Bivariate moment methods for simultaneous coagulation, coalescence and breakup

Aerosol reactors pass through regimes where subsets of the population balance terms are dominant. Initially, mixing, reaction, nucleation and accretional growth dominate. This is generally followed by a regime in which coagulation and coalescence control evolution of the particle population into sintered aggregates. Conventionally, the boundary between regimes where coalescence is or is not important is assumed to be sharp, and the collision–coalescence regime is followed by a regime dominated by coagulation and breakup which controls the growth of loosely bonded agglomerates that grow large enough to be captured in conventional gas–solids separation equipment. When this boundary is not sharp, there can be a regime in which coagulation, coalescence and breakup all occur simultaneously. This paper describes the daughter distributions required to model breakup in a bivariate population, the moment models describing simultaneous collision, coalescence and breakage, and exhibits reconstructed steady-state distributions formed when the rate kernels are size independent. A criterion is developed to guide the choice of when a bivariate formulation is necessary and when it is not.