The surface temperature of the droplet is determined from a heat and mass transfer balance at the interface between the droplet and the surrounding gas. There are two regimes of heat transfer, that is, heat transfer occurring from the inside of the droplet to the surface, qi, and heat transfer occurring from the outer gas to the surface, qo. The rate of heat transfer balances the required heat for vaporization at the surface

(6–82)

where is the latent heat of the fuel at the surface temperature, and is the mass vaporization rate.

The heat transfer from inside the droplet is modeled as a convective heat-transfer process with internal circulation taken into account. The effective heat transfer coefficient for the outer flux is determined from an approximate solution of the energy equation for the vapor phase with the effects of inter-diffusion and Stefan flow considered. An explicit equation that relates the vaporization rate, , to the temperatures of the droplet and the surrounding gas mixture can be derived as [74]

(6–83)

where h _i,eff is the heat transfer coefficient inside the droplet, which is determined from the thermal conductivity, λ, and the unsteady equivalent thickness of the thermal boundary layer, r ₀ is the droplet radius, Sh is the Sherwood number, Nu is the Nusselt number, C _p is the average specific heat of the gas mixture including fuel vapor, K is a correlation factor defined by Ra and Reitz 2003 [73] , [CA] is the inter-diffusional difference of energy flux between fuel and air, , is the average diffusion coefficient of fuel species, y _F0 and y _Fsur are the mass fractions of fuel at the interface and far away, respectively, and T _sur is the surrounding gas temperature.

The rate of mass transport at the droplet surface is calculated using the high mass transfer rate equation with Spalding’s transfer number [88]

(6–84)

where g _m is the mass transfer coefficient determined from , and B _M is Spalding’s transfer number, .