A kinetic model for droplet growth in the transition regime

A variant of the moment expansion method, used in an earlier paper to describe the flow of a gas toward an absorbing sphere, is applied to a more realistic model of a droplet condensing from a supersaturated vapor. In the simplest version a spherical droplet absorbs all incoming vapor molecules, but spontaneously emits molecules with a Maxwellian distribution at the droplet temperature and with the corresponding saturated vapor density. From a solution of the stationary linearized Boltzmann equation with these boundary conditions we obtain expressions for the heat and mass currents toward the sphere as a function of the supersaturation and the temperature difference between the droplet and the vapor at infinity. For small droplet radii the known free flow limit is obtained in a natural way. From the calculated expressions for the heat and mass current we derive evolution equations for the radius and temperature of the droplet. The temperature evolves more rapidly and can thus be eliminated adiabatically; the resulting growth curve for the radius shows a sharp transition from a kinetically controlled regime for small radii to a regime dominated by heat conduction for large radii. The effect of incomplete absorption at the surface is also studied. The actual calculations are carried out for Maxwell molecules, with parameters corresponding to argon at 0.65T_cand 100% supersaturation.