An efficient numerical technique for solving one-dimensional batch crystallization models with size-dependent growth rates

Abstract

In this work, an efficient numerical method is introduced for solving one-dimensional batch crystallization models with size-dependent growth rates. The proposed method consist of two parts. In the first part, a coupled system of ordinary differential equations (ODEs) for the moments and the solute concentration is numerically solved to obtain their discrete values in the time domain of interest. These discrete values are also used to get growth and nucleation rates in the same time domain. To overcome the issue of closure, a Gaussian quadrature method based on orthogonal polynomials is employed for approximating integrals appearing in the ODE system. In the second part, the discrete growth and nucleation rates along with the initial crystal size distribution (CSD) are used to construct the final CSD. The expression for CSD is obtained by applying the method of characteristics and Duhamel's principle on the given population balance model (PBM). The proposed method is efficient, accurate, and easy to implement in the computer. Several numerical test problems of batch crystallization processes are considered. For a validation, the results of the proposed technique are compared with those obtained using a high resolution finite volume scheme.