Efficient numerical method for solving multidimensional population balance equations in batch cooling crystallization--颗粒学报PARTICUOLOGY

Li, C., Diao, Z., Cheng, Y., Shan, B., Xu, Q., Wang, X., & Zhang, F. (2026). Efficient numerical method for solving multidimensional population balance equations in batch cooling crystallization. Particuology, 110, 310-319. https://doi.org/10.1016/j.partic.2026.01.024

Efficient numerical method for solving multidimensional population balance equations in batch cooling crystallization

Chuan Li^a, Zimu Diao^a, Yongyan Cheng^b, Baoming Shan^a, Qilei Xu^a *, Xuezhong Wang^{c d}, Fangkun Zhang^a *

a College of Automation and Electronic Engineering, Qingdao University of Science & Technology, Qingdao, 266061, China
b Qilu Pharmaceutical Co., Ltd., Jinan, 250100, China
c School of Chemical and Process Engineering, University of Leeds, Leeds, LS2 9JT, United Kingdom
d Centre for Pharmaceutical and Crystallization Process Systems Engineering, School of Chemical Engineering, Bejing Institute of Petrochemical Technology, Beijing, 102617, China

10.1016/j.partic.2026.01.024

Volume 110, March 2026, Pages 310-319

Received 9 October 2025, Revised 11 January 2026, Accepted 17 January 2026, Available online 30 January 2026, Version of Record 6 February 2026.

E-mail: xuqilei@qust.edu.cn; f.k.zhang@qust.edu.cn

Highlights

• Proposed a numerical method for solving multidimensional PBEs with fourth-order accuracy in both space and time.

• Computational cost is significantly reduced while maintaining precision and stability.

• Four cases involving growth and nucleation verify the method's performance.

• It overcomes sharp CSD gradients and transient dynamics issues common in other methods.

Abstract

The efficient numerical solution of the multi-dimensional population balance models is still a hot topic and a challenging problem in crystallization. To address this issue, we develop a high-efficiency, high-accuracy scheme for two-dimensional PBEs, which integrates high-order compact difference discretization with an alternating-direction implicit strategy. The multidimensional solution problem was decomposed into two one-dimensional implicit systems to reduce computational complexity and suppress numerical dissipation. This proposed method can achieve fourth-order accuracy in space and time while circumventing stability constraints of explicit schemes by integrating compact fourth-order spatial discretization with alternating direction implicit time integration. Four cases were used to verify the performance by considering crystal size-independent growth, dependent growth, and nucleation in crystallization processes. The proposed method demonstrated superior accuracy and efficiency compared to the high-resolution finite volume method and high-order compact difference. In addition, stability and convergence analyses further confirm its robustness, particularly in capturing transient nucleation dynamics and steep gradients. This work is of great value and significance for modeling and optimal crystallization process control.

Graphical abstract

Keywords

Population balance equation; Crystallization; High-order compact difference; Numerical dissipation; Alternating direction implicit method

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