Analytical solution to the discretized population balance equation for pure breakage with application to kernel identification

Particle breakage is a key process in industries such as pharmaceuticals, mining, and materials processing, where controlling particle size distribution is critical for optimizing product properties. The evolution of particle size during breakage is often described by the population balance equation (PBE) with mechanistic and empirical models for breakage. While numerical methods are commonly used to solve PBE, they are computationally intensive and prone to instabilities, and existing analytical solutions are typically limited to specific kernel forms. In this work, we present an analytical solution for discretized PBEs applicable to both linear and nonlinear breakage kernels. Our method utilizes Sylvester’s expansion to solve the PBE and compute eigenvectors of the milling matrix, enabling generalized, efficient, and accurate predictions of particle size distributions. Verification against Ziff and McGrady’s analytical solution for specific kernels demonstrated excellent agreement, while experimental data from IKA Magic Lab® and Quadro Ytron® wet milling processes confirmed the model’s kernel identification capabilities, robustness, and versatility across different operating conditions. By incorporating Austin et al.'s breakage kernels, the model effectively captures the influence of milling parameters on particle fragmentation. This analytical solution offers a robust framework for understanding and optimizing breakage processes across diverse systems. Its adaptability to various kernels and minimal computational complexity makes it a valuable tool for both researchers and process development engineers. The insights gained through this work can guide the design of optimal particle processing technologies, with potential applications extending beyond milling to include grinding and other comminution systems. • Analytical solution to the population balance equations (PBEs) applicable to various breakage kernels is developed. • The eigenvalues and eigenvectors of the milling matrix are expressed using a novel graph theoretic approach. • Right eigenvector elements reflect net jump rates towards smaller sizes. • Left eigenvector elements reflect net jump rates towards larger sizes, incorporating microscopic reversibility. • This solution offers mechanistic insights into milling and helps estimate breakage kernels from experimental data.