Thiele modulus analysis

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. 

We would be remiss if we did not indicate that some authors have defined yet another Thiele-type modulus ( ) for this problem as <. = 0s/3 and they have developed their analysis according to this parameter. In using plots of f] versus S9 one must be careful to determine which of the two alternative definitions has been adopted for s. Otherwise his value for rj may be considerably in error. A variety of other symbols for dimensionless groups akin to the Thiele modulus have been employed by different authors. 

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 above analysis and Fig. 19-25 provide a theoretical foundation similar to the Thiele-modulus effectiveness factor relationship for fluid-solid systems. However, there are no generalized closed-form expressions of E for the more general case ofa complex reaction network, and its value has to be determined by solving the complete diffusion-reaction equations for known intrinsic mechanism and kinetics, or alternatively estimated experimentally. 

In the most general case, i.e. when intraparticlc and interphase transport processes have to be included in the analysis, the effectiveness factor depends on five dimensionless numbers, namely the Thiele modulus the Biot numbers for heat and mass transport Bih and Bim, the Prater number / , and the Arrhenius number y. Once external transport effects can be neglected, the number of parameters reduces to three, because the Biot numbers then approach infinity and can thus be discarded. 

The analysis of the effects of transport on catalysis has focused on a comparison of the availability of reacting species by diffusion to the rate of reaction on the catalytic sites. High-surface-area catalysts are usually porous. Comparison of transport to reaction rates has usually been based on Knudsen diffusion (by constricted collision with the pore walls) as the dominant mode of transport. DeBoer has noted that for small pores surface diffusion may dominate transport (192). Thiele modulus calculations may therefore not be valid if they are applied to systems where surface diffusion can be significant. This may mean that the direct participation of spillover species in catalysis becomes more important if the catalysts are more microporous. Generalized interpretations of catalyst effectiveness may need to be modified for systems where one of the reactants can spill over and diffuse across the catalyst surface. 

For most hydrocarbons burning in air, coo,oe/v is only about 0.07, and WF.eff is about 2, so that the pseudo-homogeneous source term in Equation 6 depends only weakly on wq. For this case, i/r plays the role of a zero-order Thiele modulus (13). Thus, the present continuum analysis clearly reveals that the dimensionless parameter which dictates the onset of incipient group combustion is simply a Damkohler niunber (Thiele modulus) which takes on a critical value (Equation 15) dependent only on the ambient oxidizer mass fraction, on the fuel vapor mass fraction at the droplet surface, and on the stoichiometric oxidizer/fuel mass ratio. 

The implications of severe diffusional resistance on observed reaction kinetics can be determined by simple analysis of this more general Thiele modulus. The observed rate of reaction can be written in terms of the intrinsic rate expression and the effectiveness factor as ... 

Dimensional analysis of the coupled kinetic-transport equations shows that a Thiele modulus (4> ) and a Peclet number (Peo) completely characterize diffusion and convection effects, respectively, on reactive processes of a-olefins [Eqs. (8)-(14)]. The Thiele modulus [Eq. (15)] contains a term ( // ) that depends only on the properties of the diffusing molecule and a term ( -) that includes all relevant structural catalyst parameters. The first term introduces carbon number effects on selectivity, whereas the second introduces the effects of pellet size and pore structure and of metal dispersion and site density. The Peclet number accounts for the effects of bed residence time effects on secondary reactions of a-olefins and relates it to the corresponding contribution of pore residence time. 

The probability of readsorption increases as the intrinsic readsorption reactivity of a-olefins (k,) increases and as their effective residence time within catalyst pores and bed interstices increases. The Thiele modulus [Eq. (15)] contains a parameter that contains only structural properties of the support material ( <>, pellet radius Fp, pore radius 4>, porosity) and the density of Ru or Co sites (0m) on the support surface. A similar dimensional analysis of Eqs. (l9)-(24), which describe reactant transport during FT synthesis, shows that a similar structural parameter governs intrapellet concentration gradients of CO and H2 [Eq. (25)]. In this case, the first term in the Thiele modulus (i/>co) reflects the reactive and diffusive properties of CO and H2 and the second term ( ) accounts for the effect of catalyst structure on reactant transport limitations. Not surprisingly, this second term is... 

The rate processes of diffusion and catalytic reaction in simple square stochastic pore networks have also been subject to analysis. The usual second-order diffusion and reaction equation within individual pore segments (as in Fig. 2) is combined with a balance for each node in the network, to yield a square matrix of individual node concentrations. Inversion of this 2A matrix gives (subject to the limitation of equimolar counterdiffusion) the concentration profiles throughout the entire network [14]. Figure 8 shows an illustrative result for a 20 X 20 network at an intermediate value of the Thiele modulus. The same approach has been applied to diffusion (without reaction) in a Wicke-Kallenbach configuration. As a result of large and small pores being randomly juxtaposed inside a network, there is a 2-D distribution of the frequency of pore fluxes with pore diameter. 

Figure 4 shows the usual representation of eflFectiveness factor vs. Thiele modulus modified in the present case by the square root of average activity at T = 503 K. This figure presents the evolution of both parameters with time. The shape remains approximately similar to that of a non-deactivated pellet, at least for the smaller ki values. For the highest ki selected, t initially decreases. This fact can be explained by considering eqn. (10) and making a simultaneous analysis of the activity and CO profiles. 

Dimensional analysis of the equations relates the concentration profile, both along the lumen and the radius of the fiber, to seven dimensionless parameters Thiele modulus, X 2, dimensionless length, z, dimensionless Michaelis constant, O and the following dimensionless quantities b/a, d/a, Di/D3 and Di/tt Dj. 

Tphe most effective way to utilize a given quantity of catalytic material - is to deposit it on a layer of porous support. The classical Thiele analysis demonstrated that, for a first order reaction, it is preferable to concentrate the active ingredients in a thin layer to minimize diffusion effects this conclusion remains valid for any positive order kinetics. This diffusion effect causes a decline in reactant concentration toward the interior of a porous catalytic layer, leading to a decline in reaction rate in the interior. When the Thiele modulus is sufficiently large, such as when the reaction rate is fast and when diffusion through a porous layer is slow, only a very thin layer on the exterior is contributing to the reaction rate. 

For the single-reaction cases, we performed dimensional analysis and found a dimensionless number, the Thiele modulus, which measures the rate of production divided by the rate of diffusion of some component. A complete analysis of the first-order reaction in a sphere suggested a general approach to calculate the production rate in a pellet in terms of the rate evaluated at the pellet exterior surface conditions. This motivated the definition of the pellet effectiveness factor, which is a function of the Thiele modulus. 

Obviously it does not pay to try to guess an appropriate Thiele modulus for these more complex rate expressions. An asymptotic analysis as presented here is required to find the appropriate scaling. This idea appears to have been discovered independently by three chemical engineers in 1965. To quote from Aris [2, p. 113]... 

The payoff for this analysis is shown in Figures 74 3 and 7.14, If we use our first guess for the Thiele modulus, Equation 7.51, we obtain Figure 7.13 in which the various values of

Thiele modulus defined in Equation 7.58, we obtain the results in Figure 7.14. Figure 7.14 displays things more clearly. First, notice from the mass balance, Equation 7.54, that the dimensionless reaction rate is... 

Another approach, which requires a bit of foreknowledge concerning the magnitude of the Thiele modulus likely to be encountered, employs the direct analysis of Liu given in equations (7-28) and (7-33). For example, if we have a first-order nonisothermal reaction system that fits the restrictions accompanying equation (7-33), then... 

In a more realistic case we may have the reaction first-order in both A and B, where the analysis is complicated by the fact that the Thiele modulus becomes a function of Bj, which is changing with time. This complication is tractable, however, and the final time-conversion relationship is... 

The Wheeler-Robell analysis envisions the main reaction to be dilfusion-con-trolled, but not the poisoning reaction. Whether or not this is so is a question of relative dimensions of molecules, but dual diffusion control would seem more typical. The analysis has been extended to diffusion-controlled poisoning by Haynes [H.W. Haynes, Jr., Chem. Eng. Sci., 25, 1615 (1970)], who used a shell model as an approximation for rapid poisoning in a Type I system. The Thiele modulus for the poisoning reaction is and shell model assumption to be valid. For a spherical catalyst particle the fraction of original activity, s, is related to the radius of the poison-free zone by... 

This basic analysis is commonly attributed to Thiele (1939) [114] and the dimensionless parameter 0 is commonly called the Thiele modulus, although essentially the same analysis was published many years earlier, in 1909, by Jiittner [115]. 

In a zeolite catalyst diffusional limitations may occur at either the particle scale or the crystal scale. In the latter case the basic analysis remains the same, but since the rate constant is defined with respect to the concentration of reactant in the vapor phase while the intracrystalline diffusivity is defined with respect to the adsorbed phase concentration, the Thiele modulus must be redefined to introduce the dimensionless adsorption equUibriimi constant (K) ... 

The concept of diffusion is used whenever one is dealing with transport within a phase as a function of time and position. For example, when a chemical reaction occurs in a catalyst pellet, the reactant has to diffuse through the catalyst and react while it is still diffusing. Thus, in any rational analysis of such a situation, we (chemists or chemical engineers) are concerned with diffusion. As we shall see in Chapter 7, the Thiele modulus, which is central to the analysis of catalytic reactions, is based on the joint use of diffusion and reaction coefficients in a single dimensionless group. 

Lee and Reilly (1981) defined a more rigorous form of the Thiele modulus based on the generalized modulus of Bischolf and Aris (see Chapter 7) which is particularly useful in analyzing the role of diffusion in deactivation. Their analysis shows that, in reactant-independent deactivation, the presence of a strong dilfusional limitation lowers the rate of deactivation to half the diffusion-free value. Thus, surprisingly, diffusion seems to have a favorable effect on the performance of a deactivating immobilized enzyme catalyst. 

For the silica granules the distribution of the Mn concentration as a function of position (Figure 6.4), however, clearly reveals a diffusion limitation for the final deposition reaction. From a kinetic analysis in combination with diffusion (Thiele modulus, (p) De Jong has concluded that the rate constant for the final deposition reaction greatly exceeded the rate constants shown in Table 6.2. An analysis was proposed that involved as the final deposition reaction... 

The effects of geometry can generally be neglected. This allows for the generalization of the analysis of the effectiveness factor and its dependence on the Thiele modulus. 

For exothermic reactions, > 0, and the P curves generated at each Ys value show that may indeed exceed unity, depending on whether or not the rate increase caused by temperature rise offsets the decrease resulting from the fall in concentration. When // > 1, -Ra)p > -Ra)s> hut this situation may not always be beneficial because the increase in temperature toward the center of the particle may induce deactivation or may promote undesirable reactions with an overall decrease in the selectivity for the desirable product. At high values of the Thiele modulus, t] becomes inversely proportional to 4>s. as in the isothermal case therefore, the reactant is rapidly consumed as it penetrates the particle so that the reaction takes place in a thin shell just underneath the particle surface while the interior is more or less isothermal at a higher temperature. On the other hand, at low values of Os and for highly exothermic reactions, for example, > 0.3, the value of r] is not uniquely defined by the three dimensionless parameters of the analysis. Three possible values of // exist, each representing a different set of conditions at which the rate of heat release balances the rate of heat removal the... 

---