Brennan 2007

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Multiphase modelling of hydrocyclones prediction of cut-sizeM.S. Brennan a,*, M. Narasimha b, P.N. Holtham aaJulius Kruttschnitt Mineral Research Centre, The University of Queensland, Isles Road, Indooroopilly 4068, Queensland, AustraliabR&D Division, Tata Steel, Jamshedpur, Jharkhand 831 007, IndiaReceived 1 August 2006; accepted 6 October 2006Available online 28 November 2006AbstractA comprehensive multiphase model of cyclone separators using Computational Fluid Dynamics is under development. The model iscapable of predicting velocity proles, ow splits, air core position and eciency 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 Hsiehs [Hsieh, K.T., 1988. A phenomenological model of the hydrocyclone, Ph.D. thesis, University ofUtah] data show a very good prediction of the cyclone eciency curve. Whilst further model development is needed, the approach isshowing promise as a cyclone design tool. 2006 Elsevier Ltd. All rights reserved.Keywords: Hydrocyclones; Classication; Computational uid dynamics; Large eddy simulation; Mixture model1. IntroductionHydrocyclones rely on the centrifugal forces thatdevelop under the swirling ow to eect separation of par-ticles in slurries and particle laden ows and can classify ondensity or particle size. Modelling the ow by computa-tional uid dynamics (CFD) is a key to understandinghow cyclone separators behave. However the ow is multi-phase, usually turbulent and an air core usually forms if thecyclone outlets are exposed to the atmosphere. These fac-tors inuence the separation behavior and must beaccounted for even though they complicate the CFDmodeling.Turbulent ows 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-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 ows in cyclone sepa-rators with reasonable accuracy for single phase gas ows(Slack et al., 2000; Shalaby et al., 2005) and the basictwo phase ow (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 eect of particle loading onthe ow 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 eciency) curve has been con-structed from the time averaged ow rates of each sizerange out the underow and overow boundaries. The0892-6875/$ - 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: m.brennan@uq.edu.au (M.S. Brennan).This article is also available online at:www.elsevier.com/locate/minengMinerals Engineering 20 (2007) 395406predicted distribution of each size in the distribution insidethe cyclone body are shown.2. Previous workCFD of hydrocyclones has primarily focused on model-ing the basic ow 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 (orDierential 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 classication and if this goal is to beachieved then the interactions between the particulatephases and the uid must be modeled. In CFD there aretwo approaches to simulating particulate phases in uid 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 uid 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 uid 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 ow 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 owapproach 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 eective viscous stresses in the dis-persed phase momentum equation.A simplied Eulerian approach is the Mixture model(Manninen et al., 1996) where the momentum and continu-NomenclatureTransport 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 uid phase)Ak particle cross-sectional area of phase kCd drag coecientCs Smagorinsky Lilly constantekk coecient of restitution of phase kdk diameter of phase kgi i component of gravityh distance from wallk turbulent kinetic energyLs length scale of the sub grid scale stressesMk,i i component of interphase momentum transferto phase kpk granular pressure of phase kSij mean strain ratet timeui i component of mixture velocityukc,i i component of velocity of phase k relative tomixture (drift velocity)ukm,i i component of velocity of phase k relative tocontinuous phase (slip velocity)Vg volume of grid nite volumeVk particle volume of phase kxi i co-ordinatey+dimensionless distance from wallak volume fraction of phase kdij kronecker deltaj Von Karmanns 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 kl eective molecular viscosity of the mixturels sub grid scale eddy viscosity of the mixtureSubscriptsc continuous phase ci, j components in i and j directionsk phase km mixture396 M.S. Brennan et al. / Minerals Engineering 20 (2007) 395406ity 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 modied to incorporate someof the features of granular ow theory and these modica-tions are discussed in the next section.The advantage of the Lagrangian multiphase approachis that particleparticle and particleuid 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 nite volume in the CFD grid. Thusthe Eulerian approach innately involves averaging of somesort, which implies more modelling. However the computa-tional requirements of the