Polydisperse multiphase flows arise in many applications, and thus there has been considerable interest in the development of numerical methods to find solutions to the kinetic equations used to model such flows. However, the direct numerical solution of the kinetic equations is intractable for most applications due to the large number of independent variables. A useful alternative is to reformulate the problem in terms of the moments of the number density function (NDF), yet the resulting moment transport equations are not closed for flows away from the equilibrium limit. To attain closure, Quadrature-based moment methods (QBMM) is proposed. QBMM reconstruct NDF from a set of moments, which is the key step and named moment-inversion algorithm, then use NDF to close the moment transport equations.

By different function approximation, two types of moment-inversion algorithm can be determined. The first type is approximate NDF by Dirac delta function. Quadrature method of moments (QMOM) has been proposed to handle three-dimensional problem for this type of approximation. However, the positivity of NDF cannot be guaranteed by QMOM. Therefore, a novel moment-inversion algorithm, based on 1-D adaptive quadrature of conditional velocity moments, is introduced and shown to yield NDF which is always promise positivity. This conditional quadrature method of moments (CQMOM) can be used to compute exact N-point quadratures for multi-valued solutions, and provides optimal approximations of continuous distributions. In order to control numerical errors arising in volume averaging and spatial transport, an adaptive 1-D quadrature algorithm is formulated for use with CQMOM. The use of adaptive CQMOM in the context of QBMM for the solution of kinetic equations is illustrated by applying it to problems involving particle trajectory crossing, Riemann problem, and granular flow.

The drawback of Dirac delta function approximation has two fold, one is when large numbers of nodes are required to achieve the desired accuracy, the moment-inversion problem can become ill-conditioned. Another is value of NDF cannot be provided in QMOM or CQMOM when it is necessary in some applications. To conquer these disadvantages, a new generation of quadrature algorithm is introduced that uses an explicit form for the distribution function. This extended quadrature method of moments (EQMOM) approximates the distribution function by a sum of classical weight functions, which allow unclosed source terms to be computed with great accuracy by increasing the number of quadrature nodes independent of the number of transported moments. EQMOM is used to solve a population balance equations with evaporation, aggregation and breakage terms and compare the results with analytical solutions.

This novel quadrature methods EQMOM is then applied to simulate bubbly flow. Bubble-column reactors are widely used in the biological, chemical and petrochemical industries. The accurate design of these reactors depends largely on the complex fluid dynamics of gas-liquid two-phase flows that still remains inadequately understood. Modeling of the fluid dynamics of gas-liquid bubble columns is therefore a challenging task. The Euler-Euler method is widely used in industry to simulate bubble columns. However, accurately predicting polydisperse bubbly flow is a nontrivial task due to the complexity of the bubble number density function, which can involve up to four internal coordinates including size and velocity. To describe polydisperse bubbles, a joint velocity-mass NDF for bubbles is adopted. QBMM is applied to solve the kinetic equation of the joint velocity-mass NDF using EQMOM. It is coupled with an incompressible Navier-Stokes solver for the liquid phase. In this model, transport equations for the joint velocity-mass moments are derived from a kinetic equation for the NDF and closure is attained using a monokinetic NDF, which is valid in the limit of small bubble Stokes number. The pure moments of mass are used to reconstruct mass NDF with EQMOM, while the joint moments determine the conditional velocity. Forces including buoyancy, drag, virtual-mass and lift are accounted for. The injection with a narrow bubble size distribution cases are used to validate the model with experimental data from the literature. Other cases with a continuous size distribution injection show the ability of new model to handle polydisperse bubbles. Results demonstrate that the onset of segregation is sensitive to the bubble size distribution and, thus, an accurate solution for the size-dependent fluxes is required when simulating polydisperse bubbly flows.