Mixing velocity field development

Fig. 10.20 Axial location of the two planes perpendicular to the counterrotating screws, where velocity fields were calculated. Plane (I) is at the middle of the side, and plane (II) at the middle of the calender gaps. [Reprinted by permission from T. Katziguara, Y. Nagashima, Y. Nakano, and K. Funatsu, Numerical Study of Twin Screw Extruders by 3-D Flow Analysis - Development of Analysis Technique and Evaluation of Mixing Performance for Full Flight Screws, Polym. Eng. Sci., 36, 2142 (1996).]...

These values must be compared with the experimental measurements. The results are shown in Table 7. There are some differences and two main reasons to explain some of the discrepancies (i) the fully developed laminar flow assumption of Equation (40) is not fulfilled and (ii) in the connection between both reactor lengths there is some mixing that has not been taken into account by the model. The problems occurring with the velocity field can certainly be solved without difficulties, substituting Equation (40) by a more realistic velocity distribution inside the reaction space resorting to available CFD programs. 

The application of CFD in the modeling of solid-liquid mixing is fairly recent. In 1994, Bakker et al. developed a two-dimensional computational approach to predict the particle concentration distribution in stirred vessels. In their model, the velocity field of the liquid phase is first simulated taking into account the flow turbulence. Then, using a finite volume approach, the diffusion-sedimentation equation along with the convective terms is solved, which includes Ds, a... 

Figure 3.3 Nonidealities in a PFR boundary layer development. The nonuniformities in the velocity fields cause mixing problems, giving rise to axial and/or radial dispersion effects.

We see that the ratio between tube wall concentration and cup mix concentration is highest for undeveloped flow and lowest for a fully developed velocity profile. It further follows that under our experimental conditions (no 1, 2, 3 Table I) for all three flow conditions the average concentration is roughly two times as hi as the wall concentration. This implies that also the rate constant roughly spoken has to be two times as high as the experimental value. In the preceeding paragraph we saw that for the fully developed velocity field this indeed is the case. 

As an example of what is needed, consider the models developed by McKelvey et al. (1975) using kinetic data from Toor (1969) and Mao and Toor (1971). McKelvey et al. (1975) measnred the velocity field and mixing characteristics in exactly the same multinozzle pipe reactor that was used by Toor and co-workers to measure the kinetics. McKelvey et al. had two objectives. The first was to... 

The classical CRE model for a perfectly macromixed reactor is the continuous stirred tank reactor (CSTR). Thus, to fix our ideas, let us consider a stirred tank with two inlet streams and one outlet stream. The CFD model for this system would compute the flow field inside of the stirred tank given the inlet flow velocities and concentrations, the geometry of the reactor (including baffles and impellers), and the angular velocity of the stirrer. For liquid-phase flow with uniform density, the CFD model for the flow field can be developed independently from the mixing model. For simplicity, we will consider this case. Nevertheless, the SGS models are easily extendable to flows with variable density. 

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. 

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