First order reaction, dispersed plug flow model

Fig. 28. Comparison of performance of reactors for the plug flow and dispersed plug flow models. Reaction is of first order, aA- products, and constant density, occurring in a closed vessel (L14, L15).

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. 

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

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 ... 

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. 

FIG. 23-15 Chemical conversion by the dispersion model, (a) First-order reaction, volume relative to plug flow against residual concentration ratio, (h) Second-order reaction, residual concentration ratio against kC t. 

To incorporate mixing by the dispersed plug flow mechanism into the model for the bubble column, we can make use of the equations developed in Chapter 2 for dispersed plug flow accompanied by a first-order chemical reaction. In the case of the very fast gas-liquid reaction, the reactant A is transferred and thus removed from the gas phase at a rate which is proportional to the concentration of A in the gas, i.e. as in a homogeneous first-order reaction. Applied to the two-phase bubble column for steady-state conditions, equation 2.38 becomes ... 

For pneumatic conveying all the particles are evenly dispersed in the gas. This makes contacting ideal or close to ideal. The plug flow model is thus well suited for the dilute transport reactors, but has also been used for the denser fast fluidization regime neglecting gradients in the solids distribution. For first order reactions the model can be written as ... 

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-... 

A few reactor models have recently been proposed (30-31) for prediction of integral trickle-bed reactor performance when the gaseous reactant is limiting. Common features or assumptions include i) gas-to-liquid and liquid-to-solid external mass transfer resistances are present, ii) internal particle diffusion resistance is present, iii) catalyst particles are completely externally and internally wetted, iv) gas solubility can be described by Henry s law, v) isothermal operation, vi) the axial-dispersion model can be used to describe deviations from plug-flow, and vii) the intrinsic reaction kinetics exhibit first-order behavior. A few others have used similar assumptions except were developed for nonlinear kinetics (27—28). Only in a couple of instances (7,13, 29) was incomplete external catalyst wetting accounted for. 

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