Beyond traditional time discretization: An efficient methodology for multidimensional weighted finite volume fragmentation equations

Multidimensional population balance equations (MPBEs) model the dynamics of populations such as particles, cells, or droplets, considering properties like size, shape, liquid-to-solid ratio, and composition. Coupling MPBEs with computational fluid dynamics (CFD) enables the study of systems from micro to macro scales. However, the high dimensionality of MPBEs poses significant computational challenges. This paper presents a novel semi-analytical approach to address these challenges by approximating multidimensional weighted finite volume breakage PBEs [Saha et al. “Numerical solutions for multidimensional fragmentation problems using finite volume methods,” Kinet. Relat. Mod. 12(1), 79-103 (2019)]. The existing weighted finite volume scheme (WFVS) relies on the fourth-order Runge-Kutta method for solving discretized ordinary differential equations, where accuracy depends on grid type and the number of cells. The proposed approach eliminates the need for a time discretization, offering a more efficient alternative to traditional schemes. The time-free discretization nature of this method ensures seamless integration with CFD tools. To assess its effectiveness for two-dimensional (2D) and three-dimensional (3D) PBEs, nine combinations of selection functions, breakage kernels, and initial conditions are considered. As analytical solutions for the number density functions are unavailable, validation is performed using zeroth- and first-order integral moments. Results show that the new approach maintains the accuracy of existing methods while significantly improving computational efficiency—reducing runtime by approximately 60% for 2D PBEs and 80% for 3D PBEs. This efficiency makes the proposed approach a promising option for CFD applications.