CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer

8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer

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© 2013 ANSYS, Inc. 4-1 Release 14.5

14. 5 Release

Multiphase Flow Modeling

in ANSYS CFX

Interphase Momentum and

Heat Transfer


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Overview

• Interphase Momentum Transfer― Drag Force

― Non Drag Forces

• Lift Force

• Wall Lubrication Force

• Virtual Mass Force• Turbulent Dispersion Force

• Interphase Heat Transfer


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Interphase Momentum Transfer


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• The Multiphase equation is weighted by volume fraction rα and contains twoextra terms.

• The term (ΓαβUβ- ΓβαUα) represents momentum transfer induced by interphasemass transfer .

• The term Mαβ

represents the total interfacial force acting on phase α due to

phase β. This may arise from several independent physical effects:

= +

+ +

+

where D : Interface drag force, L : Lift force, WL : Wall lubrication force

VM : Virtual mass, TD : Turbulence dispersion force

Momentum Equation

P P N N

T

M U U

r pr r r t

11

][)()(

U U U U U

][)()( T pt

U U U U U

Single Phase

Multiphase Phase


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Interphase Drag

• Consider gas bubbles rising through a liquid such as you might see in a

bubble column or a glass of soda:

• Expressions for the interphase drag are needed in order to solve themomentum equations for the two phases.

• The bubbles rise through the liquid. This

difference in velocities causes interphase

drag or transfer of momentum between

the phases:

– The bubbles are slowed by the liquid.

– The liquid is accelerated by the bubbles


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Drag Force for Single Particle

• Drag force exerted by a single particle of phase β on the continuous phase

(α):

where AP is the cross-sectional area of particle and is given by

• Drag coefficient (CD) depends on particle Reynolds number (ReP) which is

defined based on the relative speed (Uβ – Uα) , the continuous phase

properties, and the particle diameter (dP) :

=| |

A = π

4

=

| |( )


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8/59© 2013 ANSYS, Inc. 4-8 Release 14.5

Interphase Drag Modeling ()

• The term represents the drag force per unit volume exerted by dispersed

phase (β) on continuous phase (α). It is modelled as function of relative speed(Uβ – Uα) as :

where constant cαβ is known as momentum transfer/exchange coefficient

• Comparing with with :

=

cαβ (UβUα) =

3

4

CD

dPrβ ρα|Uβ Uα|(Uβ Uα)

cαβ =

3

4

CD

dPrβ ρα|Uβ Uα|

cαβ =

CD

8Aαβρα|Uβ Uα |

Aαβ (interfacial area density ) is

related to volume fraction (rβ) and

particle diameter (dP): =

(Particle Model)

CD for particles, bubbles and

drops is found using correlations



Drag Models for Fluid Particles(Solid Spherical Particle & Drops)



Spherical Particle Drag Regimes

Stokes Transitional Newton Supercritical

CD


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Spherical Particle Drag Regimes

• Stokes – 0 < ReP < 0.2

– Viscous forces

– CD =

• Transitional

– 0.2 < ReP < 1 103

– Viscous and inertia forces

– CD =

1 + 0.15.

(Schiller –Naumann)

• Newton – 1 103 < ReP < 1 10

5

– Mainly inertia forces

– Independent of particle Reynoldsnumber

– CD = 0.44

• Supercritical

– ReP > 1 105

– Transition from laminar to turbulent

boundary layer

– Separation on particle surface

further downstream

– Drag reduction


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Drag Correlations for Particles

• CFX modifies the Schiller-Naumann drag law this to ensure the

correct limiting behavior in the inertial regime by taking:

CD = max

24

Re 1 + 0.15

. , 0.44

• Modified Schiller-Naumann drag law covers Stoke, Transitional

and Newton drag regimes only


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• Bubble shapes depend on size, surface tension, particle Reynolds number,

density difference, …

• Small bubbles spherical bubble shape

• Large bubbles ellipsoidal & spherical cap bubble shape

Bubble Regimes

Bubble size variation Ellipsoidal shape Spherical Cap


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Clift, Grace, Weber: Bubbles, Drops and

Particles. Academic Press, 1978

Bubble Regimes

• Eotvos number:

– ratio of buoyancy force to surfacetension force

=Δ

• Morton number:

– function of physical properties of fluid

=

• Reynolds number: – ratio of inertia force to viscous force

=| |


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Drag Correlations for Bubbles

Regimes Ishii Zuber Grace

SphericalRegime

CD =24

Re 1 + 0.15

.

(Schiller-Naumann)

CD =24

Re 1 + 0.15

.

(Schiller-Naumann)

Ellipsoidal

Regime =4

3

Δ

∞ =

4

3

Δ

∞

Drag coefficient is found by balance between buoyancy force and

drag force

∞ = 2

Δ

∞ =

−.9 0.857

= (, )

Spherical

Cap

Regime

=8

3 =

8

3


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• CFX automatically takes into account the bubble regime change by

setting:

CD = max [CD (sphere), min ( CD (ellipse), CD (cap) ) ]

Automatic Regime Detection

• Larger diameter bubbles

– the distorted bubble regime

min ( CD (ellipse), CD (cap) ) > CD (sphere)

CD = min (CD (ellipse), CD (cap))

• Smaller diameter bubbles:

– the viscous regime

CD (sphere) > min ( CD (ellipse), CD (cap) )

CD = CD (sphere)


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correlation for spherical regime only

Grace correlation

Source: Grace & Weber, 1982

EQUIVALENT DIAMETER / mm

T E R M I N A L V E L O C I T Y

/ c m / s

Grace Correlation for Bubbles


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Non-Drag Forces


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Non-Drag Forces

• The term Mαβ represents the total interfacial force acting on phase α due to phase β. It is sum of drag and non drag forces :

= + + + +

= + + + +

• Such forces are fundamental to the physics of phase distribution inmultiphase flows. Implemented for Continuous-Dispersed Phase Pairs

Only.

P P N N

T

M U U

r pr r r

t

11

][)()(

U U U U U

Lift Wall

LubricationVirtual

MassTurbulent

Dispersion


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• Transverse to flow direction

• Physical mechanism: acts on particles, droplets and bubbles in

shear flows

– due to liquid velocity gradients

– due to asymmetric wake

– due to bubble shape changes

• Significant for: – Large continuous-dispersed phase density ratios, e.g. bubbly

flows

– Large shear e.g. flow in pipes, where pipe diameter iscomparable to bubble diameter

Lift Force


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• Lift coefficient CL=0.5 for inviscid flow around a sphere (Drew, Lahey,Auton et al.).

• For viscous flow, CL varies from 0.01 to 0.15.• In general CL is correlated as a flow-regime dependent function of other

non-dimensional variables:

)Re ,Eo ,Re( P L L C C

Formulation of Lift Force

Particle Reynolds NumberVorticity Reynolds Number Eotvos number

=| |

Reω =

× U d

μ Eo =

gΔρd

σ

ccd cd Ld L U U U r C F


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• Small bubbles migrate towards the wall and large bubblesmigrate towards the core

• Change of sign of CL due to change in bubble shape asbubble size increases

• For small bubbles CL is function of ReP but for intermediateand large bubbles C

L

is function of Eo

Lift force on small and large bubbles

largeellipsoidal

bubble

lift

force

smallspherical

bubble

lift

force

Lift coefficient for air-water system under atmospheric pressure and room

temperature (Tomiyama, Tamai, et al)

CL

fluid vel.


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Lift Force Formulations

• Tomiyama Model

– Well validated model for bubbly flow. – Takes into account change of sign of lift force due to change in bubble shape as

bubble size increases.

3 2

min 0.288 tanh(0.121 Re ), ( ) 4

( ) 0.00105 0.0159 0.0204 0.474 4 10.0

0.27 10.0

P d d

L d d d d d

d

f Eo Eo

C f Eo Eo Eo Eo Eo

Eo

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 2 4 6 8 10

Bubble diameter [mm]

L i f t F o r c e C o e f f .

C_

L

[ - ]

Tomiyama C_L (u_slip=0.01 m/s)


C_L (Tomiyama), 0


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Lift Force Formulations

• Saffman Mei

– Applicable to rigid spheres. – Generalises Saffman’s anaytical model to extend

applicability to higher particle Reynolds numbers.

• Legendre Magnaudet – Applicable to small spherical liquid droplets.

– Takes account of induced circulation inside drops.


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Surface tension prevents bubbles from approaching solid

walls very closely

• Effect is modelled by a wall force, pushing bubbles

away from walls

• Results in near wall area of low gas void fraction

wall lubr.

force

fluid vel.

gas void fraction

Wall Lubrication Force

FWL = CWL r ρ U U

nW

nW : unit normal pointing away from the wall

CWL : Wall Lubrication coefficient

Formulation :

• Antal model : Good for fine mesh only

• Tomiyama model : Restricted to pipe geometries.Works well for pipes

• Frank model : Removes dependency of Tomiyama

model on pipe diameter


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• For bubbly flow it important to use

both lift and wall lubrication force topredict accurate flow field.

• For vertical cocurrent upflow in a pipe,bubbles tend to be pushed towards the

wall. In conjunction with the walllubrication force, gives a void fraction

peak close to but away from the wall.

• For vertical cocurrent downflow in a

pipe, both lift and lubrication forces actaway from the wall leading to a large

flat void fraction profile in the centre

of the pipe (void coring).

Lift Force + Wall Lubrication Force

Vertical Cocurrent

Upflow

Vertical Cocurrent

Downflow


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Virtual Mass Force

• “Virtual Mass” effect occurs when dispersed phase accelerates relative to

continuous phase• Due to viscous interaction, fluid particles have to accelerate

some of the surrounding fluid. The inertia of this mass

exerts a opposing force on the fluid particles

• CVM=0.5 for inviscid flow around an isolated sphere.

In general, CVM depends on shape and particle concentration.

• Potentially significant for: – Large continuous-dispersed phase density ratios, e.g. bubbly flows

– Transient Flows – can affect period of oscillating bubble plume.

– Strongly Accelerating Flows e.g. bubbly flow through narrow constriction.

Dt

U D

Dt

U D

C r F ccd d

VM cd

d

VM

Maliska and Paladino etal.


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Turbulent Dispersion Force

• Leads to dispersion of dispersed phase from high volumefraction to low volume fraction due to turbulentfluctuations

• Equalizes dispersed phase volume fraction• Formulation :

• Favre Averaged Drag Model (Burns, et al.) – Turbulent dispersion = action of turbulent eddies

via interphase drag

– derivation via mass weighted (Favre) averaging ofthe drag term:

– cαβ : momentum transfer coefficient for the interface drag force

– νtc and σtc : turbulent viscosity and turbulent Schmidt number of continuous phase

turb.

dispersion

force

fluid vel.

gas void fraction

FTD = CTD cαβ

νt

σt

r

r

r

r


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2. Lopez de Bertodano Model

– kc : turbulent kinetic energy of continuous phase

–

CTD = 0.1 to 0.5 good for medium sized bubbles in ellipsoidal flow regime. – CTD up to 500 required for small bubbles.

cccTD

d

TD r k C F

Turbulent Dispersion Force


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Non-Drag Forces Validation

• Bubbly flow in a vertical pipe• Forschungszentrum Dresden (FZD) MT-Loop test facility.

– Wiremesh sensor with 24x24 electrodes.

– Database to test CFD predictions.

– Length, L = 4 m, Inner Diameter, D = 51.2 mm.

• Air-Water at atmospheric pressure, and 30 C.

• Measurements carried out for stationary flows of various superficialvelocity ratios.

– 10 different cross sections located between L/ D = 0.6 and 59.2 from gas

injection. – Select test cases in bubbly flow regime with a near-wall peak in gas volume

fraction.


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Bubbly Flow in Vertical Pipe NDF Validation

Test d p [mm]

U l ,sup[m/s]

U g,sup[m/s]

017 4.8 0.405 0.0040

019 4.8 1.017 0.0040

038 4.3 0.225 0.0096

039 4.5 0.405 0.0096

040 4.6 0.641 0.0096

041 4.5 1.017 0.0096

042 3.6 1.611 0.0096

074 4.5 1.017 0.0368


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Bubbly Flow in Vertical Pipe NDF Validation

• SST turbulence model for continuous phase.

• Sato model for particle induced turbulence.

• Simple algebraic turbulence model for dispersed phase turbulence:

• Grace model for drag force.

• Tomiyama models for the lift and wall lubrication force.

• FAD and Lopez de Berterdano (RPI Model) for the turbulencedispersion force.

• Virtual Mass Force neglected.

/t t 1


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Bubbly Flow in Vertical Pipe Validation Data


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Bubbly Flow in Vertical Pipe Validation Data


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Mixture Model vs Particle Model

• In Particle Model user need to provide particle diameter (dp) which isused in calculation of

• Interfacial Area Density

• Interphase transfer term

• Particle Model is used for

• Gas-liquid bubbly flows

• Droplet flows in gas or immiscible liquid

• Fluid-particle flows

• But for complex interfacial boundaries, gas-liquid flows with flow

regime transition, Mixture model is used:

• Plug flow

• Slug flow

• Annular flow

• Churn flow

Mixture Model


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Mixture Model

• Treats both phases symmetrically. It requires both phases to becontinuous.

• Fluid properties are calculated as volume averaged mixtures

• The term represents the drag force per unit volume exerted

by phase β on phase α.

• dαβ (interfacial length scale) and CD (drag coefficient) are to be

provided by user

=

= | |( )

ρα = rαρα + rβρβ


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Free Surface Model

• Similar to Mixture Model• Difference in the calculation of the Interfacial Area Density

= ||

= | |( )



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Interphase Heat Transfer


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• eα , λα , Tα : internal energy, thermal conductivity and temperature of phase α

• The Multiphase equation is weighted by volume fraction rα and contains two

extra terms.• The term (Γαβeβ- Γβαeα) represents heat transfer induced by interphase mass

transfer

• The term Q αβ represents interphase heat transfer to phase α across interfaceswith phase β

Thermal Energy Equation

P P N N

Qee

r r r r t

11

:)()()(

U T e U e

:)()()( U T e U e

t Single Phase

Multiphase Phase


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Interphase Heat Transfer

• Q is the heat transferred per unit time per unit volume, from to .

• A is the interfacial area per unit volume

• h is the interfacial heat transfer coefficient (also known as overall heattransfer coefficient).

• h depends on Nusselt Number (Nu)

Nu = h dp/c

where dp = particle diameter

c = thermal conductivity of the continuous phase

)( T T AhQ


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Heat Transfer Rate Options

•Specified overall heat transfer coefficient

• Specified Nusselt number

• Specified interphase heat transfer flux

• Correlations for overall heat transfer coefficient• Two Resistance Model


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• Available for Continuous phase – Dispersed phase only (Particle Model)

• The Nusselt number depends upon the surrounding fluid Prandtl number( Pr = cpα µα/λα ) as well as the particle Reynolds number (ReP)

–Ranz- Marshall (0 < ReP < 200, 0 < Pr < 250)

–Hughmark (0 < Pr < 250)

Correlations for Overall Heat Transfer

Coefficient

3.05.06.02 Pr Re Nu P

)06.776(,27.02

)06.7760(,6.0233.062.0

33.05.0

P P

P P

Re Pr Re Nu

Re Pr Re Nu


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• Available for both the particle and mixture models:

–for Continuous phase – Dispersed phase

– for Continuous phase – Continuous phase

• There are special situations where the use of an overall heat

transfer coefficient is not sufficient to model the interphaseheat transfer process.

• A more general class of models considers separate heat

transfer processes either side of the phase interface.

• This is achieved by using two heat transfer coefficients definedon each side of the phase interface.

The Two Resistance Model


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• The heat flux from the interface to phase α and phase β

• Overall heat balance, qα + qβ = 0, this condition determines interfacialtemperature (Ts)

• The overall heat transfer coefficient (h

• Fluid specific Nusselt Number

Nuα = h d /

where d is the interfacial length scale for the mixture model (themean particle diameter for the Particle Model )

The Two Resistance Model

)( T T hq s )( T T hq s

hh

T hT hT s

)(- T T Ahqq hhh

11

1


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Heat Transfer Coefficient Options

Continuous side (α )

• Zero resistance

• Specified heat transfer coefficient


• Correlations :

(available only for Particle Model)

– Ranz-Marshall

– Hughmark

Continuous

T

hT ,

T

T

sT T h


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Heat Transfer Coefficient Options

Dispersed side (β)

• Zero resistance

• Specified heat transfer coefficient


Continuous

hT ,

hT ,

hT ,

T

sT T h


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Distributing Boundary Heat Transfer

• At wall and fluid-solidinterface boundaries, the heat

transfer must be distributed

between the individual phases

– The default behavior is forthe partitioning to be based

on the phasic volume

fraction – It is possible to over-ride

this default and directly set

the contact area fraction for

the individual phase


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Appendix


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Mixture Model

• Gas-liquid flows with flow regime transition like plug flow, slugflow, annular flow, churn flow

• Treats both phases symmetrically. It requires both phases to

be continuous.

• The term

represents the drag force per unit volumeexerted by phase β on phase α.

• dαβ (interfacial length scale) and CD (drag coefficient) are to be

provided by user

=

= | |( )



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Terminalbubble

rise

velocity

Terminal Rise Velocity for Bubbles


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Lift Force - Saffman Formulation

• Applicable to dilute concentrations of spherical particles

• Saffman (0


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Lift Force - Tomiyama Formulation

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 2 4 6 8 10

Bubble diameter [mm]

L i f t F o r c

e C o e f f .

C_

L

[ - ]



C_L (Tomiyama), 0


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Wall Lubrication Force

• Tomiyama Modification

– Like Tomiyama lift force, depends on Eotvos number, hence accounts for

dependence of wall lubrication force on bubble shape.

– In conjunction with Tomiyama lift force, produces excellent results for bubble

flow in vertical pipes.

– However, requires pipe diameter (D) as input parameter, hence geometrydependent.

• Frank Modification

– Generalises Tomiyama’s model to be geometry independent.

– Model constants calibrated and validated for bubbly flow in vertical pipes

C WC = 10, C WD = 6.8, p = 1.7


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Virtual Mass Force Numerics

• Virtual Mass Force

– Proportional to difference in phasic accelerations.• Implementation in ANSYS CFX

– Upwind linearisation of acceleration terms

– May choose first order upwind, or upwind scheme compatible with chosenadvection discretisation (expert parameter)

– Coupled implicit treatment of upwind acceleration terms. – Consistent account of VMF terms in Rhie-Chow interpolation.

– Inclusion of VMF often improves convergence compared to no VMF

– However, rarely alters converged results


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Virtual Mass Force Validation

• Flow in the converging part of a converging diverging nozzle to evaluate the

effect of flow curvature.

• Drag is set to zero, there is no buoyancy.

• As the flow accelerates a transverse pressure gradient is set up by thecontinuum, water, which accelerates the dilute disperse phase, air, towards

the axis.


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Virtual Mass Force Validation

z

Particle Tracking Model solution

(no Virtual Mass force)

Eulerian Fluid Model Solution with

Virtual Mass Force

VMF

Vi t l M F R b t


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Virtual Mass Force Robustness

No VMF VMF, High Res diff. VMF, UDS diff.


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GGI Conjugate Heat Transfer

• Conjugate heat transfer with GGI fluid-solid interfaces has been available since

the 12.0 release

– Previously a feature matrix gap as 1:1 GGI connections were required for multiphaseflows with conjugate heat transfer to solids

– Gives more flexibility in meshing as meshes in fluid and solid regionsare no longer required matched at interfaces

– Conjugate Additional Variable transfer also supported

• GGI numerics are often more robust and give better answers than 1:1 numericsfor CHT problems, and are often preferred

– ‘Automatic’ mesh connections now use GGI numerics at fluid-solid interfaces andsolid-solid domain interfaces (connecting separate domains)

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CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer