Analytical solutions of 1D population balance equation at steady-state

Due to their effectiveness in separation and purification, two-phase flow columns (liquid-liquid, gas-liquid, and solid-liquid) are extensively utilized in the chemical industries. PBE has recently been recognized as an appropriate tool for modeling this kind of column owing to its ability to describe both the hydrodynamics and the mass transfer of the dispersed phase. In this work, we solved analytically one-dimensional PBE at steady-state using the Adomian decomposition method and the Method of moments. Analytical solutions are provided for pure growth, pure breakup, breakup with growth, pure aggregation, and breakup with growth with aggregation. The obtained results encourage extending the applicability of both methods to solve 1D PBE. Highlights Analytical solutions of the one-dimensional steady-state Population Balance Equation (PBE) are presented. The Adomian Decomposition Method (ADM) is applied to solve cases involving breakup, aggregation, and growth. The Method of Moments (MOM) is used to handle combined growth, breakup, and aggregation processes. Both space-dependent and volume-dependent particle velocity models are considered. The results enhance understanding of dispersed phase behavior in two-phase flow columns and confirm the effectiveness of ADM and MOM.