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Chemically Reacting Flows
Published in Greg F. Naterer, Advanced Heat Transfer, 2018
The reactor contains particles that have spent different times inside the fluidized bed so there is a distribution of conversion rates. Population balance equations (PBEs) are often used to describe this evolution of a population of particles in a fluidized bed. PBEs may be derived as an extension of the Smoluchowski coagulation equation (Smoluchowski 1916),which describes the coalescence of particles. However PBEs are more general as they define how populations of separate particles develop over time. They are given by a set of integral/differential equations that identify the mean-field behavior of a population of particles from the analysis of the behavior of a single particle. The particle systems are characterized by various stages of a particle transformation from its initial formation to its subsequent decomposition from the chemically reacting flow. The PBE represents a balance of the number of particles of a particular state. Monte Carlo methods, discretization methods, and moment methods have been used to solve population balance equations. Further detailed analysis of population balances was presented by Ramkrishna (2000).
Enhancing Gravity Thickener Feedwell Design and Operation for Optimal Flocculation through the Application of Computational Fluid Dynamics
Published in Mineral Processing and Extractive Metallurgy Review, 2021
P. D. Fawell, T. V. Nguyen, C. B. Solnordal, D. W. Stephens
The mathematical equations that make up a PB sub-model ultimately derive from the classical Smoluchowski (1917) coagulation equation. This usefully describes spherical aggregates that maintain their form throughout the process, but not those that have an open, irregular structure and also change their shape over time, as is observed in practice. The many modifications and refinements proposed giving greater relevance to practical systems, in particular in a solid-liquid separation context, have been reviewed by Jeldres, Fawell and Florio (2018), while the derivation of the PB sub-model for polymer-bridging flocculation as used here and the process for estimating fitted parameters were described in detail by Heath et al. (2006b, 2006c)).