(256c) A Graph-Theoretic Solution to the Population Balance Equation for Pure Breakage

Comminution or milling is an essential particulate processing technique in numerous applications, including mineral processing, pharmaceuticals, and agrochemicals. Population balance equations (PBEs) with mechanistic breakage kernels are frequently employed to characterize the evolution of particle size distribution during milling. Although a number of numerical methods have been devised to solve such PBEs, analytical solutions are restricted to PBEs with linear breakage kernels. In this work, we devise a graph-theoretic method for obtaining a generalized analytical solution for PBEs with complex breakage kernels. The discretized PBE solution can be expanded using Sylvester's formula, where the eigenvalues and eigenvectors of the milling matrix are determined using a graph-theoretic approach. The i^th components of the j^th eigenvector are determined by the transition rates of particles of size j to size i. This transition rate may comprise multiple size jumps from j to i. In a similar fashion, the elements of left eigenvectors are obtained by contemplating microscopic reversibility between sizes j and i. The results presented in this work is the first attempt to investigate the generalized analytical solution which can consider all types of kernels (linear or non-linear). To validate, we compared the graph-theoretic solution to an existing, well-known analytical solution for a linear fracture kernel, and the maximum error was about 3% for the discretization of 100 points for large time scales. We present a general case with the potential for future work. Such generalized solutions can provide mechanistic insights into the milling process as well as the means to estimate milling matrix or fracture kernels from experimental data. The method presented in this work gives a solution that involves less than 100 lines of code and is automated for use in studying the breakage problem or uses it for solving similar problems.