Plug flow design first-order kinetics

In Chapter 3, the analytical method of solving kinetic schemes in a batch system was considered. Generally, industrial realistic schemes are complex and obtaining analytical solutions can be very difficult. Because this is often the case for such systems as isothermal, constant volume batch reactors and semibatch systems, the designer must review an alternative to the analytical technique, namely a numerical method, to obtain a solution. For systems such as the batch, semibatch, and plug flow reactors, sets of simultaneous, first order ordinary differential equations are often necessary to obtain the required solutions. Transient situations often arise in the case of continuous flow stirred tank reactors, and the use of numerical techniques is the most convenient and appropriate method. 

For the plug-flow reactor there is a formal analogy with the batch reactor equations derived in Chapter 1 if there is no volume change on reaction. As an example, consider the PFR design equation, (4-45), written for first-order irreversible kinetics (A B). 

The approximate overall effectiveness of Eq. 7.24 above can be used for reactor design for first-order reactions. For nonlinear kinetics, however, it is difficult to obtain an analytical expression for the overall effectiveness factor that is accurate for partially wetted pellets. Even when a relatively accurate effectiveness factor is available, the presence of different bulk and interfacial concentrations in both phases reduces the utility of the overall effectiveness factor except for limiting cases. Therefore, an internal effectiveness factor applicable to both wetted and nonwetted parts of the pellet is used here instead. For the low liquid velocities being considered, only dispersion in the liquid phase need be considered, and the gas phase may be treated as though it were in plug-flow (Levee and Smith 1976). 

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