Numerical calculation of the moments of the population balance equation

The combined CFD–PBM (population balance models) are computationally intensive, so a possibility is to calculate only a few moments of the probability density function (PDF) of the PBM minimizing the computational costs. However, this formulation results in an excess of unknowns with respect to equations which is referred to as a closure problem. One approach for dealing with this closure problem is to apply a numerical quadrature approximation. On the other hand, a different possibility is to compute the PDF and from this, the moments of interest if required.In this work, the two mentioned approaches are discussed and numerical experiments are used to show the capability of the methods for predicting the moments of the PBE. In particular, the quadrature method of moments and a time–space least squares spectral method will be discussed.