The spherical-droplet problem of evaporation and condensation in a vapour-gas mixture

Abstract

The behaviour of a binary mixture of a vapour and an inert gas around the spherical condensed-phase droplet is studied analytically using kinetic theory. By the singular-perturbation method, the linearized Boltzmann equation of B–G–K type is first solved for problems with spherical symmetry under the diffusive boundary condition when the Knudsen number of the problem is small. The macroscopic equations and the appropriate boundary conditions in the form of the temperature and partial-pressure jumps on the interface between the droplet and the gas phase, which enable us to treat the problems at the level of ordinary fluid dynamics, are derived together with the Knudsen-layer structure formed near the interface. Then the velocity, temperature and pressure fields around the droplet are explicitly obtained, as well as the mass, heat and energy flows from it. The results obtained are capable of describing the transition from the diffusion-control to the kinetic-control mechanism in the mass-transfer process. The negative-temperature-gradient phenomenon, a common phenomenon for pure-vapour cases (absence of inert gas), is also possible, manifesting itself more easily as the kinetic-control mechanism becomes prevalent and the critical condition for its existence is given. The present analysis could be applied to other problems with spherical symmetry as well.