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An overview of the basic formulation and conceptual ideas needed for modeling polydisperse multiphase systems is provided. Special emphasis is given to systems exhibiting polydispersity in more than one internal coordinate. Such systems are described by a multivariate population balance equation, governing a number density function, which can be solved using sectional or moment methods. When the particle velocity is treated as a fluctuating quantity, the corresponding number density function is the one-point velocity density function used in kinetic theory. For this special case, a generalized population balance equation is employed to describe polydispersity in the velocity and other internal coordinates (such as the particle size.) Here, due to their flexibility in treating inhomogeneous flows, we focus on quadrature-based moment methods and show how moment transport equations can be derived from the generalized population balance equation for polydisperse multiphase flows. An example application to the one-dimensional spray equation is used to illustrate the modeling concepts.

An overview of different moment methods for the solution of the generalized population balance equation is presented and in particular the use of methods based on quadrature approximations is discussed. Firstly the quadrature method of moments is derived for a simple spatially homogeneous, single-phase system, and some mathematical issues related to the algorithm to derive the quadrature approximation and its degree of accuracy are discussed. Then the method is extended and implemented for the simulation of spatially heterogeneous multi-phase flows. Eventually the direct quadrature method of moments is derived and its use for the solution of mono-, bi- and multi-variate population balance equations is explained. Particular attention is also devoted to how to couple these methods with multi-phase models, such as the mixture model and the multi-fluid model.

In this contribution we propose a presentation of Eulerian multi-fluid models for polydisperse evaporating sprays. The purpose of such a model is to obtain a Eulerian-type description with three main criteria: to take into account accurately the polydispersity of the spray as well as size-conditioned dynamics and evaporation; to keep a rigorous link with the Williams spray equation at the kinetic, also called mesoscopic, level of description, where elementary phenomena such as coalescence can be described properly; to have an extension to take into account non-resolved but modeled fluctuating quantities in turbulent flows. We aim at presenting the fundamentals of the model, the associated precise set of related assumptions as well as its implication on the mathematical structure of solutions, robust numerical methods able to cope with the potential presence of singularities and finally a set of validations showing the efficiency and the limits of the model.

An overview of modeling and simulation of flow processes in gas/particle and gas/liquid systems are presented. Parcular emphasis is given to computation fluid dynamics (CFD) models that use the multi-dimensional multi-fluid techniques. Turbulence modeling strategies for gas/particle flows based on the kinetic theory for granular flows are given. Sub models for the interfacial transfer processes and chemical kinetics modeling are presented. An overview of a well established numerical solution method used is also given. Examples are shown for several gas/particle systems including flow and chemical reaction in risers as well as gas/liquid systems including bubble columsns and stirred tanks.

Cellular automata for mimicking physical systems, the lattice gas and lattice Boltzmann automata, fluid dynamics with the lattice-Boltzmann method, practicalities of the lattice-Boltzmann method, DNS of solid-liquid suspensions, scaling in single-phase turbulence, direct numerical and large-eddy simulations, subgrid-scale modeling in LES, solid-liquid flow: point particles in LES, passive and reactive scalar transport in turbulent flow, filtered density function approach to turbulent reactive flow.

Numerical procedures to describe the dispersion, evaporation and combustion of a polydisperse liquid fuel in a turbulent oxidizer are presented. Direct Numerical Simulation (DNS) allows one to describe accurately the evolution of the fully compressible gas-phase coupled with a Lagrangian description in order to describe two-phase flows. Standard coupling is used for the Eulerian/Lagrangian system while some practical issues related to the reactive source terms are addressed by suggesting a fast single-step Arrhenius law allowing one to capture the main fundamental properties of the flame whatever the local equivalence ratio. Then some basic procedures to describe spray preferential segregation in a turbulent reactor are described. Eventually spray combustion is addressed by first demonstrating the complex interactions caused by the presence of an evaporating liquid phase: definition of various equivalence ratios, apparition of flame instabilities for a unit Lewis number, etc. Then a history of the development of the existing spray combustion diagrams is presented to display the possible flame structures and combustion regimes encountered in spray combustion.