(206g) A Numerical Method for Population Balance Equation Based on the Moving Boundary Condition and Its Extensions to MP-PIC-PBE Model

The Population Balance Equation (PBE) is a mathematical model that accounts for fluid flow as well as physicochemical phenomena such as particle nucleation, growth, aggregation, and breakage to predict the particle size distribution in particulate systems. Conventional approaches to solving the PBE rely on discretizing the internal coordinate about the particle size, using fixed grids. However, the discretization-based approach introduces several numerical challenges. First, a fine grid is required to capture sharp changes in particle size distribution, which significantly increases computational cost. Conversely, using a coarse grid to reduce computation time leads to poor resolution and numerical diffusion. In addition, the fixed grid often fails to accurately capture the dynamic changes in particle size distribution that occurs fast birth and death of particles. Although flux limiter methods [1] have been proposed to address numerical diffusion, they may degrade solution resolution and cause excessive smoothing, especially near steep gradients or discontinuities. To overcome these limitations, a discretization method that can flexibly respond to the temporal changes in particle size distribution can alleviate unnecessary expansion of the computational domain and accuracy degradation caused by coarse grids, thereby addressing the shortcomings of conventional PBE solvers.

To this end, the present study proposes a novel numerical approach based on the moving boundary method, a representative numerical technique used to handle problems with time-varying boundary conditions. Originally, the moving boundary method tracks the boundary position using normalized domain and incorporates it into the governing equations to represent the domain as if a fixed region is moving [2,3]. Inspired by this idea, the proposed numerical scheme adjusts the boundaries of particle size distribution based on the particle information, aiming to maintain high accuracy while minimizing computational cost.

The novel method, termed Flexible Domain Population Balance Equation (FlexPBE), is built upon the Second-Order Upwind scheme and introduces a dynamic adjustment of the particle size domain. The domain boundaries are updated in real-time using Z-score-based formulation relative to the particle size distribution, enabling the solver to focus computational resources where they are most needed. FlexPBE has been validated through comparison with traditional Second-Order Upwind and HR-Vans schemes, demonstrating faster convergence to the analytical solution and a significant reduction in computational time. Furthermore, the proposed method has been extended to the MP-PIC-FlexPBE framework [4,5], which integrates the PBE into the multiphase CFD environment. Given the five-dimensional nature of MP-PIC-FlexPBE (3D space, time, and particle distribution), computational efficiency becomes critical. The incorporation of FlexPBE improves numerical stability while reducing computational overhead.

In conclusion, FlexPBE offers a practical and scalable alternative to fixed-grid PBE solvers, especially for time-evolving particulate systems such as crystallization. In addition, Its adaptability makes it well-suited for integration into high-dimensional CFD simulations, contributing to more accurate and efficient predictions, and supporting the design and optimization of complex multiphase processes.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant number RS-2024-00361307).

[1] SWEBY, P. K., et al., (1985), Flux limiters, Numerical Methods for the Euler Equations of Fluid Dynamics, 48-65

[2] Xiaolei Yang, Xing Zhang, Zhilin Li and Guo-Wei He, (2009), A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations, Journal of Computational Physics, 228, 7821-7836

[3] Chuan-Chieh Liao, Yu-Wei Chang, Chao-An Lin and J.M. McDonough, (2010), Simulating flows with moving rigid boundary using immersed-boundary method, Computers & Fluids, 39, 152-167

[4] Shin Hyuk Kim, Jay H. Lee and Richard D. Braatz, (2020), Multi-phase particle-in-cell coupled with population balance equation(MP-PIC-PBE) method for multiscale computational fluid dynamics simulation, Computers and Chemical Engineering, 134

[5] Shin Hyuk Kim, Jay H. Lee and Richard D. Braatz, (2021), Multi-scale fluid dynamics simulation based on MP-PIC-PBE method for PMMA suspension polymerization, Computers and Chemical Engineering, 152