Bivariate extensions of the Extended Quadrature Method of Moments (EQMOM) to describe coupled droplet evaporation and heat-up

Abstract

In this study, four Eulerian moment methods based on the Extended Quadrature Method of Moments (EQMOM) (Yuan, Laurent, & Fox, 2012) are applied to model the evaporation of droplets. EQMOM enables the number density function (NDF) to be reconstructed from a set of moments of a droplet ensemble. Knowledge of the NDF allows us to describe the disappearance flux of the smallest droplets accurately, which is a critical issue in moment methods (Fox, Laurent, & Massot, 2008; Yuan et al., 2012). As EQMOM is restricted to univariate distributions, initially only the droplet size is considered as an internal coordinate. The description is then extended to the bivariate models mono-thermal Extended Conditional Quadrature Method of Moments (mt-ECQMOM), whose formulation is based on the semi-kinetic approach (Vié, Laurent, & Massot, 2013) and Extended Conditional Quadrature Method of Moments (ECQMOM, Marchisio & Fox, 2013). Based on ECQMOM and the general principle of a sectional method, a third bivariate approach is proposed, called the Extended Conditional Sectional Method of Moments (ECSQMOM). The resulting three bivariate approaches allow us to describe the coupling of heat-up and evaporation using the internal coordinates droplet diameter and temperature.The results based on EQMOM, mt-ECQMOM, ECQMOM and ECSQMOM are evaluated against a direct numerical solution of the population balance equation (PBE) investigating a test setup based on experimental studies for iso-octane sprays. Results indicate that in the limiting case of a mono-thermal distribution, the ECQMOM model implementation referenced above is not applicable to describe evaporating sprays accurately, whereas the mt-ECQMOM leads to much better results. ECSQMOM is also found to yield good results, even if the accuracy of mt-ECQMOM is not reached. However, since ECSQMOM offers the potential to capture distributions with nonzero conditional variance, it can be considered a suitable approach to describe the evolution of droplet distribution in sprays in future applications.