Theoretical Modeling of Levitated Clusters of Water Droplets Stabilized by Infrared Irradiation

Abstract

This paper shows how clusters of radiation-stabilized water droplets levitated in an upward flow of air and water vapor above a heated water surface can be modeled using Spalding's self-similarity theory of heat and mass transfer and Stefan flow. The model describes equilibrium droplet states, including stability conditions, as well as nonequilibrium (quasi-steady) transient evolution. Equilibrium states are shown to exist when Stefan-flow supersaturation, which has a quadratic-like variation with height above the water surface, and radiation-stabilized equilibrium supersaturation, which is nearly constant with height, are equal. The latter can be predicted by a fundamentally derived function of absorbed radiant flux (linear), droplet radius (linear if opaque), continuum thermal conductivity, and thermodynamic properties. In fact, all of the experimentally observed droplet behavior can be predicted using simple analytical results based on quasi-steady droplet energy and continuum transport. Unsteady droplet energy, Knudsen-layer transport, numerical solutions, and curve-fitting of numerical computations, as used previously in modeling this behavior, are not necessary. An interesting reversal of the usual effect of mass transfer on droplet drag in low-Re flow when levitated droplets are irradiated asymmetrically by significant infrared radiation is also postulated, which relates to the relative importance of normal (pressure) and tangential (shear stress) drag. This theory of radiation-augmented droplet evaporation, condensation, and relative motion in a moving gas has application to conditions in clouds, wherein droplets can experience either net radiative heating or cooling and fluctuating updrafts or downdrafts.