Population Balances Involving Aggregation and Breakage Through Homotopy Approaches

Abstract

Homotopy techniques in nonlinear problems are getting increasingly popular in engineering practice. The main reason is because the homotopy method deforms continuously a difficult problem under study into a simple problem, which then can be easy to solve. This study explores several homotopy approaches to obtain semi- or approximate analytical solutions for various cases involving mechanistic phenomena such as aggregation and breakage. The well-established approximate analytical methods namely, the Homotopy Perturbation Method (HPM), the Homotopy Analysis Method (HAM), and the more recent forms of homotopy approaches such as the Optimal Homotopy Asymptotic Method (OHAM) and the Homotopy Analysis Transform Method (HATM) have been used to solve using a general mathematical framework based on population balances. In this study, several test cases have been discussed such as conditions in which the aggregation kernel is not only constant, but also sum or product dependent. Furthermore cases involving pure breakage, pure aggregation and a combined aggregation-breakage have been studied to understand the sensitivity of these homotopy-based methods in solving PBM. In all these cases, the solutions have been analytically studied and compared with literature. Using symbolic computation and carefully chosen perturbation parameters, the approximate analytical solutions are compared with each other and with the available analytical solution. A convergence analysis of the solution methods is made in comparison to the available solution. The case studies indicate that OHAM performs slightly better than both HATM and HPM in solving nonlinear equations such as the PBEs.