(654a) A Graph-Theoretic Approach for Generalized Analytical Solutions of Population Balance Equations with Complex Breakage Kernels in Particle Milling

Particle size reduction through milling is pivotal in diverse industries like mineral processing, pharmaceuticals, and agrochemicals. Understanding how particle size distribution changes during milling often relies on population balance equations (PBEs) with mechanistic breakage kernels. While numerical methods are common for solving these equations, analytical solutions are usually limited to PBEs with linear breakage kernels. In this study, we introduce a new graph-theoretic approach to develop a generalized analytical solution for PBEs with complex breakage kernels. By breaking down the PBE solution and using Sylvester's formula, we identify the eigenvalues and eigenvectors of the milling matrix based on graph theory principles. The transition rates of particles between sizes, as shown by the elements of eigenvectors, capture the intricate nature of particle breakage, including multiple size changes. This method marks a comprehensive attempt to devise a generalized analytical solution capable of handling all types of breakage kernels, regardless of their linearity. To confirm our method’s effectiveness, we used the particle size distribution (PSD) of small molecule crystals from wet-milling to predict the breakage kernels. Further, these breakage kernels were used to validate the PSD at different conditions. We also compared our graph-theoretic solution with a recognized analytical solution for a linear fracture kernel, observing a maximum error of around 3% for a 100-point discretization over extensive time scales. Our study also outlines a broader framework for potential future research. These generalized solutions offer crucial mechanistic insights into milling processes and facilitate the estimation of milling matrix or fracture kernels from experimental data. Importantly, our method is computationally efficient, requiring fewer than 100 lines of code, and can be automated for exploring breakage-related phenomena across various fields.