BED model

Analytical determination of the hydraulic resistance of the medium is difficult. However, for the simplest filter medium structures, certain empirical relationships are available to estimate hydraulic resistance. The relationship of hydraulic resistance of a cloth of monofilament fiber versus fiber diameter and cloth porosity can be based on a fixed-bed model. 

Most biochemical reactors operate with dilute reactants so that they are nearly isothermal. This means that the packed-bed model of Section 9.1 is equivalent to piston flow. The axial dispersion model of Section 9.3 can be applied, but the correction to piston flow is usually small and requires a numerical solution if Michaehs-Menten kinetics are assumed. 

Figure 3 shows calibration plots of log (particle diameter) vs. elution voliame difference (AV) between marker and particle using three different monodisperse latexes at a low eluant ionic strength of 1.29 mM SLS. These results illustrate the featiire of universal calibration behavior predicted by the capillary bed model as mentioned earlier. Of note also is the fact that the curve deviates from linearity for the 38 nm particle and begins to approach the origin as also indicated by the model calculations. 

Illustration 12.7 indicates how to estimate an effective thermal conductivity for use with two-dimensional, pseudo homogeneous packed bed models. 

Fitzgerald, T. J., Bushnell, D., Crane, S., and Shieh, Y., Testing of Cold Scaled Bed Modeling for Fluidized-bed Combustors, Powder Technol., 38 107 (1984)... 

The most important parts of creating a segment model are the application of the physical boundary conditions and the positioning of the internals to allow for the symmetry and periodic boundary conditions. Without properly applying boundary conditions the simulation results cannot be compared to full-bed results, both as a concept and as a validation, since the segment now is not really a part of a continuous geometry. Our approach was to apply symmetry boundaries on the side planes parallel to the main flow direction, thereby mimicking the circumferential continuation of the bed, and translational periodic boundaries on the axial planes, as was done in the full-bed model. 

When simulations are done in a WS model, the results need to be validated against a full-bed model. The main reason for this is not only to see if the WS model results are representative for a full bed but also to check that the symmetry boundaries, which are relatively close to all parts of the segment model, do not influence the solution. 

A section in the full-bed models was isolated that was comparable to the WS model. The layout of these different sections was identical, except that the WS model had a two-layer periodicity and the full-bed models had a six-layer periodicity. To be able to make direct comparisons of velocity profiles, several sample-points needed to be defined. In the three different models seven tangential planes were defined and on each plane three axial positions were defined. This reduced the data to single radial velocity profiles at corresponding positions in all three models, as shown in Fig. 10, for the WS model. Identical planes were defined in the full-bed models. Some spheres and sample planes 4 and 5 are not displayed to improve the visibility of the sample planes and lines. In the right-hand part of the figure, plane 4 is shown with the axial positions at which data were taken and compared. 

The velocity profile plots show interruptions in the velocity profile, where the solid packing was located. In general, the data of the three different cases agreed very well qualitatively velocity highs and lows are shown at the same points in the bed. Quantitatively, the data of the two full-bed models are practically identical, indicating that the solutions were completely mesh independent. 

The data from the WS model in some cases deviated slightly from the full-bed models. This could be explained by the slightly different layout of the WS model. Some spheres had to be relocated in the WS model to create a two-layer periodicity from the six-layer periodicity in the full-bed models. The differences in velocity magnitudes were mainly found in the transition area between the wall layers and the center layers. The effect of slightly larger gaps between spheres from the nine-sphere wall layers and the three-sphere central layers, due to the sphere relocations, had a noticeable effect on the velocity profile. Differences were also found in the central layer area where the sphere positions were not identical. 

A one-parameter model, termed the bubbling-bed model, is described by Kunii and Levenspiel (1991, pp. 144-149,156-159). The one parameter is the size of bubbles. This model endeavors to account for different bubble velocities and the different flow patterns of fluid and solid that result. Compared with the two-region model, the Kunii-Levenspiel (KL) model introduces two additional regions. The model establishes expressions for the distribution of the fluidized bed and of the solid particles in the various regions. These, together with expressions for coefficients for the exchange of gas between pairs of regions, form the hydrodynamic + mass transfer basis for a reactor model. 

The rise velocity of bubbles is another important parameter in fluidized-bed models, but it can be related to bubble size (and bed diameter, D). For a single bubble, the rise... 

Figure 23.6 Bubbling-bed model representation of (a) a single bubble and (b) regions of a Auidized bed (schematic)...

In the following sections, we discuss reactor models for fine, intermediate, and large particles, based upon the Kunii-Levenspiel (KL) bubbling-bed model, restricting ourselves primarily to first-order kinetics. Performance for both simple and complex reactions is considered. Although the primary focus is on reactions within the bed, we conclude with a brief discussion of the consequences of reaction in the freeboard region and near the distributor. 

Extension of the Kunii-Levenspiel bubbling-bed model for first-order reactions to complex systems is of practical significance, since most of the processes conducted in fluidized-bed reactors involve such systems. Thus, the yield or selectivity to a desired product is a primary design issue which should be considered. As described in Chapter 5, reactions may occur in series or parallel, or a combination of both. Specific examples include the production of acrylonitrile from propylene, in which other nitriles may be formed, oxidation of butadiene and butene to produce maleic anhydride and other oxidation products, and the production of phthalic anhydride from naphthalene, in which phthalic anhydride may undergo further oxidation. 

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

Figure 23.9 Schematic representation of regions of bubbling-bed model for intermediate-sized particles...

Using the Kunii-Levenspiel bubbling-bed model of Section 23.4.1 for the fluidized-bed reactor in the SOHIO process for the production of acrylonitrile (C3H3N) by the ammoxidation... 

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