Improved Rate Laws and Population Balance Simulation Methods; CRE Applications, Including the Combustion Synthesis of Valuable Nano-Particles

Abstract

Using ‘flame-synthesized’ nanoparticles (nps) as one prototypical application, we illustrate our recent progress in two broad areas of current CRE-interest, viz., the development of: 1. Improved rate laws/transport coefficients for next-generation Eulerian, multi-(state) variable population-balance formulations, and 2. Quadrature-based multi-variate moment methods (hereafter QMOM) suitable for articulation with evolving Eulerian CFD simulation methods Admittedly, in previous work much insight was obtained by introducing deliberately (over-) simplified rate laws (for nucleation, Brownian coagulation, vapor growth/evaporation, sintering, thermophoresis, ) into the generally nonlinear integro-partial differential equation called the ‘population balance’ equation (PBE). However, despite the complexity of this equation, and the need to satisfy it along with many other local PDE-balance principles in multi-dimensional CRE environments, in our view current requirements for reactor design, as well as the frequent need to infer meaningful physico-chemical parameters based on laboratory measurements on populations rather than individual ‘particles’, make the introduction of more accurate rate/transport laws essential for next-generation particle synthesis reactor models. Our present examples are motivated both by measurements/calculations of the structure of laminar counterflow flames synthesizing Al2O3 nps and/or the predicted performance of well-mixed steady-flow devices in which sintering or sublimation occurs. Corresponding illustrative results, which focus on the rate laws for sphere dissolution or aggregate Brownian coagulation support our contentions that: i) systematic introduction of more accurate rate laws (including nucleation, sintering, growth, )/transport coefficients will be essential to meet the quantitative demands of next-generation PBE-based CRE-simulation models for high-value particulate synthesis equipment, and, ii) QMOM is able to incorporate realistic rate laws and faithfully generate their effects on important ‘moments’ characterizing the product joint distribution functions.