(627b) Sectional Method of Moments: An Efficient Method to Solve Multidimensional Population Balance Equation Models

Population Balance Equations (PBEs), formulated as partial differential equations (PDEs), are widely used to model particulate systems in fields such as crystallisation, aerosols, and granulation. In crystallisation, PBEs describe the evolution of particle size and shape distributions (PSSDs), which are critical in dictating the product quality and processing efficiency. However, efficient solution of PBEs remains challenging, particularly in systems involving multiple mechanisms (e.g., nucleation, size-dependent growth) and anisotropic particles (e.g., needle-like crystals), where multidimensional modelling (size and shape) is required.[1]

In this work, we developed a method and computational tool for efficient PBE simulation. The method, which we call the Sectional Method of Moments (SecMoM), analogous to the standard Method of Moments, reformulates the PBE as a system of ordinary differential equations (ODEs). [2] In this method, rather than characterising the entire system using moments, average properties of the PSSD as done in standard MoM, we preserve the underlying distribution by segmenting it into sections. By dividing the distribution into sections, and solving the moment equations in each, while coupling them with the mass balance, we reconstruct the distribution at each time step. Our version of SecMoM provides a unique twist in terms of transforming the PSSD into moment nodes and then back into the PSSD, allowing for a robust solution that is intuitive to understand. It accommodates multidimensional models and a variety of mechanistic pathways (e.g., nucleation, growth, dissolution), making it suitable for processes where particle shape evolution is to be characterised. Although developed with crystallisation as the primary application, it is potentially applicable to other systems.

The accompanying tool implements SecMoM with GPU acceleration using JAX [3] (library developed by Google for Scientific Machine Learning, which natively leverages GPU acceleration), enabling substantial reductions in computation time without sacrificing accuracy. We compare the simulation speed with the traditional Finite Volume Method, GPU and CPU computing times, and accuracy against analytical baseline solutions. This tool provides a flexible and practical platform for modelling complex particulate systems, supporting both research and industrial use. By combining a generalisable and interpretable PBE solution strategy with modern computational techniques, this work enables rapid, high-fidelity simulation of crystallisation and other distribution-driven processes. [4]

(1) Ramkrishna, D., Population balances: theory and applications to particulate systems in engineering; Academic Press: San Diego, CA, 2000.
(2) Sun, F.; Liu, T.; Nagy, Z. K.; Ni, X. Chemical Engineering Science 2022, 254,
117625.
(3) Bradbury, J.; Frostig, R.; Hawkins, P.; Johnson, M. J.; Leary, C.; Maclaurin, D.; Necula, G.; Paszke, A.; VanderPlas, J.; Wanderman-Milne, S.; Zhang, Q. JAX: composable transformations of Python+NumPy programs, version 0.3.13,
2018.
(4) Szilagyi, B.; Nagy, Z. K. In Computer Aided Chemical Engineering, Gernaey, K. V., Huusom, J. K., Gani, R., Eds.; Elsevier: 2015; Vol. 37, pp 947–952.