Mathematical modelling of heating and evaporation of a spheroidal droplet

Abstract

Most of the currently used models for droplet heating and evaporation are based on the assumption that droplets are perfect spheres. At the same time the shapes of many observed droplets in engineering applications are far from spherical. We have studied the influence of droplet non-sphericity on their heating and evaporation, approximating droplet shapes as prolate and oblate spheroids. The previously developed exact solutions to the heat and mass transfer equations for the gas phase surrounding a spheroidal droplet have been used as boundary conditions for the solutions to these equations in the liquid phase. The temperature gradients inside and at the surface of the droplets, and the changes in their shape during the heating and evaporation process have been taken into account. The effects of surface tension and droplet motion on droplet heating and evaporation are ignored. The results are applied to the analysis of heating and evaporation of an n-dodecane fuel droplet in Diesel engine-like conditions. The effect of droplet nonsphericity is shown to be relatively small for the evaporation time of these droplets with initial eccentricities 2/3 ≤ ǫ ≤ 1.5.

Modelling of spheroidal drop evaporation with non-uniform temperature conditions

“…The case ∆T =0 is the one with uniform surface temperature and for this case the function Nu(η) can be found analytically (see [14]), showing that the surface curvature itself influences this parameter. The results for ∆T ≠0 show that the surface heat flux is strongly influenced by the non-uniform temperature distribution (see figure 3a).…”

“…The choice of a prolate drop is related to the fact that for such shape it was shown [14] that the uneven heat flux caused by the variable curvature produces, during drop heating and evaporation, an uneven distribution of temperature over the surface. However, similar results can be obtained for the case of an oblate drop and even for the spherical drop.…”

“…The different behaviour is due to a combination of the effects of boiling temperature and latent heat of vaporisation. These peculiarities should be considered when approximating the heat transfer coefficient from the analytic relation that can be deduced from the uniform surface temperature analysis [9,14]. The drop deformation has an effect on the deviation of the local Nusselt number from the values for the uniform temperature case, as reported in figure 5.…”

“…However a more accurate simulation can be obtained by using the concept of effective conductivity, firstly introduced by [10], to account for the effect of recirculation (see also [11] and [12]) and, although this cannot properly describe the temperature field inside the droplet, it can give a better estimation of the droplet surface temperature [13] . Recent modelling of heating and evaporation of spheroidal droplets [14] revealed that the uneven distribution of fluxes on the drop surface causes a corresponding uneven distribution of temperature on the drop surface, during most of the drop lifetime. This non-uniform temperature distribution affects the heat and vapour flow fields in a non neglectful way.…”

Kinetic and MD modelling of automotive fuel droplets heating and evaporation: recent results

“…The kinetic model for non-spherical droplets has not yet been developed, to the best of our knowledge. The results of development of a hydrodynamic model of spheroidal droplet heating and evaporation are described in [8].…”