Effectiveness factor generalized approximations

The generalized modulus defined in Equation (6.3.51) has been normalized so that the effectiveness factor is approximately 1/< q large values of Q, as illustrated in Figure 6.3.9. 

This example illustrates calculation of the rate of a surface reaction from an intrinsic-rate law of the LH type in conjunction with determination of the effectiveness factor (rj) from the generalized Thiele modulus (G) and Figure 8.11 as an approximate representation of the 7]-Q relation. We first determine G, then 17, and finally (—rA)obs. 

Equations (14-37) and (14-42) represent two different ways of obtaining an effective factor, and a value of Ae obtained by taking the reciprocal of S, from Eq. (14-42) will not check exactly with a value of A, derived by substituting At = 1/Si and A2 = 1/S2 into Eq. (14-37). Regardless of this fact, the equations generally give reasonable results for approximate design calculations. 

Unlike the Thiele modulus related to a reaction order n for porous catalyst pallets, co and co will maintain keep their form in this nonlinear case, due to the dimensionless concentrations used in the general B-V equations (Eqs. 41 and 84). The approximate general solutions of Eq. (18) can be used for Eqs. (86) and (87) as long as the reaction order n and Thiele modulus are replaced by q and co or co respectively. The expressions for the concentration A and the effectiveness factor for a r/th-ordcr reaction by using m term approximating can be obtained from Eqs. (25) and (26). 

So far the generalized modulus of Bischoff has only been applied for a single reaction. Since two reactions are occurring simultaneously, each with their effectiveness factor, (18) can only be conveniently applied when a simple relationship exists between dCj and dC2. The experimental program led to a fairly constant ratio of CO/COj over a rather broad range of total conversion, confirming previous observations by Akers and Camp (18). Therefore, the ratio of dCj/dC2 was approximated by (cf-cf T/ (cl-C ), which is the more accurate the more equilibrium is approached. 

When the effective reaction rate is controlled by pore diffusion, then the asymptotic solution of the catalyst effectiveness factor as a function of the generalized Thiele modulus can be utilized (cq 108). This (approximate) relationship has been derived in Section 6.2.3.1. It is valid for arbitrary order of reaction and arbitrary pellet shape. 

For rather general kinetics the effectiveness factor definition can be approximated as... 

These formulae, together with the generalized approximation 6.59, enable the effectiveness factor for nondiluted bimolecular reactions to be calculated. 

Since An < 0, approximation 6.59 cannot be used. To calculate the effectiveness factor exactly involves solving partial differential equations, which is very time consuming. The effectiveness factor is therefore estimated as follows construct an infinite slab in such a way, that for an exothermic zeroth-order reaction, it has the same Aris numbers as given above. Since the Aris numbers are generalized the hollow cylinder under consideration and the constructed slab will have almost the same effectiveness factor. Calculation of the effectiveness factor for a slab is relatively easy. Hence an estimate for the effectiveness factor for the hollow cylinder is obtained relatively easily. 

On the other hand, approximate calculations of ti can also be used [85].In this way, cumbersome solutions of the differential equations are not required if one merely wishes to obtain a general idea of substrate-related diffusional restrictions. A profound insight into effectiveness factors is given by Kasche [86] where a good correlation between calculated and experimental data is demonstrated. 

Since b values for simple electron-transfer-controlled processes are approximately of the correct magnitude at 298 K, taking P — 0.5, it is clear that the temperature factor in the experimental behavior must be entering the electrochemical Arrhenius expression in more or less the conventional way, i.e., as a (kT) term. However, since b is often found to be independent of 7, it is clear that there must be another compensating temperature-dependent effect, namely an approximately linear dependence of a or j8 on temperature in the Tafel slope, b = RT/a T)F. The experimental results for a variety of reactions, summarized in Section III, show that this is a general effect. Reduction of C2H5NO2 is an exception while reduction of other nitro compounds takes place with substantial potential dependence of a ... 

Cylindrical catalyst pellets can be prepared by extrusion or by pelletization. For short, stubby cylinders L/dp = 1), the effectiveness factor equation for a sphere is generally used with R equal to the cylinder radius. (The surface-to-volume ratio for the cylinder is i /3, the same as for the sphere, though the cylinder has 30% more surface area than a sphere of the same volume.) For long cylinders, the effectiveness factor is less than for a sphere of the same radius, since the surface-to-volume ratio is lower. Numerical solutions are available for different L/dp ratios, but the solution for spheres can be used for approximate values of if i in the modulus is replaced with. 2R for L/d = 2, 1.33 R for L/d = 4, and 1.5i for a very long cylinder. 

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