Numerical solutions of population balance models in dispersed systems

Abstract

This work focuses on the modeling and numerical approximation of population balance models (PBMs) for simulating dispersed systems, especially the batch crystallization pro- cess. Apart from applying the existing numerical schemes, new numerical techniques are introduced for solving these models efficiently and accurately. The effects of nucleation, growth, aggregation, breakage, and fines dissolution phenomena on the crystal size distri- bution (CSD) are investigated. An alternative quadrature method of moments (QMOM) is introduced for solving the single-variate length-based PBM incorporating simultaneous nucleation, growth, aggregation and breakage phenomena. In the proposed QMOM, or- thogonal polynomials, formed by lower order moments, are used to find the quadrature points and weights. To ensure better accuracy of the scheme, a third order orthogonal polynomial, utilizing the first six moments, is selected to calculate the quadrature points (abscissas) and corresponding quadrature weights. Therefore, at least a six moment sys- tem is needed to solve. This choice of polynomial gives a three-point Gaussian quadrature rule which generally yields exact results for polynomials of degree five or less. A mathe- matical model is derived for simulating batch crystallization process incorporating crystals nucleation, size-dependent growth and dissolution of small nuclei below certain critical size in a recycling pipe. Moreover, a time delay in the dissolution unit is also incorporated in the model. The dissolution of small crystals (fines dissolution) is helpful to further improve the product CSD. It withdraws and dissolve excessive fines from the quiescent zone of crystallizer which are generated during periods of high supersaturation. This ef- fectively shifts the CSD towards right and often makes the distribution narrow. A new numerical scheme is introduced for simulating this model. The method of characteristics, the Duhamel’s principle, and the QMOM are employed together to devise the proposed numerical scheme. Several test problems are considered and the numerical results are val- idated against available analytical solutions and the finite volume scheme (FVS). It was found that the suggested numerical methods have capability to solve the given models efficiently and accurately.