Numerical treatment of the population balance equation using a Spline-Galerkin method

Abstract

This paper describes a numerical technique for solving the Lifshitz-Slyozov equation of continuity which applies to certain mass transfer proceses. The Lifshitz-Slyozov equation which describes a mechanism involving the transfer of atoms (or undissociated molecules) from smaller particles to larger particles dispersed in a supersaturated medium (Ostwald ripening) is also considered together with collision and subsequent coalescence of particles (ripening plus collection). Special attention is given to the case in which the net growth rate appearing in the Lifshitz-Slyozov equation is a nonsmooth function of the type I( υ) α υ β, where υ is the particle volume and O<β<1. The basic numerical approach is to perform a spatial discretization of the equations using a projection technique on a space of cubic splines. A Galerkin technique and a θ-method for solving systems of ordinary differential equations is used to determine the expansion coefficients. The performance of the numerical method is investigated by solving equations that arise in population balance. It is shown that in the case of a combination of ripening plus collection that a single initial particle size distribution can evolve into a double distribution of particle sizes.