Effectiveness, catalyst Thiele modulus

Effect of Thiele modulus on the normalized concentration profiles in a spherical catalyst particle with first-order reaction. The external surface of the particle is located at /Rp = 1. 

The solid phase could be a reactant, product, or catalyst. In general the decision on the choice of the particle size rests on an analysis of the extra-and intra-particle transport processes and chemical reaction. For solid-catalyzed reactions, an important consideration in the choice of the particle size is the desire to utilize the catalyst particle most effectively. This would require choosing a particle size such that the generalized Thiele modulus < gen, representing the ratio of characteristic intraparticle diffusion and reaction times, has a value smaller than 0.4 see Fig. 13. Such an effectiveness factor-Thiele modulus analysis may suggest particle sizes too small for use in packed bed operation. The choice is then either to consider fluidized bed operation, or to used shaped catalysts (e.g., spoked wheels, grooved cylinders, star-shaped extrudates, four-leafed clover, etc.). Another commonly used procedure for overcoming the problem of diffu-sional limitations is to have nonuniform distribution of active components (e.g., precious metals) within the catalyst particle. 

The results of simulation of these equations show that the effects of Thiele modulus and Biot number are similar to those for a solid catalyst. Thus for low values of < (< 1), the reaction is chemically controlled, while for higher values > 2) it is diffusion controlled within the sphere. External mass transfer is not important at high Biot numbers. The concentration of the active form of catalyst within the sphere is an imporant determinant of the rate. High values of

reaction rates, with the active catalyst concentrated close to the surface. More important, for a given set of parameter values, there appears to be (in most cases) an optimum time at which the active catalyst concentration within the sphere reaches a maximum. 

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). 

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

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). 

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. 

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. 

The term in brackets is a dimensionless group that plays a key role in determining the limitations that intraparticle diffusion places on observed reaction rates and the effectiveness with which the catalyst surface area is utilized. We define the Thiele modulus hT as... 

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. 

The reactor feed mixture was "prepared so as to contain less than 17% ethylene (remainder hydrogen) so that the change in total moles within the catalyst pore structure would be small. This reduced the variation in total pressure and its effect on the reaction rate, so as to permit comparison of experiment results with theoretical predictions [e.g., those of Weisz and Hicks (61)]. Since the numerical solutions to the nonisothermal catalyst problem also presumed first-order kinetics, they determined the Thiele modulus by forcing the observed rate to fit this form even though they recognized that a Hougen-Watson type rate expression would have been more appropriate. Hence their Thiele modulus was defined as... 

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. 

In order to demonstrate the selective effect of pore-mouth poisoning, it is instructive to consider the two limiting cases of reaction conditions corresponding to large and small values of the Thiele modulus for the poisoned reaction. For the case of active catalysts with small pores, the arguments of all the hyperbolic tangent terms in equation 12.3.124 will become unity and... 

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 data derived from modeling at different conversion degrees (X = 5, 40, and 80%) were also compared to the results obtained from the calculation of the classical Thiele modulus. The calculated (by the Thiele modulus) and modeled (by Presto Kinetics) effectiveness factors showed comparable values. Hence, the usage of simulation software is not required to get a first impression of the diffusion limitations in a Fischer-Tropsch catalyst pore. Nevertheless, modeling represents a valuable tool to better understand conditions within a catalyst pore. 

The presence (or absence) of pore-diffusion resistance in catalyst particles can be readily determined by evaluation of the Thiele modulus and subsequently the effectiveness factor, if the intrinsic kinetics of the surface reaction are known. When the intrinsic rate law is not known completely, so that the Thiele modulus cannot be calculated, there are two methods available. One method is based upon measurement of the rate for differing particle sizes and does not require any knowledge of the kinetics. The other method requires only a single measurement of rate for a particle size of interest, but requires knowledge of the order of reaction. We describe these in turn. 

Another way in which catalyst deactivation may affect performance is by blocking catalyst pores. This is particularly prevalent during fouling, when large aggregates of materials may be deposited upon the catalyst surface. The resulting increase in diffu-sional resistance may dramatically increase the Thiele modulus, and reduce the effectiveness factor for the reaction. In extreme cases, the pressure drop through a catalyst bed may also increase dramatically. 

Problem P7.06.02 reproduces one result from the literature. There it is apparent that in some ranges of the parameters, effectiveness can be much greater than unity, and also that at low values of Thiele modulus several steady states are possible. When both external and internal adiabatic diffusion occur, moreover, other studies find that half a dozen or more steady states can exist. Those kinds of findings involve much computer work. A book by Aris (Mathematical Theory of Diffusion and Reaction in Permeable Catalysts,... 

For reasons of simplicity, the Thiele modulus will be defined and calculated for a catalyst plate with pore access at both ends of the plate and not at the bottom or top. Note that for most cases in real-life applications the assumptions have to be modified using polar coordinates for the calculations. The Thiele modulus q> is therefore defined as the product of the length of the catalyst pore, /, and the square root of the quotient of the constant of the speed of the reaction, k. divided by the effective diffusion coefficient DeS ... 

Co catalysts, metal crystallite size and support effects, 39 242-246 Ru catalysts, metal crystallite size and support effects, 39 237-242 Thiele modulus effect, 39 275 reaction-transport models, 39 222-223 readsorption probability, 39 264-265 secondary chain growth, hydrogenation, and depolymerization reactions, 39 224—225... 

Fig. 2. Effectiveness factor as a function of Thiele modulus for an isothermal catalyst pellet.

The concentration and temperature Tg will, for example, be conditions of reactant concentration and temperature in the bulk gas at some point within a catalytic reactor. Because both c g and Tg will vary with position in a reactor in which there is significant conversion, eqns. (1) and (15) have to be coupled with equations describing the reactor environment (see Sect. 6) for the purpose of commerical reactor design. Because of the nonlinearity of the equations, the problem can only be solved in this form by numerical techniques [5, 6]. However, an approximation may be made which gives an asymptotically exact solution [7] or, alternatively, the exponential function of temperature may be expanded to give equations which can be solved analytically [8, 9]. A convenient solution to the problem may be presented in the form of families of curves for the effectiveness factor as a function of the Thiele modulus. Figure 3 shows these curves for the case of a first-order irreversible reaction occurring in spherical catalyst particles. Two additional independent dimensionless paramters are introduced into the problem and these are defined as... 

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... 

Another problem—catalyst formulators (those magicians) have been very successful in creating better and better catalysts, those that give higher and higher rates of reaction. But to use the whole of the catalyst volume effectively we must keep the Thiele modulus... 

Figure 7-13 Plots of effectiveness factor 17 versus Thiele modulus

Thus we have expressions for the effectiveness factor for different catalyst geometries. The Thiele modulus can be computed from catalyst geometry and surface area parameters. The characteristic size is 21 for a porous slab and 2R for a cylinder or sphere. While the expressions for r)(0) appear quite different, they are in fact very similar when scaled appropriately, and they have the same asymptotic behavior,... 

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. 

In Figure 7.4 the effectiveness factor is plotted against the Thiele modulus for spherical catalyst particles. For low values of 0, Ef is almost equal to unity, with reactant transfer within the catalyst particles having little effect on the apparent reaction rate. On the other hand, Ef decreases in inverse proportion to 0 for higher values of 0, with reactant diffusion rates limiting the apparent reaction rate. Thus, decreases with increasing reaction rates and the radius of catalyst spheres, and with decreasing effective diffusion coefficients of reactants within the catalyst spheres. 

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