A Robust Moment Method for Evaluation of the Disappearance Rate of Evaporating Sprays

Abstract

Abstract. 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 include 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. (PBE, also called Williams equation in the spray community). 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 coordinate, and its influence on moment dynamics results in a difficulty from both a modeling and a numerical point of view. We obtain in the present contribution 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 should lead to a low computational cost in multi-dimensional configurations.

A high order moment method simulating evaporation and advection of a polydisperse liquid spray

“…Let us remark that, if the last equation of (5) had been written on m 0 u, then the drag term would not have been integrable (the NDF n has generally a non zero value at S = 0 [37]). Concerning the evaporation term, the resolution of the corresponding system will be detailed in section 4 with an algorithm developed in [37].…”

“…Concerning the evaporation term, the resolution of the corresponding system will be detailed in section 4 with an algorithm developed in [37].…”

“…Let us notice that this does not involve any loss of generality since a dedicated scheme for diffusion can be further added in the context of an operator splitting algorithm. For the sake of legibility, the evaporation coefficient is assumed to be independent of the droplet size, referring to [37] for more complex evaporation laws.…”

“…An exact computation of the fluxes is obtained using the fact that the equations on moments can be derived from a kinetic equation as originally proposed 2 High order moment models and associated properties 4 in Bouchut [3]. By coupling this approach to a recently designed algorithm for evaporation which also preserves the moment space [37], polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the EMSM (Eulerian Multi Size Moment) model.…”

“…Consequently, further studies have been undertaken and have lead to a proper treatment of the evaporation term. The high order moment method together with the associated numerical scheme presented in [37] is a breakthrough in terms of accuracy, and can be effectively used for quantitative predictions of an evaporating spray. The second problem is related to transport in physical space.…”