Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization

Abstract

The population balance equations (PBEs) serve as the primary governing equations for simulating the crystallization process. Two-dimensional (2D) PBEs pertain to crystals that exhibit anisotropic growth, which is characterized by changes in two internal coordinates. Because PBEs are the hyperbolic equations, it becomes imperative to establish a high-resolution scheme to reduce numerical diffusion and numerical dispersion, thereby ensuring accurate crystal size distribution. This paper uses Euler’s first-order explicit (EE) method–Peridynamic Differential Operator (PDDO) to solve 2D PBE, namely, the EE method for discretizing the time derivative and the PDDO for discretizing the internal crystal-size derivative. Five examples, including size-independent growth with smooth and non-smooth distributions, size-dependent growth, nucleation, and size-independent/dependent growth for batch crystallization are considered. The results show that the EE–PDDO method is more accurate than the HR method and that it is as good as the fifth-order Weighted Essential Non-Oscillatory (WENO) method in solving 2D PBE. This study extends the EE–PDDO method to the simulation of 2D PBE, and the advantages of the EE-PDDO method in dealing with discontinuous and sharp front problems are demonstrated.

Keywords

• Peridynamic Differential Operator
• population balance equation
• crystallization
• anisotropic