Computational Study of Liquid-Liquid Dispersion in a Rotating Disc Contactor A. Vikhansky and M....

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.