Analytical treatment of fragmentation‐diffusion population balance

Abstract

AbstractThe quest for advanced models to realistically simulate various processes involving populations of “particles” (aerosols, dispersions of all kinds, minerals, cultures of microorganisms) necessitates the extension of the classical population balance formulation, which is of zero dimension (lumped) in external and one‐dimensional (1‐D) (single particle‐characterization variable) in internal coordinate space, to more dimensions in both spaces. In order to contribute in this direction, this work is focused on a spatially distributed fragmentation equation, the spatial nonuniformity being governed by a diffusion mechanism and by deposition of particles on solid boundaries. The combined effect of diffusion and fragmentation on the particle‐size distribution is of interest here. The linear form of the fragmentation‐diffusion equation allows a convenient analytical manipulation, based on decomposition of solutions in spatial and temporal modes. Generalized solutions are derived for the above equation for arbitrary spatial domains, fragmentation functions, deposition rates and particle sources. The above approach can be combined with the well‐known approaches for the numerical solution of the lumped fragmentation equation to derive solutions for the fragmentation‐diffusion equation under arbitrary conditions. This study, as a first step, is restricted to obtaining explicit results for some rather simple cases, for example, power law fragmentation and deposition rates, fragmentation in equal sizes and simple shapes of the domain under consideration. © 2004 American Institute of Chemical Engineers AIChE J, 50: 1746–1759, 2004