Solution of Multidimensional Population Balance Equation Models via the Sectional Method of Moments

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Population balance equations (PBEs) provide a rigorous framework for modelling particulate processes, but their numerical solution becomes increasingly demanding for multidimensional systems. This work introduces the Sectional Method of Moments (SecMoM). Through an extension of the widely used Method of Moments (MoM), the population distribution is divided into moving sections and local moments are calculated within each section, providing distribution-level information with moderate additional computational overhead. By tracking only the populated region of the internalcoordinate domain, SecMoM avoids requiring a fixed global grid. The formulation is extended to multiple internal coordinates, property-dependent transport, nucleation, and complete dissolution. SecMoM is demonstrated for one-and two-dimensional crystallisation PBEs involving size-dependent growth, nucleation, dissolution, and temperature cycling. Comparisons with finite volume method (FVM) solutions show excellent agreement in liquid-phase concentration, total particle number, average particle dimensions, and population-distribution evolution, with mean relative absolute deviations of 1.26×10 −3 or lower. The method captures distribution broadening, the formation and removal of fine particles, and anisotropic growth in two dimensions. For the 2D crystallisation case, the corresponding FVM data structure contained approximately 14 times more stored values than the SecMoM representation. For the simplified benchmark, both codes were implemented in JAX, enabling execution on CPU and GPU hardware. SecMoM was approximately 1.4 times faster on the CPU and 2.5 times faster on the GPU than the optimised FVM implementation. SecMoM therefore combines distribution-level resolution with efficient computation and compact data storage, making it well suited to multidimensional PBEs and repeated-simulation applications.

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