Simplifying population balance models to promote broader use in industry

Abstract

The approach outlined in this paper takes advantage of the existence of stationary states and similarity solutions to reduce a one-dimensional population balance model with particle volume as the internal dimension from a partial integro-differential equation to a single ordinary differential equation for the evolution of the mean size. The mechanisms of accretional growth, collisional growth and breakage, either individually or in all possible combinations, are considered. Power-law rate kernels are employed and a self-similar fragment distribution is assumed for the breakage term. When expressed in terms of the evolution of the scaled number-mean particle volume, the model parameters consist of kernel orders and the characteristic times associated with each mechanism. Equations for the scaled moments of the stationary states and similarity solutions are exhibited for all possible mechanism combinations as are equations for the trajectories of scaled number-mean particle volume. Stability of the stationary states and similarity solutions is addressed in an appendix. It should be clear that when this level of simplification is possible, the subject of population balance modeling becomes considerably easier to teach while its use becomes much easier to promote in industry. The use of this simplified approach for extraction of model parameters from experiment is demonstrated in an example, followed by a brief discussion of using the results for scale up. To close out the paper, the same approach, but now in two internal dimensions (particle volume and surface area), is used to demonstrate the capability of the model to unify a set of operating conditions for carbon black manufacture across multiple grades and two different types of synthesis reactors. This is done by extracting the characteristic coalescence time at multiple temperatures and correlating the results in an Arrhenius plot. The model matches known values of primary particle size, and predicts a key indicator of the product’s fractal aggregate structure, namely the number of primary particles per aggregate. The model results are then pushed to yield insight into the nature of the possible collision and coalescence mechanisms.