On the solution of population balance equations by discretization .1. A fixed pivot technique

A new framework for the discretization of continuous population balance equations (PBEs) is presented in this work. It proposes that the discrete equations for aggregation or breakage processes be internally consistent with regard to the desired moments of the distribution. Based on this framework, a numerical technique has been developed. It considers particle populations in discrete and contiguous size ranges to be concentrated at representative volumes. Particulate events leading to the formation of particle sizes other than the representative sizes are incorporated in the set of discrete equations such that properties corresponding to two moments of interest are exactly preserved. The technique presented here is applicable to binary or multiple breakage, aggregation, simultaneous breakage and aggregation, and can be adapted to predict the desired properties of an evolving size distribution more precisely. Existing approaches employ successively fine grids to improve the accuracy of the numerical results. However, a simple analysis of the aggregation process shows that significant errors are introduced due to steeply varying number densities across a size range. Therefore, a new strategy involving selective refinement of a relatively coarse grid while keeping the number of sections to a minimum, is demonstrated for one particular case. Furthermore, it has been found that the technique is quite general and yields excellent predictions in all cases. This technique is particularly useful for solving a large class of problems involving discrete-continuous PBEs such as polymerization-depolymerization, aerosol dynamics, etc.