On a multivariate population balance model to describe the structure and composition of silica nanoparticles

The aim of this work is to present the mathematical description of a detailed multivariate population balance model to describe the structure and composition of silica nanoparticles. Silica nanoparticles are formed by the interaction of silicic acid monomers (Si(OH)4)in the gas-phase. A detailed numerical study of a stochastic particle algorithm for the solution of the multidimensional population balance model is presented. Each particle is described by its constituent primary particles and the connectivity between these primaries. Each primary, in turn, has internal variables that describe its chemical composition, i.e., the number of Si, free O and OH units. A particle undergoes transformations due to different particle processes such as surface reactions, coagulation, sintering, and intra-particle reactions. The algorithms used to solve the population balance equations and to couple the population balance model to gas-phase chemistry are described. Numerical studies are then performed for a number of functionals calculated from the model to establish the convergence with respect to the numerical parameter that determines the number of computational particles in the system. A brief numerical investigation of convergence with respect to the splitting time step has also been undertaken. The computational times (for runs that provide acceptable statistical errors) are determined to be sufficiently small to facilitate the application of this detailed multidimensional model to simulate industrial scale systems.