A new solution to a weakly non-linear heat conduction equation in a spherical droplet: Basic idea and applications

Abstract

A new analytical solution to a non-linear heat transfer equation in a spherically-symmetric droplet is suggested. All thermophysical properties inside the droplet are considered to be close to their average values. This allows us to consider the non-linearity of this equation as weak. The solution is presented as T=T0+T1, where T0 is the solution to a linear heat conduction equation, and T1≪T0. The equation for T1 is presented as a linear heat conduction equation with a source term depending on the distribution of T0 and its spatial derivatives inside the droplet. The latter equation is solved analytically alongside the linear equation for T0, and the final solution is presented as T=T0+T1. The predictions of the numerical code in which this solution was implemented are verified based on a comparison of those predictions with the predictions of COMSOL Multiphysics code using input parameter values that are typical for nanofluid (water and SiO2 nanoparticles) droplet evaporation in atmospheric conditions. It is demonstrated that for these experiments T1≪T0 which justifies the applicability of the linear heat conduction equation used for the analysis of this process. Small differences in the temperatures predicted by both non-linear and linear models lead to a much more noticeable difference in integral characteristics such as time before the start of the formation of the cenosphere when the mass fraction of nanoparticles at the droplet surface reaches about 40%.