(232s) Modeling of Aggregation and Breakup of Colloidal Particles in a Turbulent Stirred Tank through a Multizonal PBE Model
Processing of particulate systems often involves aggregation and breakup phenomena driven by turbulent fluid motions. These phenomena may either be desired, e.g., flocculation to facilitate solid-liquid separation, dispersion of a powder, etc., or they may be an unwanted side effect, e.g., coagulation in emulsion polymerization, agglomeration in solution crystallization, etc. In this work we address the aggregation and breakup of small colloidal particle clusters in a diluted turbulent stirred tank reactor for which we develop a multizonal population balance equation (PBE) model.
The PBE model combines a detailed aggregation rate model  with a comprehensive breakage rate model . The aggregation rate model  accounts for colloidal and hydrodynamic interactions among the colliding clusters. For the former, the van der Waals force between the primary particles in the two approaching clusters that are the nearest is considered. To model the hydrodynamic interactions, the porous clusters are described as uniformly permeable spheres where Brinkman's equation is adopted to describe intra-aggregate flow. For the breakage rate model , it is assumed that there exists a critical hydrodynamic stress above which a cluster of given mass breaks up. Furthermore, it is assumed that the breakup of an individual cluster is instantaneous once the stress exceeds the critical stress. In this framework, the breakage kinetics is governed by turbulent fluctuations that lead to a fluctuating hydrodynamic stress acting on the clusters. To describe the critical hydrodynamic stress, a recently developed scaling model for dense fractal clusters is used  whereas turbulent fluctuations on a local scale are described through a multifractal model [2,4].
Finally, to account for the spatial variation of the flow properties in the stirred tank reactor, i.e., the large scale heterogeneity, a multizonal approach is used. Thereby, the flow domain is divided into a small number of zones where in each of them the flow is assumed homogenous. Regarding the complexity of both the aggregation and breakage model, this multizonal approach presents a fair compromise between numerical load and accuracy. Two implementations of the multizone model are investigated. In the first one, the zones are predefined  and CFD is used to determine the zone properties and interzonal fluxes. In the second one, the zone properties are determined empirically and the interzonal fluxes are obtained by fitting to experiments. The model is compared to the aggregation of a fully destablized polystyrene latex in a stirred tank reactor [6,7]. Good agreement for the cluster mass distribution at steady state at various stirring speeds and various solid volume fractions is found in the case of the empirical zone model whereas the CFD model is insufficient in describing the variation in the solid volume fraction.
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