Integrated Framework for the Numerical Solution of Multicomponent Population Balances. 1. Formulation, Representation, and Growth Mechanisms

Abstract

Current numerical solutions of population balance problems including growth are limited by the Courant condition time step constraint. As a result, particle size ranges must be restricted and limited growth rate mechanisms can be solved. A thorough analysis of growth rate mechanisms and dynamics is presented that relates the growth rate mechanisms and particle size ranges to the time step limit. These relationships reveal a method for producing the optimal time step for any growth rate model, which can increase the time step by a factor of 107. When these methods are combined with previous approaches, new hybrid methods are developed that allow the user to flexibly choose the optimal time step given the resolution requirements of the problem, producing solutions that are 104−107 times faster than conventional solutions. The numerical accuracy of these solutions compares favorably against analytical solutions, demonstrating their applicability across a wide range of problems.

Solution of the bivariate dynamic population balance equation in batch particulate systems: Combined aggregation and breakage

“…The initial condition of the PBE is n(v, x, 0) = n 0 (v, x) and the boundary conditions are n v (0, x, t) = n 1 (x, t) and n x (v, 0, t) = n 2 (v, t). Different numerical methods have been used to solve the bivariate PBE (Rosner et al, 2003;Obrigkeit and McRae, 2004). These include the finite element method (Kim and Seinfeld, 1992), the sectional method (Kim and Seinfeld, 1990), MonteCarlo techniques (Gillespie, 1975), the method of moments (McGraw, 1997;Marchisio et al, 2003), and other methods.…”

Solution of the Population Balance Equation for Two-Component Aggregation by an Extended Fixed Pivot Technique

“…Simulating the dynamics of multicomponent aggregation is of interest in a number of fields, such as chemical engineering, , aerosol and atmospheric science, medical research, and so forth. The simplest case one may encounter, but nevertheless important, is obviously that of a two-component system.…”

“…The simplest case one may encounter, but nevertheless important, is obviously that of a two-component system. If the system is spatially homogeneous, the aggregation process is described by the following population balance equation (PBE) , where n ( x , y , t ) d x d y is the number of particles of state ( x , y ) per unit volume at time t and β( x , y ; x ‘, y ‘; t ) is the aggregation rate coefficient. The internal coordinates x and y denote the amount (mass, moles, etc.)…”

“…The resolution of PBEs such as eq 1 still raises a number of difficulties. The reader is referred to Obrigkeit et al , for an up-to-date review of methods for solving multicomponent PBEs. Basically, one may distinguish between methods capable of determining the full size-composition distribution and simpler methods based on approximate representations of the variations in particle composition.…”