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366 Progress in Computational Fluid Dynamics, Vol. 10, Nos. 5/6, 2010

Turbulence model analysis of flow inside ahydrocyclone

D.W. Stephens* and K. Mohanarangam

Parker Centre,CSIRO Mathematics Informatics and Statistics,Minerals Down Under National Flagship,Clayton, Victoria 3169, AustraliaE-mail: [email protected]: [email protected]*Corresponding author

Abstract: Turbulence analysis of flow inside a hydrocyclone is carried out usingcommercially available CFD software ANSYS CFX (release 11.0) (ANSYS Inc., 2007).CFD software(s) and their turbulence models have come a long way in accurately predictingthe flow inside a hydrocyclone. This paper shows, among various turbulence models tested,a two-equation Shear Stress Transport (SST) turbulence model coupled with curvature

correction can accurately predict the mean flow behaviour. The same level of accuracy wasonly found with a SSG Reynolds stress model with a penalty of solving an additional fivetransport equations. Experimental data of Hsieh (1988) and Monredon et al. (1992) wasused to validate our CFD models.

Keywords: hydrocyclones; computational fluid dynamics; turbulence; curvature correction.

Reference to this paper should be made as follows: Stephens, D.W. and Mohanarangam, K.(2010) Turbulence model analysis of flow inside a hydrocyclone, Progress inComputational Fluid Dynamics, Vol. 10, Nos. 5/6, pp.366373.

Biographical notes: D.W. Stephens completed a Bachelor of Mechanical Engineering atJames Cook University, Townsville, Australia in 1996. In 2001, he was awarded aPhD in Mechanical Engineering from James Cook University, Townsville, Australia for

investigating the circulation and heat transfer occurring in sugar crystallisation vesselsand development of a computational fluid dynamics model of the system. He is presentlyworking as a Principal Research Engineer in the CSIRO Division of MathematicsInformatics and Statistics, Victoria, Australia. His areas of interest are the simulation ofcomplex multi-phase flows and optimisation.

K. Mohanarangam completed his Bachelor of Aeronautical Engineering at the Universityof Madras in India. During his Undergraduate D, he was awarded the Young EngineersFellowship Programme (YEFP) from the Indian Institute of Science (IISc), Bangaloreto carry out research in the area of supersonic and hypersonic flows. He undertook hisDoctoral studies at RMIT University, Melbourne completing in 2008. His Doctoral Thesisconsisted of looking at the numerical simulation of Turbulence Modulation (TM) in two-phase flows of varying density regimes. He is currently a Postdoctoral research fellow in the

CSIRO Division of Mathematics Informatics and Statistics, Victoria, Australia.

1 Introduction

Hydrocyclones have found use in numerous industriesfor over half a century. They are also called the liquidcyclone or the hydraulic cyclone, due to inherent use ofwater as its primary phase. As they aid in the mechanicalseparation of dispersed solid particles from a suspension,they have found key applications in the mineral,chemical and pharmaceutical industries, to name a few.

In comparison to other mechanical separation devices,hydrocyclones just need energy to overcome the pressuredrop they encounter during their operation, making

them a cheap alternative (Schuetz et al., 2004). They arealso essentially passive devices, with a short residencetime making them the ideal device for classification ofparticles (Brennan, 2006). They work on the principle ofcentrifugal forces that develop due to the swirling flowinside the cyclone body that effect particle separationbased on their density characteristics. In addition, theyare simple in design with low maintenance; this alliedwith the capacity to handle high throughputs make them

the preferred units in many industrial applications. Thevery first hydrocyclone models that appeared in theliterature were empirical and were capable of predicting

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Turbulence model analysis of flow inside a hydrocyclone 367

the size-classification curve within a reasonable degreeof accuracy for a specific flow calibration. While theywere good for most flow-sheeting purposes, a CFDmodel was necessary to study the effect of flow structurewith changes in geometry and flow rates. A numberof CFD studies were carried out inside a hydrocyclone(Pericleous and Rhodes, 1986; Hsieh and Rajamani,1991; Dyakowski and Williams, 1993; Malhotra et al.,1994; Rajamani and Devulapalli, 1994). Due to theinherent presence of high swirl and very large curvatureof streamlines within the flow, modelling such flow poseda challenge. The high level of turbulence encounteredin simulating these flows was not properly representedby the conventional turbulence models (Pericleousand Rhodes, 1986; Malhotra et al., 1994). However,Pericleous and Rhodes (1986), Hsieh and Rajamani(1991) applied a modified Prandtl mixing-length modelfor turbulent transport. Their governing equations wereformulated in terms of vorticity, stream function and

angular velocity, with mixing length being a function ofposition within the hydrocyclone.

Dyakowski and Williams (1993) proposed a revisedapproach to modelling turbulent flow in a small diameterhydrocyclone by taking into account the anisotropy ofturbulent viscosity, as well as the non-linear interactionbetween mean vorticity and mean strain rate, by utilisinga k- model coupled with equations for calculatingnormal components of Reynolds stresses. Alternatively,Malhotra et al. (1994) tried to capture the turbulenceaccurately using an altered form of dissipation equation.However, both these works considered that the flow

was axi-symmetrical and therefore a two-dimensionalequation was solved. The above assumption is nottrue in the case of industrial hydrocyclones and theerror induced in such computations was investigated byHe et al. (1999), who concluded that two-dimensionalaxi-symmetric inlets are far from accurate comparedto full three-dimensional ones. The same author alsostudied the flow pattern inside a hydrocyclone using amodified k- model involving a curvature correction termrelated to the turbulence Richardson number within thedissipation equation.

With increases in computational power, Large EddySimulations (LES) are being used as a turbulent closure

for resolving the fluid flow inside a hydrocyclone.The advantage is that all of the large eddies are resolvedwhile the small eddies are modelled. The LES approacheliminates the explicit empiricism that is imposed in thek- model. Since only the sub-grid scales are modelledin LES, the anisotropy is largely taken care of in thisapproach (Delgadillo and Rajamani, 2007). Some of theLES studies of hydrocyclones include Brennan (2006),Mainza et al. (2006), Brennan et al. (2007) and Delgadilloand Rajamani (2007, 2009).

In this paper, we have shown that in an industrialenvironment a two-equation model can still be used to

depict the turbulence encountered inside a hydrocyclone.It is actually the proper choice of turbulence modeland the type of mesh that governs the accuracy of the

underlying solution. In this regard, both the tetrahedraland hexahedral meshes are trialled, along with varyingmesh distributions to study the effect of turbulencewithin these passive devices. In order to verify thevalidity of the numerical simulations, the results werecompared against the experimental data of Monredon etal. (1992).

2 Model description

A single-phase numerical model was used to simulatethe flow in a 75 mm diameter hydrocyclone, thedimensions of which are detailed in Monredon etal. (1992). The model is based on the ReynoldsAveraged Navier-Stokes equations using the eddyviscosity hypothesis:

Continuity equation:

(U) = 0 (1)

Momentum equation:

(UU) = p + U+UTu u) + B (2)

where U is the fluid velocity vector, the fluid density,p the modified pressure, the viscosity, u u theReynolds stresses and B is the body force.

2.1 Two-equation turbulence models

Two-equation turbulence models are widely used in theCFD modelling of many industrial applications; theyoffer a good compromise between numerical effort andcomputational accuracy. They derive their name fromthe fact they solve both the velocity and length scalefrom two separate transport equations. The k- and k-based two-equation models use the gradient hypothesisto relate the Reynolds stresses to the mean velocitygradients and the turbulent viscosity.

u u = t U+UT . (3)The turbulent viscosity is defined as the product ofa turbulent velocity and the turbulent length scale.In two-equation models, the turbulence velocity scaleis computed from the turbulence kinetic energy fromthe solution of a transport equation. The turbulentlength scale is estimated from two properties of theturbulence field, usually the turbulence kinetic energyand its dissipation rate. The dissipation rate of theturbulence kinetic energy is provided from the solutionof its own transport equation.

2.1.1 k- turbulence model

Based on the above formulation, the values of turbulencekinetic energy (k) and turbulence dissipation () areobtained by solving differential transport equations and

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368 D.W. Stephens and K. Mohanarangam

are given by Equations (4) and (5), respectively

(Uk) =

+tk

k

+ Pk (4)

(U) =

+t

+

k(C1Pk

C2) . (5)

The turbulent viscosity in Equation (3) is computed usingthe formulation

t = Ck2

. (6)

Here C, C1, C2, k and are constants. Pk isthe turbulence production due to viscous and buoyancyforces, which is modelled using

Pk = t

U

U+

U

T

2

3 U3t

U

23

k U. (7)

2.1.2 Shear Stress Transport (SST) turbulencemodel

The second set of two-equation turbulence models usedin our current study is the k- based (SST) modelof Menter (1994). The transport equations for k andturbulence frequency () are given by Equations (8)and (9), respectively

(Uk) = + tk3

k + Pk k (8)

(U) =

+t

3

+ 3

kPk 32

+ (1 F1) 22

k. (9)

The combined k- and k- models do not account for thetransport of the turbulent shear stress, which results in anover-prediction of eddy-viscosity, and ultimately leads toa failed attempt in predicting the onset and the amountof flow separation from smooth surfaces. The proper

transport behaviour can be obtained by a limiter to theformulation of the eddy-viscosity and is given by

t =a1k

max(a1,SF2); t =

t

. (10)

Readers are advised that further information regardingthe value of constants and blending functions can befound in the ANSYS CFX (release 11.0) user manual(ANSYS Inc., 2007).

2.2 SST with curvature correction

One of the weaknesses of the eddy-viscosity models isthat they are insensitive to streamline curvature andsystem rotation, which play a significant role in our

current hydrocyclone modelling. A modification of theturbulence production term is available to sensitisethe standard eddy-viscosity models to these effects. Amultiplier is introduced into the production term and isgiven by Pk Pkfr, where

fr = Cscale max(min(frotation, 1.25) , 0.0) . (11)

The empirical functions suggested by Spalart and Shur(1997) to account for these effects are given by

frotation = (1 + cr1)2r

1 + r

1 cr3 tan1 (cr2r)

cr1. (12)

r =S

; r = ccij

2ikmag

/D0.5 (13)

ccij = Sjk

DSij

Dt+ (imnSij + jmnSin)

rotm /D (14)

ij = 0.5

uixj

ujxi

+ 2mjirotm (15)

D = maxS2, 0.092

; S2 = 2SijSij (16)

2 = 2ijij ; mag =

212 + 213 +

223

0.5. (17)

2.2.1 Reynolds stress model

Reynolds Stress or Second Moment Closure (SMC)models are applicable where the eddy-viscosityassumption is no longer valid and the results of eddyviscosity models might be inaccurate. They include

the solution of transport equations for the individualcomponents of the Reynolds stress tensor and thedissipation rate. The increased number of equationsusually leads to reduced numerical robustness, increasedcomputational time and restrictions for usability incomplex flows. The standard Reynolds Stress model inANSYS CFX is based on the -equation. The CFX-Solver solves the following equations for the transportof the Reynolds stresses:

xk(Ukuiuj) =

xk

+

2

3Cs

k2

uiuj

xk

+ Pij

23 ij + ij (18)

where ij is the pressure-strain correlation and the exactproduction term Pij is given by

Pij = uiuk Ujxk

ujuk Uixk

. (19)

The most important term in the Reynolds Stress modelis the pressure-strain correlation ij as it acts to driveturbulence towards an isotropic state by redistributingthe Reynolds stresses. It can be split into two parts ij =

ij,1 + ij,2, where ij,1 is the slow term, also knownas return-to-isotropy term, and ij,2 is called the rapidterm. For the SSG model that we used in our current

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study, these are given by

ij,1 = Cs2

aikakj 13

amnamnij

Cs1aij (20)ij,2 = Cr1P aij + Cr2kSij

amnamn

+Cr4k

aikSij + ajkSik 23

aklSklij

+Cr5k (aikjk + ajkik) . (21)

As the turbulence dissipation appears in the individualstress equations, an equation for it is still required andtakes the form

xk(Uk) =

k(C1P C2)

xk

+

t,RS

xk

. (22)

The constants in the equations take the form C,RS =0.1, ,RS = 1.36, Cs1 = 1.7, Cs2 = 1.05, Cr1 = 0.9,Cr2 = 0.8, Cr3 = 0.65, Cr4 = 0.625, Cr5 = 0.2.

3 Numerical procedure

The conservation equations of mass, momentum andturbulence given above were solved using a finite volumemethod in order to determine the single-phase liquidvelocity for comparison against the experimental data. It

is not possible to arrive at a solution with these equationsusing analytical approaches; consequently ANSYS CFX(release 11.0) is used to solve them on hexahedral andtetrahedral grids. Rhie and Chow (1983) interpolationis used to avoid chequer-board oscillations in the flowfield. Coupling between velocity and pressure is handledimplicitly by a coupled solver. Advection terms arediscretised using the High Resolution Scheme which issecond-order accurate. In the current study, the air corewas not modelled; rather it was imposed as a free-slipboundary condition along the centre of the hydrocyclone,the width of which was derived from the experiments ofHsieh (1988) and Monredon et al. (1992). As a normal

CFD practice and a strategy to save computational time,the converged results of the k- model was used asan initial guess for the SST model and these resultswere then used for SST with curvature correction.The converged results from the two-equation turbulencemodel was not good enough to start the solution processof the SSG model, thereby LRR-IP (a simpler ReynoldsStress model in ANSYS CFX) was run and its resultswere used as an initial guess for the SSG model. Usingthese these initialisation procedures the SSG modelrequired approximately twice the computational timeas the SST-CC model to reach a converged solution.

Scalable wall functions within ANSYS CFX were usedfor k- turbulence model, while automatic wall functionswere used for SST and its variants and SSG. For the

hexahedral meshes the y+ values were typically between2400, while for the tetrahedral/prism meshes the valueswere typically between 7200.

4 Results

In this section, the experimental results of Hsieh(1988) and Monredon et al. (1992) are comparedagainst our numerical findings. Axial and tangentialvelocities are compared at three different locations(60mm, 120mm and 180mm from the top of thecyclone). Two different mesh types, pure hexahedraland mixed tetrahedral/prism meshes, where used for thesimulations. For each of the mesh types, three differentmesh densities were used to assess the effect of meshresolution on the simulation results. The number ofnodes contained in each of the mesh types and densitiesare shown in Table 1. The results on the finest mesh

densities were found to be slightly better than the

Table 1 Node number for mesh independency tests

Coarse Medium Fine

Hex 173,000 617,000 1,480,000Tet/Prism 205,000 728,000 2,800,000

Figure 1 Medium density computational meash for: (a)hexahedral and (b) tetrahedral/prismmesh types

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medium mesh density for both mesh types considered.However, the simulations on the finest mesh density cameat an enormous computational cost. Figure 1 shows thecomputational mesh for the medium density hexahedraland mixed tetrahedral/prism mesh types.

Figures 2 and 3 show the axial and tangentialvelocities for coarse, medium and fine mesh usinghexahedral elements with the SST curvature correctionturbulence model. It could be seen that there is hardlyany change from the medium density mesh to the finemesh. In fact, the coarse mesh in our simulations seemsto ably replicate the experimental data. Comparisonbetween turbulence models was performed only on themedium density meshes.

Figure 2 Axial velocities at various vertical locations in thehydrocyclone for varying node numbers ofhexahedral elements using SST with curvaturecorrection

Figures 4 and 5 show the predicted axial and tangentialvelocities compared against experimental data using the

Figure 3 Tangential velocities at various vertical locations inthe hydrocyclone for varying node numbers ofhexahedral elements using SST with curvaturecorrection

hexahedral mesh, respectively. Here the axial velocitiesare plotted in the 90270 plane, whereas the tangential

velocities are plotted in the 0180 plane. The 0180

plane is normal to the feed port. Four differentturbulence models, namely k-, SST, SST with curvaturecorrection (SST-CC) and SSG, are plotted against theexperimental data shown in solid dots.

From the plot of axial velocity at 60 mm from thetop, it can be seen that the k- and SST models failto capture the flow trend throughout while the SST-CCand SSG models are very good in capturing the vortexflow at the far end and also at the ends of the air core.At 120 mm, the same behaviour is observed, however,the SSG model falls short of capturing the magnitude

of velocity near the air core, while SST-CC is able toaccurately replicate the experimental data. At 180 mm,SST-CC is closer to experimental data than SSG, but

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Figure 4 Comparison of axial velocities at various verticallocations in the hydrocyclone for the fourturbulence models tested on the medium densityhexahedral mesh

the asymmetry in the flow structure observed in the

experiments is not captured by any of the turbulencemodels. This could potentially be due to the assumedridgid, straight boundary condition for the air core-waterinterface.

Tangential velocities at the 60 mm section show avery good comparison for SST-CC and SSG, however,the magnitude of the maximum velocity is not quitereplicated by the SSG model. At the 120 mm sectionthe SSG model again fails to compare well with themaximum magnitude while the SST-CC shows a verygood match. While SST-CC shows a good match furtherbelow at 180 mm, the asymmetry between the velocities

is still not clearly predicted by the model.The results presented show neither the k- nor

SST without curvature correction models perform well.

Figure 5 Comparison of tangentiall velocities at variousvertical locations in the hydrocyclone for the fourturbulence models tested on the medium densityhexahedral mesh

The major reason is the absence of a curvature term

which is important for the strong swirling flows inhydrocyclones. This correction appears in the form of amultiplier on the production term in the SST-CC model,whereas in SSG this happens through the pressure-straincorrelation which redistributes the Reynolds stresses inthe turbulence equations.

4.1 Effect of mesh types

The long standing issue of whether hexahedral meshesprovide better results over tetrahedral/prism meshes isinvestigated here with the hydrocyclone geometry. It is

true that its rather tedious to obtain pure hexahedralmeshes within complex geometries in comparison totetrahedral/prism meshes.

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Figure 6 Comparison of axial velocities with hexahedral andtetrahedral/prism meshes for the SST-CCturbulence model using the medium density meshes

Figures 6 and 7 show a plot of the axial as well as

tangential velocities for three locations from the top(60, 120 and 180 mm) for the two mesh types usingSST-CC, respectively. Axial velocities at location 60 mmshow a good comparison for both mesh types butas one moves down to the lower conical section ofthe hydrocyclone the hexahedral mesh outperforms thetetrahedral/prism mesh type. For sections 120mm and180 mm the maximum velocity that the tetrahedral/prismmesh predicts is truncated. As mentioned above, theasymmetry in velocities is not well resolved by themodels.

For the tangential velocities, sections 60 mm and

120 mm show a good match for the tetrahedral/prismmeshes in comparison to the 180 mm section, wherethe discrepancy is greatest. Throughout various locations

Figure 7 Comparison of tangential velocities with hexahedraland tetrahedral/prism meshes for the SST-CCturbulence model using the medium density meshes

the tetrahedral/prism mesh was not able to predict themaximum velocity magnitude observed experimentally.

4.2 Pressure drop and underflow rate comparison

There is a need to establish how well CFD modellingof hydrocyclones predicts the performance characteristicswhich are of interest to plant designers and operators,such as the pressure drop vs. flow behaviour, therecovery to underflow and the classification efficiencyfor particular cyclone designs (Brennan et al., 2009).Table 2 shows the comparison between experimentaldata and the predictions from the various turbulencemodels for the pressure drop and recovery to underflow.

In Table 2 the error percentage is expressed as thedifference between experimental and simulated data as apercentage of experimental data.

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Table 2 Pressure drop and mass split ratio results

Mass split Error Pressure Error underflow, (%) (%) drop, (kPa) (%)

Exp. 4.90 46.70 k- model 12.12 147.6 27.59 41.0SST 11.95 144.1 24.30 48.0

SST-CC 5.54 13.2 44.77 4.1SSG 5.66 15.5 35.51 24.0

Table 2 clearly shows that the SST-CC model errors aremuch fewer in comparison to the other models. The nextclosest model to the SST-CC is the more computationallyexpensive SSG model.

5 Conclusion

Analysis of different turbulence models was carried

out for a turbulent flow inside a 75 mm diameterhydrocyclone. Four different turbulence models weretested of which three belong to the two-equation modelclass and one to the Reynolds Stress class. ANSYS CFX(release 11.0) (ANSYS Inc., 2007) was used to solve thegoverning set of partial differential equations for flowand turbulence. Experimental results of Monredon etal. (1992) were used to compare our numerical findings.For the different turbulence models tested, SST-CCgave the best prediction. The only other model tocompete with these predictions is the SSG ReynoldsStress model, with the penalty of solving an additional

five equations for varying stresses. Mesh independencytests were carried out to minimise any errors caused dueto the underlying mesh. In addition to this, hexahedraland tetrahedral/prism meshes were compared againsteach other to check the performance of each mesh type incapturing the flow physics within a hydrocyclone. It canbe ascertained from this study that hexahedral meshis still a superior option to tetrahedral/prism mesh insimulating the flow behaviour within a hydrocyclone.

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