COMBINED METHOD OF INTEGRAL TRANSFORMS FOR THE SPHERICALLY SYMMETRIC DROPLET HEATING PROBLEM

Abstract

In this paper, we analysed the spherically symmetric heat diffusion equation, which governs the temperature distribution inside a heated but non-evaporating droplet. The spherical droplet, with an initial uniform temperature, is assumed at rest in an unsteady gas environment. The classical Fourier sine integral transform (FSIT) and the unilateral Laplace integral transform (LIT) are successively used to solve the resulting initial-boundary value problem, first reduced in a dimensionless form. An explicit solution in the Laplace domain is obtained for the temperature inside the droplet. Then, depending on the time-varying temperature of the gas environment at the immediate vicinity of the droplet, an exact series solution and an approximate analytical solution in short time limits are derived for the droplet internal temperature. In the case of steady gas environment at constant temperature, the standard series solution obtained in the literature for the symmetrical problem of heating or cooling of a solid spherical body, is recovered. The results may be useful for time step analysis in droplets and sprays vaporization models.