(167b) Efficient Evaluation of Multi-Dimensional Source Term Integrals in Population Balance Models

The solution of a population balance equation is a function F(t,r,x) describing a population density of particles of property vector x at time t and space r. Depending on the application, the population balance equation contains additional source terms involving integral operators. The dimensionality of the vector x in turn determines the number of integrals. For instance, granulation processes should ideally be characterized by three internal coordinates (i..e., solid, liquid and gaseous dimensions) [1], which results in a triple integral in the aggregation and breakage source terms.

In this study, novel semi-analytical solutions are proposed for the aggregation and breakage source terms. This results in casting the complex triple integrals into simpler addition and multiplication terms, major portions of which are computed once a priori to the start of the simulation. These semi-analytical solutions are derived based on the assumption of uniform population density within each finite volume. Via a finite volume discretization technique, the overall integro-partial differential equation as represented by the population balance equation is reduced to a system of ordinary differential equations in terms of the rates of aggregation and breakage. A first order Euler explicit predictor-corrector technique in combination with an error control mechanism incorporating the novel solution of the source terms is compared with an implicit solution technique that solves the full triple integral during the course of the simulation. Results show that the explicit technique with the proposed novel source term solutions result in significant savings in computational tine without compromising the accuracy of the solution. To reduce computational time further, a parallel programming framework was implemented using a message passing interface (MPI) that essentially distributes the computational burden of numerically evaluating the triple integral, across multiple nodes. Results show that parallel processing decreases simulation time further without any numerical inconsistencies seen in the solutions.

References:

1. D. Verkoeijen, G. A. Pouw, G. M. H. Meesters and B. Scarlett, ?Population balances for particules processes ? a volume approach?, Chemical Engineering Science, 57, 2287-2303.