Effectiveness factors calculation

At the limit of Knudsen diffusion control it is not reasonable to expect that any of the proposed approximation methods will perform well since, as we know, percentage variations in pressure are quite large. Nevertheless it is interesting to examine their results, which are shown in Figure 11 4 At this limit it is easy to check algebraically that equations (11.54) and (11.55) become the same, while (11.60) differs from the other two. Correspondingly the values of the effectiveness factor calculated using the approximation of Kehoe and Aris coincide with the results of Apecetche et al., and with the exact solution, ile Hite and Jackson s effectiveness factors differ substantially. 

If pore diffusion is controlUng, we repeat the effectiveness factor calculations in Chapter 10. Equation (10.29) has the form of Equation (11.48), and it includes both film resistance and pore diffusion. 

Effectiveness Factors Calculated with the Simulation Program Presto Kinetics (Lp0re = 6.75-10-4 m -+... 

The effectiveness factors calculated by the Thiele modulus as well as the findings obtained from the simulation are shown in Table 12.4. 

Under excess of the second reactant (in automobile exhaust gas typically H20, C02 and for lean-burn engines exhaust specifically also 02), the effectiveness factor calculation can be simplified by approximating the reaction rate Rj by a pseudo-first-order rate law with respect to the component using new rate constant kiefj (evaluated from the original rate law)... 

The effectiveness factors calculated for the parallel, equilibrium-restrained reaction systems are not able to predict the catalyst internal-surface utilization accurately. Therefore, the intraparticle distributions of the temperature, the concentrations of species and so on should be taken into account. 

The effectiveness factors calculated in this study are under the experimental conditions utilized in this study and give an idea of the magnitude of pore diffusion problem in the case of the Nalcomo 474 catalyst when Synthoil liquid is processed. On the other hand, the Monolith catalyst shows promise in this regard and warrants further investigation regarding its activity under different compositions of the catalyst and different reactor operating conditions. 

Effectiveness factors calculated in this way are of the order of 0.02-0.04 for both reactions. Van Hook (1 9) and Rostrup-Nielsen (20) published values which are of the same order of magnitude, but the latter were obtained assuming first order kinetics for the methane conversion and equilibrium for the watergas shift. The corresponding reaction layer amounts to 0.65-1.3 10 A, and to a surface area still exceeding Sp by a factor IOOO. 

This tutorial paper begins with a short introduction to multicomponent mass transport in porous media. A theoretical development for application to single and multiple reaction systems is presented. Two example problems are solved. The first example is an effectiveness factor calculation for the water-gas shift reaction over a chromia-promoted iron oxide catalyst. The methods applicable to multiple reaction problems are illustrated by solving a steam reformer problem. The need to develop asymptotic methods for application to multiple reaction problems is apparent in this example. 

A simplification is often employed for effectiveness factor calculations in the asymptotic limit of strong intraparticle diffusion resistance (12,13). In this situation, an alternative form of the key component mass balance can be written as follows ... 

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. 

The results for Na versus reactor length using 25 collocation points for the pellet are shown in Figure AJ. Also shown are the simplified effectiveness factor calculations for this problem from Example 7.5. A magnified view is shown in Figure A.8. Notice the effectiveness factor approach gives a good approximation for the bed performance. It is not exact because the reaction is second order. ... 

Notice that the geometric factor a is unity in the effectiveness factor calculation given by equation (18-17) for all catalyst geometries when the characteristic length L is V cataiyst/6 extemai- Specifically for spheres, the dimensionless independent spatial variable rj ranges from 10 near the center of the catalyst to 3 at the external surface. 

Effectiveness factor calculations summarized in Tables 19-1 to 19-5 are consistent with Langmuir-Hinshelwood kinetics, as discussed in this chapter. E is larger and approaches 1 asymptotically in the reaction-controlled regime where the intrapellet Damkohler number is small, and E decreases in the diffusion-controlled regime at large values of A a- These trends are verified by simulations provided in Table 19-1. 

However, the void area fraction is equivalent to the void volume fraction, based on equation (21-76) and the definition of intrapellet porosity Sp at the bottom of p. 555. Effectiveness factor calculations in catalytic pellets require an analysis of one-dimensional pseudo-homogeneous diffusion and chemical reaction in a coordinate system that exploits the symmetry of the macroscopic boundary of a single pellet. For catalysts with rectangular symmetry as described above, one needs an expression for the average diffusional flux of reactants in the thinnest dimension, which corresponds to the x direction. Hence, the quantity of interest at the local level of description is which represents the local... 

As illustrated in Sections 30-1 and 30-2, all intrapellet resistances can be expressed in terms of f-A, surface a, intrapeiiet and Ea mtrapeiiet approaches zero near the central core of the catalyst when the intrapellet Damkohler number is very large. For small values of the intrapellet Damkohler number, effectiveness factor calculations within an isolated pellet allow one to predict Ca, intrapeUet in terms of CA,sur ce via the dimensionless molar density profile. All external transport resistances can be expressed in terms of Ca, buit gas — Ca, surface, and integration of the plug-flow mass balance allows one to calculate the bulk gas-phase concentration of reactant A. The critical step involves determination of Ca, surface via effectiveness factor formalism. Finally, a complete reactor design strategy is... 

FIGURE 23.24 Effect of bed entrance pressure upon the conversion of diethyl sulphide in a fixed bed reactor. Lines represent fixed bed reactor simulations with catalyst effectiveness factor calculated from the random pore network model, including the influence of capillary condensation. 

Gottifredi JC, Gonzo EE. On the effectiveness factor calculation for a reaction—diffusion process in an immobilized biocatalyst pellet. Biochemical Engineering Journal 2005 24 235-242. 

Effectiveness factor calculation methods in catalytic washcoats... 

Table 8.7 Accuracy of effectiveness factor calculation methods for nonuniform geometry (circle-in-square shape with a first-order reaction).

Papadias D, Edsberg L, Bjdrnbom P. Simplified method of effectiveness factor calculations for irr ular geometries of washcoats a general case in a 3D concentration field. Catalysis Today 2000 60 11-20. 

Papadias D, Edsberg L, Bjombom P. Simplified method for effectiveness factor calculations in irregular geometries of washcoats. Chemical Engineering Science 2000 55 1447-1459. 

Values of the intraparticle effectiveness factors, calculated according to Eq. (6.69), and of maximum temperature gradients within the catalyst particles for the two beds of Fig. 6.2 are reported in Table 6.8. 

Fig. 10.16 Effectiveness factor for immobilized enzyme catalysts with Michaelis-Menten kinetics. (From DJ. Fink, T. Na, J.S. Schulte Effectiveness factor calculations for immobilized enzyme catalysts. Biotechnology and Bioengineering, 15 (1973) 879-888. Copyright 1973 Wiley).

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