A quadrature-based moment method for the evolution of the joint size-velocity number density function of a particle population

Abstract

A quadrature-based moment method for the approximate solution of the generalized population balance equation (GPBE) governing the evolution of the joint size-velocity number density function (NDF) of a particle population is formulated and tested. The proposed method relies on the third-order hyperbolic conditional quadrature method of moments developed for velocity distribution transport. This approach is combined with the conditional quadrature method of moments to incorporate the dependency of the NDF on particle size, leading to an efficient, stable, quadrature method that uses an analytical solution to determine the size-conditioned velocity moments. The incorporation of source terms accounting for aggregation, breakup, and collisions, as well as acceleration terms such as gravity and drag, is performed using a realizable ODE solver. The approach is then demonstrated by considering zero-dimensional cases to verify the correct integration of the source terms. A set of one-dimensional cases involving droplet evaporation and coalescence is used to validate the velocity-dependent source terms. A two-dimensional case of crossing jets of particles with different sizes is used to demonstrate the proposed method in the case of a polydisperse flow with inertial particles. All work has been implemented in the open-source framework OpenQBMM, based on OpenFOAM®.