Forward and inverse integration of population balance equations with size-dependent growth kinetics

A method for forward and inverse integration of a class of population balance equations with a size-dependent growth rate is contributed in this article. A unique differential transformation of the independent time and internal coordinates is introduced, leading to straight line characteristics with constant values for the density function. The evolution of the density function is then given by transporting the initial and boundary conditions. For the integration of the temporal behavior of the boundary conditions, a generalization of the standard method of moments is introduced, resulting in integro-differential equations involving convolution integrals. The solution to the inverse integration problem involves a pre-computation of given correlation/convolution integrals. The usability of the method is illustrated in a case study of a batch crystallization process with size-dependent growth rate kinetics. The proposed method is compared to a high resolution finite volume scheme using a numerical example.