Solution of population balance equations

Abstract

This chapter addresses the solution methods for population balance equations. Laplace transforms are particularly suitable for obtaining analytical solutions for certain forms of population balance equations. In aggregating systems, the population balance equation in particle mass (or volume) features a convolution integral in the source term, which makes it amenable to solution by Laplace transforms. The calculation of moments of the number density function can occasionally be accomplished by directly taking moments of the population balance equation producing a set of moment equations. In order for the moment equation to generate only terms containing moments, the breakage rate must be a polynomial function. However, even this requirement leads to a set of unclosed moment equations because the differential equation in any moment involves higher moments. The chapter also discusses the method of orthogonal collocation in which the residual is exactly equated to zero at a finite number of points , normally the zeroes of the polynomial of the smallest degree not included in the expansion. This procedure produces a number of equations equal to the number of expansion coefficients to be estimated in the unknown trial solution.