Solution of population balance equations using the direct quadrature method of moments

The implementation of a population balance equation (PBE) in computational fluid dynamics (CFD) represents a crucial element in the simulation of multiphase flows. Some of the available methods, such as classes methods (CM) and Monte Carlo (MC) methods, are computationally expensive and simulation of real cases of practical interest requires intractable CPU times. On the other hand, other methods such as the method of moments (MOM) are computationally affordable but have proven to be inaccurate for a number of cases. In recent work a new closure, the quadrature method of moments (QMOM), has been introduced, applied and validated. In our earlier work, QMOM was shown to be an efficient and accurate method for tracking the moments of the particle size distribution (PSD) in a CFD simulation. However, QMOM presents two main disadvantages: (i) if applied to multi-variate distributions it loses simplicity and efficiency, and (ii) by tracking only the moments of the PSD, it does not represent realistically polydisperse systems with strong coupling between the internal coordinates and phase velocities. In order to address these issues, in this work the direct quadrature method of moments (DQMOM) is formulated, validated, and tested. DQMOM is based on the idea of tracking directly the variables appearing in the quadrature approximation, rather than tracking the moments of the PSD. Nevertheless, for monovariate cases we show that QMOM and DQMOM yield identical results. In addition, we show how it is possible to extend the DQMOM to multivariate cases and some of relevant theoretical and numerical issues are discussed. These issues are discussed in the present work for homogeneous and one-dimensional flows. References to recent CFD applications of DQMOM to multiphase flows are provided as further proof of the utility of the method.