Transient heating of an evaporating droplet

Two new solutions to the heat conduction equation, describing transient heating of an evaporating droplet, are suggested. The first solution is the explicit analytical solution to this equation, while the second one reduces the solution of the differential transient heat conduction equation to the solution of the Volterra integral equation of the second kind. Both solutions take into account the effect of the reduction of the droplet radius due to evaporation, assuming that this radius is a linear function of time. This approach can be considered as the generalisation of the approach currently used in all research and commercial CFD codes known to us (e.g. KIVA, FLUENT, PHOENICS), in which it is assumed that droplet radius is constant during the timestep. The new analytical solution has been incorporated into a zero-dimensional CFD code and applied to the analysis of Diesel fuel droplet heating and evaporation in typical engine conditions. The results have been compared with those which follow from the conventional (traditional) approach to modelling droplet heating and evaporation, based on the assumption that the droplet radius is constant over the timestep (but changes from one step to another). It has been pointed out that the new approach predicts lower droplet surface temperatures and slower evaporation rates compared with the traditional approach. The relative difference between evaporation times is shown to be practically independent of the values of the initial droplet radii and to increase with increasing gas temperatures. Larger timesteps can be used in the case of the new approach compared with the conventional one to achieve the same accuracy of calculation. It is recommended that the effect of a moving boundary on droplet heating is taken into account in modelling droplet heating and evaporation in CFD codes.