Part II: Dynamic evolution of the particle size distribution in particulate processes undergoing simultaneous particle nucleation, growth and aggregation

The present study provides a comprehensive investigation on the solution of the dynamic population balance equation (PBE) for particulate processes undergoing simultaneous particle nucleation, growth and aggregation. The general PBE was numerically solved in both the continuous and its equivalent discrete form using the orthogonal collocation on finite elements and the discretized PBE method, respectively. A detailed investigation on the effect of particle nucleation rate on the calculated particle size distribution (PSD) was carried out over a wide range of variation of dimensionless aggregation, nucleation and growth times. The performance (i.e., accuracy and stability) of the two numerical methods was assessed by a direct comparison of predicted PSDs and/or their respective moments to available analytical solutions. For combined aggregation and nucleation problems, the numerical error scaled with the product of the dimensionless aggregation and nucleation times. On the other hand, for combined growth and nucleation problems, the numerical error scaled only with the dimensionless growth time. For particulate systems with minimal particle growth, constant particle nucleation rate and Brownian aggregation, the total particle number approached a “steady-state” value characterized by the equilibrium of particle aggregation and nucleation rates. When the particle nucleation rate followed a pulse-like function, the PSD converged to a self-similar distribution after the end of particle nucleation. Moreover, for particulate systems exhibiting a constant particle nucleation rate and a Brownian-type particle aggregation kernel, an increase in the particle growth rate resulted in a decrease in the final total number of particles. On the other hand, for a constant particle nucleation rate and an electrostatically stabilized Brownian aggregation kernel, an increase in the particle growth rate can lead to an increase in the final total number of particles.