Plug-Flow Reactor with Distributed Feed

2 PLUG-FLOW REACTOR WITH DISTRIBUTED FEED 

To derive the design equation of a plug-flow reactor with a distributed feed, we write a species balance equation for any species, say species j, that is not injected along the reactor over reactor element dV. Since the species is not fed or withdrawn, its molar balance equation is 

We follow the same procedure as the one used in Chapter 4 to derive the reaction-based design equation of a plug-flow reactor and obtain 

2 is the differential design equation for a plug-flow reactor with a distributed feed, written for the wth-independent chemical reaction. Note that it is identical to the design equation of a plug-flow reactor. The only difference between the two is the way the volumetric flow rate and the species concentrations vary along the reactor. 

To reduce the design equation to a dimensionless form, we select a reference stream that combines the inlet steam and the injection stream. Hence, its total volumetric flow rate is 

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Figure 9.2 Plug-flow reactor with distributed feed.

PLUG-FLOW REACTOR WITH DISTRIBUTED FEED 405 From overall material balance. 

Example 9.3 Valuable product V is produced in a plug-flow reactor with a distributed feed. The following simultaneous, gas-phase chemical reactions take place in the reactor ... 

For adiabatic plug-flow reactor, we solve (r), (s), and (u) simultaneously, sub-jeet to the initial conditions that Zi(0) = Z2(0) = 0, 0(0) = 1. Figure E9.3.3 shows a comparison of the V production between the adiabatic distributed-feed reactor and the adiabatic plug flow reactor with = 300°C. Figure E9.3.4 compares the production of V. 

Unlike the situation in a plug flow reactor, the various fluid elements mix with one another in a CSTR. In the limit of perfect mixing, a tracer molecule that enters at the reactor inlet has equal probability of being anywhere in the vessel after an infinitesimally small time increment. Thus, all fluid elements in the reactor have equal probability of leaving in the next time increment. Consequently, there will be a broad distribution of residence times for various tracer molecules. The character of the distribution is discussed in Section 11.1. Because some of the molecules have short residence times, there is a rapid response at the reactor outlet to changes in the reactor feed stream. This characteristic facilitates automatic control of the reactor. 

This is a partial differential equation, as we should expect from a plug-flow tubular reactor with a single reaction. We note in passing that the solution requires the specification of an initial distribution and a boundary, or feed, value. These are both functions (the first of z because t = 0 the second of t because z = 0) in the distributed system. Of the corresponding quantities, c0 and cin, in the lumped system, the latter is embodied in the ordinary differential equation itself and the former is the initial value. 

The idealized plug-flow and bateh reactors are the only two classes of reaetors in which all the atoms in the reaetors have the same residenee time. In all other reactor types, the various atoms in the feed spend different times inside the reactor that is, there is a distribution of residenee times of the material within the reactor. For example, eonsider the CSTR the feed introduced into a CSTR at any given time beeomes eompletely mixed with the material already in the reactor. In other words, some of the atoms entering the CSTR leave it almost immediately, because material is being continuously withdrawn from the reactor other atoms remain in the reactor almost forever because all the material is never removed from the reactor at one time. Many of the atoms, of eourse, leave the reactor after spending a period of time somewhere in the vieinity of the mean residence time. In any reactor, the distribution of residence times ean significantly affect its performance. 

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