Coupled Population Balance - Fluid Flow Models

Coupled Population Balance - Fluid Flow Models 

Some approximate methods have been applied previously to deal with the inhomogeneous nature of flow patterns in stirred tanks [67, 68]. Koh [67] divided the stirred tank into three compartments, the impeller zone, the bulk zone and a dead space, and assigned different shear rates for each compartment. Furthermore, Koh et al. [69] ignored the dead space, but split the impeller zone into impeller tip zone and impeller zone. Ducoste [68] essentially followed the same approach, dividing the suspension volume into two zones, the impeller discharge zone and the bulk zone. 

Instead of assigning different shear rates, he employed different breakage rate expressions for the two zones. The problem of coupling population balance models with fluid flow models has received some attention recently and coupled PB-CFD models have been developed for a wide variety of processes such as fluidization [70], gas-liquid reactions in bubble columns [71] and nanoparticle synthesis in flame aerosol reactors [72]. Complete description of aggregation in turbulent environments requires simultaneous solution of basic balance equations for mass, momentum, energy and concentration of species present along with population balances for particles/aggregates of different size classes. 

In one of the earliest attempts, Schuetz and Piesche [73] calculated the flow field in a stirred tank first to determine the local energy dissipation rate and then solved the PBEs using the finite volume method [74] to predict the local aggregate size distribution. Heath and Koh [75] have solved the population balances as scalar equations in the commercial CFD software CFX for simulating flocculation of suspensions by polymers. They employed 35 discrete sectional equations to represent the aggregate size distribution. 

Clearly, current aggregation models can reliably predict simple effects of salt and some effects of polymers, but additional work is warranted to develop the capability to predict aggregation and dispersion of particulates of all types (solid, gas and liquid) under real-life conditions of varying concentrations of flocculants or dispersants, hydrodynamic perturbation and local pressure and temperature variation in addition to the effects of any magnetic or sonic fields. 

---

Schreiner etal. (2001) modelled the precipitation process of CaC03 in the SFTR via direct solution of the coupled mass and population balances and CFD in order to predict flow regimes, induction times and powder quality. The fluid dynamic conditions in the mixer-segmenter were predicted using CFX 4.3 (Flarwell, UK). 

These models require information about mean velocity and the turbulence field within the stirred vessels. Computational flow models can be developed to provide such fluid dynamic information required by the reactor models. Although in principle, it is possible to solve the population balance model equations within the CFM framework, a simplified compartment-mixing model may be adequate to simulate an industrial reactor. In this approach, a CFD model is developed to establish the relationship between reactor hardware and the resulting fluid dynamics. This information is used by a relatively simple, compartment-mixing model coupled with a population balance model (Vivaldo-Lima et al., 1998). The approach is shown schematically in Fig. 9.2. Detailed polymerization kinetics can be included. Vivaldo-Lima et a/. (1998) have successfully used such an approach to predict particle size distribution (PSD) of the product polymer. Their two-compartment model was able to capture the bi-modal behavior observed in the experimental PSD data. After adequate validation, such a computational model can be used to optimize reactor configuration and operation to enhance reactor performance. 

The interfacial and turbulence closures suggested in the literature also differ considering the anticipated importance of the bubble size distributions. It thus seemed obvious for many researchers that further progress on the flow pattern description was difficult to obtain without a proper description of the interfacial coupling terms, and especially on the contact area or projected area for the drag forces. The bubble column research thus turned towards the development of a dynamic multi-fluid model that is extended with a population balance module for the bubble size distribution. However, the existing models are still restricted in some way or another due to the large cpu demands required by 3D multi-fluid simulations. 

---