The simulation of polymer aggregation/breakage at high solid fraction in turbulent flow by population balance

A mathematical model has been developed to describe the size and settling rate of aggregates formed by flocculation in thickeners/clarifiers used in the mineral processing industry. The aggregation process was simulated with a population balance model, which calculates the aggregate size distribution through time as a function of the competing processes of aggregation and breakage. The population balance was written specifically to describe aggregation by high molecular weight polymer flocculants that have now largely replaced coagulant salts in mineral processing operations, and the model includes terms to describe flocculant/slurry mixing and irreversible aggregate breakage (Table 1). The model was written to form part of a full computational fluid dynamics (CFD) model of a thickener/clarifier, allowing the eventual simulation and optimisation of the full-scale unit. In addition to the effects of fluid shear and residence time normally described by population balance models, additional process variables have been considered, with the model correctly accounting for changes in the flocculant dosage, primary particle size and solid fraction. The model has also been extended to describe the hindered settling rate under the same range of conditions, considerably increasing its usefulness by forming a link between the aggregation kinetics in the feedwell and the subsequent setting rate as the aggregates enter the settling zone of the thickener. The model was fitted to experimental data from a turbulent pipe reactor. A variety of pipe sizes, lengths, and flow rates were used to give a range of fluid shear rates and mean residence times, with the aggregate size distribution measured by an on-line sizing probe placed at the end of the pipe, immediately before a settling column. Experimental data was collected under a sparse matrix of experimental conditions, with the fluid shear rate (G), flocculant dosage (θf), primary particle size (dp) and solid fraction (φ) varied independently away from a common baseline. Additional data was collected from conditions in the gaps of the experimental matrix, and was used to check the predictivity of the model. The population balance model is based on the discretised balance by Hounslow et al. (1988) and Spicer and Pratsinis (1996), in this case using 35 channels covering the size 4 range: 0.2-3500 μm. The aggregation rate is described by Saffman and Turner’s (1956) turbulent collision kernel, used in conjunction with a capture efficiency term. The capture efficiency is initially taken to be zero (no successful collisions) before flocculant addition, but increases to unity (all collisions successful) as described by a flocculant-suspension mixing term. The breakage rate is described as a function of the aggregate size, energy dissipation rate, suspension viscosity and effective flocculant surface coverage. The effective flocculant coverage is taken to decrease through time, reflecting the loss of flocculant activity caused by repeated aggregation/breakage. Aggregate porosity is also included, using fractal geometry, and is used to calculate the aggregate capture radius and effective suspension solid volume fraction. The solid volume fraction is used to determine the suspension viscosity, accounting for experimental data showing the pressure drop in the pipe is a function of the aggregate size. The model equations were solved numerically as an initial value problem with a commercial dynamic simulation package (gPROMS), which was also used to estimate the unknown model parameters. The model was found to be robust and stable, and gave good predictions of the additional experimental data sets. The extension of the model to also describe the hindered settling rate allowed a dynamic optimisation to give the maximum settling flux, and hence unit throughput. Various optimisation strategies were investigated, in particular the use of recycle to find the optimal feed solid fraction. The inclusion of the model within a full CFD simulation will allow further optimisation strategies to be investigated, in particular changes to the feedwell geometry to give good mixing but without subjecting fully formed aggregates to regions of destructive high fluid shear.