Augmented Quadrature Method of Moments for Solving Population Balance Equations of Industrial Crystallization Processes

Quadrature method of moments (QMOM) is a promising method to solve population balance models of industrial crystallization processes. The existing techniques for solving the differential algebraic equations (DAEs) arising in the QMOM formulation suffer from poor numerical robustness. In this paper, the fundamental formulation of QMOM is revisited by using automatic differentiation (AD) for computation to improve its performance. A closed-form solution to the algebraic equations resulting from the quadrature approximation of moments is derived by augmenting the original variable set with an extra set of variables. Based on this closed-form solution, two algorithms, namely, the augmented ODE and augmented AD-QMOM, are proposed. Numerical case studies show that both of the proposed approaches can effectively improve numerical robustness such that a much higher number of moments can be solved compared to the existing approaches. Moreover, the augmented AD-QMOM approach is shown to be computationally more efficient than the augmented ODE approach.