Direct quadrature spanning tree method for solution of the population balance equations

A new method for solution of the multivariate population balance equations (PBEs) is presented in this work. The method uses M quadrature abscissas and weights to close the PBE. Unlike other similar methods, e.g., the direct quadrature method of moments (DQMoM) and the quadrature method of moments (QMoM), the proposed method neither inverts a badly scaled matrix nor solve a time-consuming eigenvalue problem. The method does not use any a priori discretization of the phase space and as the particlesʹ size distribution (PSD) widens and shifts toward large characteristic sizes due to coagulation, the abscissas adaptively follow the PSD. The particles are linked by a spanning tree; the tree works as a pipeline redistributing the mass across the system and ensuring that each computational particle accounts for a prescribed fraction of the total mass and therefore the method is given the name: direct quadrature spanning tree method (DQST).