A Class of Solutions to the Steady-State, Source-Enhanced, Kinetic Coagulation Equation

Abstract

Analytical solutions to the steady-state kinetic coagulation equation, including sources, are presented for the case in which the collection kernel is of the form K = K0u?v?, where K0 and ? am constants (0 ≤ ? < 1), and u and v represent aerosol particle volumes. Particle sources are represented by gamma distributions. The solutions look like modulated power law spectra, and rapidly approach their asymptotic power law form with increasing size. For, ? = 0 the solution is equivalent to Friedlander's quasi-stationary distribution in the Brownian coagulation regime, and for the case of a constant input rate of single particles with beta; = 0 it is in excellent agreement with the corresponding numerical solution of Quon and Mockros. For ? = 2/3 the collection kernel provides an approximate description of gravitational collection, and the corresponding power law solution conforms fairly well with observations of the tropospheric aerosol spectrum under clear air conditions for 1 ? r ? 102 ?m. The characteristic time to approach this solution is investigated, and appears to he short enough for the solution to he applicable to the real atmosphere. Finally, an approximate analysis is provided to give an indication of the effect of truncating the spectra at some specified maximum size. The truncated spectra are found to be little changed from the original forms.