Closed Form Equations for the Evaporation Rate and Droplet Size of Knudsen Droplets

Abstract

The flux-matching method is reexamined and found to be a necessary but not a sufficient condition at the flux-matching surface. The equation of continuity in the quasi-stationary approximation requires, that the flux density also be continuous across the boundary surface. When both flux-matching and flux-density matching are considered, it is found that the traditional flux-matching equation of Fuchs is only valid for Kn « 1 where Kn is the Knudsen number. A new evaporation rate equation is developed valid for all Kn. The equation reduces to Maxwell's diffusional equation in the limit of small Kn. Under appropriate conditions in the limit of large Kn the equation reduces to the free molecule regime's standard rate of evaporation in a vacuum. When Kn and the droplet radius are small, the equation reduces to both Schafer's equation and Fuchs' equation. The evaporation rate equation is integrated to give a second closed form equation which describes droplet size as a function of time. This equation is also valid in the Knudsen regime and reduces, under appropriate conditions, to either the equation for the continuum or the free molecule regime. This second equation is applied to the DBS/N2 data of Davis and Ray (1978). Excellent agreement is found throughout the Knudsen regime of 0.07 Kn < 0.2, for which the data applied.