Modelling collision frequencies and predicting bi-variate agglomerate size distributions for bi-disperse primary particle systems

Agglomerate growth and the evolution of the agglomerate size distribution is determined by the collision frequencies between the agglomerates of the different size classes. For size distributions that can be parameterized by the agglomerates size only, expressions for the collision kernels exist and agglomerate growth can be predicted with sufficient accuracy. In the case of systems with polydisperse primary particles such as the agglomeration of soot or of systems with several components such as the flame synthesis of nanoparticles with taylor-made catalytic properties, a bi- or polydisperse size distribution is needed to account for the effects of the different primary particle sizes. In the present paper, collision frequency are obtained from a large series of Langevin dynamics (LD) simulations that are largely "model-free". Bi-disperse primary particle systems are investigated where the size ratios of the primary particles are varied from unity to a factor of up to six. An analytic expression for an effective collision radius is suggested and accounts for functional dependencies on agglomerate size, composition and fractal dimension. Independent simulations for the evolution of the population balance equation (PBE) for the bi-variate agglomerate size distribution are conducted and assessed by comparison with corresponding Langevin dynamics simulations. The agreement between PBE solution and LD simulation results is generally very good indicating sufficient accuracy in modelling the collision kernel. Additional PBE simulation for mono-variate size distributions notably underpredict collision rates and errors of up to 140% in the total number of agglomerates can be expected by the end of a simulation for the larger size ratios. Errors are small for size ratios of two, but overall, a bi-variate parameterization of the population size distribution is needed to accurately predict agglomerate growth if the size ratio between the primary particles is notably larger than two.