Computational Study of LiquidLiquid Dispersion in a Rotating

Computational Study of Liquid-Liquid Dispersion in a Rotating Disc Contactor A. Vikhansky and M. Kraft Department of Chemical Engineering, University of Cambridge, UK M. Simon, S. Schmidt, H. -J. Bart Department of Mechanical and Process Engineering, Technical University of Kaiserslautern, Germany

Rotating disc contactor Department of Mechanical and Process Engineering, Technical University of Kaiserslautern, Germany

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 Population balance equation

Smoluchowski's equation

Identification procedure 1. Formulate a model. 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.

n n A Monte Carlo method for sensitivity analysis of population balance equations x x Stochastic particle system:

n n A Monte Carlo method for sensitivity analysis of population balance equations x x Stochastic particle system:

A Monte Carlo method for sensitivity analysis of population balance equations Stochastic particle system:

Acceptance-rejection method 1. generate an exponentially distributed time increment with parameter 2. choose a pair to collide according to the distribution 3. the coagulation is accepted with the probability 4. or reject the coagulation and perform a fictitious jump that does not change the size of the colliding particles with the probability

Calculation of parametric derivatives of the solution of the coagulation equation A disturbed system:

Evolution of the disturbed system is the same as the undisturbed one, while the factors have to be recalculated as if the coagulation is accepted, or as if the coagulation is rejected

The model: Breakage of the droplets

The model: Collision and coalescence

The model: Transport

Operational conditions

Identified parameters and residuals

Experimental vs. numerical results fitted unfitted

Coefficients of sensitivity 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.