Analytical Solutions to Simple Models of Condensation and Coalescence

R. C. Srivastava Department of Geophysical Sciences, The University of Chicago, Chicago IL 60637

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R. E. Passarelli Department of Geophysical Sciences, The University of Chicago, Chicago IL 60637

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Abstract

The kinetic equation for the evolution of particle size spectra by condensation and coalescence is considered. The condensation rate, the rate of increase of the particle mass ẋ, is taken as (i) a(t)x and (ii) a(t), where a(t) is an arbitrary non-negative function of the time. In case (i) it is shown that, for a homogeneous kernel, the solution of the kinetic equation for condensation and coalescence can be reduced to that for pure coalescence by simple transformations. In case (ii) the solution is expressed as an infinite series, the terms of which involve convolutions of arbitrary order of the initial distribution and a function of the condensation rate. The central limit theorem of probability theory is used to obtain an expansion for the convolutions, and an approximate analytical expression for the sum of the infinite series is obtained for large x. A few numerical evaluations of the solutions are presented.

Abstract

The kinetic equation for the evolution of particle size spectra by condensation and coalescence is considered. The condensation rate, the rate of increase of the particle mass ẋ, is taken as (i) a(t)x and (ii) a(t), where a(t) is an arbitrary non-negative function of the time. In case (i) it is shown that, for a homogeneous kernel, the solution of the kinetic equation for condensation and coalescence can be reduced to that for pure coalescence by simple transformations. In case (ii) the solution is expressed as an infinite series, the terms of which involve convolutions of arbitrary order of the initial distribution and a function of the condensation rate. The central limit theorem of probability theory is used to obtain an expansion for the convolutions, and an approximate analytical expression for the sum of the infinite series is obtained for large x. A few numerical evaluations of the solutions are presented.

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