Quadrature-based moment methods for the population balance equation: An algorithm review

The dispersed phase in multiphase flows can be modeled by the population balance model(PBM). A typical population balance equation(PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods(QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function(NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments(QMOM), direct quadrature method of moments(DQMOM),extended quadrature method of moments(EQMOM), conditional quadrature method of moments(CQMOM),extended conditional quadrature method of moments(ECQMOM) and hyperbolic quadrature method of moments(Hy QMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics(CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.

A Quadrature-Based Kinetic Model for Dilute Non-Isothermal Granular Flows

A moment method with closures based on Gaussian quadrature formulas is proposed to solve the Boltzmann kinetic equation with a hard-sphere collision kernel for mono-dispersed particles.Different orders of accuracy in terms of the moments of the velocity distribution function are considered,accounting for moments up to seventh order.Quadrature-based closures for four different models for inelastic collisionthe Bhatnagar-Gross-Krook,ES-BGK,the Maxwell model for hard-sphere collisions,and the full Boltzmann hard-sphere collision integral-are derived and compared.The approach is validated studying a dilute non-isothermal granular flow of inelastic particles between two stationary Maxwellian walls.Results obtained from the kinetic models are compared with the predictions of molecular dynamics(MD)simulations of a nearly equivalent system with finite-size particles.The influence of the number of quadrature nodes used to approximate the velocity distribution function on the accuracy of the predictions is assessed.Results for constitutive quantities such as the stress tensor and the heat flux are provided,and show the capability of the quadrature-based approach to predict them in agreement with the MD simulations under dilute conditions.