Gradients and Profiles

By and large we can describe the results of the analysis of distributed parameter systems (i.e., flow reactors other than CSTRs) in terms of the gradients or profiles of concentration and temperature they generate. To a large extent, the analysis we shall pursue for the rest of this chapter is based on the one-dimensional axial dispersion model as used to describe both concentration and temperature fields within the nonideal reactor. The mass and energy conservation equations are coupled to each other through their mutual concern about the rate of reaction and, in fact, we can use this to simplify the mathematical formulation somewhat. Consider the adiabatic axial dispersion model in the steady state. 

The original balance equation now becomes a function of a single dependent variable. 

While all this simplifies the problem, we still need a numerical solution. 

For the steady state of a nonisothermal PFR we have, via the simplification of equations (6-97) and (6-98), and adding a heat-transfer term. 

The immediate question is whether it is possible for the PFR to demonstrate multiplicity in the same way as for the CSTR. Intuitively, we can approach this question by going all the way back to the visualization of the httle traveling batch reactors discussed in Chapter 1. Thus we can replace the PFR with an assembly of traveling batch reactors and ask whether it is possible for a batch reactor to 

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Fig. 2.3.1 A schematic diagram of GARField magnet pole pieces and the field pattern they produce together with a magnified sketch of the sample and sensor mounting showing the relative field, gradient and profile [/(r)] orientations.

Consider diffusion in the geometries shown in Figure 1.14Aand Figure 1.14B. Give a qualitative description of the concentration gradients and profiles thaf result. 

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