Reactor point effectiveness

The reactor point effectiveness of Eq. 4.83 is in a form suitable for direct inclusion in reactor conservation equations. However, the pellet center concentration Q has to be specified. As discussed earlier, C can be set to zero when G 3. For G less than 3, Q can be calculated in principle using Eq. 4.74, but this involves trial and error procedures, which are cumbersome. Instead, an approximate solution for Q can be based on the fact that the value of the integral / in Eq. 4.84 is rather insensitive to errors in the estimated value of Q. If we let AQ be the error in C, the corresponding maximum relative error in the integral I for a second-order reaction, for instance, is 3 AQC /Cg. A 30% error in Q will result in a maximum error of less than 10% for = 1. 

For reactor design purposes, these equations have to be solved along with the pellet conservation equations 4.27 through 4.29 and the relationships at the pellet-bulk fluid interface (Eqs. 4.30 and 4.31). With the use of reactor point effectiveness, however, solution of these equations can be avoided. The global rate is ... 

So far, only a single reaction has been considered. While the reactor point effectiveness cannot be expressed explicitly for a reversible reaction, the internal effectiveness factor can readily be obtained analytically using the generalized modulus (see Problem 4.23). For complex multiple reactions, however, it is not possible to obtain analytical expressions for the global rates and one has to solve the conservation equations numerically. The numerical solution of nonlinear, coupled diffusion equations with split boundary conditions is by no means trivial and often presents convergence difficulties. In this section, the same approach is taken as was used for the reactor point effectiveness. This enables the global rates to be obtained in a straightforward manner and the diffusion equations to be solved as an initial value problem (Akella 1983). 

Much more rigorous tests of the importance of diffusional effects can be made if the intrinsic kinetics are known on the basis of the generalized internal effectiveness factor and reactor point effectiveness. Using the rate expression of Eq. 4.2 in Eq. 4.73 for 4>c 8 =... 

Since the fraction of catalyst deactivated is uniform throughout the pellet, the reactor point effectiveness is simply that for a diffusion limited reaction (Eq. 4.87) with k replaced by the effective rate constant k(l — y)-. 

Table 5.3 Global Rates and Reactor Point Effectiveness...

Now that the kinetics are known for the change of exposed surface area due to sintering, the results of Section 6-2 can be used to arrive at the reactor point effectiveness. The overall pdlet effectiveness derived earlier can be used for this purpose. However, the term Sr/So has to be expressed in terms of known quantities before it can be used for the reactor point effectiveness. 

The reactor point effectiveness can now be readily determined from the overall pellet effectiveness developed earlier together with the known time dependence of the change of active surface area. Under the assumptions of negligible external mass transfer resistance and an isothermal pellet, the reactor point effectiveness is simply the pellet effectiveness multiplied by (ks/ki,), where ks and kj, are the rate constants evaluated at the pellet surface and bulk-fluid temperatures ... 

The reactor point effectivenesses thus obtained for various cases are summarized in Table 6.4. Take as an example the first entry in Table 6.4, which is for diffusion-free reactions. Since the only transport resistance to consider is in heat transfer across the pellet-bulk fluid interface, the reactor point effectiveness is simply given by ... 

In many reactions, sintering is as important as chemical deactivation. For those reactions affected by both sintering and chemical deactivation, reactor point effectivenesses have been developed for use in reactor design, which is treated in Part III. 

Derive the reactor point effectiveness given in the s x>nd row of Table 6.4 for a reaction affected by diffusion, uniform chemical deactivation, and sintering. 

In this chapter, the classical approach to the design and analysis of fixed-beds is considered. Because of the complexity involved in using the classical approach, attention here is limited to reactions affected by diffusion, and the treatment of reactions affected by simultaneous diffusion and deactivation are postponed to Chapter 10, where the concept of reactor point effectiveness is used. 

The approach of reactor point effectiveness, however, does require some approximations. The adequacy of these approximations will be examined in detail. This will be followed by a section on the equivalence between the plug-flow model and the axial dispersion model, and that between the plug-flow and the radial dispersion model. With these equivalences established, it is possible to concentrate on the plug-flow model. Using this model, the design and analysis problem for reactions affected by diffusion, those affected by both diffusion and chemical deactivation, and those affected by catalyst sintering will be treated in detail. Detailed design and analysis procedures result from this treatment. 

The simplification made possible by the use of the reactor point effectiveness allows a close examination of the design of a reactor affected by catalyst deactivation, the characteristic of which is time-dependence. This leads to the optimal design for a reactor whose performance is time-dependent. Finally, reactor design involving multiple reactions is considered, using essentially the approach of reactor point effectiveness. 

The basic approximations made in arriving at the reactor point effectiveness are (1) isothermal pellet, (2) negligible external mass transfer resistance, and (3) estimation of the pellet center concentration by a simple relationship when the reaction is not severely diffusion-limited. The first two approximations are quite adequate in view of the fact that the mass Biot number is of the order of hundreds under realistic reaction conditions. Both theoretical and experimental justifications for these approximations have been given in Chapter 4. The first approximation will be relaxed when reactions affected by pore-mouth poisoning are considered since a definite temperature gradient then exists within the pellet. An additional approximation is the representation of the difference between the Arrhenius exponentials evaluated at the pellet surface and the bulk-fluid temperatures by a linear rela-... 

The problem posed by Eqs. 10.16 through 10.18 and Eqs. 10.30 through 10.41 can be solved in principle for the analysis of a fixed-bed reactor. However, it is quite complicated and the numerical problem involved is not simple by any means, especially when it involves the design accounting for the time dependence of conversion. The reactor point effectiveness in Chapter 5 is now used to reduce this problem to a simpler one. 

Figure 10.15 Profiles of reactor point effectiveness shell-progressive deactivation.

Here, Sr and So are the active surface area and total surface area, respectively, for fresh catalyst, and the overbar denotes evaluation at the pellet surface. Without loss of generality, the functional dependence of the rate constant on surface area h(Sr/So) can be set to Sr/So for structure-insensitive reactions. This reactor point effectiveness for a certain smooth activity profile has been developed in Chapter 6 and is given in Table 6.5 along with the reactor point effectivenesses for the other cases. Combining Eqs. 10.64 and 10.65 and rearranging, there results ... 

Figure 10.25 Profiles of reactor point effectiveness for the reaction system of Table 10.9. (Reprinted from Lee and Ruckenstein 1983, by courtesy of Marcel Dekker, Inc.)...

For the reaction system given in Table 10.5, plot the reactor point effectiveness against y at the reactor inlet. 

Multiphase reactors, 431. See also Slurry reactor Trickle-bed Multiple steady state for effectiveness factor, 109 in interfacial transport (reactor point effectiveness), 123... 

It has been shown that, under realistic reaction conditions, the pellet can be considered isothermal and the external mass transfer resistance can be neglected. Under these conditions, the reactor point effectiveness can be written for the arbitrary kinetics of Eq. 4.2 as ... 

This equation together with Eq. 4.72 can be used in Eq. 4.78 to arrive at the reactor point effectiveness expressed in terms of bulk-fluid quantities ... 

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