Transient heating of an evaporating droplet with presumed time evolution of its radius

New solutions to the heat conduction equation, describing transient heating of an evaporating droplet, are suggested, assuming that the time evolution of droplet radius Rd(t) is known. The initial droplet temperature is assumed to be constant or allowed to change with the distance from the droplet centre. Since Rd(t) depends on the time evolution of the droplet temperature, an iterative process is required. Firstly, the time evolution of Rd(t) is obtained using the conventional approach, when it remains constant during the timestep, but changes from one timestep to another. Then these values of Rd(t) are used in the new solutions to obtain updated values of the time evolution of the distribution of temperatures inside the droplet and on its surface. These new values of droplet temperature are used to update the function Rd(t). This process continues until convergence is achieved, which typically takes place after about 15 iterations. The results of these calculations are compared with the results obtained using the previously suggested approach when the droplet radius was assumed to be a linear function of time during individual timesteps for typical Diesel engine-like conditions. For sufficiently small timesteps the time evolutions of droplet temperatures and radii predicted by both approaches coincided. This suggests that both approaches are correct and valid. Similarly to the case when droplet radius is assumed to be a linear function of time during the timestep, the new solutions predict lower droplet temperature and slower evaporation when the effects of the reduction of Rd are taken into account. It is shown that in the case of constant droplet initial temperature, models both taking and not taking into account the changes in the initial droplet temperature with the distance from the droplet centre predict the same results. This indicates that both models are correct.