A numerical algorithm for kinetic modelling of evaporation processes

Abstract

A numerical algorithm for kinetic modelling of droplet evaporation processes is suggested. This algorithm is focused on the direct numerical solution of the Boltzmann equations for two gas components: vapour and air. The physical and velocity spaces are discretised, and the Boltzmann equations are presented in discretised forms. The solution of these discretised equations is performed in two steps. Firstly, molecular displacements are calculated ignoring the effects of collisions. Secondly, the collisional relaxation is calculated under the assumption of spatial homogeneity. The conventional approach to calculating collisional integrals is replaced by the integration based on random cubature formulae. The distribution of molecular velocities after collisions is found based on the assumption that the total impulse and energy of colliding molecules are conserved. The directions of molecular impulses after the collisions are random, but the values of these impulses belong to an a priori chosen set. A new method of finding the matching condition for vapour mass fluxes at the outer boundary of the Knudsen layer of evaporating droplets and at the inner boundary of the hydrodynamic region is suggested. The numerical algorithm is applied to the analysis of three problems: the relaxation of an initially non-equilibrium distribution function towards the Maxwellian one, the analysis of the mixture of vapour and inert gas confined between two infinite plates and the evaporation of a diesel fuel droplet into a high pressure air. The solution of the second problem showed an agreement between the results predicted by the widely used Bird’s algorithm and the algorithm described in this paper. In the third problem the difference of masses and radii of vapour and air molecules is taken into account. The kinetic effects predicted by the numerical algorithm turned out to be noticeable if the contribution of air in the Knudsen layer is taken into account.