Köhler equation for finite systems: A simple estimation of possible condensation mechanisms in aircraft contrails

A generalization of the Köhler equation for systems with a limited amount of available water vapor is presented. Following the classical nucleation theory, a condensation process on soluble particles is discussed under isothermal‐isochoric conditions. The resulting generalization of the Köhler equation deviates from the classical approach for droplets with radius larger than the critical radius. An additional minimum in the Köhler curve can be determined describing the effect of water limitation. The presented theory is applied for the estimation of possible water condensation processes during the formation of aircraft contrails. In particular, three different kinds of condensation centers (soot, soot covered with sulfuric acid, and pure H2O/H2SO4 droplets) are compared with respect to the critical conditions which are necessary for their activation and with respect to their suitability for water uptake during condensation processes in the plume. Based on such investigations, soot particles with or without sulfuric acid can be easier activated than small H2O/H2SO4 droplets for atmospheric situations where jets get supersaturated with respect to water and, consequently, may lead to sufficiently large frozen particles. If the plume is undersaturated with respect to water, there is no difference, at least in view of the water condensation processes, between the highly concentrated H2O/H2SO4 droplets and soot particles with or without sulfuric acid.