Effectiveness factor definition

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

Effective core potential, 269 Effective double layer characterization of, 189 isotherm, 306, 315 kinetic expressions, 316 observations of with STM, 259 stability of, 225, 351, 503 Effectiveness factor of promotion computation of, 505 definition of, 505 Electrocatalysis... 

Using this definition of the Thiele modulus, the reaction rate measurements for finely divided catalyst particles noted below, and the additional property values cited below, determine the effectiveness factor for 0.5 in. spherical catalyst pellets fabricated from these particles. Comment on the reasons for the discrepancy between the calculated value of rj and the ratio of the observed rate for 0.5 in. pellets to that for fine particles. 

The definition of the particle effectiveness factor 77 involves the intrinsic rate of reaction, ( rA)int> for reaction A - products, at the exterior surface conditions of gas-phase concentration (cAs) and temperature (Ts). Thus, from equation 8.55,... 

The ratio of these rates (actual rate/ideal rate) is the fraction by which the rate is reduced by pore diffusion limitations, which we call rj, and this is the definition of the effectiveness factor. 

The reaction rate expressed in terms of surface concentrations provides the relationship between Cs and CL. From the definition of the effectiveness factor, we may express the required equality of mass transfer and reaction rates as... 

From the definition of the effectiveness factor and for a fust-order reaction with respect to A... 

This study employed conventional diffusion-reaction theory, showing that with diffusion-limited reactions the internal effectiveness factor of a heterogeneous catalyst is inversely related to the Thiele modulus. Using a standard definition of the Thiele modulus [100], the observed reaction rate of an immobilized-enzyme reaction will vary with the square root of the immobilized-enzyme concentration in a diffusion-limited system. In this case, a plot of the reaction rate versus the enzyme loading in the catalyst formulation will be nonlinear. 

According to the definition, the effectiveness factor E of the porous electrode is given by... 

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

Comparing eqs 56 and 27, and recalling the definition of the effectiveness factor according to cq 40, yields the following simple relationship between the Thiele modulus and the Weisz modulus ... 

The maximum effective reaction rate is obtained for the limiting value of cs = 0. This means that the product of the effectiveness factor and the second Damkohler number can never exceed unity. A comparison of the definition of the Weisz modulus (eq 56) with the definition of Dan (eq 78) gives the equivalence... 

Combining eq 108 with the definition of the effectiveness factor (eq 40), the following expression for the effective reaction rate is obtained ... 

Ca relates the concentration difference over the film to procurable quantities and is therefore a so-called observable [4]. A criterion for the absence of extra-particle gradients in the rate data can be derived from the definition of an effectiveness factor for a particle. This should not deviate more than 5% from unity ... 

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

Figure 12 shows the effectiveness factor as a function of the Wheeler-Weisz modulus for different reaction orders, indicating that criterion (33) holds for the generalized Thiele modulus. Due to the definition of L it is fairly independent of the catalyst geometry. 

This equation, when is used in the definition of the effectiveness factor based on the fluid bulk concentration a leads to the combined resistance of fluid and particle... 

In the definitions of >7 the effectiveness factor of a reaction has been mentioned. It should be realized that for a simple reaction all components must have the same effectiveness factor, which therefore can be called the effectiveness factor of the reaction. For example, for the reaction of Equation 6.21 ... 

Then by matter of definition in the low rj region, the effectiveness factor ij must equal... 

An0 can only have positive values, whereas An, can have any real value, thus it may also become negative. This follows directly from the definitions of An0 and An,. Since the effectiveness factor cannot become imaginary, An0 cannot become negative (Equation 6.23). Since the effectiveness factor can have values both larger than one and smaller than one, it follows that An, can have both negative and positive values respectively (Equation 6.24). From Equation 6.38 we see that An, will become negative if... 

Then, from the definition of An, it follows that effectiveness factors larger than one can be expected if criterion 6.52 is adhered to. 

Since we did not find adverse effects on zinc balance by the higher fiber diet containing spinach until the fourth week of study 3, balance studies should be carried out for longer periods of time to determine if negative zinc balances persist. Perhaps in the earlier studies on effects of spinach on calcium balance of human subjects, more definitive effects would have been noted if the spinach had been fed along with a higher fiber diet. Length of study period is also a likely factor, as some of the studies were carried out for less than 2 weeks. 

According to the above definitions, the effectiveness factor for any of the above shapes can adequately describe simultaneous reaction and diffusion in a catalyst particle. The equation for the effectiveness factor in a slab is the simplest in Table 6.3.1 and will be used for all pellet shapes with the appropriate Thiele modulus ... 

Since the concentration profile is determined by Equation (6.4.32), evaluation of the overall effectiveness factor, rj , is straightforward. By definition, is the observed rate divided by the rate that would be observed at conditions found in the bulk fluid. Recall... 

We will now proceed to relate this measured reaction order n to the tme reaction order n. Using the definition of the effectiveness factor, note that the actual rate is the product of t and the rate of reaction, evaluated at the external surface, k, S Cl ... 

It has been demonstrated by numerical simulations [9] that, with this definition, eq. 2 provides a reasonable order-of-magnitude estimate of the effectiveness factor also in the case of single-file diffusion. While in the case of ordinary diffnsion the intracrystalline mean life time may be easily correlated with the crystal size and the internal mobility [11], similar analytical expressions for single-file diffusion have not been established. The rule-of-thumb given in Ref. [10] on the basis of a few first numerical simulations turned out to be of rather limited validity in recent more refined considerations [12]. 

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