Formulation of a Mass-Based Population Balance Equation: Insights Into Derivation, Mass Transfer, and Nondimensionalization

Population balance equations (PBEs) are well-suited for modeling properties of dispersed phases such as bubbles, droplets and solid particles in processes. Unlike commonly used momentum and energy balance equations, PBEs transcend external spatial coordinates, incorporating internal property dimensions such as length, shape, and concentration. This study focuses on transforming the traditional number-based PBEs into a mass-based formulation, offering superior characteristics and insights in engineering applications. The mass-based formulation results in a conservative equation which allows the evaluation of breakage and coalescence kernels as well as the numerical method for mass conservation. The transformation introduces a new source term describing mass increase from outside the balanced system, confirming an approach origin from kinetic gas theory published by other authors. The study addresses challenges in discretizing the coalescence birth term denominator by nondimensionalization of the PBE. This research enhances our understanding of PBE modeling, particularly with the inclusion of mass transfer, contributing valuable insights for engineering applications.