in a packed bed with two internal obstacles

and a gas point feed from top-left side at an equivalent feed superficial velocity of 0.5m/s

(Re' = 28.5) (Dimensionless pressure drop, ψρG

G gPZ

=

1 ∆∆

is plotted). Two obstacle

plates are placed in this bed, one is located at Z/dp of 66 (at the left side), another is at

Z/dp of 30 (at the right side). The width of the obstacle plate (i.e. the length that it

protrudes into the bed) is half of the width of bed (4 cells).

69

3.6 Concluding Remarks A discrete cell approach for modeling single phase flow in packed beds was

analyzed by considering the contributions of the individual terms in the equation for the

rate of energy dissipation. The inertial term in the energy dissipation rate per unit cell is

negligible compared to the kinetic term and viscous term except in the regions of

structural obstacles. Even in the presence of obstacles the overall inertial term in the total

energy dissipation rate is still not important. Reynolds stress term can be ignored due to

negligible contribution of this term to cell-scale (i.e. particle diameter scale) fluid

velocity distribution. DCM can also be applied for liquid up-flow prediction and gas flow

at high operating pressure. A numerical comparison study based on DCM and CFDLIB

approaches for the non-parallel flow field has been carried out to verify the DCM

approach which rests on the assumption that flow is governed by the minimum rate of

total energy dissipation in packed beds. A reasonable agreement between predictions of

these two methods is achieved over a wide range of Reynolds numbers for gas flow. It

was found that the local superficial velocities track the local bed porosity well. Lower

flow resistance produces higher local superficial velocity. The cell superficial velocity

with respect to the cross-sectionally averaged superficial velocity varies in the range of

0.8 to 1.2 for the case considered in this study.

It is not our intent to advocate the use of DCM instead of CFD. The purpose of

this study was to indicate that the model so frequently used by engineers, which is based

on minimization of the total rate of energy dissipation, indeed works for flows in packed

beds in the sense that it provides acceptable solutions of engineering accuracy. The

method is relatively simple to use and contains only those physical terms that are deemed

important in a particular situation. Using the Ergun equation to describe the pressure drop

velocity relation at the cell level is apparently successful enough in describing flows in

packed beds for a wide range of Reynolds numbers, fluid densities and velocities. That is

the main message of this paper. It should also be clear that our intent was not to obtain

the refined more precise flow field in the presence of internal obstacles in the packed

beds, which could be done by local mesh refinement in direct numerical simulation

70

(DNS), but rather to describe the gross flow pattern, which is of interest for quickly

evaluating packed-bed reactor performance, at the same level of discretization via DCM

and CFD. Only such comparisons are reported.

An agreeable comparison of numerical simulations (DCM and CFDLIB) and

experimental velocity data inside a bed is achieved both qualitatively and quantitatively.

Additional experimental efforts in obtaining the experimental data for multi-dimensional

porosity and fluid velocity distributions are needed to further verify these numerical

models and enhance our understanding of flow distribution within beds with complex

internal structural nonuniformities.

71

Chapter 4

Discrete Cell Model Approach

Revisited: II. Two Phase Flow Modeling 4.1 Introduction

Trickle bed reactors with gas-liquid cocurrent downflow have been widely used in

hydrogenation, hydrodesulfurization and other hydrotreating processes. One of the major

challenges in the design and operation of this type of reactor is the prevention of liquid

flow maldistribution which causes portions of the bed to be incompletely wetted by the

flowing liquid. This results in an underutilized catalyst bed and, hence, reduces reactor

performance and productivity, particularly for liquid limited reactions at low liquid mass

velocities. Consequently, conventional reactor models that assume a uniform wetting

efficiency throughout the reactor are found to over-predict the reaction rate (Funk et al,

1990). The solution to this problem requires a quantitative understanding of flow

maldistribution at different scales in trickle beds.

A number of models of the liquid distribution have been developed in the past two

decades based on different concepts or governing principles (Herskowitz et al., 1979;

Crine et al., 1980; Stanek et al, 1981; Zimmerman and Ng, 1986; Ahtchi-Ali and

Pedersen, 1986; Fox, 1987; Melli and Scriven, 1991; Marchot et al., 1992). Although

these efforts have provided insights into liquid flow distribution at a certain level, these

models cannot simulate a number of experimental observations. For example, they cannot

account for prewetting of the bed, which is known to have a marked influence on liquid

72

flow distribution (Lutran et al., 1991; Ravindra et al., 1997a). Thus, there is a need to

incorporate such effects in an engineering model that can reflect these experimental

observations in the model predictions.

The objective of this study is to develop a phenomenological, user friendly, model

for prediction of liquid and gas flow distribution in trickle-bed reactors. The developed

model should be able to capture the experimental observations, and have acceptable

engineering accuracy. Since the trickle bed is treated as a number of inter-connected

cells, the flow distribution model developed in this study is called 'discrete cell model'

(DCM). The gas and liquid distribution is assumed to be governed by the minimization of

total energy dissipation rate (Holub, 1990). The interactions between phases can be

incorporated into the model in terms of the capillary pressure and the particle surface

wetting factor, etc. The motivation for this study was provided by the fact that

minimization of the total energy dissipation rate has been used frequently in engineering

models, yet the results from such an approach were always accepted with a degree of

scepticism as not being based on fundamentals. In this study, our goal is also to compare

the results obtained from the application of the total rate of energy dissipation "principle"

(DCM) to those that arise from solution of more fundamental momentum and mass

balance formulations (i.e. Computational Fluid Dynamics, CFD) and, of course, to

experimental evidence (i.e. photo images by Ravindra et al., 1997a). The intent is not to

replace computational fluid dynamics (CFD) simulations by the minimization of total

energy dissipation rate, but to examine whether an alternative of acceptable engineering

accuracy exists to CFD in flow modeling in trickle beds and can be used more

convenient.

Before presenting the strategy involved in the DCM development, and discussing

the superiority of DCM to other models, it is necessary to summarize the previous liquid

distribution models in the literature with reference to their governing principles and

spatial scales considered.

73

4.1.1 Spatial Scales in Trickle Beds Since different spatial scales exist in a packed-bed (i.e., bed-scale, cell-scale,

particle-scale), there is no question that a flow distribution model based on different

spatial scales will require computation at different levels. At one extreme, time-

consuming computations required to determine liquid flow on the particle-scale

(Zimmerman and Ng., 1986), limit the application of this model to a small size bed,

although this model can reflect partial catalyst wetting on the particle scale. On the other

hand, a bed-scale model, which divides the bed into several regions (i.e., central region

and wall region etc.), is too simplistic to capture the important features of the flow field.

Therefore, Holub (1990) assumed that a packed bed could be represented as a number of

interconnected cells. Each cell consists of a few particles, and each cell has uniform

structure and physical properties. The cell size has to be small compared to the bed scale

(i. e., bed diameter), to obtain the desired resolution of bed properties and phase

distribution, but large compared to the particle scale (i.e., particle diameter), to apply the

existing phenomenological hydrodynamic models developed in lab-scale packed-beds

(i.e. two phase Ergun equations, Holub et al., 1992, 1993). The appropriate cell

dimensions to satisfy these criteria were discussed by Vortmeyer (1984) who concluded

that homogeneous models of packed bed heat transfer failed in beds with a tube to

particle diameter ratio less than three. While this conclusion was not reached for the exact

situation considered here, a minimum linear dimension of three particle diameters for

each cell can be considered appropriate (Holub, 1990). Figure 4-1 represents a typical

two dimensional cell (which consists of nine particles) with the velocity convention and

coordinate system used in DCM formulations. The nature of such a discrete cell model

allows us to obtain flow distribution information (at few particles scale) with reasonable

computational efforts.

4.1.2 Governing Principles for Flow Distribution Because of the complexity of two phase flow distribution in trickle-bed reactors, a

number of models have been developed, in which different governing principles for flow

distribution have been assumed. The ‘diffusion model’ assumed that the irrigation flux

74

satisfies the diffusion equation, which is then solved for a variety of inlet flow

distributions (Stanek et al., 1981; Stanek, 1994). This model was not capable of

predicting phase separation phenomena that occur in trickle beds. The ‘percolation

approach’ has been used by many researchers to model the flow in trickle beds (Crine et

al. 1979, 1980; Larson et al. 1981; Melli and Scriven, 1991; Marchot et al., 1992). The

model assumed that the flow distribution in the bed is due to a stochastic process. The

liquid was distributed on the network by randomly choosing the bonds in the structure

that have flowing liquid. While the model had the merit of representing the liquid

structure as discontinuous in the bed, the predictive ability is questionable for the small

size of particles since a direct relationship does not exist between the network and bed

structure. In a computer generated packed bed of equally sized spheres, the ‘sphere-pack

model’ predicts the liquid distribution based on developed wetting criteria (Zimmerman

and Ng, 1986). The model was able to predict liquid coring, but gas flow was not

included in the model. The model was also limited to the case of initially dry spherical

particle surface. The effect of particle prewetting could not be accounted for. In the

discrete interconnected cells model (DCM) (Holub, 1990) addressed in this study, it was

assumed that the flow can be determined by the minimum rate of total energy dissipation

in the packed-bed (i.e. flow follows the path of the least resistance). The known porosity

variation in the bed could readily be incorporated into DCM by inputting cell porosity

values. The type of liquid and gas distributors (i.e. point source; uniform source; irregular

source for two fluids) is accounted for by setting two phase velocity values at inlet cell

boundaries. To consider the effect of particle wetting state on the liquid distribution (i.e.

prewetted bed and nonprewetted bed), which has been observed in experimental studies

(Lutran et al., 1991; Ravindra et al., 1997a), the contribution of capillary pressure has

been incorporated into the original DCM, and reported as an extended DCM in this study.

The superiority of the extended DCM to other flow distribution models can be attributed

to its ability to consider (i) the effect of bed structure nonuniformity (two dimensional

porosity variations; internals in packed bed); (ii) the effects of gas and liquid distributors;

(iii) the effect of particle prewetting.

75

4.2 Extended Discrete Cell Model The discrete cell model (DCM) for packed beds was originally proposed and

formulated by Holub (1990). The full analysis and detailed implementation of individual

aspects of DCM for single phase modeling (gas flow or liquid upflow) have been

presented in Chapter 3. In this Chapter, DCM is applied to two phase flow modeling in

trickle beds. Since the detail formulation is available in Chapter 3 and Holub (1990), only

the key model equations and the parts pertinent to two phase flow are presented here.

The equation for the macroscopic mechanical energy balance for phase α in the

jth unit cell can be expressed in continuum form by Equation (4-1), and is based on the

following key assumptions: (i). Each unit cell of the bed has a uniform porosity (εj, which

can vary from cell to cell), and constant phase holdup as well as constant phase

properties; (ii). Steady-state flow distribution is considered in the entire bed and fluids are

incompressible; (iii). No phase change occurs at the gas-liquid interface. The contribution

of chemical reaction to the flow distribution is not accounted for in this model.

−

Φ+

+

⋅

∑

=inj

iiii s

sVs

sVPs

sV

sV

,

4

1

2

21

ααα

ρρ

0,,

4

1,

2

=

−

Φ+

+

⋅

∑

= αααεε

ρεεε

ρjVj

outj

iiiEsVsVPsVV (4-1)

76

VZ, j, αααα

VX, j, αααα VX+∆∆∆∆X, j, ααααCell j

∆∆∆∆X=3dP

VZ+∆∆∆∆Z, j, αααα

∆∆∆∆Z=3dPX

Z

Figure 4-1. The coordinate system and velocity conventions for the α phase in the jth

cell

For a 2D rectangular cell (j) as depicted in Figure 4-1, the mechanical energy

dissipation rate of phase α, EV, j , α , can be written in the discretized form as below

−+−=

∆+∆+∆+∆+ α

α

αα

α

αα ε

ρε

ρ,

3332,

,

332,

,,1

2 jZZZZZZj

jXXXXXXj

jV SVSVSVSVE

( ) ( )[ ]( ) jjXXjXXjjjXjXjjj

VolVVbaVVba 2,,,,,

2,,,,,

3

,

1αααααα

αε ∆+∆++++

+

( ) ( )[ ]( ) jjZZjZZjjjZjZjjj

VolVVbaVVba 2,,,,,

2,,,,,

3

,

1αααααα

αε ∆+∆++++

+

( ) ( )[ ]( ) − + + ++ +g V V g V V VolX X X X j X Z Z Z Z j Z j∆ ∆, ,cos cos

α αγ γ2 2 2 (4-2)

A discretized form of the macroscopic mass balance equation can be similarly written as

( ) ( ) 0,, =−+− ∆+∆+∆+∆+ αα ρρρρ jZZZZZZjXXXXXX SVSVSVSV (4-3)

A detailed derivation of each term in Equation (4-2) is available in Holub (1990).

To simulate the flow distribution, the two phase velocities at each cell interface are

obtained by minimization of the total energy dissipation rate over the entire bed domain.

77

This is essentially a nonlinear, multivariable minimization problem as given in Equation

(4-4) subject to the mass balance constraint (Equation 4-3) for each phase in cell j, and

additional constraints to reflect gas and liquid velocities at the bed inlet and at other

boundaries of the bed (i.e., phase velocities in the cell adjacent to the wall are zero in the

direction normal to the wall).

Minimize: ∑∑= =

=2

1 1,,,

αα

N

jjVbedV EE (4-4)

The subroutine DN0ONF from the International Mathematical Statistics Library

(IMSL) was used to solve this nonlinear multivariable minimization problem.

To get phase velocities from the above equations, we have also to solve for cell

phase holdup (εj,α) corresponding to a set of assumed phase velocities. This can be done

by equating pressure drops in the gas and liquid phase (in absence of capillary pressure).

∆

∆=

∆

∆L

PL

P jLjG ,, (4-5a)

Then pressure drops are expressed in terms of dimensionless pressure drop

functions (ψG for gas, ψL for liquid).

( )1,, −Ψ=

∆∆

jGLjG g

LP

ρ (4-5b)

( )1,, −Ψ=

∆∆

jLLjL g

LP

ρ (4-5c)

Substitution of Equations (4-5b) and (4-5c) into equation (4-5a) yields equation (4-6)

( )11 ,, −Ψ+=Ψ jGL

GjL ρ

ρ (4-6)

To relate these pressure drop functions to cell flow velocities, two phase flow

Ergun equations (Holub et al., 1992, 1993) as given in equations (4-6a) and (4-6b) are

used. This is equivalent to utilizing the concept of relative permeability discussed by Saez

and Carbonell (1985).

+

−

=ΨjG

jG

jG

jG

jLj

jjG Ga

EGa

E

,

2,2

,

,1

3

,,

ReReεε

ε (4-6a)

78

+

=Ψ

jL

jL

jL

jL

jL

jjL Ga

EGa

E

,

2,2

,

,1

3

,,

ReReεε

(4-6b)

Thus, substitution of ψG, j and ψL, j (from Eq 4-6a and 4-6b) into Equation (4-6)

yields a nonlinear equation in terms of phase holdups (εj,α) and cell phase velocities

which can be readily solved if flow velocities are known.

The essential part of extended DCM is the treatment of the drag which takes into

account capillary pressure. The pressure difference in the gas and liquid phase were

correlated with the capillary pressure (Grosser et al., 1988) and a particle wetting factor,

f, as

( )fPPP jCjLjG −=− 1,,, (4-7)

where the capillary pressure, Pc,j, in packed bed can be written as Equation (4-8) in terms

of the well-known Leverett’s J-function (Leverett, 1941) as suggested by Grosser et al.

(1988).

( ) ( )jWPj

jjC sJE

dP ,

5.01,

1σ

εε−

= (4-8a)

( )

−+=

jW

jWjW s

ssJ

,

,,

1ln036.048.0 (4-8b)

When complete external particle wetting occurs (f = 1) the pressure difference between

the gas and liquid phase disappears. This is the case treated in the original DCM (Holub,

1990). The pressure difference reaches a maximum (equal to the capillary pressure, Pc,j)

when the wetting factor f is equal to zero (completely non-prewetted case). Depending on

the cell-scale, liquid velocity and cell porosity the f value is somewhere in the range of

zero to one, which can be exactly calculated by the correlation for the particle external

wetting efficiency (Al-Dahhan and Dudukovic, 1995).

Thus, phase holdup is solved for by equating the difference of pressure drops in

gas and liquid phases to the capillary pressure times the factor (1-f) as given in Equation

(4-9).

79

( ) ( ) ( )LsJ

fEdL

PL

P jW

Pj

jjLjG

∆∆

−−

+∆

∆=

∆∆ ,5.0

1,, 1

1σ

εε

(4-9)

Similarly to Equation (4-6), we can get Equation (4-10) in terms of dimensionless

pressure drop functions, and from it we can solve for liquid holdup.

[ ] ( ) ( ) ( )( )LsJ

ssbfE

djW

jWjWPjL

jjG

L

GjL ∆

∆

−

−−

+−Ψ+=Ψ ,

,,

5.01,, 1

11

11 σερ

ερρ (4-10)

when ψG, j and ψL, j are obtained from Equations 4-6a and 4-6b as before.

4.3 Modeling Results and Discussion The modeling results are presented for a '2D' rectangular bed, 7.2 cm (width) ×

28.8 cm (height) × 0.9 cm (thickness), referred henceforth as “model bed”, which has an

average bed porosity of 0.415 corresponding to the value measured experimentally. To

examine the cell porosity effect on flow distribution, the internal porosity profiles were

specially designed by using a pseudo- ‘random’ porosity distribution generated by a

computer program with given constraints (i.e. porosity is kept in the range of 0.36 to 0.44

with an average of 0.406 for the inner bed region away from the walls while higher

porosity (0.44) is assigned to the wall region: 3 particles next to the wall) (see Figure 4-

2a). Two transverse locations with low average porosity were deliberately designed as

plotted in Figure 4-3b. This bed is divided into 8 cells in width (nC) and 32 cells in length

(nR) and is 1 cell thick (nT). Each cell has a size of 3 dp × 3 dp × 3 dp (0.9 cm × 0.9 cm ×

0.9 cm) as depicted in Figure 4-1. The bed walls are considered to be impermeable

boundaries. The liquid inlet distribution was assigned as: uniform, single point source and

as two points source to simulate different liquid distributors. The inlet distribution for the

gas phase was assigned as uniform in all the case studies.

80

Figure 4-2a. Local porosity distribution in model bed; Random internal porosity (0.36 ~

0.44). Higher porosity of 0.44 at the walls. Darker color corresponds to higher porosity.

0.390.400.410.420.430.440.45

0 8 16 24 32 40 48 56 64 72 80 88 96

x/dp or z/dp

poro

sity

axial (Z direction)radial (X direction)

Figure 4-2b. Average porosity profiles in X and Z directions in model bed.

81

To quantify the liquid flow maldistribution, it is necessary to compare the

deviation from a uniform velocity profile in term of the liquid maldistribution factor, mf,

defined as

∑=

−=

N

i

ii

VV

AAmf

1

2

00

1 (4-11)

When the liquid flow distribution is uniform over the bed cross-section, mf is equal to

zero and mf increases as the distribution becomes less uniform. The effect of different

parameters (i.e. a state of particle prewetting, liquid distributor type, particle size) on

flow distribution can be quantitatively described by the value of mf. The axial mf profile

reflects the effect of bed depth on the flow distribution.

4.3.1 Comparison of DCM and CFD Simulations The main assumption of DCM is that the flow is governed by the minimum rate

of total energy dissipation in the bed. The complete theoretical justification for this

assumption has been provided for linear relationships between fluxes and driving forces

and rests on the principle of entropy maximization (Jaynes, 1980). In Chapter 3, for non-

linear systems, particularly non-parallel gas flow or liquid up-flow in the packed beds

(where the local phase holdup is equal to the local porosity), agreeable numerical

comparisons of DCM and CFD (using CFDLIB code as described below) have been

achieved (Jiang et al., 1998). The difference between DCM and CFDLIB simulations was

found to be always within 10 % over a wide range of Reynolds numbers. Nevertheless, it

is desirable to compare the predictions of these two methods for the gas-liquid two phase

flow system which is of interest in trickle flow. For this purpose, the Computational Fluid

Dynamics code, CFDLIB developed by Los Alamos National Laboratory (Kashiwa et al.,

1994), has been used to obtain the results for comparison with the DCM. The governing

equations that serve as the basis for the CFDLIB codes and drag closures used in the

simulation are given below

82

Equation of continuity:

>αρ<=ρ∇+∂

∂ρkkkk

k u.t

! (4-12a)

(Accumulation) (Convection) (Mass source)

Equation of momentum:

gp'u'u.uu.tu

kkkk0kkkkkk ρ+∇θ−>ρα<−∇=ρ∇+

∂∂ρ

(Accumulation) (Convection) (Reynolds stress) (Mean pressure) (Body force)

>αρ<+>α∇τ−−<+ k00k00 u].I)pp[( !!

(Exchange term) (Mass source)

)pp(. 0k0k −θ∇−>τα<∇+ (4-12b)

(Average stress) (Non-equilibrium pressure)

This set of equations is exact with no approximations other than the ensemble

averaging used in the two-fluid model approach. One of advantages of CFDLIB is that it

treats the packed bed case specifically and has options for user defined drag force

formulation. Boundary conditions for inflow, outflow, and free/no slip at the reactor

walls can be directly specified (Kumar, 1994). In this study, a user defined drag

formulation is incorporated in simulating the drag between the stationary solid phase and

each of the flowing phases in terms of phase fractions and relative velocity given for any

combination of phases k and l as given below

( ) ( )lkkllklkD uuXF −=− θθ (4-13)

where the Xkl is modeled by the modified Ergun equation (Holub et al., 1992, 1993) with

universal Ergun constants in this study. The drag between flowing phases has been

ignored. This drag form is the same as that used in DCM simulation. To keep the

consistency with the discrete cell approach used in DCM, free-slip boundary conditions

are used for the reactor walls in CFD simulation. The spatial discretization of the model

bed is also the same in both methods.

For a given set of operating conditions, Figures 4-4a ~ 4-4d display the predicted

relative gas flow interstitial velocity profiles by CFD and DCM at different heights (Z/dp)

83

in the bed. Comparison of the complete data set at all heights is plotted in Figure 4e.

Quantitatively, the agreement between the two model predictions for gas flow is good,

and the differences in prediction of gas flow in all the cells by CFD and DCM are less

than 13%.

0

0.3

0.6

0.9

1.2

1.5

0 3 6 9 12 15 18 21 24X/dp

Rel

ativ

e ite

rstit

ial

velo

city

CFDDCM

Z/dp = 15

0

0.3

0.6

0.9

1.2

1.5

0 3 6 9 12 15 18 21 24X/dp

Rel

ativ

e in

ster

stiti

al v

eloc

ity

CFDDCM

Z/dp = 30

(4-3a) (4-3b)

0

0.3

0.6

0.9

1.2

1.5

0 3 6 9 12 15 18 21 24X/dp

Rel

ativ

e in

ters

titia

l vel

ocity

CFDDCM

Z/dp = 45

0

0.3

0.6

0.9

1.2

1.5

0 3 6 9 12 15 18 21 24X/dp

Rel

ativ

e su

perfi

cial

vel

ocity

CFDDCM

Z/dp = 75

(4-3c) (4-3d)

Figures 4-3a, 3b, 3c and 3d. Comparison of the predicted gas interstitial velocities

(relative) at the specific axial level by DCM and CFD. Ul = 0.00148 m/s (UF); Ug = 0.05

m/s (UF); Completely prewetted packed bed. (The relative interstitial velocity is defined

as the local interstitial velocity (Vi) divided by the overall interstitial velocity (V0). The

value of V0 in this case is equal to 0.1205 m/s).

84

0.5

0.7

0.9

1.1

1.3

1.5

0.5 0.7 0.9 1.1 1.3 1.5Relative velocity by CFD simulation

Rel

ativ

e ve

loci

ty b

y D

CM

mod

elin +13%

-13%

Figure 4-3e. Comparison of the predicted gas interstitial velocities (relative) for all the

cells by DCM and CFD. Inlet superficial velocities (uniform): Ul = 0.00148; Ug =

0.05m/s; Completely prewetted packed bed.

85

0.00

0.05

0.10

0.15

0.20

0.25

0 3 6 9 12 15 18 21 24

X/dp

Liqu

id h

oldu

p"Z/dp=74""Z/dp=62""Z/dp=50""Z/dp=74""Z/dp=62""Z/dp=50"

Solid line: CFDLIBDash line: DCM

Figure 4-4. Comparison of predicted liquid holdup at specific levels by DCM and CFD.

Single point source liquid inlet: U L = 0.00148 m/s (Ul (PS1)= 0.01184 m/s);

Ug = 0.05m/s; Non-prewetted packing.

The comparison of predicted liquid holdup at the specific levels of bed is shown

in Figure 4-4 for the case of a single point source liquid inlet (PS). From the engineering

point of view, the comparison of predicted liquid holdup by both methods is reasonable

particularly in the central core. The difference in the prediction of liquid holdup by two

methods (CFD and DCM) at locations far from the central flow indicates that the DCM

seems to be more sensitive to local porosity values than CFD, and also predicts more

liquid spreading. These numerical results can be qualitatively compared to experimental

observations presented in Figure 2-8, which illustrates that rivulet flow is affected by

variations in the local porosity which causes it not to flow straight down through the bed.

86

4.3.2 Effect of Liquid Distributor

Three types of liquid inlet distributors: single point source (PS1), two points

source (PS2) and uniform distributor (UF) have been used to demonstrate the effect of

liquid distributors on the liquid distribution in a non-prewetted packed bed. The boundary

(inlet) values of the liquid superficial velocities at the top cell layer of the bed were

assigned to keep the same volumetric liquid feed rate for all types of liquid distributors

studied. With liquid point source inlets, as shown in Figure4-5a for single inlet and

Figure 4-5b for two inlets, it was found that the number of liquid channels (rivulets)

formed in the non-prewetted packed bed corresponded to the number of liquid point

sources (e.g one for PS1, and two for PS2). This observation is qualitatively reflected in

the result shown in Figure 2-8. With the uniform liquid inlet, as shown in Figure 4-6c,

however, uniform liquid distribution occurs only in the entrance region, then channel

(rivulet) flow forms in the downstream region due to the nonuniform porosity and

capillary pressure effect. Under the chosen set of operating conditions (in Table 4.1), an

onset of formation of liquid channels (i.e., phase segregation) is seen at a depth of 2 cm

and formation of distinct rivulets occurred at a bed depth of 8 cm. These rivulets

meandered, merged, and split as experimentally observed by Ravindra et al. (1997). It

can be concluded that liquid rivulet flow is typical of non-prewetted beds. These DCM

simulation results are qualitatively comparable with direct flow visualizations (Figure 4-

2a and Ravindra’s et al photo observations, 1997). A comparison of the calculated liquid

maldistribution factor (mf) along the bed for different distributors is presented in Figure

4-5d. The effect of liquid distributor on liquid flow maldistribution is significant in the

upper half of the bed (in non-prewetted beds) and is less pronounced at depths exceeding

50 particle diameters (15 cm) for total bed length of 96 particle diameters.

87

Table 4.1 Summary of operating conditions used in flow simulations

Ul = 0.00148 m/s Ug = 0.050 m/s (superficial velocity)

Completely prewetted bed Completely nonprewetted bed

Gas inlet Liquid inlet Gas inlet Liquid inlet

UF PS1, PS2; UF UF PS1

UF: uniform

PS1: single point source located at top layer* at No. 5 cell

PS2: two points source located at top layer at No. 3 and No. 6 cell

*There are 8 cells on the top layer from No.1 to No.8

Figure 4-5a. Liquid holdup distribution with single liquid point source inlet (located at

No. 5 cell from left) by DCM. U L = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05m/s;

Non-prewetted packing.

88

Figure 4-5b. Liquid holdup distribution with two liquid points source inlet (located at

No. 3 cell and No. 6 cell from left) by DCM. U L = 0.00148 m/s (Ul (PS2)=0.00592 m/s);

Ug = 0.05m/s; Non-prewetted packing.

89

Figure 4-5c. Liquid holdup distribution in whole domain of the non-prewetted packed

bed with uniform liquid distributor by DCM. Ul = 0.00148 m/s; Ug = 0.05m/s.

90

0.0

0.51.0

1.5

2.02.5

3.0

0 9 18 27 36 45 54 63 72 81 90

z/dp (from top)

mf

PS1(non-prewetted)PS2(non-prewetted)PS1(prewetted)UF(non-prewetted)

Figure 4-5d. Comparison of liquid flow maldistribution calculated by DCM along the

bed for different liquid distributors. U L = 0.00148 m/s; Ug = 0.05m/s.

4.3.3 Effect of Particle Prewetting Experimental observations have corroborated the fact that the effect of particle

prewetting on liquid distribution is significant and causes more liquid spreading in 3D

rectangular beds compared to non-prewetted bed (Lutran et al., 1991; Ravindra et al.,

1997a). The CCD video images in Figure 2-8 also confirm the same finding in a pseudo-

2D bed. It is known that lower capillary pressure (by lower liquid surface tension, lower

contact angle (θ) at the three-phase contact line) causes more particle wetting in the bed,

and accordingly, causes an increase in overall liquid holdup (Levec, et al., 1986). In order

to predict these experimental observations, we have incorporated a particle surface etting

factor (f) into DCM as described earlier. Figure 4-6a shows the liquid holdup distribution

in the entire completely prewetted bed with a single point liquid inlet (PS1) (actually one

cell inlet). For further comparison, Figures 4-6b and 4-7c show the predicted liquid

holdup at the specific levels (Z/dp = 94.5; 73.5; 61.5; 49.5 from bottom) in the completely

prewetted bed (f = 1) and in non-prewetted bed (f = 0), respectively. More liquid

91

spreading is evident in the prewetted bed whereas the effect of capillary pressure on

liquid holdup distribution is apparent in the non-prewetted bed, where the pressure

difference between the gas and liquid phase exists and prevents liquid from spreading.

This is the reason for liquid channel (rivulet) flow formation in the non-prewetted bed. If

the whole bed is pre-wetted with liquid, thin liquid films will be formed around the

particle surfaces, in addition to the pores of the particles being filled by liquid, even when

the liquid is drained off. These liquid films nullify the effect of capillary pressure and

help spreading of the incoming liquid. Therefore, as expected, more liquid spreading in a

prewetted bed is observed as shown by DCM simulation in Figure 4-6a. The overall

liquid holdup in the prewetted bed is 6% higher than in the non-prewetted bed at the same

operating conditions (as seen in Figures 4-6b and 4-6c). The increase in predicted overall

liquid holdup by DCM is in agreement with Levec's et al experimental finding (1986).

It is also of interest to consider Figure 4-6d, which shows that the liquid flow

distribution with one point source liquid inlet in the prewetted bed is better in most of the

bed (except the inlet region) than that obtained by two points source liquid inlet in the

non-prewetted bed. This also corroborates the evidence of better reactor performance in

prewetted beds. The only detrimental consequence of prewetting is liquid ‘wall flow’

which occurs in the case of the completely prewetted bed with uniform liquid inlet

(Figure 4-5c), since liquid spreads more easily until it reaches the wall and then continues

along it resulting in the observed wall flow. If we consider one cell (three particles in this

case) next to the wall as the wall zone, the magnitude of the wall flow is about 20-30%

higher than the central region flow. These results for wall flow are only slightly different

if ghost cells are created next to the wall to set a zero slip velocity at the wall. This has

also been tested through the CFD simulations with slip boundary and with no-slip

boundary conditions.

92

Figure 4-6a. Liquid holdup distribution in the whole domain of the completely prewetted

packed bed (f = 1). U L = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05 m/s; Point

liquid distributor (PS1).

93

0.0500.070

0.0900.110

0.1300.150

0.1700.190

0.2100.230

0.0 6.0 12.0 18.0 24.0

X/dp

Liqu

id h

oldu

p

Z/dp=94.5Z/dp=73.5Z/dp=61.5Z/dp=49.5

Figure 4-6b. Liquid holdup distribution at specific levels (Z/dp) in the completely

prewetted packed bed (f = 1), U L = 0.00148 m/s (Ul (PS1)= 0.01184 m/s); Ug = 0.05 m/s;

Point liquid distributor (PS1); Overall liquid holdup = 0.0758.

0.0500.070

0.0900.1100.130

0.1500.1700.190

0.2100.230

0.0 6.0 12.0 18.0 24.0

X/dp

Liqu

id h

oldu

p

Z/dp=94.5Z/dp=73.5Z/dp=61.5Z/dp=49.5

Figure 4-6c. Liquid holdup distribution at specific levels (Z/dp) in the completely non-

prewetted packed bed (f = 0), U L = 0.00148 m/s (Ul (PS1) = 0.01184 m/s); Ug = 0.05 m/s;

Point liquid distributor (PS1); Overall liquid holdup = 0.0716.

94

Although two extremes of external particle wetting were considered here, in

reality this parameter takes the value between two limiting cases (f = 0 and f = 1.0)

depending on the particle surface and fluid properties of system. It is expected that the

value of surface wetting factor, f, is associated with three phase interfacial-phenomena,

such as liquid-solid contact angle, liquid surface tension, particle internal porosity etc.

Local liquid vaporization may also cause local particle wetting non-uniformity, and

further affect the value of f. It is also believed that the differences between non-porous

and porous particles are reflected in different values of f, and thus cause different liquid

distribution as observed experimentally by Ravindra et al., (1997a). The surface wetting

factor (f) used here can be evaluated through the correlation of the particle external

wetting efficiency which has been widely used in the literature (Al-Dahhan and

Dudukovic, 1995). The flow simulations based on two limiting values of f (zero and one)

essentially cover the range of possible liquid distribution at given operating conditions

(see Figures 4-5a and 4-6a), which is valuable in examining possible trickle-bed scale-up

and design.

4.4 Conclusions and Final Remarks An extended discrete interconnected cell model (DCM) was developed for

simulation of two phase flow in trickle-bed reactors. Due to the nature of DCM, structural

nonuniformities and different liquid inlet distributors can be readily incorporated into the

model. Particle wetting characteristics are accounted for in the model by introducing the

particle wetting factor (f) which allow us to distinguish between the flow patterns in

prewetted and non-prewetted beds. The model predicted results are quantitatively

comparable with those obtained from computational fluid dynamic codes (CFDLIB).

Simulated liquid holdup distribution data qualitatively agree with the flow visualization

experiments, which has not been achieved by other available models. Two bounds

(corresponding to the completely prewetted and completely non-prewetted catalyst) of

the liquid flow distribution at given operating conditions can be provided by the DCM

model. The effect of liquid distributor on liquid flow distribution is significant in the

95

upper half of the bed. In regard to the computational efficiency of DCM, which is

essentially formulated as a non-linear multi-variable optimization problem, more

effective optimization algorithms are desirable for industrial scale problems with a large

number of cells (as compared with only 256 cells used for our model bed).

The advantage of DCM will become more apparent when we utilize it to compute

not only just the flow distribution but also reactor performance. At this point, we re-

emphasize that DCM is not suggested as a replacement for CFD. However, it is shown

here that when one is interested only in the coarse structure of the flow pattern, DCM can

provide answers comparable to those obtained by CFD.

96

Chapter 5

Computational Fluid Dynamics

(CFD): I. Modeling Issues 5.1 Introduction and Background

5.1.1 CFD Applied to Multiphase Reactors The performance of multiphase reactors, in principle, can be predicted by solving

the conservation equations for mass, momentum and (thermal) energy in combination

with the constitutive equations for species transport, chemical reaction and phase

transition. However, because of the incomplete understanding of the physics, plus the

nature of the equations- highly coupled and nonlinear, it is difficult to obtain the

complete solutions unless one has reliable physical models, advanced numerical

algorithms and sufficient computational power. Hence, in the past several decades,

‘Residence Time Distribution’ (RTD) together with the ‘macromixing’ and

‘micromixing’ models have been the primary tool in reactor modeling used to

characterize the nonideal flow pattern and mixing in the reactor without solving the

complete flow velocity field (Levenspiel, 1972). The disadvantage of such an approach is

that it cannot be adopted well to serve as a diagnostic tool for operating units, which

normally need to be operated under conditions not amenable to the above simplified

analysis. To achieve this goal, one has to solve the complete multi-dimensional flow

equations coupled with chemical species transport, reaction kinetics, and kinetics of

97

phase change. Fortunately, computational fluid dynamics (CFD) has made great progress

during the last few years, and has been applied to chemical processes (Trambouze, 1993;

Kuipers and van Swaaij, 1998). In particular, one of the promising methods is the so-

called full Probability Density Function (PDF) model for single-phase reactive-flow

systems (Pope, 1994; Fox, 1996). For most multiphase reactive-flow systems, however,

the challenge still exists in both numerical technique and physical understanding.

Reasonable progress has been made for multiphase cold-flow systems and few reactive-

flow systems via CFD modeling. The features and the problems encountered in the

current CFD modeling of multiphase reactors have been summarized in Table 5.1, which

clearly indicates that more effort is needed in applications of CFD in gas-liquid stirred

tanks, gas-liquid-solid packed-beds (e.g., trickle-beds), gas-liquid-solid fluidized beds

and slurry reactors. For a detailed discussion of these topics, one is encouraged to consult

the recent comprehensive review by Kuipers and van Swaaij (1998).

Based on the growing applications of CFD in multiphase flow systems, it is

expected that the role of CFD in the future design of multiphase reactors will increase

substantially and become common engineering practice. So far, a consensus emerges

with regard to the following issues:

• It is unrealistic to hope for a universal CFD code that applies to all multiphase

flow problems (Johnson, 1996). Even for one type of multiphase reactor such as bubble

column, a ‘hierarchy of models’ is more likely to have successful impact (Delnoij et al.,

1997).

• Experimental validation of CFD results for several benchmark multiphase flows is

essential to the widespread acceptance of CFD in multiphase reaction engineering

(Kuipers and van Swaaij, 1998; Dudukovic et al., 1999).

• Properly formulated two-fluid model is able to capture most of large-structure

characteristics of multiphase flow (Lahey and Drew, 1999; Pan et al., 2000)

• The solutions from direct numerical simulation (DNS) (Joseph, 1998) and Lattice

Boltzmann simulation (LB) (Sankaranarayanan et al., 1999; Manz et al., 1999) can

provide an improved understanding of flow microstructure, and are a tool for obtaining

98

closures for averaged equation models used to predict large scale flows in industrial

reactors.

99

Tabl

e 5.

1 C

urre

nt st

atus

of C

FD m

odel

ing

in m

ultip

hase

reac

tors

R

eact

ors

Cur

rent

Fea

ture

s and

Fut

ure

Cha

lleng

es

Prog

ress

Sa

mpl

e C

FD W

ork

Bub

ble

Col

umns

Tw

o-flu

id E

uler

ian

mod

el

Mix

ed E

uler

ian-

Lagr

ange

mod

els

Vol

ume

of F

luid

(VO

F) m

odel

for s

ingl

e ga

s bub

ble

risin

g M

ostly

lim

ited

to b

ubbl

y flo

w

Futu

re c

halle

nge:

chu

rn-tu

rbul

ent f

low

mod

elin

g.

Rea

sona

ble

!!!

Soko

lichi

n &

Ei

genb

erge

r, 19

94

Del

noij

et a

l., 1

997

Pan

et a

l., 2

000

G-L

Stir

red

Tank

s Tw

o-flu

id m

odel

(Sna

psho

t app

roac

h, M

RF

mes

h, S

lidin

g m

esh)

Fu

ture

ch

alle

nges

: A

ccur

ate

mod

elin

g of

th

e im

pelle

r; av

aila

bilit

y of

loca

l flo

w d

ynam

ic i

nfor

mat

ion

and

the

rang

e of

di

sper

sed

phas

e ho

ldup

from

exp

erim

ents

; tur

bule

nce

mod

elin

g

Littl

e !

R

anad

e &

van

den

Akk

er,

1994

R

anad

e &

D

eshp

ande

, 19

99

G-S

Flu

idiz

ed B

eds

e.g.

bub

blin

g, so

lid

riser

s

Two-

fluid

mod

el w

ith si

mpl

e so

lid rh

eolo

gy

Two-

fluid

mod

el w

ith k

inet

ic th

eory

D

iscr

ete

parti

cle

appr

oach

Fu

ture

cha

lleng

es: R

efin

ed m

odel

for p

artic

le-p

artic

le, p

artic

le-

wal

l int

erac

tions

; cou

ple

with

reac

tion;

Pre

dict

ion

of fl

ow re

gim

e tra

nsiti

on.

Goo

d !!!!

Si

ncla

ir &

Jack

son,

198

9 D

ind

& G

idas

pow

, 199

0 N

ieuw

land

et a

l., 1

996

G-S

or L

-S P

acke

d B

eds

Two-

fluid

mod

el w

ith 3

D m

esh

for i

nter

stiti

al d

omai

n Tw

o-flu

id m

odel

s with

rand

om p

oros

ity d

istri

butio

n St

ruct

ural

pac

king

with

hea

t tra

nsfe

r Fu

ture

ch

alle

nges

: G

eom

etric

al

com

plex

ity;

avai

labi

lity

of

expe

rimen

tal d

ata

for v

alid

atio

n.

Littl

e !

Lo

gten

berg

&

D

ixon

, 19

98

Cha

pter

3 in

this

thes

is

G-L

-S P

acke

d B

eds

e.g.

tric

kle

beds

Tw

o-flu

id m

odel

with

rand

om p

oros

ity d

istri

butio

n Fu

ture

ch

alle

nges

: G

eom

etric

al

com

plex

ity;

parti

al

wet

ting

conc

ern;

flo

w h

isto

ry d

epen

denc

e; a

vaila

bilit

y of

exp

erim

enta

l da

ta fo

r val

idat

ion

Ver

y Li

ttle

Cha

pter

4 in

this

thes

is

To b

e di

scus

sed

in t

his

Cha

pter

Sl

urry

reac

tors

e.

g. G

-L-S

stirr

ed

tank

s, sl

urry

bub

ble

colu

mn

Stan

dard

k-ε

turb

ulen

ce m

odel

(FLO

W3D

)-Stir

red

tank

Tw

o-flu

id m

odel

Fu

ture

cha

lleng

es: A

vaila

bilit

y of

exp

erim

enta

l dat

a fo

r va

lidat

ion;

diff

icul

t to

capt

ure

the

mic

rosc

opic

phe

nom

ena

(e.g

., pa

rticl

e ac

cum

ulat

ion

near

the

G-L

inte

rfac

e).

Ver

y lit

tle

Ham

ill e

t al.,

199

5

G-L

-S fl

uidi

zed

beds

No

99

100

5.1.2 CFD and Other Modeling Approaches to Multiphase Flow in

Packed Beds Packed-beds have been extensively used in petroleum, petrochemical and

biochemical applications (Dudukovic et al., 1999). The stationary packing in the columns

can be either active catalyst for chemical reaction systems or an absorbent in a separation

column. Depending upon the application there are multiple configurations available for

packed beds with gas and liquid flows: cocurrently downward (i.e., trickle-bed),

cocurrently upward (i.e., packed bubble column) and counter-currently flows (e.g.,

catalytic distillation column). The criteria for choosing the proper flow direction have

been established, and the evaluation of the effect of flow direction on reactor

performance has also been performed (Wu et al., 1996; Khadilkar et al., 1996). Since

most of these models rely on assumed ideal flow patterns and are one dimensional, the

accurate prediction of multiphase flow pattern (i.e. spatial and temporal distributions) in

the packed beds is still an unresolved issue, which is an obstacle to advanced reactor

model development.

Multiphase flow modeling in packed beds is a challenging task because of the

difficulty in incorporating the complex geometry (e.g. tortuous interstices) into the flow

equations, and the difficulty in accounting for the fluid-fluid (gas-liquid) interactions in

presence of complex fluid-particle (e.g., partial wetting) contacting. Moreover, until

recently, the lack of noninvasive experimental techniques suitable for validating the

numerical results was also a detrimental factor in numerical model development due to

lack of reasonable validation.

The earliest flow models of packed beds focused on the bed-scale flow pattern

without considering the detailed heterogeneities of the bed structure. The 'diffusion'

model (Stanek et al., 1974) and porous media model (Anderson and Sapre, 1991) are

examples of such an approach. To account for the statistical nature of the bed structure, a

'percolation based' model was adopted to predict the flow pattern in packed beds (Crine et

al., 1979). These models provided certain predictions of the overall quantities that were

found comparable with the experiments; however, they could not yield much insight into

the flow distribution in the beds. A discrete cell model (DCM) approach evolved from the

101

assumption made by Holub (1990) that flow distribution is governed by the minimum

total energy dissipation rate. In the recently updated DCM, as presented in Chapter 4, a

statistical assignment of cell porosity values and the incorporation of the interfacial

tension force related to the particle wetting and inflow distributors has been accomplished

for two-phase flow in trickle beds. The quantitative predictions of liquid upflow in

packed beds by the DCM approach compare well with the available experimental data

and other independent numerical methods as presented in Chapter 3. However, the

numerical scheme of multivariable non-linear minimization used in DCM often leads to

low computational efficiency when dealing with a large packed bed with small cell

dimensions.

Direct numerical simulation (DNS) on single particle and single void scale

requires complete characterization of solids boundaries and voids configuration, which is

difficult to obtain for a massive packed bed. Statistic implementation of porosity

distribution for a large size packed bed is proper for modeling of the macroscopic flow

field. For example, to consider the interactions between fluid and particles a global flow

model in packed beds, a k-fluid model, resulting from the volume averaging of the

continuity and momentum equations, has been developed and solved for a one-

dimensional representation of the bed at steady state, and at isothermal non-reaction

conditions (Attou et al., 1999). It provided reasonable predictions for global

hydrodynamic quantities such as liquid holdup and pressure-drop. A similar k-fluid

model, based on the relative permeability concept, was used to compute the two-

dimensional flow without considering porosity variation and without solving for the solid

phase. The simulated liquid flow pattern qualitatively agreed with experimental

observation (Anderson and Sapre, 1991). It seems that the Eulerian-Eulerian two-fluid

model is a rational choice for flow simulation in packed beds if good closures for fluid-

fluid and fluid-particle interactions can be found. Moreover, the geometrical complexity

of packed beds can in a certain sense be avoided in a two-fluid model, since there is no

need to deal with the exact boundaries of particles, and since one treats the solid phase as

a penetrated continuum. A study has been made to resolve the flow field at fine scale and

CFD simulations were conducted of heat-transfer in a tubular fixed-bed using a 3-D fine-

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mesh within the pore space (Logtenberg and Dixon, 1998). These simulations were

limited to the tube with very low column to particle diameter ratio (Dr/dp = 2~3) and

with large particle size (e.g., 5cm). Obviously, it is impossible to adopt such an approach

for a massive commercial packed bed, or even for a bench-scale trickle bed packed with

small particles (e.g., 0.5~3 mm). Hence, one has to discover an efficient way to

implement the bed structure into the flow model. It is most desirable to retain all the

statistical characteristics of the pore space but without introducing the real pore structure,

since the exact 3-D interstitial pore-structure varies with repacking the bed, even with the

same particles and using the same packing method, although the mean porosity may

retain the same value.

In this work, we introduce a statistical description of the bed structure into a k-

fluid model framework. In order to properly consider the effect of the solid phase on gas

and liquid flows, the k-fluid model is applied to the gas, liquid and solid phase

simultaneously while turning off the momentum equation for the solid phase, so that the

initial volume fraction distribution of the solid phase is retained.

The work accomplished is presented in two subsequent Chapters. In this Chapter

(Chapter 5) the various issues related to the k-fluid model implementation in packed-beds

were discussed, and the current state of the art for closures is presented. The multi-scale

and statistical nature of flow is illustrated and the choice of the grid size and boundary

conditions is discussed. Chapter 6 presents selected numerical simulation results based on

the model presented in Chapter 5, and discusses the comparison of the numerical results

with available experimental data and recommends the future research-focus and a

methodology for utilizing the modeling results in packed-bed analysis and design.

5.2 Spatial and Temporal Characteristics of Flow in

Packed Beds There are many structure and flow parameters responsible for the flow

distribution in packed beds such as porosity distribution in the bed and the inlet flow

velocity distribution (see Chapter 4). For example, in a system saturated with single-

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phase flow (e.g., gas flow, liquid upflow case), the spatial variation of porosity is the

essential parameter in determining the spatial distributions of fluid velocity and volume-

fraction variations. In a system with gas and liquid two-phase flow, the additional

parameters affecting the liquid distribution are the state of particle external wetting, the

interaction between phases, distributor design, etc. It is believed that there exists a

quantitative relation between flow, bed structure and operating conditions of the system.

Moreover, since the flow distribution/maldistribution can be observed at different spatial

scales (Melli et al., 1990; Wang et al., 1998), it has been suggested that different scales

be used to describe the corresponding flow phenomena. This so-called ‘multiscale’ nature

of the flow in packed beds results from the multiscale heterogeneities of bed structure.

In packed beds, two complementary spaces coexist: the grain space and the

porous space (i.e. cavity). The pore size, defined by the radius of the largest sphere,

which can be put inside that cavity, depends on particle size, shape and packing method.

For porosities of 0.36 to 0.4 obtained from monosize spheres, the pore size is in the

range of 0.38R to 0.44R (R: radius of particle). As reported in several studies on packed

beds, the mean porosity is reproducible for a given packing method with the standard

deviation of only 0.0016 (Cumberland and Crawford, 1987). The longitudinally

averaged radial porosity profile follows a certain oscillatory pattern due to the confines

of walls, which can be predicted in terms of particle size, shape and column to particle

diameter ratio (Benenati and Brosilow, 1962; Mueller, 1991; Bey and Eigenberger,

1997). Although typical bed structural information such as the above is available, it is

not sufficient to predict the complete spatial distribution of flow in packed beds.

Additional information on porosity distribution in 3-D or at least 2-D, wall effect,

entrance and exit effect on flow are needed before going on to flow simulation.

For both steady state and dynamic flow simulation in packed beds, the temporal

behavior of flow has to be considered. In a two-phase flow trickle-bed in which gas is

the continuous phase whereas the liquid is trickling down through the packing (i.e.

trickle bed) at low superficial velocity, so-called ‘trickle-flow regime’ in the literature,

the bed-scale liquid flow pattern is rather stable whereas the local scale interstitial flow

within the pore space still fluctuates in a chaotic motion. Once the gas and liquid

104

superficial velocities increase to a certain level, and the flow reaches so-called ‘high

interaction pulsing regime’, even the macroscale liquid pattern becomes unstable: liquid

rich-zone and gas-rich zone move alternately through the bed with a certain frequency.

Such a macroscale flow fluctuation pattern can be also generated through the periodic

input of the flows, which has been shown to enhance the reactor performance (Khadilkar

et al., 1999).

The experimental exploration of these spatial and temporal flow variations in

packed beds are definitely important, but it is difficult for a single technique to capture

both spatial and temporal behavior of flow simultaneously with high resolutions

(Reinecke et al., 1998). For example, the magnetic resonance imaging (MRI) can provide

a good spatial resolution of 0.02-0.3 mm, but it is not suitable to measure dynamics of the

flow such as these encountered in pulsing flow regime due to the temporal resolution

problem. The electric capacitance tomography (ECT) gives a temporal resolution of a

millisecond, but with relatively poor spatial resolution at this stage (Reinecke et al.,

1998). On the numerical flow modeling side, a similar trend exists. We do not expect to

use a single model to obtain information on a variety of spatial and temporal scales of

flow, but we are to obtain one level of flow information through one particular model. In

these two Chapters, we focus on the macroscale flow pattern at steady state operating

conditions. We do explore the dynamic flow behavior of large-scale structures under

periodic operating condition by including flow modulation to examine the possible

improvement of the liquid distribution, but we do not intend to model the flow dynamics

in the natural pulsing flow regime, which involves complex flow dynamic mechanism

(Tsochatzidis and Karabelas, 1995).

5.3 Structure Implementation The implementation of porosity distribution in flow simulation increases the level

of difficulty in packed beds as compared to other multiphase reactors. So far, this issue

has been tackled in a deterministic and simplified manner to a large extent. For example,

either uniform porosity or the radial porosity variation is considered in the model of the

105

bed (Bey and Eigenberger, 1997; Yin et al., 2000). In some cases, a multi-zone porosity

assignment was used (Stanek, 1994). Since the 3-D interstitial pore space varies with

repacking the bed, the porosity distribution possesses a statistical nature (Wijngaarden

and Westerterp, 1992) and the use of statistical description of the porosity structure in the

flow model has considerable potential for success (Crine et al., 1992). In this Section we

discuss how to partition the 3-D pore space into sections and what will be the type of the

section porosity distribution, since in the flow simulation of a volume-averaged k-fluid

model, one needs to assign the initial solid phase volume fraction to each section.

Depending on the section size chosen for the partition, the section porosity values follow

a certain probability density function (p.d.f). That means that the p.d.f. is section size

dependent. For example, the measured section porosity data from a cylindrical column

packed with 3-mm monosize spheres has exhibited a Gaussian distribution at a section

size of 3 mm (Chen et al., 2000). However, a nearly binomial type of section porosity

distribution was found by MRI measurement at a section size of 180 µm (Sederman,

2000).

In principle, a quantitative relationship of the section size and the variance of

section porosity distribution, σB, can be developed through extensive MRI measurements

of packed beds. Obviously, this relationship varies with particle shape and packing

method. Thus, for a certain size of a section, a set of pseudo random section porosities

can be generated based on the following constraints

• Mean porosity (measurable)

• Longitudinally averaged radial porosity profiles (correlation available)

• Correlation of section size, lv and the variance of section porosity distribution, σB

(obtainable by MRI, Sederman, 2000).

Figure 5-1a shows a sample contour plot of 2-D section porosity distribution in r-z

coordinates, which was generated under the constraints of a mean porosity of 0.35, a

longitudinal averaged radial porosity profile (see Fig.5-1b) measured by Stephenson and

Stewart (1986) and a pseudo Gaussian distribution with a variance of 12% mean porosity

(see Fig.5-1c). The tail shown at high porosity range in Figure 5-1c indicates the effect of

106

walls on porosity. Such porosity generation process under constraints has certain

analogue to particle repacking process in practice.

It is noted that based on the mean porosity and the radial profile of section

porosity, for a given section size, many possible probability density functions exist. The

third constraint is definitely needed for generating a realistic porosity distribution.

Moreover, although the example used provided an illustration for a 2-D distribution case,

the approach is applicable for the 3-D porosity distribution case.

107

(a)

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.0 1.0 2.0 3.0 4.0r (cm)

sect

iona

l por

osity

(b)

108

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

5

10

15

20

25

30

35

40

45

50

(c)

Figure 5-1. Generated pseudo-Gaussian distribution of porosity under three constraints:

(1) ε0 = 0.36; (2) Longitudinally averaged radial porosity profile (white filled circles)

reported by Stephenson and Stewart (1986). (Dr = 7.6 cm, dp = 0.703 cm, Section size =

0.05R = 0.19 cm). (a)-contour plot; (b)-radial profiles; (c)-histogram (standard deviation

of porosity, σB = 12% ε0).

5.4 k-Fluid Approach and CFDLIB Code

5.4.1 Eulerian k-Fluid Model In the Eulerian k-fluid approach, the different phases or materials are treated

mathematically as interpenetrating continua (Ishii, 1975). The derivation of the

conservation equations for mass, momentum and enthalpy is performed by using either

the volume averaging or ensemble averaging technique to describe the time-dependent

motion of fluids and track volume fraction distribution of each phase. In the ensemble

averaging technique, the probability of occurrence of any one phase in multiple

realizations of the flow is given by the instantaneous volume fraction of that phase at that

point. Sum total of all volume fractions at a point is identically unity (Anderson and

Jackson, 1967).

109

The microscopic flow structure such as local detailed flow structure is achievable

only by direct numerical simulation (DNS), which is limited to relatively low Reynolds

and Schmidt numbers (Kuipers and van Swaaij, 1998). Since DNS is not possible for

most industrial gas-liquid flows, several authors (Sokolichin and Eigenberger, 1994;

Ranade, 1995; Pan et al., 2000) have used the Eulerian k-fluid model for simulation of

the dynamic gas-liquid flow by modeling subgrid/local phenomena and simulating the

large-scale phenomena. It is well known that the successful applications of such

simulations in multiphase flow are mainly dependent on the appropriate closure laws for

the inter-phase transport of mass, momentum and energy (Delnoij et al., 1997; Kuipers

and van Swaaij, 1998). For modeling multiphase flow in packed beds, additional effort is

needed, as discussed above, to properly implement the porosity structure of packed beds

into the model equations.

In this work, the packaged computational fluid dynamics code, CFDLIB,

developed by Los Alamos National Laboratory (Kashiwa et al., 1994), has been adopted

as a transient multiphase flow simulation tool using a k-fluid model for the simulation of

multiphase phase flows in packed beds. The key aspects of the k-fluid model and the

main features of CFDLIB code are described below.

5.4.2 k-Fluid Model in CFDLIB CFDLIB is a library of hydrocodes that share a common numerical solution

algorithm, and a common data format (i.e., ‘block-structured’). There is a hierarchy to the

codes in CFDLIB, which depends on the complexity of the systems dealt with (from

multispecies, multiphase and compressible system to single species, single phase and

incompressible system). The time dependent mass, momentum and energy conservation

equations were derived using ensemble-averaging technique, and were cast in the integral

form, and the solution is based on the finite-volume method (FVM) (Kashiwa et al.,

1994).

In the finite-volume approach, the physical domain is subdivided into small

volumes, and the dependent variables are evaluated either at the center of the volumes

(cells) or at the corners (vertices) of the volumes. In the CFDLIB codes, the physical

110

domain is divided into the main computational subunits: blocks, sections and cells. A

block is a logical rectangular portion of the meshes, having left, right, and bottom and top

boundaries. In each mesh direction within the block there are several sections. In each

section, the material data is constant and is specified in the input file before initializing

the calculation. Therefore, the material data distribution information can be introduced

into the computational domain through each section. Moreover, the code allows unequal

section size within a single block. All these features allow inputting a detailed porosity

distribution data as an initial condition in the input file. The code can discretize each

section into cells; each cell has four vertices and four faces for 2D simulation treated

here. Each interior face is common to two cells, and each interior vertex is common to

four cells. Figure 5-2 illustrates how to assign the logical block, sections and cells from

the physical block in the 2D CFDLIB code, respectively. It is noted that in the two-

dimensional CFD computation in CFDLIB code, the 2-D cells still represent three-

dimensional volumes. For example, in 2-D Cartesian coordinate, the cells have a nominal

depth of ∆z = 1 length unit; in 2-D cylindrical coordinate, the cells have a normal depth

of ∆ ϕ =1 radian.

For the system with more complicated geometry, several blocks are needed,

which can be linked together in logical space by boundaries that are completely

transparent to the flow (Johnson et al., 1997). This finite-volume method used in

CFDLIB has an obvious advantage over a finite-difference method if the physical domain

is highly irregular and complicated, since arbitrary volumes can be utilized to subdivide

the physical domain. Since the integral equations are solved directly in the physical

domain, no coordinate transformation is required. Also the mass, momentum, and energy

are automatically conserved, since the integral forms of the governing equations are

solved (Tannehill et al., 1997).

The ensemble-averaged conservation equations that serve as the basis for k-fluid

model in CFDLIB are:

111

Mass equation (continuity):

kkk u

tρ

ρ⋅∇+

∂∂

>=< kαρ !0 (5-1)

The terms on the left hand side of Equation (5-1) constitute the rate of change in mass of

phase k at a given point, and the term on the right hand side is the source term due to

conversion of mass from one phase to the other. kα represents the net rate at which

material k is being created. In present study this term is equal to zero since no phase

exchange, reaction or mass transfer is considered at this stage.

Momentum equation:

=⋅∇+∂

∂kkk

kk uutu

ρρ

><+ ok uαρ !0 (Mass exchange source)

><∇⋅− ''0 kkk uuρα (Reynolds stress)

>∇<+ 0pkα (Pressure term)

>⋅∇<+ 0τα k (Stress term)

gkρ+ (Body force) (5-2)

After decomposing the pressure and stress terms in terms of pressure acceleration

and material stress divergence, one can get the following momentum equation:

=⋅∇+∂

∂kkk

kk uutu

ρρ

><+ ok uαρ !0 (Mass exchange source)

><∇⋅− ''0 kkk uuρα (Reynolds stress)

pk ∇−θ (Pressure term)

( )pp kk −∇− 0θ (Mean pressure)

( )[ ] kIpp ατ ∇⋅+−−<− 00 (Momentum exchange)

><∇⋅+ 0τα k (Average stress)

gkρ+ (Body force) (5-3)

112

To close Equation (5-3), the closure models for computing the Reynolds stress and the

momentum exchange terms are needed. Such closure problems can be resolved either by

phenomenological models (e.g., Ergun equation), or by the formula from the

microstructure flow element (e.g., DNS), or from an original transport equation (e.g.,

Lattice Boltzmann simulation). In the flow modeling of packed beds discussed in this

chapter, the phenomenological closure formulas are used.

(a) (b) (c)

Figure 5-2. Block, sections and cells in CFDLIB for packed bed modeling: (a) physical

block, (b) logical block consists of a number of sections, (c) a section consists of a cell or

a number of cells.

Cell Block

Section

113

5.5 CFD Modeling Issues To adopt the CFDLIB code to the present flow problem in packed beds, some

subroutines related to the closures and phase pressure calculations need to be added and

modified to carefully take into account the essential physics of the system.

5.5.1 Significance of Terms in the Momentum Balance To numerically describe the flow pattern at different scales, one has to formulate

the governing flow equations with different dependent basic force terms: inertial force,

viscous force, capillary force, gravitational force and turbulence related force, etc. For

example, one has to take Reynolds stress term into consideration in the fine-mesh CFD

modeling with high gas flow rate (Logtenberg and Dixon, 1998). In fact, the existence of

microscale turbulence in porous media has been detected by several experiments by

point-wise probes (Jolls and Hanratty, 1966; Latifi et al., 1989). For the macroscopic

flow modeling in packed beds, however, the contribution of the Reynolds stress term to

the fluid momentum equation is not important as discussed in Chapter 3 because when

averaging a number of local (random) signals within a representative elementary volume

(e.g., a cubic cell contain a cluster of particles), the microscopic turbulence is smoothed

out. More discussions of micro- and macro-scale turbulence modeling in porous media

are available elsewhere (Lage, 1998).

The size and the shape of packing elements determine the void structure of the

granular assembly, which further affects the contribution of each basic-force on flow

distribution. Table 5.2 lists the typical ranges of these force ratios in gas and liquid flow

through the packed granules. Obviously, oil displacements are capillary-dominated

creeping flows. In packed beds used as separation columns with large packing elements

(e.g., 10 ~ 30 mm Pall rings and Rasching rings), both the gravity and inertial forces are

important, whereas the liquid distribution patterns are not much sensitive to the wetability

of the packing surface (see Bemer and Zuiderweg, 1978) due to negligible capillary

force. However, in trickle beds, the particle sizes are typically in the range of 0.5 to 3

mm, all the forces contribute to the flow distribution, and the influence of particle

114

external wetting on liquid distribution is significant (Lutran et al., 1991; Ravindra, et al.,

1997a; also see Chapter 2). This implies that even for describing the same macroscale

flow pattern in the overall packed bed, the contribution of each basic-force may be of

different magnitude depending on the different characteristic radii of the flow passages.

Table 5.2 Typical ranges of force rations in two-phase flow in packed granular packing

(adapted from Melli et al., 1990)

Packings Particle dp, m

Re inertial/viscous

Ca viscous/capillarity

1/Bo capillarity/gravity

Porous media: oil displacement; chromatography

10-7 - 10-4 10-9 - 10-2 10-7 - 10-3 102 - 109

Trickle-beds 10-3 - 10-2 10-2 - 103 10-1 – 10 10-1 - 10 Separation columns

10-2 – 10-1 10 - 105 10 -102 10-3 - 10-1

Reynolds number: kkpP Ud µρ≡Re ; Capillary number: SLLUCa σµ≡ ;

Bond number: ( )[ ] SGLp gdBo σρρ −≡ 2

The inertial effect in modeling of flow in porous media has been the topic of

debate for many years (Stanek, 1994; Lage, 1998). The numerical modeling of single

flow phase in packed beds has shown that the contribution of the inertial term to the total

(mechanical) energy dissipation rate is negligible compared to the viscous term and to the

kinetic term except in the regions of structural obstacles which change the flow direction

sharply (see Chapter 3). This agrees with the experimental findings of gas flow through

the packed beds with obstacles (Choudhary et al., 1976). In trickle beds with two-phase

cocurrent flow, the increase in gas and liquid flow rates can generate high inertial forces

exerted in the bulk fluids, which further contributes to the growth of interfacial waves

and to the bed-scale destabilization of the trickle flow regime. On the other hand, the high

inertial force causes the gradients in liquid saturation, and further result in the capillary

force at gas and liquid interface, which contributes to the attenuation of interfacial waves

and to the stabilization of the trickle flow regime. The simulation of 1-D bed using a two-

115

fluid model has shown that the inertial forces of fluids play an important role in the

mechanism of flow transition from trickling to pulsing flow (Grosser et al., 1988; Attou

and Ferschneider, 2000).

In summary, the Reynolds stress term is not important in determining the

macroscale flow pattern in packed beds with a particle size of 10-4 to 10-2 m, however,

other basic forces (i.e., inertial, viscous, gravity and capillary forces) are normally of a

similar order of magnitude so that they have to be taken into account in the flow

equations in a proper way.

5.5.2 Closures for Multiphase Flow Equations The volume averaging technique for equations of motion leads to the well known

closure issue for some of the terms associated with fluctuating variables and source terms

in which some of the forces acting on a representative permeable volume need to be

modeled. For single-phase flow through porous media, several studies used the effective

viscosity idea of Brinkman, and lumped the forces acting on the fluid phase of the

permeable medium into an effective viscous force (Bey and Eigenberger, 1997). In this

work, we compute the drag forces due to fluid-particle and fluid-fluid interactions based

on the phenomenological models developed in bench-scale hydrodynamics experiments.

Moreover, the magnitude of the drag force is expressed as a product of a user defined

exchange coefficient, Xkl, phase volume fractions, θl, θk and relative interstitial velocity

of the two phases k and l as below

( ) ( )lkkllklkD uuXF −=− θθ (5-4)

Clearly, the essential part of Eq (5-4) is to determine the exchange coefficient values, Xkl.

Thus far, there are several models capable of providing Xkl values: namely, the

relative permeability model (Saez and Carbonell, 1985), the single slit model (Holub, et

al., 1992), the two-fluid interaction model (Attou et al., 1999) as tabulated in Table 5.3.

Note that the interaction force between the gas and the liquid phase was neglected in both

single slit and relative permeability models indicating zero shear stress at the gas-liquid

interface. This may be true only for the case where gas and liquid flows are both low, and

116

the flow system is located ‘deep’ in the trickling flow regime. Otherwise, one does need

to take into account this force in the hydrodynamics computations because experimental

studies of Larachi et al (1991) and Al-Dahhan and Dudukovic (1994) have shown that

gas flow can exert considerable influence on the hydrodynamics of the trickle-bed

reactor, especially at high operating pressure and/or high gas velocity. Hence, Attou et al

(1999) included the gas-liquid interaction force (see Equation 5-9) in the 1-D two-fluid

model based on an ‘annular flow’ model in which the gas and liquid phases are separated

by a smooth and stable interface. Good predictions of liquid holdup and pressure-drop

were claimed in their paper. In our earlier CFD simulations of two phase flow (see

Chapter 4), either no interaction was assumed or a drag formula of a single-sphere in

fluid was used as an approximation for the momentum gas-liquid exchange coefficient,

Xgl. In this work, the gas-liquid interface drag formula for Xgl developed by Attou et al

(1999) has been chosen due to more appropriate physical basis.

The expressions for other two-phase momentum exchange coefficients, Xkl, are

written in similar format in Table 5.3. The comparisons of these expressions for given

sectional porosity and particle size are shown in Figure 5-3. For a given gas superficial

velocity of 6.0 cm/s, an increase in liquid superficial velocity causes Xgs to increase and

Xls to decrease. Moreover, the relative permeability model gives relatively higher values

of Xgs and Xls than either the single slit model and the two-fluid interaction model.

Hence, the relative permeability model gives lower predictions of liquid holdup than the

single slit model at the same flow conditions as shown in Figure 5-4. The two-fluid

interaction model yields the same values of Xls as the slit model, and provides

intermediate values of Xgs, between the slit model and the relative permeability model.

The effect of liquid velocity on the Xgl value is not significant. Due to the lack of detailed

flow velocity and volume-fraction distribution data together with known bed structure at

certain section scale, we are not able to establish the best drag expression for Xgl at the

present time. However, we can appreciate how these drag expressions affect the

predictions of the liquid holdup and pressure gradient at bed scale, which will be

presented in Chapter 6. The full validation of the best drag expression, in principle, will

be possible by using MRI technique to obtain the needed experimental data.

117

0

5

10

15

20

0 1 2 3 4 5 6 7 8

Ul, superficial velocity, kg/m2/s

X gs,

X gl

x1.0

4

0.0

2.0

4.0

6.0

X ls

x1.0

6

Xgs(A) Xgs (H) Xgs (SC)

Xgl (A) Xls (H) Xls (SC)

Xls (A)

Figure 5-3. Comparison of Xkl values from different models [Ug = 6 cm/s]: A- Two-fluid

interaction model (Attou et al., 1999); H- Single slit model (Holub et al., 1992); SC-

Relative permeability model (Saez and Carbonell, 1985).

The overall particle external wetting efficiency can be calculated by the

correlation of Al-Dahhan and Dudukovic (1995) based on the superficial velocities of gas

and liquid as well as particle parameters etc. As shown in Figure 5-4, at the bed scale, the

particle external partial wetting does exist at a low liquid superficial mass velocity (L

<5.0 kg/m2/s). Therefore, the drag formulations derived from the double-slit model in

which two interconnected slits: wet and dry slit, are assumed (see Iliuta et al., 2000),

could provide a reasonable alternative, because it is possible then to account for particle

wetting in the multiphase drag calculations. However, the computation may become

cumbersome due to too many equations involved in the double-slit model. Furthermore,

118

how much improvement can be gained is still uncertain, because based on the

comparisons of these drag models in predictions of global hydrodynamics quantities

Larachi et al (1999) and Iliuta et al (2000) concluded that all of these models fit the

experimental data to about the same degree of accuracy.

0.0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8

Ul, kg/m2/s

Liqu

id h

oldu

p

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Wet

ting

effic

ienc

y

hp: S & C modelhp: Holub modelwt: S & C modelwt: Holub model

Figure 5-4. Effect of liquid superficial mass velocity on liquid holdup (hp) and particle

external wetting efficiency (wt) at a gas superficial velocity of 6 cm/s. Holub model

(Single slit model, see Holub et al., 1992); S & C model (Relative permeability model,

see Saez and Carbonell, 1985). Particle external wetting efficiency values (wt) were

calculated by the correlation of Al-Dahhan and Dukovic (1995). wt-S & C model means

the pressure-drop value used in calculating wt value was from S & C model; wt-Holub

model means the pressure-drop value used in calculating the wt value was from Holub

model.

119

Table 5.3 Models for drag coefficients Single Slit Model (Holub et al, 1992) Fluid-Particle: (Xls and Xgs)

( )( ) kkkSkkkSkS u

UBUAXε

ρµ α −+=

112 (5-5)

( )23

2

11

pkkS d

EAϑ

ε−= ; ( )pk

kS dEB 32

1ϑ

ε−=

Relative Permeability Model (Saez and Carbonell, 1985) Fluid-Particle: (Xls and Xgs)

( )( ) kkkSkkkSkS u

UBUAXε

ρµ α −+=

112 (5-6)

( )28.4

8.121180pg

gS dA

ϑεε−= ; ( )

pggS d

B 8.4

8.118.1ϑ

εε−=

( ) 43.2

0

0

23

21180

−−−=

ll

l

pllS d

Aθθθε

ϑε ; ( ) 43.2

0

0

3

18.1

−−−=

ll

l

pllS d

Bθθθε

ϑε

Two-Fluid Interaction Model (Attou et al., 1999) Fluid-Particle: (Xls and Xgs)

( )( ) llllslllslS u

UBUAXε

ρµ−

+=1

12 (5-7)

( )23

21180pl

lS dA

ϑε−= ; ( )

pllS d

B 3

18.1ϑ

ε−=

( )( ) g

gggsgggsg

gS uUBUAX

ερµ

εθ

−+=

112 (5-8)

( ) 32

23

2

111

180

−−−

=gpg

ggS d

Aθε

ϑθ

; ( ) 31

3 111

8.1

−−−

=gpg

ggS d

Bθε

ϑθ

Fluid-Fluid: (Xgl)

( )( ) lg

rggsrgglg

gl uuUBUAX

−−+=

ερµ

εθ

112 (5-9)

gsgl AA = ; gsgl BB = ; lggr uuU −=θ

120

200

240

280

320

360

0 0.1 0.2 0.3liquid holdup

Cap

illary

pre

ssur

e, P

a

J functionAttuou

Figure 5-5. Comparison of the calculated capillary pressure values from two different

expressions, Eq 5-17a and 5-17b for air-water system (dp = 0.003m; θs =0.63)

5.5.3 Interfacial Tension Effect, Wetting Correction Direct and indirect liquid flow visualizations have shown that the effect of pre-

wetting of the packing on liquid distribution is significant (Lutran et al., 1991; Ravindra,

et al., 1997; see also Chapter 2). The analysis of basic forces outlined in Section 5.1 also

confirms this result. In general, liquid holdup and particle wetting efficiency are reduced,

and liquid rivulets are favorably formed when introducing trickle flow into the dry

packing. Moreover, the liquid distribution in packed beds is a function of flow history

(Lutran et al., 1991; Ravindra, et al., 1997a). It means that the wetting state of the particle

surface affects the upcoming flow distribution significantly. Experimental studies have

established that the gas and liquid interfacial tension forces and the packing wettability

are not only responsible for the liquid flow maldistribution (see Lutran et al., 1991), but

are also responsible for the hysteresis observed in the pressure-drops and liquid holdup

measured during cocurrent and countercurrent flow in packed beds (see Levec et al.,

121

1988). To numerically capture these flow phenomena, one must consider the interfacial

tension effect and packing wettability on the pressure calculations.

When two immiscible fluids (e.g., gas and liquid) are in contact with each other,

interfacial tension causes the fluids to have different pressures. This discontinuity in

pressure between fluids is know as the capillary pressure, Pc. Specifically, we define the

capillary pressure between gas (G) and liquid (L) as:

CLG ppp =− (5-10)

At a pore scale the capillary pressure can be expressed by

+=

21

112dd

p SC σ (5-11)

The characteristic lengths d1 and d2 are further described in terms of particle diameter,

porosity and the minimum equivalent diameter of the area between three particles in

contact as well as pressure factor, F, as given in Eq 5-12b (see Attou and Ferschneider,

2000).

+

−

=L

G

pG

SSC F

ddp

ρρ

θθ

σmin

3111

12 (5-12a)

L

G

L

GFρρ

ρρ

1.881+=

for ρG/ρL< 0.025 (5-12b)

The capillary pressure also could be expressed through the permeability concept together

with the correlation of the experimental data in various porous media. Grosser et al.,

(1988) proposed the following expression for calculating the capillary pressure

( )5.01

,

−

=k

Jp SSLC

θθθσ (5-13)

where σ is the surface tension, k is the permeability of the porous media, which is related

to the Ergun constant (E1) and the equivalent particle diameter (de) for viscous flow in

packed beds; J is a dimensionless function obtained from the experimental data of

various sand samples with air and water (Leverett, 1941) as given below

122

( ) eS

SS

dE

k θθθ−

=

−

11 5.0

15.0

(5-14)

( )

−−+=

L

LSSLJ

θθθ

θθ 1ln036.048.0, (5-15)

Therefore, the capillary pressure is a function of liquid holdup, and the gradient of the

capillary pressure depends on the gradient of liquid holdup in the packed bed. From the

experimental observation of liquid distribution in prewetted and non-prewetted beds (see

Chapter 2), complete pre-wetting of the particle surface can greatly reduce the gradient of

liquid holdup in the packed bed, which considerably reduces the capillary pressure effect

on liquid distribution. For the modeling of macroscale flow, Equation (5-10) is further

modified by incorporating the particle wetting factor, f, or external particle wetting

efficiency (a fraction of external particle area wetted by liquid) as given below

( ) CGL pfpp −−= 1 (5-16)

By substituting Equation (5-12), or Equations (5-13), (5-14) and (5-15), into Equation (5-

16), one gets Equations (5-17a) and (5-17b), respectively.

( )

+

−

−−=L

G

pG

SSGL F

ddfpp

ρρ

θθ

σmin

3111

112 (5-17a)

( ) ( )

−−+

−−−=

L

LS

eS

SSGL d

Efpp

θθθ

θθ

σ1

ln036.048.01

15.0

1 (5-17b)

It means that the macroscale capillary effect is negligible when the particles are

completely externally wetted (f = 1.0) whereas this effect could be significant when the

particle surfaces are completely dry (f = 0), as one can see from the experiment results

with the non-prewetted beds in Chapter 2.

In the CFD flow simulation the value of particle wetting factor at each cell scale

can be evaluated based on the cell scale flow velocities and local pressure-gradient using

the correlation of Al-Dahhan and Dudukovic (1995) for the external wetting efficiency of

particles in trickle beds.

123

9/1

3/1 1

1Re104.1

+∆

−=

L

LL

GagZP

f ρε

(5-18)

The simulations reveal that in cases of partial particle wetting (f < 1.0), the contribution

of macroscale capillary pressure on liquid flow distribution is significant. It is also

expected that the hysteresis observed in the pressure-drops and global liquid holdup at

bed scale is due to the hysteresis in liquid flow distribution. This in turn is caused by the

capillary pressure hysteresis and different particle wetting status in liquid imbibition and

drainage experiments.

The difference of the calculated capillary pressure values by two expressions,

Equations (5-17a) and (5-17b) does exist even for the same system of air-water as shown

in Figure 5-5. It is noted that Equation (5-17b) was originally derived based on the

experimental data for air-water flow through consolidated porous media (such as sands,

Leverett, 1941). This expression was suggested for flow through packed beds (Grosser et

al., 1988). The derivation of Equation 5-17a was based on the local linear momentum

balance law, applied to the gas-liquid interface, in which the effect of gas density was

incorporated through F(ρG/(ρL), and was claimed to be suitable for elevated pressure

system (see Attou and Ferschneider, 2000). In fact, there have been no direct experiments

designed for validation of these two expressions for trickle beds. For the present time, the

J-function expression (i.e. Eq 5-17b) is used for the air-water system, and the expression

of Attou and Ferschneider (2000) can be used for other systems, particularly at elevated

operating pressures.

5. 5.4 Effect of Mesh Size on Computed Results In general, the grid is a discrete representation of the continuous field phenomena

that one wants to model. The accuracy and numerical stability of the simulation depends

on the choice of the grid (Tannehill et al., 1997). In CFDLIB code the finite volume

method is used to discretize the conservation equations. At this stage, we utilized two

different 2-D coordinate systems for the grid cells: Cartesian and Cylindrical coordinates.

124

In the 2-D cylindrical coordinates, we assume that there is no dependence on the θ-

direction. Although the flow in a cylindrical column does distribute in three-dimensions,

the radial and vertical distributions of the flow play a more important role than the θ-

direction distribution in determining the reactor performance (Stanek, 1994). In addition

to discretization in space, an explicit temporal discretization scheme is utilized in the

code, so the solution proceeds with respect to a sequence of discrete time, t n, where n is

the cycle number (n = 0, 1, 2, …). The time step ∆t n = t n+1 - t n varies from cycle to

cycle.

As discussed in Section 3, there is certain relationship between the section size

and the standard deviation of sectional porosities. After the section size is chosen, the cell

size needs to be specified by further space discretization. Once a converged solution has

been obtained it is essential to assess the invariance of the computated results with

respect to the section discretization. Particularly for the flow in packed beds, various

flow scales and structural scales exist which make the selection of the grid size important

to generating meaningful computational results. To test the dependence of the solution on

the grid size, two simulation runs, one with a fine grid, and another with coarse grid size

were performed. Figures 5-6 and 5-7 show the comparison of the steady state liquid

upflow interstitial velocity components (Vx, Vz) in the forms of contour and transversal

profiles. Apparently, the flow patterns do no vary with changing the cell size from 1.0 cm

to 0.5 cm. However, the relatively detailed flow characters are obtained in the fine-cell

simulation. Figure 5-8 shows the initial solid volume fraction at section size of 1 cm for

two-phase cocurrent down-flow simulation at the gas and liquid superficial velocities of

0.001 m/s and 0.05 m/s, respectively. If one zooms in a specific square area, say x = 0 ~

4, and z = 4 ~ 8, one can see the sectional flow patterns are similar but the cell scale flow

texture becomes more detailed when the cell size is reduced from 1.0 cm to 0.25 cm as

shown in Figure 5-9. The mesh size independent gas holdup at section scale are shown in

Figure 5-10.

125

XZ

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

CFDLIB97. 2

T=8.994E+01

N=60001

X

Z

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

U243.532.521.510.50

-0.5-1-1.5-2-2.5-3

CFDLIB97.2

T=8.994E+01

N=60001

CFDLIB97.2

T=4.498E+01

N=60001

V=10cm/sLiquidupflow Ux

(a)

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

0 1 2 3 4 5 6 7 8 9 10

X

Vx (I

nter

stiti

al v

eloc

ity),

cm/s

z=10 cm (mesh1) z =11 cm (mesh1) z1=10 cm (mesh2)z2=10.5 cm (mesh2) z3=11 cm (mesh2)

(b)

Figure 5-6. Simulated liquid upflow velocity component, Vx contour (a) and profiles (b)

using mesh1 (10 × 20) and mesh2 (20× 40) at a superficial velocity of 10 cm/s.

VX

126

X

Z

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

CFDLIB97. 2

T=8.994E+01

N=60001

X

Z

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20V2

-2-3.04167-4.08333-5.125-6.16667-7.20833-8.25-9.29167-10.3333-11.375-12.4167-13.4583-14.5-15.5417-16.5833-17.625-18.6667-19.7083-20.75-21.7917-22.8333-23.875-24.9167-25.9583-27

CFDLIB97.2

T=8.994E+01

N=60001

CFDLIB97.2

T=4.498E+01

N=60001

V=10cm/sLiquidupflow

Uz

(a)

-30.0-29.0-28.0-27.0-26.0-25.0-24.0-23.0-22.0-21.0-20.0

0 1 2 3 4 5 6 7 8 9 10

X

Vz (I

nter

stiti

al v

eloc

ity),

cm/s

z=10 cm (mesh1) z =11 cm (mesh1) z1=10 cm (mesh2)z2=10.5 cm (mesh2) z3=11 cm (mesh2)

(b)

Figure 5-7. Simulated liquid upflow velocity component, Vz contour (a) and profiles (b)

using mesh1 (10 × 20) and mesh2 (20× 40) at a superficial velocity of 10 cm/s.

VZ

127

THE10.650.6442860.6385710.6328570.6271430.6214290.6157140.610.6042860.5985710.5928570.5871430.5814290.5757140.57

CFDLIB97.2

T=9.000E+01

N=91354

Figure 5-8. Initial solid volume fraction distribution at 10 ×15 section-discretization

(section size = 1.0 cm) for gas-liquid cocurrent downflow simulation

(zoom: x = 0 ~ 4; z = 4 ~ 8).

Solid volume-fraction

128

X0 1 2 3 4

CFDLIB97.2

T=9. 000E+01

N=91354

CFDLIB97.2

T=6. 000E+01

N=226091

CFDLIB97.2

T=5. 000E+01

N=80586

CFDLIB97.2

T=9. 000E+01

N=91354

X0 1 2 3 4

CFDLIB97. 2

T=9.000E+01

N=91354

CFDLIB97. 2

T=6.000E+01

N=226091

CFDLIB97. 2

T=5.000E+01

N=80586

CFDLIB97. 2

T=9.000E+01

N=91354

CFDLIB97. 2

T=5.000E+01

N=80586 (a) (b)

X0 1 2 3 4

THE30.30.2857140.2714290.2571430.2428570.2285710.2142860.20.1857140.1714290.1571430.1428570.1285710.1142860.1

CFDLIB97. 2

T=9.000E+01

N=91354

CFDLIB97. 2

T=6.000E+01

N=226091

CFDLIB97. 2

T=5.000E+01

N=80586

CFDLIB97. 2

T=9.000E+01

N=91354

CFDLIB97. 2

T=5.000E+01

N=80586

CFDLIB97. 2

T=6.000E+01

N=226091 (c) Figure 5-9. Gas phase holdup contours and gas interstitial velocity vectors in the 4 × 4 cm

zone marked in Figure 5-9 (a) cell size =1.0 cm; (b) cell size = 0.5 cm; (c) cell size = 0.25

cm (zoom: x = 0 ~ 4; z = 4 ~ 8).

Gas volume-fraction

129

0.210

0.220

0.230

0.240

0.250

0.260

0 1 2 3 4 5

gas

hold

up

mesh-(a)mesh-(b)mesh-(c)

Figure 5-10. Effect of the mesh sizes (a, b, c) on the cell-scale gas holdup values

5.5.5 Boundary Conditions

It is well known that the quality of flow distribution at the top boundary can have

a profound influence on the bed dynamics (Christensen et al., 1986). Szady and

Sundaresan (1991) experimentally examined the effects of the top boundary and the

bottom boundary on the hydrodynamics in a pilot-scale trickle bed. They found that both

the top and bottom boundaries affect the flow characteristics in the trickling regime of

flow, such as overall pressure-gradient and liquid saturation. They also affect the onset of

pulsing. In the flow simulations of interest in this work, special care has been taken in

setting all the boundary conditions. The stationary boundary conditions are used in most

flow simulations with steady state feed conditions in packed beds. This includes specified

inflow velocities of fluids, zero velocity-gradient for outflow, reflective boundary

condition (or symmetry) at the center, and no-slip condition for the wall(s). However, for

the case of periodic liquid feed or gas feed case (so called ‘on-off’ flow modulation), the

130

inflow velocity is specified to vary with time. For a 2-D cylindrical coordinate (a

cylindrical bed), a reflective-wall boundary is used for the left side of the logical block

(i.e., the center-line of the column); for a 2-D Cartesian coordinate (a rectangular bed), a

no-slip wall condition is used for the right side of the block. All the boundaries are

treated as Eulerian boundaries since these boundaries are stationary in space even in the

periodic operation. In Chapter 6, we present the simulated flow distributions at both

steady state and unsteady state flow feed conditions and show how the top boundaries

affect the multiphase flow distributions in the entire packed bed.

5.6 Conclusions and Remarks

The Eulerian k-fluid model has been adopted to model the macroscale multiphase

flow in packed beds in which the geometric complexity of bed structure is resolved

through statistical implementation of sectional porosities, and the complicated multiphase

interactions are evaluated using the Ergun type of expressions which were developed

based on bench-scale hydrodynamics experiments. The effect of particle wetting on flow

distribution is incorporated in the phase pressure computations. The drag formulas for

fluid-particle and fluid-fluid interactions are examined and discussed. The drag exchange

coefficients for the solid particle and fluid, Xks, is obtained based on the models of Holub

et al (1992, 1993) or Attou et al (1999) with E1 of 180 and E2 of 1.8. The drag exchange

coefficient for gas and liquid is calculated by the model of Attou et al (1999). The J-

function is used to calculate the capillary pressure term. Due to the relationship between

the section size and the variance of the sectional porosity, the selection of the section-size

has to follow a certain relation, which is expected to be available by analyzing the full 3-

D porosity distribution data of MRI (Sederman et al., 1997) and of computer tomography,

CT of high spatial resolution. The dependence of the simulated sectional scale flow

pattern on the cell size has been examined for both liquid-solid and gas-liquid-solid and it

was demonstrated that a grid independent macroscopic flow structure can be obtained by

the k-fluid model although the more detailed flow field may be obtained with a finer grid

(i.e., small cell size).

131

Chapter 6

Computational Fluid Dynamics

(CFD): II. Numerical Results &

Comparison with Experimental Data 6.1 Introduction

In recent years, computational fluid dynamics (CFD) has become an important

tool in studies of multiphase flows. It is expected that CFD modeling will become more

pervasive in the design of multiphase reactors as researchers in both academic and

industrial communities are intensifying their efforts in this area. So far the CFD approach

has been used to simulate single phase flow within relatively simple geometries and to

compare the results to those obtained from experiments (Kuipers and van Swaaij, 1998;

Lahey and Drew, 1999; Pan et al., 2000). More complex systems such as multiphase flow

in packed beds has not been studied in detail by the CFD approach due to the complex

geometry of the tortuous pore space and the complicated fluid-fluid and fluid-particles

interactions. A new strategy for flow modeling in packed beds is devised by

implementing the statistical description of the bed structure into the CFD model and by

using the drag forces that have been developed and discussed in Chapter 5. A

multidimensional k-fluid Eulerian model has been adopted and executed in the

framework of the CFDLIB package from Los Alamos National Laboratory. The details of

132

this code library are available elsewhere (Kashiwa et al., 1994; Johnson, 1996; Johnson et

al., 1997).

Since the void space and the flow distribution in random packed beds are

intrinsically statistical in nature, a statistical approach to porosity distribution description

in the bed certainly has advantage over the conventional deterministic mean porosity

assignment everywhere in the bed and/or the use of longitudinal-averaged radial porosity

profile (Stanek, 1994; Bey and Eigenberger, 1997; Yin et al., 2000). Up to date, there

have been few comparisons between the CFD results and the measured flow data in the

bed with randomly packed particles, with different liquids in the upflow mode

(Stephenson and Stewart, 1986). Those comparisons, however, were limited to the use of

longitudinal-averaged radial velocity profiles at different particle Reynolds numbers (Rep

= 5, 80), and no statistical quantities of the two-dimensional velocity field were reported

in Chapter 3. In this Chapter, we expand the above comparison study to include the

statistical quantity comparison such as the probabilities of sectional liquid velocity

distribution.

The entrance (or feed) distribution of fluid(s) is controlled by the distributor

design and the top layer of the packing. The effects of feed distribution on the

macroscopic flow structure were found significant in experiments (Christensen et al.,

1986; Szady and Sundaresan, 1991) and numerical simulations (Anderson and Sapre,

1991). The use of inert large particles as the top layer is rather common in commercial

packed columns. In fact, Moller et al (1996) found that compact ceramic cylindrical

tablets, TK-10, on the top of the packed beds of 1/16 inch cylindrical extrudates have a

positive effect on the liquid distribution (e.g., the top layer compensates for the poor inlet

distribution). However, a negative effect on liquid distribution (e.g., enhancing rivulet

flow) was reported by Szady and Sundaresan (1991) in the packed bed of 3 mm glass

spheres topped with a 10 cm layer of 6 mm Rashig rings. The discrepancy in the effect of

the top layer might be due to the different particle structures used in the two studies for

the top layer, but, definitely, the down-stream flow distribution is sensitive to the upper

boundary of flow, particularly in the trickling flow regime (Szady and Sundaresan, 1991).

Any uneven feed distribution due to the distributor or the top layer can cause a change in

133

the downstream flow pattern. In this Chapter, we intend to explore such flow phenomena

numerically by computing the flow pattern based on the k-fluid CFD model.

This Chapter has been organized in following manner. First, we present a

comparison of CFD predictions and experimental data for liquid upflow in packed beds.

Then we report some comparisons of CFD computations and the measured liquid holdup

and pressure-drop in a pilot-scale trickle beds with gas and liquid cocurrent downflow.

We report only global quantities due to the lack of data on spatial distribution of these

quantities. The second part presents some simulation results regarding the effects of feed

distribution, at steady state conditions, on downstream flow distribution. A summary and

conclusions follow at the end of the Chapter.

6.2 Comparison of CFD and Experimental Results Since the CFD approach to flow modeling presented in Chapter 5 has to be based

on known porosity distribution at a certain scale, a full-comparison of CFD predictions

and experimental data is possible only if the data for distribution of porosity, flow

velocity, and phase volume-fraction are available on the same scale. The lack of such sets

of experimental data in packed beds has made the validation of the current CFD model

impossible. While magnetic resonance imaging (MRI) has recently shown some promise

in providing volume- and velocity-distribution data in packed beds (Sederman et al.,

1997), and it has been illustrated well for the validation of Lattice Boltzmann simulation

for single phase flow (Martz et al., 1999), for multiphase flow of interest in this work,

there is no suitable data in the open literature. What we found in the literature so far is

few experimental results, which could be used for partial validation of our CFD

simulation results. For example, Stephenson and Stewart (1986) presented the

longitudinally averaged radial profile of porosity in a packed bed of spheres and the

corresponding liquid velocity profiles obtained from cylindrical packed beds using a

marker tracking method (i.e., optical measurement). Data were given for several particle

Reynolds numbers within the range from 5 to 280 in the beds with Dr/dp = 10.7 and L/dp

= 20.6. The statistical information on the axial interstitial velocity distribution was also

134

reported in the paper. In at least a partial validation of the numerical results of CFD

simulations, we reported the comparison of CFD predicted velocity profiles and the

measured profiles of Stephenson and Stewart (1986) at particle Reynolds numbers of 5

and 80 for the liquid upflow case in Chapter 3. For a multiphase flow system such as gas-

liquid flow in a packed bed, there is no suitable data for a similar comparison. What we

can do is to look at the CFD predictions of the global hydrodynamic quantities, such as

liquid holdup and pressure-gradient, and compare these quantities quantitatively with

experimental data. For the predicted distributed quantities, a qualitative comparison is the

only choice as the data for the distribution of porosity, multiphase velocities and phase

volume-fraction are available only based on computational results obtained at different

conditions.

6.2.1 Liquid Upflow in Packed Beds Since there was no 2-D porosity distribution data reported in Stephenson and

Stewart (1986), the comparison of CFD predictions for liquid upflow in packed beds was

limited to the longitudinally-averaged radial axial velocity profiles, in which only one-

dimensional (i.e., radial) variation of porosity was considered in the 2-D flow simulation.

Although velocity profiles that agree well with experimental data were achieved in

Chapter 3, the simulated velocity results based on the radial porosity profile could not

capture the reported statistical information with axial interstitial velocity reported by

Stephenson and Stewart (1986). Clearly, there is a need to use a 2D variation in porosity

in the CFD flow simulation in order to get comparable statistical results for velocity, but

unfortunately, these is no such data available in the original paper.

Continuing the discussion initiated in Chapter 5, a packed bed can be treated as a

network of interconnected sections with certain section size (see Figure 5-2 in Chapter 5).

The sectional porosities are normally distributed in a pseudo-Gaussian manner except

when the sectional size is very small, where a pseudo- binomial distribution might be

expected. In this study, based on the mean porosity and reported longitudinal-averaged

radial porosity profile with two different standard deviations, we generated two sets of

pseudo-Gaussian porosity distributions, namely, RN1 and RN2. The sectional porosities

135

together with the longitudinal-averaged porosity values from Stephenson and Stewart

(1986) are plotted in 2-D cylindrical coordinates (r, z) as shown in Figures 6-1a and 6-1b.

The heterogeneity of the RN2 bed is clearly higher than that of RN1 bed, as one can see

from the standard deviations of the two porosity distributions in Table 6.1. Figure 6-2a

shows the comparison of the predicted and the measured longitudinally-averaged radial

axial velocity profiles for RN1 section porosity assignment as well as with the CFD result

from 1-D porosity variation assignment in Chapter 3 at different Reynolds numbers.

Similarly, Figure 6-2b gives the comparison for the RN2 sectional porosity assignment

case. A similar conclusion regarding the predictions of the radial axial liquid velocity

profile can be drawn from these two figures as in Chapter 3. This is expected since the

longitudinally averaged porosity profile determines the longitudinally averaged velocity

profile. However, significant differences do exist in the predicted statistical information

of sectional liquid velocities based on two different porosity distributions RN1 and RN2

in the beds. As one can see from Figures 6-3a and 6-3b, the histogram of predicted liquid

axial velocity in the RN2 bed is much closer to the experimental data than that in the

RN1 bed. This implies that the sectional porosities in the experimental beds are much

more spread than a narrow Gaussian distribution and are much closer to the RN2 bed

shown in Figure 6-1b. This is most likely caused by the fact that cylindrical particles

were used in the experiments of Stephenson and Steward (1986), and larger spread in the

porosity distribution is expected with cylindrical particle than with spherical particles

(Bey and Eigenberger, 1997). Thus, if we know the mean porosity and the statistics of the

porosity distribution, we can predict the statistics of the velocity distribution. In fact, our

CFD simulations reveal that if we fix the mean porosity and the variance of the porosity

distribution, we get the same variance of the flow velocity if we keep the same operating

conditions.

136

Table 6.1. Statistical description of porosities and CFD simulated velocities Two random packed beds RN1 RN2 Porosity mean = 0.3527

S.D. = 0.0420 mean = 0.3534 S.D. = 0.0916

Axial interstitial velocity, Vz, cm/s S.D. / mean (Re = 5) S.D. / mean (Re = 280)

2.0012/ 6.6740 6.9313/ 29.8377

3.8640 / 7.0915 12.0708 / 31.3548

Radial interstitial velocity, Vx, cm/s S.D. / mean (Re = 5) S.D. / mean (Re = 280)

0.4379/ 0.1029 1.2752/ 0.2645

1.8790/ (-0.2034) 7.7352/ (-1.5758)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0r, cm

sect

iona

l por

osity

Figure 6-1a. Generated sectional porosities (RN1) plotted in the radial direction (blank

diamonds) and the longitudinally averaged radial porosity profile of Stephenson &

Stewart (1986) (blank circles). Statistics of the RN1 distribution are given in Table 6.1.

137

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0r, cm

sect

iona

l por

osity

Figure 6-1b. Generated sectional porosities (RN2) (blank diamonds) plotted in the radial

direction and the longitudinally averaged radial porosity profile of Stephenson & Stewart

(1986) (blank circles). Statistics of the RN2 distribution are given in Table 6.1.

138

0.3

0.6

0.9

1.2

1.5

1.8

0 0.76 1.52 2.28 3.04 3.8r (cm)

Rel

ativ

e su

perf

icia

l vel

ocity

0.20

0.30

0.40

0.50

0.60

0.700.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

r/dp

poro

sity

Uz/U0 (Exp.) Uz/U0 (Re =280) RN1Uz/U0 (Re=5) RN1 Uz/U0 (Re=5) PAporosity (Exp.)

Figure 6-2a. Comparison of longitudinally averaged radial velocity profiles at different


Statistics of the RN1 bed are available in Table 6.1; PA bed: sectional porosities are only

varying in the radial direction.

139

0.3

0.6

0.9

1.2

1.5

1.8

0 0.76 1.52 2.28 3.04 3.8r (cm)

Rel

ativ

e su

perf

icia

l vel

ocity

0.20

0.30

0.40

0.50

0.60

0.700.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

r/dp

poro

sity

Uz/U0 (Exp.) Uz/U0 ( Re =280) RN2Uz/U0 (Re=5) PA Uz/U0 ( Re=5)RN2porosity (Exp.)

Figure 6-2b. Comparison of longitudinally averaged radial velocity profiles at different


Statistics of the RN2 bed are available in Table 6.1; PA bed: sectional porosities are only

varying in the radial direction.

140

0.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

0.225

-5.0 0.0 5.0 10.0 15.0 20.0 25.0

Axial Interstitial Velocity Vz, cm/s

n/nt

RN1RN2Exp. (Re = 5)

Figure 6-3a. Frequency distribution of axial interstitial velocity (Re = 5):

RN1-CFD simulation based on random porosity set 1

RN2-CFD simulation based on random porosity set 2

Exp. –Experimental data reported by Stephenson and Stewart (1986).

141

0.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

0.225

-10.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0Axial Interstitial Velocity Vz, cm/s

n/nt

RN1RN2Exp. (Re = 280)

Figure 6-3b. Frequency distribution of axial interstitial velocity (Re = 280):

RN1-CFD simulation based on random porosity 1 (ε: S.D./µ = 0.0916/0.3534; Vx: S.D./µ

= 1.879/0.2034; Vz: S.D./µ = 3.864/7.0915).

RN2-CFD simulation based on random porosity 2 (ε: S.D./µ = 0.0916/0.3534; Vx: S.D./µ

= 1.879/0.2034; Vz: S.D./µ = 3.864/7.0915).

Exp. –Experimental data reported by Stephenson and Stewart (1986).

142

We conclude, based on the above simulations, for single phase flow, that the k-

fluid model can predict not only the longitudinally averaged radial profiles of axial

velocity but also provide the statistical information on fluid velocity distribution provided

that the following information on bed structure are all available.

(i) Mean porosity

(ii) Longitudinally averaged radial porosity

(iii) Sectional porosity distribution type and variance

The mean porosity and the longitudinally averaged radial porosity profile are

obtainable by experiments and are also predictable by various empirical correlations in

the literature (Benenati and Brosilow, 1962; Muller, 1991; Bey and Eigenberger, 1997).

The statistical properties of the porosity distribution are functions of particle size, shape,

column diameter as well as the packing method, which can, in principle, be developed

through 3D sphere-packing computer simulations (see Tory et al., 1973) and extensive

MRI or CT measurements of packed-bed structures (see Baldwin et al., 1996; Sederman

et al., 1997).

6.2.2 Gas and Liquid Cocurrent Downflow in Trickle Beds One should note that the above comparisons are limited to the system which

consists of a fixed solid phase and a saturated flowing fluid, typical examples of which

are (i) gas flowing through fixed beds; (ii) liquid upflow through packed beds (e.g.,

Stephenson and Stewart, 1986). For the packed beds with gas and liquid two-phase flows

(e.g. gas-liquid cocurrent downflow in trickle beds), the competition of gas and liquid for

the fixed cavities between solid particles makes the liquid distribution much more

complicated than the saturated single phase flow distribution. In this Section, we intend

to partially validate the CFD two-phase flow predictions by comparing them with the

experimental data of overall liquid saturation and overall pressure gradient. The case of

two-phase flow in a pilot-scale trickle bed with relatively high gas superficial velocity

(e.g. 0.22 m/s) is chosen to assess the model capability in scaling up trickle bed reactors.

The simulations were based on a cylindrical column having an internal diameter (I.D.) of

0.152 m. The packing part is 1.50 m high and consists of 3 mm spherical particles. The

143

measured voidage of ~0.37 was used as the mean porosity. The working fluids were air

for gas phase and water for the liquid phase. The experimental data was obtained from

the paper by Szady and Sundaresan (1991), in which the overall liquid saturation and

pressure gradient data were reported in packed beds with similar dimensions in both

trickling and pulsing flow regimes.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.8 1.6 2.4 3.2 4 4.8 5.6 6.4 7.2

(R-r), cm

Poro

sity

sectional valueMueller correlation

Figure 6-4a. Discretization of the radial porosity profile into sectional porosity values

(dp = 3mm) From the wall to the center: sectional mean = 0.411, 0.363, 0.363, 0.365,

0.362, 0.362, 0.363,.364, 0.362, 0.366; sectional S.D./mean = 20%, 15%, 10%, 10%,

10%, 10%, 10%, 10%, 10%, 10%.

Before computing the two-phase flow using the k-fluid CFD model, one needs to

generate a multi-dimensional porosity distribution at a certain sectional size as discussed

in Chapter 5. For an axisymmetric cylindrical column, a two-dimensional porosity

distribution, ε(r, z), is needed for flow simulation. Based on the measured mean-porosity

and the longitudinally averaged radial porosity profile curve calculated by Mueller’s

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.3 0.6 0.9 1.2

144

model (Mueller, 1991), one can discretize along the radius, r, into several annular

sections, and calculate the sectional porosities by integrating the radial porosity profile

curve as shown in Figure 6-4a. It is known that the oscillation of the porosity profile is

pronounced in the wall zone which is about 3~4 particle diameters distance from the

wall. The variance of porosities in the wall region is expected to be higher than that in the

core region. Table 6-2 lists the parameters used in the discretization of the annular

sections, and Figure 6-4b displays the solid volume-fraction distribution of the generated

packed bed, in which the porosities of 1500 (=10 × 150) sections are represented by a 2D

pseudo-Gaussian distribution with the standard deviations given in Table 6-2. There are

two annular sections in the radial direction (r) in the wall region with relatively small

section size. The section size in longitudinal direction (z) is 0.01 m. At the top boundary

of the bed, it was assumed that a uniform gas and liquid feed distribution is attained as

claimed in the experiments (Szady and Sundaresan, 1991). Figures 6-5a and 6-5b show

the simulated liquid and gas volume-fraction distribution at gas superficial velocity of

0.22 m/s and liquid superficial velocity of 0.0045 m/s. Relatively high liquid and gas

holdups appear in the wall region where the porosities are high due to the interference of

the wall.

In Figures 6-6 and 6-7, we compare the CFD predictions of the overall liquid

saturations and pressure gradients with the experimental data of Szady and Sundaresan

(1991) at different liquid superficial velocities. We also plotted the calculated values

from the single-slit model (Holub et al., 1992) and the relative permeability model (Saez

and Carbonell, 1985). As discussed in Chapter 5, there is a need to predetermine the two

Ergun values (i.e. E1 and E2) experimentally for using Holub’s model to calculate the

overall liquid holdup and pressure-drop. Similarly, the static liquid holdup, εL0, has to be

determined in order to calculate εL and (∆Ρ⁄∆z) in Saez and Carbonell’s model. The

values obtained by Holub’s model shown in Figures 6-6 and 6-7 are based on the

measured values, E1 = 215 and E2 = 1.4. Two sets of data from Saez and Carbonell model

are based on both measured static liquid holdup (εL0 = 0.022) and the correlation-

estimated value (εL0 = 0.05), respectively. The mean values for liquid saturation and

pressure gradient from the CFD model are plotted. Note that there was no need for using

145

the measured Ergun constants and the measured static liquid holdup in the CFD two-fluid

simulations in which the momentum exchange coefficients, Xgs and Xls, are calculated

from Holub model with constant E1 (= 180) and E2 (= 1.8), and Xgl is from Attou et al

(1999). The detail discussion of Xkl calculations was given in Chapter 5.

0

20

40

60

80

100

120

140

THE10.6997810.6836630.6675440.6514250.6353060.6191880.6030690.586950.5708310.5547130.5385940.5224750.5063560.4902380.474119

CFDLIB97.2

T=9.500E+01

N= 96557

Figure 6-4b. Solid volume-fraction distribution generated based on the data in Table 2 in

a pilot scale packed bed.

146

0

20

40

60

80

100

120

140

THE20.1318840.1297590.1276330.1255080.1233820.1212560.1191310.1170050.1148790.1127540.1106280.1085030.1063770.1042510.102126

CFDLIB97.2

T=9.500E+01

N= 96557

0

20

40

60

80

100

120

140

THE30.4076590.3921190.3765780.3610380.3454970.3299560.3144160.2988750.2833340.2677940.2522530.2367130.2211720.2056310.190091

CFDLIB97.2

T=9.500E+01

N= 96557

(a) (b)

Figure 6-5. Simulated phase volume-fraction distribution at liquid superficial velocity of

0.45 cm/s and gas superficial velocity of 22 cm/s in a pilot-scale packed bed. (a) liquid;

(b) gas.

147

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

Liquid superficial velocity, cm/s

Liqu

id S

atur

atio

n

CFD modelHolub modelExperimentSaez & Carbonell model (exp)Saez & Carbonell model (cal)

Figure 6-6. Comparison of CFD k-fluid model and other phenomenological models

prediction of liquid saturation with the experimental data of Szady and Sundaresan

(1991) (gas superficial velocity is 22 cm/s). The f values used in CFD modeling are

evaluated by the particle external wetting efficiency correlation by Al-Dahhan and

Dudukovic (1995). exp- use measured static liquid holdup (0.022) in Saez & Carbonell

model; cal- use the correlation-estimated value (0.05) in Saez & Carbonell model.

148

0

4

8

12

0 0.2 0.4 0.6 0.8 1

Liquid superficial velocity, cm/s

Pres

sure

Gra

dien

t (kP

a/m

)

decreasing Vlincreasing VlCFD modelHolub modelSaez & Carbonell model (exp)Saez & Carbonell model (cal)

Figure 6-7. Comparison of CFD k-fluid model and phenomenological models prediction

of pressure gradient with the experimental data of Szady and Sundaresan (1991) (gas

superficial velocity is 22 cm/s) The f values used in CFD modeling are evaluated by the

particle external wetting efficiency correlation by Al-Dahhan and Dudukovic (1995).

exp- use measured static liquid holdup (0.022) in Saez & Carbonell model; cal- use the

correlation-estimated static liquid holdup (0.05) in Saez & Carbonell model.

149

Both bed-scale models (Holub model, Saez and Carbonell model with calculated

εL0) give unsatisfactory predictions of the pressure gradient. The k-fluid CFD model and

Saez and Carbonell’s model with measured εL0 provide more reasonable predictions for

the pressure gradient and better prediction for liquid saturation as one can see from

Figures 6-6 and 6-7. The comparison of the k-fluid CFD model predictions with

additional experimental data for overall liquid holdup and pressure gradient can be

performed in a similar way to fully assess how good the k-fluid CFD model is. The onset

of the natural pulsing was observed experimentally at the liquid superficial velocity of 0.8

cm/s at the given gas superficial velocity of 22 cm/s. It seems reasonable that the k-fluid

CFD model should produce agreeable predictions of the overall hydrodynamics quantities

only in the trickling regime (< 0.6 cm/s) for which proper closures were well provided.

As we discussed in Chapter 5, the interactions between the fluid and particles, fluid and

fluid become very complicated at flow transition regime and in pulsing flow regime

which remain a challenge for researchers (Szady and Sundaresan, 1991).

Table 6.2. Parameters used in the discretization of the radial porosity profile, and in the

generation of 2D porosity distribution Two Regions Wall Core Section number and size in r

2, 0.6 cm 8, 0.8 cm

Section number and size in z

150, 1 cm 150, 1 cm

Radial position (from center to wall), cm

7.6, 7.0 6.4, 5.6, 4.8, 4.0, 3.2, 2.4, 1.6, 0.8

Longitudinal-averaged sectional porosity

0.411, 0.363 0.363, 0.365,0.362,0.362,0.363,0.364,0.362,0.366

Ratio of std/mean 0.2, 0.15 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1

6.3 Simulating Feed Distribution Effects There have been many discussions about the role of feed distribution on the flow

distribution inside packed beds in the literature, particularly for gas liquid cocurrent

downflow in the trickling flow regime where well designed gas and liquid distributors are

important for achieving good flow distribution. Beyond this general concern, however,

150

there have been some discrepancies reported on how the feeding of the liquid and gas

affects the downstream flow pattern, particularly in the quantitative sense. In fact, there

are many parameters, which contribute to the feed distribution effect. In most absorption

columns packed with relative large elements (~ 2 - 10 cm), the inertial force and gravity

play an important role in causing significant wall flow, but the particle wetting seems not

to be a significant factor for large-size packing (Stanek, 1994). However, in most trickle

beds with relative small porous particles (~0.5 – 5.0 mm), the capillary force and particle

partial external wetting become important in determining the flow distribution so that

significant feed distribution effects on flow and wetting were found.

As a preliminary study of this topic, we provide a set of numerical results that

describe the simulated flow distribution at steady state liquid and gas feed. Three types of

liquid inlet distributors: single point source, two-point source and uniform distributor

have been tested in numerical simulations using the discrete cell model (DCM) approach

based on the minimization of total energy dissipation rate, as presented in Chapter 4. The

effect of liquid feed distribution was observed to be significant in the upper half of the

bed, and less pronounced at depths exceeding 50 particle diameters (15 cm) for total bed

length of 96 particle diameters. Since the steady-state simulations in Chapter 4 were

limited to a bench scale 2D rectangular packed bed, it is desirable to see if those effects

are retained in a cylindrical pilot-scale packed bed. Hence, we test the effect of a

nonuniform steady state liquid feed distribution in the pilot-scale trickle bed used above

(see Figures 6-4a and 6-4b) on downstream two phase flow distribution using the CFD k-

fluid simulation. The averaged feed superficial velocity for the top ten sections is 0.295

cm/s for liquid and 22 cm/s for gas. Table 6-3 lists the sectional velocities and volume-

fractions of the top layer ten sections for flow simulation with a nonuniform feed

distribution. Such uneven liquid feed distribution might result from the improper design

of the liquid distributor or due to the improper use of the packing top-layer.

151

Table 6.3. Feed velocities and holdups at top ten sections from the center to the wall

Section 1 2 3 4 5 6 7 8 9 10

Vl

εl

0.885

0.363

0

0

0

0

0.738

0.373

0.738

0.391

0

0

0

0

0

0

0.295

0.150

0.295

0.100

Vg

εg

0

0

33.0

0.389

33.0

0.362

0

0

0

0

36.67

0.360

36.67

0.370

36.67

0.364

22.0

0.218

22.0

0.391

CFDLIB97.2

T=8.500E+01

N=86718

CFDLIB97.2

T=9.500E+01

N=96643

126

128

130

132

134

136

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140

142

144

146

148

150THE2

0.20.1892860.1785710.1678570.1571430.1464290.1357140.1250.1142860.1035710.09285710.08214290.07142860.06071430.05

CFDLIB97.2

T=8.500E+01

N=86718 Figure 6-8a. Comparison of liquid holdup distribution under nonuniform (left) and

uniform (right) feed conditions at Ul0 = 0.295 cm/s, Ug0 = 22 cm/s.

152

126

128

130

132

134

136

138

140

142

144

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148

150THE3

0.430.4135710.3971430.3807140.3642860.3478570.3314290.3150.2985710.2821430.2657140.2492860.2328570.2164290.2

CFDLI B97. 2

T=8.500E+01

N=86718

CFDLIB97. 2

T=8.500E+01

N=86718

CFDLIB97. 2

T=9.500E+01

N=96643 Figure 6-8b. Comparison of gas holdup contour and gas interstial velocity vector plot

under nonuniform (left) and uniform (right) feed conditions at Ul0 = 0.295 cm/s, Ug0 = 22

cm/s.

The gas and liquid flow maldistribution are detected at downstream locations for

a nonuniform gas and liquid feed distribution, listed in Table 6-3, and show that the

overall liquid holdup decreases by ~ 11% in case of a nonuniform distribution (from

0.1084 for uniform inlet to 0.0968 for nonuniform inlet). Moreover, the maldistribution is

more significant in the upper 25 cm portion of the packed bed, although the effect does

propagate throughout the whole packed bed. Figure 6-8a exhibits the comparison of the

liquid holdup distribution at nonuniform feed condition (left plot) and uniform feed

153

condition (right plot). Figure 6-8b displays the gas holdup contour plot and gas interstitial

velocity plot at nonuniform and uniform feed condition. At high gas superficial velocity,

more gas flow is encountered in the wall region due to the higher porosity as shown in

Figure 6-8b (right plot). The nonuniform feed of gas and liquid make the gas

maldistribution worse as one can see from Figure 6-8b (left plot).

6.4 Conclusions The comparison of the k-fluid CFD simulation and the experimental results has

been performed for both liquid upflow and gas-liquid cocurrent downflow in packed

beds. The effects of feed flow distributions have been simulated for a packed bed at

steady state flow conditions. The following conclusions are reached:

(1) The k-fluid CFD model can capture the longitudinally averaged radial axial liquid

velocity profile and the statistical features of the 2-D sectional velocity

distribution provided that the following information on bed structure are all

available: (i) mean porosity, (ii) longitudinally-averaged radial porosity and (iii)

sectional porosity distribution type and its variance.

(2) For two phase flow system reported in Szady and Sundaresan (1991), the

predictions of the k-fluid CFD model on overall liquid saturation and pressure

gradient are comparable with experimental data and with phenomenological

hydrodynamics models developed for reactors of modest scale.

(3) The k-fluid CFD model can simulate the feed distribution effects on flow

distribution at downstream locations.

washington university sever institute of ...crelonweb.eec.wustl.edu/theses/jiang/jiang thesis...

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