Theoretical Analysis of Water Droplet Condensation and Evaporation in Prescribed Environment with Radiation

Abstract A quasi-explicit algebraic solution of the problem of quasi-steady water droplet heat and mass transfer with condensation or evaporation in a prescribed environment including radiation has been obtained. The solution is based on the mass fraction form of Fick’s law of species diffusive mass transport, which allows accounting for variation in gas density through the diffusion layer surrounding the droplet. A new term, the average vapor supersaturation through the diffusion layer, has been included that allows the model to be extended to environmental conditions significantly outside those for which it is theoretically derived (dilute or near saturation), even to conditions of relatively high temperature, low relative humidity, and low pressure. The latter two have atmospheric significance because evaporative cooling can induce a significant decrease in droplet temperature and possibly induce freezing. Basic modeling assumptions are confirmed by comparison with time-dependent temperature and evaporation rate measurements of a laboratory laser-heated droplet. The accuracy of the new model is assessed by comparison with exact equations. For temperatures between −30° and 0°C and negligible radiation, relative error magnitudes in droplet temperature and evaporation rate are less than 5%, even for relative humidity down to 10% and pressure down to 60 kPa. This represents an improvement in accuracy over the conventional model, in which property variations of density and supersaturation through the diffusion layer are not taken into account. Differences in predicted evaporation rates between the mass fraction and the more common partial density Fick’s laws, which can be significant (>10% above 70% relative humidity) in coupled implicit equations, are shown to be made negligibly small (<0.5%) by linearizing about the saturation state.