Plug flow design with axial dispersion

The approach of reactor point effectiveness, however, does require some approximations. The adequacy of these approximations will be examined in detail. This will be followed by a section on the equivalence between the plug-flow model and the axial dispersion model, and that between the plug-flow and the radial dispersion model. With these equivalences established, it is possible to concentrate on the plug-flow model. Using this model, the design and analysis problem for reactions affected by diffusion, those affected by both diffusion and chemical deactivation, and those affected by catalyst sintering will be treated in detail. Detailed design and analysis procedures result from this treatment. 

It has been shown in Section 10.3 that the solution of the plug-flow model leads directly to that of the one-dimensional model with axial dispersion. These results can be applied to reactions affected by diffusion. Use of the results leads to the design equations for the fixed-beds with axial dispersion given in Table 10.3. Note that Co is the solution for the plug-flow model and that Eq. (E) in Table 10.3 is used for kb in Eq. (B), whereas Eq. (G) is in Eq. (F) (see Problem 10.9 for the case of = 0). 

In most adsorption processes the adsorbent is contacted with fluid in a packed bed. An understanding of the dynamic behavior of such systems is therefore needed for rational process design and optimization. What is required is a mathematical model which allows the effluent concentration to be predicted for any defined change in the feed concentration or flow rate to the bed. The flow pattern can generally be represented adequately by the axial dispersed plug-flow model, according to which a mass balance for an element of the column yields, for the basic differential equation governing llie dynamic behavior,... 

In Chapter 2, the design of the so-called ideal reactors was discussed. The reactor ideahty was based on defined hydrodynamic behavior. We had assumedtwo flow patterns plug flow (piston type) where axial dispersion is excluded and completely mixed flow achieved in ideal stirred tank reactors. These flow patterns are often used for reactor design because the mass and heat balances are relatively simple to treat. But real equipment often deviates from that of the ideal flow pattern. In tubular reactors radial velocity and concentration profiles may develop in laminar flow. In turbulent flow, velocity fluctuations can lead to an axial dispersion. In catalytic packed bed reactors, irregular flow with the formation of channels may occur while stagnant fluid zones (dead zones) may develop in other parts of the reactor. Incompletely mixed zones and thus inhomogeneity can also be observed in CSTR, especially in the cases of viscous media. 

It has been shown that the solutions of the plug-flow model lead directly to near-accurate solutions of the axial dispersion model and that the plug-flow model is equivalent to the radial dispersion model with some simplifying assumptions. Radial gradients usually exist when there is a significant heat effect (Chapter 9). For highly exothermic reactions, the usual design practice is to select a small... 

The simplest packed bed design arises with a single dilute adsorbate in a carrier fluid when it can be assumed that the process is isothermal, that there is plug flow, and that there are no mass transfer resistances. In such a situation, instantaneous equilibrium exists at all points in the system. Without the axial dispersion term and taking the velocity outside the partial differential term for the convective flow, equation (6.19) is simplifled to ... 

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