Effectiveness factor approach

Figure 12.2 is a plot of the effectiveness factor r] versus the Thiele modulus hT. For low values of hT (slow reaction, rapid diffusion), the effectiveness factor approaches unity. For values of the Thiele modulus above 2.0, tanh hT 1 and the effectiveness factor may be approximated by... 

This relation is plotted as curve Bin Figure 12.11. Smith (66) has shown that the same limiting forms for are observed using the concept of effective dififusivities and spherical catalyst pellets. Curve B indicates that, for fast reactions on catalyst surfaces where the poisoned sites are uniformly distributed over the pore surface, the apparent activity of the catalyst declines much less rapidly than for the case where catalyst effectiveness factors approach unity. Under these circumstances, the catalyst effectiveness factors are considerably less than unity, and the effects of the portion of the poison adsorbed near the closed end of the pore are not as apparent as in the earlier case for small hr. With poisoning, the Thiele modulus hp decreases, and the reaction merely penetrates deeper into the pore. 

To assess whether a reaction is influenced by intraparticle diffusion effects, Weisz and Prater [11] developed a criterion for isothermal reactions based upon the observation that the effectiveness factor approaches unity when the generalised Thiele modulus is of the order of unity. It has been shown that the effectiveness factor for all catalyst geometries and reaction orders (except zero order) tends to unity when... 

Values of effectiveness factors in washcoat layers with non-uniform thickness around the channel perimeter have been studied by Hayes et al. (2005). However, the applicability of (even the generalized) effectiveness factor approach is quite limited in complex systems with competing reactions, surface deposition of reaction components, non-linear rate laws and under transient operating conditions (e.g. periodically operated NSRC). Typically, the effectiveness factor method can be used for more accurate prediction of CO, H2 and HC oxidation light-off and conversions in DOC. 

In assessing whether a reactor is influenced by intraparticle mass transfer effects WeiSZ and Prater 24 developed a criterion for isothermal reactions based upon the observation that the effectiveness factor approaches unity when the generalised Thiele modulus is of the order of unity. It has been showneffectiveness factor for all catalyst geometries and reaction orders (except zero order) tends to unity when the generalised Thiele modulus falls below a value of one. Since tj is about unity when 0 < ll for zero-order reactions, a quite general criterion for diffusion control of simple isothermal reactions not affected by product inhibition is < 1. Since the Thiele modulus (see equation 3.19) contains the specific rate constant for chemical reaction, which is often unknown, a more useful criterion is obtained by substituting l v/CAm (for a first-order reaction) for k to give ... 

Figure 5 shows the dependence of the effectiveness factor on the Thiele modulus for the different pellet shapes. At small values of 4> the effectiveness factor approaches unity in all cases. Here, the chemical reaction constitutes the rate determining step—the corresponding concentration profiles over the pellet cross-section arc flat (sec Fig. 4). This situation may occur at low catalyst activity (k is small), large pore size and high porosity (Dc is large), and/or small catalyst pellets (R is small, i.c. in fluidized bed reactors R is typically around 50 /im).

First to be considered are the limiting forms of this dependence. In the kinetic regime, that is without any diffusion limitations (2 1, and hence usually An0 1 and Anx 1), the effectiveness factor approaches unity and the mean reaction rate according to Equation 8.1 is proportional to the specific surface area ... 

It aliows heuseof ryjujejsaUlyjI.garljsks, which can give an effectiveness factor approaching unity. This is especially important if diffusion limitations cause rapid catalytic deactivation or poorer selectivity. 

We observe that as the particle diameter becomes very small. decreases, so that the effectiveness factor approaches 1 and the reaction is surface-reaction-limited. On the other hand, when the Thiele modulus < ) is large (- 30), the internal effectiveness factor v) is small (i.e., < 1), and the reaction is diffu-... 

For complex reactions involving many species, we cannot use the simple Thiele modulus and effectiveness factor approach, and must solve numerically the complete reaction-diffusion problem. These problems are challenging because of the steep pellet profiles that are possible. 

Finally, we showed several ways to couple the mass and energy balances over the fluid flowing through a fixed-bed reactor to the balances within the pellet. For simple reaction mechanisms, we were still able to use the effectiveness factor approach to solve the fixed-bed reactor problem. For complex mechanisms, we solved numerically the full problem given in Equations 7.84-7.97, We solved the reaction-diffusion problem in the pellet coupled to the mass and energy balances for the fluid, and we used the Ergun equation to calculate the pressure in the fluid. 

Solve the model exactly and compare the numerical solution to the solution obtained with the approximate Thiele modulus and effectiveness factor approaches shown in Figure 7.26. 

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. ... 

It is rather straightforward to employ numerical methods and demonstrate that the effectiveness factor approaches unity in the reaction-rate-controlled regime, where A approaches zero. Analytical proof of this claim for first-order irreversible chemical kinetics in spherical catalysts requires algebraic manipulation of equation (20-57) and three applications of rHopital s rule to verify this universal trend for isothermal conditions in catalytic pellets of any shape. 

Since tanh

sufficiently large Thiele moduli, it follows from Eq. 29 that the effectiveness factor approaches the reciprocal value of the Thiele modulus. In the classical Thiele concept, this leads to the well-known reduction in the apparent activation energy for transport-controlled reactions, since the effective reactivity is now proportional to Vk rather than to fc, as in the absence of any transport limitation. 

The following figure (Fig. 1.6) shows the log-log plot of 17 versus the dimensionless group mLj. We note that the effectiveness factor approaches unity when mL2 is much less than unity and it behaves like /mL2 as mL2 is very large. 

Effectiveness factor approach (rj-approach) accounts for diffusion limitations in the washcoat. rj-approach is based on the assumption that one target species determines overall reactivity (Deutschmann, 2008). An effectiveness factor for a first-order reaction is calculated for the chosen species based on the dimensionless Thiele modulus (< ) (Hayes et al., 2007, 2012), and all reaction rates are multipfied by this factor at the species governing equation at the gas-surface interface. is calculated as... 

The inlet conditions for the numerical simulations are based on the experimental conditions. The simulations are performed with the three different models for internal diffusion as given in Section 2.3 to analyze the effect of internal mass transfer limitations on the system. The thickness (100 pm), mean pore diameter, tortuosity (t = 3), and porosity ( = 60%) of the washcoat are the parameters that are used in the effectiveness factor approach and the reaction-diffusion equations. The values for these parameters are derived from the characterization of the catalyst. The mean pore diameter, which is assumed to be 10 nm, hes in the mesapore range given in the ht-erature (Hayes et al., 2000 Zapf et al., 2003). CO is chosen as the rate-limiting species for the rj-approach simulations, rj-approach simulations are also performed with considering O2 as the rate-hmiting species. 

Glaoguen F, Durand R. Simulations of PEFC cathodes an effectiveness factor approach. J Appl Electrochem 1997 27 1029-35. 

From the standpoint of a high effective reaction rate, it is desirable to use very fine particles as we then have a pore effectiveness factor approaching unity. However, we have to consider the pressure drop in a technical reactor (Figure 6.1.8). 

Effectiveness factor approaches Macrohomogeneousmodel of ionomer-bound CL Structural (percolation) model of ionomer-bound CL Structural model coupled with water balance in pores Thin-film morphology of ionomer in CL Hierarchical Model, coupling of meso-and macroscale... 

Simple pore models (Srinivasan et al., 1967), thin-film models (Srinivasan and Hurwitz, 1967), macrohomogeneous models, and refined variants of agglomerate models are still being applied and further developed for the present generation of ionomer-bound composite catalyst layers in PEFCs (Gloaguen and Durand, 1997 Jaouen et al., 2002 Karan, 2007 Kulikovsky, 2002a, 2010b Sun et al., 2005). Effectiveness factor approaches have been elaborated as quantitative tools to compare... 

Gloaguen, F. and Durand, R. 1997. Simulations of PEFC cathodes An effectiveness factor approach. J. Ami. Electrochem.. 27(9). 1029-1035. 

A property of the effectiveness factor is that the product of the Thiele modulus and effectiveness factor approaches unity as the value of the Thiele modulus approaches infinity. In this asymptotic region of strong diffusion effects, the pellet center concentration approaches zero. Therefore, a generalized Thiele modulus can be defined as ... 

Thus, the intrinsic rate is reduced by a factor of l/ when diffusion limitations are severe. Properties of the effectiveness factor emerge from Eqs. 4.5 and 4.6 the effectiveness factor approaches unity when is small (diffusion-free reaction) and the product of 17 and approaches unity when is large (diffusion-limited reaction). It is clear from the definition of the Thiele modulus that a higher intrinsic rate of reaction, a larger pellet, or a lower effective diffusivity results in a higher Thiele modulus and thus a relatively lower global rate. 

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