Chapter 23. Modeling Multiphase FlowsThis chapter discusses the general multiphase models that are available in FLUENT.Section 23.1: Introduction provides a brief introduction to multiphase modeling, Chap-ter 22: Modeling Discrete Phase discusses the Lagrangian dispersed phase model, andChapter 24: Modeling Solidication and Melting describes FLUENTs model for solidi-cation and melting. Section 23.1: Introduction Section 23.2: Choosing a General Multiphase Model Section 23.3: Volume of Fluid (VOF) Model Theory Section 23.4: Mixture Model Theory Section 23.5: Eulerian Model Theory Section 23.6: Wet Steam Model Theory Section 23.7: Modeling Mass Transfer in Multiphase Flows Section 23.8: Modeling Species Transport in Multiphase Flows Section 23.9: Steps for Using a Multiphase Model Section 23.10: Setting Up the VOF Model Section 23.11: Setting Up the Mixture Model Section 23.12: Setting Up the Eulerian Model Section 23.13: Setting Up the Wet Steam Model Section 23.14: Solution Strategies for Multiphase Modeling Section 23.15: Postprocessing for Multiphase Modelingc Fluent Inc. September 29, 2006 23-1Modeling Multiphase Flows23.1 IntroductionA large number of ows encountered in nature and technology are a mixture of phases.Physical phases of matter are gas, liquid, and solid, but the concept of phase in a mul-tiphase ow system is applied in a broader sense. In multiphase ow, a phase can bedened as an identiable class of material that has a particular inertial response to andinteraction with the ow and the potential eld in which it is immersed. For example,dierent-sized solid particles of the same material can be treated as dierent phases be-cause each collection of particles with the same size will have a similar dynamical responseto the ow eld.23.1.1 Multiphase Flow RegimesMultiphase ow regimes can be grouped into four categories: gas-liquid or liquid-liquidows; gas-solid ows; liquid-solid ows; and three-phase ows.Gas-Liquid or Liquid-Liquid FlowsThe following regimes are gas-liquid or liquid-liquid ows: Bubbly ow: This is the ow of discrete gaseous or uid bubbles in a continuousuid. Droplet ow: This is the ow of discrete uid droplets in a continuous gas. Slug ow: This is the ow of large bubbles in a continuous uid. Stratied/free-surface ow: This is the ow of immiscible uids separated by aclearly-dened interface.See Figure 23.1.1 for illustrations of these regimes.Gas-Solid FlowsThe following regimes are gas-solid ows: Particle-laden ow: This is ow of discrete particles in a continuous gas. Pneumatic transport: This is a ow pattern that depends on factors such as solidloading, Reynolds numbers, and particle properties. Typical patterns are duneow, slug ow, packed beds, and homogeneous ow. Fluidized bed: This consists of a vertical cylinder containing particles, into whicha gas is introduced through a distributor. The gas rising through the bed suspendsthe particles. Depending on the gas ow rate, bubbles appear and rise through thebed, intensifying the mixing within the bed.23-2 c Fluent Inc. September 29, 200623.1 IntroductionSee Figure 23.1.1 for illustrations of these regimes.Liquid-Solid FlowsThe following regimes are liquid-solid ows: Slurry ow: This ow is the transport of particles in liquids. The fundamentalbehavior of liquid-solid ows varies with the properties of the solid particles relativeto those of the liquid. In slurry ows, the Stokes number (see Equation 23.2-4) isnormally less than 1. When the Stokes number is larger than 1, the characteristicof the ow is liquid-solid uidization. Hydrotransport: This describes densely-distributed solid particles in a continuousliquid Sedimentation: This describes a tall column initially containing a uniform dispersedmixture of particles. At the bottom, the particles will slow down and form a sludgelayer. At the top, a clear interface will appear, and in the middle a constant settlingzone will exist.See Figure 23.1.1 for illustrations of these regimes.Three-Phase FlowsThree-phase ows are combinations of the other ow regimes listed in the previous sec-tions.c Fluent Inc. September 29, 2006 23-3Modeling Multiphase Flowsslug ow bubbly, droplet, orparticle-laden owstratied/free-surface ow pneumatic transport,hydrotransport, or slurry owsedimentation uidized bedFigure 23.1.1: Multiphase Flow Regimes23-4 c Fluent Inc. September 29, 200623.2 Choosing a General Multiphase Model23.1.2 Examples of Multiphase SystemsSpecic examples of each regime described in Section 23.1.1: Multiphase Flow Regimesare listed below: Bubbly ow examples include absorbers, aeration, air lift pumps, cavitation, evap-orators, otation, and scrubbers. Droplet ow examples include absorbers, atomizers, combustors, cryogenic pump-ing, dryers, evaporation, gas cooling, and scrubbers. Slug ow examples include large bubble motion in pipes or tanks. Stratied/free-surface ow examples include sloshing in oshore separator devicesand boiling and condensation in nuclear reactors. Particle-laden ow examples include cyclone separators, air classiers, dust collec-tors, and dust-laden environmental ows. Pneumatic transport examples include transport of cement, grains, and metal pow-ders. Fluidized bed examples include uidized bed reactors and circulating uidized beds. Slurry ow examples include slurry transport and mineral processing Hydrotransport examples include mineral processing and biomedical and physio-chemical uid systems Sedimentation examples include mineral processing.23.2 Choosing a General Multiphase ModelThe rst step in solving any multiphase problem is to determine which of the regimesprovides some broad guidelines for determining appropriate models for each regime, andhow to determine the degree of interphase coupling for ows involving bubbles, droplets,or particles, and the appropriate model for dierent amounts of coupling.c Fluent Inc. September 29, 2006 23-5Modeling Multiphase Flows23.2.1 Approaches to Multiphase ModelingAdvances in computational uid mechanics have provided the basis for further insightinto the dynamics of multiphase ows. Currently there are two approaches for the nu-merical calculation of multiphase ows: the Euler-Lagrange approach (discussed in Sec-tion 22.1.1: Overview) and the Euler-Euler approach (discussed in the following section).The Euler-Euler ApproachIn the Euler-Euler approach, the dierent phases are treated mathematically as inter-penetrating continua. Since the volume of a phase cannot be occupied by the otherphases, the concept of phasic volume fraction is introduced. These volume fractions areassumed to be continuous functions of space and time and their sum is equal to one.Conservation equations for each phase are derived to obtain a set of equations, whichhave similar structure for all phases. These equations are closed by providing constitutiverelations that are obtained from empirical information, or, in the case of granular ows,by application of kinetic theory.In FLUENT, three dierent Euler-Euler multiphase models are available: the volume ofuid (VOF) model, the mixture model, and the Eulerian model.The VOF ModelThe VOF model (described in Section 23.3: Volume of Fluid (VOF) Model Theory) isa surface-tracking technique applied to a xed Eulerian mesh. It is designed for two ormore immiscible uids where the position of the interface between the uids is of interest.In the VOF model, a single set of momentum equations is shared by the uids, and thevolume fraction of each of the uids in each computational cell is tracked throughout thedomain. Applications of the VOF model include stratied ows, free-surface ows, lling,sloshing, the motion of large bubbles in a liquid, the motion of liquid after a dam break,the prediction of jet breakup (surface tension), and the steady or transient tracking ofany liquid-gas interface.The Mixture ModelThe mixture model (described in Section 23.4: Mixture Model Theory) is designed for twoor more phases (uid or particulate). As in the Eulerian model, the phases are treated asinterpenetrating continua. The mixture model solves for the mixture momentum equationand prescribes relative velocities to describe the dispersed phases. Applications of themixture model include particle-laden ows with low loading, bubbly ows, sedimentation,and cyclone separators. The mixture model can also be used without relative velocitiesfor the dispersed phases to model homogeneous multiphase ow.23-6 c Fluent Inc. September 29, 200623.2 Choosing a General Multiphase ModelThe Eulerian ModelThe Eulerian model (described in Section 23.5: Eulerian Model Theory) is the most com-plex of the multiphase models in FLUENT. It solves a set of n momentum and continuityequations for each phase. Coupling is achieved through the pressure and interphase ex-change coecients. The manner in which this coupling is handled depends upon the typeof phases involved; granular (uid-solid) ows are handled dierently than nongranular(uid-uid) ows. For granular ows, the properties are obtained from application of ki-netic theory. Momentum exchange between the phases is also dependent upon the typeof mixture being modeled. FLUENTs user-dened functions allow you to customize thecalculation of the momentum exchange. Applications of the Eulerian multiphase modelinclude bubble columns, risers, particle suspension, and uidized beds.23.2.2 Model ComparisonsIn general, once you have determined the ow regime that best represents your multiphasesystem, you can select the appropriate model based on the following guidelines: For bubbly, droplet, and particle-laden ows in which the phases mix and/ordispersed-phase volume fractions exceed 10%, use either the mixture model (de-scribed in Section 23.4: Mixture Model Theory) or the Eulerian model (describedin Section 23.5: Eulerian Model Theory). For slug ows, use the VOF model. See Section 23.3: Volume of Fluid (VOF) ModelTheory for more information about the VOF model. For stratied/free-surface ows, use the VOF model. See Section 23.3: Volume ofFluid (VOF) Model Theory for more information about the VOF model. For pneumatic transport, use the mixture model for homogeneous ow (describedin Section 23.4: Mixture Model Theory) or the Eulerian model for granular ow(described in Section 23.5: Eulerian Model Theory). For uidized beds, use the Eulerian model for granular ow. See Section 23.5: Eu-lerian Model Theory for more information about the Eulerian model. For slurry ows and hydrotransport, use the mixture or Eulerian model (described,respectively, in Sections 23.4 and 23.5). For sedimentation, use the Eulerian model. See Section 23.5: Eulerian ModelTheory for more information about the Eulerian model. For general, complex multiphase ows that involve multiple ow regimes, selectthe aspect of the ow that is of most interest, and choose the model that is mostappropriate for that aspect of the ow. Note that the accuracy of results will notbe as good as for ows that involve just one ow regime, since the model you usewill be valid for only part of the ow you are modeling.c Fluent Inc. September 29, 2006 23-7Modeling Multiphase FlowsAs discussed in this section, the VOF model is appropriate for stratied or free-surfaceows, and the mixture and Eulerian models are appropriate for ows in which the phasesmix or separate and/or dispersed-phase volume fractions exceed 10%. (Flows in whichthe dispersed-phase volume fractions are less than or equal to 10% can be modeled usingthe discrete phase model described in Chapter 22: Modeling Discrete Phase.)To choose between the mixture model and the Eulerian model, you should consider thefollowing guidelines: If there is a wide distribution of the dispersed phases (i.e., if the particles varyin size and the largest particles do not separate from the primary ow eld), themixture model may be preferable (i.e., less computationally expensive). If thedispersed phases are concentrated just in portions of the domain, you should usethe Eulerian model instead. If interphase drag laws that are applicable to your system are available (eitherwithin FLUENT or through a user-dened function), the Eulerian model can usuallyprovide more accurate results than the mixture model. Even though you can applythe same drag laws to the mixture model, as you can for a nongranular Euleriansimulation, if the interphase drag laws are unknown or their applicability to yoursystem is questionable, the mixture model may be a better choice. For most caseswith spherical particles, then the Schiller-Naumann law is more than adequate. Forcases with nonspherical particles, then a user-dened function can be used. If you want to solve a simpler problem, which requires less computational eort, themixture model may be a better option, since it solves a smaller number of equationsthan the Eulerian model. If accuracy is more important than computational eort,the Eulerian model is a better choice. Keep in mind, however, that the complexityof the Eulerian model can make it less computationally stable than the mixturemodel.FLUENTs multiphase models are compatible with FLUENTs dynamic mesh modelingfeature. For more information on the dynamic mesh feature, see Section 11: ModelingFlows Using Sliding and Deforming Meshes. For more information about how other FLU-ENT models are compatible with FLUENTs multiphase models, see Appendix A: FLUENTModel Compatibility.Detailed GuidelinesFor stratied and slug ows, the choice of the VOF model, as indicated in Section 23.2.2: ModelComparisons, is straightforward. Choosing a model for the other types of ows is lessstraightforward. As a general guide, there are some parameters that help to identify theappropriate multiphase model for these other ows: the particulate loading, , and theStokes number, St. (Note that the word particle is used in this discussion to refer toa particle, droplet, or bubble.)23-8 c Fluent Inc. September 29, 200623.2 Choosing a General Multiphase ModelThe Effect of Particulate LoadingParticulate loading has a major impact on phase interactions. The particulate loading isdened as the mass density ratio of the dispersed phase (d) to that of the carrier phase(c): = ddcc(23.2-1)The material density ratio = dc(23.2-2)is greater than 1000 for gas-solid ows, about 1 for liquid-solid ows, and less than 0.001for gas-liquid ows.Using these parameters it is possible to estimate the average distance between the indi-vidual particles of the particulate phase. An estimate of this distance has been given byCrowe et al. [68]:Ldd=

61 +

1/3(23.2-3)where = . Information about these parameters is important for determining how thedispersed phase should be treated. For example, for a gas-particle ow with a particulateloading of 1, the interparticle space Lddis about 8; the particle can therefore be treatedas isolated (i.e., very low particulate loading).Depending on the particulate loading, the degree of interaction between the phases canbe divided into the following three categories: For very low loading, the coupling between the phases is one-way (i.e., the uidcarrier inuences the particles via drag and turbulence, but the particles have noinuence on the uid carrier). The discrete phase (Chapter 22: Modeling DiscretePhase), mixture, and Eulerian models can all handle this type of problem correctly.Since the Eulerian model is the most expensive, the discrete phase or mixture modelis recommended. For intermediate loading, the coupling is two-way (i.e., the uid carrier inuencesthe particulate phase via drag and turbulence, but the particles in turn inuencethe carrier uid via reduction in mean momentum and turbulence). The discretephase(Chapter 22: Modeling Discrete Phase) , mixture, and Eulerian models areall applicable in this case, but you need to take into account other factors in orderto decide which model is more appropriate. See below for information about usingthe Stokes number as a guide.c Fluent Inc. September 29, 2006 23-9Modeling Multiphase Flows For high loading, there is two-way coupling plus particle pressure and viscousstresses due to particles (four-way coupling). Only the Eulerian model will handlethis type of problem correctly.The Signicance of the Stokes NumberFor systems with intermediate particulate loading, estimating the value of the Stokesnumber can help you select the most appropriate model. The Stokes number can bedened as the relation between the particle response time and the system response time:St = dts(23.2-4)where d = dd2d18cand ts is based on the characteristic length (Ls) and the characteristicvelocity (Vs) of the system under investigation: ts = LsVs.For St 1.0, the particle will follow the ow closely and any of the three models (discretephase(Chapter 22: Modeling Discrete Phase) , mixture, or Eulerian) is applicable; youcan therefore choose the least expensive (the mixture model, in most cases), or the mostappropriate considering other factors. For St > 1.0, the particles will move independentlyof the ow and either the discrete phase model (Chapter 22: Modeling Discrete Phase)or the Eulerian model is applicable. For St 1.0, again any of the three models isapplicable; you can choose the least expensive or the most appropriate considering otherfactors.ExamplesFor a coal classier with a characteristic length of 1 m and a characteristic velocity of10 m/s, the Stokes number is 0.04 for particles with a diameter of 30 microns, but 4.0for particles with a diameter of 300 microns. Clearly the mixture model will not beapplicable to the latter case.For the case of mineral processing, in a system with a characteristic length of 0.2 m and acharacteristic velocity of 2 m/s, the Stokes number is 0.005 for particles with a diameterof 300 microns. In this case, you can choose between the mixture and Eulerian models.(The volume fractions are too high for the discrete phase model (Chapter 22: ModelingDiscrete Phase), as noted below.)Other ConsiderationsKeep in mind that the use of the discrete phase model (Chapter 22: Modeling DiscretePhase) is limited to low volume fractions. Also, the discrete phase model is the only mul-tiphase model that allows you to specify the particle distribution or include combustionmodeling in your simulation.23-10 c Fluent Inc. September 29, 200623.2 Choosing a General Multiphase Model23.2.3 Time Schemes in Multiphase FlowIn many multiphase applications, the process can vary spatially as well as temporally. Inorder to accurately model multiphase ow, both higher-order spatial and time discretiza-tion schemes are necessary. In addition to the rst-order time scheme in FLUENT, thesecond-order time scheme is available in the Mixture and Eulerian multiphase models,and with the VOF Implicit Scheme.i The second-order time scheme cannot be used with the VOF ExplicitSchemes.The second-order time scheme has been adapted to all the transport equations, includ-ing mixture phase momentum equations, energy equations, species transport equations,turbulence models, phase volume fraction equations, the pressure correction equation,and the granular ow model. In multiphase ow, a general transport equation (similarto that of Equation 25.3-15) may be written as()t + (

V ) = + S (23.2-5)Where is either a mixture (for the mixture model) or a phase variable, is the phasevolume fraction (unity for the mixture equation), is the mixture phase density, V isthe mixture or phase velocity (depending on the equations), is the diusion term, andS is the source term.As a fully implicit scheme, this second-order time-accurate scheme achieves its accuracyby using an Euler backward approximation in time (see Equation 25.3-17). The generaltransport equation, Equation 23.2-5 is discretized as3(pppV ol)n+14(pppV ol)n+ (ppp)n12t = (23.2-6)[Anb(nbp)]n+1+ SUn+1Spn+1pn+1Equation 23.2-6 can be written in simpler form:App =Anbnb + S (23.2-7)whereAp = Anbn+1+ Spn+1+ 1.5(ppV ol)n+1tS = SUn+1+ 2(pppV ol)n0.5(pppV ol)n1tc Fluent Inc. September 29, 2006 23-11Modeling Multiphase FlowsThis scheme is easily implemented based on FLUENTs existing rst-order Euler scheme.It is unconditionally stable, however, the negative coecient at the time level tn1, of thethree-time level method, may produce oscillatory solutions if the time steps are large.This problem can be eliminated if a bounded second-order scheme is introduced. How-ever, oscillating solutions are most likely seen in compressible liquid ows. Therefore,in this version of FLUENT, a bounded second-order time scheme has been implementedfor compressible liquid ows only. For single phase and multiphase compressible liquidows, the second-order time scheme is, by default, the bounded scheme.23.2.4 Stability and ConvergenceThe process of solving a multiphase system is inherently dicult, and you may encountersome stability or convergence problems. If a time-dependent problem is being solved, andpatched elds are used for the initial conditions, it is recommended that you perform afew iterations with a small time step, at least an order of magnitude smaller than thecharacteristic time of the ow. You can increase the size of the time step after performinga few time steps. For steady solutions it is recommended that you start with a smallunder-relaxation factor for the volume fraction, it is also recommended not to start witha patch of volume fraction equal to zero. Another option is to start with a mixturemultiphase calculation, and then switch to the Eulerian multiphase model.Stratied ows of immiscible uids should be solved with the VOF model (see Sec-tion 23.3: Volume of Fluid (VOF) Model Theory). Some problems involving small volumefractions can be solved more eciently with the Lagrangian discrete phase model (seeChapter 22: Modeling Discrete Phase).Many stability and convergence problems can be minimized if care is taken during thesetup and solution processes (see Section 23.14.4: Eulerian Model).23.3 Volume of Fluid (VOF) Model Theory23.3.1 Overview and Limitations of the VOF ModelOverviewThe VOF model can model two or more immiscible uids by solving a single set ofmomentum equations and tracking the volume fraction of each of the uids throughoutthe domain. Typical applications include the prediction of jet breakup, the motion oflarge bubbles in a liquid, the motion of liquid after a dam break, and the steady ortransient tracking of any liquid-gas interface.23-12 c Fluent Inc. September 29, 200623.3 Volume of Fluid (VOF) Model TheoryLimitationsThe following restrictions apply to the VOF model in FLUENT: You must use the pressure-based solver. The VOF model is not available witheither of the density-based solvers. All control volumes must be lled with either a single uid phase or a combinationof phases. The VOF model does not allow for void regions where no uid of anytype is present. Only one of the phases can be dened as a compressible ideal gas. There is nolimitation on using compressible liquids using user-dened functions. Streamwise periodic ow (either specied mass ow rate or specied pressure drop)cannot be modeled when the VOF model is used. The second-order implicit time-stepping formulation cannot be used with the VOFexplicit scheme. When tracking particles in parallel, the DPM model cannot be used with the VOFmodel if the shared memory option is enabled (Section 22.11.9: Parallel Processingfor the Discrete Phase Model). (Note that using the message passing option, whenrunning in parallel, enables the compatibility of all multiphase ow models withthe DPM model.)Steady-State and Transient VOF CalculationsThe VOF formulation in FLUENT is generally used to compute a time-dependent solution,but for problems in which you are concerned only with a steady-state solution, it ispossible to perform a steady-state calculation. A steady-state VOF calculation is sensibleonly when your solution is independent of the initial conditions and there are distinctinow boundaries for the individual phases. For example, since the shape of the freesurface inside a rotating cup depends on the initial level of the uid, such a problemmust be solved using the time-dependent formulation. On the other hand, the ow ofwater in a channel with a region of air on top and a separate air inlet can be solved withthe steady-state formulation.The VOF formulation relies on the fact that two or more uids (or phases) are notinterpenetrating. For each additional phase that you add to your model, a variable isintroduced: the volume fraction of the phase in the computational cell. In each controlvolume, the volume fractions of all phases sum to unity. The elds for all variables andproperties are shared by the phases and represent volume-averaged values, as long asthe volume fraction of each of the phases is known at each location. Thus the variablesand properties in any given cell are either purely representative of one of the phases, orc Fluent Inc. September 29, 2006 23-13Modeling Multiphase Flowsrepresentative of a mixture of the phases, depending upon the volume fraction values.In other words, if the qthuids volume fraction in the cell is denoted as q, then thefollowing three conditions are possible: q = 0: The cell is empty (of the qthuid). q = 1: The cell is full (of the qthuid). 0 < q < 1: The cell contains the interface between the qthuid and one or moreother uids.Based on the local value of q, the appropriate properties and variables will be assignedto each control volume within the domain.23.3.2 Volume Fraction EquationThe tracking of the interface(s) between the phases is accomplished by the solution of acontinuity equation for the volume fraction of one (or more) of the phases. For the qthphase, this equation has the following form:1q

t(qq) + (qqvq) = Sq +np=1( mpq mqp) (23.3-1)where mqp is the mass transfer from phase q to phase p and mpq is the mass transfer fromphase p to phase q. By default, the source term on the right-hand side of Equation 23.3-1,Sq, is zero, but you can specify a constant or user-dened mass source for each phase.See Section 23.7: Modeling Mass Transfer in Multiphase Flows for more information onthe modeling of mass transfer in FLUENTs general multiphase models.The volume fraction equation will not be solved for the primary phase; the primary-phasevolume fraction will be computed based on the following constraint:nq=1q = 1 (23.3-2)The volume fraction equation may be solved either through implicit or explicit timediscretization.23-14 c Fluent Inc. September 29, 200623.3 Volume of Fluid (VOF) Model TheoryThe Implicit SchemeWhen the implicit scheme is used for time discretization, FLUENTs standard nite-dierence interpolation schemes, QUICK, Second Order Upwind and First Order Upwind,and the Modied HRIC schemes, are used to obtain the face uxes for all cells, includingthose near the interface.n+1q n+1q nqnqt V +f(n+1q Un+1f n+1q,f ) =

Sq +np=1( mpq mqp)V (23.3-3)Since this equation requires the volume fraction values at the current time step (ratherthan at the previous step, as for the explicit scheme), a standard scalar transport equationis solved iteratively for each of the secondary-phase volume fractions at each time step.The implicit scheme can be used for both time-dependent and steady-state calculations.See Section 23.10.1: Choosing a VOF Formulation for details.The Explicit SchemeIn the explicit approach, FLUENTs standard nite-dierence interpolation schemes areapplied to the volume fraction values that were computed at the previous time step.n+1q n+1q nqnqt V +f(qUnf nq,f) =

np=1( mpq mqp) + SqV (23.3-4)where n + 1 = index for new (current) time stepn = index for previous time stepq,f = face value of the qthvolume fraction, computed from the rst-or second-order upwind, QUICK, modied HRIC, or CICSAM schemeV = volume of cellUf = volume ux through the face, based on normal velocityThis formulation does not require iterative solution of the transport equation during eachtime step, as is needed for the implicit scheme.i When the explicit scheme is used, a time-dependent solution must be com-puted.When the explicit scheme is used for time discretization, the face uxes can be interpo-lated either using interface reconstruction or using a nite volume discretization scheme(Section 23.3.2: Interpolation near the Interface). The reconstruction based schemesavailable in FLUENT are Geo-Reconstruct and Donor-Acceptor. The discretization schemesavailable with explicit scheme for VOF are First Order Upwind, Second Order Upwind,CICSAM, Modied HRIC, and QUICK.c Fluent Inc. September 29, 2006 23-15Modeling Multiphase FlowsInterpolation near the InterfaceFLUENTs control-volume formulation requires that convection and diusion uxes throughthe control volume faces be computed and balanced with source terms within the controlvolume itself.In the geometric reconstruction and donor-acceptor schemes, FLUENT applies a spe-cial interpolation treatment to the cells that lie near the interface between two phases.Figure 23.3.1 shows an actual interface shape along with the interfaces assumed duringcomputation by these two methods.actual interface shapeinterface shape represented bythe donor-acceptor schemeinterface shape represented bythe geometric reconstruction(piecewise-linear) schemeFigure 23.3.1: Interface Calculations23-16 c Fluent Inc. September 29, 200623.3 Volume of Fluid (VOF) Model TheoryThe explicit scheme and the implicit scheme treat these cells with the same interpo-lation as the cells that are completely lled with one phase or the other (i.e., usingthe standard upwind (Section 25.3.1: First-Order Upwind Scheme), second-order (Sec-tion 25.3.1: Second-Order Upwind Scheme), QUICK (Section 25.3.1: QUICK Scheme,modied HRIC (Section 25.3.1: Modied HRIC Scheme), or CICSAM scheme), ratherthan applying a special treatment.The Geometric Reconstruction SchemeIn the geometric reconstruction approach, the standard interpolation schemes that areused in FLUENT are used to obtain the face uxes whenever a cell is completely lledwith one phase or another. When the cell is near the interface between two phases, thegeometric reconstruction scheme is used.The geometric reconstruction scheme represents the interface between uids using apiecewise-linear approach. In FLUENT this scheme is the most accurate and is applicablefor general unstructured meshes. The geometric reconstruction scheme is generalizedfor unstructured meshes from the work of Youngs [411]. It assumes that the interfacebetween two uids has a linear slope within each cell, and uses this linear shape forcalculation of the advection of uid through the cell faces. (See Figure 23.3.1.)The rst step in this reconstruction scheme is calculating the position of the linear in-terface relative to the center of each partially-lled cell, based on information aboutthe volume fraction and its derivatives in the cell. The second step is calculating theadvecting amount of uid through each face using the computed linear interface repre-sentation and information about the normal and tangential velocity distribution on theface. The third step is calculating the volume fraction in each cell using the balance ofuxes calculated during the previous step.i When the geometric reconstruction scheme is used, a time-dependent solu-tion must be computed. Also, if you are using a conformal grid (i.e., if thegrid node locations are identical at the boundaries where two subdomainsmeet), you must ensure that there are no two-sided (zero-thickness) wallswithin the domain. If there are, you will need to slit them, as described inSection 6.8.6: Slitting Face Zones.c Fluent Inc. September 29, 2006 23-17Modeling Multiphase FlowsThe Donor-Acceptor SchemeIn the donor-acceptor approach, the standard interpolation schemes that are used inFLUENT are used to obtain the face uxes whenever a cell is completely lled withone phase or another. When the cell is near the interface between two phases, a donor-acceptor scheme is used to determine the amount of uid advected through the face [144].This scheme identies one cell as a donor of an amount of uid from one phase andanother (neighbor) cell as the acceptor of that same amount of uid, and is used toprevent numerical diusion at the interface. The amount of uid from one phase thatcan be convected across a cell boundary is limited by the minimum of two values: thelled volume in the donor cell or the free volume in the acceptor cell.The orientation of the interface is also used in determining the face uxes. The interfaceorientation is either horizontal or vertical, depending on the direction of the volumefraction gradient of the qthphase within the cell, and that of the neighbor cell that sharesthe face in question. Depending on the interfaces orientation as well as its motion, uxvalues are obtained by pure upwinding, pure downwinding, or some combination of thetwo.i When the donor-acceptor scheme is used, a time-dependent solution mustbe computed. Also, the donor-acceptor scheme can be used only withquadrilateral or hexahedral meshes. In addition, if you are using a con-formal grid (i.e., if the grid node locations are identical at the boundarieswhere two subdomains meet), you must ensure that there are no two-sided(zero-thickness) walls within the domain. If there are, you will need to slitthem, as described in Section 6.8.6: Slitting Face Zones.The Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM)The compressive interface capturing scheme for arbitrary meshes (CICSAM), based onthe Ubbinks work [376], is a high resolution dierencing scheme. The CICSAM scheme isparticularly suitable for ows with high ratios of viscosities between the phases. CICSAMis implemented in FLUENT as an explicit scheme and oers the advantage of producingan interface that is almost as sharp as the geometric reconstruction scheme.23-18 c Fluent Inc. September 29, 200623.3 Volume of Fluid (VOF) Model Theory23.3.3 Material PropertiesThe properties appearing in the transport equations are determined by the presence ofthe component phases in each control volume. In a two-phase system, for example, ifthe phases are represented by the subscripts 1 and 2, and if the volume fraction of thesecond of these is being tracked, the density in each cell is given by = 22 + (1 2)1 (23.3-5)In general, for an n-phase system, the volume-fraction-averaged density takes on thefollowing form: =qq (23.3-6)All other properties (e.g., viscosity) are computed in this manner.23.3.4 Momentum EquationA single momentum equation is solved throughout the domain, and the resulting velocityeld is shared among the phases. The momentum equation, shown below, is dependenton the volume fractions of all phases through the properties and .t(v) + (vv) = p +

v +vT

+ g + F (23.3-7)One limitation of the shared-elds approximation is that in cases where large velocitydierences exist between the phases, the accuracy of the velocities computed near theinterface can be adversely aected.Note that if the viscosity ratio is more than 1x103, this may lead to convergence di-culties. The compressive interface capturing scheme for arbitrary meshes (CICSAM)(Section 23.3.2: The Compressive Interface Capturing Scheme for Arbitrary Meshes(CICSAM)) is suitable for ows with high ratios of viscosities between the phases, thussolving the problem of poor convergence.c Fluent Inc. September 29, 2006 23-19Modeling Multiphase Flows23.3.5 Energy EquationThe energy equation, also shared among the phases, is shown below.t(E) + (v(E + p)) = (keT) + Sh (23.3-8)The VOF model treats energy, E, and temperature, T, as mass-averaged variables:E =nq=1qqEqnq=1qq(23.3-9)where Eq for each phase is based on the specic heat of that phase and the sharedtemperature.The properties and ke (eective thermal conductivity) are shared by the phases. Thesource term, Sh, contains contributions from radiation, as well as any other volumetricheat sources.As with the velocity eld, the accuracy of the temperature near the interface is limited incases where large temperature dierences exist between the phases. Such problems alsoarise in cases where the properties vary by several orders of magnitude. For example, if amodel includes liquid metal in combination with air, the conductivities of the materialscan dier by as much as four orders of magnitude. Such large discrepancies in propertieslead to equation sets with anisotropic coecients, which in turn can lead to convergenceand precision limitations.23.3.6 Additional Scalar EquationsDepending upon your problem denition, additional scalar equations may be involved inyour solution. In the case of turbulence quantities, a single set of transport equations issolved, and the turbulence variables (e.g., k and or the Reynolds stresses) are sharedby the phases throughout the eld.23.3.7 Time DependenceFor time-dependent VOF calculations, Equation 23.3-1 is solved using an explicit time-marching scheme. FLUENT automatically renes the time step for the integration of thevolume fraction equation, but you can inuence this time step calculation by modifyingthe Courant number. You can choose to update the volume fraction once for each timestep, or once for each iteration within each time step. These options are discussed inmore detail in Section 23.10.4: Setting Time-Dependent Parameters for the VOF Model.23-20 c Fluent Inc. September 29, 200623.3 Volume of Fluid (VOF) Model Theory23.3.8 Surface Tension and Wall AdhesionThe VOF model can also include the eects of surface tension along the interface betweeneach pair of phases. The model can be augmented by the additional specication of thecontact angles between the phases and the walls. You can specify a surface tensioncoecient as a constant, as a function of temperature, or through a UDF. The solverwill include the additional tangential stress terms (causing what is termed as Marangoniconvection) that arise due to the variation in surface tension coecient. Variable surfacetension coecient eects are usually important only in zero/near-zero gravity conditions.Surface TensionSurface tension arises as a result of attractive forces between molecules in a uid. Con-sider an air bubble in water, for example. Within the bubble, the net force on a moleculedue to its neighbors is zero. At the surface, however, the net force is radially inward, andthe combined eect of the radial components of force across the entire spherical surfaceis to make the surface contract, thereby increasing the pressure on the concave side ofthe surface. The surface tension is a force, acting only at the surface, that is requiredto maintain equilibrium in such instances. It acts to balance the radially inward inter-molecular attractive force with the radially outward pressure gradient force across thesurface. In regions where two uids are separated, but one of them is not in the formof spherical bubbles, the surface tension acts to minimize free energy by decreasing thearea of the interface.The surface tension model in FLUENT is the continuum surface force (CSF) model pro-posed by Brackbill et al. [39]. With this model, the addition of surface tension to theVOF calculation results in a source term in the momentum equation. To understand theorigin of the source term, consider the special case where the surface tension is constantalong the surface, and where only the forces normal to the interface are considered. It canbe shown that the pressure drop across the surface depends upon the surface tension co-ecient, , and the surface curvature as measured by two radii in orthogonal directions,R1 and R2:p2p1 =

1R1+ 1R2

(23.3-10)where p1 and p2 are the pressures in the two uids on either side of the interface.In FLUENT, a formulation of the CSF model is used, where the surface curvature iscomputed from local gradients in the surface normal at the interface. Let n be thesurface normal, dened as the gradient of q, the volume fraction of the qthphase.n = q (23.3-11)c Fluent Inc. September 29, 2006 23-21Modeling Multiphase FlowsThe curvature, , is dened in terms of the divergence of the unit normal, n [39]: = n (23.3-12)where n = n|n| (23.3-13)The surface tension can be written in terms of the pressure jump across the surface. Theforce at the surface can be expressed as a volume force using the divergence theorem. Itis this volume force that is the source term which is added to the momentum equation.It has the following form:Fvol =pairs ij, i 0), the ow is known to besubcritical where disturbances can travel upstream as well as downstream. In thiscase, downstream conditions might aect the ow upstream. When Fr = 1 (thus Vw = 0), the ow is known to be critical, where upstreampropagating waves remain stationary. In this case, the character of the ow changes. When Fr > 1, i.e., V > gy (thus Vw > 0), the ow is known to be supercriticalwhere disturbances cannot travel upstream. In this case, downstream conditionsdo not aect the ow upstream.Upstream Boundary ConditionsThere are two options available for the upstream boundary condition for open channelows: pressure inlet mass ow ratePressure InletThe total pressure p0 at the inlet can be given asp0 = 12( 0)V2+ ( 0)|g |( g (b a )) (23.3-21)where b and a are the position vectors of the face centroid and any point on the freesurface, respectively, Here, free surface is assumed to be horizontal and normal to thedirection of gravity. g is the gravity vector, |g | is the gravity magnitude, g is the unitvector of gravity, V is the velocity magnitude, is the density of the mixture in the cell,and 0 is the reference density.From this, the dynamic pressure q isq = 02 V2(23.3-22)and the static pressure ps isps = ( 0)|g |( g (b a )) (23.3-23)c Fluent Inc. September 29, 2006 23-27Modeling Multiphase Flowswhich can be further expanded tops = ( 0)|g |(( g b ) + ylocal) (23.3-24)where the distance from the free surface to the reference position, ylocal, isylocal = (a g) (23.3-25)Mass Flow RateThe mass ow rate for each phase associated with the open channel ow is dened by mphase = phase(Areaphase)(V elocity) (23.3-26)Volume Fraction SpecicationIn open channel ows, FLUENT internally calculates the volume fraction based on theinput parameters specied in the Boundary Conditions panel, therefore this option hasbeen disabled.For subcritical inlet ows (Fr < 1), FLUENT reconstructs the volume fraction values onthe boundary by using the values from the neighboring cells. This can be accomplishedusing the following procedure: Calculate the node values of volume fraction at the boundary using the cell values. Calculate the volume fraction at the each face of boundary using the interpolatednode values.For supercritical inlet ows (Fr > 1), the volume fraction value on the boundary can becalculated using the xed height of the free surface from the bottom.Downstream Boundary ConditionsPressure OutletDetermining the static pressure is dependent on the Pressure Specication Method. Usingthe Free Surface Level, the static pressure is dictated by Equation 23.3-23 and Equa-tion 23.3-25, otherwise you must specify the static pressure as the Gauge Pressure.For subcritical outlet ows (Fr < 1), if there are only two phases, then the pressure istaken from the pressure prole specied over the boundary, otherwise the pressure istaken from the neighboring cell. For supercritical ows (Fr >1), the pressure is alwaystaken from the neighboring cell.23-28 c Fluent Inc. September 29, 200623.4 Mixture Model TheoryOutow BoundaryOutow boundary conditions can be used at the outlet of open channel ows to modelow exits where the details of the ow velocity and pressure are not known prior tosolving the ow problem. If the conditions are unknown at the outow boundaries, thenFLUENT will extrapolate the required information from the interior.It is important, however, to understand the limitations of this boundary type: You can only use single outow boundaries at the outlet, which is achieved by set-ting the ow rate weighting to 1. In other words, outow splitting is not permittedin open channel ows with outow boundaries. There should be an initial ow eld in the simulation to avoid convergence issuesdue to ow reversal at the outow, which will result in an unreliable solution. An outow boundary condition can only be used with mass ow inlets. It is notcompatible with pressure inlets and pressure outlets. For example, if you choosethe inlet as pressure-inlet, then you can only use pressure-outlet at the outlet. If youchoose the inlet as mass-ow-inlet, then you can use either outow or pressure-outletboundary conditions at the outlet. Note that this only holds true for open channelow. Note that the outow boundary condition assumes that ow is fully developedin the direction perpendicular to the outow boundary surface. Therefore, suchsurfaces should be placed accordingly.Backow Volume Fraction SpecicationFLUENT internally calculates the volume fraction values on the outlet boundary by usingthe neighboring cell values, therefore, this option is disabled.23.4 Mixture Model Theory23.4.1 Overview and Limitations of the Mixture ModelOverviewThe mixture model is a simplied multiphase model that can be used to model multiphaseows where the phases move at dierent velocities, but assume local equilibrium overshort spatial length scales. The coupling between the phases should be strong. It canalso be used to model homogeneous multiphase ows with very strong coupling and thephases moving at the same velocity. In addition, the mixture model can be used tocalculate non-Newtonian viscosity.c Fluent Inc. September 29, 2006 23-29Modeling Multiphase FlowsThe mixture model can model n phases (uid or particulate) by solving the momentum,continuity, and energy equations for the mixture, the volume fraction equations for thesecondary phases, and algebraic expressions for the relative velocities. Typical applica-tions include sedimentation, cyclone separators, particle-laden ows with low loading,and bubbly ows where the gas volume fraction remains low.The mixture model is a good substitute for the full Eulerian multiphase model in severalcases. A full multiphase model may not be feasible when there is a wide distribution ofthe particulate phase or when the interphase laws are unknown or their reliability canbe questioned. A simpler model like the mixture model can perform as well as a fullmultiphase model while solving a smaller number of variables than the full multiphasemodel.The mixture model allows you to select granular phases and calculates all properties ofthe granular phases. This is applicable for liquid-solid ows.LimitationsThe following limitations apply to the mixture model in FLUENT: You must use the pressure-based solver. The mixture model is not available witheither of the density-based solvers. Only one of the phases can be dened as a compressible ideal gas. There is nolimitation on using compressible liquids using user-dened functions. Streamwise periodic ow with specied mass ow rate cannot be modeled whenthe mixture model is used (the user is allowed to specify a pressure drop). Solidication and melting cannot be modeled in conjunction with the mixturemodel. The LES turbulence model cannot be used with the mixture model if the cavitationmodel is enabled. The relative velocity formulation cannot be used in combination with the MRF andmixture model (see Section 10.3.1: Limitations). The mixture model cannot be used for inviscid ows. The shell conduction model for walls cannot be used with the mixture model. When tracking particles in parallel, the DPM model cannot be used with the mix-ture model if the shared memory option is enabled (Section 22.11.9: Parallel Pro-cessing for the Discrete Phase Model). (Note that using the message passing option,when running in parallel, enables the compatibility of all multiphase ow modelswith the DPM model.)23-30 c Fluent Inc. September 29, 200623.4 Mixture Model TheoryThe mixture model, like the VOF model, uses a single-uid approach. It diers from theVOF model in two respects: The mixture model allows the phases to be interpenetrating. The volume fractionsq and p for a control volume can therefore be equal to any value between 0 and1, depending on the space occupied by phase q and phase p. The mixture model allows the phases to move at dierent velocities, using theconcept of slip velocities. (Note that the phases can also be assumed to moveat the same velocity, and the mixture model is then reduced to a homogeneousmultiphase model.)The mixture model solves the continuity equation for the mixture, the momentum equa-tion for the mixture, the energy equation for the mixture, and the volume fraction equa-tion for the secondary phases, as well as algebraic expressions for the relative velocities(if the phases are moving at dierent velocities).23.4.2 Continuity EquationThe continuity equation for the mixture ist(m) + (mvm) = 0 (23.4-1)where vm is the mass-averaged velocity:vm =nk=1kkvkm(23.4-2)and m is the mixture density:m =nk=1kk (23.4-3)k is the volume fraction of phase k.c Fluent Inc. September 29, 2006 23-31Modeling Multiphase Flows23.4.3 Momentum EquationThe momentum equation for the mixture can be obtained by summing the individualmomentum equations for all phases. It can be expressed ast(mvm) + (mvmvm) = p +

m

vm +vTm

+mg + F +

nk=1kkvdr,kvdr,k

(23.4-4)where n is the number of phases, F is a body force, and m is the viscosity of the mixture:m =nk=1kk (23.4-5)vdr,k is the drift velocity for secondary phase k:vdr,k = vkvm (23.4-6)23.4.4 Energy EquationThe energy equation for the mixture takes the following form:tnk=1(kkEk) +nk=1(kvk(kEk + p)) = (keT) + SE (23.4-7)where ke is the eective conductivity (k(kk +kt)), where kt is the turbulent thermalconductivity, dened according to the turbulence model being used). The rst term onthe right-hand side of Equation 23.4-7 represents energy transfer due to conduction. SEincludes any other volumetric heat sources.In Equation 23.4-7,Ek = hk pk+ v2k2 (23.4-8)for a compressible phase, and Ek = hk for an incompressible phase, where hk is thesensible enthalpy for phase k.23-32 c Fluent Inc. September 29, 200623.4 Mixture Model Theory23.4.5 Relative (Slip) Velocity and the Drift VelocityThe relative velocity (also referred to as the slip velocity) is dened as the velocity of asecondary phase (p) relative to the velocity of the primary phase (q):vpq = vpvq (23.4-9)The mass fraction for any phase (k) is dened asck = kkm(23.4-10)The drift velocity and the relative velocity (vqp) are connected by the following expression:vdr,p = vpqnk=1ckvqk (23.4-11)FLUENTs mixture model makes use of an algebraic slip formulation. The basic assump-tion of the algebraic slip mixture model is that to prescribe an algebraic relation for therelative velocity, a local equilibrium between the phases should be reached over shortspatial length scale. Following Manninen et al. [229], the form of the relative velocity isgiven by:vpq = pfdrag(pm)pa (23.4-12)where p is the particle relaxation timep =pd2p18q(23.4-13)d is the diameter of the particles (or droplets or bubbles) of secondary phase p, a is thesecondary-phase particles acceleration. The default drag function fdrag is taken fromSchiller and Naumann [320]:fdrag =

1 + 0.15 Re0.687Re 10000.0183 Re Re > 1000 (23.4-14)and the acceleration a is of the forma = g (vm )vm vmt (23.4-15)c Fluent Inc. September 29, 2006 23-33Modeling Multiphase FlowsThe simplest algebraic slip formulation is the so-called drift ux model, in which the ac-celeration of the particle is given by gravity and/or a centrifugal force and the particulaterelaxation time is modied to take into account the presence of other particles.In turbulent ows the relative velocity should contain a diusion term due to the disper-sion appearing in the momentum equation for the dispersed phase. FLUENT adds thisdispersion to the relative velocity:vpq =(pm)d2p18qfdraga mpDq (23.4-16)where (m) is the mixture turbulent viscosity and (D) is a Prandtl dispersion coecient.When you are solving a mixture multiphase calculation with slip velocity, you can directlyprescribe formulations for the drag function. The following choices are available: Schiller-Naumann (the default formulation) Morsi-Alexander symmetric constant user-denedSee Section 23.5.4: Interphase Exchange Coecients for more information on these dragfunctions and their formulations, and Section 23.11.1: Dening the Phases for the MixtureModel for instructions on how to enable them.Note that, if the slip velocity is not solved, the mixture model is reduced to a homogeneousmultiphase model. In addition, the mixture model can be customized (using user-denedfunctions) to use a formulation other than the algebraic slip method for the slip velocity.See the separate UDF Manual for details.23.4.6 Volume Fraction Equation for the Secondary PhasesFrom the continuity equation for secondary phase p, the volume fraction equation forsecondary phase p can be obtained:t(pp) + (ppvm) = (ppvdr,p) +nq=1( mqp mpq) (23.4-17)23-34 c Fluent Inc. September 29, 200623.4 Mixture Model Theory23.4.7 Granular PropertiesSince the concentration of particles is an important factor in the calculation of the eec-tive viscosity for the mixture, we may use the granular viscosity (see section on Euleriangranular ows) to get a value for the viscosity of the suspension. The volume weightedaveraged for the viscosity would now contain shear viscosity arising from particle mo-mentum exchange due to translation and collision.The collisional and kinetic parts, and the optional frictional part, are added to give thesolids shear viscosity:s = s,col + s,kin + s,fr (23.4-18)Collisional ViscosityThe collisional part of the shear viscosity is modeled as [119, 363]s,col = 45ssdsg0,ss(1 + ess)

s

1/2(23.4-19)Kinetic ViscosityFLUENT provides two expressions for the kinetic viscosity.The default expression is from Syamlal et al. [363]:s,kin = sdsss6 (3 ess)1 + 25 (1 + ess) (3ess1) sg0,ss

(23.4-20)The following optional expression from Gidaspow et al. [119] is also available:s,kin = 10sdss96s (1 + ess) g0,ss1 + 45g0,sss (1 + ess)

2(23.4-21)c Fluent Inc. September 29, 2006 23-35Modeling Multiphase Flows23.4.8 Granular TemperatureThe viscosities need the specication of the granular temperature for the sthsolids phase.Here we use an algebraic equation derived from the transport equation by neglectingconvection and diusion and takes the form [363]0 = (psI + s) : vss + ls (23.4-22)where(psI + s) : vs = the generation of energy by the solid stress tensors = the collisional dissipation of energyls = the energy exchange between the lthuid or solid phase and the sthsolid phaseThe collisional dissipation of energy, s, represents the rate of energy dissipation withinthe sthsolids phase due to collisions between particles. This term is represented by theexpression derived by Lun et al. [221]m = 12(1 e2ss)g0,ssds s2s3/2s (23.4-23)The transfer of the kinetic energy of random uctuations in particle velocity from the sthsolids phase to the lthuid or solid phase is represented by ls [119]:ls = 3Klss (23.4-24)FLUENT allows you to solve for the granular temperature with the following options: algebraic formulation (the default)This is obtained by neglecting convection and diusion in the transport equation(Equation 23.4-22) [363]. constant granular temperatureThis is useful in very dense situations where the random uctuations are small. UDF for granular temperature23.4.9 Solids PressureThe total solid pressure is calculated and included in the mixture momentum equations:Ps,total =Nq=1pq (23.4-25)where pq is presented in the section for granular ows by equation Equation 23.5-4823-36 c Fluent Inc. September 29, 200623.5 Eulerian Model Theory23.5 Eulerian Model TheoryDetails about the Eulerian multiphase model are presented in the following subsections: Section 23.5.1: Overview and Limitations of the Eulerian Model Section 23.5.2: Volume Fractions Section 23.5.3: Conservation Equations Section 23.5.4: Interphase Exchange Coecients Section 23.5.5: Solids Pressure Section 23.5.6: Maximum Packing Limit in Binary Mixtures Section 23.5.7: Solids Shear Stresses Section 23.5.8: Granular Temperature Section 23.5.9: Description of Heat Transfer Section 23.5.10: Turbulence Models Section 23.5.11: Solution Method in FLUENT23.5.1 Overview and Limitations of the Eulerian ModelOverviewThe Eulerian multiphase model in FLUENT allows for the modeling of multiple sepa-rate, yet interacting phases. The phases can be liquids, gases, or solids in nearly anycombination. An Eulerian treatment is used for each phase, in contrast to the Eulerian-Lagrangian treatment that is used for the discrete phase model.With the Eulerian multiphase model, the number of secondary phases is limited onlyby memory requirements and convergence behavior. Any number of secondary phasescan be modeled, provided that sucient memory is available. For complex multiphaseows, however, you may nd that your solution is limited by convergence behavior. SeeSection 23.14.4: Eulerian Model for multiphase modeling strategies.FLUENTs Eulerian multiphase model does not distinguish between uid-uid and uid-solid (granular) multiphase ows. A granular ow is simply one that involves at leastone phase that has been designated as a granular phase.c Fluent Inc. September 29, 2006 23-37Modeling Multiphase FlowsThe FLUENT solution is based on the following: A single pressure is shared by all phases. Momentum and continuity equations are solved for each phase. The following parameters are available for granular phases: Granular temperature (solids uctuating energy) can be calculated for eachsolid phase. You can select either an algebraic formulation, a constant, auser-dened function, or a partial dierential equation. Solid-phase shear and bulk viscosities are obtained by applying kinetic the-ory to granular ows. Frictional viscosity for modeling granular ow is alsoavailable. You can select appropriate models and user-dened functions forall properties. Several interphase drag coecient functions are available, which are appropriatefor various types of multiphase regimes. (You can also modify the interphase dragcoecient through user-dened functions, as described in the separate UDF Man-ual.) All of the k- turbulence models are available, and may apply to all phases or tothe mixture.LimitationsAll other features available in FLUENT can be used in conjunction with the Eulerianmultiphase model, except for the following limitations: The Reynolds Stress turbulence model is not available on a per phase basis. Particle tracking (using the Lagrangian dispersed phase model) interacts only withthe primary phase. Streamwise periodic ow with specied mass ow rate cannot be modeled whenthe Eulerian model is used (the user is allowed to specify a pressure drop). Inviscid ow is not allowed. Melting and solidication are not allowed. When tracking particles in parallel, the DPM model cannot be used with the Eule-rian multiphase model if the shared memory option is enabled (Section 22.11.9: Par-allel Processing for the Discrete Phase Model). (Note that using the message pass-ing option, when running in parallel, enables the compatibility of all multiphaseow models with the DPM model.)23-38 c Fluent Inc. September 29, 200623.5 Eulerian Model TheoryTo change from a single-phase model, where a single set of conservation equations formomentum, continuity and (optionally) energy is solved, to a multiphase model, addi-tional sets of conservation equations must be introduced. In the process of introduc-ing additional sets of conservation equations, the original set must also be modied.The modications involve, among other things, the introduction of the volume fractions1, 2, . . . n for the multiple phases, as well as mechanisms for the exchange of momen-tum, heat, and mass between the phases.23.5.2 Volume FractionsThe description of multiphase ow as interpenetrating continua incorporates the conceptof phasic volume fractions, denoted here by q. Volume fractions represent the spaceoccupied by each phase, and the laws of conservation of mass and momentum are satisedby each phase individually. The derivation of the conservation equations can be doneby ensemble averaging the local instantaneous balance for each of the phases [10] or byusing the mixture theory approach [36].The volume of phase q, Vq, is dened byVq =

VqdV (23.5-1)wherenq=1q = 1 (23.5-2)The eective density of phase q is q = qq (23.5-3)where q is the physical density of phase q.c Fluent Inc. September 29, 2006 23-39Modeling Multiphase Flows23.5.3 Conservation EquationsThe general conservation equations from which the equations solved by FLUENT arederived are presented in this section, followed by the solved equations themselves.Equations in General FormConservation of MassThe continuity equation for phase q ist(qq) + (qqvq) =np=1( mpq mqp) + Sq (23.5-4)where vq is the velocity of phase q and mpq characterizes the mass transfer from the pthto qthphase, and mqp characterizes the mass transfer from phase q to phase p, and youare able to specify these mechanisms separately.By default, the source term Sq on the right-hand side of Equation 23.5-4 is zero, but youcan specify a constant or user-dened mass source for each phase. A similar term appearsin the momentum and enthalpy equations. See Section 23.7: Modeling Mass Transfer inMultiphase Flows for more information on the modeling of mass transfer in FLUENTsgeneral multiphase models.Conservation of MomentumThe momentum balance for phase q yieldst(qqvq) + (qqvqvq) = qp + q + qqg+np=1(

Rpq + mpqvpq mqpvqp) + (

Fq + Flift,q + Fvm,q) (23.5-5)where q is the qthphase stress-strain tensorq = qq(vq +vTq ) + q(q 23q) vqI (23.5-6)Here q and q are the shear and bulk viscosity of phase q, Fq is an external body force,

Flift,q is a lift force, Fvm,q is a virtual mass force, Rpq is an interaction force betweenphases, and p is the pressure shared by all phases.vpq is the interphase velocity, dened as follows. If mpq > 0 (i.e., phase p mass is beingtransferred to phase q), vpq = vp; if mpq < 0 (i.e., phase q mass is being transferred tophase p), vpq = vq. Likewise, if mqp > 0 then vqp = vq, if mqp < 0 then vqp = vp.23-40 c Fluent Inc. September 29, 200623.5 Eulerian Model TheoryEquation 23.5-5 must be closed with appropriate expressions for the interphase force Rpq.This force depends on the friction, pressure, cohesion, and other eects, and is subjectto the conditions that Rpq =

Rqp and Rqq = 0.FLUENT uses a simple interaction term of the following form:np=1

Rpq =np=1Kpq(vpvq) (23.5-7)where Kpq (= Kqp) is the interphase momentum exchange coecient (described in Sec-tion 23.5.4: Interphase Exchange Coecients).Lift ForcesFor multiphase ows, FLUENT can include the eect of lift forces on the secondary phaseparticles (or droplets or bubbles). These lift forces act on a particle mainly due to velocitygradients in the primary-phase ow eld. The lift force will be more signicant for largerparticles, but the FLUENT model assumes that the particle diameter is much smallerthan the interparticle spacing. Thus, the inclusion of lift forces is not appropriate forclosely packed particles or for very small particles.The lift force acting on a secondary phase p in a primary phase q is computed from [88]

Flift = 0.5qp(vqvp) (vq) (23.5-8)The lift force Flift will be added to the right-hand side of the momentum equation forboth phases (

Flift,q =

Flift,p).In most cases, the lift force is insignicant compared to the drag force, so there is noreason to include this extra term. If the lift force is signicant (e.g., if the phases separatequickly), it may be appropriate to include this term. By default, Flift is not included.The lift force and lift coecient can be specied for each pair of phases, if desired.i It is important that if you include the lift force in your calculation, youneed not include it everywhere in the computational domain since it iscomputationally expensive to converge. For example, in the wall boundarylayer for turbulent bubbly ows in channels, the lift force is signicantwhen the slip velocity is large in the vicinity of high strain rates for theprimary phase.c Fluent Inc. September 29, 2006 23-41Modeling Multiphase FlowsVirtual Mass ForceFor multiphase ows, FLUENT includes the virtual mass eect that occurs when asecondary phase p accelerates relative to the primary phase q. The inertia of the primary-phase mass encountered by the accelerating particles (or droplets or bubbles) exerts avirtual mass force on the particles [88]:

Fvm = 0.5pq

dqvqdt dpvpdt

(23.5-9)The term dqdt denotes the phase material time derivative of the formdq()dt = ()t + (vq ) (23.5-10)The virtual mass force Fvm will be added to the right-hand side of the momentum equationfor both phases (

Fvm,q =

Fvm,p).The virtual mass eect is signicant when the secondary phase density is much smallerthan the primary phase density (e.g., for a transient bubble column). By default, Fvm isnot included.Conservation of EnergyTo describe the conservation of energy in Eulerian multiphase applications, a separateenthalpy equation can be written for each phase:t(qqhq)+(qquqhq) = qpqt +q : uqqq+Sq+np=1(Qpq+ mpqhpq mqphqp)(23.5-11)where hq is the specic enthalpy of the qthphase, qq is the heat ux, Sq is a source termthat includes sources of enthalpy (e.g., due to chemical reaction or radiation), Qpq isthe intensity of heat exchange between the pthand qthphases, and hpq is the interphaseenthalpy (e.g., the enthalpy of the vapor at the temperature of the droplets, in the caseof evaporation). The heat exchange between phases must comply with the local balanceconditions Qpq = Qqp and Qqq = 0.23-42 c Fluent Inc. September 29, 200623.5 Eulerian Model TheoryEquations Solved by FLUENTThe equations for uid-uid and granular multiphase ows, as solved by FLUENT, arepresented here for the general case of an n-phase ow.Continuity EquationThe volume fraction of each phase is calculated from a continuity equation:1rq

t(qq) + (qqvq) =np=1( mpq mqp)

(23.5-12)where rq is the phase reference density, or the volume averaged density of the qthphasein the solution domain.The solution of this equation for each secondary phase, along with the condition that thevolume fractions sum to one (given by Equation 23.5-2), allows for the calculation of theprimary-phase volume fraction. This treatment is common to uid-uid and granularows.Fluid-Fluid Momentum EquationsThe conservation of momentum for a uid phase q ist(qqvq) + (qqvqvq) = qp + q + qqg +np=1(Kpq(vpvq) + mpqvpq mqpvqp) +(

Fq + Flift,q + Fvm,q) (23.5-13)Here g is the acceleration due to gravity and q, Fq, Flift,q, and Fvm,q are as dened forEquation 23.5-5.Fluid-Solid Momentum EquationsFollowing the work of [7, 51, 79, 119, 198, 221, 267, 363], FLUENT uses a multi-uidgranular model to describe the ow behavior of a uid-solid mixture. The solid-phasestresses are derived by making an analogy between the random particle motion arisingfrom particle-particle collisions and the thermal motion of molecules in a gas, taking intoaccount the inelasticity of the granular phase. As is the case for a gas, the intensity of theparticle velocity uctuations determines the stresses, viscosity, and pressure of the solidphase. The kinetic energy associated with the particle velocity uctuations is representedc Fluent Inc. September 29, 2006 23-43Modeling Multiphase Flowsby a pseudothermal or granular temperature which is proportional to the mean squareof the random motion of particles.The conservation of momentum for the uid phases is similar to Equation 23.5-13, andthat for the sthsolid phase ist(ssvs) + (ssvsvs) = sp ps + s + ssg +Nl=1(Kls(vlvs) + mlsvls mslvsl) +(

Fs + Flift,s + Fvm,s) (23.5-14)where ps is the sthsolids pressure, Kls = Ksl is the momentum exchange coecientbetween uid or solid phase l and solid phase s, N is the total number of phases, and

Fq, Flift,q, and Fvm,q are as dened for Equation 23.5-5.Conservation of EnergyThe equation solved by FLUENT for the conservation of energy is Equation 23.5-11.23.5.4 Interphase Exchange CoefcientsIt can be seen in Equations 23.5-13 and 23.5-14 that momentum exchange between thephases is based on the value of the uid-uid exchange coecient Kpq and, for granularows, the uid-solid and solid-solid exchange coecients Kls.Fluid-Fluid Exchange CoefcientFor uid-uid ows, each secondary phase is assumed to form droplets or bubbles. Thishas an impact on how each of the uids is assigned to a particular phase. For example,in ows where there are unequal amounts of two uids, the predominant uid should bemodeled as the primary uid, since the sparser uid is more likely to form droplets orbubbles. The exchange coecient for these types of bubbly, liquid-liquid or gas-liquidmixtures can be written in the following general form:Kpq = qppfp(23.5-15)where f, the drag function, is dened dierently for the dierent exchange-coecientmodels (as described below) and p, the particulate relaxation time, is dened asp =pd2p18q(23.5-16)23-44 c Fluent Inc. September 29, 200623.5 Eulerian Model Theorywhere dp is the diameter of the bubbles or droplets of phase p.Nearly all denitions of f include a drag coecient (CD) that is based on the relativeReynolds number (Re). It is this drag function that diers among the exchange-coecientmodels. For all these situations, Kpq should tend to zero whenever the primary phase isnot present within the domain. To enforce this, the drag function f is always multipliedby the volume fraction of the primary phase q, as is reected in Equation 23.5-15. For the model of Schiller and Naumann [320]f = CDRe24 (23.5-17)whereCD =

24(1 + 0.15 Re0.687)/Re Re 10000.44 Re > 1000 (23.5-18)and Re is the relative Reynolds number. The relative Reynolds number for theprimary phase q and secondary phase p is obtained fromRe = q|vpvq|dpq(23.5-19)The relative Reynolds number for secondary phases p and r is obtained fromRe = rp|vrvp|drprp(23.5-20)where rp = pp + rr is the mixture viscosity of the phases p and r.The Schiller and Naumann model is the default method, and it is acceptable forgeneral use for all uid-uid pairs of phases. For the Morsi and Alexander model [252]f = CDRe24 (23.5-21)whereCD = a1 + a2Re + a3Re2 (23.5-22)c Fluent Inc. September 29, 2006 23-45Modeling Multiphase Flowsand Re is dened by Equation 23.5-19 or 23.5-20. The as are dened as follows:a1, a2, a3 =

0, 24, 0 0 < Re < 0.13.690, 22.73, 0.0903 0.1 < Re < 11.222, 29.1667, 3.8889 1 < Re < 100.6167, 46.50, 116.67 10 < Re < 1000.3644, 98.33, 2778 100 < Re < 10000.357, 148.62, 47500 1000 < Re < 50000.46, 490.546, 578700 5000 < Re < 100000.5191, 1662.5, 5416700 Re 10000(23.5-23)The Morsi and Alexander model is the most complete, adjusting the function def-inition frequently over a large range of Reynolds numbers, but calculations withthis model may be less stable than with the other models. For the symmetric modelKpq = p(pp + qq)fpq(23.5-24)wherepq = (pp + qq)(dp+dq2 )218(pp + qq) (23.5-25)andf = CDRe24 (23.5-26)whereCD =

24(1 + 0.15 Re0.687)/Re Re 10000.44 Re > 1000 (23.5-27)and Re is dened by Equation 23.5-19 or 23.5-20. Note that if there is only onedispersed phase, then dp = dq in Equation 23.5-25.The symmetric model is recommended for ows in which the secondary (dispersed)phase in one region of the domain becomes the primary (continuous) phase inanother. Thus for a single dispersed phase, dp = dq and (dp+dq)2 = dp. For example,if air is injected into the bottom of a container lled halfway with water, the airis the dispersed phase in the bottom half of the container; in the top half of thecontainer, the air is the continuous phase. This model can also be used for theinteraction between secondary phases.23-46 c Fluent Inc. September 29, 200623.5 Eulerian Model TheoryYou can specify dierent exchange coecients for each pair of phases. It is also possibleto use user-dened functions to dene exchange coecients for each pair of phases. If theexchange coecient is equal to zero (i.e., if no exchange coecient is specied), the owelds for the uids will be computed independently, with the only interaction beingtheir complementary volume fractions within each computational cell.Fluid-Solid Exchange CoefcientThe uid-solid exchange coecient Ksl can be written in the following general form:Ksl = ssfs(23.5-28)where f is dened dierently for the dierent exchange-coecient models (as describedbelow), and s, the particulate relaxation time, is dened ass = sd2s18l(23.5-29)where ds is the diameter of particles of phase s.All denitions of f include a drag function (CD) that is based on the relative Reynoldsnumber (Res). It is this drag function that diers among the exchange-coecient models. For the Syamlal-OBrien model [362]f = CDResl24v2r,s(23.5-30)where the drag function has a form derived by Dalla Valle [73]CD =

0.63 + 4.8

Res/vr,s

2(23.5-31)This model is based on measurements of the terminal velocities of particles inuidized or settling beds, with correlations that are a function of the volume fractionand relative Reynolds number [305]:Res = lds|vsvl|l(23.5-32)where the subscript l is for the lthuid phase, s is for the sthsolid phase, and ds isthe diameter of the sthsolid phase particles.c Fluent Inc. September 29, 2006 23-47Modeling Multiphase FlowsThe uid-solid exchange coecient has the formKsl = 3sll4v2r,sdsCD

Resvr,s

|vsvl| (23.5-33)where vr,s is the terminal velocity correlation for the solid phase [113]:vr,s = 0.5

A 0.06 Res +

(0.06 Res)2+ 0.12 Res (2B A) + A2

(23.5-34)withA = 4.14l (23.5-35)andB = 0.81.28l (23.5-36)for l 0.85, andB = 2.65l (23.5-37)for l > 0.85.This model is appropriate when the solids shear stresses are dened according toSyamlal et al. [363] (Equation 23.5-64). For the model of Wen and Yu [396], the uid-solid exchange coecient is of thefollowing form:Ksl = 34CDsll|vsvl|ds2.65l (23.5-38)whereCD = 24lRes

1 + 0.15(lRes)0.687

(23.5-39)and Res is dened by Equation 23.5-32.This model is appropriate for dilute systems.23-48 c Fluent Inc. September 29, 200623.5 Eulerian Model Theory The Gidaspow model [119] is a combination of the Wen and Yu model [396] andthe Ergun equation [96].When l > 0.8, the uid-solid exchange coecient Ksl is of the following form:Ksl = 34CDsll|vsvl|ds2.65l (23.5-40)whereCD = 24lRes

1 + 0.15(lRes)0.687

(23.5-41)When l 0.8,Ksl = 150s(1 l)lld2s+ 1.75ls|vsvl|ds(23.5-42)This model is recommended for dense uidized beds.Solid-Solid Exchange CoefcientThe solid-solid exchange coecient Kls has the following form [361]:Kls =3 (1 + els)

2 + Cfr,ls28

ssll (dl + ds)2g0,ls2 (ld3l + sd3s) |vlvs| (23.5-43)whereels = the coecient of restitutionCfr,ls = the coecient of friction between the lthand sthsolid-phase particles (Cfr,ls = 0)dl = the diameter of the particles of solid lg0,ls = the radial distribution coecientNote that the coecient of restitution is described in Section 23.5.5: Solids Pressureand the radial distribution coecient is described in Section 23.5.5: Radial DistributionFunction.c Fluent Inc. September 29, 2006 23-49Modeling Multiphase Flows23.5.5 Solids PressureFor granular ows in the compressible regime (i.e., where the solids volume fraction is lessthan its maximum allowed value), a solids pressure is calculated independently and usedfor the pressure gradient term, ps, in the granular-phase momentum equation. Becausea Maxwellian velocity distribution is used for the particles, a granular temperature isintroduced into the model, and appears in the expression for the solids pressure andviscosities. The solids pressure is composed of a kinetic term and a second term due toparticle collisions:ps = sss + 2s(1 + ess)2sg0,sss (23.5-44)where ess is the coecient of restitution for particle collisions, g0,ss is the radial distribu-tion function, and s is the granular temperature. FLUENT uses a default value of 0.9for ess, but the value can be adjusted to suit the particle type. The granular temperatures is proportional to the kinetic energy of the uctuating particle motion, and will bedescribed later in this section. The function g0,ss (described below in more detail) is adistribution function that governs the transition from the compressible condition with < s,max, where the spacing between the solid particles can continue to decrease, tothe incompressible condition with = s,max, where no further decrease in the spacingcan occur. A value of 0.63 is the default for s,max, but you can modify it during theproblem setup.Other formulations that are also available in FLUENT are [363]ps = 2s(1 + ess)2sg0,sss (23.5-45)and [226]ps = sss[(1 + 4sg0,ss) + 12[(1 + ess)(1 ess + 2fric)]] (23.5-46)When more than one solids phase are calculated, the above expression does not take intoaccount the eect of other phases. A derivation of the expressions from the Boltzmanequations for a granular mixture are beyond the scope of this manual, however there isa need to provide a better formulation so that some properties may feel the presence ofother phases. A known problem is that N solids phases with identical properties should beconsistent when the same phases are described by a single solids phase. Equations derivedempirically may not satisfy this property and need to be changed accordingly withoutdeviating signicantly from the original form. From [118], a general solids pressureformulation in the presence of other phases could be of the formpq = qqq +Np=13g0,pqd3qpnqnp(1 + eqp)f(mp, mq, p, q) (23.5-47)23-50 c Fluent Inc. September 29, 200623.5 Eulerian Model Theorywhere dpq = dp+dq2 is the average diameter, np, nq are the number of particles, mp and mqare the masses of the particles in phases p and q, and f is a function of the masses of theparticles and their granular temperatures. For now, we have to simplify this expressionso that it depends only on the granular temperature of phase qpq = qqq +Np=12d3pqd3q(1 + epq)g0,pqqpqq (23.5-48)Since all models need to be cast in the general form, it follows thatpq = qqq + (Np=1d3pqd3qpc,qp)qq (23.5-49)where pc,qp is the collisional part of the pressure between phases q and p.The above expression reverts to the one solids phase expression when N = 1 and q = pbut also has the property of feeling the presence of other phases.Radial Distribution FunctionThe radial distribution function, g0, is a correction factor that modies the probabilityof collisions between grains when the solid granular phase becomes dense. This functionmay also be interpreted as the nondimensional distance between spheres:g0 = s + dps (23.5-50)where s is the distance between grains. From Equation 23.5-50 it can be observed thatfor a dilute solid phase s , and therefore g0 1. In the limit when the solid phasecompacts, s 0 and g0 . The radial distribution function is closely connectedto the factor of Chapman and Cowlings [51] theory of nonuniform gases. is equalto 1 for a rare gas, and increases and tends to innity when the molecules are so closetogether that motion is not possible.c Fluent Inc. September 29, 2006 23-51Modeling Multiphase FlowsIn the literature there is no unique formulation for the radial distribution function. FLU-ENT has a number of options: For one solids phase, use [267]:g0 =

1

ss,max131(23.5-51)This is an empirical function and does not extends easily to n phases. For twoidentical phases with the property that q = 1 + 2, the above function is notconsistent for the calculation of the partial pressures p1 and p2, pq = p1 + p2. Inorder to correct this problem, FLUENT uses the following consistent form