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.
Highlights
IntroductionCrystallization operations are widely used in chemical and pharmaceutical industries
Crystallization operations are widely used in chemical and pharmaceutical industries.Crystallization kinetics is the basis for the development of crystallization theory
In order to weaken the hyperbolic property of population balance equation (PBE), a diffusive term is added
Summary
Crystallization operations are widely used in chemical and pharmaceutical industries. Crystallization kinetics is the basis for the development of crystallization theory. Studies on crystalline kinetics guide the design of crystallizers and the quality control of crystalline products [1]. The population balance equation (PBE) is a widely accepted equation to model the dynamic crystallization behavior [2]. It is of great significance to accurately solve the PBE and to achieve accurate crystal size distribution (CSD). There are a many numerical results on the study of PBE. In terms of numerical methods, they can be roughly divided into the following five categories [1,2]: (1) the method of moments [3];
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have