1.2 The Standard Method of Moments

The standard method of moments (SMM) is an efficient alternative to the discrete population balance approach. In this approach, the population balance equation is transformed into a set of transport equations for moments of the distribution. The $i$th moment is defined by integrating the number density throughout the particle space weighted with the particle property raised to its $i$th power. It is generally sufficient to solve only a few moment equations, typically three to six. This may provide a significant reduction in the number of equations to be solved compared with the discretized approach. Apart from the computational advantage, the SMM approach is useful when the entire distribution is not needed and certain average and total quantities are sufficient to represent the particle distribution. Typically, the zeroth moment represents the total number density, the second moment represents the total surface area per unit volume, and the third moment represents the total mass density.

In the SMM approach, no assumptions are made about the size distribution, and the moment equations are formulated in a closed form involving only functions of the moments themselves. However, this exact closure requirement poses a serious limitation, as aggregation (with the exception of the constant aggregation kernel) and breakage phenomena cannot be written as functions of moments.