(358e) Dynamic Evolution of Multi-Variate Particle Size Distributions in Particulate Processes: a Population Balance Perspective

An important property of many industrial particulate processes is the particle size distribution (PSD) that controls key aspects of the process as well as the end-use properties of the product. In many processes, the dynamic PSD involves the distribution of one or more internal particle variables, i.e., number of radicals, amount of adsorbed species, porosity, etc. The quantitative determination of the evolution of such multi-variate PSDs in processes such as particulate reactors is a rather complex problem for it requires good knowledge of the particle nucleation, growth, breakage, and aggregation mechanisms in addition to the reaction kinetics (Ramkrishna and Mahoney, 2002). The evolution of the PSD in particulate processes is commonly obtained by the solution of a population balance equation (PBE) which has been utilized in a diverse range of problems (Ramkrishna, 1985). The distribution of particles is considered to be continuous over the volume and is described by a number density function, n(v,t), that represents the number of particles, per unit volume, in a differential particle volume, v to v+dv. For multi-variate problems additional distributed properties must be included. For example, the bi-variate number density function, n(v,x,t), represents the number of particles, per unit volume, in a differential particle volume, v to v+dv, and internal property, x to x+dx, size range. The rate of change of the multi-variate particle number density is described by the multi-variate PBE which is a multi-dimensional nonlinear integro-differential equation. The numerical solution of the PBE generally requires the discretization of the v and x particle domain into a small number of elements leading to a system of nonlinear ODE's or DAE's which is usually stiff. The PBE can be solved in its continuous form, e.g. by collocation on finite elements (Alexopoulos et al., 2004) or it can be solved in a discretized form (Kumar and Ramkrishna, 1996). Despite the increased interest in solving such dynamic multi-variate PBEs, the majority of the numerical methods described in the open literature have not been adequately tested. Only recently has a comprehensive investigation of the uni-variate PBE been completed (Alexopoulos et al. 2004). In the present study the Galerkin finite elements method (GFEM), the sectional grid technique (SGT), and a Monte-Carlo (MC) stochastic method are employed to solve the dynamic bi-variate PBE under the combined action of particle growth, breakage, aggregation and nucleation (Kumar and Ramkrishna, 1996; Alexopoulos and Kiparissides, 2005). The numerical techniques are first validated by solving the bi-variate PBE for a number of simple test problems including aggregation, combined particle aggregation and breakage, combined particle nucleation and aggregation, as well as combined particle growth and aggregation. The calculated solutions are compared to the corresponding analytical solutions for the bi-variate PSDs. For problems where analytical solutions are not available the numerical solutions are evaluated by comparison of the calculated to the analytical PSD moments. (Alexopoulos and Kiparissides, 2005). Two general cases of particulate processes are considered: The first case involves processes where particle aggregation and breakage are the dominant mechanisms. This first case corresponds to many important particulate processes involving droplet suspensions and particle flocs. The dynamic behavior of the PSD depends not only on particle size but on other internal variables such as reactant concentration, temperature, etc. The second case involves particulate reactors in which particle aggregates are formed and then fuse together with time. These type of processes are often found in aerosols and the gas-phase production of nanoparticles. The internal variables that can influence the evolution of the PSD are porosity, total surface area, temperature, etc. Case 1: Dominant Aggregation and Breakage. The simplest test case is the ?tracer-experiment? where the internal property x is the amount of an inert species distributed unevenly among the particle population. Even this simple case displays a surprising wealth of dynamic behavior depending on the initial distribution and the breakage and aggregation rate functions. For example, for an aggregation-dominant case with an initial distribution consisting of two populations of particles with different concentrations of a passive species, the dynamic bi-variate PSD evolves through a wide range of multi-peak distributions eventually converging to the steady-state PSD at an intermediate concentration. Although this is a simple physical problem the numerical solution is difficult as the initial and the final steady-state bi-variate PSDs are singular with respect to the concentration. Different formulations of the bi-variate PBE are considered that allow a more efficient treatment of these types of singularities. More realistic particle processes are also considered by employing aggregation and breakage functions that depend on the property x. Case 2: Dominant Aggregation, Fusion and Growth. A general case of particulate reactors is examined in which particle aggregates are formed and then fuse together with time. The surface area of the particle aggregate is taken as the internal property in the bi-variate number density function. The dynamic bi-variate PBE is then solved assuming simple expressions for the particle aggregation rate kernel and the particle growth function. The particle fusion process is assumed to follow the phenomenological expression of Koch and Friedlander (1989). The dependence of the aggregation rate kernel on the porostiy of the aggregate is also considered and the effect on the dynamic bi-variate PSD is determined. For processes with dominant particle growth the moving-grid formulation of the PBE is employed to solve the bi-variate PBE. When the internal variable is chosen to be the reacting monomer the effect of the bi-variate nature of the PSD on the final distribution can be determined.