Self-similar solution of a nonsteady problem of nonisothermal vapour condensation on a droplet growing in diffusion regime

TL;DR

This paper derives an exact self-similar solution for the nonsteady diffusion and heat transfer during vapour condensation on a droplet, establishing the droplet temperature and growth law based on initial vapour-gas medium parameters.

Contribution

It introduces a mathematically exact self-similar solution for coupled vapour diffusion and heat transfer during droplet growth, including an analytical solution for droplet temperature under strong heat effects.

Findings

Droplet temperature is constant and uniquely defined during growth.

The solution describes vapour and temperature profiles around the droplet.

The droplet growth follows a t^1/2 law after nucleation.

Abstract

This paper presents a mathematically exact self-similar solution to the joint nonsteady problems of vapour diffusion towards a droplet growing in a vapour-gas medium and of removal of heat released by a droplet into a vapour-gas medium during vapour condensation. An equation for the temperature of the droplet is obtained; and it is only at that temperature that the self-similar solution exists. This equation requires the constancy of the droplet temperature and even defines it unambiguously throughout the whole period of the droplet growth. In the case of strong display of heat effects, when the droplet growth rate decreases significantly, the equation for the temperature of the droplet is solved analytically. It is shown that the obtained temperature fully coincides with the one that settles in the droplet simultaneously with the settlement of its diffusion regime of growth. At the obtained temperature of the droplet the interrelated nonsteady vapour concentration and temperature profiles of the vapour-gas medium around the droplet are expressed in terms of initial (prior to the nucleation of the droplet) parameters of the vapour-gas medium. The same parameters are used to formulate the law in accordance with which the droplet is growing in diffusion regime, and also to define the time that passes after the nucleation of the droplet till the settlement of diffusion regime of droplet growth, when the squared radius of the droplet becomes proportionate to time. For the sake of completeness the case of weak display of heat effects is been studied.