Multiphase Flow

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CFX-5 Solver and Solver Manager

Multiphase Flow

• Introduction p. 382

• Specifying Fluids for Multiphase Flow p. 387

• Multifluid Model p. 389

• Phasic Hydrodynamic Equations p. 392

• Interphase Momentum Transfer Models p. 394

• Heat Transfer p. 407

• Turbulence Modelling in Multiphase Flow p. 411

• Multicomponent Multiphase Flow p. 415

• Additional Variables in Multiphase Flow p. 416

• Sources in Multiphase Flow p. 421

• Boundary Conditions in Multiphase Flow p. 422

• Modelling Advice for Multiphase Flow p. 426

• Restrictions p. 429

Page 381 CFX-5.5.1


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Introduction

Multiphase flow refers to the situation where more than one fluid ispresent. Each fluid may possess its own flow field, or all fluids mayshare a common flow field. Unlike multicomponent flow, the fluids arenot mixed on a microscopic scale in multiphase flow. CFX-5 includes avariety of multiphase models to allow the simulation of processeswhich transport and bring into direct contact multiple fluid streams toeffect mixing, reaction, and separation.

Multiphase flow in CFX-5 uses a Eulerian–Eulerian multiphasemodel. Two different sub-models are available for Eulerian-Eulerianmultiphase flow:

The Homogeneous Model

This is the simplest model, in which all fluids share the same flow field.It is suitable for modelling:

• Free surface flows where the interface is well defined.

• Dispersed flows of extremely small particles.

This model is discussed separately, see Homogeneous Multiphaseand Free Surface Flow (p. 431) for details.

The Inter-fluid Transfer or Inhomogeneous Model

Each fluid possesses its own flow field and the fluids interact viainterphase transfer terms. Two different sub-models are availablewhich differ in the way they model the interphase transfer terms. Theseare:

• The Particle Model . This is suitable for modelling simple dispersedmultiphase flow problems, for example:

• the dispersion of gas bubbles in a liquid,• the dispersion of liquid droplets in a gas or in immiscible liquid,• the dispersion of solid particles in a gas or in a liquid.

• The Mixture Model . This is suitable for modelling more complexmultiphase flow problems, for example:

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• Gas-liquid flow containing both continuous and dispersed gasand liquid phases.

• Churn flow.

This chapter describes how multiphase flow is modelled and alsoincludes modelling advice.

The corresponding information for single-phase, single-componentflow can be found in the following:

• Details of the models used for single-phase flow can be found inPhysical Models, Fluid Properties and Boundary Conditions inCFX-5 (p. 207).

• The mathematical implementation of the single-phase flow can befound in Mathematical Models and Solution Algorithms (p. 249) .

• Advice on modelling can be found in Advice on Flow ModellingWith CFX-5 (p. 469) .

You are advised to read the relevant sections in the single-phasedocumentation before reading this chapter, since this chapter extendsthe information given in the single-phase documentation.

The notation used in this chapter is defined in List of Symbols (p. 250)and in Multiphase Notation (p. 385) .

TerminologyThe following terms are used in CFX-5 multiphase flow.

Multiphase Flow

Multiphase flow is a flow in which more than one fluid is present. Ingeneral, the fluids consist of different chemical species, e.g. air-water.In some applications, they may represent different thermodynamicphases of the same species, e.g. steam-water.

It is important to distinguish between multicomponent and multiphaseflow. A multicomponent fluid is assumed to consist of a mixture ofchemical species which are mixed at the molecular level. In this case,we solve for single mean velocity and temperature fields etc. for thefluid. Examples are gaseous mixtures, and solutes in liquids.

83 Introduction CFX-5.5.1


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The fluids in a multiphase flow are assumed mixed at macroscopiclength scales, much larger than molecular. Examples are gas bubblesin a liquid, liquid droplets in a gas or another immiscible liquid etc. Inthis case. it is necessary to solve for different velocity and temperaturefields etc. for each fluid. These may interact with each other by meansof interfacial forces and heat and mass transfer across the phaseinterfaces.

For example, if cold wet particles are injected into a fast flowing streamof hot air, the particles will be accelerated by interphase drag, they willbe heated up by heat transfer across the phase boundary, and they willbe dried by evaporation of water into water vapour at the phaseboundary.

Multicomponent Multiphase Flow

It is possible to combine the notions of multicomponent and multiphaseflows. In this case, more than one fluid is present, and each such fluidmay be a mixture of chemical species mixed at molecular lengthscales. An example is air bubbles in water in which ozone gas isdissolved in both the gaseous and liquid phases. In this case, masstransfer of common species may occur by diffusion across the phaseinterface.

Morphology

Morphology is used to describe the “connectivity” or “distribution” of thefluid – whether it forms a single continuous medium, or whether it ispresent, for example, in small droplets which are not connected.

Dispersed Phase

A dispersed solid or dispersed fluid is one which is present in discreteregions which are not connected. Examples are water droplets in air,bubbles, and solid particulates.

Continuous Phase

A continuous phase or continuous fluid is one which forms a connectedcontinuous region. An example is air, when modelling rain drops in air.



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Volume Fraction

Multiphase modelling employs the notion of interpenetratingcontinua . Although phases are mixed at length scales much largerthan molecular, they are also assumed to be mixed at length scalessmaller than we wish to resolve.

Thus, each phase is assumed to be present in principle in each controlvolume, and assigned a volume fraction equal to the fraction of thecontrol volume occupied by that phase.

Eulerian-Eulerian

The Eulerian-Eulerian model is the multiphase model that has beenimplemented in CFX-5. Future versions of CFX-5 will support othermultiphase models, such as the Lagrangian Tracking model.

Within the Eulerian-Eulerian model, the interphase transfer terms canbe modelled using either the Particle Model , the Mixture Model or theHomogeneous Model .

Multiphase NotationDifferent phases of fluids are denoted using lowercase Greek letters α,β, γ, etc. In general, a quantity subscribed with α, β, γ, etc., refers to thevalue of the quantity for that particular phase. For example, the volumefraction of α is denoted rα. Thus, the volume Vα occupied by phase αin a small volume V around a point of volume fraction rα is given by:

It is important to distinguish between the material density of fluid α ρα,the mixture density of a fluid, and the effective density of fluid α. Theeffective density is defined as:

The material density, ρα, is the density of the fluid if it is the only phasepresent, i.e. the mass of α per unit volume of α. The effective densityis the actual mass per unit volume of phase α, given that phase α onlyoccupies a fraction of the volume, i.e. the mass of α per unit volume ofthe bulk fluid.

Vα rαV=

ρ̃α rαρα=



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The mixture density is given by:

ρm ραrαα∑=



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Specifying Fluids for Multiphase Flow

Currently, you are allowed to select five fluids for multiphase flow. Toselect the fluids for a multiphase flow, set the number of fluids on thedomains form, and select distinct fluids from the library of fluids whichhave been defined under the fluids editor.

In general each fluid possesses its own velocity, temperature andvolume fraction field, although it shares a common pressure field withother fluids. If more than one fluid is used in a simulation, then thevolume fractions of the fluids must sum to unity at any computationalpoint in space and in time.

Fluid specific attributes are defined on the Fluid Details form inCFX-Build - see Fluid Details (p. 1082 in CFX-Build: Chapter 4) .

• Morphology : see Morphology (p. 1040 in CFX-Build:Chapter 4) .

• Mean Diameter : see Mean Diameter (p. 1040 in CFX-Build:Chapter 4)

• Minimum Volume Fraction : see Minimum Volume Fraction(p. 1041 in CFX-Build: Chapter 4) .

• Maximum Packing : see Maximum Packing (p. 1041 inCFX-Build: Chapter 4)

Multiphase ExamplesThe following are examples of multiphase flow.

Water droplets in air

Water droplets in air constitute two different fluids which are mixed atthe macroscopic level and not the microscopic level. Hence you needto use a multiphase model and define two distinct fluids, Water and Air .Each fluid contains only one component comprising of one material (airor water), whose properties can be defined. Air is the ContinuousPhase , and Water is the Dispersed Phase .

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Air bubbles in water

As above, you need to use a multiphase model and define two distinctfluids, Water and Air . In this case. Air is the Dispersed Phase , andWater is the Continuous Phase .

Gas-solid and liquid-solid flow

It is possible to model the motion of a large number of solid particles ina gas or a liquid as an Eulerian-Eulerian two phase flow. Examplesoccur in pneumatic conveying, sedimentation in rivers, and fluidisedbeds.

For example, the two phases may be Water , the Continuous Phase ,and Sand , the Dispersed Phase .

In such problems, you should assign the solid phase a smallinsignificant molecular viscosity. This is permissible, as the physics isdominated by inter-phase drag and turbulence effects. The solid phaseshould be assigned free slip boundary conditions at walls.

Three-phase flow

It is possible to have more than one dispersed phase in a continuousphase. For example, certain regimes of water-oil-gas flow in an oilpipeline may involve both oil droplets and gas bubbles immersed in acontinuous water phase.

Poly-dispersed flow

The above dispersed flow examples assume a single mean particlediameter for the dispersed phases. Poly-dispersed flows involvedispersed phases of different mean diameters. In these cases, it isnecessary to define a different phase for each particle size, as theparticle size has a strong influence on interphase transfer. Forexample, to model the flow of air bubbles in water of differentdiameters, you need to create three fluids, Air1, Air2, Air3 say, eachwith the same material properties as air. Then set the number of fluidsequal to four, and select Water, Air1, Air2 and Air3 as the four fluids.Next, under Fluid Models, designate Water as the continuous phase,and the other three fluids as dispersed phases of desired meandiameters.

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Multifluid Model

In the multifluid model there is one solution field for each separatephase. Transported quantities interact via interphase transfer terms.For example, two phases may have separate velocity and temperaturefields, but there will be a tendency for these to come to equilibriumthrough interphase drag and heat transfer terms.

Phases are labelled using Greek indices α, β, γ. The total number ofphases is NP. The volume fraction of each phase is denoted rα, whereα = 1 to NP.

Interfacial Area DensityInterfacial transfer of momentum, heat and mass is directly dependenton the contact surface area between the two phases. This ischaracterised by the interfacial area per unit volume between phase αand phase β, known as the interfacial area density, Aαβ. Note that it hasdimensions of one over length.

Interfacial transfer can be modelled using either the particle or mixturemodels. These essentially provide different algebraic prescriptions forthe interfacial area density.

The Particle Model

The particle model for interfacial transfer between two phasesassumes that one of the phases is continuous (phase α) and the otheris dispersed (phase β). The surface area per unit volume is thencalculated by assuming that phase β is present as spherical particlesof Mean Diameter dβ. Using this model, the interphase contact area is:

Non-dimensional interphase transfer coefficients may be correlated interms of the particle Reynolds number and the fluid Prandtl number .These are defined using the particle mean diameter, and thecontinuous phase properties, as follows:

Aαβ6rβdβ--------=

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where µα, CPα and λα are the viscosity, specific heat capacity andthermal conductivity of the continuous phase α.

The Mixture Model

This is a very simple model which treats both phases α, βsymmetrically. It may be appropriate as a first approximation or as thebasis of user supplied interfacial transfer models for a calculation ofnon-disperse liquid-liquid or gas-liquid two phase flow. It does notmake any assumptions about the morphology of the phases. Thesurface area per unit volume is calculated from

where dαβ is an interfacial length scale, which you must specify. This isdone on the Multiphase Options (p. 1088 in CFX-Build: Chapter 4)form in CFX-Build.

By way of example, suppose we have oil-water flow in which we mayhave either water droplets in continuous oil, or oil droplets incontinuous water, in the limits , respectively. Then, asimple model for interfacial area density which has the correctbehaviour in these two limits is given by:

Non-dimensional interphase transfer coefficients may be correlated interms of the mixture Reynolds number and Prandtl number definedas follows:

Reαβρα Uβ Uα– dβ

µα--------------------------------------=

PrαβµαCPα

λα------------------=

Aαβrαrβdαβ-----------=

rα 0→ rβ 0→

Aαβ6rαrβ

rαdβ rβdα+-------------------------------= dαβ

rαdβ rβdα+

6-------------------------------=⇒

Reαβραβ Uβ Uα– dαβ

µαβ--------------------------------------------=



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where ραβ, µαβ, CPαβ and λαβ are the density, viscosity, specific heatcapacity and thermal conductivity of the mixture respectively, definedby:

etc.

PrαβµCPαβ

λαβ-----------------=

ραβ rαρα rβρβ+=

µαβ rαµα rβµβ+=



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Phasic Hydrodynamic Equations

The following is a summary of the equations of momentum and masstransfer for inhomogeneous multiphase flow in CFX-5. The equivalentfor single-phase flow can be found in Transport Equations (p. 262) .The equivalent for homogeneous multiphase flow can be found inHomogeneous Hydrodynamic Equations (p. 434) .

Momentum Equations

where:

• SMα describes momentum sources due to external body forces, anduser defined momentum sources, see User Specified Sources(p. 270).

• Mα describes the interfacial forces acting on phase α due to thepresence of other phases. See Interphase Momentum TransferModels (p. 394) for the models available for interfacial forces.

• The above momentum equations are valid for fluid phases only. Fordispersed solid phases, additional terms are present representingadditional stresses due to particle collisions.

Continuity Equations

(Eq. Sol.-74)

where:

• SMSα describes user specified mass sources, see User SpecifiedSources (p. 270) .

Volume Conservation Equation

This is simply the constraint that the volume fractions sum to unity:

t∂∂

rαραUα( ) rα ραUαUα( )( )∇•+

rα pα∇–= rαµα Uα∇ Uα∇( )T+( )( )∇• SMα M α+ + +

∂∂t----- rαρα( ) ∇ rαραUα( )•+ SMSα=

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(Eq. Sol.-75)

Pressure Constraint

The complete set of hydrodynamic equations represent 4NP + 1equations in the 5NP unknowns UαVα, Wα, rα, Pα. We need NP -1 moreequations to close the system. These are given by constraints on thepressure, namely that all phases share the same pressure field:

rαα 1=

NP

∑ 1=

pα p for all α 1,......, NP= =

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Interphase Momentum Transfer Models

Interphase momentum transfer, Mαβ, occurs due to interfacial forcesacting on each phase α due to interaction with another phase β. Thetotal force on phase α due to interaction with other phases is denotedMα, and is given by:

Note that interfacial forces between two phases are equal andopposite, so the net interfacial forces sum to to zero:

The total interfacial force acting between two phases may arise fromseveral independent physical effects:

The forces indicated above respectively represent the interphase dragforce, lift force, wall lubrication force, virtual mass force, turbulencedissipation force and solids pressure force (for dense solid particlephases only).

CFX-5.5.1 provides a wide range of physical models for the mostimportant of these forces, namely interphase drag. These models aredescribed in Interphase Drag (p. 394) .

CFX-5.5.1 also provides a simple model for the turbulent dispersionforce. See Interphase Turbulent Dispersion Force (p. 404) .

Models for other forces will implemented in future releases of CFX-5.

Interphase DragThe following general form is used to model interphase drag forceacting on phase α due to phase β.

M α M αββ α≠∑=

M αβ M– βα=( ) M αα∑ 0=⇒

M αβ M αβD

M αβL

M αβLUB

M αβVM

M αβTD

M S ....+ + + + + +=

M α cαβd( )

Uβ Uα–( )=

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Note that cαα = 0 and cαβ = cβα. Hence the sum over all phases of allinterphase transfer terms is zero.

In this section, we describe how the coefficients may be computedfrom a knowledge of dimensionless drag coefficients. We also describethe range of models available for drag coefficients.

For low Mach number flows, the drag exerted on an immersed body bya moving fluid arises from two mechanisms only . The first is due to theviscous surface shear stress, and is called skin friction. The secondis due to the pressure distribution around the body, and is called theform drag. We denote the continuous phase by α and the dispersedphase by β. The total drag force is most conveniently expressed interms of the dimensionless drag coefficient ,

where ρ is the fluid density, ( Uα - Uβ ) is the relative speed, D is themagnitude of the drag force and A is the projected area of the body inthe direction of flow.

Interphase Drag for the Particle ModelFor spherical particles, the coefficients may be derivedanalytically. The area of a single particle projected in the flow direction,Ap, and the volume of a single particle Vp are given by:

where d is the mean diameter. The number of particles per unit volume,np is given by:

The drag exerted by a single particle on the continuous phase is,

cαβd( )

CDD

12---ρα Uα Uβ–( )2

A--------------------------------------------=

cαβd( )

Apπd

2

4---------= Vp

πd3

6---------=

np

rβVp-------

6rβ

πd3

---------= =

Dp12---CDραAp Uβ Uα– Uβ Uα–( )=



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Hence the total drag per unit volume on the continuous phase is,

Comparing with the momentum equation for phase α, where the dragforce per unit volume is:

we get,

which can be written as:

This is the form implemented in CFX-5.

For a particle of simple shape, immersed in a Newtonian fluid andwhich is not rotating relative to the surrounding free stream, the dragcoefficient, CD, depends only on the particle Reynolds number. Thefunction CD ( Reαβ ) may be determined experimentally, and is knownas the drag curve .

CFX-5 offers several different models for the drag curve, and alsoallows you to specify the drag coefficients directly.

This section describes drag correlations specific to dispersedmultiphase flow.

Specifying a Drag Coefficient

You can choose to specify the dimensionless Drag Coefficients CDdirectly. This is done by selecting to use the Drag Coefficient option onthe Multiphase Options form in CFX-Build, and entering theappropriate Drag Coefficient (see Multiphase Options (p. 1088 inCFX-Build: Chapter 4)

Dαβ npDp34---

CD

d--------rβρα Uβ Uα– Uβ Uα–( )= =

Dαβ cαβd( )

Uβ Uα–( )=

cαβd( ) 3

4---

CD

d--------rβρα Uβ Uα–=

cαβd( ) CD

8--------Aαβρα Uβ Uα–=



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You can supply your own drag coefficient correlation as an expressionand are free to define your own interfacial Reynolds number, or anyother dimensionless group.

Dilute Solid Particles

At low particle Reynolds numbers (the viscous regime), the dragcoefficient for flow past spherical particles may be computedanalytically. The result is Stokes’ law:

For particle Reynolds numbers which are sufficiently large for inertialeffects to dominate viscous effects (the inertial or Newton’s regime),the drag coefficient becomes independent of Reynolds number:

In the transitional region between the viscous and inertial regimes,0.1 < Re < 1000for spherical particles, both viscous and inertial effectsare important. Hence, the drag coefficient is a complex function ofReynolds number, which must be determined from experiment.

This has been done in detail for spherical particles . Several empiricalcorrelations are available. The one available in CFX-5.5.1 is due toSchiller and Naumann (1933) [7] . It can be written as follows:

Schiller Naumann Drag Model

CFX-5 modified this to ensure the correct limiting behaviour in theinertial regime by taking:

This should only be used for solid spherical particles, or for fluidparticles that are sufficiently small that they may be consideredspherical. For non-spherical particles, the user should supply the dragcurve from experiment.

CD24Re------,= Re 1«

CD 0.44,= 1000 Re 1 2x105

–≤ ≤

CD24Re------ 1 0.15Re

0.687+( )=

CD max24Re------ 1 0.15Re

0.687+( ) 0.44,

=



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You can select this Drag Curve by selecting to use the SchillerNaumann Drag Model on the Multiphase Models form in CFX-Build(see Multiphase Options (p. 1088 in CFX-Build: Chapter 4) ).

Dense Solid Particles

As the Schiller Naumann correlation is derived for flow past a singlespherical particle, it is only valid in the dilute limit of very small solidphase volume fractions.

On the other hand, the Wen Yu correlation is valid for solid phasevolume fractions at least up to 0.2, and possibly higher.

Wen Yu Drag Model

Note that this has the same functional form as the Schiller Naumanncorrelation, with a modified particle Reynolds number, and a power lawcorrection, both functions of the continuous phase volume fraction rc.

You can select this Drag Curve by selecting to use the Wen Yu DragModel on the Multiphase Models form in CFX-Build (see MultiphaseOptions (p. 1088 in CFX-Build: Chapter 4) ).

You may also change the Volume Fraction Correction Exponentfrom its default value of -1.65, if you wish.

For very dense gas-solid or liquid-solid flows, such as occur in fluidisedbed applications, the Gidaspow correlation is recommended. This usesthe Wen Yu correlation for low solid volume fractions rd < 0.2, andswitches to Ergun’s law for flow in a porous medium for larger solidvolume fractions.

Gidaspow Drag Model

CD rc1.65–

max24Re′-------- 1 0.15Re'

0.687+( ) 0.44,

= Re' r cRe=

CD CD Wen Yu( ),= r c 0.8>

cαβd( )

1501 r c–( )2µc

rcdp2

--------------------------- 74---

1 r c–( )ρc Uc Ud–

dp------------------------------------------------ ,+= r c 0.8<



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You can select this Drag Curve by selecting to use the Gidaspow DragModel on the Multiphase Models form in CFX-Build (see MultiphaseOptions (p. 1088 in CFX-Build: Chapter 4) ). You may also changethe Volume Fraction Correction Exponent of the Wen Yu part of thecorrelation from its default value of -1.65, if you wish.

Dilute Fluid Particles (drops and bubbles)

At sufficiently small particle Reynolds numbers (the viscous regime ),fluid particles behave in the same manner as solid spherical particles.Hence the drag coefficient is well approximated by the Schiller-Naumann correlation described above.

At larger particle Reynolds numbers, the inertial or distorted particleregime , surface tension effects become important. The fluid particlesbecome, at first, approximately ellipsoidal in shape, and finally,spherical cap shaped.

In the spherical cap regime, the drag coefficient is well approximatedby:

Several correlations are available for the distorted particle regime.CFX-5 uses the Ishii Zuber and Grace and correlations.

Ishii-Zuber Drag Model

This is applicable to general fluid particles (drops and bubbles), for anypair of phases.

In the distorted particle regime, the drag coefficient is approximatelyconstant, independent of Reynolds number, but dependent on particleshape through the dimensionless group known as the Eotvosnumber , which measures the ratio between gravitational and surfacetension forces:

CD sphere( ) max24Re------ 1 0.15Re

0.687+( ) 0.44,

=

CD cap( ) 83---=

Eog∆ρdp

2

σ-----------------=



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Here, ∆ρ is the density difference between the phases, g is thegravitational acceleration, and σ is the surface tension coefficient.

The Ishii-Zuber correlation gives:

You can select this Drag Curve by selecting to use the Ishii-ZuberModel on the Multiphase Models form in CFX-Build (see MultiphaseOptions (p. 1088 in CFX-Build: Chapter 4) ).

In this case, CFX-5 automatically takes into account the sphericalparticle and spherical cap limits by setting:

The Ishii Zuber Model also automatically takes into account dense fluidparticle effects. See Dense Fluid Particles (p. 401) for full details.

Grace Drag Model

This model was developed using air-water data and produces betterresults for air-water systems. Here the drag coefficient in the distortedparticle regime is given by:

where the terminal velocity UT is given by:

where:

and:

CD ellipse( ) 23---Eo

1 2⁄=

CD CD sphere( )= if CD sphere( ) CD ellipse( )≥

CD min CD ellipse( ) CD cap( ),( )= if CD sphere( ) CD ellipse( )<

CD ellipse( ) 43--- gd

UT2

--------∆ρρc-------=

UT

µcρcdp------------M

0.149–J 0.857–( )=

Mµc

4g∆ρ

ρ2σ3----------------- Morton Number= =



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µref = 0.0009 kg m-1 s-1 is the molecular viscosity of water at somereference temperature and pressure.

You can select this Drag Curve by selecting to use the Grace Model onthe Multiphase Models form in CFX-Build (see Multiphase Options(p. 1088 in CFX-Build: Chapter 4) ).

In this case, CFX-5 automatically takes into account the sphericalparticle and spherical cap limits by setting:

Availability

Both the Ishii Zuber and Grace Drag Models make explicit use of thegravity vector and surface tension. Hence, both are only available forbuoyant multiphase flows with specified surface tension. The fluidmorphologies must be “Continuous” and “Dispersed Fluid”respectively.

Dense Fluid Particles

Ishii-Zuber Drag Model

The Ishii Zuber drag laws automatically take into account denseparticle effects. This is done in different ways for different flow regimes.

In the viscous regime , where fluid particles may be assumed to beapproximately spherical, the Schiller Naumann correlation is modifiedusing a mixture Reynolds number based on a mixture viscosity.

J0.94H

0.7512 H 59.3≤<

3.42H0.441

H 59.3>

=

H43---EoM

0.149– µc

µref---------

0.14–

=





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Dense Spherical Particle Regime (Ishii Zuber)

Here, rdm is the user defined Maximum Packing value. This is defaultedto unity for a dispersed fluid phase.

In the distorted particle regime, the Ishii Zuber modification takes theform of a multiplying factor to the single particle drag coefficient.

Dense Distorted Particle Regime (Ishii Zuber)

Dense Spherical Cap Regime (Ishii Zuber)

The Ishii Zuber correlation, as implemented in CFX-5, automaticallyselects flow regime as follows:

Automatic Regime Selection (Ishii Zuber):

You can select this Drag Curve by selecting to use the Ishii ZuberModel on the Multiphase Models form in CFX-Build (see MultiphaseOptions (p. 1088 in CFX-Build: Chapter 4) ).

CD sphere( ) 24Rem---------- 1 0.15Rem

0.687+( )=

Rem

ρc Ud Uc– dp

µm------------------------------------=

µm

µc------- 1

rd

rdm---------–

2.5r dµ*–

= µ*

µd 0.4µc+

µd µc+--------------------------=

CD ellipse( ) E rd( )CD∞,= CD∞23---Eo

1 2⁄=

CD∞23---Eo

1 2⁄=

E rd( )1 17.67f r d( )6 7⁄

+( )18.67f rd( )

-------------------------------------------------- ,= f r d( )µc

µm------- 1 rd–( )1 2⁄

=

CD cap( ) 1 rd–( )2CD∞,= CD∞

38---=





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Grace Drag Model

The Grace drag model, as described in Dilute Fluid Particles (dropsand bubbles) (p. 399) , is formulated for flow past a single bubble.

You can select this Drag Curve by selecting to use the Grace Model onthe Multiphase Models form in CFX-Build (see Multiphase Options(p. 1088 in CFX-Build: Chapter 4) ).

For high bubble volume fractions, it may be modified using a simplepower law correction:

Here, is the single bubble Grace drag coefficient. For the dilutelimit, leave the Volume Fraction Correction Exponent at its defaultvalue of zero. For non-dilute flows, set a non-zero value, depending onthe bubble size as discussed below.

Small bubbles tend to rise more slowly at high void fraction, due to anincrease in the effective mixture viscosity. To capture this effect, anegative exponent p < 0 should be used. The Ishii Zuber correlationgives p = -1 in this limit. p = -0.5 has also been used successfully bysome investigators.

Large bubbles, on the other hand, tend to rise faster at high voidfractions, because they are dragged along by the wakes of otherbubbles. This effect may be modelled using a positive exponent, p > 0should be selected. The Ishii Zuber correlation uses p = 2 in thisregime. The value p = 4 has been used successfully by someinvestigators [46] .

Interphase Drag for the Mixture ModelIn the mixture mode, we define a non-dimensional drag coefficientCD, as follows:

where is the total drag exerted by phase β on phase α per unitvolume.

The mixture density ραβ is given by:

CD rcpCD∞=

CD∞

Dαβ CDραβAαβ Uβ Uα– Uβ Uα–( )=

Dαβ



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and the interfacial area per unit volume Aαβ is given by:

where is a user-specified mixture length scale.

The mixture model is primarily a tool to permit advanced users tospecify their own interfacial transfer models for complex situations.Hence the only drag model available is the specified Drag Coefficientmodel. This may be a constant, or a function defined using expressionlanguage or user fortran.

Interphase Turbulent Dispersion ForceCFX-5 implements the model for turbulent dispersion force due toLopez de Bertodano (1991) [34] .

Lopez de Bertodano Turbulent Dispersion Model

Unfortunately, there does not exist a universally valid value of the non-dimensional Turbulent Dispersion Coefficient CTD. Values of0.1 - 0.5 have been used successfully for bubbly flow with bubblediameters of order a few millimetres. See Lopez de Bertodano(1998) [35] . for a general discussion on recommended values of CTD.

You can select this model by selecting to use the Lopez de BertodanoTurbulent Dispersion Model on the Multiphase Models form inCFX-Build (see Multiphase Options (p. 1088 in CFX-Build:Chapter 4) ). Under this option, you may set the Turbulent DispersionCoefficient as a constant or as an expression.

Solid Particle Collision ForcesForces due to solid collisions may be taken into account by introducingadditional solids pressure and solids stress terms into the solid phasemomentum equations:

ραβ rαρα rβρβ+=

Aαβrαrβdαβ-----------=

dαβ

McTD

M– dTD

CTDρckc rc∇–= =



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Here, Pf denotes the fluid phase pressure, Ps and τs denote the solidspressure and stress tensor respectively due to inter-particle collisions,SM describes momentum sources due to external body forces and userdefined momentum sources acting on the solid phase, and Msdescribes interfacial forces acting on the solid phase due tointeractions with other phases (e.g. drag).

CFX-5 implements the model due to Gidaspow [32] . This ignores thesolids stress tensor, and assumes that the solids pressure may beexpressed as an empirical function of solid volume fraction,Ps = Ps( rs ), so that:

Bouillard et al (1989) [31] assumed the following functional form of theGeneralised Solids Elasticity Modulus :

You can select this model by selecting to use the Gidaspow Model forsolids pressure on the Fluid Models form in CFX-Build for dispersedsolid phases. (see Fluid Details (p. 1082 in CFX-Build: Chapter 4) ).

Under this option, you must set the Reference Elasticity Modulus ,G0, and the Compaction Modulus , c. There are no universallyaccepted values for these. Values used by Bouillard et al [31] are:

The parameter rsm is the Maximum Packing Parameter , seeMaximum Packing (p. 1041 in CFX-Build: Chapter 4) .

Fortunately, results tend to be insensitive to the details of the solidspressure model. The solids pressure gradient is only activated inregions close to the maximum packing, where its tendency is toprevent solid volume fractions from becoming too large.

Using this simple model, in conjunction with the Gidaspow drag model(see Dense Solid Particles (p. 398) ), it is possible to model the largescale features of bubbling fluidised beds.

t∂∂

r sρsUs( ) r s ρsUsUs( )( )∇•+ r s Pf∇– Ps∇ r sτs( )∇• SM M s+ + + +=

Ps∇ G rs( ) r s,∇= G rs( )∂Ps

∂r s---------=

G rs( ) G0exp c rs rsm–( )–( )=

G0 1 Pa,= c 20 to 600=



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There is a more sophisticated class of models which use the KineticTheory of Granular Flow to model the solids pressure and stresses.These will be implemented in future releases of CFX-5.



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Heat Transfer

In the multiphase model, there are separate enthalpy and temperaturefields for each phase. The phases are not in general in thermalequilibrium, so heat is transferred across phase interfaces viainterphase transfer terms.

Phasic EquationsHeat transfer is governed by the multiphase thermal energy equationsfor sensible enthalpy:

where:

• hα, Tα, λα, denote the sensible enthalpy, the temperature, and thethermal conductivity of phase α.

• Sα describes external heat sources, see User Specified Sources(p. 270).

• Qα denotes interphase heat transfer to phase α across interfaceswith other phases. See Interphase Heat Transfer Models (p. 407) .

Note that the total energy equation option is not permitted formultiphase flow in CFX-5.5.1.

Interphase Heat Transfer ModelsInterphase heat transfer occurs due to thermal non-equilibrium acrossphase interfaces. The total heat per unit volume transferred to phase αdue to interaction with other phases is denoted Qα, and is given by:

where:

t∂∂

rαρhα( ) rα ραUαhα λα Tα∇–( )( )∇•+ Sα Qα+=

Qα Qαββ α≠∑=

Qαβ Q– βα= Qαα∑ 0=⇒

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Heat transfer across a phase boundary is usually described in terms ofa heat transfer coefficient hαβ, which is the amount of heat energycrossing a unit area per unit time per unit temperature difference. Thus,the rate of heat transfer, Qαβ, per unit time across a phase boundary ofinterfacial area per unit volume Aαβ, from phase β to phase α, is:

This may be written in a form analogous to momentum transfer:

where the interphase heat transfer coefficient, , is modelled usingthe correlations described below.

Particle Model Correlations

For particle model the interphase heat transfer coefficient is modelledas:

Hence, in order to specify the interphase heat transfer coefficient (andsubsequently the interphase heat transfer term), it is necessary tospecify the interfacial area per unit volume and the heat transfercoefficient hαβ.

The interfacial area density calculation is described in The ParticleModel (p. 389) .

It is often convenient to express the heat transfer coefficient in terms ofa dimensionless Nusselt number:

In the particle model, the thermal conductivity scale λ is taken to be thethermal conductivity of the continuous phase, and the length scale d istaken to be the mean diameter of the dispersed phase:

Qαβ hαβAαβ Tβ Tα–( )=

Qαβ cαβh( )

Tβ Tα–( )=

cαβh( )

cαβh( )

hαβAαβ=

hλNu

d-----------=

hαβλαNuαβ

dβ---------------------=



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For laminar forced convection around a spherical particle, theoreticalanalysis shows that Nu = 2. For a particle in a moving incompressibleNewtonian fluid, the Nusselt number is a function of the particleReynolds number Re and the surrounding fluid Prandtl numberPr = µα CPα / λα.

The available models in CFX-5.5.1 are given below. They can beselected in the Multiphase Options form in CFX-Build (seeMultiphase Options (p. 1088 in CFX-Build: Chapter 4) ).

• Ranz-Marshall Correlation . The most well tested correlation forflow past a spherical particle is that of Ranz and Marshall [8] :

This is based on boundary layer theory for flow past a sphericalparticle. Its restriction to particle Reynolds number should be noted.

• Hughmark Correlation . Hughmark [9] proposed the followingempirical correlation for flow past a spherical particle.

It extends the Ranz Marshall correlation to a wider range ofReynolds numbers. The Reynolds number cross over point ischosen to guarantee continuity.

• Nusselt Number . Specify the Nusselt Number directly. This shouldbe based on empirical correlations, if available, for your application.

• Heat Transfer Coefficient . You specify the Heat TransferCoefficient directly. This should be based on empirical correlations,if available.

• Interface Flux . This is an advanced option which permitsexperienced users to implement interphase heat transfer modelswhich are not of the simple form of a heat transfer coefficientmultiplied by a bulk temperature difference.

The user specifies directly the interfacial heat flux F12 from fluid 1 tofluid 2 of a specified fluid pair. This is the rate of heat transfer perunit time per unit interfacial area from phase 1 to phase 2. Hence theheat transferred to fluid 2 from fluid 1 per unit volume is given by:

Nu 2 0.6Re0.5

Pr0.3

+= 0 Re 200<≤ 0 Pr 250<≤

Nu2 0.6Re

0.5Pr

0.33+ 0 Re 776.06<≤ 0 Pr 250<≤

2 0.27Re0.62

Pr0.33

+ 776.06 Re≤ 0 Pr 250<≤=



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F12 may be given as a constant or an expression.

Typically, F12 will be a function of the fluid 1 and fluid 2 temperaturefields, and possibly other variables. In this case, the user mayaccelerate convergence of the coupled solver by also specifyingoptional fluid 1 and fluid 2 heat flux coefficients.

For numerical stability these coefficients should be positive (orzero). The partial derivatives need not be computed exactly; it issufficient for the specified coefficients to simply approximate thepartial derivatives. Specification of heat flux coefficients only affectsthe convergence rate to the solution of the coupled heat transferequations, it does not affect the accuracy of the converged solution.

For example, the simple model using a heat transfer coefficientmultiplied by a bulk temperature difference my be recovered using:

Mixture Model Correlations

If you are using the mixture model, interphase heat transfer may bespecified using either:

• A specified heat transfer coefficient

• A specified Nusselt number.

• The interface flux model

In this case, the Nusselt number is defined in terms of a mixtureconductivity scale and the mixture length scale:

Q21 Q12– A12F12= =

h1

∂F12∂T1------------ 0≥ ,≈ h2

∂F21∂T2------------≈ 0≥

F12 F21– h T1 T2–( ),= = h1 h2 h= =

hαβλαβNuαβ

dαβ------------------------,= λαβ rαλα rβλβ+=



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Turbulence Modelling in Multiphase Flow

This section describes the extension of the single-phase turbulencemodels to multiphase simulations. You are encouraged to readTurbulence and Turbulence Models (p. 211 in CFX-5 Solver andSolver Manager) , which describes single-phase turbulence modelling,before reading this section.

Phase-Dependent Turbulence ModelsAll models available for single-phase calculations are also available formultiphase calculations. It is even possible to use different models indifferent phases. For example, a recommended model for dilutedispersed two-phase flow uses a k-ε model for the continuous phase,and an algebraic eddy viscosity model for the dispersed phase, whichsimply sets the dispersed phase viscosity proportional to thecontinuous phase eddy viscosity.

The Eddy Viscosity Hypothesis

The eddy viscosity hypothesis is assumed to hold for each turbulentphase. Diffusion of momentum in phase α is governed by an effectiveviscosity:

Algebraic Models

If using the zero-equation algebraic turbulence model, there areseveral options:

• The default zero-equation model uses a formula based ongeometric length scale and the mean solution velocity. This shouldbe used with caution for multiphase flow, as it is correlated forsingle-phase turbulent pipe flow. The turbulence viscosity ismodelled as the product of a turbulent velocity scale, Utα, and aturbulence length scale, ltα, as proposed by Prandtl andKolmogorov,

µαeff µα µtα+=

µtα ρα f µUtαl tα=

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where fµ is a proportionality constant. The velocity scale is calculatedto be the maximum velocity in phase α. If you specify a value for thevelocity scale it will be used for all phases. The length scale is derivedusing the formula:

where VD is the fluid domain volume.

• You may supply your own prescription for eddy viscosity, using aconstant or an expression.

• For dispersed phases only, you may invoke the dispersed phasealgebraic turbulence model.

Note: The algebraic equation model is only available for the disperse fluid whenthe continuous fluid is set to use a turbulence model (i.e. not laminar)

The parameter σ is a turbulent Prandtl number relating the dispersedphase kinematic eddy viscosity νtd to the continuous phase kinematiceddy viscosity νtc.

In situations where the particle relaxation time is short compared toturbulence dissipation time scales, e.g. bubbles or very small solidparticles, you may safely use the default value σ = 1.

If the particle relaxation time is long compared to turbulence dissipationtime scales, e.g. large solid particles in a gas, it may be better to use avalue of σ > 1. This is highly model dependent. Several models areavailable in the literature.

Two-Equation Models

The k-ε Model

For the k-ε model the turbulent viscosity is modelled as:

l tαVD

13---

7-------=

νtd

νtcσ

-------= µtd

ρd

ρc------

µtcσ

-------=⇒



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The transport equations for k and ε in a turbulent phase are assumedto take a similar form to the single-phase transport equations:

For definitions of the terms, please see The k-e model in CFX-5(p. 279).

The additional terms and represent interphase transfer for kand ε respectively. These are omitted in CFX-5.5.1, though they maybe added as user sources.

Reynolds Stress Models

As for two equation models, the multiphase versions of Reynoldsstress models are equivalent to the single phase version, with all fluxand volumetric source terms multiplied by volume fractions. SeeReynolds Stress Models (RSM) (p. 286) for the single phaseversions.

By default, no additional exchange terms are added, though they maybe added as user sources.

Turbulence EnhancementIn dispersed two phase flows, large particles tend to increaseturbulence in the continuous phase due to the presence of wakesbehind the particles. This is known as Particle Induced Turbulence .

Sato [36] successfully modelled this for bubbly flow using an enhancedcontinuous phase eddy viscosity:

µtα Cµραkα2

εα------

=

t∂∂

rαραkα( ) rα ραUαkα µµtασk--------+

kα∇–

∇•+ rα Pα ραεα–( ) Tαβk( )

+=

t∂∂

rαραεα( ) rαραUαεα µµtασε--------+

εα∇–

∇•+ rαεαkα------ Cε1Pα Cε2ραεα–( ) Tαβ

ε( )+=

Tαβk( )

Tαβε( )



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where µ ts is the usual Shear Induced Eddy Viscosity , and µ tp is anadditional Particle Induced Eddy Viscosity :

This may be selected in the Multiphase Models form in CFX-Build(see Multiphase Options (p. 1088 in CFX-Build: Chapter 4) ).

µtc µts µtp+=

µtp Cµpρcrddp Ud Uc–=



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Multicomponent Multiphase Flow

CFX-5.5.1 allows the modelling of multiphase multicomponent flows.Please see Multicomponent Flow (p. 3-367) for details onmulticomponent, single-phase flows.

Transport equations for the mass fractions of components YAα areassumed to take a similar form those used for single-phasemulticomponent flow:

One component in each phase must be calculated using a Constraintin the same way as for single-phase multicomponent flow – refer toMulticomponent Flow (p. 367) . The limitations that currently apply tosingle-phase multicomponent flow also apply to multiphasemulticomponent flow, and you should note that the molecular diffusioncoefficients ΓAα are given by ΓAα = ραDAα, where DAα is the KinematicDiffusivity set on the Component Models form in CFX-Build (seeComponent Models (p. 1086 in CFX-Build: Chapter 4) ).

If you need to model interfacial mass transfer of components betweenthe phases, you are recommended to use Additional Variables tomodel your flow.

Source terms in multicomponent multiphase flow behave the same asmulticomponent mass sources, but on a per fluid basis. See UserSpecified Sources (p. 270) .

t∂∂

rαραYAα( ) rα ραUαYAα ΓAα YAα∇( )–( )( )∇•+ 0=

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Additional Variables in Multiphase Flow

Additional Variables can exist in all fluids or only a limited number offluids in a multiphase simulation. In general, interphase transfer ofadditional variables can occur between any two phases. When usingthe homogeneous model, additional variables can only exist separatelyin each fluid and interphase transfer is not permitted.

For the particle and mixture models, the treatment of additionalvariables in multiphase flow is very general. Given an additionalvariable Φ and a fluid phase α, any one of the following choices ispermissible:

• Φ exists in phase α, and is governed by a transport equation.

• Φ exists in phase α, and is governed by an algebraic equation.

• Φ does not exist in phase α.

When Φ exists in phase α, the corresponding field variable is denotedΦα. If it obeys a transport equation, it is given by:

(Eq. Sol.-76)

Note:

• Φα is the conserved quantity per unit mass of phase α.

• φα is the conserved quantity per unit volume of phase α, whereΦα = ραφα

• is the Kinematic Diffusivity for the scalar in phase α. This maybe set for each Additional Variable and each phase separately.

• is a the external volumetric source term in phase α, with unitsof conserved quantity per unit volume per unit time. These areexplained in Additional Variable Source (p. 1096) .

• represents the total source to φα due to inter-phase transferacross interfaces with other phases.

t∂∂

rαραφα( ) rαραUαφα( )∇• rα ραDαφ( ) µtα

Sctα-----------+

φα∇

∇•–+ SαΦ( )

TαΦ( )

+=

DαΦ( )

SαΦ( )

TαΦ( )

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Interphase Additional Variable Transfer ModelsIt is possible for an additional variable Φα to be coupled to a differentadditional variable Ψβ across a phase interface between fluids α and β.Such a situation may arise, for example, when modelling theevaporation of water in a solid phase to water vapour in a gaseousphase. The only restriction is that Φα and Ψβ have the same physicaldimensions.

The total source to Φα per unit volume due to interaction with otherphases is given by:

where:

The simplest models for interphase transfer between Φα and Ψβ takethe driving force to be proportional to the difference in bulk additionalvariable values across the phase interface:

The first of these is used if the additional variable is defined per unitmass. The latter is used if the additional variable is defined per unitvolume.

The coefficients are defined by analogy with heat transfer, asdescribed in Interphase Heat Transfer Models (p. 407) .

Transfer of an additional variable across a phase boundary isdescribed by an additional variable transfer coefficient ταβ, which isanalogous to the heat transfer coefficient. It is the amount of Φαcrossing a unit area per unit time per unit difference in Φα across thephase boundary. Thus:

TαΦ( )

TαβΦ( )

β α≠∑=

TαβΦ( )

T– αβΦ( )

= TαΦ( )

α∑⇒ 0=

TαβΦ( )

cαβφ( ) ψβ φα–( )=

TαβΦ( )

cαβΦ( ) Ψβ Φα–( )=

cαβΦ( )



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So, we have:

It is often convenient to express the additional variable transfercoefficient in terms of a dimensionless Sherwood number Sh,analogous to the Nusselt number in heat transfer.

The diffusivity scale Γ is the kinematic diffusivity D for a volumetricvariable, and the dynamic diffusivity ρD for a specific variable.

Particle Model Correlations

In the particle model, the diffusivity scale Γ is that of the continuousphase, and the length scale d is the mean diameter of the dispersedphase:

For laminar forced convection around a spherical particle, theoreticalanalysis shows that Sh = 2. For a particle in a moving incompressibleNewtonian fluid, the Sherwood number is a function of the particleReynolds number Re and the additional variable Prandtl numberPr = µα / (ρD)α.

The available models in CFX-5.5.1 for the particle model are givenbelow. They can be selected in the Multiphase Models form inCFX-Build (see Multiphase Options (p. 1088 in CFX-Build:Chapter 4) ).

• Ranz-Marshall Correlation :

TαβΦ( ) ταβAαβ ψβ φα–( ),= for specific variables

TαβΦ( ) ταβAαβ Ψβ Φα–( ),= for volumetric variables

cαβΦ( ) ταβAαβ=

τ ΓShd

----------=

ταβΓαShαβ

dβ--------------------=

Nu 2 0.6Re0.5

Pr0.3

+= 0 Re 200<≤ 0 Pr 250<≤



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• Hughmark Correlation .

• Sherwood Number . You specify the Sherwood Number directly.This should be based on empirical correlations, if available, for yourapplication.

• Additional Variable Transfer Coefficient . You specify theAdditional Variable Transfer Coefficient directly. This should bebased on empirical correlations, if available.

• Interface Flux :

This is an advanced option which permits experienced users toimplement interphase additional variable transfer models which arenot of the simple form of a transfer coefficient multiplied by a bulkadditional variable difference.

The user specifies directly the interfacial flux F12 from additionalvariable Φ1 in fluid 1 to additional variable Ψ2 in fluid 2 of a specifiedfluid pair. This is the rate of additional variable transfer per unit timeper unit interfacial area from phase 1 to phase 2. Hence the amountof additional variable transferred to fluid 2 from fluid 1 per unitvolume is given by:

F12 may be given as a constant or an expression.

Typically, F12 will be a function of the fluid 1 and fluid 2 additionalvariable fields, and possibly other variables. In this case, the usermay accelerate convergence of the coupled solver by alsospecifying optional fluid 1 and fluid 2 additional variable fluxcoefficients.

Nu2 0.6Re

0.5Pr

0.33+ 0 Re 776.06<≤ 0 Pr 250<≤

2 0.27Re0.62

Pr0.33

+ 776.06 Re≤ 0 Pr 250<≤=

T21 T12– A12F12= =

c1

∂F12∂Φ1------------ 0≥ ,≈ c2

∂F21∂Ψ2------------≈ 0,≥ if volumetric

c1

∂F12∂φ1------------ 0≥ ,≈ c2

∂F21∂ψ2------------≈ 0,≥ if specific



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These should be positive (or zero), for numerical stability. Thepartial derivatives need not be computed exactly; it is sufficient forthe specified coefficients to simply approximate the partialderivatives. Specification of additional variable flux coefficients onlyaffects the convergence rate to the solution of the coupled transferequations, it does not affect the accuracy of the converged solution.

For example, the simple model using a transfer coefficient multipliedby bulk specific additional variable differences my be recoveredusing:

Mixture Model Correlations

If you are using the mixture model, interphase transport for additionalvariables may be specified using either:

• A specified interphase transfer coefficient

• A specified Sherwood number.

• The interface flux model

In this case, the Sherwood number is defined in terms of a mixturediffusivity scale and the mixture length scale:

F12 F21– c φ1 ψ2–( ),= = c1 c2 c= =

ταβΓαβShαβ

dαβ-----------------------,= Γαβ rαΓα rβΓβ+=



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Sources in Multiphase Flow

Sources in multiphase flow are very similar to those in single phaseflow, except that they are implemented on a phase basis.

It is important to note that phasic sources do not automatically includea volume fraction term. If the intended behaviour of the source is for itto reduce to zero as the phase volume fraction reduces to zero, thenyou should multiply the source by the phase volume fraction.

The single-phase source documentation can be found in UserSpecified Sources (p. 270) . Also see Sub-domain Sources (p. 1027in CFX-Build: Chapter 4) .

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Boundary Conditions in Multiphase Flow

The implementation of boundary conditions for multiphase flow is verysimilar to that for single-phase flow. This is described in BoundaryConditions (p. 230) (physical description) and Boundary Conditions(p. 309) (mathematical models). The main differences are:

• Boundary conditions need to be specified for both fluids for allvariables except the shared pressure field.

• Volume fractions of both phases must be specified on Inlet andOpening boundary conditions. These must sum to unity.

• For multiphase flow, pressure boundary conditions at Inlets, Outlets,and Openings are always defined in terms of static pressure.

• There is a new option, applicable only to dispersed two phase flow:

Degassing Boundary ConditionFor a discussion of the degassing boundary condition, see DegassingCondition (p. 238 in CFX-5 Solver and Solver Manager)

Wall Boundaries in MultiphaseFor multiphase flows, it is convenient to allow two distinct methods forthe user to specify wall boundary conditions:

• Bulk wall boundary conditions are defined as boundary conditionattributes of the wall, and subsequently shared amongst the phasesin a well defined manner. The algorithm for sharing wall fluxesamongst fluids depends on the concept of a wall contact model, tobe explained shortly.

• Alternatively, wall boundary conditions may be defined on a perfluid basis . This is a more advanced option, permitting the usercomplete flexibility in the definition of multiphase wall boundaryconditions.

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Bulk Wall Boundary Conditions

This is the simplest and most useful option. In most applications, wallboundary conditions are known as attributes of the wall. It is then partof the modelling procedure to decide how the consequent wall fluxesare to allocated amongst the phases in contact with the wall.

For example, for bulk heat transfer , the user may specify either:

• a bulk heat flux to the wall, Qwall, or

• a bulk heat transfer coefficient hwall and outer temperature To, or

• a bulk wall temperature, Twall.

A specified bulk wall heat flux Qwall determines heat fluxes Qα to eachphase α as follows:

Here, Aα is the contact area fraction of phase α at the wall. This isassumed by default to be equal to the volume fraction rα of phase α inthe control cell adjacent to the wall. However, this default may be over-ridden, as explained below.

A specified bulk heat transfer coefficient hwall determines heat fluxes toeach phase as follows:

A specified wall temperature Twall is allocated to all phases in contactwith the the wall, and resulting wall heat fluxes are partitioned amongstphases using the contact area fraction.

Area Contact Model

As explained above, a sensible default model for the contact areafraction Aα of phase α at the wall is that it be identical to the volumefraction in the control volume adjacent to the wall:

Qα AαQwall=

Qα Aαhwall To Tα–( )=

Aα rα=

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However, there are situations in which an alternative model may berequired. For example, for annular gas-liquid flow, the liquid forms athin film at the wall. so it is sensible to assign:

Similarly, in dispersed gas-solid flows with spherical solid particles, thewall contact area of the dispersed phase may be considered to benegligible relative to the continuous phase.

Hence, CFX-5 includes a facility for the user to over-ride the defaultwall contact area model by providing a value or an expression for thecontact area fraction for each phase. Note:

• Contact area fractions must sum to unity over all phases. Tofacilitate this, a constraint option has been included which may beapplied to at most one phase at each wall boundary condition.

• Specified contact area fractions are applied to all transportprocesses at the wall, that is, wall stresses, heat fluxes, additionalvariable fluxes, etc.

• You should only use the simple model:

in situations where phase α does not vanish at the wall. Otherwise,it is possible that you will be applying a non-zero wall flux to a regionof zero volume of phase α, which could cause the calculation tooverflow. For the same reason, your initial guess should not set thefluid volume fraction to be zero at a wall when that phase uses anon-zero area fraction.

• In general it is desirable, though not essential, to express contactarea fractions as functions of volume fraction such that:

This ensures that finite wall fluxes do not enter regions of zero volumeof phase α.

Aliquid 1= Agas 0=

Aα 1= Aβ 0=

Aα 0→ as rα 0→



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Fluid Dependent Wall Boundary Conditions

This option is included to give the advanced user complete flexibility inassigning wall boundary conditions to individual phases. For example,for heat transfer , the user may specify either:

• the temperature, Tα of phase α, or

• the heat flux Qα to phase α, or

• phase specific heat transfer coefficient hα outer temperature Toα, inwhich case:

Note:

• These options may be mixed arbitrarily amongst the phases.

• Heat fluxes arising from phase specific isothermal boundaryconditions are automatically multiplied by contact area fraction.

• However, specified heat fluxes Qα are not automatically multipliedby contact area fractions Aα. You should include these asmultiplicative factors in your expressions for Qα or hα, if this is theeffect that you require.

• Care should be exercised to ensure:

to prevent overflow errors due to finite sources being applied to zerofluid volumes.

Qα hα Toα Tα–( )=

Qα 0→ as rα 0→



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Modelling Advice for Multiphase Flow

Turbulence Models

• For flows in which a dilute phase is dispersed in a continuous phase,we recommend using the k-ε model for the continuous fluid. For thedispersed phase, we recommend the Dispersed Phase ZeroEquation turbulence model using the Free Slip option for WallInfluence On Flow. This type of flow can occur in bubble columns.

• For a liquid phase dispersed in a continuous liquid in which thephases tend to separate, the k-ε model is recommended for bothfluids. This type of flow occurs in oil-water separators.

• The Turbulent Prandtl Number for an algebraic model should beunity, except for large heavy particles, in which case it should begreater than unity.

Minimum Volume Fraction Setting

• The Default option can almost always be used. If this causesnumerical problems, then set the value to be several orders ofmagnitude less than the minimum expected physical value, butlarger than machine zero.

Buoyancy

See Buoyancy (p. 217 in CFX-5 Solver and Solver Manager) .

Initial Conditions

• Always set slightly different initial velocity fields for each phase toensure a non-zero slip velocity. The difference should be a fractionof the expected terminal slip velocity.

• We recommend you avoid using the default values for the initialFluid Volume Fraction. A sensible initial condition can be a suitablefraction of an inlet condition. For bubble columns which include ariser and a downer, you should use CEL to ensure the riser has a

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non-zero bubble volume fraction, while the downer should havezero bubble volume fraction. Note that you can leave one fluid to usethe Automatic option — the remaining volume fraction will beassigned to this fluid such that the volume fractions sum to unity.

Timestepping

• For bubble columns, use a fraction of the bubble rise time for a totalof one rise time, then increase the timestep to a fraction of the fluidcirculation time.

• For dilute particle cases, use a fraction of the dynamical timescalefor the momenta. This is essentially a fluid travel time through oraround the region of interest, but is not always trivial. If volumefraction convergence is slow, then the timescale for these equationscan be increased suitably.

Convergence

• If convergence stalls completely or becomes very slow, check theconservation and imbalances at the end of the output file. If theseare acceptable, plot the residual fields. It may be that the residualfields are good almost everywhere except for a few ‘hot spots’. Inthis case the solution is valid. You can also monitor some importantquantity, for example in a bubble column you can monitor if gas holdup has reached steady-state.

• For a case with a very dilute phase, the continuous phase will havea volume fraction very close to unity everywhere. Rounding errorswill affect the residuals at a much higher level for the continuousphase in this case. By default the volume fraction residual target is10 times the global target residual.

Transient Simulations

• The CFX-Build default of 3 coefficient loops per time step is rarelyadequate to resolve the strong non-linearities present in multiphaseflows, especially during the initial transient. You should set a largemaximum, say 50. Eventually, the code should settle down to onlyusing a small number of coefficient loops per time step.



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• If possible, a transient should be started from a small perturbation ofa converged steady state result. This may reduce the number ofcoefficient loops required during the initial transient.



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Restrictions

Various restrictions apply to the implementation of multiphase flow inCFX-5.5.1. These are summarised here.

• Interphase mass transfer is not supported for CFX-5.5.1.

• Conjugate heat transfer is not supported with multiphase flow.

• Radiation and multiphase flow is not supported.

• If you specify more than two phases, then CFX-Build requires youto set one phase to be continuous, and the other phases to bedispersed. Phase pair interactions are only allowed for continuous-disperse phase pairs. This restriction does not apply to CCL.

• Additional variables are not supported in CFX-Build for more thantwo phases. This restriction does not apply to CCL.

• Total Pressure and Total Temperature are not calculated andwritten to the results file for multiphase cases.

Page 429 Restrictions CFX-5.5.1


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Multiphase Flow

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