Improved High-Resolution Algorithm for Solving Population Balance Equations

Population balance equations (PBEs) in conjunction with appropriate kinetic expressions have been extensively employed as modeling tools to elucidate the dynamic behavior of particulate processes such as precipitation, crystallization, aerosolization, microbial growth, and cell population growth. It is still challenging to ensure the stability and accuracy of solutions since the PBE for a crystallization system represents a hyperbolic integral-partial differential equation that describes the temporal and spatial evolution of the crystal shape and size distribution. An improved high-resolution (HR) algorithm along with new flux limiters to suppress spurious oscillations and improve accuracy was proposed based on batch crystallization in this paper. Through systematic research and analysis of the influence of the limiter function on the HR algorithm, a new limiter function was designed to improve solution accuracy and stability. Simulation cases considering crystal size-dependent and -independent growth with and without nucleation were used to test the effectiveness of the presented algorithm. Results demonstrated that the new flux limiter presents marked superiority compared with the classical Van Leer flux limiter, and the limiter functions should be reasonably selected under different Courant-Friedrichs-Lewy conditions to improve the accuracy of the HR algorithm. In addition, an ameliorated gradient ratio was further designed to solve the problem of decreased accuracy at discontinuities.