Modeling and Simulation of Population Balances for Particulate Processes

This work focuses on the modeling and numerical approximations of population balance equations (PBEs) for the simulation of different phenomena occurring in process engineering. The population balance equation (PBE) is considered to be a statement of continuity. It tracks the change in particle size distribution as particles are born, die, grow or leave a given control volume. In the population balance models the one independent variable represents the time, the other(s) are property coordinate(s), e.g., the particle volume (size) in the present case. They typically describe the temporal evolution of the number density functions and have been used to model various processes such as granulation, crystallization, polymerization, emulsion and cell dynamics. The semi-discrete high resolution schemes are proposed for solving PBEs modeling one and two-dimensional batch crystallization models. The schemes are discrete in property coordinates but continuous in time. The resulting ordinary differential equations can be solved by any standard ODE solver. To improve the numerical accuracy of the schemes a moving mesh technique is introduced in both one and two-dimensional cases. A model is derived for the batch preferential crystallization of enantiomers with fines dissolution unit. The model is further elaborated by considering the isothermal and nonisothermal conditions. In this model, the crystallization of the preferred enantiomers is assumed to take place in a single crystallizer with a fines dissolution loop. The model is further extended to a coupled batch preferential crystallization process with isothermal and non-isothermal conditions. In this setup, the crystallization of two enantiomers is assumed to take place in two separate crystallizers, coupled by their fines dissolution loops. The finite volume scheme for one-component pure aggregation model is extended to twocomponent aggregation model. For this purpose the integro-ordinary differential equation for two-component aggregation is reformulated to a partial differential equation (PDE) coupled with integral equations. The resulting PDE is then solved by a semi-discrete finite volume scheme which also employs the geometric grid discretization technique. Analogous procedure is used for the numerical solution of one-component breakage problem. The proposed numerical schemes are further investigated by solving PBEs with simultaneous nucleation, growth, aggregation and breakage processes. Two methods are proposed for this purpose. In the first method, a method of characteristics (MOC) is used for growth process while a finite volume scheme for aggregation and breakage processes. In the second method, a semi-discrete finite volume scheme (FVS) is used for all processes. Various numerical test problems are considered for the underlying models. The numerical results are also validated against available analytical solutions.