Numerical Methods for the Solution of Population Balance Equations Coupled with Computational Fluid Dynamics

This review article discusses the solution of population balance equations, for the simulation of disperse multiphase systems, tightly coupled with computational fluid dynamics.Although several methods are discussed,the focus is on quadrature-based moment methods (QBMMs) with particular attentiontothequadraturemethodofmoments,theconditionalquadrature method of moments, and the direct quadrature method of moments. The relationship between the population balance equation, in its generalized form, and the Euler-Euler multiphase flow models, notably the two-fluid model, is thoroughly discussed. Then the closure problem and the use of Gaussian quadratures to overcome it is analyzed. The review concludes with the presentation of numerical issues and guidelines for users of these modeling approaches.