Dispersed plug flow model with first order reaction

Dispersed Plug-Flow Model with First-Order Chemical Reaction... 

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

We will consider a dispersed plug-flow reactor in which a homogeneous irreversible first order reaction takes place, the rate equation being 2ft = k, C. The reaction is assumed to be confined to the reaction vessel itself, i.e. it does not occur in the feed and outlet pipes. The temperature, pressure and density of the reaction mixture will be considered uniform throughout. We will also assume that the flow is steady and that sufficient time has elapsed for conditions in the reactor to have reached a steady state. This means that in the general equation for the dispersed plug-flow model (equation 2.13) there is no change in concentration with time i.e. dC/dt = 0. The equation then becomes an ordinary rather than a partial differential equation and, for a reaction of the first order ... 

FIG. 19-2 Chemical conversion by the dispersion model, (a) Volume relative to plug flow against residual concentration ratio for a first-order reaction. (b) Residual concentration ratio against kCQt for a second-order reaction, (c) Concentration profile at the inlet of a closed-ends vessel with dispersion for a second-order reaction with kC01 = 5. 

Note that this equation has no physical significance It is only for first order reactions that these two models pr ict the same conversion for the same mean residence time. There is, however, an important physical difference between the two models in the cascade model there is no bacl xing from reactor number N to reactor number N-1, whereas in the model for plug flow with axial dispersion there is only one discontinuity, that is at the reactor entrance. 

FIGURE 4.33 Reactor size predicted by an axial dispersion model compared with the size predicted by a plug flow model. First-order reaction, — ta = aca-... 

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