01/45
Interface Tracking and Multi-Fluid
Simulations of Bubbly Flows
in Bubble Columns
Akio Tomiyama and Kosuke Hayashi
Kobe University
CFD2011
June 21 – 23, 2011, Trondheim, Norway
Kobe University
02/45
Numerical Methods for Bubbly Flow Simulation
Interface Tracking
Method (ITM)
Bubble Tracking
Method (BTM)
d > 10∆x
d ≅ ∆x
Two-Fluid Model
(TFM)
Multi-Fluid Model
(MFM)
d m < ∆x
( m = 1,L , N )
d < ∆x
Hybrid CMFD (Computational Multi-Fluid Dynamics)
BTM
BTM & MFM
ITM
MFM
HCMFD
BTM
ITM & MFM
ITM
MFM
ITM & BTM
BTM
Kobe University
03/45
Arbitrary Combination of Different Functions
Low Res. ITM+MFM
High Res.
Mid. Res. ITM+BTM ITM+BTM+MFM
ITM
2
1
0
-20
-10
0
x (mm)
10
20
Measured
Predicted
20
Void fraction (%)
Measured
Predicted
Void fraction (%)
Void fraction (%)
3
10
0
-20
-10
0
x (mm)
10
10
0
-20
20
Measured
Predicted
20
-10
0
x (mm)
10
Color means void
Fraction of tiny
bubbles
20
Good predictions by using an appropriate combination!
Kobe University
04/45
Examples of Interface Tracking Simulations
104
logM=-11.6, µ*=31.5
logM= -6.0, µ*=1.85
logM= -4.8, µ*=0.94
logM=-3.0, µ*=0.37
logM=-1.0, µ*=0.11
simulation
103
102
Re
2
Experiment
Simulation
1
CL
101
100
0
-1
10-1
10-2
10-1
100
Eo
101
102
Comparison between measured and
predicted drop Re numbers
2
4
6
Eo
Comparison between measured
& predicted bubble lift coefficients
Mixing by Ωshaped wake
When applying ITM to elementary phenomena in bubble columns,
functions for simulating mass transfer, chemical reaction etc. are desired.
Kobe University
05/45
Interface Tracking Simulation of
Mass Transfer through or onto
Gas-Liquid Interface
Mass Transfer through bubbles at high Reynolds numbers
Adsorption and Desorption of Surfactant
Kobe University
06/45
Interface Tracking Methods for Mass Transfer
Boundary-Fiited Coordinate Method
Ponoth & McLaughlin, 2000
Sugiyama et al., 2003
Ponoth &
McLaughlin
Front Tracking Method
Koynov et al., 2005, 2006
Tryggvason et al., 2010
Darmana et al., 2007
Darmana
Volume of Fluid and Level Set Methods
VOF: Davidson & Rudman, 2002
Bothe et al., 2004, 2010, 2011
Onea et al., 2006, 2009
LS:
Yang & Mao, 2005
Wang et al., 2006
Bothe
Yang &
Mao
Few methods can deal with volume change and bubbles at high Sc and high Re.
Kobe University
07/45
Bubbles at High Re and Sc Numbers
CO2 bubble in downward water flow
Schmidt number: 530
Reynolds number: 1500
Pipe diameter: 12.5mm
Initial bubble diameter: 29.0 mm
Initial Ingredients: CO2 only
Final bubble diameter: 7.5 mm
Final ingredients: N2 (80%), O2 (20%)
Requirements for simulating bubbles at high Re and Sc numbers
(1) Accurate conservation of species moles
(2) Accurate evaluation of interfacial mass transfer
(3) To capture a thin concentration boundary layer at high Sc
Kobe University
08/45
Field Equations
Mass & Momentum Equations (One-Fluid Formulation)
∂ρ
+ ∇ ⋅ ρV = 0
∂t
∂V
1
1
σκnδ
+ V ⋅ ∇V = − ∇P + ∇ ⋅ µ[∇V + (∇V )T ] + g +
∂t
ρ
ρ
ρ
Conservation of Species Molar Concentration Ck
∂C k
+ V ⋅ ∇ C k = ∇ ⋅ Dk ∇ C k
∂t
Liquid
ρL, µL, CL
(k = G , L)
n
Gas
ρ G , µG , C G
Jump Conditions for Ck
CG = mC L
m: distribution coefficient
(Henry’s law)
j ⋅ n |int = − DG
∂CG
∂n
= − DL
int
∂C L
∂n
interface
CL
CG
int
Kobe University
09/45
Popular Method: Single-Variable Formulation
Variable Transformation
Interface
⎧mC x ∈ L
Φ=⎨ L
⎩ CG x ∈ G
CL
CG
Φ
Φ
Single-Variable Formulation
∂Φ
+ V ⋅ ∇Φ = ∇ ⋅ Dˆ ∇Φ
∂t
Φ Gint = Φ Lint
− DG
∂Φ
∂n
= −( D L / m)
int
∂Φ
∂n
int
Solutions, CL & CG, are obtained by solving only the single equation for Φ.
This formulation has been often adopted.
(Bothe et al., 2004; Yang & Mao, 2005; Onea et al., 2006; 2009; Francois & Carlson, 2010)
Kobe University
Defect of Single-Variable Formulation
10/45
Example: 20mm CO2 Bubble without Any Mass Transfer (No Volume Change)
Yan & Mao’s method (Level set, 5th order WENO, 3rd order TVD, RK)
Φ field
0s
d=20mm
D=25mm
CO2-water
center line
0.5 s
2.0 s
Strong shape deformation causes
very large numerical diffusion
CFD2011: Fleckenstein & Bothe stopped using this for simulating high Re bubbles
Kobe University
11/45
Our Method : Two-Variable Formulation
Two-Variable Formulation for Moles, MG and ML
∂M k
+ ∫S C kV ⋅ dS = ∫S Dk ∇C k ⋅ dS
k
k
∂t
M
Ck = k
Θk
Mk: mole in phase k [mol]
Ck: molar concentration [mol/m3]
Θk: volume of phase k [m3]
Accurate Advection of Mk
M kn +1 = M kn − ∆t ∫S C kV ⋅ dS
k
Transferred mole = Transferred fluid volume times C
Transferred volume: Volume Tracking Method based on
NSS (Non-uniform Subcell Scheme, Hayashi et al., 2006)
+ EI-LE scheme (Aulisa et al., 2003)
Accurate Conservation of Fluid Volume & Species Moles
Kobe University
12/45
Validation: Conservation of Moles
Comparison between single- and two-variable formulations
20 mm CO2 bubble in water (pipe diameter = 25 mm)
Single-variable
Two-variable
(LSM + Φ)
(Proposed)
2.0 s
2.0 s
Two-variable formulation gives no numerical diffusion
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13/45
Computation of Interfacial Mass Transfer
Trend: Use of a Solution based on Boundary Layer Approximation
Tryggvason et al., 2010, Bothe et al., 2010
⎛ CGi +1 j
C L (ξ, η) = ⎜⎜
⎝ m
⎛
⎞⎡
ξ
⎟ ⎢1 − erf ⎜
⎟⎢
⎜ 4D η / V
⎠⎣
L
B
⎝
⎞⎤
⎟⎥
⎟⎥
⎠⎦
⎡ ∂C ⎤
j ⋅ n |int = − DL ⎢ L ⎥
⎣ ∂n ⎦ int
This may not hold for actual boundary layers (wavy, flow separation, ..).
Evaluation of Interfacial Mass Flux using Single-Variable & Harmonic Mean
CL*
ΦL*
CG*
Φ G*
∆x/2
∆x/2
*: interpolated value
j ⋅ n int = − DG
Φ int − Φ*G
Φ* − Φ int
= −( DL / m) L
∆x / 2
∆x / 2
Elimination of Φint
Φ *L − Φ *G
ˆ
j ⋅ n |int = − D
∆x
Peskin’s kernel
2 Dˆ G Dˆ L
Dˆ =
Dˆ G + Dˆ L
Φ G*
Φ L*
Φint
⎧ D / m for k = L
Dˆ k = ⎨ L
⎩ DG for k = G
Kobe University
14/45
Validation: Mass Diffusion with Interface Jump
CG=1 & CL=0 at t=0
m = 0.2 (CG=mCL, CL>CG at interface)
DG/DL = 10
No volume change
(x, y) = (128, 128 cells)
1.5
1.4
1.2
1
CL, CG
1
CL
CG
Theory
Liquid
0.8
Bubble
interface
0.6
0.4
0
0.2
0
0
0.2
0.4
x
0.6
0.8
10,000 step
Interfacial mass transfer is accurately computed.
Kobe University
1
15/45
Validation: Mass Transfer from Free Rising Bubble
Case 1: Eo=1, M=1x10-4
Case 2: Eo=40, M=9.2x10-3
Case 3: Eo=3.1, M=5x10-7
Sc=1
m = 1 (no jump in C), CG=constant
No volume change
(R, H) = (40, 160 cells)
Mass transfer coefficient kL
1
(C Ln +1 − C Ln )dV
∫
V ∈L
A∆t
kL =
15
1/2
1/2
1/2
Sh=(2/π )[1−2.89/Re ]Re Sc
for high Re bubbles
1
1/2
10
Sh
Sh =
kLd
DL
5
Sh=1+(1+0.564Re2/3Sc2/3)3/4
for creeping flow
0 0
10
101
Re
0
Case 1
Re=5.5
Case 2
Re=31
102
Mass transfer from a bubble
at low Re and low Sc is well
predicted
Case 3
Re=97
Kobe University
16/45
Accuracy of Computation of Volume Change
300
0.7
200
∆ΘG [mm3]
CL [mol/m3]
Bubble: d=15 mm, CO2 100% at t = 0 sec
Pipe: D= 18.6 mm
Re = 2700, Sc = 530
Water: equilibrium concentration of N2 & O2
Volume Change : Considered
100
0
0
0
T= 10 s
40 s
70s
90 s
Calculated from volume fraction αG
∆ΘG=ΣijΘcellαG(t=0)−ΣijΘcellαG(t)
Calculated from total mass flux
∆ΘG=[∆MCO2−∆MN2−∆MO2]RT/P
10
20
30
40
50
t [s]
The volume change due to mass transfer is accurately computed.
Kobe University
Simulation of 8 mm CO2 Bubble in Stagnant Water
17/45
Re=1200, Sc=530, D = 18.2 mm
1.5
1.5
0
CO2
CL
[mol/m3]
Interface oscillation plays an important
role in mass transfer process.
Boundary layer approximation would not
hold in separated regions
85 cells for the bubble radius
Kobe University
Comparison with Measured Bubble Dissolution Process
1.5
5.3
0
CL
CL [mol/m3] CO2
18/45
0
CL
CG [mol/m3] N2
8
d [mm]
7.9
7.8
7.7
7.6
7.5
0
8mm CO2 Bubble
Correlation
dini = 8.0 mm
D = 18.2 mm
Prediction
2
Sh =
(0.6λ2 + 0.49λ + 1) Re1/ 2 Sc1/ 2
π
0.02
0.04
0.06
t [s]
0.08
Cell size = 47 µm
(170 cells for d)
0.1
Kobe University
19/45
Adaptive Mesh Refinement for Concentration Field
Sc = 500
Reduction of cell numbers 98.4%
An example of AMR
Coarse cell
for V and P
Finer cells
for Moles
near the
interface
High Resolution only for Concentration Boundary Layer
Kobe University
20/45
Effects of Surfactant on Bubble & Drop Dynamics
1. Reduction of surface tension σ
2. Marangoni effect (tangential force)
3. Increase in damping coefficient of capillary wave
µ-regime
Terminal velocity VT [cm/s]
50
σ-regime
i-regime
Air bubble in water
10mm
clean
bubble
clean
30
20
10
Fully-contaminated
10mm
dirty
bubble
5
0.5
1
2 3
5
10
Bubble diameter d [mm]
20 30
Decrease in VT only in some bubble diameter range
Simulations have been performed only for µ-regime
No effects of surfactant on VT at High EoD (Well-known fact)
Increase in VT at low EoD (Almatroushi & Borhan, 2004)
∆ρgD 2
EoD =
σ
Kobe University
21/45
Interface Tracking Methods for Contaminated Bubbles
Takagi
Boundary-Fiited Coordinate Method
Tukovic
Cuenot et al.,1997 (spherical bubble)
Takagi et al., 2003 (spherical bubble)
Tukovic & Jasak, 2008 (deformed bubble)
Front Tracking Method
Muradoglu & Tryggvason, 2008
Muradoglu &
Tryggvason
(low Re spheroidal bubble)
Volume of Fluid and Level Set Methods
Alke & Bothe
VOF: Renardy et al., 2002
James & Lowengrub, 2004
Alke & Bothe, 2009
LS:
Xu & Shao, 2003
Xu et al., 2006
Few methods can simulate highly distorted bubbles at high Re numbers
Kobe University
22/45
Field equations & Numerical Methods
Mass & Momentum Equations (Projection Method)
Ghost fluid
∇ ⋅V = 0
method CSF model
∂V
1
1
1
+ V ⋅ ∇V = − ∇P + ∇ ⋅ µ[∇V + (∇V )T ] + g + [σκn + ∇ S σ]δ
∂t
ρ
ρ
ρ
Bulk & Surface Concentrations of Surfactant
Liquid: ρL, µL, C
Two step method
∂C
+ V ⋅ ∇C = ∇ ⋅ DB∇C + S&Γ δ (Muradoglu & Tryggvason)
∂t
∂Γ
+ ∇S ⋅ ΓVS = ∇S ⋅ DS ∇ S Γ − S&Γ
∂t
n ⋅ ∇C = 0
Γ
n
Bubble
ρG, µG
Extrapolation method (Xu & Zhao)
Adsorption-Desorption Kinetics (Langmuir model)
S&Γ = − n ⋅ ( DC ∇C |int ) = k [C S (Γmax − Γ ) − β Γ ]
Advection of Interface
∂φ
+ V ⋅ ∇φ = 0
∂t
Level set method
(Susuman)
k: adsorption constant
β: desorption constant
(Frumkin & Levich)
Surface Tension
⎛
Γ
σ = σ 0 + RTΓmax ln⎜⎜1 −
⎝ Γmax
⎞
⎟
⎟
⎠
Kobe University
23/45
Small Air Bubble in Contaminated Water (µ-Regime)
0.565
Γ*=Γ/Γmax
Fluids: air and water
Surfactant:1-pentanol
k=5.08 m3/mol⋅s
β=21.7 mol/m
Γmax=5.9x10-6 mol/m2
Bubble diameter: 0.6-0.98 mm
0.585
100 [kN/m3]
Center line
d=0.72mm, C=30mol/m3
d=0.72mm
3
d [mm]
0.98
0.85
0.72
0.60
2
8
C =30, 20, 10 mol/m3
2
1
0.5
C =30, 20, 10 mol/m3
8
CD
Mei et al. (1994)
Moore (1965)
Mei (1993)
CD=(4/3Re2)[Eo3/M]1/2
2
1
1
0.5
0.1
C =0 mol/m3
8
0.1
3
8
C =0 mol/m
Center line
Pure
0.5
mol/m3
30
0
0
mol/m3
50
100
150
Re
200
250
300
Kobe University
24/45
4 mm air bubble in clean water
4 mm air bubble in contaminated
water (C=10mol/m3)
Distorted Air Bubble in Contaminated Water (σ-Regime)
100
0.20
Γ*
CD=8Eo/3(Eo+4)
z
0.37
0.10
10-1
Correlation
Clean (Tomiyama et al., 1998)
Contaminated (Tomiyama et al., 1998)
Γ*
z
d=4mm
y
Clean
0.43
C=10 mol/m3
Measured
Clift et al. (1978)
(Data depicted from figure in Tomiyama et al., 1998)
10-2 0
10
Predicted
Clean
3
C = 10 mol/m
8
d=2mm
VT [m/s]
y
101
d [mm]
Kobe University
25/45
Taylor Bubbles in Air-Water System (σ and i-Regimes)
∆ρgD 2
EoD =
σ
No effects of surfactant on VT at High EoD (Well-known fact)
Increase in VT at low EoD (Almatroushi & Borhan, 2004)
0
-10
-20
8
Clean
C = 30 mol/m3
-30
0
5
10
-2
EoD=9.7
(D=8.5mm)
-4
-6
Clean
C = 30 mol/m3
8
EoD=120
(D=30mm)
Distance from bubble nose [mm]
Distance from bubble nose [mm]
0
-8
-8.5
15
0
1
r [mm]
2
3
4 4.25
r [mm]
Clean Contam.
VT(m/s) 0.187 0.187
Clean Contam.
VT(m/s) 0.049 0.056
Kobe University
Distributions of Surfactant, Marangoni Force & Velocity
EoD=120
26/45
50 [kN/m3]
EoD=9.7
Γ*=Γ/Γmax
50 [kN/m3]
σ0=0.073
0.47
z
x
Γ*
σ
0.91
CL
0.038
σ0=0.073
0.48
z
x
Γ*
σ
0.82
Bubble
0.40 m/s
Bubble
0.47
Γ*
0.91
0.048
0.10 m/s
CL
0.48
Γ*
0.82
Marangoni force acts only at bottom edge.
Surfactant distribution strongly depends on velocity profile. Kobe University
27/45
Cause of VT Increase at Low EoD
0.4
Fr = 0.351 (Dumitrescu, 1943)
Good predictions
for the clean system
Fr
VT
Fr =
(ρ L − ρG ) gD
0.3
ρL
Contaminated case
high Fr at low Eo
same Fr at high Eo
0.2
Negligible effect of
Marangoni force
White & Beardmore
(1962)
M < 9.1x10−11
Clean
C = 30 mol/m33
C = 30 mol/m
without Marangoni force ∇ S σ
C = 30 mol/m3
plotted against EoD using σ at nose
0.1
0
4
EoD =
102
10
∆ρgD 2
σ nose
Surface tension at
bubble nose is
the key parameter
300
EoD
The nose surface tension determines the rise velocity of a low M Taylor Bubble
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28/45
Multi-Fluid Simulation of
Bubbly Flow in
a Bubble Column
Validation of Bubble Coalescence and Breakup Model
Effects of Fine Particles on Bubble Coalescence
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29/45
Closure Relations of MFM for Poly-Dispersed Flow
Dispersed Phases
(m = 0,L, N )
G
∂nm
+ ∇ ⋅ nmVm = ∑ (Γm '→m − Γm→m ' ) − Rm
∂t
m '≠ m
DVm
α m ρG
= −∇P + g − (M int m + M Γm + M Rm )
Dt
RGm
mass
transfer
Continuous Gas & Liquid Phases
release &
entrainment
turbulence
Γm
τ Re
M
int m
coalescence
& breakup interfacial forces
L
N
∂α G
∂α L
+ ∇ ⋅ (α GVC ) = ∑ Rm
+ ∇ ⋅ (α LVC ) = 0
∂t
∂t
m =0
N
DVC
α C ρC
= −∇P + FS + α C ρC g + ∑ M int m + M Rm + ∇ ⋅ α C ( τ + τ Re )
Dt
m =0
(
)
Experiments on single bubbles and drops in infinite liquids or in pipes
Drag and lift correlations for various bubbles and drops
Experiments on mass transfer from single bubbles in vertical pipes
Sherwood number correlations
Experiments on turbulent poly-dispersed bubbly pipe flows
Validation of two-phase turbulence model
Experiments on poly-dispersed bubbly flow in a bubble column
Validation of bubble coalescence and breakup models
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30/45
Experimental Setup & Conditions
D=200mm
z=800mm
z/D=4
z=600mm
z/D=3
H x W x D : 1400 x 200 x 200 [mm]
Fluids : Air-water at 25 oC and 1 atm
Gas flow rate : 1.44 x 10-3 m3/s (JG= 0.036 m/s)
Initial liquid level : 1 m
Void distribution: Conductivity probe
Bubble sizes: Images (2500 bubbles/elevation)
Gas diffusers
z=400mm
z/D=2
dhole = 2 mm
∆x
∆y ∆x=25mm
∆y=25mm
200mm
z=200mm
z/D=1
200mm
Nhole=9
Qhole [cc/s]
dexp [mm]
160
27.3
Nhole=25
57.6
18.1
Nhole=49
29.4
13.9
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31/45
Flow Patterns (High-Speed Video Images)
z ~ 800 mm
L/D~4
z ~ 200 mm
D=200 mm
L/D~1
z ~ 75 mm
Breakup dominates
Coalescence dominates
L/D~0.4
Nhole=9, d=27.3mm
Nhole=25, d=18.1mm
Nhole=49, d=13.9mm
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32/45
Bubble Coalescence & Breakup Models
Source Sink
∂nm
+ ∇ ⋅ nmVm = ∑ (Γm '→m − Γm→m ' )
∂t
m '≠ m
1
∞
θ
Source: ∑ Γm '→m = ∫ θm rB (θm , θm ' ) f (θm ' ) dθm ' + ∫ 0 m rC (θm ' , θm ) f (θm ' ) f (θm − θm ' ) dθm '
2
m '≠ m
Coalescence birth
Breakup birth
Sink:
θ
∑ Γm→m ' = ∫ 0 m θm 'rB (θm ' , θm )dθm '
m '≠ m
Breakup
Breakup death
f (θ m ) ∞
+ ∫ 0 rC (θm ' , θm ) f (θm ' )dθm ' f (θm )
θm
Coalescence death
Luo & Svendsen
Collision of turbulent eddies
Coalescence
Random collision
VB difference
Prince & Blanch
Shear slip
Wake entrainment
Wang
Kobe University
33/45
Comparison of Bubble Size Distributions
Void Fraction
1
0.05
0.20
0.8
3
4
5
1
6
Breakup dominates
0.4
PDF (Volume ratio:θm / θtotal )
0.2
z/D = 4
z/D = 2
0
0
0.8
0.8
z/D ~ 2
0.6
0.4
0.2
0
0.8
z/D ~ 1
0.6
0.4
4
5
6
z/D ~ 4
Coalescence dominates
0.4
0.2
0
0.8
0.6
0.2
0
measured
calculated
0.8
z/D ~ 0.25
0.6
0.4
0.4
0.2
0.2
0
0
z/D ~ 1
0.4
0
measured
calculated
z/D ~ 2
0.6
0.2
0.6
z/D = 0.25
3
0.2
0.8
z/D = 1
2
dinlet = 14mm Nhole = 49
0.6
z/D ~ 4
0.4
1
0.8
dinlet = 27mm Nhole = 9
0.6
6 groups
din=const.
2
PDF (Volume ratio:θm / θtotal )
0.0
1
10
20
30
Bubble diameter [mm]
z/D ~ 0.25
0
0
40
10
20
30
Bubble diameter [mm]
40
Kobe University
34/45
Comparison
Void Distributions
5.4of
計算結果
0.15
Breakup dominates
0.15
dinlet = 18mm Nhole = 25
dinlet = 27mm Nhole = 9
z/D = 4
0.1
0.1
0.05
0
0
0
z/D = 2
0.1
0.05
αB
αB
0
0
z/D = 2
0.1
0.05
0.05
0
0
measured
calculated
z/D = 1
0.1
0
x [mm]
100
z/D = 2
0.1
0
measured
calculated
z/D = 1
0.1
0
-100
0
0.05
0.05
0.05
z/D = 3
0.1
0.05
0.05
0
-100
z/D = 3
0.1
z/D = 4
0.1
0.05
z/D = 3
Coalescence dominates
dinlet = 14mm Nhole = 49
z/D = 4
0.05
0.1
αB
0.15
measured
calculated
z/D = 1
0.1
0.05
0
x [mm]
100
0
-100
0
x [mm]
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100
35/45
Experimental Setup and Condition
Silica particle: d=60, 100, 150 µm,
ρ=1320kg/m3
hydrophilic, spherical
D=200 mm
200 mm
Locations of void
measurement
Particle concentration CS : 0, 20, 40 vol.%
Gas volume flux JG: 0.02, 0.034 m/s
Initial water level: 1 m
1200 mm
z
Inlet bubble diameter: 10.9, 12.6 mm
z/D = 3
Void distribution: Conductivity probe
z/D = 2
z/D = 1
25mm
y
Water
Bubble
25mm
200mm
x
Stainless tube
φf =1.4mm
L = 300 mm
49 holes
φ = 1.4 mm
Air
Air
200mm
Air chamber
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36/45
Effect of Particles on Flow Patterns
JG = 0.02 m/s
CS = 0 % 20 %
200mm
dS = 100 µm
JG = 0.034 m/s
C
=
0
% 40 %
20 %
S
Particles seem to promote bubble coalescence.
40 %
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Effects of Particle Concentration on Void Distributions
10
JG = 0.02 m/s dS = 100 µm
CS = 0 % 20 % 40 %
20
5
0
z/D = 2
5
z/D = 3
Bubble Frequency [Hz]
Void Fraction [%]
z/D = 3
10
0
z/D = 2
15
10
5
0
-100
0
x [mm]
Void fraction decreases with increasing CS.
Bubble velocity increases with CS.
100
0
-100
0
x [mm]
100
Number of bubbles decreases with increasing CS.
Particles promote coalescence.
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38/45
Observation of Particle Effects
CS = 0%
CS = 20%
CS = 40 %
300mm
100mm
Air
3mm
Bubble in narrow
parallel plates
dS = 100 µm
“Time required for coalescence” decreases with increasing CS.
Increase in coalescence probability
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Measurement of Time required for Coalescence tC
0s
0.0769 s
1
tC = 230 msec
0
0
0.1
0.2
0.3
4
Distance between bubbles [mm]
2
Distance between bubbles [mm]
Distance between bubbles [mm]
3
0.0788 s
Film breakup
tC = 0.0788 – 0.0769 = 0.0019 s
Distance between bubbles
4
0.0779 s
3
2
1
0
0
0.02
t [s]
0.04
t [s]
0.06
0.08
4
3
2
1
0
0
0.02
0.04
t [s]
0.06
0.08
(a) Long contact: tC = 230 msec (b) Short contact: tC = 15 msec (c) Instantaneous coales. tC = 0
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40/45
Effect of Particle Size & Concentration on tC
dS=60 µm
20%
dS=100 µm
20%
40%
dS=150 µm
40%
20%
40%
50
CS=0%
PDF (%)
40
20%
40%
30
20
10
0
0
0.05
0.1
0.15
0
0.05
0.1
t C (s)
0.15
0
0.05
0.1
0.15
Coalescence probability increases with decreasing dS and increasing CS.
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41/45
Introduction of a Particle Effect Multiplier to tC
0.1
dS [µm]
60
100
150
tC [s]
0.08
0.06
tC decreases with increasing CS.
tC decreases with dS.
0.04
tC reaches a minimum value at CS=40%.
0.02
Instantaneous coalescence
0
0
10
20
30
CS [%]
40
50
Coalescence efficiency: PC = exp(−β tC / τ)
β: particle effect multiplier
τ: contact time
β=1 at CS = 0% (Gas-liquid)
β=0 for CS > 40% (Instantaneous coalescence)
β~0.5 at CS = 20% (Experimental data)
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Comparison for CS = 0 %
JG = 0.02 m/s
β=1
Volume ratio ( θm/θtotal )
Calculated Measured
1
z/D = 3
0.8
0.6
0.4
0.2
0
0
10 20 30 40 50 60
Bubble diameter [mm]
20
z/D = 3
10
Void Fraction [%]
15
10
5
0
z/D = 2
15
10
5
0(%)
Void Fraction
near the wall
Void Fraction
on the center plane
0
-100
0
x [mm]
100
Kobe University
Comparison for CS = 20 %, dS = 100 µm
JG = 0.02 m/s
β = 0.5
43/45
Volume ratio ( θm/θtotal )
Calculated Measured
1
z/D = 3
0.8
0.6
0.4
0.2
0
0
10 20 30 40 50 60
Bubble diameter [mm]
20
z/D = 3
10
23
0(%)
Void Fraction
near the wall
Void Fraction [%]
15
10
5
0
z/D = 2
15
10
5
0(%)
Void Fraction
Particle Fraction
on the center plane
0
-100
0
x [mm]
100
Kobe University
Comparison for CS = 40 %, dS = 100 µm
JG = 0.02 m/s
β=0
44/45
Volume ratio ( θm/θtotal )
Calculated Measured
1
z/D = 3
0.8
0.6
0.4
0.2
0
0
10 20 30 40 50 60
Bubble diameter [mm]
20
z/D = 3
10
45
0(%)
Void Fraction
near the wall
Void Fraction
Particle Fraction
on the center plane
0(%)
Void Fraction [%]
15
10
5
0
z/D = 2
15
10
5
0
-100
0
x [mm]
100
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45/45
Summary
Accurate interface tracking simulation of contaminated bubbles and
drops for a wide range of fluid properties and Reynolds number is
possible, provided that physical properties for the adsorption and
desorption kinetics are available.
Interface tracking simulation of mass transfer from a bubble for a
wide range of Sc and Re numbers is also feasible, provided that
HPCs are available.
Fine particles promote bubble coalescence and the effects of particles
can be reasonably predicted by introducing a multiplier to the time
required for coalescence.
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