Brennan 2007

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www.elsevier.com/locate/mineng

Minerals Engineering 20 (2007) 395–406

Multiphase modelling of hydrocyclones – prediction of cut-size

M.S. Brennan a,*, M. Narasimha b, P.N. Holtham a

a Julius Kruttschnitt Mineral Research Centre, The University of Queensland, Isles Road, Indooroopilly 4068, Queensland, Australiab R&D Division, Tata Steel, Jamshedpur, Jharkhand 831 007, India

Received 1 August 2006; accepted 6 October 2006Available online 28 November 2006

Abstract

A comprehensive multiphase model of cyclone separators using Computational Fluid Dynamics is under development. The model iscapable of predicting velocity profiles, flow splits, air core position and efficiency curves in classifying hydrocyclones. The modelapproach uses the Mixture model with the granular options and large eddy simulation (LES) to resolve the turbulent mixing of the par-ticles. Multiphase simulations of Hsieh’s [Hsieh, K.T., 1988. A phenomenological model of the hydrocyclone, Ph.D. thesis, University ofUtah] data show a very good prediction of the cyclone efficiency curve. Whilst further model development is needed, the approach isshowing promise as a cyclone design tool.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Hydrocyclones; Classification; Computational fluid dynamics; Large eddy simulation; Mixture model

1. Introduction

Hydrocyclones rely on the centrifugal forces thatdevelop under the swirling flow to effect separation of par-ticles in slurries and particle laden flows and can classify ondensity or particle size. Modelling the flow by computa-tional fluid dynamics (CFD) is a key to understandinghow cyclone separators behave. However the flow is multi-phase, usually turbulent and an air core usually forms if thecyclone outlets are exposed to the atmosphere. These fac-tors influence the separation behavior and must beaccounted for even though they complicate the CFDmodeling.

Turbulent flows in industrial processes have mostly beenmodeled by CFD using RANS based turbulence models.However previous studies (Slack and Wraith, 1997; Suasn-abar, 2000; Cullivan et al., 2003; Delgadillo and Rajamani,2005) have shown that the turbulence in hydro-cyclones istoo anisotropic to model with k � e models and the veloc-

0892-6875/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mineng.2006.10.010

* Corresponding author. Tel.: +61 7336 55888; fax: +61 7336 55999.E-mail address: [email protected] (M.S. Brennan).

ities are predicted more accurately with a 2nd momentclosures.

In recent years large eddy simulation (LES) has becomeavailable in commercial CFD packages. Whilst computa-tionally more intensive than a RANS simulation, LESresolves the largest turbulent scales without modeling andhas been shown to predict the mean flows in cyclone sepa-rators with reasonable accuracy for single phase gas flows(Slack et al., 2000; Shalaby et al., 2005) and the basictwo phase flow (water feed plus air core) in hydrocyclones(Delgadillo and Rajamani, 2005; Brennan, 2006). In princi-ple LES should also be able to resolve the turbulent mixingof particles, provided that the effect of particle loading onthe flow and turbulence is accounted for in the simulation.

In this paper we report the results of multiphase CFDstudies of the Hsieh cyclone (1988) with a limestone feed,where the turbulence has been resolved using large eddysimulation (LES) and the limestone has been simulatedusing the Mixture model (Manninen et al., 1996) with thesize distribution approximated by six phase transport equa-tions. The tromp (cyclone efficiency) curve has been con-structed from the time averaged flow rates of each sizerange out the underflow and overflow boundaries. The

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Nomenclature

Transport equations in this paper have generally used cartesian tensor notation. Velocities, densities, viscosities,pressure and stress tensor terms associated with the equations of motion refer to the mixture unless denoted withthe subscript k (dispersed phase k) or c (continuous fluid phase)

Ak particle cross-sectional area of phase k

Cd drag coefficientCs Smagorinsky Lilly constantekk coefficient of restitution of phase k

dk diameter of phase k

gi i component of gravityh distance from wallk turbulent kinetic energyLs length scale of the sub grid scale stressesMk,i i component of interphase momentum transfer

to phase k

pk granular pressure of phase k

Sij mean strain ratet timeui i component of mixture velocityukc,i i component of velocity of phase k relative to

mixture (drift velocity)ukm,i i component of velocity of phase k relative to

continuous phase (slip velocity)Vg volume of grid finite volume

Vk particle volume of phase k

xi i co-ordinatey+ dimensionless distance from wallak volume fraction of phase k

dij kronecker deltaj Von Karmann’s constant = 0.41q mixture densitysd,ij drift stress tensor of mixturess,ij turbulent or sub grid scale stress tensor of mix-

tureslij viscous stress tensor of mixtureHk granular temperature of phase k

l effective molecular viscosity of the mixturels sub grid scale eddy viscosity of the mixture

Subscripts

c continuous phase c

i, j components in i and j directionsk phase k

m mixture

396 M.S. Brennan et al. / Minerals Engineering 20 (2007) 395–406

predicted distribution of each size in the distribution insidethe cyclone body are shown.

2. Previous work

CFD of hydrocyclones has primarily focused on model-ing the basic flow behavior, which is understandable aseven a basic two phase water/air core simulation using aRANS turbulence model still requires between one andtwo days to reach convergence using current hardware.As noted by Slack and Wraith (1997), Suasnabar (2000)and Cullivan et al. (2003), the turbulence anisotropy inhydrocyclones means that a RANS based CFD solutionof a hydrocyclone needs to use a 2nd moment closure (orDifferential Reynold Stress Turbulence Model, DRSM)where the transport equations for the individual Reynoldsstresses are solved. However both Delgadillo and Rajamani(2005) and Brennan (2006) have shown that the Launderet al. 2nd moment model (1975) under predicted the tan-gential velocities in simulations of the Hsieh cyclone(1988) and that large eddy simulation gave more accuratepredictions of the mean velocities.

The goal of a CFD simulation of a classifying hydrocycl-oner is to predict classification and if this goal is to beachieved then the interactions between the particulatephases and the fluid must be modeled. In CFD there aretwo approaches to simulating particulate phases in fluid sys-tems; these are the Lagrangian and Eulerian approaches.

A Lagrangian simulation runs ‘‘on top’’ of a singlephase CFD simulation and simulates the paths of individ-ual particles through the fluid by a double integration ofthe particle acceleration, which is calculated from a forcebalance on the particle. The particle force balance includesdrag, buoyancy forces and other body forces and the dragforce is calculated from the local fluid velocity as predictedby the single phase CFD calculation. The particle path cal-culation starts from an initial position, which is usually thefeed boundary condition and terminates when the particlehas left the domain via an outlet boundary condition orreaches a stagnant zone.

Eulerian CFD techniques treat the dispersed phases as apseudo continuum and solve transport equations for thephase concentrations. Flows containing dispersed solidphases at high concentrations and where inter-particle col-lisions result in additional stresses can be simulated using afull Eulerian granular flow approach (Ding and Gidaspow,1990; Gidaspow et al., 1992) where momentum and conti-nuity equations are solved for both the dispersed phasesand the continuous phases. The full Eulerian granular flowapproach also solves transport equations for the continu-ous phase turbulence and also solves transport equationsfor the granular temperature and granular pressure whichare used to calculate effective viscous stresses in the dis-persed phase momentum equation.

A simplified Eulerian approach is the Mixture model(Manninen et al., 1996) where the momentum and continu-

M.S. Brennan et al. / Minerals Engineering 20 (2007) 395–406 397

ity equations are only solved the mixture are solved and atransport equation, which contains an equilibrium slipvelocity, is solved for the volume fraction of each phase.The Mixture model has been modified to incorporate someof the features of granular flow theory and these modifica-tions are discussed in the next section.

The advantage of the Lagrangian multiphase approachis that particle–particle and particle–fluid interactions arecalculated dynamically for every particle present in the sys-tem based on the instantaneous velocity of the particle. Bycomparison the Eulerian approach only calculates a phasevelocity, a phase volume fraction and overall stresses asso-ciated with the average behavior of an ensemble of phaseparticles in a each finite volume in the CFD grid. Thusthe Eulerian approach innately involves averaging of somesort, which implies more modelling. However the computa-tional requirements of the Lagrangian approach scaledirectly with the number of particles, whereas the computa-tional requirements of the Eulerian approach only scalewith the number of resolved phases. In recent simulations(Boivin et al., 2000), 885,000 particles have been simulatedbut even with this number of particles, the particle loadingis still dilute and is miniscule compared to the number ofparticles present in a small cyclone separator at reasonableparticle loadings (ca � 109 particles). For higher particleloadings the Eulerian approach is computationally morepractical.

Hsieh (1988) used the Lagrangian approach to predictclassification of limestone in a 75 mm hydrocyclone. Hsiehused the mean velocity field from his 2D axi-symmetricCFD calculation of the water phase and did not incorporateany effects due to the turbulence. The Tromp (or cycloneefficiency) curve was constructed from repeated simulationsfor different particle sizes. Hsieh’s simulations predicted thed50 of the 75 mm cyclone for limestone very well, but theshort circuiting of large particles was not predicted andthe cut above the d50 was much sharper than was measured.Hsieh (1988) attributed this shortcoming to the 2D axi-sym-metric grid which used what was effectively a ring inlet andHsieh suggested that the effects of turbulence and particle/particle interactions should be incorporated in the simula-tion. Devulapalli (1994,1996,1997) extended Hsieh’sLagrangian approach, and incorporated the effects of tur-bulence by using a Lagrangian cloud model developed byBaxter and Smith (1993), where the mean position of acloud of particles is calculated and where the cloud wasreleased from the cyclone inlet. The spread of the cloudabout the mean position is calculated by a probability dis-tribution function (defined as a function of the turbulence)as the cloud moves through the domain. Devulapalli (1997)used the Lagrangian cloud approach to predict Trompcurves for a variety of 250 mm hydrocyclones (using severaldifferent vortex finders and spigots) passing limestone andcopper ore. Again the d50 was predicted well, but againthe cut above the d50 was sharper than was observed fromreal experiments using a cyclone of the same geometry.Devullapalli suggested that a full 3D grid of the cyclone,

with better turbulence models and other particle effects wereneeded to improve these predictions.

The Lagrangian approach has been used to model mul-tiphase flows using large eddy simulation for the continu-ous fluid phase (Boivin et al., 2000; Portela andOliemans, 2001; Millelli et al., 2001). These studies havefocused on simulating the damping of the turbulence bysmall dense suspended particles and have used two-waycoupling where the momentum exchange with the particlesby drag is incorporated as a source term in the fluidphase momentum equation. The studies by Portela andOliemans (2001) and Millelli et al. (2001) have also mod-eled the effect of the dispersed phases on the LES sub gridscale model.

The Eulerian approach has been used to simulate med-ium segregation in dense medium cyclones by Suasnabar(2000). Both the full Eulerian multiphase granular flowmodel (Ding and Gidaspow, 1990; Gidaspow et al., 1992)and the simplified Mixture model (Manninen et al., 1996)were used to model the medium and a single medium par-ticle size was used. The Eulerian Granular flow model andthe Mixture model predicted similar velocities and mediumconcentrations. The Eulerian granular flow model was feltto be more fundamentally correct but the Mixture modelwas computationally more economical.

The Mixture model has been the mainstay of the multi-phase cyclone CFD studies at the JKMRC. In our initialwork (Brennan, 2003) the medium in a dense mediumcyclone was modeled with a single particle size. It wasfound that medium segregation was over predicted com-pared to experimental measurements of the slurry densityin a 350 mm DSM cyclone by gamma ray tomography(Subramanian, 2000). This was attributed to the fact thatonly a single medium particle size was used. It was alsofound that the medium concentration was generally highestat the wall, whereas the measurements suggested that inapex the density reached a peak in a central zone betweenthe wall and the air core. In later work (Narasimha et al.,2006) the medium distribution was simulated and the Mix-ture model slip velocity was modified to incorporate a liftforce near the wall based on the Saffman (1965, 1968)(expression), and the mixture viscosity was modified touse the model of Ishii and Mishima (1984) which increasesthe mixture viscosity to infinity as the dispersed phasepacking density was approached. These modificationsimproved the predictions of the density profiles consider-ably, but the lift force coefficient needed to be increasedsubstantially over the value used by Saffman to achieve thisimprovement.

2.1. CFD approach

The flow in hydrocyclones is turbulent and is a multi-phase flow. In this work the turbulence has been resolvedusing large eddy simulation (LES) for the mixture andphase segregation has been modeled using the Mixturemodel (Manninen et al., 1996).


2.2. LES equations

The principle behind LES is that the larger turbulenteddies are resolved as part of a dynamic integration ofthe equations of motion but the eddies that are smallerthe grid are not resolved and the momentum transfer dueto these sub grid scale eddies is modeled. Thus LESresolves the momentum transfer by the largest turbulenteddies without any modeling. In many flows these largerturbulent scales carry out most of the turbulent mixing ofmomentum and mass. In principle this means that anLES should give more accurate mean flow predictions thana RANS based simulation without needing to resort to anyad-hoc tuning of the turbulence model since the problem ofmatching the predictions to experimental data becomes pri-marily a matter of refining the grid so that enough scalesare resolved and a simple sub grid scale model is adequate.The disadvantage of LES is that the technique requires afiner grid than is required for a RANS simulation andthe technique is intrinsically a time integration whereas atime averaged steady flow RANS solution can be obtainedfor a statistically steady turbulent flow. LES is of interest inmultiphase flows because the largest (and resolved) turbu-lent scales carry out most of the turbulent mixing of theparticles.

In an LES the equations of motion that are solved arethe unsteady filtered form, which for this problem are fora variable density slurry mixture:

oqotþ oðquiÞ

oxi¼ 0 ð1Þ

o

otðquiÞ þ

o

oxjðquiujÞ ¼ �

o

oxipi þ

o

oxjðsl;ij þ sd;ij � ss;ijÞ þ qgi

ð2Þ

The viscous stress tensor in Eq. (2) is

sl;ij ¼ 2lSij ð3Þ

LES requires a sub grid scale (SGS) model to calculate thetransfer of momentum by turbulent scales which are smal-ler than the grid and which appears in Eq. (2) as the subgrid scale stress tensor ss,ij. In this work the basic Smago-rinsky Lilly SGS model (Lilly) was used, where the SGSstresses are modelled using an SGS eddy viscosity ls:

ss;ij �1

3ss;kkdij ¼ �2lsSij ð4Þ

and the SGS eddy viscosity is calculated algebraically froma length scale Ls and the mean local strain rate:

ls ¼ qL2s jSj ð5Þ

Ls is normally equal to a third power of the finite volumesize at each grid point in regions of high turbulence butthe FluentTM implementation also makes Ls a function ofthe distance from the wall h in wall bounded regions:

Ls ¼ minðjh;CsV 1=3g Þ ð6Þ

Cs is the SGS calibration constant which in this work was0.1.

Millelli et al. (2001) has proposed a modification to thebasic Smagorinsky Lilly SGS model which corrects theSGS eddy viscosity in the presence of dispersed phases.This modification has been derived from dimensional con-siderations and is of the form:

ls ¼ qLsjSj þ qLsð1� acÞjuc � ukj ð7ÞThe multiphase correction given in Eq. (7) has been testedfor the multiphase flows reported here and in Hsieh’s(1988) experimental work but was found to very small.

2.3. Mixture model equations

The Mixture model (Manninen et al., 1996) is a simpli-fied Eulerian approach to modeling dispersed multiphaseflows, where the equations of motion are solved only forthe mixture. The mixture model assumes that the primaryphase is the continuous fluid phase and solves transportequations for the volume fractions of each distinct dis-persed phases present in the mixture:

o

otak þ

o

oxiðakuiÞ þ

o

oxiðakukm;iÞ ¼ 0

ukm;i ¼ uk;i � ui

ð8Þ

The Mixture model can simulate a slurry containing solidswith a distribution of size or density by solving a transportequation for the average size or average density over a sub-range within the distribution.

Phase segregation is accounted for in the phase trans-port equation (Eq. (8)) by the drift velocity of the phasek relative to the mixture ukm,I, which is calculated fromthe slip velocities, ukc,i which are the velocities of the dis-persed phases relative to the continuous fluid phase:

ukm;i ¼ ukc;i �Xn

l¼1

akqk

qulc;i

ukc ¼ uk � uc

ð9Þ

Phase segregation results in an additional source ofmomentum transfer in the mixture momentum equation(Eq. (2)), known as the drift tensor sd,ij, which can calcu-lated exactly as:

sd;ij ¼Xn

k¼1

akqkukm;iukm;j ð10Þ

The premise behind the Mixture model is that the slipvelocities can be modeled algebraically without the needto solve the dispersed phase momentum equations. Manni-nen et al. (1996) derived a model for ukc,i by subtracting themomentum equation for a dispersed phase k from themomentum equation for the mixture and substituting aconstitutive drag law for the interphase momentum trans-fer. The combined expression was then simplified byassuming that the dispersed phase k accelerates to its localterminal slip velocity very rapidly when compared to the


time scale of the flow. This simplification was called a localequilibrium assumption. Manninen et al.’s (1996) generalexpression for the slip velocity ukc,i is:

ak3

4Cdk

qc

dkjukcjukc;i ¼ akðqk � qÞ gi �

o

otui � uj

o

oxjui

� �

� ako

oxjðsl;ij þ sd;ij þ ss;ijÞ

þ o

oxj� ðakskT ;ijÞ þMki ð11Þ

Eq. (11) includes forces due the various stress tensors in themixture and dispersed phase momentum equations. Theimplementation of the Mixture model used here assumesthat the forces due to these stress terms are negligible intheir effect on the slip velocity when compared to thehydrodynamic and gravity forces:

3

4Cdk

qc

dkjukcjukc;i ¼ ðqk � qÞ gi �

o

otui � uj

o

oxjui

� �ð12Þ

Eqs. (11) and (12) can be derived using Manninen et al.’s(1996) formalism but using the filtered LES equations forthe mixture. In this case the mixture velocities that appearon the RHS are the instantaneous mixture velocities andwill include the resolved turbulent fluctuations. Thus theMixture model when used in conjunction with an LESshould resolve the turbulent mixing of the dispersed phasesby the resolved scales.

2.4. Granular enhancement to the mixture model

The Mixture model used in this work is an enhancedform of the Manninen et al. (1996) model which incorpo-rates the granular viscosity, the granular pressure and thedrag law formulations from the kinetic theory of granularflows. These granular terms require the granular tempera-ture Hk which in the Mixture model is calculated algebra-ically from Eq. (13)

ð�pkI þ skÞ : ruk � cHk þ /kl ¼ 0 ð13Þ

Eq. (12) is derived by simplifying the transport equation forHk as used in the full Eulerian granular model by assumingthat the convection and accumulation and diffusion termsare negligible compared to production, dissipation andinterphase transfer.

In the Mixture model, the effective molecular viscosityof the mixture is treated as the weighted sum of the phaseviscosities (including the continuous phase viscosity):

l ¼Xn

k¼1

aklk ð14Þ

With granular options enabled, each dispersed phase vis-cosity is treated as the sum of a collisional viscosity anda kinetic viscosity (Gidaspow et al., 1992):

lk ¼ lk;col þ lk;kin

lk;col ¼4

5akqkdkg0;kkð1þ ekkÞ

Hk

p

� �1=2

lk;kin ¼10qkdk

ffiffiffiffiffiffiffiffiffiHkpp

96akð1þ ekkÞg0;kk1þ 4

5g0;kkakð1þ ekkÞ

� �2

ð15Þ

The phase granular viscosity (Eq. (15)) is a function of thegranular temperature Hk, which is a function of shear viaEq. (13) and thus is a non-newtonian model.

g0,kk is the radial distribution function and is a correc-tion factor that modifies the probability of collisionsbetween grains when the solid granular phase becomesdense and is modeled by Eq. (16):

g0;kk¼ 1� 1� ac

1� ac;min

� �1=3" #�1

þ 1

2dk

XNk

k¼1

ak

dkð16Þ

g0,kk appears in the denominator for the expression for thekinetic viscosity lk,kin in Eq. (15) and has the main functionof numerically limiting the total dispersed phase concentra-tion to less than the packing limit. This is effected by mak-ing lk,kin (and hence the total mixture viscosity) approachinfinity as the dispersed phase packing limit (=1 � ac,min)is approached. This feeds back into the slip calculationand reduces the slip velocities of all phases to zero as thepacking limit is approached, which eliminates phasesegregation.

The pressure of the mixture is assumed to be the sum ofthe continuous phase pressure and the granular pressureassociated with each phase (Syamlal et al., 1993):

P s;t ¼XNk

k¼1

pk

pk ¼ akqkHk þ 2qkð1þ ekkÞa2kg0;kkHk

ð17Þ

The granular enhancements to the mixture model also per-mit a number of different drag laws and in this work, theGidaspow et al. (1992) drag law has been used, with thislaw the slip velocity calculation is:

3

4Cdka

�2:65c

qc

dkjukcjukc;i

¼ ðqk � qÞ gi �o

otui � uj

o

oxjui

� �ac P 0:8

1501� ac

ac

lc

d2k

þ 1:75qc

dkjukcj

!ukc;i

¼ ðqk � qÞ gi �o

otui � uj

o

oxjui

� �ac < 0:8

Cd ¼24

Rekð1þ 0:15ðacRekÞ0:687Þ

Rek ¼qcdkjukcj

lc

ð18Þ


2.5. Grid, boundary conditions and problem set up

The simulations were conducted on 3D body fitted gridsof Hsieh’s (1988) 75 mm geometry, which were generated inGambitTM were unsteady LES simulations. The approachused was to generate an initial steady water/air two phasecase on a coarse grid of 56,000 nodes using our existingmethodology (Brennan, 2006). A steady multiphase casestudy using the Mixture model was then generated usingthe coarse grid water/two phase case as an initial conditionand then a steady multiphase case using a finer grid of450,000 nodes was generated by adapting the coarse gridmultiphase case using the Fluent adaptation algorithm.

Axial Velo

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-0.04 -0.03 -0.02 -0.01Radial p

Vel

oci

ty -

m.s

-1

Fig. 1. Time averaged axial velocities from water/air LES simulations on coarLDA measurements on the same cyclone geometry, at 60 mm below the top o

Tangential Ve

0

1

2

3

4

5

6

7

-0.04 -0.03 -0.02 -0.01

Radial po

Vel

oci

ty -

m.s

-1

Fig. 2. Time averaged tangential velocities from water/air LES simulations on c1 LDA measurements on the same cyclone geometry, at 60 mm below the top

Steady in this context meant that the overall material bal-ance of each dispersed phase had reached a time averagesteady state during the LES simulation over approximately0.6 s of simulation time which is roughly the residence timeof the cyclone.

A fine grid water/air simulation was also generated byadapting the coarse grid water/air simulation.

The Fluent grid adaptation algorithm divides everyfinite volume element selected for adaptation in a 3D gridin eight volume elements. Grid adaptation is a numericallyeconomical method of grid refinement because the adapta-tion algorithm uses a converged case study using a coarsegrid and avoids the process of remeshing the geometry

city - 60 mm

0 0.01 0.02 0.03 0.04osition - m

Hsieh 0-180

Coarse

Fine

se and fine grids at a feed rate of 1.116 kg s�1, with Hsieh’s (1988) series 1f the cyclone body.

locity - 60 mm

0 0.01 0.02 0.03 0.04

sition - m

Hsieh 0-180

Fine

Coarse

oarse and fine grids at a feed rate of 1.116 kg s�1, with Hsieh’s (1988) seriesof the cyclone body.


and generating a converged case from scratch. In this workthe entire grid was adapted.

All simulations were run using Fluent 6.2.16TM withlarge eddy simulation with a fixed time step of1.0 · 10�4 s. Presto discretization was used for pressureand quick discretization was used for the phase transportequations. Both bounded central differencing and the 3rdorder muscle discretization schemes were used on themomentum equation. Simple was used for the pressurevelocity coupling. A velocity inlet boundary conditionwas used for the feed where the inlet turbulence was gener-ated using the Spectral Synthesiser option. The overflowand underflow used pressure outlet boundary conditionswith an air back-flow volume fraction of 1. The feed airvolume fraction was set to zero.

The Mixture model was run using eight phase transportequations in the multiphase simulations, where the primaryphase was water and the dispersed phases were air and sixlimestone phases of density 2700 kg m�3 with sizes 7 lm,9 lm, 13 lm, 17 lm, 25 lm and 35 lm. The volume frac-tion of each limestone size in the feed boundary conditionwas set so that the cumulative limestone size distributionand the total limestone volume fraction matched thoseused in the feed in Hsieh’s experimental runs. The mixtureslip velocity was disabled for the air phase which meant ineffect that the air phase was being solved using the VOFmodel with quick discretization.

The coarse grid simulations were run using the Windowsversion of Fluent on a desktop PC using a P4 dual coreprocessor, whilst the fine grid simulations were run onSGI Altix servers using the parallel solver with eight pro-cessors. These fine grid simulations took approximately167 h per second of simulation time.

Fig. 3. Contours of the ratio of SGS turbulent kinetic energy to theresolved turbulent kinetic energy for the coarse grid.

Fig. 4. Contours of the ratio of SGS turbulent kinetic energy to theresolved turbulent kinetic energy for the fine grid.

3. Results

3.1. Two phase water/air results

Case studies using the coarse and fine grids and withonly two phases (air and water), were run with a water feedrate of 1.116 kg s�1. This is a feed rate equal to Hsieh’s(1988) series 1 single phase experimental results where thewater velocities were measured using laser doppleranemometry.

Fig. 1 shows a comparison between the time averagedtangential velocities as predicted by the LES from thesetwo phase studies and Hsieh’s series 1 measurements at60 mm below the top of the cyclone in the 0–180� plane,whilst Fig. 2 shows a comparison between the axial veloc-ities at the same location. The results from both grids com-pare well from a radial position of 0.01 m outwards, butthe fine grid gives better agreement with Hsieh’s databetween 0.005 m (approximately the air/water free surfacecore position) and 0.01 m. The third order muscle discreti-zation scheme give better predictions of the velocities thandoes bounded central differencing on the coarse grid, but

on the finer grids both discretization schemes give similarvelocity predictions.

One test of an LES is to estimate the ratio of the turbu-lent energy carried by the sub grid scales to the turbulentkinetic energy of the resolved scales. This ratio can be esti-mated by the formula in Eq. (15), where the u0i are the timeaveraged values of the resolved velocity fluctuations:

ksgs

kres

¼ðlsgs=LsqCsÞ2

12ðu01 þ u02 þ u03Þ

2ð19Þ

Fig. 3 shows contours of the ratio ksgs/kres for the coarsegrid, whilst Fig. 4 shows contours for ksgs/kres for the fine

SGS k to Resolved k - 60 mm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04Radial position - m

Fine

Coarse

Fig. 5. Ratio of SGS turbulent kinetic energy to the resolved turbulent kinetic energy for fine and coarse grids at 60 mm below the top of the cyclone.

35 μm Limestone flow - LES

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

0.011

0.012

9.9 10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

Time - s

kg/s

Overflow

Underflow

Fig. 6. Instantaneous mass flow rate of 35 lm limestone size out overflow and underflow for coarse grid multiphase LES. Feed mass flow rate of 35 lmlimestone = 6.84 · 10�3 kg s�1. 3rd Order muscle on momentum equation.


grid and Fig. 5 shows a plot of ksgs/kres for both grids at the60 mm position. ksgs/kres for the coarse grid is �0.3 which istoo high which indicates that the coarse grid is too coarsefor an LES. ksgs/kres for the fine grid is more satisfactoryat �0.05, although further refinement may be desirable toreduce the ratio to below 0.01, however this would resultin a grid of 3 · 106 nodes if the entire grid was adapted.

It seems reasonable to propose that turbulent momen-tum transfer in hydrocyclones in the region between theair water free surface and the peak in tangential velocityis being carried by turbulent scales which have a structurewhich approaches the two component limit (where velocity

fluctuations normal to the free surface are damped com-pared to velocity fluctuations in other directions) as the freesurface is approached. To properly resolve these highlyanisotropic eddies the LES grid needs to be fine in thisregion at least in the radial direction, so further refinementin this region alone may be satisfactory.

3.2. Multiphase results

The multiphase simulations were run with a feed waterflow rate of 1.117 kg s�1 and a total feed limestone flowrate of 0.0574 kg s�1, (4.88% limestone by weight) which


matches Hsieh’s (1988) series VII data. Coarse grid simula-tions were run with bounded central differencing and 3rdorder muscle on the momentum whilst fine grid simulationswere run with bounded central differencing. The 3rd ordermuscle scheme was found to give more accurate predictionsof limestone classification behavior on the coarse grid butthere seemed to be only small differences between the twodiscretization schemes on the fine grid and only the resultsfor bounded central differencing on the fine grid arereported here.

Water fl

0.00

0.20

0.40

0.60

0.80

1.00

1.20

Tim

kg/s

9.9 10 10.1 10.2 10.3 1

Fig. 7. Instantaneous mass flow rate of water out overflow and underflow for3rd Order muscle on momentum equation.

Cyclone

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15

Particle

Fra

ctio

n t

o u

nd

er f

low

Fig. 8. Predicted cyclone efficiency curve. Coarse grid multiphase LES. Ferate = 0.0574 kg s�1. BCD – Bounded central differencing on momentum. 3rdunderflow flow rate. OF – Fraction to underflow by difference from overflow

Fig. 6 shows a plot over 1 s starting at 9.9 s of the instan-taneous overflow and underflow 35 lm limestone flowrates, whilst Fig. 7 shows the instantaneous water flowrates over the same period for the coarse grid simulations.These plots show a number of interesting features of thesimulation. Firstly the simulation is predicting a significantclassification of the 35 lm size to the underflow (The pre-dicted tromp or cyclone efficiency curves are shown onFig. 8 and are discussed further below). It is evident thatthe LES is predicting a relatively steady water flow rate

ow - LES

e - s

Overflow

Underflow

0.4 10.5 10.6 10.7 10.8 10.9

coarse grid multiphase LES. Feed mass flow rate of water = 1.117 kg s�1.

Efficiency

20 25 30 35 40

size - μm

Hsieh Series VII

Coarse OF BCD

Coarse UF BCD

Coarse OF 3rd O

Coarse UF 3rd O

ed mass flow rate of water = 1.117 kg s�1. Feed Limestone mass flowO – 3rd Order muscle on momentum. UF – Fraction to underflow fromflow rate.


but there is a fairly large fluctuation in the instantaneousflow rates of the 35 lm limestone.

Figs. 8 and 9 shows the predicted cyclone efficiencycurve, where the fraction of each limestone size reportingto underflow as a function of particle size is plotted.Fig. 8 shows curves from the coarse grid with bounded cen-tral differencing and 3rd order muscle on momentum.Fig. 9 shows the efficiency curve for the fine (adapted grid)with bounded central differencing on momentum. The frac-tion of each size to underflow was calculated by taking thetime average of the instantaneous flow rates out of thecyclone over a 1 s time interval. The open points were cal-culated from the underflow flow rates and the closed pointswere calculated by difference from the overflow flow rates.

The fraction of each size to underflow as calculateddirectly from the underflow rate and as calculated by differ-ence from the overflow rate are similar which indicates thatthe simulations are at steady flow and the overall materialbalance is being closed. The cyclone efficiency curves on allsimulations are being predicted with quite good accuracy,except that the simulations using bounded central differ-encing on the momentum equation predict more short cir-cuiting of the 25 and 35 lm size fractions to the overflow.The finer grid seems to improve the predictions at the smal-ler size range, but has a small effect on the short circuiting,however short circuiting has been reduced by using the 3rdorder muscle scheme on the momentum equation on thecoarse grid.

Fig. 10 shows the contours the instantaneous volumefraction for each size fraction of limestone inside thecyclone in elevation for the fine grid simulation withbounded central differencing on momentum. Note thatthe contours on these graphs are plotted on a logarithmicscale with an upper bound of 0.1 on volume fraction.The central region on each plot, which is bright red, is

Cyclone

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15

Particle

Fra

ctio

n t

o u

nd

erfl

ow

Fig. 9. Predicted cyclone efficiency curve. Fine grid multiphase LES. Feerate = 0.0574 kg s�1. BCD – Bounded central differencing on momentum. 3rdunderflow flow rate. OF – Fraction to underflow by difference from overflow

the air core. These plots show three distinct characteristicsin the distribution of limestone inside the cyclone as theparticle size increases; (i) The concentration increases atthe wall in the bottom of the apex as the particle sizeincreases. (ii) There is a sharp drop in the concentrationand a region of very low concentration forms around theair core and this region become larger as the particle sizeincreases. (iii) A region of low concentration forms aroundthe vortex finder and this region of low concentrationbecomes larger as the particle size increases.

Fig. 10 shows that there is distinct short circuiting of thelarger sizes to the overflow, but contrary to accepted wis-dom, the short circuiting is not due the larger particlesmigrating down the outer wall of the vortex finder. Ratherfor the particle sizes ranges being simulated, the larger par-ticles simply have not had sufficient time to classify in theupper vortex finder region before they encounter the shortcircuiting flow. There are also fairly distinct local varia-tions in the concentration of the larger particle sizes whichis indicative that turbulent mixing of the particles is beingresolved as part of the simulation and this must have aneffect on the short circuiting flow as well.

Fig. 11 shows a comparison between the coarse gridwith bounded central differencing on momentum, thecoarse grid with 3rd order muscle on momentum and thefine grid with bounded central differencing on momentum.It is evident that the higher order discretization reducesshort circuiting. The fine grid appears to reduce short cir-cuiting but this is not reflected in an improvement in theefficiency. This seems to be because the smaller width ofthe short circuiting flow that occurs with the finer grid isoffset by a larger concentration of limestone.

It is difficult to identify the cause of the discrepancybetween modeled and measured efficiency at 35 lm, how-ever the difference may be due to differences between the

Efficiency

20 25 30 35 40

size - μm

Hsieh Series VII

Fine OF BCD

Fine UF BCD

d mass flow rate of water = 1.117 kg s�1. Feed Limestone mass flowO – 3rd Order muscle on momentum. UF – Fraction to underflow fromflow rate.

Fig. 10. Contours of instantaneous limestone (plus air) volume fraction inside cyclone in xz plane. The central region in red is the air core. Fine gridsimulation, large eddy simulation with bounded central differencing on momentum. Note that contours are plotted on a log scale. (For interpretation ofthe references in colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. Instantaneous distribution of 35 lm limestone volume fraction inside cyclone in xz plane. BCD – Bounded central differencing, 3O – 3rd Ordermuscle.


modeled geometry and the real geometry or possiblybecause the simulations modeled only six size ranges inthe feed size distribution. Alternatively there may havesome measurement error.

4. Conclusions

CFD simulations of the Hsieh cyclone (1988) using largeeddy simulation and the mixture model with the granularoptions in FluentTM have been used to predict classificationand the distribution of limestone inside the cyclone. The

cyclone efficiency curve is predicted with good accuracyeven on coarse grids, but refining the grid and using ahigher order discretization scheme improves the predic-tions by reducing the simulated short circuiting of the lar-ger sized limestone fractions.

The simulated distribution of the limestone particlessuggests that short circuiting of the 35 lm occurs not bymigration down the outer wall of the vortex finder butrather because this size has not had sufficient time to clas-sify before some portion is entrained in the up flow throughthe vortex finder.


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