Lecture 4: Gas-Liquid Flows 15.0 Release
Advanced Multiphase Course
Outline •
Introduction
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Conservation equations
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Modelling strategies : Euler-Lagrangian and Eulerian
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Interfacial Forces •
Drag
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Non-Drag Forces
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Turbulence Interaction
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Mixture Model
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Validation example
Introduction •
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Gas –liquid flows occur in many applications. The motion of bubbles in a liquid as well as droplets in a conveying gas stream are examples of gas – liquid flows. Bubble columns are commonly used in several process industries Atomization to generate small droplets for combustion is important in power generation
Rain/Hail Stones
Combustion
Spray Drying
Bubble Column
Distillation Process
Boiling Process Absorption Process
Why Study Gas-Liquid Flows The main interests in studying gas-liquid flows, in devices like bubble columns or stirred tank reactors, are:
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Design and scale-up
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Fluid dynamics and regime analysis
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Hydrodynamic parameters
Bubble Columns •
To design bubble column reactors, the following hydrodynamic parameters are required: •
Specific gas –liquid interfacial area ( )
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Sauter mean bubble diameter, ( )
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Axial and radial dispersion coefficients of the gas and liquid, ( )
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Heat and mass transfer coefficients, (,)
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Gas holdup, ( )
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Physicochemical properties of the liquid medium, (,)
Regime Analysis Two types of flow regimes are commonly observed in bubble columns:
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The bubbly flow regime,
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< / Bubbles are of relatively uniform small sizes ( = )
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Rise velocity does exceed 0.025m/s
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Holdup shows linear dependence with the flo w
< . < . /
Regime Analysis •
The churn turbulent flow regime •
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> . > . /
> / Bubble are Large bubbles ( > ) and show wide size distribution Rise velocity is in the range of 1-2m/s
Most frequently observed flow regime in industrialsize, large diameter columns
Photographic Representation of Bubbly and ChurnTurbulent Flow Regimes
Bubbly Flow Regime
Churn Turbulent Flow Regime
Design and Scale-up of Bubble Column Reactors •
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Bubble have significant effect on hydrodynamics well as heat and mass transfer coefficients in a bubble columns The average bubble size and rise velocity in a bubble column is found to be affected by:
Euler-Lagrangian Method •
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In this approach, a single set of conservation equations is solved for a continuous phase
Eulerian Cell Gravity
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
Buoyancy Liquid Flow
Eulerian Approach •
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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
Conservation Equations Continuity equation: q q t
m n
q q v q
pq
mqp S q
p 1
so ur ce
mass transfer
Momentum equation: Drag Force s 2 αq ρq v q αq ρq v q αq p τ q αq ρq g K pq v p v q m pq v pq mqp v qp t p 1 Friction Pr essure Bouyancy mass trans fer Interfacial Force Non Drag Force s F q F lift,q F wl,q F vm,q F td,q external Lift Wall Lubrication Virtual Ma ss Turbulent Dispersion n
Interphase Momentum Exchange A key question is how to model the interphase momentum exchange
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This is the force that acts on the bubble and takes into account:
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Effect of multi-bubble interaction
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Gas holdup
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Turbulent Interaction •
Turbulent Dispersion
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Turbulent Interaction
Drag
Virtual Mass
Lift Interphase Momentum Exchange
Turbulent Interaction
Turbulent Dispersion
Drag Force •
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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 •
For a single spherical bubble, rising at steady state, the drag force is given by: F D
C D
A p
drag coefficien t
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q
2
v p
v q v p v q
slip velocity
For a swarm of bubbles the drag, in absence of bubble-bubble interaction, is given by: 6 p q F NF C A v v q v p v q D , swarm D d p3 D p 2 p q 3 p v p v q v p v q C D 4 d
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: F D , swarm
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3 p q 4
q v v q v p v q C D d p p
In Fluent
F D , swarm K pq v p v q
p p
p p
18 d d 6 A f v q
2 p
p
i
p
vq
p 18 q d p C D Re Ai v p v q d 2 p p 6 24
Drag Force •
To estimate the drag force bubble diameter, ,is needed
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The is often taken as ‘the mean bubble size’
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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
Drag Coefficient •
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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
Glycerol
Water /
Bubble Shape 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
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Number
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Eo
3mm air bubble rising in tap water
gd p2
Ratio of bouncy force and surface tension force and essentially gives a measure of the volume of the bubble
Lorond Eotvos Number
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Mo
Ratio of physical properties
g q4 2
3
q
Constant for a given incompressible two-phase system. Water has a Morton number of . −
Bubble Regime Map
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
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Schiller and Naumann (1978)
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CD
1 0.15 Re Re 24
0.68 7
C D 0.44
for : Re 1000 for : Re 1000
Morsi and Alexander (1972)
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C D a1 Symmetric Drag Model:
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a2 Re
a3 Re 2
Re
q
v p v q d p q
When Reynolds number is small ( < 1) these correlations essentially reduce to the well known Stokes drag law = 24
The density and the viscosity are calculated from volume averaged properties and is given
Drag Laws for Variable Bubble Sizes •
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For all other flow rate, bubble size and shapes varies with the flow
Larger bubbles - ellipsoidal
Consequently, different drag correlations are needed Several drag correlation are found in literature •
Grace drag law
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Tomiyama drag law
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Universal drag law
As bubble size increases, spherical caps may be formed
Terminal Rise Velocity for Bubbles Spherical Bubble Correlation
The drag correlations for large bubbles are very different from those for spherical particles
Grace Correlation
Bubble Regimes
Viscous Regime
Distorted Bubble Regime
Viscous and inertial forces are important • the function is given by an empirical correlation e.g. SN •
Bubbles follow zig-zag paths • is proportional to the size of bubble • is independent of viscosity •
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Cap Regime
Drag coefficient Reaches a constant value
24 1 0.15 Re0 .68 7 ,0.44 Re
C D max
The drag coefficient on the Reynolds number decreases with increasing values of the Reynolds number
C D ,
2
C D
8
3
3
d p
g
Automatic Regime Detection •
Flow regime automatically determined from continuity of drag coefficient C D , viscous C D , distorted C D ,viscous
C C C min C , C D , viscous D , distorted D D , viscous D , distorted
The determined by choosing minimum of vicious regime and capped regime
3cm/s
35cm/s
Drag Laws for Variable Bubble Sizes •
Universal Drag Law (for Bubbly Flow) •
Viscous regime C D
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0 .75
ρq
Re
v q v p d p μe
; μe
μq
1 α p
Distorted regime C D
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1 0.1Re ; Re 24
2 3
d p
g 1 17.67 f 6 / 7
18.67 f
2
f (1 p )1.5
Capped regime •
As the bubble size increases the bubble become spherical caped shaped C D
1 - 3
8
p
2
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
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Distorted regime
CD
1 0.15 Re Re 24
0.68 7
g C d p D 2 3 q v t 0.757 , 0.94 H J 0.441 , 3.42 H 4
q 4 H EoMo 0.149 3 ref •
Capped regime
C D
8
q Mo 0.149 ( J 0.857) q d p
v t
2 H 59.3 H 59.3
- 0.14
, ref 9 x10
4
kg / ms
Drag Laws for Variable Bubble Sizes •
Tomiyama Model (1998)
24
C D max min
Re p
(1 0.15 Re 0.687 ),
Viscous Regime •
72 8 Eo , Re 3 Eo 4
Distorted Regime
Cap Regime
Like the Grace et al model and universal drag model the Tomiyama model is well suited to gasliquid flows in which the bubbles can have a range of shapes
Non-Drag Forces •
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For gas –liquid 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:
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Wall Lubrication Force •
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Lift Force
Virtual Mass Force
Turbulence Dispersion Force
Lift Force When the liquid flow is non-uniform or rotational, bubbles experience a lift force
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This lift force depends on the bubble diameter, the relative velocity between the phases, and the vorticity and is given by the following form
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F lift C L p q v q v p v q •
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The lift coefficient, , often is approximately constant in inertial flow regime and ( < < ) and, 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
Lift Coefficients: Saffman Mei Model •
Saffman and Mei developed an expression for lift force constant by combining the two lift forces: “Classical” aerodynamics lift force resulting from interaction
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between bubble and liquid shear Lateral force resulting from interaction between bubbles and vortices shed by bubble wake
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CL
Known as wake effect 3 2 Re
Shear Lift Force
2
C L' ; Re
q d p q
q
1 - 0.3314 1 Re e( 0.1Re) 0.3314 1 Re ; 2 Re 2 Re ' C L 6.46 Re ; 0.0524 2
for : 40 Re 100
Vorticity induced Lift Force
Suitability
for : Re 40
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Mainly spherical rigid particles Could be applied to small liquid drops
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
0.0767 Re Re Re Re C L 0.12 0.2e 36000 e 3e -0.6353 ω
7
for Re Re
ω
6000
ω
for 6000
Re Re
for Re Re
5 107
ω
ω
5 107 Suitability •
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Mainly spherical rigid particles Could be applied to small liquid drops
Lift Coefficients: Legendre and Magnaudet Model •
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Legendre and Magnaudet proposed an expression for the lift coefficient that correlated with the product of bubble Reynolds number and dimensionless shear rate This model accounts of induced circulation inside bubbles CL
C L2,low Re C L2,high Re ,
C L ,low Re C L ,high Re J '
6
for 0.1 Re 500 , Sr 2 1
0. 5 ' Re Sr J 2
Suitability
1 1 16 Re 1
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2 1 29 Re 1 2.255 3
1 0.1
2 2
,
2 Re
,
1 Re 2 Re
, Re
q d p2 q
q
Mainly small spherical bubbles and liquid drops
Lift Coefficients – Tomiyama Model •
Tomiyama et al correlated the lift coefficient for larger bubbles with a modified Eötvös number and accounts for bubble deformation
min 0.288 tanh 0.121Re, f Eo' ' C L f Eo 0.27
f Eo 0.00105 Eo - 0.0159 Eo '
Eo '
'3
g
q
2 p d H
,
'2
for Eo 10 '
10 Eo'
0.474
d H d p 1 0.163 Eo
Suitability •
for Eo' 4
All shape and size of bubble and drops
, 1
0.757
3
Eo
g
q
p d p2
Wall Lubrication Force This is a force that prevents the bubbles from touching
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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:
F WL
C WL p q v p
vq
nw ||
Slip velocity component parallel to the wall
gas void fraction
Wall Lubrication Coefficient: Antal et al Model Antal et al. (1991) proposed a wall lubrication force coefficient according to:
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C C max 0, W 1 W 2 d p y w C W 1 0.01 C W 2 0.05 y w distance to nearest wall C WL
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Only active in thin region near wall where:
Suitability •
C W 2 d b 5d b C W 1
yw
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Mainly small bubbles Requires Fine Mesh
As a result, the Antal model will only be active on a sufficiently fine mesh
Wall Lubrication Coefficient: Tomiyama Model •
Modified the Antal model for special case of pipe flow and accordingly: C WL
C W
d p 2
1 1 2 yw D y 2 w
0.47 0.933 Eo 0.179 e C W 0.00599 Eo 0.0187 0.179 D Pipe Diameter •
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for E o 1
for 1 Eo 5 for 5 Eo 33 for 33 Eo
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 Suitability •
Viscous Fluids and all bubble size and shapes
Wall Lubrication Coefficient: Frank Model •
Generalised Tomiyama model to be geometry independent
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Model constants calibrated and validated for bubbly flow in vertical pipes yw 1 1 C WC d b C WL C W max 0, m 1 C yw WD y w C d WC b for E o 1 0.47 0.93 3 Eo0.179 for 1 Eo 5 e C W for 5 Eo 33 0.00599 Eo 0.0187 0.179 for 33 Eo C WD Distance to nearest wall 6.8 m 1.7
Suitability •
Viscous Fluids and all bubble size and shapes in vertical pipe flows Could be used for low air-water system
Wall Lubrication Coefficient: Hosokawa Model •
Hosokawa et al. (2002) investigated the influence of the Morton number and developed a new correlation for the coefficient:
C WL
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7 max 1.9 ,0.0217 Eo Re
Includes the effects of Eotvos number and bubble relative Reynolds number on the lift coefficient
Suitability •
All bubble size and shapes
Turbulent Dispersion forces •
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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: F TD
C TD q k q p
turb. dispersion force gas void fraction fluid vel.
Turbulent Dispersion Models •
Lopez de Bertodano Model, Default CTD =1 •
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Burns et al. Model Default C TD =1 •
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The defaults value of C TD are appropriate for bubbly flows
Simonin Model Default C TD =1 •
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CTD = 0.1 to 0.5 good for medium sized bubbles in ellipsoidal flow regime. However, CTD up to 500 required for small bubbles
Same as Burns et al. Model
Diffusion in VOF Model : Models •
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
Turbulent Interaction Turbulence in bubbly flows are very complex due:
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Bubble-induced turbulence Interaction between bubble-induced and shear –induced turbulences Direct interaction between bubbles and turbulence eddies and
Turbulence Dispersion Models in Fluent •
Sato
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Simonin •
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Only available when dispersed and per phase turbulence models are enabled
Troshko and Hassan •
Alternative to Simonin Mode l
Virtual Mass Force •
The virtual mass force represents the force due to inertia of the dispersed phase due to relative acceleration
f vm
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Dv q Dv p ; C VM 0.5 C VM p q Dt Dt
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.
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
Mixture Multiphase Model
Introduction The mixture model, like the Eulerian model, allows the phases to be interpenetrating. It differs from the Eulerian model in three main respects:
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Solves one set of momentum equations for the mass averaged velocity and tracks volume fraction of each fluid throughout domain
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Particle relaxation times < 0.001 - 0.01 s
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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
Underlying Equations of the Mixture Model •
Solves one equation for continuity of mixture
m mum 0 t
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Solves one equation for the momentum of the mixture n um T mumum p eff um um mg F k k uk r uk r t k 1
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Solves for the transport of volume fraction of each secondary phase t
( p p ) .( p p um ) .( p pu pr )
Constitutive Equations •
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Average density
Mass weighted average velocity
Drift velocity
n m k k k 1
um
Slip Velocity
u k 1 k k k
m
uk r uk um
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n
u pq
u p uq
n
u
r k
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Relation between drift and slip velocities
u pq
k 1
k k
uqk
Relative Velocity •
If we assume the particles follows the mixture flow path, then, the slip velocity between the phases is
u pq
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a v p f drag
m
p
um a g um um t
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:
u pq
a p f
p m m q
Validation of the Multiphase Flow in Rectangular Bubble Column, 15.0 Release
Objectives •
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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
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Lift force
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Turbulent dispersion
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Turbulence Interaction
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Turbulence models
Computational Geometry Duct Dimensions:
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Height: 1.0 m Width: 0.1 m Depth: 0.01 m
Outlet: Degassing or Pressure Outlet
Bubbles are introduced at the bottom
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L W 0.02 0.01 m
Inlet: Velocity or mass
Fluid Materials and Phase Setup Materials Setups Gas Bubble
FLUENT Fluid Materials:
air
Water
FLUENT Fluid Materials:
water-liquid (h2o)
Phases Setup Phase Specification
Primary Phase: Secondary Phase:
water (Material: water) gas bubble (diameter: 3mm with Material: air)
Phase Interaction
Drag: Lift: Wall Lubrication: Turbulent Dispersion Turbulent Interaction Surface Tension Coeff.:
Grace Drag Force Tomiyama lift force Antal et al (default coeff.) Burns et al. (cd=0.8) Sato Model (default coeff.) 0.072
Boundary Conditions Boundary Patch Inlet
Properties Type: Gas Bubble:
Water:
Outlet
Type: Degassing outlet:
Walls
No Slip
Mass flow inlet 2.37E-05 kg/s Gas Volume Fraction (VF): 1.0 Turbulence Intensity 10% Viscosity Ratio 10 mass flow rate: 0 kg/s Water VF: 0.0
Degassing Symmetry for water Sink for air
Solution Methods and Control Solution Methods Pres.-Vel. Coupling
Coupled Scheme
Spatial Discretization
Gradient: Momentum: Volume Fraction: TKE:
Least Squared Cell Based QUICK QUICK 1st Order Upwind Bounded 2nd Order Implicit
Transient Formulation
Solution Controls Courant No.
200
Explicit Relax. Factors
Momentum:
0.75 Pressure:
Under-Relax. Factors
Density: 1 Body Forces: Volume Fraction: 0.5 TKE: Specific. Diss. Rate: 0.8 Turb. Viscosity:
0.75 0.5 0.8 0.5
Instantaneous Gas Volume Fraction kω-SST-Sato
Gas volume fraction at 25s, 35s, 45s
k-ε Troshko-Hassan
Gas volume fraction at 20s, 30s
Turbulence Validation, Sato Model
Mean gas volume fraction distribution at plane y=0.63m
Mean gas volume fraction distribution at plane y=0.08m
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
Summary and Conclusions •
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It was found that the most appropriate drag which is in good accordance with the measurements is the Grace Drag law The k-ε turbulence model combined with the Sato Model reproduced well the experiments with no fundamental differences to the k- ω SST plus the Sato Model. This may indicate that the bubble induced turbulence is quite significant in this bubble column The Troshko-Hassan k-ε 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-ω SST plus the Sato Model
Numerical Schemes and Solution Strategies
Numerical schemes for multiphase flows Equations describing the two fluid model are strongly linked and source terms dominated
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Drag forces create large source terms in the conservation equations. Without decoupling a numerical solution may not be obtained Lift forces create extra coupling among velocity components
Three algorithms available for solving the pressure-velocity coupling
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Phase coupled SIMPLE (PC- SIMPLE)
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Pressure Coupled (Volume Fraction solved in a segregated manner )
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Full multiphase coupled (Volume Fraction solved along with pressure and momentum)
Multiphase coupled solver •
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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 methodology is more efficient than segregated methodology For transient problems the efficiency of coupled 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.
Solution Strategies •
Solution controls for PC-SIMPLE – Conservative solution control settings are shown – If convergence is slow, try reducing URFs for volume fraction and turbulence.
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Tighten the multi-grid settings for pressure (lower it by two orders of magnitude) . Default is 0.1 – Use gradient stabilization (BCGSTAB) Try using W cycle for pressure
