Equilibrium particle distribution in the disperse system with coagulation and aggregate disintegration

Abstract

Kinetic equations were formulated, which describe coagulation–fragmentation process in a low concentrated suspension flow at a low shear rate. In such a system dispersed phase divided into fine and coarse fractions as the system is brought to equilibrium. Kinetic equations of two-fraction model were formulated. An approximate solution and, in one particular case, the exact solution of these equations were obtained for the equilibrium state. Detailed analysis of equilibrium particle distribution over the mass m was performed for an exponential coagulation kernel β = β0mν and an degenerated disintegration kernel γ = γ1γ2, in which the disintegration frequency is an exponential function of aggregate mass γ1 = γ0mν + α, and the probability of the fragment detachment from an aggregate is independent ofm and decreases exponentially with an increase in mass μ of a fragment: γ2 = μ0–1exp(–μ/μ0). The equilibrium distribution was shown to exist only at α > 0, and in particular, it is described at ν = α = 1 by the f (μ) = λγ0β0–1exp(–μ/μ0) and F(m) = Cx–1(x + 1)2λ – 1e–λx functions for the particles of fine and coarse fractions (x = m/m0, λ = m0/μ0, m0 and μ0 are the characteristic masses of coarse and fine fractions, respectively). The particle distribution for the fine fraction at λ ≫ 1 is well approximated by the Gaussian distribution exp[–(m – m0)2/(4α–1m0μ0)].