Integral formulation of the population balance equation: Application to particulate systems with particle growth

Abstract

Numerical solution of the population balance equation (PBE) is widely used in many scientific and engineering applications. Available numerical methods, which are based on tracking population moments instead of the distribution, depend on quadrature methods that destroy the distribution itself. The reconstruction of the distribution from these moments is a well-known ill-posed problem and still unresolved question. The present integral formulation of the PBE comes to resolve this problem. As a closure rule, a Cumulative QMOM (CQMOM) is derived in terms of the monotone increasing cumulative moments of the number density function, which allows a complete distribution reconstruction. Numerical analysis of the method show two unique properties: first, the method can be considered as a mesh-free method. Second, the accuracy of the targeted low-order cumulative moments depends only on order of the CQMOM, but not on the discrete grid points used to sample the cumulative moments.