Analytical solution of the 1-D population balance equation using Laplace transformation and Adomian Decomposition Method

The population balance equation (PBE) is a fundamental tool in chemical engineering for modeling particulate processes where particle size distribution significantly impacts product quality and process performance. It finds extensive applications in crystallization, polymerization, granulation, L-L extraction and aerosol dynamics. In this study, we present an analytical technique for solving the one-dimensional population balance equation (PBE). This technique combines Laplace transformation with the Adomian Decomposition Method (LADM), where the Laplace transformation is applied with respect to time, and the Adomian Decomposition Method (ADM) is used to address the resulting equation along the spatial coordinate. We examine specific cases involving pure convection, pure growth, pure breakage, and the combined processes of growth and breakage. This methodology provides analytical solutions involving the transport and particle-particle interaction phenomena of the dispersed phase.