A Coupled Stochastic-Deterministic Method for the Numerical Solution of Population Balance Systems

Abstract

In this thesis, a new algorithm for the numerical solution of population balance systems is proposed and applied within two simulation projects. The regarded systems stem from chemical engineering. In particular, crystallization processes in fluid environment are regarded. The descriptive population balance equations are extensions of the classical Smoluchowski coagulation equation, of which they inherit the numerical difficulties introduced with the coagulation integral, especially in regard of higher dimensional particle models. The new algorithm brings together two different fields of numerical mathematics and scientific computing, namely a stochastic particle simulation based on a Markov process Monte—Carlo method, and (deterministic) finite element schemes from computational fluid dynamics. Stochastic particle simulations are approved methods for the solution of population balance equations. Their major advantages are the inclusion of microscopic information into the model while offering convergence against solutions of the macroscopic equation, as well as numerical efficiency and robustness. The embedding of a stochastic method into a deterministic flow simulation offers new possibilities for the solution of coupled population balance systems, especially in regard of the microscopic details of the interaction of particles. In the thesis, the new simulation method is first applied to a population balance system that models an experimental tube crystallizer which is used for the production of crystalline aspirin. The device is modeled in an axisymmetric two-dimensional fashion. Experimental data is reproduced in moderate computing time. Thereafter, the method is extended to three spatial dimensions and used for the simulation of an experimental, continuously operated fluidized bed crystallizer. This system is fully instationary, the turbulent flow is computed on-the-fly. All the used methods from the simulation of the Navier—Stokes equations, the simulation of convection-diffusion equations, and of stochastic particle simulation are introduced, motivated and discussed extensively. Coupling phenomena in the regarded population balance systems and the coupling algorithm itself are discussed in great detail. Furthermore, own results about the efficient numerical solution of the Navier—Stokes equations are presented, namely an assessment of fast solvers for discrete saddle point problems, and an own interpretation of the classical domain decompositioning method for the parallelization of the finite element method.