International Max Planck Research School for Analysis, Design and Optimization in Chemical and Biochemical Process Engineering Magdeburg "Mathematical Theory for the Dynamics of Coagulation-Fragmentation Equations for Process Engineering"

We are considering coagulation-fragmentation equations which are a type of partial integro-differential equations. For these we are considering typical questions of mathematical and numerical analysis. The coagulation-fragmentation equations model the dynamics of cluster growth and describe the time evolution of a system of clusters under the combined effect of coagulation and fragmentation. Each cluster is identified by its size (or volume) which is assumed to be a positve real number. From a physical point of view the basic mechanisms taken into account are the coalescence of two clusters to form a larger one and the breakage of clusters into smaller ones. These models are of subtantial interest in many areas of science: colloid chemistry, aerosol physics, astrophysics, polymer science, oil recovery dynamics, fluidized bed granulation processes, mathematical biology etc. Several researcher derived existence and uniqueness results for solutions to coagulation equations with binary fragmentation. However, the case of multiple fragmentation was mostly neglected. We established the existence of solutions to coagulation equations with multiple fragmentation for a large class of kernels which relies on the weak L1 compactness methods applied to suitably chosen approximating equations. The question of uniqueness was also considered and a new result was established. Recently, we gave the convergence analysis of the fixed pivot technique given by S. Kumar and Ramkrishna for solving the nonlinear coagulation population balance equations. In a sequel to this work, we also study the convergence analysis of the cell average technique given by J. Kumar et al. for nonlinear coagulation population balance equation and compared the mathematical and numerical observations with those for the fixed pivot technique. It is observed that the cell average technique gives a better performance than the fixed pivot technique on non-uniform grids. The doctorate was successfully completed in November 2010.