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1 2015 ANSYS, Inc. April 24, 2015 ANSYS Confidential
16.0 Release
Lecture 4:
Gas-Liquid Flows
Multiphase Modeling using
ANSYS Fluent
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Outline Introduction
Conservation equations
Modelling strategies : Euler-Lagrangian and Eulerian
Interfacial Forces
Drag Non-Drag Forces Turbulence Interaction
Mixture Model
Validation example
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Gasliquid flows occur in many applications. The motion of bubbles in a liquid as well as droplets in a conveying gas stream are examples of gasliquid flows
Bubble columns are commonly used in several process industries
Atomization to generate small droplets for combustion is important in power generation systems
Introduction
Bubble Column
Rain/Hail Stones
Spray Drying
Distillation Process
Absorption
Process
Boiling Process
Combustion
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Why Study Gas-Liquid Flows
The main interests in studying gas-liquid flows, in devices like bubble columns or stirred tank reactors, are:
Design and scale-up
Fluid dynamics and regime analysis
Hydrodynamic parameters
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Bubble Columns
To design bubble column reactors, the following hydrodynamic parameters are required:
Specic gasliquid interfacial area ()
Sauter mean bubble diameter, ()
Axial and radial dispersion coefcients of the gas and liquid, ()
Heat and mass transfer coefcients, (, )
Gas holdup, ()
Physicochemical properties of the liquid medium, (, )
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Two types of ow regimes are commonly observed in bubble columns:
The bubbly flow regime, Gas velocity < 5cm/s
Bubbles are of relatively uniform small sizes (db =2 to 6 mm)
Rise velocity does exceed 0.025m/s
Holdup shows linear dependence with the flow
Regime Analysis < . < . /
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The churn turbulent flow regime Gas velocity > 5cm/s
Bubble are Large bubbles ( > ) and show wide size distribution
Rise velocity is in the range of 1-2m/s
Regime Analysis > . > . /
Most frequently observed flow regime in industrial-size, large diameter columns
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Photographic Representation of Bubbly and Churn-Turbulent Flow Regimes
Bubbly Flow Regime Churn Turbulent Flow Regime
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Design and Scale-up of Bubble Column Reactors
Bubble have significant effect on hydrodynamics well as heat and mass transfer coefcients in a bubble columns
The average bubble size and rise velocity in a bubble column is found to be affected by:
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In this approach, a single set of conservation equations is solved for a continuous phase
The dispersed phase is explicitly tracked by solving an appropriate equation of motion in the lagrangian frame of reference through the continuous phase flow field
The interaction between the continuous and the dispersed phase is taken into account with separate models for drag, and non-drag forces
Euler-Lagrangian Method
Eulerian Cell
Gravity
Buoyancy
Liquid Flow
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Eulerian Approach
In the Eulerian approach, both the continuous and dispersed phases are considered to be interpenetrating continua
The Eulerian model describes the motion for each phase in a macroscopic sense
The flow description therefore consists of differential equations describing the conservation of mass, momentum and energy for each phase separately
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Conservation Equations
Continuity equation:
Momentum equation:
sDrag ForceNon
ForceDispersionTurbulent
td,q
ForcessVirtual Ma
vm,q
ForcecationWall Lubri
wl,q
ForceLift
lift,q
Forceexternal
q
n
p
sDrag Force
fermass trans
qpqppqpq
l ForceInterfacia
qppq
Bouyancy
Friction
q
essure
qqqqqqq
FFFFF
vmvmvvK gpvvt 1
Pr
2
source
q
transfermass
n
p
qppqqqqqq Smmvt
1
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A key question is how to model the inter-phase momentum exchange
This is the force that acts on the bubble and takes into account:
Effect of multi-bubble interaction
Gas holdup
Turbulent modulation
Turbulent Dispersion
Turbulent Interaction
Interphase Momentum Exchange
Interphase Momentum
Exchange
Drag
Lift
Turbulent Dispersion
Turbulent Interaction
Virtual Mass
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We can think of drag as a hydrodynamic friction between the liquid phase and the dispersed phase
We can also think of drag as a hydrodynamic resistance to the motion of the particle through the water. The source of this drag is shape of particle
Drag Force
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Drag Force
For a single spherical bubble, rising at steady state, the drag force is given by:
For a swarm of bubbles the drag, in absence of bubble-bubble interaction, is given by:
4
3
2
63,
qpqp
p
q
D
p
qpqpq
pD
p
p
DswarmD
vvvvd
C
vvvvACd
NFF
qpqpq
pDD vvvvACF velocitysliptcoefficien drag
2
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Drag Force
In order to ensure that the interfacial force vanishes in absence any dispersed phase, the drag force needs to multiplied by as shown:
In Fluent
4
3, qpqp
p
q
D
qp
swarmD vvvvd
CF
24
ReC
6
18
6
18
D
2
2,
qpi
p
pp
q
pp
qpi
p
pp
q
ppqppqswarmD
vvAd
d
vvfAd
dvvKF
= Interfacial Area Density, m2/m3
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Drag Force
To estimate the drag force bubble diameter, ,is needed
The is often taken as the mean bubble size
For bubble columns operating at low gas superficial velocities (< 5 cm/s) works reasonably well
For bubble columns operating at higher gas superficial velocities (> 5 cm/s), bubble breakup and coalesce dominate and bubble size is no longer uniform and mean bubble size approach may not be adequate
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The drag coefficient is likely to be different for a single bubble and a bubble swarm. This is because the shape and size of a bubble in a swarm is different than that of an isolated bubble
When the bubble size is small ( < 1mm in water): bubble is approximately spherical
When the bubble size is large ( > 18mm in water): bubble is approximately a spherical cap
When the bubble of intermediate size: bubbles exhibit complex shapes
Drag Coefficient Water Glycerol /
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We can use the Eotvos number () together with the Morton number () to characterize the shape of bubbles or drops moving in a surrounding fluid or continuous phase
Number Ratio of bouncy force and surface tension force and
essentially gives a measure of the volume of the bubble
Number Ratio of physical properties
Constant for a given incompressible two-phase system. Water has a Morton number of .
Bubble Shape
2pgdEo
32
4
q
qgMo
Lorond Eotvos
3mm air bubble rising in tap water
Bubble Regime Map
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Drag Laws for Small and Constant Bubble Sizes
At low flow rates bubbles assume an approximately spherical shape while they rise in a rectilinear path
Schiller and Naumann (1978)
Morsi and Alexander (1972)
Symmetric Drag Model: The density and the viscosity are calculated from volume averaged properties and is given by
Schiller Naumann model
1000 Re :for 44.0C
1000 Re :for Re15.01Re
24C
D
687.0
D
2
321D
ReReC
aaa
q
pqpq dvv
Re
When Reynolds number is small ( < 1) these correlations essentially reduce to the well known Stokes drag law =
24
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For all other flow rate, bubble size and shapes varies with the flow
Consequently, different drag correlations are needed
Several drag correlation are found in literature Grace drag law
Tomiyama drag law
Universal drag law
Drag Laws for Variable Bubble Sizes Larger bubbles - ellipsoidal
As bubble size increases, spherical caps may be formed
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Terminal Rise Velocity for Bubbles The drag correlations for
large bubbles are very different from those for spherical particles
Grace Correlation
Spherical Bubble Correlation
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Bubble Regimes
Viscous and inertial forces are important
the function is given by an empirical correlation e.g. SN
Viscous Regime
Bubbles follow zig-zag paths
is proportional to the size of bubble
is independent of viscosity
Distorted Bubble Regime
Drag coefficient Reaches a constant value Cap Regime
.C .D
44.0,Re1501
Re
24max 6870
gdC pD
3
2,
3
8DC
The drag coefficient on the Reynolds number decreases with increasing values of the Reynolds number
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Flow regime automatically determined from continuity of drag coefficient
Automatic Regime Detection
The determined by choosing minimum of vicious regime and capped regime
CCCCCCCC
distortedDviscousDDdistortedDviscousD
viscousDdistortedDviscousD
,,,,
,,,
,min
3cm/s
35cm/s
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Drag Laws for Variable Bubble Sizes
Universal Drag Law (for Bubbly Flow) Viscous regime
Distorted regime
Capped regime
As the bubble size increases the bubble become spherical caped shaped
)1( 67.18
67.171
3
2 1.52
7/6
ppD ff
fgdC
-13
8C
2
pD
1
ReRe101Re
24 750
;
dvv; .C
p
q
e
e
ppqq.
D
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Drag Laws for Variable Bubble Sizes
Grace Drag Law The flow regime transitions between the viscous and distorted particle flow and can
expressed as follows.
Viscous regime
Distorted regime
Capped regime
/10x9, 3
4H
59.3H ,42.3
59.3H2 ,94.0
)857.0( 3
4
4
0.14-
ref
149.0
441.0
757.0
149.0
q
q
2
mskgEoMo
H
HJ
JMod
vv
gdC
ref
q
p
t
tq
pD
Re15.01Re
24C 687.0D
3
8C D
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Drag Laws for Variable Bubble Sizes
Tomiyama Model (1998)
Like the Grace et al model and universal drag model the Tomiyama model is well suited to gas-liquid flows in which the bubbles can have a range of shapes
43
8,
Re
72),Re15.01(
Re
24minmax 687.0
Eo
EoC
p
D
Viscous Regime
Distorted Regime
Cap Regime
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Non-Drag Forces For gasliquid flows, non-drag forces have a profound influence on the flow characteristics,
especially in dispersed flows
Bubbles rising in a liquid can be subject to a additional forces including:
Lift Force
Wall Lubrication Force
Virtual Mass Force
Turbulence Dispersion Force
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Lift Force When the liquid flow is non-uniform or rotational, bubbles experience a lift force
This lift force depends on the bubble diameter, the relative velocity between the phases, and the vorticity
and is given by the following form
The lift coefficient, , often is approximately constant in inertial flow regime ( < < ), Following the recommendations Drew and Lahey, it is
set to 0.5
Lift forces are primarily responsible for inhomogeneous
radial distribution of the dispersed phase holdup and could be important to include their effects in CFD simulations
qpqqpLlift vvvCF
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Saffman and Mei developed an expression for lift force constant by combining the two lift forces:
Classical aerodynamics lift force resulting from interaction between bubble and liquid shear
Lateral force resulting from interaction between bubbles and vortices shed by bubble wake
Lift Coefficients: Saffman Mei Model
100Re40 :for ;2
Re0.0524
40 Re :for ;Re
Re
2
13314.0
Re
Re
2
10.3314-1
46.6C
Re;Re2
3C
Re)1.0(
'
L
2
'
L
e
dC q
q
pq
L
Suitability Mainly spherical rigid particles Could be applied to small liquid
drops
Shear Lift Force Vorticity induced Lift Force
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Lift Coefficients: Moraga et al Model
Moraga et al. (1999) proposed an al alternative expression for the lift coefficient that correlated with the product of bubble and shear Reynolds numbers
7
73
ReRe
36000
ReRe
L
105ReRe63530
105ReRe6000 20120
6000ReRe07670
C7
e
for .-
for ee..
for .
Suitability Mainly spherical rigid particles Could be applied to small liquid
drops
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Lift Coefficients: Legendre and Magnaudet Model
Legendre and Magnaudet proposed an expression for the lift coefficient that is a function of bubble Reynolds number and dimensionless shear rate
This model accounts of induced circulation inside bubbles
q
q
pq
highL
lowL
highLlowL
dJ
C
JSrC
CC
2
2
32
'
1
1
Re,
'5.0
2Re,
2
Re,
2
Re,L
Re ,Re
Re
2
1 ,
Re
2 ,
1.01
255.2
Re291
Re161
2
1
Re 6
12Sr , 500Re0.1for ,C
Suitability Mainly small spherical bubbles
and liquid drops
33 2015 ANSYS, Inc. April 24, 2015 ANSYS Confidential
Tomiyama et al correlated the lift coefficient for larger bubbles with a modified Etvs number and accounts for bubble deformation
Lift Coefficients Tomiyama Model
g ,163.01 ,
g
0.4740.0159-0.00105
Eo10 27.0
10Eofor
4Eofor Re,121.0tanh288.0min
C
2
q3
1757.0
2
q'
2'3''
'
''
''
L
pp
pH
Hp dEoEodd
dEo
EoEoEof
Eof
Eof
Suitability All shape and size of bubble
and drops
Dependence of lift coefficient on bubble diameter
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This is a force that prevents the bubbles from touching The main effect of this force is to ensure zero void
fraction (found experimentally) near vertical walls
Wall lubrication force is normally correlated with slip velocity and can be expressed as force is defined as:
Wall Lubrication Force
wqpqpWLWL nvvCF||
gas void fraction
Slip velocity component parallel to the wall
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Wall Lubrication Coefficient: Antal et al Model
Antal et al. (1991) proposed a wall lubrication force coefficient according to:
Only active in thin region near wall where:
As a result, the Antal model will only be active on a sufficiently fine mesh
llnearest wa todistance
05.0
01.0
,0max
2
1
21
w
W
W
w
W
p
WWL
y
C
C
y
C
d
CC
bb
W
Ww dd
C
Cy 5
1
2
Suitability Mainly small bubbles Requires Fine Mesh
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Wall Lubrication Coefficient: Tomiyama Model
Modified the Antal model for special case of pipe flow and accordingly:
Coefficients were developed on a single air bubble in a glycerol solution but results have been extrapolated to air-water system
Depends on Eotvos number, hence accounts for dependence of wall lubrication force on bubble shape
eter Pipe DiamD
Eo for .
Eofor .Eo.
Eo for e
o for E .
C
yDy
dCC
.Eo.
W
ww
p
WWL
331790
33501870005990
51
1470
11
2
17909330
22
Suitability Viscous Fluids and all bubble size and shapes Could be used for low air-water system
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Wall Lubrication Coefficient: Frank Model
Generalised Tomiyama model to be geometry independent Model constants calibrated and validated for bubbly flow in vertical pipes
1.7m
6.8llnearest wa toDistance
331790
33501870005990
51
1470
11
,0max
17909330
1
WD
.Eo.
W
m
bWC
ww
bWC
w
WD
WWL
C
Eo for .
Eofor .Eo.
Eo for e
o for E .
C
dC
yy
dC
y
CCC
Suitability Viscous Fluids and all bubble size and
shapes in vertical pipe flows Could be used for low air-water system
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Wall Lubrication Coefficient: Hosokawa Model
Hosokawa et al. (2002) investigated the influence of the Morton number and developed a new correlation for the coefficient:
Includes the effects of Eotvos number and bubble relative Reynolds number on the lift coefficient
EoCWL 0217.0,
Re
7max
9.1
Suitability All bubble size and shapes Could be used for low air-water system
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The turbulent dispersion force accounts for an interaction between turbulent eddies and particles
Results in a turbulent dispersion and homogenization of the dispersed phase distribution
The simplest way to model turbulent dispersion is to assume gradient transport as follows:
Turbulent Dispersion forces
turb.
dispersion
force
fluid vel.
gas void fraction pqqTDTD kCF
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Turbulent Dispersion Models
Lopez de Bertodano Model, Default CTD = 1 CTD = 0.1 to 0.5 good for medium sized bubbles in ellipsoidal flow regime. However, CTD up to 500
required for small bubbles
Burns et al. Model Default CTD = 1 The defaults value of CTD are appropriate for bubbly flows
Simonin Model Default CTD = 1 Same as Burns et al. Model
Diffusion in VOF Model Instead of modelling the turbulent dispersion as an interfacial momentum force in the phase
momentum equations, we can model it as a turbulent diffusion term in the phasic continuity equation
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Turbulence in bubbly flows are very complex due:
Bubble-induced turbulence
Interaction between bubble-induced and shearinduced turbulences
Direct interaction between bubbles and turbulence eddies and
Turbulence Dispersion Models in Fluent Sato
Simonin
Only available when dispersed and per phase turbulence models are enabled
Troshko and Hassan
Alternative to Simonin Model
Turbulent Interaction
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The virtual mass force represents the force due to inertia of the dispersed phase due to relative acceleration
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.
Virtual Mass Force
5.0;
VM
pq
qpVMvm CDt
vD
Dt
vDCf
Dip your palms into the water and slowly bring them together. Such a movement will
require small effort. Now try to clap your hands frequently. The speed of hands now is
low and will require considerable effort
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Mixture Multiphase Model
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Introduction
The mixture model, like the Eulerian model, allows the phases to be interpenetrating. It differs from the Eulerian model in three main respects:
Solves one set of momentum equations for the mass averaged velocity and tracks volume fraction of each fluid throughout domain
Particle relaxation times < 0.001 - 0.01 s
Local equilibrium assumption to model algebraically the relative velocity
This approach works well for flow fields where both phases generally flow in the same direction and in the absence of sedimentation
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Underlying Equations of the Mixture Model
Solves one equation for continuity of mixture
Solves one equation for the momentum of the mixture
Solves for the transport of volume fraction of each secondary phase
0
mm
m ut
rkrkkn
k
km
T
mmmmmm uuFguupuut
u
1
eff
).().()( rpppmpppp uut
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Constitutive Equations
Average density
Mass weighted average velocity
Drift velocity
Slip Velocity
Relation between drift and slip velocities
n
kkkm
1
m
n
k kkk
m
uu
1
mk
r
k uuu
qppq uuu
qk
n
k m
kkpq
r
k uuu
1
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Relative Velocity
If we assume the particles follows the mixture flow path, then, the slip velocity between the phases is
In turbulent flows, the relative velocity should contain a diffusion term in the momentum equation for the disperse phase. FLUENT adds this dispersion to the relative velocity as follows:
q
Dp
m
p
mpp
pqf
au
drag
p
mpvpq
f
au
drag
t
uuuga mmm
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16.0 Release
Validation of the Multiphase Flow in
Rectangular Bubble Column
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Investigate air-water bubbly flow in a rectangular bubble column as investigated at HZDR by Krepper et al., Experimental and numerical studies of void fraction distribution in rectangular bubble columns, Nuclear Engineering and Design Vol. 237, pp. 399-408, 2007
Validation of Momentum Exchange Models for disperse bubbly flows accounting: Drag force
Lift force
Turbulent dispersion
Turbulence Interaction
Turbulence models
Objectives
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Duct Dimensions: Height: 1.0 m
Width: 0.1 m
Depth: 0.01 m
Bubbles are introduced at the bottom LW 0.020.01 m
Computational Geometry
Outlet: Degassing or Pressure Outlet
Inlet: Velocity or mass inlet
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Fluid Materials and Phase Setup
Phases Setup
Phase Specification Primary Phase: water (Material: water) Secondary Phase: gas bubble (diameter: 3mm with Material: air)
Phase Interaction Drag: Grace Drag Force Lift: Tomiyama lift force Wall Lubrication: Antal et al (default coeff.) Turbulent Dispersion Burns et al. (cd=0.8) Turbulent Interaction Sato Model (default coeff.) Surface Tension Coeff.: 0.072
Materials Setups
Gas Bubble FLUENT Fluid Materials: air
Water FLUENT Fluid Materials: water-liquid (h2o)
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Boundary Conditions
Boundary Patch Properties
Inlet
Type: Mass flow inlet Gas Bubble: 2.37E-05 kg/s Gas Volume Fraction (VF): 1.0 Turbulence Intensity 10% Viscosity Ratio 10 Water: mass flow rate: 0 kg/s Water VF: 0.0
Outlet
Type: Degassing Degassing outlet: Symmetry for water Sink for air
Walls
No Slip
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Solution Methods and Control
Solution Methods
Pres.-Vel. Coupling Coupled Scheme
Spatial Discretization Gradient: Least Squared Cell Based Momentum: QUICK Volume Fraction: QUICK TKE: 1st Order Upwind
Transient Formulation Bounded 2nd Order Implicit
Solution Controls
Courant No. 200
Explicit Relax. Factors Momentum: 0.75 Pressure: 0.75
Under-Relax. Factors Density: 1 Body Forces: 0.5 Volume Fraction: 0.5 TKE: 0.8 Specific. Diss. Rate: 0.8 Turb. Viscosity: 0.5
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Instantaneous Gas Volume Fraction
Gas volume fraction at 25s, 35s, 45s
k-SST-Sato k- Troshko-Hassan
Gas volume fraction at 20s, 30s
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Turbulence Validation, Sato Model
Mean gas volume fraction distribution at plane y=0.63m
Mean gas volume fraction distribution at plane y=0.08m
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Turbulence Validation, Troshko-Hassan Model
Mean gas volume fraction distribution at plane y=0.63m
Mean gas volume fraction distribution at plane y=0.08m
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Summary and Conclusions
It was found that the most appropriate drag which is in good accordance with the measurements is the Grace Drag law
The k- turbulence model combined with the Sato Model reproduced well the experiments with no fundamental differences to the k- SST plus the Sato Model. This may indicate that the bubble induced turbulence is quite significant in this bubble column
The Troshko-Hassan k- turbulence model performed well, particularly near the injection point, a region of interest as it seemed to be problematic when the validations were carried out with ANSYS CFX using k- SST plus the Sato Model
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Numerical Schemes and Solution Strategies
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Numerical schemes for multiphase flows
Three algorithms available for solving the pressure-velocity coupling Phase coupled SIMPLE (PC-SIMPLE)
Pressure Coupled (Volume Fraction solved in a segregated manner)
Full multiphase coupled (Volume Fraction solved along with pressure and momentum)
A possibility of solving all primary and secondary phase volume fractions directly rather than solving only the secondary phases directly
Ability to use the Non-Iterative Time Advancement (NITA)
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Multiphase coupled solver Simultaneous solution of the equations of a multiphase system offers a more robust
alternative to the segregated approach
Can be extended to volume fraction correction (Full multiphase coupled)
For steady state problems the coupled based methodology is more efficient than segregated methodology
For transient problems the efficiency is not as good as for steady, particularly for small time steps. Solver efficiency increases with increase in time steps used for discretization of the transient terms.
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Solution controls for PC-SIMPLE Conservative solution control settings are shown
If convergence is slow, try reducing URFs for volume fraction and turbulence
Tighten the multi-grid settings for pressure (lower it by two orders of magnitude). Default is 0.1
Use gradient stabilization (BCGSTAB)
Try using F (or W) cycle for pressure
Solution Strategies
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For steady state problems using coupled multiphase solver is effective
Use lower courant numbers for steady state and higher URFs for momentum and pressure
Recommended values
Courant number = 20
URF pressure and momentum = 0.5 - 0.7
URF volume fraction = 0.2 - 0.5
For transient problems the efficiency of coupled not as good as for steady, particularly for small time steps.
Use larger time steps and high courant numbers (1E7) for coupled solvers and high URFs (> 0.7)
Solution Strategies