Observable Thiele modulus

A practical problem in applying the Thiele modulus is that the values of the kinetic constants Vmax and Km are frequently unknown. Therefore an observable Thiele modulus was introduced which depends on the measurable overall rate, but not on kinetic parameters ... 

Let us analyze what is the relationship between the Thiele modulus and observable Thiele modulus 0. For that we will set S -S0. At these conditions the observed rate is equal to the rate in the absence of internal diffusion, thus... 

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 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... 

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. 

Generalizati on. It has been observed that all plots of effectiveness against Thiele modulus are similar and that a single plot can represent the nine main cases fairly adequately by defining a generalized modulus 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... 

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. 

When the Thiele modulus is large Cam is effectively zero and the maximum difference in temperature between the centre and exterior of the particle is (- AH)DeCAJke. Relative to the temperature outside the particle this maximum temperature difference is therefore 0. For exothermic reactions 0 is positive while for endothermic reactions it is negative. The curve in Fig. 3.6 for 0 = 0 represents isothermal conditions within the pellet. It is interesting to note that for a reaction in which -AH- 10 kJ/kmol, ke= lW/mK, De = 10 5m2/s and CAa> = 10 1 kmol/m3, the value of Tu - Tx is 100°C. In practice much lower values than this are observed but it does serve to show that serious errors may be introduced into calculations if conditions within the pellet are arbitrarily assumed to be isothermal. 

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 ... 

Internal diffusional limitations are possible any time that a porous immobilized enzymatic preparation is used. Bernard et al. (1992) studied internal diffusional limitations in the esterification of myristic acid with ethanol, catalyzed by immobilized lipase from Mucor miehei (Lipozyme). No internal mass diffusion would exist if there was no change in the initial velocity of the reaction while the enzyme particle size was changed. Bernard found this was not the case, however, and the initial velocity decreased with increasing particle size. This corresponds to an efficiency of reaction decrease from 0.6 to 0.36 for a particle size increase from 180 pm to 480 pm. Using the Thiele modulus, they also determined that for a reaction efficiency of 90% a particle size of 30 pm would be necessary. While Bernard et al. found that their system was limited by internal diffusion, Steytler et al. (1991) found that when they investigated the effect of different sizes of glass bead, 1 mm and 3 mm, no change in reaction rate was observed. 

To have a quantitative idea of the problem of intraparticle diffusion, effectiveness factors for the two catalysts were calculated from the observed second order rate constants (based on surface area) using the "triangle method" suggested by Saterfield (4). The effectiveness factors for Monolith and Nalcomo 474 catalysts on Synthoil liquid at 371°C (700 F) were calculated to be 0.94 and 0.216, respectively. In applying the relationship between the "Thiele Modulus," 4>> and the "effectiveness factor," n> the following simplifying assumptions were made ... 

The difficulty in determining the Thiele modulus is that the intrinsic rate constant (without diffusion limitations) is required, while the Thiele modulus is used to determine if diffusion limitations are present in the first place. This problem can be avoided by using the iterative procedure described by van Donk et al39, where the observed rate constant can be used for determination of the Thiele modulus and effectiveness. 

In Table 7 the effectiveness and corresponding Thiele modulus for the different support materials is given. The particle size for the ASA, SiC>2 and HT supports was taken equal to the sieve fraction. This is a worst-case scenario, since it is far more likely that the particles in the sieve fraction are constructed of several crystallites which contain the relevant pores and Pt particles. Between those crystallites, the pore radii will be very large compared to the pore radius in the support material. Even in this worst case scenario, the effectiveness is still high, close to unity, for all catalysts. This demonstrates that the observed reaction kinetics reflect the intrinsic catalyst properties, since internal diffusion limitations are absent. 

Table 7 the effective diffusion De, Thiele modulus < ) and effectiveness T as a function of support at 75°C. For the observed rate constant k0bs the most active catalyst for each support material was taken. 

The calculated time-dependent effectiveness factor as a function of Thiele modulus along with experimental observation is shown in Fig, 2. The solid lines represent the computed effectiveness factor using Eq, 4 and the broken line represents experimental values. When compared with experimental data, we sec that the mathematical model represents the actual behaviour quite well. 

However, in many practical situations the problem exists that effective rate constants and activation energies have been derived on the basis of laboratory experiments. The question arises then as to whether or not these parameters arc influenced by transport effects. With the relations given so far, this question cannot be answered yet, since according to its definition by cq 27 the Thiele modulus is based on the intrinsic rate constant k. This problem can be solved by introducing a new modulus, which in contrast to only contains observable (effective) quantities, and thus can... 

This effect will be particularly emphasized at small values of the Thiele modulus where the intrinsic rate of reaction and the effective rate of diffusion assume the same order of magnitude. At large values of , the effectiveness factor again becomes inversely proportional to the Thiele modulus, as observed under isothermal conditions (Section 6.2.3.1). Then the reaction takes place only within a thin shell close to the external pellet surface. Here, controlled by the Arrhenius and Prater numbers, the temperature may be distinctly higher than at the external pellet surface, but constant further towards the pellet center. 

However, whereas effectiveness factors above unity under nonisothcrmal conditions can be explained quite easily, the observation of multiple steady states is a new and unexpected feature. These arise at small values of provided the reaction is substantially exothermic and, additionally, has a high activation energy. This means that, for a single value of the Thiele modulus, several possible solutions for the steady state overall effectiveness factor may exist (operating points), usually up to three. The middle operating point is normally unstable. Whenever the temperature and/or the... 

As stated above, instead of plotting the effectiveness factor against the Thiele modulus which contains the unknown, intrinsic rate constant, it is often more convenient to relate it to the observable Weisz modulus i/f. This leads to the representation given in Fig. 14 for the same situation as depicted in Fig. 13. The dashed portions of the curves indicate the regions in which a unique solution of the effectiveness factor docs not exist, corresponding to the regions of multiple solutions in Fig. 13. 

Olson and Haag [80] in 1983 showed that the yield of p-xylene observed during the disproportionation of toluene on various modified and unmodified ZSM-5 catalysts is actually influenced by product shape selectivity. The authors attributed the observed effects to an interaction of diffusion and reaction, characterized by means of a dimensionless modulus similar to the classical Thiele modulus . The mathematical treatment of shape selectivity in zeolite catalysts, which will be applied in this section, is largely based on the theory of Olson and Haag [80], although some modifications and extensions to this are given. 

We can observe (Table 4) that the concentration effect on Thiele modulus is very poor. Due to the Thiele modulus values, we can say that in SC CO2 an intermediate rate between the reactional and diffiisional rates was apparent. The kinetic parameters, Vmax and KM, in both reaction media are very different is higher in n-hexane (13.6 mmol.(s.kg ) 1) than in SC... 

We have demonstrated for the first time that we could apply the theory of generalized Thiele modulus to an enzymatic reaction both in n-hexane and SC CO2. The comparison between the two reaction media is not so clear in n-hexane the real reaction velocity is higher than that obtained in SC C02. Nevertheless, the Thiele modulus values indicates a limitation due to the internal mass transfer rate g 1. Thus we observed, in the hexane case, a diffusional control, while in SC C02 an intermediate rate between the reactional and diffusional rates was apparent. It therefore, seems that SC CO2 should be the solvent of choice in reactions catalyzed by immobilized enzymes, since it reduces problems with internal mass transfer. An other advantage is that the value of the inhibition constant is 43 mM in n-hexane and 120 mM in SC CO2 [14], so SC CO2 should be more convenient if we have to work with higher ethanol concentration. The economic feasibility of an industrial scale lipase catalyzed reaction on C02 may depend upon possible costs for high-pressure equipment. 

Since the conversion rate depends on a and e, the effectiveness factor will be determined by three parameters, namely a, e and a Thiele modulus. This is illustrated in Figure 6.4 [18]. For values of a larger than zero (exothermic reaction) an increase in the effectiveness factor is found, since the temperature inside the catalyst pellet is higher than the surface temperature. For endothermic reactions (a < 0) a decrease of the effectiveness factor is observed. 

The dependence of the reaction rate on the pore size as calculated from Equations 8.1, 8.9 and 8.12 for arbitrary reaction regimes is shown in Figure 8.1 for typical values of the reference Thiele modulus The results are presented as a ratio of the reaction rate (ra) to its maximum value (RAa) observed in the limit of infinitely small pore diam-... 

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