Limited Coalescence

Abstract

In aerosol dynamics one models the evolution of the number density n(x, t) of particles of volume x at time t by the equation

$$ \begin{gathered} \frac{{\partial n(x,t)}} {{\partial t}} = - n(x,t)\int_0^\infty {\phi (x,\xi )n(\xi ,t)d\xi } \hfill \\ + \tfrac{1} {2}\int_0^x {\phi (x - \xi ,\xi )n(x - \xi ,t)n(\xi ,t)d\xi } \hfill \\ \end{gathered} $$

(1)

where φ(x, ξ) is the collision rate between particles of sizes x and ξ; here the first term on the right-hand side expresses loss of particles of size x due to coalescence with particles of any size ξ, and the second integral expresses the gain of particles of size x through coalescence of particles of sizes ξ and x — ξ with ξ ≤ x — ξ; the factor ½ is introduced when we remove the restriction ξ ≤ x — ξ.