(653f) Numerical Algorithms for Solving Population Balance Equations Using Quadrature Based Moment Methods

Population Balance Equations (PBEs) are an appropriate framework for simulating mesoscopic processes. In recent years, there has been great interest in coupling population balance submodels to macroscopic engineering simulations. Large scale simulations of PBEs to describe industrial systems require efficient numerical methods. A further requirement is that for emerging applications, multivariate population balance equations need to be solved. These equations describe the evolution of a pdf in a high dimensional space and traditionally Monte Carlo methods have been used for simulation. Monte Carlo simulations of very large scale systems can be extremely costly. Moment methods are a means of simplification in which only the moments of the pdf are evolved. A drawback of moment methods is that equations for the moments are generally unclosed and closure schemes are necessary. In this talk, we will investigate numerical moment closure schemes based on Gaussian quadrature. We will discuss the theory and algorithm behind the univariate Quadrature Method of Moments (QMOM). This technique has been shown to provide an extremely accurate moment closure scheme for a large number of applications. Fundamental problems encountered in the extension of this algorithm to the case of multivariate moments will be discussed. A closely related method for solving multivariate PBEs is the Direct Quadrature Method of Moments (DQMOM). DQMOM suffers from problems due to ill-conditioning of matrices and the lack of clear guidelines on the selection of moment sets. We will present an algorithm that can be used in DQMOM to order and select a set of moments that lead to non-singular matrices. Applications of the algorithms to problems in aerosol science and turbulent mixing with chemical reaction will be presented to illustrate issues in moment closure and optimal choice of moment sets.