phase change simulation

Chapter 8. Phase Change Simulations

This chapter describes the phase change model available in FLU-

ENT and the commands you use to set up a phase change problem.Information is organized into the following sections:

� Section 8.1 : Overview of Phase Change Modeling

� Section 8.2 : Phase Change Modeling Theory

� Section 8.3 : User Inputs for the Phase Change Model

� Section 8.4 : Solution Strategies for Phase Change Problems

8.1 Overview of Phase Change Modeling in FLUENT

FLUENT can be used to solve uid ow problems involving phasechange taking place at one temperature (e.g., in pure metals) orover a range of temperature (e.g., in binary alloys). Instead ofexplicitly tracking the liquid-solid front as the phase change occurs,which requires a moving mesh methodology, an enthalpy-porosityformulation is used where the ow and enthalpy equations are solvedwith extra source terms on the �xed grid.

Marangoni shear, due to the variation of surface tension with tem-perature, is important in many industrial uid ow situations in-volving phase change. The phase change model in FLUENT providesthe ability to specify the Marangoni gradient at a sloping surface,as well as an arbitrary shear at a boundary coinciding with oneof the curvilinear grid lines. The model also allows you to specifythe convective heat transfer, radiation, and heat ux at a wall aspiecewise linear pro�les, polynomials, or harmonic functions.

FLUENT provides the following phase change modeling options:

� Calculation of liquid-solid phase change in pure metals as wellas in binary alloys.

� Modeling of continuous casting processes (i.e., \pulling" ofsolid material out of the domain).

8-2 Chapter 8 | Phase Change Simulations

� Ability to specify an arbitrary shear at a curved boundary asa piecewise linear pro�le or polynomial in terms of one of theCartesian coordinates.

� Modeling of Marangoni convection due to the variation of sur-face tension with temperature.

� Modeling of the thermal contact resistance between frozenmaterial and the wall (e.g., due to an air gap).

� Ability to patch a momentum source in each Cartesian direc-tion and/or a heat source (to simulate magnetic force �elds orheat generation in the domain, for example).

� Display and patching of latent heat content, pull velocities incontinuous casting and other relevant variables.

These modeling capabilities allow FLUENT to simulate a wide rangeof phase change problems including melting, solidi�cation, crystalgrowth and continuous casting. The physical equations used forthese phase change calculations are described in the following sec-tions.

As mentioned above, the phase change formulation in FLUENTLimitations of the

Phase Change

Model

can be used to model the melting/freezing of pure materials, aswell as alloys. The liquid fraction versus temperature relationshipused in FLUENT is the lever rule|i.e., a linear relationship (Equa-tion 8.2-3). Other relationships are possible [124], but not availablein FLUENT.

The following FLUENT features cannot be used in conjunction withthe phase change model:

� Radiation

� Combustion

� Speci�ed periodic mass ow

� Cylindrical velocities

In order for you to enable the phase change model, the energy equa-Overview of Phase

Change Modeling

Procedures

tion must be active. You are then required to supply additionalphysical constants pertaining to the phase change problem (liquidusand solidus temperature, latent heat of freezing, etc.). You may in-voke one of the enhanced boundary conditions that are available

c Fluent Inc. May 10, 1997

8.2 Phase Change Modeling Theory 8-3

in the phase change model (applied shear, free surface with sur-face tension gradient, or mixed heat transfer). Before solving thecoupled uid ow and heat transfer problem, you should patch aninitial temperature and latent heat distribution or solve the steadyconduction problem as an initial condition. The coupled problemcan then be solved as either steady or unsteady. Because of the non-linear nature of these problems, however, in most cases an unsteadysolution approach is preferred.

8.2 Phase Change Modeling Theory

The enthalpy-porosity technique [137]{[140] is used in FLUENT formodeling the phase change process. In this technique, the melt in-terface is not tracked explicitly. Instead, a quantity called the liquidfraction is associated with each control volume in the domain andthe liquid fraction is computed at each iteration. Both isothermaland non-isothermal phase change may be computed; the mushy zoneis simply a region in which the liquid fraction lies between zero andunity. The mushy zone is modeled as a \pseudo" porous mediumin which the porosity decreases from 1 to 0 as the material solid-i�es. Thus, the uid velocities in the mushy zone are attenuatedand reach values near zero when the material is fully solidi�ed. Inthis section, an overview of the phase change theory is given. Pleaserefer to [137] for details on the enthalpy-porosity method.

The energy equation is written in terms of the sensible enthalpy, h,The Energy

Equation de�ned as

h = href +Z T

Tref

cpdT (8.2-1)

where href = reference enthalpyTref = reference temperaturecp = speci�c heat at constant pressure

The total enthalpy, H, is therefore

H = h+�H (8.2-2)

where �H is the latent heat content. The latent heat content mayvary between zero (solid) and L (liquid), the latent heat of thematerial. Thus, the liquid fraction, �, can be de�ned as

� =�H

L= 0 if T < Tsolidus



� =�H

L= 1 if T > Tliquidus

� =�H

L=

T � TsolidusTliquidus � Tsolidus

if Tsolidus < T < Tliquidus (8.2-3)

For phase change problems, the energy equation is written in termsof the total enthalpy as

@

@t(�h) +

@

@t(��H) +

@

@xi(�uih) +

@

@xi(�ui�H) =

@

@xi(krT ) + S

(8.2-4)

where � = densityui = uid velocityxi = Cartesian coordinate directionsk = thermal conductivityS = source term

Note that there are two extra terms containing �H on the left-handside. The �rst is an unsteady latent heat content term; the secondresults in latent heat release when there is steady advection, as incontinuous casting.

There are two unknowns in Equation 8.2-4: h and �H (or, al-ternately, the liquid fraction, �). The value of �H depends on thetemperature, as shown by Equation 8.2-3; �H, in turn, changes thetemperature through Equation 8.2-4. Thus, the additional equa-tion required to obtain �H is the �H(T ) relation given in Equa-tion 8.2-3. Note that the �H(T ) relationship is di�erent for di�er-ent alloy systems. Equation 8.2-3 is the \lever" rule.

The solution for temperature is essentially an iteration betweenSolution Procedure

Equations 8.2-3 and 8.2-4. FLUENT uses the iterative procedureoutlined in [137]. First, a guess is made for the temperature andliquid fraction (or the previously-calculated values are used). Thesevalues will, in general, be inconsistent in that both Equations 8.2-3and 8.2-4 will not simultaneously be satis�ed. The solution tech-nique employed is then to alternate between Equations 8.2-3 and8.2-4, manipulating the liquid fraction according to

�Hn+1 = �Hn + �Hcp(T � T �) (8.2-5)


8.2 Phase Change Modeling Theory 8-5

where �Hn+1 = current solution for latent heat content�Hn = latent heat content from previous iteration�H = underrelaxation factorT � = (Tliquidus � Tsolidus) � + Tsolidus

Essentially, Equation 8.2-5 nudges �H in the right direction, mak-ing �H and the temperature consistent. You have control overthe underrelaxation factor, �H , used in Equation 8.2-5. The la-bel for this underrelaxation factor appears as DELH in the EXPERT

UNDERRELAX-1 table or the Underrelaxation panel.

The implementation in the enthalpy-porosity technique in FLUENT

applies Equation 8.2-5 to each control volume many times duringan iteration. The uid ow and temperature are solved over thedomain, assuming a certain value of the liquid fraction. Then sev-eral inner iterations of Equation 8.2-5 are performed for each pointin the domain, and the liquid fraction is updated. An update ofproperties completes one outer iteration.

The procedure described above yields the temperature �eld and theVelocity Treatment

liquid fraction. When the liquid fraction in a control volume is lessthan unity, as in the mushy zone, a large sink is applied to eachof the momentum equations to damp the velocity. The sink in themomentum equations takes the form

(1� �)2

(�3 + �)Amush(ui � up;i) (8.2-6)

where Amush = mushy zone constantup;i = pull velocity� = small number to prevent division by zero

Amush is a constant that measures the amplitude of the damping;values between 104 and 107 are recommended for most computa-tions. The higher this value, the steeper the damping curve be-comes, and the faster ui drops to zero as the material freezes. Verylarge values may cause the solution to oscillate as control volumesalternately freeze and thaw with minor perturbations in liquid frac-tion.

The term up;i is the pull velocity vector and it appears when con-tinuous casting processes are being modeled. The presence of thisterm in Equation 8.2-6 allows newly solidi�ed material to move atthe pull velocity. A description of this feature is given in the nextsection.



8.3 User Inputs for the Phase Change Model

To set up a phase change problem in FLUENT, you must �rst de�nethe problem dimensionality, physical domain size and grid size inthe usual way (i.e., either by using the De�ne Domain panel or thecommands in the DEFINE-DOMAIN menu, or by reading a grid �le).The additional inputs required for de�ning a phase change problemare described below.

You enable the phase change model by setting ACTIVATE PHASEActivating the

Phase Change

Model

CHANGE MODEL to YES in the EXPERT OPTIONS table:

MAIN �! EXPERT �! OPTIONS

(MODELING OPTIONS)

NO ALLOW LINK SETTING

NO ALLOW PROFILE SETTING

NO ALLOW HEAT FLUX BOUNDARY CONDITIONS

NO ALLOW EXTERNAL HEAT TRANSFER WALLS

NO ALLOW WALL CONDUCTION

NO ENABLE CONVECTION IN CONDUCTING WALLS

NO ALLOW HEAT CONDUCTION FOR INLETS

NO INCLUDE EXTERNAL RADIATION BC

NO SET EMISSIVITY FOR INLETS/OUTLET

NO ENABLE NON-NEWTONIAN FLOW MODEL

NO ENABLE POROUS FLOW MODEL

NO ENABLE MOLE FRACTION INPUTS (OTHERWISE MASS FRACTION)

NO ENABLE FAN/RADIATOR MODEL

NO ALLOW FIXED PRESSURE BOUNDARIES

NO ALLOW SETTING FLOW ANGLES FOR PRESSURE-INLETS

NO ENABLE STEADY CORIOLIS FORCE

NO ENABLE MULTIPLE ROTATING REFERENCE FRAMES

NO ENABLE SLIDING MESH CALCULATION

YES ACTIVATE PHASE CHANGE MODELLING

NO ENABLE DEFORMING MESH CALCULATION

ACTION (TOP,DONE,QUIT,REFRESH)

Note that the energy equation must be active (i.e., you must haveenabled calculation of temperature in the Models panel or in theDEFINE-MODELS menu).

The physical constants for the phase change problem are set inPhysical Constants

the SETUP-1 PHYSICAL-CONSTANTS menu. When the phase changemodel is active, you will notice the following additional choicesin this menu: SURFACE-TENSION-GRADIENT, CONTACT-RESISTANCEand PHASE-CHANGE. These options are described below.


8.3 User Inputs for the Phase Change Model 8-7

The menu selection SURFACE-TENSION-GRADIENT de�nes the vari-Surface Tension

Gradient for

Marangoni Flow

ation of surface tension with temperature, d�=dT . This param-eter is used to compute the surface-tension-gradient-driven shear(Marangoni shear) on a boundary. For example, if the temperaturegradient along an arc, s, is dT=ds, the applied shear stress is takento be

� = (d�=dT )dT

ds(8.3-1)

This shear stress is then decomposed into its Cartesian componentsand applied to the momentum equations.

The example table below shows the input of the surface tensiongradient

@�

@T= �3:6� 10�4 (8.3-2)

COMMANDS AVAILABLE FROM PHYSICAL-CONSTANTS:

DENSITY VISCOSITY

THERMAL-CONDUCTIVITY CP-SPECIFIC-HEAT

OPERATING-PRESSURE SURFACE-TENSION-GRADIENT

CONTACT-RESISTANCE PHASE-CHANGE

QUIT HELP

ENTER HELP (COMMAND) FOR MORE INFORMATION.

(PHYSICAL-CONSTANTS)-

STG

(R)- ENTER SURFACE-TENSION GRADIENT

(R)- UNITS= N/M/K ++(DEFAULT 0.0000E+00)++

-3.6E-04

The Marangoni shear boundary condition may be applied to anybounding face of the computational domain, both for external bound-aries and interior regions. However, the face must be a free surface(i.e., a Z-WALL with slip|see Z-WALL boundary conditions below).

The menu selection CONTACT-RESISTANCE de�nes an additional heatContact Resistance

transfer resistance between walls and cells with liquid fraction lessthan unity. This accounts for the presence of an air gap betweenthe walls and the frozen solid. Thus, the wall heat ux, as shownin Figure 8.3.1, is written as

q =(TI � Tw)

(l=k +Rc)(8.3-3)



Note that the contact resistance, Rc, has the same units as theinverse of the heat transfer coe�cient. The same contact resistancewill be applied at all W-WALLS and no-slip Z-WALLS that are adjacentto cells with liquid fraction less than unity.

●TW TI

l

● ● ●

l / k

TW TI

RC

Near-wallControl Volume

Wall

Figure 8.3.1: Circuit for Contact Resistance



The phase change properties for the melting/solidi�cation problemPhase Change

Properties are speci�ed through the PHASE-CHANGE command:

MAIN �! SETUP-1 �! PHYSICAL-CONSTANTS �! PHASE-CHANGE

FLUENT �rst asks you to de�ne the SOLIDUS TEMPERATURE, Tsolidusin Equation 8.2-3:

(R)- ENTER SOLIDUS TEMPERATURE

(R)- UNITS= K ++(DEFAULT 1.1000E+03)++

X

(R)- DEFAULT ASSUMED

Next, de�ne the LIQUIDUS TEMPERATURE, Tliquidus in Equation 8.2-3:

(R)- ENTER LIQUIDUS TEMPERATURE

(R)- UNITS= K ++(DEFAULT 1.2000E+03)++

X


Then specify the LATENT HEAT OF FREEZING for the material, L inEquation 8.2-3:

(R)- ENTER LATENT HEAT OF FREEZING

(R)- UNITS= J/KG ++(DEFAULT 1.0000E+05)++

X


Finally, enter the MUSHY ZONE CONSTANT, Amush in Equation 8.2-6,which determines the damping of velocities in the mushy zone.

(R)- ENTER MUSHY ZONE CONSTANT

(R)- UNITS= KG/M3/S ++(DEFAULT 1.0000E+04)++

X


One important concept to note is the de�nition and use of theEnhanced

Boundary

Conditions

Z-WALL cells. When the phase change model is enabled, these are nolonger ordinary wall cells like W-WALL cells, but are used for speci-fying the enhanced boundary conditions: applied shear, mixed heattransfer and Marangoni (surface tension gradient) shear boundary



conditions. The slip condition is the default at these Z-WALL cells.(To set boundary conditions at Z-WALLs in phase change problems,you must use the text interface; phase change boundary conditionsat Z-WALLs cannot be set through the GUI.) The dialog below il-lustrates the use of these enhanced boundary conditions for a wall,Z1. The available options are discussed in further detail below.

MAIN �! SETUP-1 �! BOUNDARY-CONDITIONS

(SETUP1)-

BC Z 1

(L)- CUT VELOCITY LINKS TO MUSHY CELLS ?

(L)- Y OR N ++(DEFAULT-YES)++

YES

(L)- ACTIVATE NO-SLIP CONDITION ?

(L)- Y OR N ++(DEFAULT-NO)++

NO

(L)- ACTIVATE SPECIFIED-TEMPERATURE BOUNDARY ?


NO

COMMANDS AVAILABLE FROM Z1-ZONE-BOUNDARY-CONDITIONS:

Z-WALL-BC QUIT HELP


(Z1-ZONE-BOUNDARY-CONDITIONS)-

In this initial dialog, you are asked three questions:

CUT VELOCITY LINKS TO MUSHY CELLS ? | The response to thisquestion determines how the wall in uences the motion of ad-jacent solid cells and mushy cells. The default response is YES;that is, the wall is frictionless. An answer of NO will cause thevelocity of the solidi�ed material to be retarded/acceleratedby the wall velocity.

ACTIVATE NO-SLIP CONDITION ? | The default (NO) is a slip con-dition, which allows you to specify a free surface if desired. Ananswer of YES will allow you to input Cartesian velocity com-ponents at the wall.

ACTIVATE SPECIFIED-TEMPERATURE BOUNDARY ? |The default isNO which enables the mixed (combined convection, radiationand speci�ed heat ux) heat transfer boundary condition. Ananswer of YES will allow you to specify a �xed temperatureat the boundary. In this case, the only option available in



the Z-WALL-BC command (see below) will be applied shear,and the �xed temperature will be requested in the Z-WALL

ZONE-BOUNDARY-CONDITIONS menu.

The mixed heat transfer boundary condition is governed by theMixed Heat

Transfer and Shear

Stress Boundary

Conditions at

Z-WALLS

following equation:

qnet = hc(T � Tenv) + ��(T 4� T 4

rad)� qapplied (8.3-4)

where qnet is the total heat ux at the boundary. You may \toggle"on and o� the various contributions to the right side of this equationby specifying nonzero or zero values for the convection heat trans-fer coe�cient hc, the wall emissivity �, and the applied heat uxqapplied. In the following dialog, only the constants for convectionheat transfer at the boundary are speci�ed. In addition, the appliedshear at the boundary is set to zero (default).


Z-WALL-BC

(R)- HEAT TRANS. COEFF.

(R)- UNITS= WATTS/M.SQ-K ++(DEFAULT 0.0000E+00)++

1.

(R)- CONVECTIVE ENV. TEMP.

(R)- UNITS= KELVIN ++(DEFAULT 0.0000E+00)++

295.

(R)- EMISSIVITY

(R)- UNITS= DIMENSIONLESS ++(DEFAULT 0.0000E+00)++

0.

(R)- RADIATIVE ENV. TEMP.

(R)- UNITS= KELVIN ++(DEFAULT 0.0000E+00)++

0.

(R)- HEAT FLUX

(R)- UNITS= WATTS/M.SQ. ++(DEFAULT 0.0000E+00)++

0.

(R)- SHEAR-STRESS

(R)- UNITS= PASCALS ++(DEFAULT 0.0000E+00)++

0.

COMMANDS AVAILABLE FROM Z1-ZONE-BOUNDARY-CONDITIONS:

Z-WALL-BC QUIT HELP



Note that the enhanced boundary conditions can also be speci�edas piecewise linear or polynomial pro�les. This option is enabled



in the usual manner, by setting ALLOW PROFILE SETTING to YES inthe EXPERT OPTIONS menu. The independent variable may be anyof the coordinate directions.

The term @@xi

(�ui�H) in Equation 8.2-4 can be included in FLU-Modeling

Continuous Casting

Using the Pull

Velocity

ENT's phase change model. It is possible, therefore, to computephase change in continuous casting applications, where a solid re-gion is \pulled" out of the computational domain, and pulls newlysolidi�ed material with it. The term @

@xi

(�ui�H) should be acti-vated by you when your model includes pull velocities. This termis activated by answering YES to the following question:

EXPERT �!PHASE-CHANGE-OPTIONS

(L)- TURN ON THE DEL.(RHO U DELH) TERM IN THE H-EQN?


FLUENT computes the pull velocities using a Laplace equation (de-tails are given below). By default, the boundary conditions are suchthat the solid will slide past all walls and will be pulled with a ve-locity equal to the inlet velocity to which the solid is attached. Youmay choose to attach the solid to a wall by not cutting the links tomushy cells.

The pull rate may be speci�ed by creating an inlet where the tem-perature is below the solidus temperature. The inlet may havea normal velocity pointing into or out of the domain (i.e., either\pushing" or \pulling" is possible). Multiple pull rates may bespeci�ed by using more than one such inlet. Rotation of the pulledmaterial can also be included by input of an angular velocity de�n-ing the rotation.

To determine which pull rate is to be associated with which newly-solidi�ed material and with what velocity the solid should move, aLaplace equation is solved for the pull velocity:

r � (�rup;i) = 0 (8.3-5)

The coe�cient � is similar to viscosity and determines whether thesolid slips along a boundary. As mentioned above, by default allwalls are slip; therefore, � is 0. At inlets, the default is no-slip andFLUENT sets � to 1. Thus, it is possible to pull the solid at a pullvelocity, V , as shown in Figure 8.3.2, without having it retarded bythe stationary walls.



Mushy Zone Solid

Wall

V

Figure 8.3.2: \Pulling" a Solid in Continuous Casting

Equation 8.3-5 is a vector equation; the pull rate may have all threecomponents. Thus, three Laplace equations are solved in three di-mensions. These equations are not underrelaxed. They are solvedusing the line-by-line solver in FLUENT. To activate the solutionof the pull velocities, you must set a nonzero number of sweeps inthe SWEEPS-OF-LGS-SOLVER menu (or in the Line Gauss Parameters

panel):

MAIN �! EXPERT �! LINEAR-EQN.-SOLVER �! SWEEPS-OF-LGS-

SOLVER

Solve �! Controls �!Line Gauss...



(LINEAR-EQN.-SOLVER)-

SW

(NUMBER OF SWEEPS/SWEEP DIRECTION)

5 PRESSURE-CORRECTION

1 U-VELOCITY

1 V-VELOCITY

10 LATENT-HEAT-ITERATIONS

0 PULL-U-VELOCITY

0 PULL-V-VELOCITY

1 ENTHALPY

1 SWEEP DIRECTION ( I=1, J=2 )

YES ALTERNATE SWEEP DIRECTION


When the number of sweeps is set equal to zero, FLUENT will notcalculate the corresponding component of the pull velocity vector.In this case, you may specify the desired value of the pull vector us-ing the PATCH command. (The pull velocities are PULL-U-VELOCITY,PULL-V-VELOCITY and PULL-W-VELOCITY.) Angular velocity can alsobe patched.

It is possible to specify additional sources of momentum and heatAdditional Sources

of Heat and

Momentum

within the domain. For example, the simulation of a plasma melt-ing process may require the speci�cation of a force �eld and heatgeneration in the domain. Another example is the use of magneticforce �elds to damp the velocities in crystal growth processes.

You can specify additional momentum sources in each control vol-Momentum

Sources ume for each momentum equation by patching the values SOURCE-X-MOM, SOURCE-Y-MOM and SOURCE-Z-MOM. The patched values remainin e�ect throughout the simulation. These momentum sources havethe units force/volume. Thus, patching of these quantities is to bedistinguished from patching of the absolute (force) quantities usingEXCHANGE-X, EXCHANGE-Y, and EXCHANGE-Z.

You can specify an additional energy source in each control volumeEnergy Source

by patching the values of SOURCE-HEAT. The patched values remainin e�ect throughout the simulation. The sign convention is that apositive value of SOURCE-HEAT represents heat added to the uid.This source has the units energy/volume-time and so is analogous topatching of EXCHANGE-HEAT. (Note the di�erence in sign convention,however.)


8.4 Solution Strategies for Phase Change Problems 8-15

8.4 Solution Strategies for Phase Change Problems

In phase change problems, the momentum, energy, and liquid frac-tion equations are tightly coupled, presenting a very nonlinear setof equations to be solved. In most cases, a transient solution ap-proach is recommended, even if the desired solution is steady-state,but phase change problems can also be simulated as steady in FLU-

ENT.

Specifying proper initial conditions is important for phase changeproblems, most importantly for the temperature �eld. In manycases, simply patching a uniform temperature �eld may su�ce. Inmore complex cases, it may be easier to solve a steady conductionproblem in the domain. To do this, turn o� time dependence (inthe Models panel or EXPERT menu) if the phase change problem hasbeen de�ned as unsteady. In the EXPERT SELECT-VARIABLES tableor the Select Equations panel, turn o� the solution of all quantitiesexcept PROPERTIES/TEMPERATURE and ENTHALPY. Set the underre-laxation factors for temperature and enthalpy to 1, and the steadytemperature �eld can be obtained quickly. Now, the coupled uid ow and heat transfer problem can be computed.

Many phase change problems involve unsteady natural convectionin enclosed cavities. When solving such a problem, FLUENT mayencounter di�culty in reducing the normalized pressure residual tothe convergence criterion. This is because the pressure residual isnormalized by the residual (or continuity imbalance) at the seconditeration. For an unsteady ow in a closed cavity (no inlet/outlet),the pressure residual at the second iteration is often very small. Thisin turn causes the normalized pressure residual to appear quite largeand may prevent FLUENT from reaching the convergence criterion.In this case, you can gauge convergence at each time step by check-ing the residuals for the momentum and enthalpy equations. Youshould then periodically check the unnormalized pressure residualfor convergence. To automate the time-stepping procedure, you canset the number of iterations per time step to a value that is adequatefor the convergence of the momentum and energy equations.

Care should be taken when de�ning the reference temperature for!enthalpy. Potentially misleading residuals for the enthalpy equationcan arise if the reference temperature is the same as the initial tem-perature in the domain. If the two temperatures are the same, theenthalpy is zero (or very small) everywhere in the domain. Sincethe enthalpy residual is normalized by the sum of APhP (see Sec-tion 16.3), it can be quite large. In this case, de�ne the reference



temperature for enthalpy to be a value di�erent from the initialtemperature of the problem.

For pure melting problems (i.e., Tliquidus = Tsolidus), the melt frontshould be one cell thick (i.e., the liquid fraction should be between0 and 1 only in a line of single cells representing the position of themelt front). If the melt front is not one cell thick, it indicates thatthe solution of the liquid fraction (Equation 8.2-5) is not converged.The number of sweeps for LATENT-HEAT-ITERATIONS should there-fore be increased in the SWEEPS-OF-LGS-SOLVER table (or in theLine Gauss Parameters panel):



1 U-VELOCITY

1 V-VELOCITY

1 W-VELOCITY

10 LATENT-HEAT-ITERATIONS

20 PULL-U-VELOCITY

20 PULL-V-VELOCITY

20 PULL-W-VELOCITY

5 ENTHALPY




The suggested value for the underrelaxation factor for Equation 8.2-5(and the enthalpy equation) is 0.5.


Chapter 9. The Eulerian Multiphase Model

The Eulerian multiphase model in FLUENT allows for the modelingof multiple separate, yet interacting phases. The phases can be liq-uids, gases, or solids in nearly any combination. An Eulerian treat-ment is used for each phase, in contrast to the Eulerian-Lagrangiantreatment that is used for the dispersed phase model.

Information about the Eulerian multiphase model is divided intothe following sections:

� Section 9.1: Overview of Multiphase Modeling in FLUENT

� Section 9.2: Features of the Eulerian Multiphase Model

� Section 9.3: Examples of Eulerian Multiphase Capabilities

� Section 9.4: Eulerian Multiphase Theory

� Section 9.5: Solution Method in FLUENT

� Section 9.6: Eulerian Multiphase Flow Modeling Strategies

� Section 9.7: Using the Eulerian Multiphase Model in FLUENT

� Section 9.8: Postprocessing Eulerian Multiphase Results

9.1 Overview of Multiphase Modeling in FLUENT

A large number of ows encountered in nature and technology area mixture of phases. Physical phases of matter are gas, liquid, andsolid, but the concept of phase in a multiphase ow system is appliedin a broader sense. In multiphase ow, a phase can be de�ned as anidenti�able class of material that has a particular inertial responseto and interaction with the ow and the potential �eld in which itis immersed. For example, di�erent-sized solid particles of the samematerial can be treated as di�erent phases because each collection ofparticles with the same size will have a similar dynamical responseto the ow �eld.

9-2 Chapter 9 | The Eulerian Multiphase Model

Some examples of complex multiphase systems are listed below:Examples ofMultiphase

Systems � Fluid- uid multiphase systems

{ Gas-liquid droplet systems:atomizers, scrubbers, dryers, absorbers, combustors, gascooling, evaporation, cryogenic pumping

{ Liquid-gas bubble systems:absorbers, evaporators, scrubbers, air lift pumps, cavita-tion, otation, aeration, nuclear reactors

{ Liquid-gas systems:boiling and nuclear reactor safety, surface waves of airover water

{ Liquid-liquid systems:extraction, emulsi�cation, separators, homogenization

� Fluid-solid multiphase systems

{ Gas-solid particle systems:cyclones, pneumatic conveyors, dust collectors, uidizedbed reactors, circulating bed reactors, uid catalytic crack-ing, metallized propellant rockets, xerography, air classi-�ers, biomedical and physiochemical uid systems, silos,dust-laden environmental ows

{ Liquid-solid particle systems:slurry transport, sedimentation, otation, suspension, andsome powder milling

9.1.1 Approaches to Multiphase Modeling

Advances in computational uid mechanics have provided the basisfor further insight into the dynamics of multiphase ows. Currentlythere are two approaches for the numerical calculation of multiphase ows: the Euler-Lagrange approach and the Euler-Euler approach.

The Lagrangian dispersed phase model in FLUENT (see Chapter 11)The LagrangianDispersed Phase

Modelfollows the Euler-Lagrange approach. The uid phase is treated asa continuum by solving the time-averaged Navier-Stokes equations,while the dispersed phase is solved by tracking a large number ofparticles, bubbles, or droplets through the calculated ow �eld. Thedispersed phase can exchange momentum, mass, and energy withthe uid phase. A fundamental assumption made in this modelis that the dispersed second phase occupies a low volume fraction,even though high mass loading ( _mparticles � _m uid) is acceptable.


9.1 Overview of Multiphase Modeling in FLUENT 9-3

The particle or droplet trajectories are computed individually atspeci�ed intervals during the uid phase calculation. This makes themodel appropriate for the modeling of spray dryers, coal and liquidfuel combustion, and some particle-laden ows, but inappropriatefor the modeling of liquid-liquid mixtures, uidized beds, or anyapplication where the volume fraction of the second phase is notnegligible.

In the Euler-Euler approach, the di�erent phases are treated math-The Euler-EulerApproach ematically as interpenetrating continua. Since the volume of a phase

cannot be occupied by the other phases, the concept of phasic vol-ume fraction is introduced. These volume fractions are assumed tobe continuous functions of space and time and their sum is equalto one. Conservation equations for each phase are derived to ob-tain a set of equations which have similar structure for all phases.These equations are closed by providing constitutive relations whichare obtained from empirical information, or, in the case of granular ows, by application of kinetic theory.

In FLUENT, two di�erent Euler-Euler multiphase models are avail-able: the VOF (Volume of Fluid) Model and the Eulerian Multi-phase Model.

In the VOF model (described in Chapter 10) a single set of trans-The VOF Modelport equations is solved, and the phases do not mix. The interfacebetween phases is de�ned by solution of a transport equation anddi�usion across the interface is prevented. Applications of the VOFmodel include strati�ed ows, free-surface ows, �lling, sloshing,and jet breakup (surface tension).

The Eulerian multiphase model solves a set of n momentum, en-The EulerianMultiphase Model thalpy, continuity andm species for each phase. Coupling is achieved

through the pressure and interphase exchange coe�cients. Themanner in which this coupling is handled depends upon the typeof phases involved; granular ( uid-solid) ows are handled di�er-ently than nongranular ( uid- uid) ows. For granular ows, theproperties are obtained from application of kinetic theory. Momen-tum exchange between the phases is also dependent upon the typeof mixture being modeled. FLUENT's user-de�ned subroutine ca-pability allows customization of the calculation of the momentumexchange and heat exchange.



9.2 Features of the Eulerian Multiphase Model

With the Eulerian multiphase model, the number of secondary phasesis limited only by memory requirements and convergence behavior.Any number of secondary phases can be modeled, provided that suf-�cient memory is available. For complex multiphase ows, however,you may �nd that your solution is limited by convergence behavior.See Section 9.6 for multiphase modeling strategies.

If all of the phases are uids (liquids, gases, or vapors), the owFluid-Fluid Flowsis characterized as uid- uid. For this class of ows, the FLUENTsolution is based on the following:

� A single pressure is shared by all phases.

� Momentum, enthalpy and continuity equations are solved foreach phase.

� The gas law can be used for the primary phase.

� Temperature-dependent properties are available for all phases.

� A simple interphase drag coe�cient is obtained for sphericaldroplets or bubbles in a bubbly regime. (You can modify theinterphase drag coe�cient through user-de�ned subroutines(see Chapter 20).)

� m species can be solved for each phase.

� Homogeneous reactions are allowed for each phase.

� Mass transfer is allowed between the phases.

� The k-� turbulence model is available for each phase.

If one phase is a uid and one or more of the phases is a solid,Granular(Fluid-Solid) Flows the ow is characterized as granular. For this class of ows, the

FLUENT solution is based on the following:

� The uid is represented by the primary phase.

� All secondary phases are solids (i.e., you cannot model a gran-ular ow that involves more than one uid phase).

� The uid pressure �eld is shared by all phases.

� A solids pressure �eld is calculated for each solid phase.


9.2 Features of the Eulerian Multiphase Model 9-5

� Momentum, enthalpy, and continuity equations are solved foreach phase.

� The gas law can be used for the primary phase.

� Granular temperature (solids uctuating energy) can be cal-culated for each solid phase.

� Solid-phase shear and bulk viscosities are obtained from ap-plication of kinetic theory to granular ows.

� Simple and uid-solid (granular) interphase drag coe�cientsare used for spherical particles. (You can modify the inter-phase drag coe�cient through user-de�ned subroutines (seeChapter 20).)

� m species can be solved for each phase.

� Homogeneous reactions are allowed for each phase.

� Mass transfer is allowed between the phases.

� The k-� turbulence model is available for each phase.

All other features available in FLUENT can be used in conjunctionLimitationswith the Eulerian multiphase model, except for the following limi-tations:

� The RNG k-� and Reynolds Stress models for turbulence can-not be used.

� Particle tracking (using the Lagrangian dispersed phase model)interacts only with the primary phase.

� Speci�ed periodic mass ow is not allowed.

� Compressible ow is not allowed.

� The Eulerian multiphase model cannot be used with slidingor deforming meshes.

� Phase change (melting, freezing) is not allowed.

� Heterogeneous reactions (i.e., reactions between phases) arenot allowed, except via user-de�ned subroutines.



The process of solving a multiphase system is inherently di�cult,Stability andConvergence and you may encounter stability or convergence di�culties. In gen-

eral, mixtures involving phases with large density di�erences (e.g.,water and air) may require more computational e�ort than mixtureswhere the densities are closer in magnitude. There are physical lim-itations when the volume fraction of a secondary phase increases toits maximum possible value (i.e., the packing limit for a solid phaseor 1.0 for a uid phase). In such cases the momentum exchangelaw may not be valid and convergence problems may appear. Strat-i�ed ows of immiscible uids can be solved more e�ciently withthe VOF model (see Chapter 10). Some problems involving smallvolume fractions can be solved more e�ciently with the Lagrangiandispersed phase model (see Chapter 11). Granular ows, solved ina transient manner, are usually stable and well behaved, whether incombination with a liquid or a gas. Many stability and convergenceproblems can be minimized if care is taken during the setup andsolution processes (see Section 9.6).

9.3 Examples of Eulerian Multiphase Capabilities

Below are two examples of problems that have been solved usingFLUENT. Brief problem descriptions are followed by �gures illus-trating the results. A tutorial that illustrates the setup and solutionprocess in more detail is included in the FLUENT Tutorial Guide.

9.3.1 Air-Water Separation in a Tee-Junction

In the following example, an air-water mixture ows upwards in aduct, then splits at a tee-junction. This problem is similar to thework of Issa and Oliveira [51]. In this 2D simulation, the ducts are25 mm in width. The inlet section of the duct is 125 mm long, andthe top and side ducts are 250 mm long. There are 3200 live controlvolumes, with 20 control volumes spanning each of the three arms.Figures 9.3.1 and 9.3.2 show the geometry outline and a close-upof the grid in the vicinity of the junction. The water and air areassigned densities of 1000 and 1.2 kg/m3, respectively. The airis assumed to form bubbles with an average diameter of 1 mm.The viscosities of water and air are 9 � 10�4 and 2 � 10�5 kg/m-s, respectively. The incoming mixture is composed of 2% air byvolume.

Most of the water (80%) is removed through the side arm of thejunction. Velocity vectors for water are shown in Figure 9.3.3, andcontours of stream function for air are shown in Figure 9.3.4. The


9.3 Examples of Eulerian Multiphase Capabilities 9-7

lower momentum of the air causes it to collect in the recirculatingregion just downstream of the junction. Figure 9.3.5 shows thevolume fraction of air in the duct.

Outline

Air-Water Separation in a Tee-Junction

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Figure 9.3.1: Outline of the Tee-Junction Geometry



Grid (91 X 91)


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Figure 9.3.2: Grid for the Tee-Junction, Showing the Junction Area

3.53E-03

9.02E-02

1.77E-01 2.64E-01

3.50E-01

4.37E-01

5.23E-01 6.10E-01

6.97E-01

7.83E-01 8.70E-01

9.57E-01

1.04E+00 1.13E+00

1.22E+00

1.30E+00

1.39E+00 1.48E+00

1.56E+00

1.65E+00 1.74E+00

1.82E+00

1.91E+00

2.00E+00 2.08E+00

2.17E+00

2.26E+00 2.34E+00

2.43E+00

2.52E+00

Max = 2.517E+00 Min = 3.533E-03

Water Velocity Vectors (M/S)


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Figure 9.3.3: Water Velocity Vectors in the Tee-Junction



-3.55E-03-3.43E-03-3.31E-03-3.20E-03-3.08E-03-2.96E-03-2.84E-03-2.72E-03-2.60E-03-2.48E-03-2.37E-03-2.25E-03-2.13E-03-2.01E-03-1.89E-03-1.77E-03-1.66E-03-1.54E-03-1.42E-03-1.30E-03-1.18E-03-1.06E-03-9.45E-04-8.26E-04-7.08E-04-5.89E-04-4.71E-04-3.52E-04-2.34E-04-1.15E-04 3.22E-06

Max = 3.222E-06 Min = -3.551E-03Air Stream Function (M2/S)Air-Water Separation in a Tee-Junction

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Figure 9.3.4: Contours of Stream Function for Air in the Tee-Junction

1.00E-06

3.29E-02

6.57E-02

9.86E-02

1.31E-01

1.64E-01

1.97E-01

2.30E-01

2.63E-01

2.96E-01

3.29E-01

3.61E-01

3.94E-01

4.27E-01

4.60E-01

4.93E-01

5.26E-01

5.59E-01

5.92E-01

6.24E-01

6.57E-01

6.90E-01

7.23E-01

7.56E-01

7.89E-01

8.22E-01

8.54E-01

8.87E-01

9.20E-01

9.53E-01

9.86E-01

Max = 9.859E-01 Min = 1.000E-06

Air Volume Fraction (Dim)


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Figure 9.3.5: Air Volume Fraction in the Tee-Junction



9.3.2 Bubbly Flow in a Fluidized Bed

The uidization of an initially stationary granular bed is modeled asa time-dependent problem in this example, which follows work doneby Gidaspow and Ettehadieh [38] and Gidaspow, Lin, and Seo [39].A rectangular domain, 0.4 m wide and 0.6 m high is modeled witha 41 � 41 grid (42 � 42 cells). The region contains air at a densityof 1.2 kg/m3, and is �lled halfway with a granular bed. The averageparticle diameter in the bed is 5 � 10�4 m, and the material densityis 2610 kg/m3. The air viscosity is 1.7 � 10�5 kg/m-sec. The gridand the initial con�guration of the bed are shown in Figure 9.3.6.

1.00E-06

3.00E-01

6.00E-01

Lmax = 5.999E-01 Lmin = 1.000E-06 Time = 0.000E+00Grid (left) and Solids Volume Fraction (right)Bubbly Flow in a Fluidized Bed


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Figure 9.3.6: Grid and Initial Bed Con�guration (in white)

At t = 0, a uniform vertical air ow of 0.284 m/s passes from thelower boundary through the bed. At t > 0, a narrow, vertical jet ofair, 0.03 m wide, is injected at the center of the bottom boundary.The velocity of the jet is 3.55 m/sec. A laminar solution is per-formed, using a time step of 10�3 sec. As the jet begins to penetratethe bed, it displaces the solids. This is illustrated in Figures 9.3.7aand 9.3.7b, where the air and solids velocities, respectively, are dis-played at t = 0:1 sec.



3.07E-01

4.26E-01

5.44E-01 6.62E-01

7.80E-01

8.98E-01

1.02E+00 1.13E+00

1.25E+00

1.37E+00 1.49E+00

1.61E+00

1.73E+00 1.84E+00

1.96E+00

2.08E+00

2.20E+00 2.32E+00

2.43E+00

2.55E+00 2.67E+00

2.79E+00

2.91E+00

3.02E+00 3.14E+00

3.26E+00

3.38E+00 3.50E+00

3.62E+00

3.73E+00

Lmax = 3.734E+00 Lmin = 3.074E-01 Time = 1.000E-01

Air Velocity Vectors (M/S)

Bubbly Flow in a Fluidized Bed

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(a) Air Velocity Vectors

0.00E+00

1.80E-02

3.59E-02 5.39E-02

7.18E-02

8.98E-02

1.08E-01 1.26E-01

1.44E-01

1.62E-01 1.80E-01

1.98E-01

2.15E-01 2.33E-01

2.51E-01

2.69E-01

2.87E-01 3.05E-01

3.23E-01

3.41E-01 3.59E-01

3.77E-01

3.95E-01

4.13E-01 4.31E-01

4.49E-01

4.67E-01 4.85E-01

5.03E-01

5.21E-01

Lmax = 5.208E-01 Lmin = 0.000E+00 Time = 1.000E-01

Solids Velocity Vectors (M/S)


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(b) Solids Velocity Vectors

Figure 9.3.7: Velocity Vectors for Bubbly Flow in a Fluidized Bed



As time passes, the air jet forms a bubble that rises through the bed,as illustrated in Figure 9.3.8, where the development of the bubblecan be seen. As the bubble rises in the bed, the distortion of theupper surface of the bed is evident. Also evident is the formationof the second bubble behind the �rst in the lower left frame.

4.00E-01

4.20E-01

4.40E-01

4.60E-01

4.80E-01

5.00E-01

5.20E-01

5.40E-01

5.60E-01

5.80E-01

6.00E-01

6.20E-01

6.40E-01

6.60E-01

6.80E-01

7.00E-01

7.20E-01

7.40E-01

7.60E-01

7.80E-01

8.00E-01

8.20E-01

8.40E-01

8.60E-01

8.80E-01

9.00E-01

9.20E-01

9.40E-01

9.60E-01

9.80E-01

1.00E+00

Lmax = 1.000E+00 Lmin = 4.001E-01


Bubble formation at (clockwise from upper left) t=0.05,0.1,0.2, and 0.3 sec

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Figure 9.3.8: Bubble Formation in a Fluidized Bed

The e�ect of the bed on the air jet is evidenced by the time ittakes for the top of the bubble to reach the top of the bed (roughly0.3 sec). An unimpeded uid element in the air jet would take(0.3 m)/(3.55 m/sec) or 0.0845 seconds to reach the same height.The work required to displace the solids in the bed slows the air jet,which in turn leads to the spreading of the jet which can be seen inFigure 9.3.8.

9.4 Theory

To change from a single-phase model, where a single set of con-servation equations for momentum and continuity is solved, to amultiphase model, additional sets of conservation equations mustbe introduced. In the process of introducing additional sets of con-servation equations, the original set must also be modi�ed. Themodi�cations involve, among other things, the introduction of the


9.4 Theory 9-13

volume fractions �1; �2; : : : �n for the multiple phases, as well as amechanism for the exchange of momentum between the phases.

9.4.1 Volume Fractions

The description of multiphase ow as interpenetrating continua in-corporates the concept of phasic volume fractions, denoted here by�q. Volume fractions represent the space occupied by each phase.The laws of conservation of mass, momentum, and energy are satis-�ed by each phase individually. The derivation of the conservationequations can be done by ensemble averaging the local instanta-neous balance for each of the phases [3] or by using the mixturetheory approach [10].

The volume of phase q, Vq, is de�ned by

Vq =ZV�qdV (9.4-1)

where

nXq=1

�q = 1 (9.4-2)

The e�ective density of the phase q is

�̂q = �q�q (9.4-3)

where �q is the physical density of phase q, which, in the case of agas, can obey the local gas law.

9.4.2 Conservation Equations

The conservation equations from which the equations solved byFLUENT for uid- uid and granular ows are derived are presentedin this section in general form. Below, in Sections 9.4.3 and 9.4.4,the equations solved by FLUENT will be explicitly presented.

The continuity equation for phase q isConservation ofMass

@

@t�q�q +r � �q�q~uq =

nXp=1

_mpq (9.4-4)



where ~uq is the velocity of phase q and _mpq characterizes the masstransfer from the pth to qth phase. From the mass conservation onecan obtain

_mpq = � _mqp (9.4-5)

and _mpp = 0

The momentum balance for phase q yieldsConservation ofMomentum

@

@t(�q�q~uq) +r � (�q�q~uq ~uq) = ��qrp+r � �� q + �q�q ~Fq +

nXp=1

(~Rpq + _mpq~upq) (9.4-6)

where �� q is the qth phase stress-strain tensor

�� q = 2�q�q��Sq + �q(�q � 2

3�q)r � ~uq��I (9.4-7)

and��Sq =

1

2(r~uq + (r~uq)T ) (9.4-8)

Here �q and �q are the shear and bulk viscosity of phase q, ~Fq is an

external body force, ~Rpq is an interaction force between phases, is a dyadic product, and p is the pressure shared by all phases.

~upq is the interphase velocity, de�ned as follows. If _mpq > 0 (i.e.,phase p mass is being transferred to phase q), ~upq = ~up; if _mpq < 0(i.e., phase q mass is being transferred to phase p), ~upq = ~uq; and~upq = ~uqp.

Equation 9.4-6 must be closed with appropriate expressions for theinterphase force ~Rpq. This force depends on the friction, pressure,cohesion, and other e�ects, and is subject to the conditions that~Rpq = �~Rqp and ~Rqq = 0.

FLUENT uses a simple interaction term of the form

nXp=1

~Rpq =nX

p=1

Kpq(~up � ~uq) (9.4-9)


9.4 Theory 9-15

whereKpq (= Kqp) is the interphase momentum exchange coe�cient(described in Sections 9.4.3 and 9.4.4).

To describe the conservation of energy in multiphase applications,Conservation ofEnergy a separate enthalpy equation can be written for each phase:

@

@t(�q�qhq) +r � (�q�q~uqhq) = �q

dpqdt

+ �� q : r~uq �r � ~qq + Sq

+nX

p=1

(Qpq + _mpqhpq) (9.4-10)

where hq is the speci�c enthalpy of the qth phase, ~qq is the heat ux,

Sq is a source term that includes sources of enthalpy (e.g., due tochemical reaction or radiation),Qpq is the intensity of heat exchangebetween the pth and qth phases, and hpq is the interphase enthalpy(e.g., the enthalpy of the vapor at the temperature of the droplets,in the case of evaporation).

The heat exchange between phases must comply with the local bal-ance conditions Qpq = �Qqp and Qqq = 0.

The closure of the equation set is completed by conservation equa-Conservation of theith Species in the

qth Phasetions for the prediction of the mass fraction of the ith species in theqth phase:

@

@t(�q�qm

iq) +r � (�q�q~uqm

iq) = r � (�q�qD

iqrmi

q) + _miq (9.4-11)

where miq is the mass fraction of the ith species in the qth phase, Di

q

is the di�usivity of the ith species in the mixture of the qth phase,and _mi

q is the rate of production/destruction of the ith species. Inaddition, an equation of state for the uid mixture and expressionsfor the thermodynamic properties are needed.

9.4.3 Description of Fluid-Fluid Multiphase Flow

The equations for uid- uid multiphase ow, as implemented inFLUENT, are presented here for the general case of an n-phase ow.

The Momentum Equations

The conservation of momentum for a uid phase q is



@

@t(�q�q~uq) +r � (�q�q~uq ~uq) = ��qrp +r � �� q + �q�q~g +

nXp=1

(Kpq(~up � ~uq) + _mpq~upq) +

~Fq (9.4-12)

Here ~g is the acceleration due to gravity and ~Fq represents additionalmomentum sources. The stress-strain tensor (�� q)ij in componentform is given by

�q;ij = �q�q(@uq;i@xj

+@uq;j@xi

)� 2

3�q�q�ij

@uq;l@xl

The momentum equations are subject to the constraint that theThe VolumeFraction volume fractions add to unity:

nXq=1

�q = 1 (9.4-13)

The volume fraction of each phase is calculated from a continuityequation:

@

@t(�qrq) +r � (�qrq~uq) =

1

�q;ref

nXp=1

_mpq (9.4-14)

where rq = �q=�q;ref and �q;ref is a constant reference density. Thesolution of this equation for each secondary phase, along with thecondition given by Equation 9.4-13, allows for the calculation ofthe primary-phase volume fraction. This treatment is common to uid- uid and granular ows.

It can be seen in Equation 9.4-12 that momentum exchange betweenThe ExchangeCoe�cient the phases is based on the value of the exchange coe�cient, Kpq. For

uid- uid ows, each secondary phase is assumed to form dropletsor bubbles. This has an impact on how each of the uids is assignedto a particular phase. For example, in ows where there are unequalamounts of two uids, the predominant uid should be modeledas the primary uid, since the sparser uid is more likely to formdroplets or bubbles. The exchange coe�cient for these types ofbubbly liquid-liquid or gas-liquid mixtures is [13]:


9.4 Theory 9-17

Kpq =3

4CD

�p�qj~up � ~uqjdp

(9.4-15)

where �q is the density of the primary phase q, dp is the droplet orbubble diameter of the secondary phase p, j~up � ~uqj is the relativephase velocity, and CD is a drag function based on the relativeReynolds number.

The drag function is obtained from

CD =24

Re

�1 + 0:15Re0:687

�(9.4-16)

for Re � 1000, and CD = 0:44 for Re > 1000, where the relativeReynolds number for the primary phase q and secondary phase p isobtained from

Re =�qj~up � ~uqjdp

�q(9.4-17)

The exchange coe�cient for secondary phases p and r is assumedto be symmetric, with the following form:

Krp =3

4CD

�rpj~ur � ~upjdrp

(9.4-18)

where �rp = �p�p+�r�r is the mixture density of the two secondaryphases, drp = 1

2(dr + dp) is the averaged particle diameter, and

j~up � ~urj is the relative velocity. The drag coe�cient is obtainedfrom Equation 9.4-16 but with a relative Reynolds number of theform

Re =�rpj~ur � ~upjdrp

�rp(9.4-19)

where �rp = �p�p + �r�r is the mixture viscosity of the phases pand r.

These exchange coe�cients are suited for the bubbly or droplet ow regimes, in which the primary phase dominates. Applicationof Equations 9.4-15 and 9.4-18 to slug or annular regimes, for exam-ple, reduces the accuracy of Eulerian multiphase simulations. Forthese ow regimes, the exchange coe�cients must be modi�ed orreplaced. Numerous models for the exchange term are present in



the literature. With user-de�ned subroutines, you can customizethe calculation of the exchange coe�cient if necessary.

It is possible to choose a value of zero for the exchange coe�cient,Disabling theExchange

Coe�cientsas discussed in Section 9.7.11. When this is done, the ow �eldsfor the uids will be computed independently, with the only \in-teraction" being their complementary volume fractions within eachcomputational cell. Setting the exchange coe�cient to zero can bea useful tool in initiating a calculation, but is generally not recom-mended for the �nal solution, where interaction between the uidsis usually important.

9.4.4 Description of Granular Multiphase Flow

Following the work of [2], [18], [30], [40], [73], [80], [92], and [127],FLUENT uses a multi- uid granular model to describe the ow be-havior of a uid-solid mixture. The solid-phase stresses are derivedby making an analogy between the random particle motion arisingfrom particle-particle collisions and the thermal motion of moleculesin a gas, taking into account the inelasticity of the granular phase.As is the case for a gas, the intensity of the particle velocity uc-tuations determines the stresses, viscosity, and pressure of the solidphase. The kinetic energy associated with the particle velocity uc-tuations is represented by a \pseudothermal" or granular tempera-ture which is proportional to the mean square of the random motionof particles.

The equations for granular multiphase ow are presented in thissection for the case of an n-phase ow.

The Momentum Equations

The conservation of momentum for the primary ( uid) phase, f , is

@

@t(�f�f~uf) +r � (�f�f~uf ~uf) = ��frp+r � �� f + �f�f~g +

nXs=1

(Ksf(~us � ~uf) + _msf~usf)

+~Ff (9.4-20)

and that for the sth solid phase is

@

@t(�s�s~us) +r � (�s�s~us ~us) = ��srp�rps +r � �� s + �s�s~g +


9.4 Theory 9-19

MXl=1

(Kls(~ul � ~us) + _mls~uls) + ~Fs +

Kfs(~uf � ~us) + _mfs~ufs (9.4-21)

where ps is the sth solids pressure, Kfs = Ksf is the uid-solidmomentum exchange coe�cient, Kls is the solid-solid momentumexchange coe�cient between solid phases l and s, and M is thetotal number of solid phases. The sth solids stress-strain tensor(�� s)ij in component form is given by

�s;ij = �s�s(@us;i@xj

+@us;j@xi

) + (�s�s � 2

3�s�s)�ij

@us;l@xl

(9.4-22)

It can be seen in Equations 9.4-20 and 9.4-21 that momentum ex-The ExchangeCoe�cients change between the phases is based on the value of the uid-solid

exchange coe�cient Ksf (= Kfs) and the solid-solid exchange coef-�cient Kls.

The uid-solid exchange coe�cientKsf has a form derived by Syam-The Fluid-SolidExchangeCoe�cient

lal and O'Brien [126] [127], and is based on measurements of theterminal velocities of particles in uidized or settling beds, withcorrelations that are a function of the volume fraction and relativeReynolds number [105]

Res =�fdsj~us � ~uf j

�f(9.4-23)

where the subscript f is for the uid phase, s is for the sth solidphase, and ds is the diameter of the sth solid phase particles.

The uid-solid exchange coe�cient has the form

Ksf =3�s�f�f4v2rsds

CD

�Resvrs

�j~us � ~uf j (9.4-24)

where vrs is the terminal velocity correlation for the solid phases [35]:

vrs = 0:5 (A� 0:06Res+

q(0:06Res)

2 + 0:12Res (2B � A) + A2

�(9.4-25)



withA = �4:14

f (9.4-26)

andB = 0:8�1:28

f (9.4-27)

for �f � 0:85, andB = �2:65

f (9.4-28)

for �f > 0:85.

The drag coe�cient has a form derived by Dalla Valle [26]:

CD =

0@0:63 + 4:8q

Res=vrs

1A

2

(9.4-29)

The solid-solid exchange coe�cient Kls has the following form [125]:The Solid-SolidExchangeCoe�cient Kls =

3 (1 + els)��2+ Cf;ls

�2

8

��s�s�l�l (dl + ds)

2 gols

2� (�ld3l + �sd3s)j~ul � ~usj (9.4-30)

whereels is the coe�cient of restitution (described later in this

section)Cf;ls is the coe�cient of friction between the lth and sth

solid-phase particles (Cf;ls = 0)dl is the diameter of the particles of solid lgols is the radial distribution coe�cient (described below)

For granular ows in the compressible regime (i.e., where the solidsThe Solids Pressurevolume fraction is less than its maximum allowed value), a solidspressure is calculated independently and used for the pressure gradi-ent term, @Ps

@xj, in the granular phase momentum equation. Because

a Maxwellian velocity distribution is used for the particles, a gran-ular temperature is introduced into the model, and appears in theexpression for the solids pressure and viscosities. The solids pres-sure is composed of a kinetic term and a second term due to particlecollisions:

Ps = �s�s�s + 2�s(1 + ess)�2sgoss�s (9.4-31)

where ess is the coe�cient of restitution for particle collisions, gossis the radial distribution function, and �s is the granular tempera-ture. FLUENT uses a default value of 0.8 for ess, but the value can be


9.4 Theory 9-21

adjusted to suit the particle type. The granular temperature �s isproportional to the kinetic energy of the uctuating particle motion,and will be described later in this section. The function goss (de-scribed below in more detail) is a distribution function that governsthe transition from the \compressible" condition with � < �s;max,where the spacing between the solid particles can continue to de-crease, to the \incompressible" condition with � = �s;max, whereno further decrease in the spacing can occur. A value of 0.6 is thedefault for �s;max. This can be changed by the user.

The radial distribution function go is a correction factor that modi-Radial DistributionFunction �es the probability of collisions between grains when the solid gran-

ular phase becomes dense. This function may also be interpretedas the nondimensional distance between spheres:

go =s+ dps

(9.4-32)

where s is the distance between grains. From Equation 9.4-32 itcan be observed that for a dilute solid phase s!1, and thereforego ! 1. In the limit when the solid phase compacts, s ! 0 andgo ! 1. The radial distribution function is closely connected tothe factor � of Chapman and Cowling's [18] theory of nonuniformgases. � is equal to unity for a rare gas, and increases and tends toin�nity when the molecules are so close together that motion is notpossible.

In the literature there is no unique formulation for the radial distri-bution function. FLUENT employs Syamlal et al. [127] expressionswhen the number of solid phases is greater than or equal to 1. Forcollisions between grains of phase l and grains of phase m, the fol-lowing expression is used:

golm =1

�f+

3dmdl�2f (dl + dm)

MXk=1

�k

dk(9.4-33)

For collisions involving only grains of phase l, Equation 9.4-33 re-duces to

goll =1

�f+

3dl2�2

f

MXk=1

�k

dk(9.4-34)

These equations were derived theoretically by Lebowitz [73] and donot approach in�nity near the packing limit. FLUENT can also use



the Ding and Gidaspow expression [30] for the radial distributionfunction:

go =3

5

241�

�s

�s;max

! 13

35�1

(9.4-35)

This equation is that proposed in [92] and corrected by a factor of35to match the measured data of [2].

When the number of solid phases is greater than 1, Equation 9.4-35is extended to

goll =3

5

241�

�l

�l;max

! 13

35�1

(9.4-36)

with

�l;max = �max �MX

k=1;k 6=l

�k (9.4-37)

and

golm =dmgoll + dlgomm

dm + dl(9.4-38)

The solids stress tensor contains shear and bulk viscosities arisingShear Stresses forthe Granular Phase from particle momentum exchange due to translation and collision.

The collisional and kinetic parts are added to give the solids shearviscosity:

�s = �s;col + �s;kin (9.4-39)

The collisional part of the shear viscosity is modeled as [40] [127]

�s;col =4

5�s�sdsgoss(1 + ess)

��s

�

�1=2

(9.4-40)

FLUENT uses two expressions for the kinetic part.

The default, from Syamlal et al. [127], is:


9.4 Theory 9-23

�s;kin =�sds�s

p�s�

6 (3� ess)

�1 +

2

5(1 + ess) (3ess � 1)�sgoss

�(9.4-41)

The following optional expression from Gidaspow et al. [40] is alsoavailable:

�s;kin =10�sds

p�s�

96�s (1 + ess) goss

�1 +

4

5goss�s (1 + ess)

�2(9.4-42)

The solids bulk viscosity accounts for the resistance of the granularparticles to compression and expansion. It has the following form:

�s =4

3�s�sdsgoss(1 + ess)

��s

�

�1=2

(9.4-43)

The granular temperature for the sth solids phase is proportionalThe GranularTemperature to the kinetic energy of the random motion of the particles. The

transport equation derived from kinetic theory takes the form [30]

3

2

"@

@t(�s�s�s) +r � (�s�s~us�s)

#= (�ps��I + �� s) : r~us

r � (k�sr�s)� �s

+

�fs + �ls (9.4-44)

where

(�ps��I + �� s) : r~us is the generation of energy by thesolid stress tensor

k�sr�s is the di�usion of energy (k�s is thedi�usion coe�cient)

�sis the collisional dissipation of energy

�fs is the energy exchange between the uid and thesth solid phase

�ls is the energy exchange between the lth and the sth

solid phases

Equation 9.4-44 contains the term k�sr�s describing the di�usive

ux of granular energy. When the default Syamlal et al. model [127]is used, the di�usion coe�cient for granular energy, k�s

is given by



k�s=

15ds�s�s

p�s�

4(41� 33�)

�1 +

12

5�2(4� � 3)�sgoss+

16

15�(41� 33�)��sgoss)

�(9.4-45)

where

� =1

2(1 + ess)

FLUENT uses the following expression if the optional model of Gi-daspow et al. [40] is enabled:

k�s=

150�sdsq(��)

384(1 + ess)goss[1 +

6

5�sgoss(1 + es)]

2

+

2�s�s2ds(1 + ess)goss

s�s

�(9.4-46)

The collisional dissipation of energy, �s, represents the rate of en-

ergy dissipation within the sth solids phase due to collisions betweenparticles. This term is represented by the expression derived by Lunet al. [80]

�m =12(1� e2ss)goss

dsp�

�s�2s�

3=2s (9.4-47)

You can also use the expression given by [40]

�m = 3(1� e2ss)�2s�sgoss�s

24 4

ds

s�s

��r � ~us

35 (9.4-48)

The transfer of the kinetic energy of random uctuations in particlevelocity to the uid phase is represented by �fs [40]:

�fs = �3Kfs�s (9.4-49)

The last term of Equation 9.4-44, �ls, accounts for the transfer ofenergy between the lth and sth solids phases and has been neglectedin FLUENT.


9.4 Theory 9-25

9.4.5 Description of Heat Transfer in Multiphase

The internal energy balance for phase q is written in terms of thephase enthalpy, Equation 9.4-10. In general, phase q can be a mix-ture of species i; (i = 1; m) and its enthalpy is de�ned by

hq =mXi=1

miqh

iq (9.4-50)

where

dhiq = cpiqdTq (9.4-51)

and cpiq is the speci�c heat at constant pressure of species i in phase

q.

The thermal boundary conditions used with multiphase ows arethe same as those for a single phase ow. See Chapter 14 for details.

The rate of energy transfer between phases is assumed to be a func-The Heat ExchangeCoe�cient tion of the temperature di�erence

Qpq = Hpq(Tp � Tq) (9.4-52)

where Hpq (= Hqp) is the heat transfer coe�cient between the pth

phase and the qth phase. The heat transfer coe�cient is related tothe pth phase Nusselt number Nup:

Hpq =6�q�pNup

d2p(9.4-53)

Here �q is the thermal conductivity of the qth phase. The Nusseltnumber is typically determined from one of the many correlationsreported in the literature. In the case of uid- uid multiphase FLU-ENT uses the correlation of Ranz and Marshall [102][103]:

Nup = 2:0 + 0:6Re12p Pr

13 (9.4-54)

where Rep is the relative Reynolds number based on the diameterof the pth phase and the relative velocity j~up�~uqj, Equation 9.4-17,and Pr is the Prandtl number of the qth phase:



Pr =cpq�q

�q(9.4-55)

In the case of granular ows (where p = s), FLUENT uses a Nusselt-number correlation by Gunn [43], applicable to a porosity range of0.35{1.0 and a Reynolds number of up to 105:

Nus = (7� 10�f + 5�2f )(1 + 0:7Re0:2s Pr

13 ) +

(1:33� 2:4�f + 1:2�2f)Re

0:7s Pr

13 (9.4-56)

The Prandtl number is de�ned as above with q = f .

The conductive heat ux for the qth phase is described by Fourier'sThermalConductivity for

the Granular Phaselaw:

~qq = ��q�qrTq

FLUENT uses a similar law for the granular model. Following Kuiperset al. [66], the solid phase conductivity �s is as follows:

�s = �f

p1� �f

(1� �f )(!A+ (1� !)�) (9.4-57)

where

� =2

(1� B=A)

"(A� 1)

(1� B=A)2B

Aln�A

B

�� (B � 1)

(1� B=A)� 1

2(B + 1)

#

(9.4-58)

andA =

�s;o�f

B = 1:25

1� �f

�f

! 109

! = 7:26� 10�3


9.4 Theory 9-27

Here ! is the contact area fraction between the solid phase s andthe primary phase f , and �s;o is the physical thermal conductivityof the solid material.

As in the Euler-Lagrange approach (see Chapter 11), it is possible toSolid Phase E�ectsin the P-1

Radiation Modelinclude the e�ect of radiation between the uid and granular phasesusing the P-1 radiation model. This modi�cation is limited to dilute ows only. When the P-1 model and the granular multiphase modelare active, the equation for radiation temperature �R for the uidhas an extra source term of the following form:

S�R= 1:5

mXp=1

emp�p(T

4p ��4

R) (9.4-59)

where empis the particle emissivity.

The enthalpy equation for the granular phase q also contains asource term of the form

Shq = 1:5emq�q�(�

4R � T 4

q ) (9.4-60)

where � is the Boltzmann's constant.

9.4.6 Description of Mass Transfer in Multiphase

The continuity equations for the pth phase contain the source term

nXp=1

_mpq

which characterizes the amount of mass transfer from the pth tothe qth phase per unit volume of mixture per unit time. A similarterm appears in the momentum and enthalpy equations. There arenumerous kinds of mass transfer processes and these can be modeledwith user-de�ned subroutines. FLUENT models are limited to twotypes:

� unidirectional mass transfer

� evaporation-condensation

The unidirectional mass transfer model de�nes a positive mass owUnidirectionalMass Transfer rate per unit volume from phase p to phase q:



_mpq = max[0; �pq]�max[0;��pq] (9.4-61)

where

�pq = _r�p�q (9.4-62)

and _r is a constant rate of particle shrinking or swelling, such asthe rate of burning of a uid droplet. In the case of granular ow,when mass transfer is from the solid phase s to the uid phase f ,�pq is modi�ed as

�sf =_r6�s�sds

(9.4-63)

and _r can represent the rate of volatilization of a particle.

The term appearing in the momentum equation is modi�ed as

_mpq~upq = max[0; �pq]~up �max[0;��pq]~uq (9.4-64)

FLUENT also contains a simple phenomenological model for a mix-Evaporation-Condensation

Modelture of two phases (liquid and vapor) [75]. The evaporation rate _mv

and the condensation rate _ml are determined from

_mv = rv�l�l(Tl � Tsat)=Tsat Tl � Tsat

= 0 Tl < Tsat (9.4-65)

_ml = rl�v�v(Tsat � Tv)=Tsat Tv � Tsat

= 0 Tv > Tsat (9.4-66)

where rv and rl are time relaxation parameters with default valuesof 0.1.

The momentum equation for the liquid contains a term due to masstransfer, _ml~uv � _mv~ul. In the momentum equation for the vapor,the negative of this term, _mv~ul � _ml~uv, is used.

In multiphase with mass transfer, it is necessary to calculate theThe ShadowMethod change of the size of the particulate phase. In many problems the

size of the particles determines the interphase coe�cients (mass,


9.4 Theory 9-29

momentum, enthalpy). FLUENT uses the Spalding [123] \shadow"method to calculate the change in particle size. This method entailsthe calculation of a \shadow" volume fraction. For the sth phaseundergoing mass transfer, for example, the shadow volume fractionwould be that of the sth phase had mass transfer not been included.Using this method, the diameter of a particle of this phase, ds,can be updated because of mass transfer according to the followingexpression:

ds = ds;o

�s

�s;o

! 13

(9.4-67)

where ds;o is the initial diameter, �s is the volume fraction of the sth

phase calculated with mass transfer, and �s;o is the volume fractioncalculated without mass transfer.

The mass balance equations for the prediction of the mass frac-Multiphase-Multicomponent tion of species for each phase do not contain a mass transfer model

between phases. Thus heterogeneous reactions involving multiplespecies and multiple phases are not permitted. Homogeneous re-actions are allowed within each phase, however, so that the totalcontribution to the qth phase continuity equation is zero. With user-de�ned subroutines, you can expand the capabilities of interphaseexchange, if desired.

9.4.7 Turbulence in Multiphase

To describe the e�ects of turbulent uctuations of velocities andscalar quantities in a single phase, FLUENT uses various types ofclosure models, as described in Section 6.3. In comparison to single-phase ows, the number of terms to be modeled in the momentumequations in multiphase ows is large, and this makes the modelingof turbulence in multiphase simulations extremely complex.

FLUENT provides two methods for modeling turbulence in multi-phase ows within the context of the k-� model:

� Dispersed turbulence model (default)

� Secondary turbulence model

The choice of model depends on the importance of the secondary-phase turbulence in your application.



Dispersed Turbulence Model

The dispersed turbulence model is the appropriate model whenthe concentration of the secondary phase is dilute. In this case,interparticle collisions are negligible and the dominant process inthe random motion of the secondary phase is the in uence of theprimary-phase turbulence. Fluctuating quantities of the secondaryphase can therefore be given in terms of the mean characteristics ofthe primary phase and the ratio of the particle relaxation time andeddy-particle interaction time.

The conditions for which the dispersed turbulence model is validare summarized below:

� The number of phases is limited to two: the continuous (pri-mary) phase and the dispersed (secondary) phase.

� The secondary phase must be dilute. That is, the domainmust not have large regions where the volume fraction for thesecondary phase approaches its maximum value.

The dispersed method for modeling turbulence in FLUENT involvesAssumptionsthe following assumptions:

� A modi�ed k-� model for the continuous phase: Turbulentpredictions for the continuous phase are obtained using thestandard k-� model supplemented with extra terms that in-clude the interphase turbulent momentum transfer.

� Tchen-theory correlations for the dispersed phase: Predictionsfor turbulence quantities for the dispersed phase are obtainedusing the Tchen theory of dispersion of discrete particles byhomogeneous turbulence [47].

� Interphase turbulent momentum transfer: In turbulent two-phase ows, the momentum exchange term contains the corre-lation between the instantaneous distribution of the dispersedphase and the turbulent uid motion. FLUENT includes amodel which takes into account the dispersion of the dispersedphase transported by the turbulent uid motion.

� A phase-weighted averaging process: The choice of averagingprocess has an impact on the modeling of dispersion in tur-bulent two-phase ows. A two-step averaging process leads tothe appearance of uctuations in the phase volume fractions.


9.4 Theory 9-31

When the two-step averaging process is used with a phase-weighted average for the turbulence, however, turbulent uc-tuations in the volume fractions do not appear. FLUENT usesphase-weighted averaging, so no volume fraction uctuationsare introduced into the continuity equations.

The eddy viscosity model is used to calculate averaged uctuatingTurbulence in theContinuous Phase quantities. The Reynolds stress tensor takes the following form:

�� 00q = �2

3(�qkq + �tqr � ~Uq)

��I + �tq(r~Uq + (r~Uq)T) (9.4-68)

The turbulent viscosity �tq is written in terms of the turbulent kineticenergy of phase q:

�tq = C�

k2q�q

(9.4-69)

and a characteristic time of the energetic turbulent eddies is de�nedas

� tq =3

2C�

kq�q

(9.4-70)

where �q is the dissipation rate. The length scale of the turbulenteddies is

Ltq =

s3

2C�

k32q

�q(9.4-71)

Turbulent predictions are obtained from the modi�ed k-� model:

@

@t(�q�qkq) +r � (�q�q ~Uqkq) = r � (�q�q

�tq�krkq) + �q�q(P � �q) +

�q�q�kq (9.4-72)

and



@

@t(�q�q�q) +r � (�q�q ~Uq�q) = r � (�q�q

�tq��r�q) +

�q�q�qkq(C1�P � C2��q) +

�q�q��q (9.4-73)

Here �kq and ��q represent the in uence of the dispersed phase pon the continuous phase q. All other terms have the same meaningas in the single-phase k-� model. The term �kq can be derived fromthe instantaneous equation of the continuous phase and takes thefollowing form:

�kq =Kpq

�q�q(< u

00

q;iu00

p;i > � < u00

q;iu00

q;i > +(Up;i�Uq;i)Vdr;i) (9.4-74)

which can be simpli�ed to

�kq =Kpq

�q�q(kpq � 2kq + ~Vpq � ~Vdr) (9.4-75)

where kpq is the covariance of the continuous and dispersed phase

velocities (calculated from Equation 9.4-83 below), ~Vpq is the relative

velocity, and ~Vdr is the drift velocity (de�ned by Equation 9.4-88below).

��q is modeled according to Elgobashi et al. [32]:

��q = C3��qkq�kq (9.4-76)

where C3� = 1:2.

Time and length scales that characterize the motion are used toTurbulence in theDispersed Phase evaluate dispersion coe�cients, correlation functions, and the tur-

bulent kinetic energy of the dispersed phase.

The characteristic particle relaxation time connected with inertiale�ects acting on the particulate phase is de�ned as

�Fpq = �p�pK�1pq (

�p�q

+ CV ) (9.4-77)


9.4 Theory 9-33

The Lagrangian integral time scale calculated along particle tra-jectories, mainly a�ected by the crossing-trajectory e�ect [25] isde�ned as

� tpq = � tq(1 + C��2)� 1

2 (9.4-78)

where

� =j~Vpqj� tqLtq

(9.4-79)

and

C� = 1:8� 1:35 cos2 � (9.4-80)

and where � is the angle between the mean particle velocity and themean relative velocity. The ratio between these two characteristictimes is written as

�r =� tpq�Fpq

(9.4-81)

Following Simonin [117] FLUENT writes the turbulence quantitiesfor the dispersed phase as follows:

kp = kq

b2 + �r1 + �r

!(9.4-82)

kpq = 2kq

b + �r1 + �r

!(9.4-83)

Dtpq =

1

3kpq�

tpq (9.4-84)

�p = Dtpq + (

2

3kp � b

1

3kpq)�

Fpq (9.4-85)

b = (1 + CV )(�p�q

+ CV )�1

(9.4-86)

and CV = 0:5 is the added-mass coe�cient.

The turbulent drag term in FLUENT in two-phase ows is modeledInterphaseTurbulent

MomentumTransfer

as follows (where p is the dispersed phase and q is the continuousphase):



Kpq(~up�~uq) = Kpq(~Up�~Uq)+Kpq(�p

�pq�pr�p� �q

�pq�qr�q) (9.4-87)

Here ~Up and ~Uq are phase-weighted velocities, �p and �q are di�u-sivities, and �pq is a turbulent Schmidt number. When using Tchentheory in two-phase ows, FLUENT assumes �p = �q = Dt

pq and thedefault value for �pq is 0.67. The second term of the above equationcontains the drift velocity:

~Vdr = �( �p�pq�p

r�p � �q�pq�q

r�q) (9.4-88)

The drift velocity results from turbulent uctuations in the volumefraction. When multiplied by the exchange coe�cient Kpq, it servesas a correction to the momentum exchange term for turbulent ows.

Secondary Turbulence Model

For turbulent multiphase ows involving more than two phases or anon-dilute secondary phase, the secondary turbulence model is theappropriate choice. When this model is used, FLUENT will solvek and � transport equations (Equations 9.4-72 and 9.4-73) for allphases.

The secondary turbulence model in FLUENT will not, by default,Limitationsaccount for the e�ect of the turbulence �eld of one phase on theother(s). Thus, the �kq and ��q terms will not appear in Equa-tions 9.4-72 and 9.4-73 and interphase turbulent momentum trans-fer is neglected. If the interaction of the multiple turbulence �eldsand interphase turbulent momentum transfer are important in yourproblem, you can supply these terms using user-de�ned subroutines.

Note also that, since FLUENT is solving two additional transportequations for each secondary phase, the secondary turbulence modelis more computationally intensive than the dispersed turbulencemodel.

9.5 Solution Method in FLUENT

The di�culties of solving single-phase ow using the primitive vari-able approach are well known. Because there is no explicit equationfor pressure, the SIMPLE algorithm [95] is adopted to obtain a pres-sure correction equation. Pressure and velocities are then corrected


9.5 Solution Method in FLUENT 9-35

so as to satisfy the continuity constraint. In multiphase ow, this isfurther complicated by the following facts: (i) there are n continu-ity equations and usually a single pressure �eld, (ii) phasic volumeequations are a new unknown in the set of governing equations,and (iii) the momentum equations are strongly coupled through theinterphase momentum exchange coe�cient.

FLUENT uses an extension of the SIMPLE algorithm (SIMPLE-The PressureCorrectionEquation

PEA) to solve for multiphase ow. The momentum equations aredecoupled by using the Partial Elimination Algorithm (PEA) [122].Using SIMPLE-PEA, the variables for each phase are eliminatedfrom the momentum equations for all other phases. For uid- uid ows, with a single pressure �eld, the pressure correction equationis obtained from the sum of all the normalized n-phase continuityequations. For incompressible multiphase ow (in Cartesian tensornotation), this equation takes the form

nXk=1

(@

@t(rk�k) +r � rk�k~u

0

k +r � rk�k~u�k �

1

�ref;k

nXl=1

_mlk

)= 0

(9.5-1)

where rk = �k=�ref;k, ~u0

k;i is the velocity correction and ~u�k;i is thevalue of ~uk;i at the present iteration. The velocity corrections arethemselves expressed as functions of the pressure corrections.

The volume fractions are obtained from the phase continuity equa-Volume Fractionstions. In discretized form, the equation of the kth volume fractionis

ap;k�k =Xnb

(anb;k�nb;k) + bk = Rk (9.5-2)

In order to satisfy the condition that all the volume fractions sumto one,

nXk=1

�k = 1 (9.5-3)

the solution of �k is obtained from the coupling of all phase conti-nuity equations. From Equation 9.5-2,

�k =Rk

ap;k(9.5-4)



so that

nXi=1

Ri

ap;i= 1 (9.5-5)

It follows that the central coe�cient ap;k can be expressed in termsof all of the other volume fraction equations:

ap;k = Rk + ap;knXi=1

Ri

ap;i(i 6= k) (9.5-6)

The resulting equation for �k implicitly satis�es Equation 9.5-3.

In granular multiphase ows, the shared pressure correction is ob-PressureCorrection forGranular Flows

tained from the uid continuity equation, as described above. When�s < �s;max (0.6 by default), the solid is treated as a compressible uid and the solids pressure is obtained as a direct function of thesolid-phase volume fraction. In the incompressible regime (where�s = �s;max), the solids pressure is obtained from a pressure correc-tion equation in the same way that the uid pressure is obtained.

The solution algorithms followed by FLUENT for uid- uid owsSummary ofSolution

Algorithmsand for granular ows are summarized below:

� Fluid-Fluid Multiphase Algorithm

1. Get initial conditions and boundary conditions.

2. Perform time-step iteration.

3. Calculate primary uid velocity (use PEA).

4. Calculate secondary uid velocities (use PEA).

5. Calculate pressure correction from all normalized conti-nuity equations.

6. Correct all velocities and uxes.

7. Calculate phasic volume fractions and update properties.

8. Calculate other scalar quantities. If not converged, go to3.

9. Advance time step and go to 2.

� Granular Multiphase Algorithm

1. Get initial conditions and boundary conditions.

2. Perform time-step iteration.


9.6 Eulerian Multiphase Flow Modeling Strategies 9-37

3. Calculate primary uid velocity (use PEA).

4. Calculate pressure correction from uid continuity equa-tion and correct uid velocity, pressure, and uid uxes.

5. Calculate phasic volume fractions.

6. Calculate solid velocities.

7. Calculate solid pressure corrections and correct solid ve-locities, uxes, and solid volume fractions. Update prop-erties and calculate granular temperature.

8. Calculate other scalar quantities. If not converged, go to3.

9. Advance time step and go to 2.

9.6 Eulerian Multiphase Flow Modeling Strategies

9.6.1 Problem Setup

Before you begin to solve a problem using the Eulerian multiphaseModel Selectionmodel, you should determine that this model is appropriate for yourparticular problem. Remember that for strati�ed or free surface ows you should use the VOF model, and for ows in which parti-cle volume fractions are less than or equal to 10% you should usethe Lagrangian dispersed phase model. When phases mix and/orparticle volume fractions exceed 10% you should use the Eulerianmultiphase model.

The required computational e�ort depends strongly on the numberComputationalE�ort of transport equations being solved and the degree of coupling. For

the Eulerian multiphase model, which has a large number of highlycoupled transport equations, computational expense will be high.Before setting up your problem, try to reduce the problem statementto the simplest form possible.

Instead of trying to solve your multiphase ow in all of its com-Simplifying theProblem Statement plexity on your �rst solution attempt, you should start with simple

approximations and work your way up to the �nal form of the prob-lem de�nition. Some suggestions for simplifying a multiphase owproblem are listed below:

� Use a simpli�ed geometry.

� Begin with a 2D approximation.

� Solve a single-phase problem, using a composite density.



� Reduce the number of phases.

You may �nd that even a very simple approximation will provideyou with useful information about your problem.

9.6.2 Convergence and Stability

Even a simpli�ed approximation of a multiphase ow problem maystill require signi�cant computational e�ort, so you will need to bepatient during the solution process. If you have di�culty obtain-ing a converged solution, you can try one or more of the followingsuggestions:

� Patch an initial guess for all variables.

� Increase the sweeps on the momentum equations for all phases.

� Lower the underrelaxation factors.

� Solve as a pseudo-transient to approach steady- ow solution,rather than attempt a steady-state calculation.

� Lower the density ratio; increase it in stages.

� Temporarily suppress the exchange coe�cient or begin a gran-ular simulation with a non-granular model, switching afterpartial convergence is obtained.

See Section 9.7.11 for additional information about solution strate-gies.

9.7 Using the Eulerian Multiphase Model in FLUENT

This section describes the steps necessary to set up and run a varietyof multiphase problems. Information is divided into the followingsections:

� Section 9.7.1: Turning on the Eulerian Multiphase Model

� Section 9.7.2: Specifying the Phases

� Section 9.7.3: Setting Boundary Conditions

� Section 9.7.4: Setting Physical Properties


9.7 Using the Eulerian Multiphase Model in FLUENT 9-39

� Section 9.7.5: Models for Granular Flows

� Section 9.7.6: Time-Dependent Simulations

� Section 9.7.7: Modeling Turbulence

� Section 9.7.8: Species Transport

� Section 9.7.9: Modeling Interphase Heat Transfer

� Section 9.7.10: Modeling Interphase Mass Transfer

� Section 9.7.11: Solution Strategies for Eulerian MultiphaseCalculations

� Section 9.7.12: Starting from a Case File Created by an EarlierRelease of FLUENT

� Section 9.7.13: Eulerian Multiphase Options

9.7.1 Turning on the Eulerian Multiphase Model

The Eulerian multiphase model can be enabled in the Models panelor in the DEFINE-MODELS menu.

To select the Eulerian multiphase model through the graphical userUsing the GUIinterface, select the De�ne/Models... menu item to open the Modelspanel, and then select Eulerian (for uid- uid ows, such as air-water separation) or Eulerian (Granular) (for uid-solid ows, suchas slurries and uidized beds) in the Model drop-down list underMultiphase.

De�ne �!Models...



In the SETUP-1 text menu, select DEFINE-MODELS. Then select MUL-Using the TextInterface TIPLE-PHASES and turn on EULERIAN-EULERIAN MULTIPHASE FLOW

(for uid- uid ows) or EULERIAN-EULERIAN GRANULAR FLOW (for uid-solid ows).

SETUP-1 �! DEFINE-MODELS �!MULTIPLE-PHASES

(SETUP1)-

DEFINE-MODELS



COMMANDS AVAILABLE FROM DEFINE-MODELS:

W-VELOCITY HEAT-TRANSFER

TURBULENCE NON-NEWTONIAN

RADIATION SPECIES-AND-CHEMISTRY

MULTIPLE-PHASES QUIT

HELP


(DEFINE-MODELS)-

MP

(MULTIPHASE MODEL (SELECT ONLY ONE))

NO EULERIAN-EULERIAN MULTIPHASE FLOW

YES EULERIAN-EULERIAN GRANULAR FLOW

NO VOF FREE SURFACE

D ACTION (TOP,DONE,QUIT,REFRESH)

COMMANDS AVAILABLE FROM MULTIPHASE:

DEFINE-PHASES MULTIPHASE-OPTIONS QUIT

HELP


(MULTIPHASE)-

If you have already turned on the Eulerian multiphase model for uid- uid ows, it is possible to enable granular ow in the EXPERTSOLUTION-PARAMETERS table.

9.7.2 Specifying the Phases

You can specify the number of phases and assign a name to eachphase using the DEFINE-PHASES command in the SETUP1 menu or inthe MULTIPHASE menu that appears after you turn on the multiphasemodel with the MULTIPLE-PHASES command.

SETUP-1 �!DEFINE-PHASES

SETUP-1 �! DEFINE-MODELS �! MULTIPLE-PHASES




DEFINE-PHASES MULTIPHASE-OPTIONS QUIT

HELP


(MULTIPHASE)-

DP

(I)- NUMBER OF SECONDARY PHASES

(I)- ++(DEFAULT 1)++

1

(PHASE NAMES)

NOTE : PHASE NAMES CANNOT CONSIST OF

A SINGLE CHARACTER C OR X

WATER PRIMARY PHASE

SAND PHASE 2


9.7.3 Setting Boundary Conditions

Once you have selected the multiphase option, the Phase selectiondrop-down list will become active in the upper left of the GUI panelsfor setting boundary conditions:



Select a phase from this list, set the appropriate boundary condi-tions, and then click Apply and go on to the next phase.

In the text interface, a PHASE-SELECTION menu will appear whenyou select a variable for which you wish to set boundary conditions.



COMMANDS AVAILABLE FROM I1-ZONE-BOUNDARY-CONDITIONS:

NORMAL-VELOCITY U-VELOCITY V-VELOCITY

VOLUME-FRACTION QUIT HELP


(I1-ZONE-BOUNDARY-CONDITIONS)-

UV

COMMANDS AVAILABLE FROM PHASE-SELECTION:

SAND WATER QUIT HELP


(PHASE-SELECTION)-

Note that you will only be asked for the secondary phase volume!fractions. FLUENT will compute the volume fraction of the primaryphase based on your input for the secondary phases. Also, a normalvelocity can be speci�ed only for the primary phase.

For example, consider an inlet supplying a slurry consisting of 25%sand to a domain. Both sand and water have an x-directed veloc-ity of 5 m/s. The appropriate boundary condition inputs are asshown below. Note that the uid phase must be modeled as theprimary phase in all granular ows, as discussed in more detail inSection 9.7.4 below.

First, enter the U velocity for WATER, as shown in the panel below,and click Apply.



Then select SAND in the Phase drop-down list, enter the sand Uvelocity and Volume Fraction, and click Apply again.



In the text interface, inputs are as follows:



COMMANDS AVAILABLE FROM I1-ZONE-BOUNDARY-CONDITIONS:

NORMAL-VELOCITY U-VELOCITY V-VELOCITY

VOLUME-FRACTION QUIT HELP



UV




(PHASE-SELECTION)-

SAND

(R)- U-VELOCITY OF SAND

(R)- UNITS= M/S ++(DEFAULT 0.0000E+00)++

5

(PHASE-SELECTION)-

WATER

(R)- U-VELOCITY OF WATER


5

(PHASE-SELECTION)-

Q


VF

(R)- VOLUME FRACTION OF SAND

(R)- UNITS= DIM ++(DEFAULT 0.0000E+00)++

.25

It is not necessary to set the volume fractions for both phases; FLU-ENT will compute the volume fraction of the primary phase basedon your input for the secondary phase. If your problem includesmore than two phases, a PHASE-SELECTION menu for volume frac-tion will appear when you select the VOLUME-FRACTION commandin the text interface, and you will specify the fractions for all phasesexcept the primary phase. In the GUI, the Volume Fraction will begrayed out for the primary phase, but available for all secondaryphases.

If you de�ne any volume fractions as pro�le functions, you must!be sure that the sum of all volume fractions will not exceed 1.0 atany position in the domain or at any time during the calculation.FLUENT will not rescale the volume fractions if the fractional sum



exceeds 1.0.

If you are not modeling secondary turbulence (i.e., if you have notTurbulenceParameters at

Inletsenabled the Secondary Turbulence or ENABLE SECONDARY TURBULENCE

option), you will set turbulence boundary conditions at inlets onlyfor the primary phase. If secondary turbulence is being modeled,you can set di�erent turbulence boundary conditions for each phase,but you can choose only one type of turbulence parameter speci�-cation. That is, turbulence boundary conditions at a given inlet forall phases must be de�ned either as intensity and length scale oras k and �; you cannot choose di�erent speci�cation methods fordi�erent phases.

Pressure boundary conditions are allowed for multiphase ows, sub-Pressure BoundaryCondition

Restrictionsject to the following restrictions:

� If you specify a nonzero gravitational acceleration, you shouldspecify a user-de�ned reference density. See Section 6.4.3 fordetails on setting an appropriate value.

� Convergence di�culties may arise if you have di�erent phasesgoing in opposite directions through the pressure boundary.

Note that wall boundary conditions for secondary phases cannotLimitations onWall Boundary

Condition Inputsbe set using the GUI. You can use the GUI to set wall boundaryconditions for the primary phase, but you must use the text interfaceto set them for the secondary phases.

The boundary conditions can be checked using the LIST-BOUNDARIESListing BoundaryConditions command. The FLOW-VARIABLES command lists the velocities of

both phases:



COMMANDS AVAILABLE FROM LIST-BOUNDARIES:

ALL FLOW-VARIABLES VOLUME-FRACTIONS

QUIT HELP


(LIST-BOUNDARIES)-

FL

- VELOCITY BOUNDARY CONDITIONS -

ZONE PHASE U-VEL. V-VEL. NORMAL

------- -------- -------- -------- --------

W1 WATER 0.00E+00 0.00E+00 N/A

SAND 0.00E+00 0.00E+00 N/A

I1 WATER 5.00E+00 0.00E+00 N/A

SAND 5.00E+00 0.00E+00 N/A

The VOLUME-FRACTION command lists the volume fraction informa-tion:

(LIST-BOUNDARIES)-

VF

- VOLUME FRACTION BOUNDARY CONDITIONS -

ZONE SAND

---------- ----------

W1 LINK CUT

I1 2.50E-01

9.7.4 Setting Physical Properties

When you enable the multiphase model, you must set the propertiesfor each phase using the SETUP1 PHYSICAL-CONSTANTS menu.

COMMANDS AVAILABLE FROM PHYSICAL-CONSTANTS:

DENSITY VISCOSITY OPERATING-PRESSURE

QUIT HELP



Other properties will appear in the menu, depending upon the over-all problem scope. Note that for granular ows the solids viscos-ity is obtained from application of kinetic theory. You can disable



this calculation in the Multiphase Parameters panel or the EULERIANMULTIPHASE OPTIONS table (see Section 9.7.13). Then you will beasked to input constant viscosities for the solids as well.

Setting Constant Densities

When you select the DENSITY command, you will be asked to supplyinformation about each of the phases. Underlying assumptions existregarding the modeling of the phases, and depend upon the kind ofmixture that is being modeled. To clarify these assumptions, thedi�erent combinations of phase types will be considered separately.

For nongranular constant-density ows, such as liquid-liquid owsNongranular(Fluid-Fluid)

Constant-DensityFlows

or gas-liquid ows where the gas law is not being used, the primary uid is assumed to be the predominant, or most plentiful phase. Inthe case of roughly equal overall volume fractions of the uids, theprimary phase should be used for the densest uid. The reason forthis is that the momentum exchange coe�cient depends upon thetypical size of a bubble that would be formed by the secondary uidas it interacts with the primary uid. The formation of bubbles ofthe secondary phase in the primary phase would most often occurin the con�gurations cited above.

For granular ows, the primary phase is always the nongranularGranular Flowsphase, independent of the density di�erence between the phases orthe relative volume fractions of the phases.

Consider, for example, a water-sand mixture with the water andExamplesand densities being 1000 and 1500 kg/m3, respectively. The inputsare for the densities and normalizing densities of the primary phase,water, and the secondary phase, sand. The NORMALIZING DENSITY

is used to normalize the density as it appears in the continuityequation, and the best value to use for each phase is the density ofthat phase at constant temperature.




DE




(PHASE-SELECTION)-

WA

(L)- USE GAS LAW FOR WATER?


N

(R)- DENSITY OF WATER

(R)- UNITS= KG/M3 ++(DEFAULT 1.0000E+03)++

1000

(R)- NORMALIZING DENSITY FOR WATER


1000

(PHASE-SELECTION)-

SAN

(R)- DENSITY OF SAND


1500

(R)- NORMALIZING DENSITY FOR SAND


1500

After you input the densities, FLUENT will ask for the mean di-ameter of the secondary phase. This input should be the averagesize of a particle of the secondary phase in the primary uid (sandin water in this example), to be used in the momentum exchangeterm. In the dialog below, a diameter of 1 mm is chosen.

(R)- MEAN DIAMETER FOR SAND

(R)- UNITS= M ++(DEFAULT 2.0000E-04)++

1e-3

If you are calculating changes in particle diameter due to mass trans-fer (as described in Section 9.7.10), you will be prompted for theINITIAL DIAMETER rather than the MEAN DIAMETER.

For a granular phase, you must also specify the coe�cient of restitu-tion. In this case, the default value for the RESTITUTION COEFFICIENT



FOR SAND (0.8) is retained. For perfect elastic collisions the coe�-cient of restitution is 1.0.

(R)- RESTITUTION COEFFICIENT FOR SAND

(R)- UNITS= DIM ++(DEFAULT 8.0000E-01)++

X


For multicomponent multiphase ows, you will specify the densitiesMulticomponentConstant-Density

Flowsfor each species in each phase:


DE


AIR WATER QUIT HELP


(PHASE-SELECTION)-

WAT

(L)- USE GAS LAW FOR WATER?


N

COMMANDS AVAILABLE FROM SPECIES-SELECTION:

SPECIES-1 SPECIES-2 QUIT HELP


(SPECIES-SELECTION)-

Using the Gas Law

For gas-liquid or gas-solid ows where the gas law is to be used, thegas must be modeled as the primary uid. Note that you do nothave to have the calculation of temperature or species enabled forthe gas law option to appear; however, the densities will be constantthroughout the domain if calculation of temperature or species isnot enabled.

Setting Constant Viscosities

In the absence of temperature or species dependence, the setting ofFluid-Fluid Flowsthe viscosities is straightforward. You can select each uid from thePHASE-SELECTION menu and enter its viscosity.

For granular ows, the viscosity is input only for the primary phase.Granular Flows



The solids bulk and shear viscosities are computed by Equations9.4-40{9.4-43. However, when you turn o� the Kinetic Theory Vis-cosity option in the Multiphase Parameters panel or answer NO tothe option USE VISCOSITY FROM KINETIC THEORY in the EULERIANMULTIPHASE OPTIONS table (see Section 9.7.13), FLUENT will askyou to input constant solids viscosities, as described above for uids.

Temperature- and/or Species-Dependent Properties

When the calculation of temperature is active, the properties of allphases can be input as polynomial, piecewise-linear, or harmonicfunctions of temperature. When multiple species are present in aphase, the temperature dependence of the properties can be speci-�ed for the individual species. De�nition of temperature-dependentproperties for multiphase ows is similar to the de�nition for single-phase ows. See Chapter 15 for details. Composition-dependentproperty inputs for multiphase ows are discussed below.

When multiple species are used for one or more of the phases,Composition-DependentProperties

the option to use composition-dependent (as well as temperature-dependent) properties is available. To activate this option, beginwith the PROPERTY-OPTIONS command in the PHYSICAL-CONSTANTSmenu. For composition-dependent viscosity, for example, respondYES to this option in the table.

PHYSICAL-CONSTANTS �!PROPERTY-OPTIONS

(PROPERTY CALCULATION OPTIONS)

YES COMPOSITION DEPENDENT VISCOSITY FOR AIR

NO COMPOSITION DEPENDENT THERMAL CONDUCTIVITY FOR AIR

NO COMPOSITION DEPENDENT SPECIFIC HEAT FOR AIR

NO IS ANY PROPERTY TO BE COMPUTED USING KINETIC THEORY

NO ENABLE THERMALLY DRIVEN MASS DIFFUSION (SORET EFFECT)


(If you have enabled the multicomponent multiphase model in orderto de�ne multiple species for secondary phases, additional lines willappear in the table to allow you to enable composition-dependentproperties for the secondary phases.)

When the viscosity is set, FLUENT will prompt you for the viscosityof each species in each multicomponent phase and, for nongranular ows, for the viscosity of each single-component secondary phase.



For granular ows, the solids viscosities are obtained from applica-tion of kinetic theory. You can disable this calculation in the Multi-phase Parameters panel or the EULERIAN MULTIPHASE OPTIONS ta-ble (see Section 9.7.13). Then you will be asked to input viscositiesfor the solids as well.

9.7.5 Models for Granular Flows

Two models for granular ows have been implemented in FLUENT.One is based on Syamlal et al. [127] and the other is based on Dingand Gidaspow [30][40]. The default model is Syamlal et al. To ac-tivate the second model, turn on the Gidaspow Model option in theMultiphase Parameters panel or respond YES to the option ENABLE

GIDASPOW MODEL in the EULERIAN MULTIPHASE OPTIONS table (seeSection 9.7.13). The main di�erences between the models are the ex-pressions for the radial distribution function, the kinetic part of thesolids viscosity, and the di�usion coe�cient of the granular temper-ature. The expressions used by each of these models are presentedin Section 9.4.4.

In some cases (e.g., a fully developed granular ow with constantFixing the Value ofGranular

Temperatureshear) you may wish to turn o� solution of the granular temperatureand set it to a �xed value. You can disable the solution of thegranular temperature equation in the SELECT VARIABLES table orin the Select Equations panel (Solve/Controls/Equations...).

EXPERT �!SELECT-VARIABLES

(SELECT VARIABLES)

YES PHASE 1 U-VELOCITY

YES PHASE 1 V-VELOCITY

YES PHASE 1 W-VELOCITY

YES PHASE 2 U-VELOCITY

YES PHASE 2 V-VELOCITY

YES PHASE 2 W-VELOCITY

YES PHASE 2 VOL. FRAC.

YES PRESSURE

NO GRANULAR TEMPERATURE

YES PROPERTIES/TEMPERATURE


Then set the appropriate value for Granular Temperature in theMultiphase Parameters panel (opened from the Models panel) orGRANULAR TEMPERATURE in the EULERIAN MULTIPHASE OPTIONS ta-ble:



EXPERT �!EULERIAN-MULTIPHASE-OPTIONS



(EULERIAN MULTIPHASE OPTIONS)

NO ENABLE MASS TRANSFER

YES STABILITY TERM IN P.D.E

1.0000E-01 FALSE TIME STEP FOR UNDER-RELAXATION (DIM)

YES ENABLE EXCHANGE COEFFICIENT

1.0000E-07 LOWER LIMIT FOR ALL VOFs (DIM)

NO ENABLE GIDASPOW MODEL

YES USE VISCOSITY FROM KINETIC THEORY

NO USE DRAG-CLUSTER CORRECTION

6.0000E-01 PACKING LIMIT FOR SOLID FRACTION (DIM)

1.0000E-04 GRANULAR TEMPERATURE (M2/S2)


Note that the default value for granular temperature is 1�10�5 m2/s2.

Cluster formation (i.e., the formation of large, dense regions ofDrag-ClusterCorrection maximally-packed granular phase particles) has been observed in

circulating uidized beds. When a cluster appears, it grows largerand denser and then sinks. Wall clusters join together creatinga down ow region (\core-annular" regime). FLUENT can includea modi�cation to the gas-particle drag correlation, introduced byO'Brien and Syamlal [91], to account for this phenomenon. Whenthe drag-cluster correction option is enabled, the computed dragwill be modi�ed to account for the size of the cluster rather thanthe size of the individual particles.

To enable this option, turn on Drag-Cluster Correction in the Mul-tiphase Parameters panel or USE DRAG-CLUSTER CORRECTION in theEULERIAN MULTIPHASE OPTIONS table (see Section 9.7.13).

9.7.6 Time-Dependent Simulations

FLUENT can solve time-dependent Eulerian multiphase ows, andin the special case of granular ows, a time-dependent simulationis strongly recommended. In some cases, the steady-state solutionis all that is of interest, while in others, it is the evolution of theprocess that is important. A choice between these two solutiontechniques should be made at the start of the problem so as togovern the time-dependent model parameters.

To enable time dependence, open the Models panel and turn on theTime Dependent Flow option:

De�ne �!Models...



Click Apply, and then click on Time Parameters... to open the TimeDependent Flow Parameters panel:



Here you can enter the time-dependent solution parameters. Thetransient model performs a certain number of iterations at eachtime step, and then moves to the next time step, and so on. Thecriterion that causes the code to advance to the next time step iseither that a level of convergence has been reached, based upon yourinput to the Minimum Residual Sum value, or that the Max Iterationsper Time Step (maximum number of iterations) has been performed.See Section 6.9 for details about time-dependent simulations.

For problems in which a steady-state solution is of primary impor-tance, it is not necessary to obtain a converged solution for eachtime step. Progression to the next time step can be accelerated bychoosing fewer iterations per time step than are required to get awell converged result. Alternatively, the size of the time step canbe increased. As a rule of thumb, if the anticipated time for a pro-cess to reach steady state can be approximated, a time step thatis 1/100th of this value might be chosen. The number of iterationsselected should result in a partially converged solution where theresiduals are in the 1� 10�2 to 1� 10�3 range and decreasing.

In cases where the details of the transient process are important,a converged solution for each step should be obtained. It may benecessary to compute the �rst time step several times, trying di�er-ent inputs for the time step and the number of iterations per timestep for each trial. Solution parameters such as underrelaxation fac-tors and sweeps of the solver can also be adjusted in the trial runsof the �rst time step so as to obtain a converged solution in thefewest number of iterations. In some instances, the pressure resid-ual at the end of the �rst time step may not meet the convergence



criterion, even though the remaining ow variables do. This occurswhen there are very low ow rates, because the normalization of thepressure residual is related to the mass ow at the start of the timestep. In some cases, the calculation can continue nonetheless, andthe convergence criterion for pressure will be met within the nextfew time steps.

When solving any transient analysis, data �les can be saved peri-odically by turning on the Auto Save Data Files option. When youturn on this option, the Time Steps Between Saves can be speci�ed:

The Autosave Files are ordinary Data Files containing ow-�eld(and other) data for both of the phases. By default, the AutosaveFiles are unformatted (binary) �les, but you can request formatted(text) �les by enabling Formatted Auto Saves.

Note that the time-dependence parameters can also be set with theEXPERT TIME-DEPENDENCE command.

9.7.7 Modeling Turbulence

If your turbulent multiphase ow involves only two phases and theconcentration of the secondary phase is dilute (i.e., there are nolarge regions where the volume fraction for the secondary phaseapproaches its maximum value), the default dispersed turbulencemodel is acceptable and no special inputs are required.

If your problem does not meet these criteria, you can enable the Sec-ondary Turbulence option in the Multiphase Parameters panel or the



ENABLE SECONDARY TURBULENCE option in the EULERIAN MULTIPHASE

OPTIONS table (see Section 9.7.13). FLUENT will then solve k and� transport equations for each phase.

See Section 9.4.7 for details about these two turbulence modelingoptions, including inherent assumptions and limitations.

When the default dispersed turbulence model is used, it is pos-sible to modify the value of �pq in Equation 9.4-87 (DISPERSIONPRANDTL in the EXPERT PHYSICAL-MODELS table) and the value ofC3� in Equation 9.4-76 (C3M TWO-PHASE TERM in the same table),but this is generally not recommended.

9.7.8 Species Transport

When the calculation of species is enabled in the Models panel orthe DEFINE-MODELS menu, the assumption is made that the mul-tiple species are in the primary phase only, unless you enable theoption for multicomponent multiphase to allow multiple species inthe secondary phases as well.

If your problem de�nition requires multiple species in the secondaryEnabling MultipleSpecies in

Secondary Phasesphase(s), you must turn on the Multicomponent-Multiphase Modeloption in the Multiphase Parameters panel or answer YES to ENABLE

MULTICOMPONENT-MULTIPHASE in the EULERIAN MULTIPHASE OPTIONS

table (see Section 9.7.13). When the multicomponent multiphasemodel is active, you will be able to de�ne multiple species for anyof the phases. Note, however, that the maximum number of speciesper phase is limited to the number of species in the primary phase.

When you are using the multicomponent multiphase model, theDe�ning theSpecies in Each

Phasede�nition of species will proceed as follows.

In the De�ne Species panel (opened with the De�ne/Species... menuitem), select the phase for which you are going to de�ne species inthe Phase drop-down list. (This list is accessible only after you haveturned on the multicomponent multiphase model.) Then specify thenumber of species and the species names in the usual manner (asdescribed in Section 7.1.3). When the species de�nition for theselected phase is complete, click Apply. Then choose the next phasein the Phase list and repeat the process.

Remember to click Apply after you �nish the species de�nition for!each phase.

In the text interface, you will specify the NUMBER-OF-SPECIES ineach phase:



(TOTAL NUMBER OF SPECIES PER PHASE)

2 NUMBER OF SPECIES IN AIR

2 NUMBER OF SPECIES IN WATER

NO ENABLE MOLE FRACTION INPUTS (OTHERWISE MASS FRACTION)


Then you can specify the species names for each phase with theDEFINE-SPECIES command:

(SPECIES NAMES)

NOTE : SPECIES NAMES CANNOT CONSIST OF

A SINGLE CHARACTER C OR X

SPECIES 1 SPECIES 1 IN WATER

SPECIES 2 SPECIES 2 IN WATER

SSPEC 11 SPECIES 1 IN AIR

SSPEC 12 SPECIES 2 IN AIR


See Section 7.1.3 for details about these inputs.

9.7.9 Modeling Interphase Heat Transfer

If you want to include heat transfer between phases in your mul-tiphase calculation, turn on Heat-Exchange Coe�cient in the Mul-tiphase Parameters panel or ENABLE HEAT-EXCHANGE COEFFICIENT

in the EULERIAN MULTIPHASE OPTIONS table (see Section 9.7.13).FLUENT will calculate interphase heat transfer using the equationspresented in Section 9.4.5.

Do not confuse the Heat-Exchange Coe�cient option with the Ex-!change Coe�cient option, which is described in Section 9.7.11.

9.7.10 Modeling Interphase Mass Transfer

To activate interphase mass transfer calculations, turn on the MassTransfer option in the Multiphase Parameters panel or the ENABLE

MASS TRANSFER option in the EULERIAN MULTIPHASE OPTIONS ta-ble (see Section 9.7.13). After you enable this option, you can setthe related parameters using the MASS-TRANSFER command in theMULTIPHASE menu that appears after you turn on the multiphasemodel with the MULTIPLE-PHASES command. (See Section 9.7.11 forimportant advice related to interphase mass transfer calculations.)

SETUP-1 �! DEFINE-MODELS �!MULTIPLE-PHASES



(DEFINE-MODELS)-

MP

(MULTIPHASE MODEL (SELECT ONLY ONE))

YES EULERIAN-EULERIAN MULTIPHASE FLOW

NO EULERIAN-EULERIAN GRANULAR FLOW

NO VOF FREE SURFACE



DEFINE-PHASES MULTIPHASE-OPTIONS MASS-TRANSFER

QUIT HELP


(MULTIPHASE)-

MT

First you will choose which of the two mass transfer models de-Choosing the MassTransfer Model scribed in Section 9.4.6 you want to use. The unidirectional mass

transfer model computes a mass ow rate from the primary phaseto each secondary phase (and from each secondary phase to sub-sequent secondary phases). The evaporation-condensation modelcomputes evaporation and condensation rates between the primaryand secondary phases. (The evaporation-condensation model cancompute mass transfer for only two uid phases.)

To use the unidirectional mass transfer model select CONSTANT MASS

FLOW RATE TRANSFER, and to use the evaporation-condensation modelselect TWO-PHASE EVAPORATION-CONDENSATION.

(MASS-TRANSFER: SELECT ONE OPTION ONLY)

NO CONSTANT MASS FLOW RATE TRANSFER

NO TWO-PHASE EVAPORATION-CONDENSATION


(Note that the TWO-PHASE EVAPORATION-CONDENSATION option willnot appear if granular ow is active or if you have not enabled heattransfer calculations.)

As discussed in Section 9.4.6, FLUENT uses the \shadow" methodEnabling ParticleDiameter

Calculationsto calculate the change in the size of the secondary-phase particlesdue to mass transfer for either mass transfer model. If you wish toinclude these particle size calculations, answer YES to the questionCALCULATE PARTICULATE DIAMETER? that appears after you choosethe mass transfer model:



(MASS-TRANSFER: SELECT ONE OPTION ONLY)

(L)- CALCULATE PARTICULATE DIAMETER?


FLUENT will compute the particle diameter using Equation 9.4-67,where ds;o is your input for the INITIAL DIAMETER during the de�-nition of density (see Section 9.7.4).

If the diameter of the particles will not change during the masstransfer (e.g., if you have particles of coal being converted to ash),answer NO to this question to suppress the particle diameter calcu-lation. The secondary-phase particles will have a constant diameterequal to the MEAN DIAMETER you specify during the density de�ni-tion.

If you choose the CONSTANT MASS FLOW RATE TRANSFER option, youParameters forUnidirectionalMass Transfer

will next see the following table:

(MASS TRANSFER FLOW RATE COEFFICIENTS)

0.0000E+00 MASS FLOW RATE COEFFICIENT FROM WATER TO VAPOR (/S)


Your entry for MASS FLOW RATE COEFFICIENT speci�es the rate ofshrinking or swelling of the particle due to mass transfer ( _r in Equa-tion 9.4-62 or 9.4-63). For example, this value may be the rate ofburning of a uid droplet or the rate of volatilization of a particle.FLUENT will compute the mass ow rate from the primary phase tothe secondary phase (or from one secondary phase to another) usingEquations 9.4-61 and 9.4-62 for uid- uid ows, or Equations 9.4-61and 9.4-63 for granular ows.

If you choose the TWO-PHASE EVAPORATION-CONDENSATION option,Parameters forEvaporation-Condensation

you will next see the following table:

EVAPORATION-CONDENSATION COEFFICIENTS)

1.0000E-01 EVAPORATION TIME RELAXATION (/S)

1.0000E-01 CONDENSATION TIME RELAXATION (/S)

3.7300E+02 SATURATION TEMPERATURE (K)


EVAPORATION TIME RELAXATION, CONDENSATION TIME RELAXATION

are the relaxation parameters rv and rl in Equations 9.4-65



and 9.4-66. These parameters represent the inherent time ofphase transitions and are usually determined by experiment.Both are set to 0.1 by default. For most cases you can retainthese default values.

SATURATION TEMPERATURE speci�es the temperature at which masstransfer calculations will be triggered (i.e., Tsat in Equations9.4-65 and 9.4-66). When the temperature in a given compu-tational cell exceeds (or drops below) this saturation tempera-ture, FLUENT will compute the evaporation (or condensation)rate in that cell.

9.7.11 Solution Strategies for Eulerian Multiphase Calculations

This section contains some general procedures for controlling thesolver during a multiphase calculation, as well as speci�c recommen-dations for calculations involving additional multiphase options.

For many multiphase problems, most of the default solver settingsSetting SolverParameters in FLUENT will be adequate for the initial calculations performed.

These settings include an alternating sweep direction and the de-fault underrelaxation factors, as shown below:

(UNDERRELAX 1)

2.0000E-01 AIR VELOCITIES (DIM)

2.0000E-01 WATER VELOCITIES (DIM)

2.0000E-01 WATER VOL. FRAC. (DIM)

5.0000E-01 PRESSURE (DIM)

2.0000E-01 VISCOSITY (DIM)

2.0000E-01 AIR ENTHALPY (DIM)

2.0000E-01 WATER ENTHALPY (DIM)

3.0000E-01 TEMPERATURE (DIM)


The volume fraction for the secondary phases can be underrelaxed.Based on the updated value for the secondary-phase volume frac-tions, a complementary value will be used for the primary-phasevolume fraction.

A recommended modi�cation to the default solver settings is anincrease in the number of sweeps of the LGS solver for the vari-ables for all phases. This is accomplished in the SWEEPS-OF-LGS-

SOLVER table, accessed in the EXPERT LINEAR-EQN.-SOLVER menu.The default inputs for this table are shown below.





1 U-VELOCITY

1 V-VELOCITY

1 U VELOCITY WATER

1 V VELOCITY WATER

1 VOL. FRAC. WATER




For the pressure-correction equation, the number of sweeps shouldbe increased to 20 or 30, depending upon the problem size. Sweepson the velocities for each phase can be increased to 5.

Supplying an initial guess for the ow �eld and volume fractions isPatching an InitialGuess recommended for all multiphase calculations. This can be accom-

plished with the PATCH command in the MAIN menu. For transientsimulations, the PATCH facility can be used to supply the initialconditions. For steady-state problems, an initial guess will lead toadded stability in the early stages of the solution.

As an example, consider the uidized bed example presented inSection 9.3.2. In this example, air is the primary phase, and thesolids constitute the secondary phase. The solids occupy the lowerhalf of the 42 � 42 grid, with a maximum packing fraction of 0.6.At the start of the problem, there is a uniform vertical velocity of airat 0.284 m/sec throughout the domain. To patch the air velocity,the entire domain is selected, and a velocity value for the primaryphase is set. In the y direction, this corresponds to the V-VELOCITY.



(*MAIN*)-

PA

(I)- ENTER 1ST I-


X

(I)- DEFAULT ASSUMED

(I)- ENTER 2ND I-

(I)- ++(DEFAULT 42)++

X


(I)- ENTER 1ST J-


X


(I)- ENTER 2ND J-

(I)- ++(DEFAULT 42)++

X


(VARIABLE-SELECTION)-

VV


AIR SOLIDS QUIT HELP


(PHASE-SELECTION)-

AIR

(R)- ENTER VALUE-


.284


Q

To create the bed region, the lower half of the domain is selected,and the volume fraction for the secondary phase (SOLIDS) is set at0.598. (Note that patching an initial volume fraction of 0.6, which isthe packing limit for the problem, is not recommended because thiscan cause some instability in the �rst few time steps of a uidizedbed calculation. A value of 0.598 is patched instead to eliminatethis problem.)

In this example, since there is only one secondary phase, you will



not need to specify the phase for which you are setting the VOLUME-FRACTION. The volume fraction speci�ed is for the solids. If therewere additional secondary phases, you would specify a volume frac-tion for each of them. FLUENT always determines the volume frac-tion of the primary phase based on your inputs for the secondaryphase(s).

(*MAIN*)-

PA

(I)- ENTER 1ST I-


X


(I)- ENTER 2ND I-

(I)- ++(DEFAULT 44)++

X


(I)- ENTER 1ST J-


X


(I)- ENTER 2ND J-

(I)- ++(DEFAULT 62)++

31


VF

PATCHING VOLUME FRACTION OF SOLIDS

(R)- ENTER VALUE-

(R)- UNITS= DIM ++(DEFAULT 0.0000E+00)++

.598


Q

The results of these patches are shown in Figure 9.7.1. The high-velocity air jet is assumed to begin penetrating the bed at t > 0.The velocity of the jet is set as a boundary condition, and willtake e�ect as soon as the calculation begins. There is no need foradditional patching to incorporate the presence of the jet in thesimulation.



1.00E-06

3.00E-01

6.00E-01

Lmax = 5.999E-01 Lmin = 1.000E-06 Time = 0.000E+00Air Velocity Vectors (left) and Solids Volume Fraction (right)Bubbly Flow in a Fluidized Bed


Dec 18 1994Y

XZ

Figure 9.7.1: Patching Initial Conditions for Primary-Phase Veloc-ity and Secondary-Phase Volume Fraction



If you want to calculate a multiphase ow starting from a single-Copying PhaseVelocities andTemperature

phase ow solution, you can copy and patch the primary-phase ve-locities (and temperature, if appropriate) to the secondary phase(s)to be used as an initial guess. To copy velocities, use theCOPY-PHASE-VELOCITY command:

(*MAIN*)-

PA X X X X X X


COPY-PHASE-VELOCITY

WHICH PHASE WILL VELOCITY BE COPIED FROM


PHASE-1 PHASE-2 PHASE-3 QUIT HELP


(PHASE-SELECTION)-

P1

WHICH PHASE WILL VELOCITY BE COPIED TO


PHASE-1 PHASE-2 PHASE-3 QUIT HELP


(PHASE-SELECTION)-

P2

All velocity components of PHASE-1 will be copied and patched asan initial guess for PHASE-2. To copy the primary-phase veloci-ties to the third phase, you would select the COPY-PHASE-VELOCITYcommand again.

The COPY-PHASE-TEMPERATURE command works the same way asthe COPY-PHASE-VELOCITY command.

Stability problems which sometimes occur in the early stages ofDisabling theExchangeCoe�cient

steady-state simulations can be reduced by temporarily disablingthe calculation of the momentum exchange term. To do so, turn o�the Exchange Coe�cient option in the Multiphase Parameters panelor respond NO to the option ENABLE EXCHANGE COEFFICIENT in theEULERIAN MULTIPHASE OPTIONS table (see Section 9.7.13). Oncethe solution starts to converge, this term can be reintroduced tothe calculation.



At an outlet, the uxes of the primary phase are corrected by per-Disabling thePrimary Phase

Mass Correction atOutlets

forming a global continuity check for extra mass creation or de-struction in the domain. In some multiphase systems there is nonet out ow of primary phase going through the outlet of the do-main, so this action will create some spurious results. For example,for a bubble column in which only gas passes through the outlet,you would disable the mass correction for the primary phase (wa-ter). To disable the correction of the primary phase, turn o� thePrimary Phase Mass Correction option in the Multiphase Parameterspanel or respond NO to the option INCLUDE PRIMARY PHASE MASS

CORRECTION in the EULERIAN MULTIPHASE OPTIONS table (see Sec-tion 9.7.13).

When FLUENT calculates an unsteady granular ow with only oneTime-DependentGranular Flows solid phase, it will report the accumulated mass of solids inside the

domain in the header for the residual report:

(*)- FLUID PHASE CALCULATION

NTIME = 3 NITER = 125 MASS = 0.10E+00 TIME = 3.000E-03 S

.................NORMALIZED RESIDUALS...............

NTIME (P) (U) (V) (H) (PH2)

3 5.106E-01 2.216E-10 5.278E-09 2.359E-05 V2 0.000E+00

3 3.520E-01 4.310E-04 2.651E-03 4.000E+00VF2 4.520E-08

For time-dependent problems in which a secondary phase is createdInterphase MassTransfer

Calculationssolely due to interphase mass transfer and no other sources of thatphase exist, you must ensure that some secondary-phase mass is cre-ated within the �rst time step. This can be accomplished either bymanipulating the time step, or by patching a small secondary-phasevolume fraction in the computational domain before beginning thecalculation.

9.7.12 Starting From a Case File Created by an Earlier Release of FLUENT

If you wish to start your multiphase calculation from an existingV4.3 Case FilesFLUENT V4.3 Case File, you should be aware of the following mod-i�cations that may a�ect your solution:

� The default dispersed turbulence model now contains in u-ence from the dispersed (secondary) phase. For this reason,turbulent multiphase solutions may di�er from those obtainedwith Version 4.3.



� The calculation of turbulent viscosity for the secondary phasehas been improved.

� The default value for the false time step (described in Sec-tion 9.7.13) was the same as the real time step in Version 4.3.The default value is now 0.1 and this may have an e�ect onthe convergence rate.

� The default minimum value for phase volume fractions is now1� 10�7. In V4.3 it was 1� 10�6.

If you wish to start your multiphase calculation from an existingV4.24 Case FilesFLUENT V4.24 Case File, you should be aware of the followingmodi�cations that may a�ect your solution:

� The default coe�cient of restitution in FLUENT V4.24 was0.6111, and the current default is 0.8. To reproduce a V4.24solution, you should set the coe�cient of restitution to 0.6111.

� In V4.24, only the Gidaspow model was available for calcula-tion of the radial distribution function, the kinetic part of thesolids viscosity, and the exchange coe�cient. Since the currentdefault model is that of Syamlal et al., you will need to enablethe Gidaspow model in the Multiphase Parameters panel or theEULERIAN MULTIPHASE OPTIONS table (see Section 9.7.13) ifyou want to reproduce a V4.24 solution.

� In V4.24, the granular temperature was set to a �xed valueof 1 � 10�4. To reproduce a V4.24 solution, you must turno� the solution of the granular temperature and set the valueyourself (see Section 9.7.5).

9.7.13 Eulerian Multiphase Options

Several optional features are available for use with the Eulerian mul-tiphase model. You can control them in the Multiphase Parameterspanel or the EULERIAN MULTIPHASE OPTIONS table. The MultiphaseParameters panel is opened by clicking on the Multiphase Parame-ters... button in the Models panel:



The EULERIAN MULTIPHASE OPTIONS table can be reached from theEXPERT menu or from the MULTIPHASE menu that appears after youselect the multiphase model with the DEFINE-MODELS MULTIPLE-

PHASES command.

EXPERT �!EULERIAN-MULTIPHASE-OPTIONS

MULTIPHASE �!MULTIPHASE-OPTIONS



(EULERIAN MULTIPHASE OPTIONS)

NO ENABLE MASS TRANSFER

NO ENABLE MULTICOMPONENT-MULTIPHASE

YES STABILITY TERM IN P.D.E

1.0000E-01 FALSE TIME STEP FOR UNDER-RELAXATION (DIM)

YES ENABLE EXCHANGE COEFFICIENT

1.0000E-07 LOWER LIMIT FOR ALL VOFs (DIM)

NO ENABLE HEAT-EXCHANGE COEFFICIENT

NO ENABLE GIDASPOW MODEL

YES USE VISCOSITY FROM KINETIC THEORY

NO USE DRAG-CLUSTER CORRECTION

6.0000E-01 PACKING LIMIT FOR SOLID FRACTION (DIM)


The contents of the EULERIAN MULTIPHASE OPTIONS table and theaccessible items in the Multiphase Parameters panel will depend onthe current model settings. All items that may appear in this ta-ble/panel are explained below:

ENABLE MASS TRANSFER (Mass Transfer) allows you to model in-terphase mass transfer in the multiphase calculation, usingthe \shadow" technique described in Section 9.4.6. See Sec-tion 9.7.10 for details.

ENABLE SECONDARY TURBULENCE (Secondary Turbulence) enables ordisables the solution of k and � transport equations for eachsecondary phase. See Section 9.7.7 for details. (This optionappears in the table only for turbulent ows.)

ENABLE MULTICOMPONENT-MULTIPHASE (Multicomponent-MultiphaseModel) enables/disables the modeling of multiple species insecondary phases. When this option is turned o� (the de-fault), only the primary phase can contain multiple species.When it is turned on, you can specify multiple species for anyphase. See Section 9.7.8 for details. (This option appears inthe table only for ows involving species calculations.)

STABILITY TERM IN P.D.E. (Stability Term in P.D.E.) enables ordisables extra underrelaxation in the momentum equations.In multiphase ows it is possible to get a singular matrix whensolving the linear discretized equations for momentum. Thisis due to the fact that a phase volume fraction can have a zeroor near-zero value in some region, making the coe�cients ofthe matrix tend to zero. To avoid this problem FLUENT in-troduces extra underrelaxation using a \false time step". This



method adds to both sides of the discretized equation a termsimilar to the time-dependent term, with a modi�able falsetime step. The smaller the false time step the more underre-laxation is added, slowing down the solution and preventingdivergence.

This option is turned on by default, and you should generallynot turn it o�.

FALSE TIME STEP FOR UNDER-RELAXATION (False Time Step for Un-derrelaxation) sets the value of the false time step to be usedin the underrelaxation method described above for STABILITYTERM IN P.D.E.. The default value is 0.1. In general, youshould not need to change this value, but if you do lower it,you should be sure to decrease the convergence criteria as well.

ENABLE EXCHANGE COEFFICIENT (Exchange Coe�cient) allows youto temporarily turn o� the exchange coe�cient to reduce sta-bility problems that can occur in the early stages of a steady-state simulation. Once the solution starts to converge, youcan turn the exchange coe�cient back on again to reintro-duce it to the calculation. Leave the check button turned onor answer YES (the default) to include the exchange coe�-cient in the calculation, and turn the button o� or answer NOto temporarily disable it.

LOWER LIMIT FOR ALL VOFs (Lower Limit for all VOF's) sets thelower limit for all phase volume fractions. A value of zerois not advisable due to round-o� errors, so the default valueis 1� 10�7.

ENABLE HEAT-EXCHANGE COEFFICIENT (Heat Exchange Coe�cient)allows you to model interphase heat transfer in the multi-phase calculation, using the equations in Section 9.4.5. SeeSection 9.7.9 for details.

ENABLE GIDASPOW MODEL (Gidaspow Model) allows you to use theDing and Gidaspow [30][40] model to calculate the radial dis-tribution function and the kinetic part of the solids viscos-ity. The default response of NO indicates that the model ofSyamlal et al. [127] is used. The expressions used by eachof these models are presented in Section 9.4.4. If you pre-fer to use the Ding and Gidaspow model, turn on the En-able Gidaspow Model check button or answer YES to ENABLE

GIDASPOW MODEL. (This option appears in the table only forgranular ows.)



USE VISCOSITY FROM KINETIC THEORY (Kinetic Theory Viscosity),when turned o�, allows you to specify constant viscosities forsolid phases. By default, FLUENT calculates the solids viscos-ity from kinetic theory. If you want to set a constant viscosityinstead (e.g., if you are modeling inviscid granular ow andneed to set viscosity to a �xed small value), you can turn o�the viscosity calculation by turning o� the Kinetic Theory Vis-cosity check button or answering NO to USE VISCOSITY FROM

KINETIC THEORY. (This option appears in the table only forgranular ows.)

USE DRAG-CLUSTER CORRECTION (Drag-Cluster Correction) allows youto turn on drag-cluster correction to account for the size oflarge, dense regions of maximally-packed granular phase par-ticles. See Section 9.7.5 for details. (This option appears inthe table only for granular ows.)

PACKING LIMIT FOR SOLID FRACTION (Packing Limit for Solid Frac-tion) sets the maximum volume fraction for the solid phases(�s;max). For monodispersed spheres the packing limit is about0.63, and FLUENT uses a default value of 0.6. In polydispersedcases, however, smaller spheres can �ll the small gaps betweenlarger spheres, so the maximum packing limit may increase.(This option appears in the table only for granular ows.)

GRANULAR TEMPERATURE (Granular Temperature) allows you to seta �xed value for the granular temperature. This option willappear in the table only if you have turned o� solution ofthe granular temperature equation in the SELECT-VARIABLEStable or the Select Equations panel (see Section 9.7.5). Thedefault value for granular temperature is 1� 10�5 m2/s2.

REVERSE THE MOMENTUM EXCH. COEFFICIENT (Reverse the Momen-tum Exch. Coe�cient) allows you to reverse the drag law, ef-fectively switching the primary and secondary phases. This isuseful if you want to use one of the models that is restrictedto the primary phase (e.g., gas law or non-Newtonian owmodels) for the secondary phase. (This option appears in thetable only for two-phase nongranular ows.)

INCLUDE PRIMARY PHASE MASS CORRECTION (Primary Phase MassCorrection) allows you to turn o� the primary phase mass cor-rection. At the outlet, the uxes of the primary phase are cor-rected by performing a global continuity check for extra masscreation or destruction in the domain. In some multiphase sys-tems there is no net out ow of primary phase going through



the outlet of the domain, so this action will create some spu-rious results. For example, for a bubble column in which onlygas passes through the outlet, you would disable the masscorrection for the primary phase (water). To disable the cor-rection of the primary phase, turn o� the Primary Phase MassCorrection check button or respond NO to the option INCLUDE

PRIMARY PHASE MASS CORRECTION. (This option appears inthe table only for nongranular ows.)

9.8 Postprocessing Eulerian Multiphase Results

There is a wide selection of options available for postprocessingwith the Eulerian multiphase model. The options can be accessedfrom the Display or Plot pull-down menu or the VIEW-GRAPHICS textmenu, or from selections in VIEW-ALPHA. In the GUI, if you select avariable (in the variable selection drop-down list) that can be plot-ted for multiple phases, the Phase drop-down list below the variableselection list will become active, and you can choose the desiredphase. When you select a variable in the VARIABLE-SELECTION

text menu, if it can be plotted for multiple phases, FLUENT willask you to select the phase from a PHASE-SELECTION menu. Belowis an outline of some of the capabilities, along with some examples.

Velocity vectors can be plotted using the Velocity Vectors panel orVelocity Vectorsthe VELOCITY-VECTORS command.

To open the Velocity Vectors panel, select the Display/Vectors... menuitem:

Display �!Vectors...


9.8 Postprocessing Eulerian Multiphase Results 9-77

Select the phase for which you want to plot vectors in the Phasedrop-down list located directly below the Scale Factor on the leftside of the panel. The other Phase list (on the right side of thepanel, below the Color By drop-down list) is used if you choose tocolor the vectors by a variable that can be plotted for more thanone phase. For example, you could plot the primary-phase velocityvectors and color them by a secondary-phase volume fraction.

By default, the Color By �eld is set to be the velocity magnitude of!the primary phase. If you choose to plot the secondary-phase veloc-ity vectors and you would like them to be colored by the magnitudeof the secondary-phase velocity, remember to change the phase se-lection in both Phase drop-down lists.

Once you have speci�ed the desired phase(s) and options, click onDisplay to plot the velocity vectors.

In the text interface, the VELOCITY-VECTORS command is availablein the VIEW-GRAPHICS menu. When you select this command, youwill be prompted in the PHASE-SELECTION menu to specify the phasefor which you want to plot vectors.



VIEW-GRAPHICS �!VELOCITY-VECTORS

(VIEW-GRAPHICS)-

VV


SAND AIR QUIT HELP


(PHASE-SELECTION)-

If you have enabled the option to color the vectors by a scalar (eitherin the VECTOR-PARAMETERS table or by modifying the default ColorBy selection in the Velocity Vectors panel), you will next be promptedto select a variable in the VARIABLE-SELECTION menu. If you selecta variable that can be plotted for more than one phase, you will seeyet another PHASE-SELECTION menu:


VF


AIR SAND QUIT HELP


(PHASE-SELECTION)-

In Figure 9.8.1, velocity vectors for air (top) and sand (bottom) areplotted for an example in which an air-sand mixture enters a ductthrough an inlet centered on the left wall. On the right side of the�gures, it can be seen that the air velocity vectors have a slightupward component, while the sand vectors have a slight downwardcomponent, as expected.



1.72E-03

5.33E-02

1.05E-01 1.56E-01

2.08E-01

2.60E-01

3.11E-01 3.63E-01

4.14E-01

4.66E-01 5.17E-01

5.69E-01

6.21E-01 6.72E-01

7.24E-01

7.75E-01

8.27E-01 8.78E-01

9.30E-01

9.82E-01 1.03E+00

1.08E+00

1.14E+00

1.19E+00 1.24E+00

1.29E+00

1.34E+00 1.39E+00

1.45E+00

1.50E+00

Lmax = 1.497E+00 Lmin = 1.719E-03 Time = 1.210E+00

Time = 1.210E+00

Velocity Vectors for Air (top) and Sand (bottom)

Fluent Inc.

Fluent 4.30

Aug 29 1994Y

XZ

Figure 9.8.1: Velocity Vectors for Air (top) and Sand (bottom)



In 2D simulations, it is often very useful to examine stream functionStream Functioncontours to get an idea of the overall ow patterns as well as mass ow rates in di�erent regions of the domain. For a multiphase ow,stream function contours can be plotted for any of the phases.

To plot contours of stream function using the GUI, open the Con-tours panel:

Display �!Contours...

Select Velocity... and Stream Function in the Contours Of drop-downlists, and choose the desired phase in the Phase drop-down list.Click Display to plot the contours.

In the text interface, the choice of phases is made in the PHASE-SE-LECTION menu when a plot (of contours or pro�les, or an alphanu-meric display) of stream function is requested from the VARIABLE-SELECTION menu, as shown below.

VIEW-GRAPHICS �!CONTOURS



(VIEW-GRAPHICS)-

CON


SF


AIR SAND QUIT HELP


(PHASE-SELECTION)-

One of the most useful tools in the evaluation of a two phase owVolume Fractionis the display of volume fraction for each of the uids. In the GUI,Volume Fraction is available in the Multiphase... category of thevariable selection drop-down list that appears in the panels openedfrom the Display and Plot pull-down menus. In the Contours panel,for example, you can select Multiphase... and Volume Fraction, andthen choose the phase for which you want to plot the volume fractionin the Phase drop-down list.



In the text interface, VOLUME-FRACTION is found in the VARIABLE-SE-LECTION menu, and it can be accessed from the VIEW-ALPHA andVIEW- GRAPHICS menus.

COMMANDS AVAILABLE FROM VARIABLE-SELECTION:

DENSITY EXCHANGE-MASS

EXCHANGE-X EXCHANGE-Y

MACH-NUMBER MOLECULAR-VISCOSITY

STATIC-PRESSURE-REL STATIC-PRESSURE-ABS

STREAM-FUNCTION TOTAL-PRESSURE-REL

TOTAL-PRESSURE-ABS U-VELOCITY

V-VELOCITY VELOCITY-MAGNITUDE

VOLUME-FRACTION MULTIPHASE-CONCENTRATION

SOLIDS-PRESSURE GRANULAR-TEMP

XMOM-SOURCE YMOM-SOURCE

XTENDED-XOPTIONS QUIT

HELP



VF

Figures 9.8.2 and 9.8.3 show volume fraction plots for the separation-tee example of Section 9.3.1. In Figure 9.8.2, �lled contours of theprimary phase (water) volume fraction are presented, and mark thebubble region where very little water is present. In Figure 9.8.3,�lled contours of the secondary phase (air) volume fraction areshown. This �gure is the inverse of Figure 9.8.2, since the bub-ble region is now shown to be the region where the volume fractionis greatest.

For granular ows, the solids (or granular) pressure as well as theSolids Pressure uid pressure can be displayed using graphics or alphanumerics.This is accomplished in the GUI by selecting Solids Pressure in theMultiphase... category of the variable selection drop-down list thatappears in the panels opened from the Display or Plot pull-downmenu. In the text interface, select the SOLIDS-PRESSURE com-mand in the VARIABLE-SELECTION menu. As an example, considera square tank with an inlet on the upper left wall and outlet onthe upper right wall. The inlet mixture carries a 30% solids-waterslurry into the domain, and the solids settle on the bottom of thetank as time increases. In Figure 9.8.4, 5 seconds after the start ofthe �lling process, velocity vectors for the water phase are shown,indicating the general ow pattern for the slurry.

The build-up of the granular phase on the tank bottom is shown in



1.41E-02

4.70E-02

7.99E-02

1.13E-01

1.46E-01

1.78E-01

2.11E-01

2.44E-01

2.77E-01

3.10E-01

3.43E-01

3.76E-01

4.08E-01

4.41E-01

4.74E-01

5.07E-01

5.40E-01

5.73E-01

6.06E-01

6.39E-01

6.71E-01

7.04E-01

7.37E-01

7.70E-01

8.03E-01

8.36E-01

8.69E-01

9.01E-01

9.34E-01

9.67E-01

1.00E+00

Max = 1.000E+00 Min = 1.414E-02

Water Volume Fraction (Dim)


Fluent Inc.

Fluent 4.30

Dec 06 1994Y

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Figure 9.8.2: Water Volume Fraction in a Tee-Junction

1.00E-06

3.29E-02

6.57E-02

9.86E-02

1.31E-01

1.64E-01

1.97E-01

2.30E-01

2.63E-01

2.96E-01

3.29E-01

3.61E-01

3.94E-01

4.27E-01

4.60E-01

4.93E-01

5.26E-01

5.59E-01

5.92E-01

6.24E-01

6.57E-01

6.90E-01

7.23E-01

7.56E-01

7.89E-01

8.22E-01

8.54E-01

8.87E-01

9.20E-01

9.53E-01

9.86E-01

Max = 9.859E-01 Min = 1.000E-06



Fluent Inc.

Fluent 4.30

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Figure 9.8.3: Air Volume Fraction in a Tee-Junction



3.72E-05

4.00E-02

8.00E-02 1.20E-01

1.60E-01

2.00E-01

2.40E-01 2.80E-01

3.20E-01

3.60E-01 4.00E-01

4.40E-01

4.80E-01 5.20E-01

5.60E-01

6.00E-01

6.40E-01 6.80E-01

7.20E-01

7.59E-01 7.99E-01

8.39E-01

8.79E-01

9.19E-01 9.59E-01

9.99E-01

1.04E+00 1.08E+00

1.12E+00

1.16E+00

Max = 1.159E+00 Min = 3.721E-05 Time = 5.000E+00

Water Velocity Vectors (M/S)

Transient Filling of a Tank with Solids

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Figure 9.8.4: Water Velocity Vectors for a Granular Settling Tank

Figure 9.8.5. The solids pressure, having a maximum at the bottomof the tank where the concentration of solids is greatest, can be seenin Figure 9.8.6.

For granular ows, a bulk and shear viscosity are computed, ac-Shear Viscosity forthe Solid Phase cording to the kinetic theory approach. For laminar ows, a report

of the shear viscosity for the solid phase is available from both thealphanumerics and graphics menus. In the GUI graphics panels, se-lect Molecular Viscosity in the Properties... category of the variableselection drop-down list. Then select the desired phase in the Phasedrop-down list. In the text interface, use the MOLECULAR-VISCOSITYcommand in the VARIABLE-SELECTION menu. FLUENT will ask youto specify the phase in the resulting PHASE-SELECTION menu. Forthe case of the uidized bed described in Section 9.3.2, the shearviscosity for the solid phase is plotted in Figure 9.8.7.

Additional postprocessing capabilities are available for alphanu-Additional Featuresmeric and graphical reporting for the individual phases. Some ofthese are listed below. As mentioned earlier, when you select avariable that can be plotted or reported for more than one phase,FLUENT will prompt you to select a phase in the Phase drop-downlist or the PHASE-SELECTION menu.



1.84E-02

3.78E-02

5.72E-02

7.66E-02

9.60E-02

1.15E-01

1.35E-01

1.54E-01

1.73E-01

1.93E-01

2.12E-01

2.32E-01

2.51E-01

2.70E-01

2.90E-01

3.09E-01

3.29E-01

3.48E-01

3.67E-01

3.87E-01

4.06E-01

4.25E-01

4.45E-01

4.64E-01

4.84E-01

5.03E-01

5.22E-01

5.42E-01

5.61E-01

5.81E-01

6.00E-01

Max = 5.999E-01 Min = 1.843E-02 Time = 5.000E+00

Sand Volume Fraction (Dim)


Fluent Inc.

Fluent 4.30

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Figure 9.8.5: Sand Volume Fraction in a Granular Settling Tank

3.88E-04

1.20E+01

2.41E+01

3.61E+01

4.81E+01

6.02E+01

7.22E+01

8.43E+01

9.63E+01

1.08E+02

1.20E+02

1.32E+02

1.44E+02

1.56E+02

1.69E+02

1.81E+02

1.93E+02

2.05E+02

2.17E+02

2.29E+02

2.41E+02

2.53E+02

2.65E+02

2.77E+02

2.89E+02

3.01E+02

3.13E+02

3.25E+02

3.37E+02

3.49E+02

3.61E+02

Max = 3.611E+02 Min = 3.878E-04 Time = 5.000E+00

Sand Solids Pressure (Pa)


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Figure 9.8.6: Sand Solids Pressure in a Granular Settling Tank



5.10E-10

2.00E-03

3.99E-03

5.99E-03

7.98E-03

9.98E-03

1.20E-02

1.40E-02

1.60E-02

1.80E-02

2.00E-02

2.20E-02

2.39E-02

2.59E-02

2.79E-02

2.99E-02

3.19E-02

3.39E-02

3.59E-02

3.79E-02

3.99E-02

4.19E-02

4.39E-02

4.59E-02

4.79E-02

4.99E-02

5.19E-02

5.39E-02

5.59E-02

5.79E-02

5.99E-02

Max = 5.987E-02 Min = 5.103E-10 Time = 2.000E-01

Molecular Viscosity Of Solids (Kg/M-S)


Fluent Inc.

Fluent 4.30

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Figure 9.8.7: Shear Viscosity for the Solid Phase in a Fluidized Bed

The following parameters are available in the VARIABLE-SELECTIONmenu in the text interface. The category in which you will �ndthem in the GUI is listed in parentheses. Note that this is not acomplete list of the parameters in the VARIABLE-SELECTION menu.See Section 18.1 for information about parameters not listed below.

� VELOCITY-MAGNITUDE (Velocity Magnitude in the Velocity... cat-egory): Velocity magnitudes for any phase.

� U-, V-, and W-VELOCITY (U, V, andW Velocity in the Velocity...category): Velocity components for any phase.

� DENSITY (Density in the Density... category): The density ofany phase.

� MULTIPHASE-CONCENTRATION (Multiphase Concentration in theMultiphase... category): E�ective density (density � volumefraction) for any phase.

� MOLECULAR-VISCOSITY (Molecular Viscosity in the Properties...category): The molecular viscosity of any uid phase or theshear viscosity computed by FLUENT for any solid phase.



(Not available for turbulent ows. See e�ective viscosity, be-low.)

� EFFECTIVE-VISCOSITY (E�ective Viscosity in the Properties...or Turbulence... category): The e�ective viscosity of any phase.

The following parameters are found in the XTENDED-XOPTIONS textmenu:

� I-, J-, and K-FACE-FLOW-RATE (I, J, and K Face Flow Ratein the Mass Fluxes... category): Face uxes (in SI units ofkg/s) in the 3 grid directions for any phase.

In VIEW-ALPHA SELECT-VARIABLE:

� Integral reports based on either the primary phase, or oneof the secondary phases. In the sample report below, theintegrated quantities reported in the �rst �ve columns aredescribed in Section 18.4. The �nal column reports the volumeof the speci�ed phase in each cell, determined by multiplyingthe volume fraction by the cell volume. The sum at the endof the report gives the total volume of the speci�ed phase.




INTEGRALS-I-DIRECTION

(BOUNDS)-

.

(L)- PRINT CELL-BY-CELL INFO (OTHERWISE SUMMARY ONLY)?

(L)- Y OR N ++(DEFAULT-YES)++

X

(L)- DEFAULT ASSUMED

(*)- SELECT PHASE FOR INTEGRALS


WATER SAND QUIT HELP


(PHASE-SELECTION)-

WATER

INTEGRATED QUANTITIES FOR CELL ZONE ' .'

I-DIRECTION COMPONENTS, WATER

AREA VELOCITY VOL. FLOW MASS FLOW DELTA P PH.- VOL.

I, J, K M2 M/S M3/S KG/S PA M3

----------- ---------- ---------- ---------- ---------- ---------- ----------

2, 51, 1 1.000E-02 9.959E-01 9.959E-03 7.030E+00 -2.527E+00 7.059E-05

3, 51, 1 1.000E-02 9.940E-01 9.940E-03 7.106E+00 -5.267E+00 7.149E-05

4, 51, 1 1.000E-02 9.932E-01 9.932E-03 7.197E+00 -3.381E+00 7.247E-05

5, 51, 1 1.000E-02 9.921E-01 9.921E-03 7.290E+00 -2.615E+00 7.348E-05

6, 51, 1 1.000E-02 9.908E-01 9.908E-03 7.380E+00 -1.935E+00 7.448E-05

7, 51, 1 1.000E-02 9.893E-01 9.893E-03 7.467E+00 -1.520E+00 7.547E-05

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

47, 2, 1 1.000E-02 -4.994E-01 -4.994E-03 -3.378E+00 4.075E+01 6.763E-05

48, 2, 1 1.000E-02 -4.503E-01 -4.503E-03 -3.064E+00 4.763E+01 6.805E-05

49, 2, 1 1.000E-02 -3.834E-01 -3.834E-03 -2.618E+00 4.944E+01 6.827E-05

50, 2, 1 1.000E-02 -2.819E-01 -2.819E-03 -1.931E+00 5.859E+01 6.852E-05

51, 2, 1 1.000E-02 -1.127E-01 -1.127E-03 -7.531E-01 7.742E+00 6.681E-05

------------------------------------------------------------------------------

TOTALS 2.500E+01 2.474E+00 1.923E+03 2.060E-01



In VIEW-ALPHA SELECT-VARIABLE XTENDED-XOPTIONS:

� You can use the TOTAL-MASS command to compute the totalmass of each phase:


XX

(EXTENDED-OPTIONS)-

TOTAL-MASS

*** TOTAL MASS SUMMARY ***

SOLIDS : 1.5593E-01 (KG)

AIR : 2.1752E+03 (KG)

This command is useful for determining the mass accumulatedat a given point in time for a time-dependent multiphase cal-culation.

� When you use the TOTAL-ZONE-INTEGRALS command (describedin detail in Section 18.1) for a multiphase calculation, all infor-mation in the report will be listed separately for each phase.


phase change simulation

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