Upload
andres-felipe-salas-villalva
View
252
Download
3
Embed Size (px)
Citation preview
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
1/59
© 2013 ANSYS, Inc. 4-1 Release 14.5
14. 5 Release
Multiphase Flow Modeling
in ANSYS CFX
Interphase Momentum and
Heat Transfer
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
2/59
© 2013 ANSYS, Inc. 4-2 Release 14.5
Overview
• Interphase Momentum Transfer― Drag Force
― Non Drag Forces
• Lift Force
• Wall Lubrication Force
• Virtual Mass Force• Turbulent Dispersion Force
• Interphase Heat Transfer
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
3/59
© 2013 ANSYS, Inc. 4-3 Release 14.5
Interphase Momentum Transfer
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
4/59
© 2013 ANSYS, Inc. 4-4 Release 14.5
• 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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
5/59
© 2013 ANSYS, Inc. 4-5 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
6/59
© 2013 ANSYS, Inc. 4-6 Release 14.5
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
=
| |( )
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
7/59
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
9/59© 2013 ANSYS, Inc. 4-9 Release 14.5
Drag Models for Fluid Particles(Solid Spherical Particle & Drops)
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
10/59© 2013 ANSYS, Inc. 4-10 Release 14.5
Spherical Particle Drag Regimes
Stokes Transitional Newton Supercritical
CD
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
11/59
© 2013 ANSYS, Inc. 4-11 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
12/59
© 2013 ANSYS, Inc. 4-12 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
13/59
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
14/59
© 2013 ANSYS, Inc. 4-14 Release 14.5
• 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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
15/59
© 2013 ANSYS, Inc. 4-15 Release 14.5
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
=| |
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
16/59
© 2013 ANSYS, Inc. 4-16 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
17/59
© 2013 ANSYS, Inc. 4-17 Release 14.5
• 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)
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
18/59
© 2013 ANSYS, Inc. 4-18 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
19/59
© 2013 ANSYS, Inc. 4-19 Release 14.5
Non-Drag Forces
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
20/59
© 2013 ANSYS, Inc. 4-20 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
21/59
© 2013 ANSYS, Inc. 4-21 Release 14.5
• 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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
22/59
© 2013 ANSYS, Inc. 4-22 Release 14.5
• 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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
23/59
© 2013 ANSYS, Inc. 4-23 Release 14.5
• 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.
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
24/59
© 2013 ANSYS, Inc. 4-24 Release 14.5
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)
Tomiyama C_L (u_slip=0.05 m/s)
C_L (Tomiyama), 0
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
25/59
© 2013 ANSYS, Inc. 4-25 Release 14.5
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.
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
26/59
© 2013 ANSYS, Inc. 4-26 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
27/59
© 2013 ANSYS, Inc. 4-27 Release 14.5
• 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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
28/59
© 2013 ANSYS, Inc. 4-28 Release 14.5
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.
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
29/59
© 2013 ANSYS, Inc. 4-29 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
30/59
© 2013 ANSYS, Inc. 4-30 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
31/59
© 2013 ANSYS, Inc. 4-31 Release 14.5
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.
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
32/59
© 2013 ANSYS, Inc. 4-32 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
33/59
© 2013 ANSYS, Inc. 4-33 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
34/59
© 2013 ANSYS, Inc. 4-34 Release 14.5
Bubbly Flow in Vertical Pipe Validation Data
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
35/59
© 2013 ANSYS, Inc. 4-35 Release 14.5
Bubbly Flow in Vertical Pipe Validation Data
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
36/59
© 2013 ANSYS, Inc. 4-36 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
37/59
© 2013 ANSYS, Inc. 4-37 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
38/59
© 2013 ANSYS, Inc. 4-38 Release 14.5
Free Surface Model
• Similar to Mixture Model• Difference in the calculation of the Interfacial Area Density
= ||
= | |( )
ρα = rαρα + rβρβ
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
39/59
© 2013 ANSYS, Inc. 4-39 Release 14.5
Interphase Heat Transfer
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
40/59
© 2013 ANSYS, Inc. 4-40 Release 14.5
• 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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
41/59
© 2013 ANSYS, Inc. 4-41 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
42/59
© 2013 ANSYS, Inc. 4-42 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
43/59
© 2013 ANSYS, Inc. 4-43 Release 14.5
• 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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
44/59
© 2013 ANSYS, Inc. 4-44 Release 14.5
• 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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
45/59
© 2013 ANSYS, Inc. 4-45 Release 14.5
• 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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
46/59
© 2013 ANSYS, Inc. 4-46 Release 14.5
Heat Transfer Coefficient Options
Continuous side (α )
• Zero resistance
• Specified heat transfer coefficient
• Specified Nusselt number
• Correlations :
(available only for Particle Model)
– Ranz-Marshall
– Hughmark
Continuous
T
hT ,
T
T
sT T h
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
47/59
© 2013 ANSYS, Inc. 4-47 Release 14.5
Heat Transfer Coefficient Options
Dispersed side (β)
• Zero resistance
• Specified heat transfer coefficient
• Specified Nusselt number
Continuous
hT ,
hT ,
hT ,
T
sT T h
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
48/59
© 2013 ANSYS, Inc. 4-48 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
49/59
© 2013 ANSYS, Inc. 4-49 Release 14.5
Appendix
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
50/59
© 2013 ANSYS, Inc. 4-50 Release 14.5
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
=
= | |( )
ρα = rαρα + rβρβ
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
51/59
© 2013 ANSYS, Inc. 4-51 Release 14.5
Terminalbubble
rise
velocity
Terminal Rise Velocity for Bubbles
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
52/59
© 2013 ANSYS, Inc. 4-52 Release 14.5
Lift Force - Saffman Formulation
• Applicable to dilute concentrations of spherical particles
• Saffman (0
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
53/59
© 2013 ANSYS, Inc. 4-53 Release 14.5
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
[ - ]
Tomiyama C_L (u_slip=0.01 m/s)
Tomiyama C_L (u_slip=0.05 m/s)
C_L (Tomiyama), 0
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
54/59
© 2013 ANSYS, Inc. 4-54 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
55/59
© 2013 ANSYS, Inc. 4-55 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
56/59
© 2013 ANSYS, Inc. 4-56 Release 14.5
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.
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
57/59
© 2013 ANSYS, Inc. 4-57 Release 14.5
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
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
58/59
© 2013 ANSYS, Inc. 4-58 Release 14.5
Virtual Mass Force Robustness
No VMF VMF, High Res diff. VMF, UDS diff.
8/19/2019 CFX Multiphase 14.5 L03 Interphase Momentum Heat Transfer
59/59
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)
e