A feasible solution technique for higher-dimensional population balance models

Abstract

This paper presents a numerical technique for the solution of (multi-dimensional) population balance models. The technique is based on an extension of a so-called hierarchical two-tier technique presented previously in the context of single-dimensional population balances [Immanuel, C. D., & Doyle III, F. J., (2003a). Computationally efficient solution of population balance models incorporating nucleation, growth, and coagulation: Application to emulsion polymerization. Chemical Engineering Science, 58, 3681–3698], and later extended to a three-dimensional problem [Immanuel, C. D., & Doyle III, F. J., (2005). Solution technique for a multi-dimensional population balance model describing granulation processes. Powder Technology, 156, 213–225]. The specific contributions of this article are two-fold: 1. Development of the technique to handle breakage/division phenomena in population balance. 2. Demonstration of the feasibility of the technique to handle modestly multi-dimensional population balance models. The technique is applied to one-dimensional, three-dimensional, and six-dimensional population balance models. In each case, the results are found to be qualitatively consistent with theory. The computational times are also found to be conducive for the intended applications (process analysis, optimisation, and control).