Chebyshev Spectral Collocation Method for Population Balance Equation in Crystallization

Abstract

The population balance equation (PBE) is the main governing equation for modeling dynamic crystallization behavior. In the view of mathematics, PBE is a convection–reaction equation whose strong hyperbolic property may challenge numerical methods. In order to weaken the hyperbolic property of PBE, a diffusive term was added in this work. Here, the Chebyshev spectral collocation method was introduced to solve the PBE and to achieve accurate crystal size distribution (CSD). Three numerical examples are presented, namely size-independent growth, size-dependent growth in a batch process, and with nucleation, and size-dependent growth in a continuous process. Through comparing the results with the numerical results obtained via the second-order upwind method and the HR-van method, the high accuracy of Chebyshev spectral collocation method was proven. Moreover, the diffusive term is also discussed in three numerical examples. The results show that, in the case of size-independent growth (PBE is a convection equation), the diffusive term should be added, and the coefficient of the diffusive term is recommended as 2G × 10−3 to G × 10−2, where G is the crystal growth rate.

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

“…Correspondingly, the PBE becomes a convection-reaction equation, which can be solved through a regular numerical scheme. In this example, in the implementation of the EE-PDDO method, the upwind weight function shown in Equation (15) and the corresponding normal weight function with ω(ξ 1 , ξ 2 ) = e The comparison of the final crystal size distributions between the exact solution and the numerical solutions obtained using different schemes are shown in Figure 5. In Figure 5d, normal weight function is used in the EE-PDDO method.…”

“…Figure 8 shows the comparison of crystal size distribution between the exact solutions and the numerical solutions obtained using different schemes at t = 160 s. The "exact" solution is obtained through the fifth-order WENO method on a fine spatial grid (∆r 1 = ∆r 2 = 0.1), and the other results are obtained with a spatial grid size of ∆r 1 = ∆r 2 = 1. As illustrated in Figure 8d, the result of the EE-PDDO method is obtained by using the upwind weight function in Equation (15). Here, we choose the polynomial degree N = 2 and the integer parameter m = 2.…”

“…. The "exact" solution is obtained through the fifth-order WENO method on a fine spatial grid ( ), and the other results are obtained with a spatial grid size of As illustrated in Figure 8d, the result of the EE-PDDO method is obtained by using the upwind weight function in Equation (15). Here, we choose the polynomial degree N = 2 and the integer parameter m = 2.…”

“…They found that the model becomes more complex and the solution exhibits strong discontinuity when PBE is coupled with computational fluid dynamics. Ruan [15] introduced the Chebyshev spectral collocation method to solve PBE. They suggested that the diffusive term should be added in cases of size-independent growth in which the PBE is a convection equation.…”

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

“…The discrete method directly solves the PBE with the size-distribution function. This method discretizes the internal coordinates and obtains the size-distribution function directly by applying one of the following methods: the finite volume method [14,15], finite element method [16,17], collocation method [18,19], or lattice Boltzmann method [20]. This study focuses on the discrete method for solving the PBE.…”