A robust moment method for evaluation of the disappearance rate of evaporating sprays

In this paper we tackle a critical issue in the numerical modeling, by Eulerian moment methods, of polydisperse multiphase systems, constituted of dispersed particles or droplets, a general class of systems which includes aerosols. Their modeling starts at a mesoscopic scale with an equation on the number density function (NDF) of particles/droplets which satisfies a population balance equation. In order to limit the computational cost, moment methods provide a system of conservation equation with an eventual closure problem, which can be solved using quadrature methods in order to retrieve the unclosed terms from the considered set of moments. However, a drift velocity, that is, the rate of change due to continuous phenomena of the internal coordinate, such as the size of the particles, has sometimes to be taken into account; it can be either positive like molecular growth, or negative such as for evaporation of droplets in aerosols or oxidation of soots. When negative, it leads to the disappearance of droplets/particles, thus creating a negative flux at zero size. Its closure requires an evaluation of the reconstructed NDF at zero size from the knowledge of a given finite set of moments. The nature of this information, pointwise in internal coordinates, and its influence on moment dynamics results is a difficulty from both a modeling and a numerical point of view. We obtain a comprehensive solution to this important issue. Since we introduce some new tools in order to resolve the flux evaluation, we also introduce a new Eulerian type of description, which will combine both the flexibility of Eulerian models for which the size phase space is discretized into "sections" (i.e., size intervals) and the efficiency of direct quadrature method of moments (DQMOM). It yields a precise and stable description of moment dynamics with a minimal number of variables, which will lead to a low computational cost in multidimensional configurations.