a review of cfd modelling for performance predictions131537/uq131537_oa.pdf · anisotropic...

Engineering Applications of Computational Fluid Mechanics Vol. 1, No. 2, pp.109–125 (2007)

Received: 10 Dec. 2006; Revised: 10 Feb. 2007; Accepted: 2 Mar. 2007

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A REVIEW OF CFD MODELLING FOR PERFORMANCE PREDICTIONS OF HYDROCYCLONE

M. Narasimha*, Matthew Brennan and P. N. Holtham

Julius Kruttschnitt Mineral Research Centre, The University of Queensland, Isles Road, Indooroopilly 4068, Queensland, Australia.

*E-Mail: [email protected] (Corresponding Author)

ABSTRACT: A critical assessment is presented for the existing numerical models used for the performance prediction of hydrocyclones. As the present discussion indicates, the flow inside a hydrocyclone is quite complex and there have been numerous numerical studies on the flows and the particle motions in hydrocyclone, with a wide range of turbulence and multiphase models tested. Two-equation k-ε and RNG k-ε models flow velocities with empirical modifications were led to poor results, especially the tangential components in comparison with experimental measurements. Most of the recent studies have utilized the Reynolds stress models (RSM) with different degrees of complexity in the pressure-strain correlation. These RSM studies showed good agreements with velocity measurements. Unfortunately, the velocity profiles were not validated in most of the RSM cases where multiphase particle tracking were applied. Finally, large eddy simulation (LES) is the most advanced turbulence model applied in recent hydrocyclone numeric studies. Besides the additional information on précising the air core correctly, LES provides an additional accuracy in predicting the velocity profiles or the grade efficiency in comparison to the RSM.

The multiphase models have been successfully applied in a hydrocyclone to model the Lagrangian motions of spherical particles. Eulerian-Eulerian model have been used to account for the particles effect on the fluid viscosity. Simplified Eulerian model (mixture) model predictions for solid transportation in cyclone were well predicted. Further, the inclusion of modified slip velocity calculation in the Mixture model improves the efficiency predictions close to the experimental data at low feed solid loadings. In future studies, the focus should be to model the three-dimensional flow in a hydrocyclone using at least the Reynolds stress model/LES. The particle tracking should at least include the effects of the turbulence on the particles. All these developed models will only applicable to low feed solid concentration levels. Since most of these models neglect the particle-particle interactions, a more comprehensive numerical method of modified Mixture model is applied for simulating solids flow in hydrocyclones for high feed solids concentration. Explicit models for accounting hindered settling and turbulent diffusion investigated for high feed solid concentrations in industrial cyclones are encouraging.

Keywords: hydrocyclone, multi-phase modelling, computational fluid dynamics, classification, turbulence, viscosity

1. INTRODUCTION

Hydrocyclones are widely used in the mining and chemical industries, mainly due to their design and operational simplicity, high capacity, low maintenance and operating cost, and small physical size. A typical hydrocyclone consists of a cylindrical section with a central tube connected to a conical section with a discharge tube. An inlet tube is attached to the top section of the cylinder. The fluid being injected tangentially into hydrocyclone causes swirling and thus generates centrifugal force within the device. This centrifugal force field brings about a rapid classification of

particulate material based on size from the medium in which it is suspended. The flow behavior in hydrocyclone is quite complex. This complexity of fluid flow in cyclone is basically due to the existence of different size particles as well as the dominance of turbulent length scales on separation. The complexity of flow processes has led designers to rely on empirical equations for predicting the equipment performance. These empirical relationships are derived from an analysis of experimental data and include the effect of operational and geometric variables. Different sets of experimental data lead to different equations for the same basic parameters. However, these models have limitations—they can

Engineering Applications of Computational Fluid Mechanics Vol. 1, No. 2 (2007)

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only be used within the extremes of the experimental data from which the model parameters were determined. In view of this shortcoming, mathematical models based on fluid mechanics are highly desirable. Alternatively they can be modelled more fundamentally by CFD (Computational Fluid Dynamics). In this paper, a comprehensive review of the CFD modelling for performance predictions in hydrocyclones is described. Previous literature reviews on the numerical modelling of hydrocyclones have been published by Narasimha et al. (2006a), Pericleous (1987), Chakraborti &Miller (1992), and Nowakowski et al. (2004). Since these reviews were published, different methods had been developed and tested independently, and an updated review is necessary to develop a comprehensive CFD model for predicting the performance of industrial hydrocyclones for a wide range of design and operating variables. Three main phenomena are of interest to the modelling of particle classification in a hydrocyclone: (i) turbulent flow in the hydrocyclone, (ii) the influence of the pulp rheology on the flow field, and (iii) multiphase flow with different size particles (Narasimha et al., 2006b; Hsieh and Rajamani, 1991; and Sevilla and Branion, 1995). Extensive work had been done in each of these areas, but combining them cohesively to develop a comprehensive model for predicting performance of hydrocyclone for industrial applications would be an enormous task. This review outlines the important development in each of these fields.

2. CFD MODELLING OF CYCLONES-TURBULENCE MODELLING

Industrial hydrocyclones typically operate at velocities where the flow is turbulent. However the strong swirl and the flow reversal and flow separation near the underflow introduce anisotropy and strain into the turbulence. Most hydrocyclones in mineral processing applications develop an air core and the free surface between the air and the water introduces further turbulence anisotropy. These characteristics make modelling hydrocyclones using CFD difficult and the addition of solids adds even more complexity. Hsieh(1988) and Devullapalli (1994) modelled hydrocyclones using a 2D axi-symmetric grid where the air core was not resolved and the

air/water interface was treated using a shear free boundary condition. Turbulence anisotropy was incorporated into the model by using a modified mixing length turbulence model where a different mixing length constant was used for each component of the momentum equation. Although the model required calibration, it was able to predict velocities measured by these authors using Laser Doppler Anemometry with reasonable accuracy. k-epilson models intrinsically make the assumption that the turbulence is isotropic because only one scalar velocity fluctuation is modelled. Further the Bousinessq approximation on which the eddy viscosity relies intrinsically implies equilibrium between stress and strain. These assumptions are known to be unrealistic for swirling turbulent flows and this would suggest that k-epilson models are not suitable for modelling turbulence in hydrocyclones and this has been shown to be the case by Ma et al. (2000), Sevilla and Branion (1997), Petty and Parks (2001), Witt et al. (1999) and Delgadillo and Rajamani (2005) and others. However Dyakowski and Williams (1993) have suggested that the k-epilson model can be used on small (<44 mm radius) hydrocyclones. To address this, other authors have used the RNG k-epison model with the swirl correction (Fraser et al., 1997; He et al., 1999; Suasnabar, 2000; Scheutz et al., 2003; Narasimha et al., 2005). However Suasnabar (2000) found that the swirl constant in the RNG model needed to be increased to improve predictions but beyond a certain point, further increases caused numerical instability. As an alternative, Suasnabar (2000) adjusted the constants in the standard k-epsilon model but acknowledged that this approach was limited. Stress transport models, in particular the full Differential Reynolds Stress model (DRSM), such as that developed by Launder et al. (1975), solve transport equations for each individual Reynolds stress. This enables stress transport models to model anisotropic turbulence and strained flows where the Bousinessq approximation is known to be flawed. Whilst more computationally intensive than k-epsilon models, stress transport models are being used to model turbulence in hydro-cyclones. Boysan et al. (1982) used an algebraic stress model but the full DRSM model has been used in more recent work. Cullivan (2000), Suasnabar (2000), Slack (2000), and Brennan (2003) have all used variants of the Launder et al. (1975) model.


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However even here the predictions are not what they could be and there is debate about appropriate modelling options. Whilst Slack (2000) found that the DRSM model gave good predictions of velocities in gas cyclones, both Brennan (2006) and Delgadillo and Rajamani (2005) found that the DRSM, where the air core was being resolved with the VOF model, under predicted tangential velocities in simulations of the 75 mm hydrocyclone (Hsieh, 1988). Cullivan (2003) has suggested that a DRSM simulation of a hydrocyclone needs to use the Quadratic Pressure Strain correlation of Spezial et al. (1991) as a minimum. However our experience is that velocity predictions from the Spezial model (1991) and the simpler linear pressure strain model of Launder et al. (1975) are much the same once the air core is established. Further we have also found that the constants in the linear pressure strain correlation need to be adjusted to match velocity predictions. This implies that even the Launder et al. (1975) DRSM has limitations for this problem. Recent advances in computational power have begun to make Large Eddy Simulation (LES) practical for engineering problems and the fact that LES resolves the large turbulent structures without modelling suggests that it should be appropriate for modelling cyclone separators. LES is intrinsically a dynamic simulation and requires a 3D grid. Slack et al. (2000) has modelled gas cyclones using LES and found good predictions of the velocities but the technique needed a finer grid than the DRSM simulation of the same geometry. De Souza and Silveria (2002) modelled the 76 mm hydro-cyclone of Debair (1983) but this was without an air core. However both Delagdillo and Rajamani (2005) and Brennan (2006) found that Large Eddy Simulation (LES) gave the best overall velocity predictions although this was at extra computational expense in part due to finer grid requirements and shorter time steps.

3. MULTIPHASE CFD MODELLING-PERFORMANCE PREDICTIONS

The flow in a hydroyclone is a multiphase flow consisting of solid particles which are dispersed throughout water. In addition there is the air core. Multiphase flows can be solved by a number of CFD techniques. These include the full Eulerian multiphase approach, simplified Eulerian

approaches such as the Mixture and VOF models and the Lagrangian approach. The full Eulerian multiphase flow approach, where a set of continuity, momentum and turbulence equations for each phase has been used for systems with very high dispersed phase concentrations, where solid/solid interactions carry a significant amount of the stress. The disadvantage of the full Eulerian multiphase modelling approach has been its high computational cost. Further implementations in commercial CFD codes have until recently been limited to using the k-epsilon/RSM models for turbulence. The Lagrangian approach where the paths of individual particles are tracked based on the velocity predicted by a CFD simulation of the fluid is suited to systems where the dispersed phases are dilute and where the particles interact mostly with the fluid without significantly changing the fluid transport properties. In particular the Lagrangian approach is well suited to systems where small numbers of large particles are encountered.

3.1 Lagrangian models for cyclones

By balancing the forces that act on a particle in motion in a carrier fluid, a particle can be tracked along its trajectory. Additionally corrections of the particle trajectory due to interaction with its surrounding environment can be included. The influence of particles on the fluid can be included by considering a source term in the governing equations of the fluid. Also turbulence dispersion of the particles can be included (Soo, 1990 and Crowe et al., 1996). Following the path of a solid particle, the general equation of motion based on the effects treated by Basset, Boussinesq, and Oseen is given (Soo, 1990) by:

e

t

t p

pfffp

pffpp

pfDppp

pp

Ft

uudddD

uudtdD

rPD

uuFDdt

duD

p

p

+−

−+

−+∂∂−

−=

∫0

))(/(6

)(3

421

34

)(3

43

4

2

33

33

τ

ττμπρ

ρππ

ρπρπ

(1)

where fu and pu are velocities of the fluid and the

solid particle, fρ and pρ are the densities of the


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fluid and solid particles, eF is the external force due to potential field, and DF is the time constant for momentum transfer due to drag force defined by:

( )pfpD

ppD uu

CDF −⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

24Re182

ρμ (2)

Finally, the drag coefficient can be expressed as a function on Reynolds number and is written as:

(Re)DD CC = (3)

The application of this equation is also limited to particle diameters much less than the local turbulence length scale (Crowe et al., 1996). In hydrocyclones, modelling, the complete equation (1) has not been considered. Most of the previous numerical studies included only the drag, centrifugal and Coriolis forces in their calculation of the particle trajectory. While the centrifugal and Coriolis forces are determined from the velocities alone, the drag force requires additional modelling. The particles are often assumed to be travelling in a Stokesian flow, where the particle Reynolds number is small. Other simulations have accounted for the history force by taking the life-time of the eddy through which the particle is travelling into account. Heieh (1988), Hsieh and Rajamani (1990), He et al. (1990), Rajamani and Milin (1992), and Monredon et al. (1992) only considered the drag and centrifugal forces. Particle dispersion due to turbulence of the fluid was neglected by both groups of researchers. In all these studies, an expression was derived for the radial direction by balancing the centrifugal force with the radial component of the drag force. Similarly, for the axial direction, the drag force was balanced with the gravity force. Additionally, for tangential direction He et al. (1999) assumed that the particles and the fluid have the same tangential velocity and expressed the particle slip axial and radial velocities as:

5.02

, 34

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

D

p

l

mgslipr rC

duu θ

ρρρ

(4)

5.0

, 34

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

D

p

l

mgslipz C

gdu

ρρρ

(5)

This assumption can only be true if the Stokes number of the particle is less than one (in fact the models were developed for particles in the size range of microns) and calculations were carried out for dilute concentration. Despite the simplicity of the model, all the calculated fractionation efficiency agreed well with experimental data. The particle trajectories from Hsieh’s simulation are shown in Fig. 1. Applying these equations, Monredon et al. (1992) calculated the grade efficiency curve which is shown in Fig. 2(a). The grade-efficiency curve is quite well predicted but the model predicts greater fractionation than the experiments at large particle diameters. Though the model is able to predict the liquid-phase velocity profiles accurately, it seems to have difficulty in predicting separation efficiency in the coarser size range. In such cases the only explanation would be that the model is perhaps not accounting for certain transport mechanisms found in the actual hydrocyclone operation. The short-circuiting flow, which carries the coarse particles to the overflow stream without experiencing proper classification, is a most likely candidate. It should be noted that the short-circuiting flow has been implicitly accounted for in the model formulation but not adequately. The cause of this disagreement is attributed to the inability of the numerical simulation to predict the short-circuit flow. In the experiment, the large particles are carried to the vortex finder with the short-circuit flow and this phenomenon would worsen the performance on the grade-efficiency curve. An involute creates less radial velocity components near the inlet and the short-circuit flow can be reduced. The collection efficiency in such a hydrocyclone could be accurately predicted in the 2DA model. The experimental results from a cyclone using an involute inlet are shown in Fig. 2(b), and better agreements between the experimental and numerical curves are observed.


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Fig. 1 Particle trajectories of different trial in the simulations showing (a) fluid streamline, (b) effects of initial position, and (c) effects of particle size (Hsieh and Rajamani, 1991).

Fig. 2 Comparison between the calculated and experimental grade efficiency from Monredon et al. (1992). (a) shows the grade-efficiency curve taken in a 75 mm hydrocyclone with a single tangential inlet with 4.98 wt. % limestone and (b) shows the grade-efficiency curve taken in a 150 mm hydrocyclone with an involute inlet tube (i.e. axisymmetric inlet flow) with 9.01 wt. % limestone.

(b)(a)

(a) (b) (c)


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The influence of the carrier turbulence on the particles trajectory can be included with stochastic models (Crowe et al., 1996) or more recently with the cloud model (Baxter and Smith, 1993). A stochastic model was implemented for gas cyclones by Boysan et al. (1982). The stochastic particle tracking technique described above was used to determine the paths of 10 particles, from 1 to 10 μm in diameter, from the same initial position, and the paths of 3 μm particles released at five different inlet positions. These simulations were performed in the geometry of a Stairmand high efficiency gas cyclone. The mean results are show in Fig. 3(a) and 3(b). Fig. 3(c) shows the random path of a 2 μm particle under the influence of turbulence fluctuation.

Fig. 3 Particle trajectories results from Boysan et al. (1982) showing (a) different particle sizes from the same initial point, (b) 3 μm particle from four different inlet positions and (c) particle trajectories in the stochastic turbulent flow field.

From Fig. 3(a) one can observe that large particles are quickly collected at the wall, while the other smaller particles follow the streamline of the flow and show pattern of recirculation. The 1 μm particle exits through the vortex finder, while the 2 μm particle is trapped in a loop and its position at the end of the calculated time interval is marked by a circle. Fig. 3(b) shows the trajectories of five 3 μm

particles from different initial positions. The particle with the inner-most position reached the vortex finder without traveling through the cyclone body. The stochastic turbulence effects on the path of a single particle is shown in Fig. 3(c), and one can observe the random velocity path due to the probabilistic turbulent oscillation that was included in the stochastic models. Further, to calculate the particle trajectory, Ma et al. (2000) had to determine the particle drag force. A method similar to that used by Boysan et al. (1982) was used, but the velocity fluctuation due to turbulence was not included in the formulation of the drag force equation. The grade-efficiency curve was determined numerically. Although the calculated particle cut-off diameter agrees with the experimental data, the grade efficiency curve shows that particles with small diameters respond very differently in numerical simulation than in experiments. Ma et al. (2000) attributed the difference to the large effects of turbulent flow oscillation on the smaller particles with smaller inertia. The inclusion of the stochastic turbulence fluctuation eliminates the sharp cut in particle diameter in terms of the fractionation efficiency. Fig. 4(a) shows the experimental and numerical efficiency curves of the high efficiency Stairmand cyclone. Boysan et al. (1982) attributed the discrepancies in the results to possible numerical deficiency in boundary layer definition, re-entrainment of particles via the apex, large intervals in experimental particles size, and experimental error in measurement of the particle diameter. Nevertheless, the numerical collection efficiency showed good agreement with the experiment. The eddy life-time method was used again in another study by Griffiths and Boysan (1996). The same multiphase model described above was applied to the same Stairmand high efficiency cyclone whose length scale was increased by 50% from the Boysan et al. (1982) study. Despite the change in geometry, the results could be compared by matching both the swirl number and Reynolds number of the cyclone. The 73-mm diameter vortex finder curve in Fig. 4(a) shows the grade-efficiency of the Stairmand cyclone with a Sw of 3.80 and a Re of 1.9×105 calculated with a ARSM model and a eddy life-time particle tracking method. Fig. 4(b) shows the grade-efficiency of the Stairmand cyclone with a swirl number of 3.80 and a Re of 2×105 calculated with a RNG k- ε model and a

(a) (b) (c)


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eddy life-time particle tracking method. The simulated velocity field, from which the particle trajectories and the grade efficiency shown in Fig. 4(b) were calculated, was not validated against experiments. However, it is able to produce a satisfactory behaviour in the grade-efficiency curve. This leads one to suspect that accurate velocity prediction, using second moment turbulence closure, is not necessary in calculating the grade-efficiency curve.

Fig. 4 Particle fractionation grade-efficiency curve calculated with the eddy life-time particle tracking method for the Stairmand high efficiency cyclone. (a) is the cyclone with a Re of 1.9×105. The curves for the vortex finder diameters of 40, 56 and 73 mm correspond to Swg of 2.1, 2.9 and 3.8. The solid line represents the experimental data from Mothes et al. (1981) and the circles are the results with a ARSM turbulence model from Boysan et al. (1982). (b) is the cyclone with a Re of 2.0×105 and a swirl number 3.8. with a RNG k-ε turbulence model from Griffiths and Boysan (1996).

Most of above studies have aimed to simulate only the flow of water and solid in a hydrocyclone to understand the flow physics; very few attempts have been made to predict the performance of cyclone. Narasimha et al. (2005) made an attempt to develop a comprehensive CFD model to predict cut-size of the separation along with water splits in a laboratory 102 mm hydrocyclone using Lagrangian stochastic modified k- ε turbulence model superimposed on continuous water phase at low feed concentrations. The model predictions are all over predicted compared to experimental values. The apparent over-prediction of the computed collection efficiencies for the larger particles may be due to the occurrence of turbulent bursts in the wall boundary layer which may cause parcels of particles to shoot in the radial direction towards the axis of the cyclone, and to re-entrainment of particles from the vortex finder. This phenomenon is associated with underestimation of swirling patterns with the k- ε turbulence models as discussed in the review of turbulence models. The discrepancies for fine sized particles were due to used model unaccountability of air-core estimations. Maxey postulated that particles would accumulate in regions with high strain rate or low vorticity (Maxey, 1987). Dyakowski and Williams (1996) observed in their simulations that the strain rates are large near the inlet and the vortex finder due to the generation of swirl components, the changing of flow direction from horizontal to vertical, and the presence of vortex motion near the vortex finder. These zones of large strain rate are in qualitative agreement with the high particle concentration in the same regions calculated by Pericleous and Rhodes (1986), Rajamani and Milin (1992), and observed by Abdullah et al. (1993). These areas of high concentration are observed by Cullivan et al. (2004). Fig. 5 shows that the large particles have a high concentration along the wall and near the apex. For the smaller particle, there are two circular zones of high concentration, and they coincide with the zones of recirculation in the same simulation.

(a)

(b)


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Fig. 5 Contour of particle concentration in the hydrocyclone. Figure on the top shows the contour for particles with a radius of 0.25 μm and figure on the bottom shows the contour for particles with a radius of 2.5 μm (Cullivan et al., 2004). (No absolute scale of the contours was given.).

Most studies simulated the particle motion as a deterministic behaviour that is a balance between drag and centrifugal forces. In contrast, Averous and Fuentes (1997) suggest that the radial turbulence fluctuation could transport the particles rapidly from the wall to the core of the hydrocyclone. Cullivan et al. (2004) concluded that there exists a preferential particle radial direction which is attenuated by the turbulent fluctuation components. They plotted the standard deviation of the difference between the deterministic path and the stochastic path of particles in a cross section of the hydrocyclone, and the results are shown in Fig. 6. The effects of including the stochastic turbulent component are particular largely visible near the wall and the cylindrical section.

Fig. 6 The standard deviation of the differences between the deterministic and stochastic Lagrangian particle path. Contours from the 0.25 μm particles are on the top and contours from the 2.5 μm particles are on the bottom (Cullivan et al., 2004). (No absolute scale of the contours was given.).

The Lagrangian approach has however been extended to modelling cyclones at large particle concentrations by Rajamani and Milin (1992). The effect of solid concentration on fluid viscosity could be directly coupled by magnitude of the solid concentration. Rajamani and Milin (1992) and Sevilla and Branion (1997) used the Thomas expression for viscosity to define the modified fluid viscosity as:

)6.16exp(00273.05.105.21 2

0vvv

m φφφμμ +++= (6)

where mμ is the modified viscosity, 0μ is the liquid viscosity and vφ is the concentration of the solids. The concentration, vφ , is proportional to the


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particle residence time in a mesh element and the size of the particle. Increasing particle residence time or increasing particle diameter would both increase the concentration scalar, vφ . The method was used to predict limestone partition curves for feeds with up to 35% by weight limestone with good accuracy. The technique also predicted limestone concentrations, but these were not compared to experimental data. The calculation is initially started with the density and viscosity of water. After one iteration, the seeded particles’ trajectory through the domain is calculated and the concentration distribution determined. The viscosity is adjusted using equation (6) before the Navier-Stokes equations are solved again. The particle trajectories are now calculated again and this process is iterated until a converged solution is obtained. An improved collection efficiency curve is obtained and shown in Fig 7.

Fig. 7 The grade-efficiency curve obtained by iteratively accounting for the non-Newtonian effects of thick limestone (Rajamani and Milin, 1992).

Further, a probability density function (PDF) model was used by Devulapalli (1996) to track particles in a Lagrangian reference frame at high solids loading. The technique involved tracking particles clouds rather than the individual particles. The distribution of the particles with the cloud is represented by a PDF. The interaction between the fluid and the particle phase were accounted for by calculating the density and viscosity for the mixture. These modified physical properties were used to calculate the mean position of the particles. The solution procedure was iterative and the particle submodel was interfaced with the fluid-phase calculation.

As shown in Fig. 8, overall, the predictions showed a much sharper efficiency than the experimental one. However, some discrepancies are observed in the fine and coarse particle size ranges. The simulated efficiency curves show slightly lower water split ration to the underflow; this was mainly due to overprediction of the air-core diameters in the model calculations, which has a strong influence on the flow splits. In case of high feed concentration (>30 wt %), some difference were observed in the particle size range of 5–30 microns, an inadequate estimation of hindered settling while calculating the Re- DC correlations. Another important issue while tracking the particle PDF through the flow domain is the interaction of the cloud with the walls. Also, turbulent diffusion which was not accounted for this modelling work may contribute to fine particle migration.

Fig. 8 The grade-efficiency curve obtained by iteratively accounting for the feed concentration effects of thick limestone with (a) 27.7% and (b)47.2 % by weight in a 250 mm hydrocyclone (Devulapalli, 1996).

(a)

(b)


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A similar model (Rajamani and Milin, 1992) has been re-investigated using LES turbulence model by Delgadillo and Rajamani (2005) and found that the prediction of particle classification follows very close to the experimental values as shown in Fig. 9. But the predictions for the high feed solid concentrations are over-estimated compare to the experimental classification data in a bigger cyclone (see Fig. 10).

Fig. 9 Classification function for a 10.5% (by weight) slurry in a 75 mm-hydrocyclone (Delgadilo and Rajmani, 2005).

Fig. 10 Classification function for a 38.4% (by weight) slurry in a 250 mm-hydrocyclone (Delgadilo and Rajmani, 2005).

3.2 Eulerian-Eulerian models for cyclones

The full Eulerian-Eulerian interaction between different phases has been modelled by Cokljat et al. (2003). He simulated the flow with one liquid phase, one air phase and four phases of particle with

different granular diameters using a fully coupled RSM. The results show, as in Fig. 11, that the 10 μm particles remained evenly suspended in the solution. The ratio of amount of particles leaving through the vortex finder and the apex is proportional to the flow split ratio in the cyclone. In contrast, most of the larger 30 μm particles are collected along the wall and leave via the apex at high concentration.

Fig. 11 The predicted volume fractions for 10 and 30 micron particle sizes in a hydrocyclone using an Eulerian-Eulerian methodology (Cokljat, 2003).

Also, Suasnabar (2000) used the full Eulerian approach with granular flow modelling for the particulate phases to model a DSM pattern dense medium cyclone. The technique has also been used more recently by Nowakowski et al. (2000, 2003 and 2004).

(a)

(b)


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3.3 Mixture models for cyclones

The Volume of Fluid model (VOF) (Hirt and Nichols, 1981) and the Mixture model (Manninen et al., 1996) are simplified Eulerian multiphase approaches where the equations of motion are solved for the mixture and additional transport equations for the volume fractions of additional phases are solved. The VOF model and the Mixture model solve significantly less transport equations than the full Eulerian approach and thus numerically more efficient. The VOF and Mixture models are implemented in commercial CFD codes such as Fluent with the option of being used for turbulent flows with the turbulence model enabled for the mixture. The basic model equations of the Mixture model are defined as:

,0, =∂

∂+

∂∂

i

immm

xu

tρρ

(7)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂

∂∂∂+

∂∂−=

∂∂

+∂

∂

i

jm

j

imm

jij

imimim

xu

xu

xxxuu

tu ,,,,, μρρρ

∑=

⋅∂∂++

n

k

uu

ijm jDkiDkkk

xg

1,,ρρ α (8)

where mρ and mμ are the mixture density and viscosity and are the sums of all the density and viscosity contributions from each phase. Dku is the drift velocity which is the difference between each individual phase’s velocity and the mass-averaged velocity. kα is the volume fraction of each phase. Pericleous and Rhodes (1986) coupled the particle and fluid equations through modifying the mixture density and the effective viscosity. The mixture density, mρ , is calculated by accounting for the effect from all phases, and is expressed as:

i

n

ii

l

n

ii

m

cc

ρρρ

∑∑== +

−= 11

11 (9)

where lρ is the liquid density, iρ is the particle or air density, and ic is the fraction of the i-th phase. The turbulence viscosity is implicitly affected by the distribution of the different phases through the variation of fluid density. Subsequently, the slip velocity of the particle in the radial and axial

directions is defined identically to equations (4) and (5). The simulation was run with a water phase, an air phase and three solid phases with diameters of 5, 10 and 15 μm. With increasing particle diameter, the build-up of particles towards the underflow increases. The particle equilibrium line, where the particle drag and centrifugal forces are equal, also moves towards the vortex finder as the particle diameter increases. The closer the equilibrium line is to the vortex finder, the more likely it is for the particle to be removed from the hydrocyclone through the vortex finder. The collection efficiency from the simulations is shown in Fig. 12 and the curve is corrected to account for the fact that small particles with small Stokes numbers could not be fractionated since they follow the fluid motion exactly. The 50d of the simulated device is 4.5 μm. The results are compared against an empirical model proposed by Pericleous and Rhodes (1986) and showed good agreement. This model implicitly having the axi-symmetric and low lever turbulence (prandtl mixing length model) assumptions, which are incapable of predicting highly swirling flow patterns completely as discussed in turbulence section.

Fig. 12 The grade-efficiency curve from Pericleous and Rhodes (1986).

Grady et al. (2003) used the algebraic Mixture model which modelled each phase’s continuity and momentum equations. As shown in Fig. 13, the model successfully predicted the fractionation


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efficiency of particles above 20 μm but over-predicted the fractionation of smaller particles.

Fig. 13 Separation purity function comparison Grady et al. (2003).

Davidson (1994) used what was effectively the Mixture model (Manninen et al., 1994) to resolve particle concentrations in simulations of Kelsall’s cyclone (1952). Bagnold (1954) and turbulent diffusion forces were incorporated into the solid phase slip velocity calculation. Tomographic measurements of medium segregation were not available at the time of Davidson’s work (1994) but Davidson’s predictions (1994) compare well with the GRT density predictions of Galvin and Smithan (1994) and Subramanian (2002) with the drop in medium concentration at the wall being predicted qualitatively correctly. Suasnabar (2000) used both the full Eulerian granular approach (Ding and Gidaspow, 1990) and the Mixture model (Manninen et al., 1996) to model the distribution of medium in 200 mm and 350 mm DSM pattern dense medium cyclones. Suasnabar (2000) found that both Eulerian techniques predicted medium segregation in a qualitatively sensible manner but the Eulerian granular flow model predicted more correctly the observed drop in medium concentration at the wall in the bottom of the apex. Suasnabar (2000) suggested that this was because the full Eulerian granular flow model simulated Bagnold (1954) forces on the medium through the gradient in solids pressure in the dispersed phase momentum equation. The Mixture model has been used to model dispersed phases in cyclone separators at the JKMRC. Initially, the Mixture model with the DRSM turbulence model was used by Brennan (2003) to simulate medium and the air core in a

350 mm DSM pattern dense medium cyclone using a single medium size, the basic Schiller and Naumann (1935) drag law and no viscosity corrections. Brennan (2003) found that medium segregation was over-predicted when compared to Subramanian’s GRT data (2002). The simulations also predicted that the highest concentration of medium was at the wall and also that a film of pure water was predicted to form just below the air core. Neither of these effects was observed in Subramanian’s GRT measurements (2002). Subsequently the modelling was extended where medium was simulated with a size distribution, and wall lift forces based on Saffman’s expression (1965) were included, the Ishii and Mishimi (1984) slurry viscosity model was used, and the CFD used Large Eddy Simulation (Narasimha et al., 2006b). In subsequent work (Narasimha et al., 2006c), a Lagrangian approach was superimposed on the Mixture model simulations (using medium) to construct the partition curves for coal particles with reasonable accuracy. In parallel to this work, the same multiphase CFD approach was used to model the classification efficiency of the 75 mm Hsieh (1988) classifying hydrocyclone (Brennan et al., 2006). As shown in Fig. 14, the cyclone efficiency curves on all simulations are being predicted with quite good accuracy, except that the simulations using Bounded central differencing on the momentum equation predict more short-circuiting of the 25 and 35 μm size fractions to the overflow. The finer grid seems to improve the predictions at the smaller size range, but has a small effect on the short-circuiting. However, short-circuiting has been reduced by using the 3rd Order Muscl scheme on the momentum equation on the coarse grid. Currently, authors are investigating the effect of additional forces like Bagnold dispersive forces additional to the saffman’s lift forces using the Mixture model superimposed with advanced turbulence models such as LES for high feed solid concentrations in hydrocyclones.


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Cyclone Efficiency

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Fig. 14 Predicted cyclone efficiency curve (Brennan et al., 2006). Fine grid multiphase LES. Feed mass flow rate of water = 1.117 kg.s-1. Feed Limestone mass flow rate = 0.0574 kg.s-1. BCD—Bounded central differencing on Momentum. 3rd O – 3rd Order Muscl on Momentum. UF—fraction to underflow from underflow flow rate. OF—fraction to underflow by difference from overflow flow rate.

3.4 Simulation of high feed solids flow in hydrocyclones

The simulations were conducted on 3D body fitted grids of Devulappalli’s (1997) 250 mm cyclone geometry, which were generated in GambitTM, were unsteady LES simulations. The modified Mixture model (Narasimha et al., 2006c) was run using eight phase transport equations in the multiphase simulations, where the primary phase was water and the dispersed phases were air and five limestone phases of density 2700 kg m-3 with sizes 4.25 μm, 13.8 μm, 27 μm, 55.4 μm, 110.34 μm. The volume fraction of each limestone size in the feed boundary condition was set so that the cumulative limestone size distribution and the total limestone volume fraction matched those used in the feed in Devulapalli’s experimental runs. In this case the feed solids concentration is about 27.2% by weight as used by Devulappalli’s (1997). The flow field in hydrocyclones has been established and well understood, but little attention has been brought up on distribution of solids concentration inside the cyclone. The typical concentration contours are shown in Fig. 15. From Fig. 15, it is observed that more or less a high solids concentration exists along the wall of the cyclone, indicating coarse particles presence at this zone. As expected, most of the coarse particles will be separated immediately after entering into the free vortex flow domain due to high centrifugal forces.

To some extent, short-circuiting of solids is also prevailing along the vortex finder wall. A unique radial segregation of solid particles can be seen in section, which is resulted from the entrapment of some relatively coarse particles in the forced vortex flow. That is why a relatively higher solids concentration appeared in the middle of the cyclone apart from the cyclone wall.

Fig. 15 Steady state contours of solids concentration in the hydrocyclone.

Fig. 16 At steady state—(a) 4.25 micron, (b)13.8 micron, (c)27 micron, (d) 55.4 micron and (e) 110.34 micron.

Overall, the classification of different size particles based on the above observed information (see Fig. 16) can be explained as follows. The fine particles are more equally distributed in the cyclone volume because of the predominated turbulent diffusion. On the other hand, coarser particles are much influenced by the centrifugal forces and segregate at the cyclone wall. Interestingly the

(a) (b) (c) (d) (e)


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intermediate particles (here 27 microns range), which are close to the cut-size, are equally distributed to overflow and underflow. Generally these particles will eventually take a long time to report to products.

Series-II, Feed solids=27.2 wt%

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Fig. 17 Comparison of predicted and measured classification functions in a 250 mm Krebs hydrocyclone.

The separation curve with a well-known S-shape represents a steady state and is predicted to be established after 5.5 sec in a 250 mm hydrocyclone. The predicted typical classification curve is compared with Devulapalli’s data (1997) and shown in Fig. 17. Fig. 17 demonstrates that the separation of particles by size in the large-scale hydrocyclone is closely simulated by modified CFD-LES model at high feed solid loadings. The predicted cut-size is very close to the experimental data. The beauty of the CFD model is that at high solids loadings, it considered the interactions between the water phase and solids phase in terms of hindered settling correction; shear forces (lift) at the wall and also the inter-particle interactions in terms of Bagnold forces based on pressure gradient of granular solids. An attempt is made in this study to simulate the fish-hook phenomena at finer size particles classification. It appears as a raised partition curve below 10 microns in Fig. 17. Further investigation on fish-hook phenomena is needed to be done numerically in the future.

4. CONCLUSIONS

The flow inside a hydrocyclone is quite complex and there have been numerous numerical studies on the flows and the particle motions in hydrocyclone,

and these studies have been tested against a wide range of turbulence and multiphase models. Two-equation k-ε and RNG k-ε models flow velocities with empirical modifications were led poor results, especially the tangential components in comparison with experimental measurements. Despite the varying degree of empiricism and accuracy in predicting the velocity profiles, all these simulations were able to predict reasonable classification-efficiency curves. It appears that accurate classification-efficiency predictions do not hinge on accurate predictions of the velocity profiles. Most of the recent studies have utilized the Reynolds stress models (RSM) with different degrees of complexity in the pressure-strain correlation. Most of the RSM studies showed good agreements with velocity measurements. Unfortunately, the velocity profiles were not validated in most of the RSM cases where multiphase particle tracking was applied. Finally, large eddy simulation (LES) is the most advanced turbulence model applied in recent hydrocyclone numeric studies. Besides the additional information on the précising air core correctly, LES provides an additional accuracy in predicting the velocity profiles or the grade efficiency in comparison to the RSM. Presently, multiphase models have been successfully applied in a hydrocyclone to model the Lagrangian motions of spherical particles. Eulerian-Eulerian model have been used to account for the particles effect on the fluid viscosity. Simplified Eulerian model (Mixture model) predictions for solid transportation in cyclone were well predicted. Further, the inclusion of modified slip velocity calculation in the Mixture model improves the efficiency predictions close to the experimental data at low feed solid loadings. In future studies, the focus should be to model the three-dimensional flow in a hydrocyclone with at least the Reynolds stress model/LES. The particle tracking should at least include the effects of the turbulence on the particles. All these developed models will only be applicable to low feed solid concentration levels, where mostly these models neglect the particle-particle interactions. A more comprehensive numerical method of modified Mixture model should be applied for simulating solids flow in hydrocyclones for high feed solids concentration. Explicit models for accounting hindered settling and turbulent diffusion


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investigated for high feed solid concentrations in industrial cyclones are encouraging.

REFERENCES

1. Abdullah MZ, Conway WF, Dyakowski T and Waterfall RC (1993). Investigation of electrode geometries for use in cyclonic separators. Workshop on Tomographic Techniques for Process Design and Operation, p.127. Manchester, England.

2. Bagnold RA (1954). Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proceedings of the Royal Society London, Series A, 225, 49–63.

3. Baxter LL and Smith PJ (1993). Turbulent dispersion of particles: the STP model. Energy and Fuels 7:852–859.

4. Boysan F, Ayers WH and Swithenbank J (1982). A fundamental mathematical modelling approach to cyclone design. Trans. Inst. Chem. Eng. 60:222–230.

5. Brennan MS (2006). CFD simulations of hydrocyclones with an air core: comparison between large eddy simulations and a second moment closure. Chemical Engineering Research and Design 84A:495–505.

6. Brennan MS, Narasimha M and Holtham PN (2006) Multiphase CFD modelling of hydrocyclones-prediction of cut size. Minerals Engineering (in press).

7. Brennan MS, Holtham PN, Rong R and Lyman GJ (2002). Computational fluid dynamic simulation of dense medium cyclones. Proceedings of the 9th Australian Coal Preparation Conference, Yeppoon, Australia, paper B3.

8. Brennan MS, Subramanian VJ, Rong R, Holtham PN, Lyman GJ and Napier-Munn TJ (2003). Towards a new understanding of the cyclone separator. Proceedings of XXII International Mineral Processing Congress, Cape Town, South Africa.

9. Chakraborti N and Miller JD (1992). Fluid flow in hydrocylones: A critical review. Mineral Processing and Extractive Metallurgy Review 11:211–244.

10. Chu Liang-Yin and Chen Wen-Mei. (1993). Research of the motion of solid particles in a

hydrocyclone. Separation Science & Technology 28(10):1875–1886.

11. Cokljat D, Slack M and Vasquez SA (2003). Reynolds-stress model for Eulerian multiphase, Proc. 4th Int. Symp on Turbulence Heat and Mass Transfer, Begell House Inc. 1047–1054.

12. Crowe CT; Troutt TR and Chung JN (1996). Numerical Models for Two-Phase Turbulent Flows. Annu. Re. Fluid Mech. 28:11–43.

13. Cullivan JC, Williams RA, Dyakowski T and Cross CR (2004). New understanding of a hydrocyclone flow field and separation mechanism from computational fluid dynamics. Minerals Engineering 17(5):651–660.

14. Cullivan JC, Williams RA and Cross CR (2003). Understanding the hydrocyclone separator through computational fluid dynamics. Trans. Inst. Chem. Eng. 81A:455–465.

15. Dabir D (1983). Mean Velocity Measurements in a 3 inches Hydrocyclone Using Laser Doppler Anemometry. PhD Thesis, Department of Chemical Engineering, Michigan State University, USA.

16. Davidson MR (1994). A numerical model of liquid-solid flow in a hydrocyclone with high solids fraction. Numerical Methods in Multiphase Flows ASME Fluids Engineering Division Summer Meeting, 29–39.

17. Delgadillo Jose A and Rajamani Raj K (2005). A comparative study of three turbulence-closure models for the hydrocyclone problem. International Journal of Mineral Processing 77(4):217–230.

18. Derksen JJ and Van der Akker HEA (2000). Simulation of Vortex Core Precession in a Reverse-Flow Cyclone. American Institute of Chemical Engineers Journal 46(7):1317–1331.

19. Devullapalli B (1994). Hydrodynamic modeling of solid-liquid flows in large-scale hydrocyclones. PhD Thesis, University of Utah.

20. Ding J and Gidaspow D (1990). A bubbling fluidization model using the kinetic theory of granular flow. AIChE Journal 36:523–538.

21. Dyakowski T and Williams RA (1996). Prediction of high solids concentration Region within a hydrocyclone. Powder Technology 87:43–47.

22. Fraser SM, Abdel Rasek AM and Abdullah MZ (1997). Computational and experimental investigations in a cyclone dust separator. Proc. Instn. Grid. Engrs. 211E:247–257.


124

23. Galvin KP and Smitham JB (1994). Use of x-rays to determine the distribution of particles in an operating cyclone. Minerals Engineering 7:1269–1280.

24. Grady SA, Wesson GD, Abdullah M and Kula EE (2003). Prediction of 10-mm hydrocyclone separation efficiency using computational fluid dynamics. Filtration & Separation 40(9):41–46.

25. Griffiths WD and Boysan F (1996). Computational fluid dynamics and empirical modelling of a number of cyclone samplers. Journal of Aerosol Science 27(2):281–304.

26. He P, Salcudean M and Gartshore IS (1999). A numerical simulation of hydrocyclones. Trans. Inst. Chem. Eng. 77A:429–441.

27. Hirt CW and Nichols BD (1981). Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics 39:201–225.

28. Hoesksta AJ, Derksen JJ and Van den Akker HEA (1999). An experimental and numerical study of turbulent swirling flow in gas cyclones. Chemical Engineering Science 2055–2065.

29. Hsieh KT (1988). A phenomenological model of the hydrocyclone. Ph D Thesis, University of Utah.

30. Hsieh KT and Rajamani RK (1991). Mathematical model of the hydrocyclone based on the physics of fluid flow. AIChE Journal 37:735–746.

31. Ishii M and Mishima K (1984). Two-fluid model and hydrodynamic constitutive relations. Nuclear Engineering and Design 82:107–126.

32. Kelsal DF (1952). A study of the motion of solid particles in a hydraulic cyclone. Transactions of the Institution of Chemical Engineers 30:87–108.

33. Launder BE, Reece GJ and Rodi W (1975). Progress in the development of a Reynolds-stress turbulence closure. Journal of Fluid Mechanics 68:537–566.

34. Ma L, Ingham DB and Wen X (2000). Numerical modeling of the fluid and particle penetration through sampling cyclones. Journal of Aerosol Science 31(9):1097–1119.

35. Manninen M, Taivassalo V and Kallio S (1996). On the mixture model for multiphase flow. Valtion Teknillinen Tutkimuskeskus, Espoo, Finland.

36. Maxey MR (1987). The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. Journal of Fluid Mechanics 174:441.

37. Monredon TC, Hsieh KT and Rajamani RK (1992). Fluid flow of the hydrocylone: An investigation of the device dimension. International Journal of Mineral Processing 35:65–83.

38. Narasimha M, Brennan MS and Holtham P (2006a). A brief review of empirical and numerical flow modelling of dense medium cyclone. Coal preparation 26(2):55–89

39. Narasimha M, Brennan M and Holtham PN (2006b). Numerical simulation of magnetite segregation in a dense medium cyclone. Minerals Engineering 19:1034–1047.

40. Narasimha M, Brennan M, Holtham PN and Napier-Munn TJ (2006c). A comprehensive CFD Model of dense medium cyclone performance. Minerals Engineering (in press).

41. Narasimha M, Sripriya R and Banerjee PK (2005). CFD modelling of hydrocyclone-prediction of cut-size. Int. J. Mineral. Process 71(1):53–68.

42. Nowakowski AF, Kraipech W, Williams RA and Dyakowski T (2000). The hydrodynamics of a hydrocyclone based on a three dimensional multi-continuum model. Chemical Engineering Journal 80(1–3):275–282.

43. Nowakowski AF and Dyakowski T (2003). Investigation of swirling flow structure in hydrocyclones. Chemical Engineering Research and Design 81(A):862–873.

44. Nowakowski AF, Cullivan JC, Williams RA and Dyakowski T (2004). Application of CFD to modelling of the flow in hydrocyclones. Is this a realizable option or still a research challenge? Minerals Engineering 17:661–669.

45. Pericleous KA and Rhodes N (1986). The hydrocyclone classifier—a numerical approach. International Journal of Mineral Processing. 17:23–43.

46. Petty CA and Parks SM (2001). Flow predictions within hydrocyclones. Filtration and Separation 38(6):28–34.

47. Rajamani RK and Milin L (1992). Fluid-flow model of the hydrocyclone for concentrated slurry classification. Hydrocyclone, Analysis and Application (ed. L Svarovsky and MI Thew), London, 95–101.


125

48. Saffman PG (1965). The lift on a small sphere in a slow shear flow. Journal of Fluid Mechanics 22(2):385–400. [Corrigendum, Journal of Fluid Mechanics 31:624].

49. Schiller L and Naumann Z (1935). Z. Ver. Deutsch. Ing. 77:318.

50. Schuetz S, Mayer G, Bierdel M and Piesche M (2004). Investigations on the flow and separation behaviour of hydrocyclones using computational fluid dynamics. Int. J. of Min. Process 73:229–237.

51. Sevilla EM and Branion R (1997). The fluid dynamics of hydrocyclones. Journal of Pulp and Paper Science 23(2):J85–J93.

52. Slack MD, Prasad RO, Bakker A and Boysan F (2000). Advances in cyclone modeling using unstructured grids. Chemical Engineering Research and Design 78(A):1098–1104.

53. Smagorinsky J (1963). General circulation experiments with the primitive equations. I. the basic experiment. Month. Wea. Rev. 91:99–164.

54. Soo SL (1990) Multiphase Fluid Dynamics, Science Press and Gower Technical, Beijing, China.

55. Speziale CG, Sarkar S and Gatski TB (1991). Modelling the pressure-strain correlation of turbulence: An invariant dynamical systems approach. Journal of Fluid Mechanics 227:245–274.

56. Suasnabar DJ (2000). Dense Medium Cyclone Performance, Enhancements via computational modeling of the physical process. PhD Thesis, University of New South Wales.

57. Subramanian VJ (2002). Measurement of medium segregation in the dense medium cyclone using gamma-ray tomography. PhD Thesis, JKMRC, University of Queensland.

58. Witt PJ, Mittoni LJ, Wu J and Shephero IC (1999). Validation of a CFD model for predicting gas flow in a cyclone. Proceedings of CHEMECA 99, Newcastle, Australia.

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