Many efforts have been made to extend Twomey's (1959) analytical estimate of the number concentration of activated cloud condensation nuclei (CCN). The gas kinetic effect, which was neglected in Twomey's original work, is now included in most of the recent studies. However, the parameterization formulas developed by these studies are usually based on one given value of condensation coefficient (accommodation coefficient of vapor molecules on water surface). Therefore, the coefficients in the parameterization formulas must be re-tuned if a different value of condensation coefficient is adopted. Given that the most appropriate value of the condensation coefficient remains uncertain (e.g., temperature and contamination), the aforementioned strategy can be somewhat ineffective. This work re-examines the governing equations and identifies the main dimensionless parameters. The condensation coefficient is explicitly expressed in one of the parameters. A series of numerical simulations of cloud formation under various conditions of temperature, pressure, updraft speed, and CCN distribution parameters were conducted and analyzed to verify if the numerical results can be uniquely determined by these dimensionless parameters.

Twomey, S., 1959: The nuclei of natural cloud formation, II, The supersaturation in natural clouds and the variation of cloud droplet concentration, Geophys. Pura Appl., 43, 243-249.