Equivalence of two different integral representations of droplet distribution moments in condensing flow

Abstract

It is proved that two different and independently derived integral representations of droplet size distribution moments encountered in the literature are equivalent and, moreover, consistent with the general dynamic equation that governs the droplet size distribution function. One of these representations consists of an integral over the droplet radius while the other representation consists of an integral over time. The proof is based on analytical solution of the general dynamic equation in the absence of coagulation but in the presence of both growth and nucleation. The solution derived is explicit in the droplet radius, which is in contrast with the literature where solutions are presented along characteristics. This difference is essential for the equivalence proof. Both the case of nonconvected vapor as well as the case of convected vapor are presented.