computational study of liquid-liquid dispersion in a rotating disc contactor a. vikhansky and m....
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Computational Study of Liquid-Liquid Dispersion in a Rotating Disc Contactor
A. Vikhansky and M. KraftDepartment of Chemical Engineering,
University of Cambridge, UK
M. Simon, S. Schmidt, H.-J. BartDepartment of Mechanical and Process Engineering,
Technical University ofKaiserslautern, Germany
Rotating disc contactor
Department of Mechanical and Process Engineering, Technical
University of Kaiserslautern,Kaiserslautern, Germany
cm 15
cQ
dQ
Flow patterns
The approach
• Compartment model
• Weighted particles Monte Carlo method for population balance equations
• Monte Carlo method for sensitivity analysis of the Smoluchowski’s equations
• Parameters fitting
Compartment model:Breakage, coalescence,
transport
n t ,x;B n; D n;
t
Population balance equation
0
00
( ) 1( ) ( ) ( )
2
( ) ( ) ( ) ; (0 ) ( ).
xn t xK x x x n t x x n t x dx
t
K x x n t x n t x dx n x n x
expH n t x H min
?
H
Smoluchowski's equation
Identification procedure
2. Assume a set of the model’s parameters.
3. Solve population balance equations.
4. Calculate the parametric derivatives of the solution.
5. Compare the solution with the experimental data and update the model’s parameters.
1. Formulate a model.
11
( ) { ( ) ( )} ( ) ( ( )).N
N n nn
x t x t x t n t x w x x t
Stochastic particle system:
n
x
n
x
1
( )( ( )).
Nn
nn
wn t xx x t
A Monte Carlo method for sensitivity analysis of population balance equations
11
( ) { ( ) ( )} ( ) ( ( )).N
N n nn
x t x t x t n t x w x x t
Stochastic particle system:
n
x
n
x
n
1
( )( ( )).
Nn
nn
wn t xx x t
A Monte Carlo method for sensitivity analysis of population balance equations
0 0
( ) ( )( ) ( )
( )( ) ( ) ; (0 ) ( ) ( ).
m t x K x x xm t x x m t x dx
t xK x x
m t x m t x dx m x m x xn xx
( ) ( )m t x xn t x
A Monte Carlo method for sensitivity analysis of population balance equations
11
( ) ( )1;
{ ( ) }
Nk l l l
N
K x x w K x x w x
x K x x w x
11
( ) { ( ) ( )} ( ) ( ( )).N
N n nn
x t x t x t m t x w x x t
Stochastic particle system:
1. generate an exponentially distributed time increment with parameter 1
ˆ ( ) ˆ1 N K x x w
x
2. choose a pair to collide according to the distribution
1
ˆ ( ) ˆˆ{ ( ) }ˆ
lk l lN
K x x xw
K x x xw
3. the coagulation is accepted with the probability
( )ˆ ( ) ˆ
k l l
lk l
K x x w
K x x w
4. or reject the coagulation and perform a fictitious jump that does not change the size of the colliding particles with the probability
( )1
ˆ ( ) ˆk l l
lk l
K x x w
K x x w
k k lx x x
ˆ ( ) ( )ˆ n nK x x K x x ww
Acceptance-rejection method
1
1
( ) ( ) , ; ( ( ));
( ) (1 ) ( ( ));
N
n nn
N
n n nn
H t m h x m t x dx w h x t
H t m w W h x t
Calculation of parametric derivatives of the solution of the coagulation equation
1
( )( ( )).
N
n n nn
m t xw W x x t
A disturbed system:
1
( ) 1 ( ( ));
( ).
N
n n nn
m t x w W x x t
K x x
Evolution of the disturbed system is the same as the undisturbed one, while
the factors kW ln( )k k lW W K W
if the coagulation is accepted, or as ln( )
ˆˆl
k k l
l l
K WW W w K
w K w K
have to be recalculated as
if the coagulation is rejected
2 2 31 1( ) ( )
2 2c cu D D 28.
The model: Breakage of the droplets
2 3
11 2
t /c
P D expD
D
D
Ds
6
2 1 3
2 3
1 /
/
u D
t D D
1/3
1 22/3 2/3 5/3exp
c
c cD D
The model: Collision and coalescence
1 3 1 22 2 2 3 2 3
1 2 1 2 1 2 3 1 2 12 24 1
/ // /,h D D u u c D D D D
2
1 21,2 4 32
1 2
exp1c c D D
cD D
1,2 1,2 1,2K h
The model: Transport
c
resrise
h
v D
1c
rise v T c v T
Qv D k v D v k v D
A
2/3 1/3
14.2 6
c d cT
c c
D Eëv D g
5Vk c
Operational conditions
Identified parameters and residuals
Experimental vs. numerical
results
fitted unfitted
Coefficients of sensitivity
ln
ln ic
Volume fraction
Mass-mean diameter
Sauter mean diameter
Conclusions
• A Monte Carlo method was applied to a population balance of droplets in two-phase liquid-liquid flow.
• The unknown empirical parameters of the model have been extracted from the experimental data.
• The coefficients identified on the basis of one set of experimental data can be used to predict the behaviour of the system under another set of operating conditions.
• The proposed method provides information about the sensitivity of the solution to the parameters of the model.