Thiele effectiveness factor

Table II.1 which depends on the pellet size, so the familiar plot of effectiveness factor versus Thiele modulus shows how t varies with pellet radius. A slightly more interesting case arises if it is desired to exhibit the variation of the effectiveness factor with pressure as the mechanism of diffusion changes from Knudsen streaming to bulk diffusion control [66,...

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

The result is shown in Figure 10, which is a plot of the dimensionless effectiveness factor as a function of the dimensionless Thiele modulus ( ), which is R.(k/Dwhere R is the radius of the catalyst particle and k is the reaction rate constant. The effectiveness factor is defined as the ratio of the rate of the reaction divided by the rate that would be observed in the absence of a mass transport influence. The effectiveness factor would be unity if the catalyst were nonporous. Therefore, the reaction rate is... 

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. 

The mass transport influence is easy to diagnose experimentally. One measures the rate at various values of the Thiele modulus the modulus is easily changed by variation of R, the particle size. Cmshing and sieving the particles provide catalyst samples for the experiments. If the rate is independent of the particle size, the effectiveness factor is unity for all of them. If the rate is inversely proportional to particle size, the effectiveness factor is less than unity and

experimental points allow triangulation on the curve of Figure 10 and estimation of Tj and ( ). It is also possible to estimate the effective diffusion coefficient and thereby to estimate Tj and ( ) from a single measurement of the rate (48). 

Diffusion effects can be expected in reactions that are very rapid. A great deal of effort has been made to shorten the diffusion path, which increases the efficiency of the catalysts. Pellets are made with all the active ingredients concentrated on a thin peripheral shell and monoliths are made with very thin washcoats containing the noble metals. In order to convert 90% of the CO from the inlet stream at a residence time of no more than 0.01 sec, one needs a first-order kinetic rate constant of about 230 sec-1. When the catalytic activity is distributed uniformly through a porous pellet of 0.15 cm radius with a diffusion coefficient of 0.01 cm2/sec, one obtains a Thiele modulus y> = 22.7. This would yield an effectiveness factor of 0.132 for a spherical geometry, and an apparent kinetic rate constant of 30.3 sec-1 (106). 

Figure 10.11. Effectiveness factor >) as a function of Thiele modulus 4> (4> = A./, for platelet, 4> = Xrc for...

The relationship between effectiveness factor p and Thiele modulus < >l may be calculated for several other regular shapes of particles, where again the characteristic dimension of the particle is defined as the ratio of its volume to its surface area. It is found that... 

The solution of this equation is in the form of a Bessel function 32. Again, the characteristic length of the cylinder may be defined as the ratio of its volume to its surface area in this case, L = rcJ2. It may be seen in Figure 10.13 that, when the effectiveness factor rj is plotted against the normalised Thiele modulus, the curve for the cylinder lies between the curves for the slab and the sphere. Furthermore, for these three particles, the effectiveness factor is not critically dependent on shape. 

Estimate the Thiele modulus and the effectiveness factor for a reactor in which the catalyst particles are ... 

Figure 11.13. Dependence of promotional effectiveness factor, r]p, on Thiele modulus 0>p and dimensionless current J.23...

The quantitative solution to the problem is given in section 11.3. The effectiveness factor T)P (< 1) which expresses the extent to which the promoting ion is fully utilized (qP=l) depends on three dimensionless parameters n, J and P n is the dimensionless dipole moment of the promoting ion, J is a dimensionless current and P, a promotional Thiele modulus, is proportional to the film thickness, L. 

II. The promotional effectiveness factor, t]p, (Chapter 11) must be significant, larger than, at least, 0.1. This requires small promotional Thiele modulus, Op, and significant dimensionless current, J, values. This implies thin (low L) catalyst films and slow kinetics of promoter destmction (low k values, Chapter 11). 

Figure 2.1 Dependence of the effectiveness factor on the Thiele modulus for a first-order irreversible reaction. Steady-state diffusion and reaction, slab model, and isothermal conditions are assumed.

Various simple equations have been available. However, each equation has its own disadvantages. For example, the equation by Wijnggaarden et al.[3] showing excellent estimates of effectiveness factors for various problems fails to provides appropriate estimates for a large Thiele modulus when f (l) is negative[2]. It is well-known that... 

Figure S.3S. Effectiveness factor e plotted as a function of the Thiele diffusion modulus Og. The effective factor is well approximated by 3/Og for Og > 10.

The internal effectiveness factor is a function of the generalized Thiele modulus (see for instance Krishna and Sie (1994), Trambouze et al. (1988), and Fogler (1986). For a first-order reaction ... 

In order for diffusional limitations to be negligible, the effectiveness factor must be close to 1, i.e. nearly complete catalyst utilization, which requires that the Thiele modulus is suffieiently small (< ca. 0.5), see Figure 3.32. Therefore, the surface-over-volume ratio must be as large as possible (particle size as small as possible) from a diffusion (and heat-transfer) point of view. There are many different catalyst shapes that have different SA/V ratios for a given size. 

Figure 3.32. Interna effectiveness factor as a function of the generalized Thiele modulus for a first order reaction.

Fig. 5.4-14. Effectiveness factor versus Thiele modulus for first-order reaction.

Microlevel. The starting point in multiphase reactor selection is the determination of the best particle size (catalyst particles, bubbles, and droplets). The size of catalyst particles should be such that utilization of the catalyst is as high as possible. A measure of catalyst utilization is the effectiveness factor q (see Sections 3.4.1 and 5.4.3) that is inversely related to the Thiele modulus (Eqn. 5.4-78). Generally, the effectiveness factor for Thiele moduli less than 0.5 are sufficiently high, exceeding 0.9. For the reaction under consideration, the particles size should be so small that these limits are met. 

Currently, benzene alkylation to produce ethylbenzene and cumene is routinely carried out using zeohtes. We performed a study comparing a zeohte Y embedded in TUD-1 to a commercial zeolite Y for ethylbenzene synthesis. Two different particle diameters (0.3 and 1.3 mm) were used for each catalyst. In Figure 41.7, the first-order rate constants were plotted versus particle diameter, which is analogous to a linear plot of effectiveness factor versus Thiele modulus. In this way, the rate constants were fitted for both catalysts. 

Quantitative analytical treatments of the effects of mass transfer and reaction within a porous structure were apparently first carried out by Thiele (20) in the United States, Dam-kohler (21) in Germany, and Zeldovitch (22) in Russia, all working independently and reporting their results between 1937 and 1939. Since these early publications, a number of different research groups have extended and further developed the analysis. Of particular note are the efforts of Wheeler (23-24), Weisz (25-28), Wicke (29-32), and Aris (33-36). In recent years, several individuals have also extended the treatment to include enzymes immobilized in porous media or within permselective membranes. The important consequence of these analyses is the development of a technique that can be used to analyze quantitatively the factors that determine the effectiveness with which the surface area of a porous catalyst is used. For this purpose we define an effectiveness factor rj for a catalyst particle as... 

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... 

Curve B of Figure 12.3 [adopted from Wheeler (38)] represents the dependence of the effectiveness factor on Thiele modulus for second-order kinetics. Values of r for first-order and zero-order kinetics in straight cylindrical pores are shown as curves A and C, respectively. Each curve is plotted versus its appropriate modulus. 

Plots of effectiveness factors versus corresponding Thiele moduli for zero-, first-, and second-order kinetics based on straight cylindrical pore model. For large hr, values of r are as follows ... 

Figure 12.7 adapted from Satterfield (40) contains a plot of the effectiveness factor for a zero-order reaction versus the Thiele modulus... 

In the limit of low effectiveness factors where tj becomes inversely proportional to the Thiele modulus, the apparent order of the reaction may differ from the true order. In this case, since the rate is proportional to the product of the effectiveness factor and the external concentration... 

We would be remiss in our obligations if we did not point out that the regions of multiple solutions are seldom encountered in industrial practice, because of the large values of / and y required to enter this regime. The conditions under which a unique steady state will occur have been described in a number of publications, and the interested student should consult the literature for additional details. It should also be stressed that it is possible to obtain effectiveness factors greatly exceeding unity at relatively low values of the Thiele modulus. An analysis that presumed isothermal operation would indicate that the effectiveness factor would be close to unity at the low moduli involved. Consequently, failure to allow for temperature gradients within the catalyst pellet could lead to major errors. 

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. 

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. 

For situations where the reaction is very slow relative to diffusion, the effectiveness factor for the poisoned catalyst will be unity, and the apparent activation energy of the reaction will be the true activation energy for the intrinsic chemical reaction. As the temperature increases, however, the reaction rate increases much faster than the diffusion rate and one may enter a regime where hT( 1 — a) is larger than 2, so the apparent activation energy will drop to that given by equation 12.3.85 (approximately half the value for the intrinsic reaction). As the temperature increases further, the Thiele modulus [hT( 1 — a)] continues to increase with a concomitant decrease in the effectiveness with which the catalyst surface area is used and in the depth to which the reactants are capable of... 

When the effectiveness factors for both reactions approach unity, the selectivity for two independent simultaneous reactions is the ratio of the two intrinsic reaction-rate constants. However, at low values of both effectiveness factors, the selectivity of a porous catalyst may be greater than or less than that for a plane-catalyst surface. For a porous spherical catalyst at large values of the Thiele modulus s, the effectiveness factor becomes inversely proportional to (j>S9 as indicated by equation 12.3.68. In this situation, equation 12.3.133 becomes... 

The analysis of the literature data shows that zeolites modified with nobel metals are among perspective catalysts for this process. The main drawbacks related to these catalysts are rather low efficiency and selectivity. The low efficiency is connected with intracrystalline diffusion limitations in zeolitic porous system. Thus, the effectiveness factor for transformation of n-alkanes over mordenite calculated basing on Thiele model pointed that only 30% of zeolitic pore system are involved in the catalytic reaction [1], On the other hand, lower selectivity in the case of longer alkanes is due to their easier cracking in comparison to shorter alkanes. 

The Thiele modulus and the effectiveness factor, respectively, were calculated for the three CO conversions X = 5,40, and 80%. The H2, CO, and H20 gas phase concentrations as well as the respective H2 concentration at the gas-wax phase boundary were taken from Table 12.3. The value of the diffusion coefficient Dm, is listed in Table 12.1. 

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

Comparison of the Effectiveness Factors Modeled with Presto Kinetics and Calculated by Means of the Thiele Modulus Lpore = 6.75-10-4 m -%... 

In Figure 12.4, we clearly see that the effective reaction rate is smaller in the simulation than that calculated by the Thiele modulus, which causes a higher effectiveness factor (see also Equations 12.11 and 12.20). 

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