Catalytic Thiele modulus

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. 

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). 

In the case that the chemical reaction proceeds much faster than the diffusion of educts to the surface and into the pore system a starvation with regard to the mass transport of the educt is the result, diffusion through the surface layer and the pore system then become the rate limiting steps for the catalytic conversion. They generally lead to a different result in the activity compared to the catalytic materials measured under non-diffusion-limited conditions. Before solutions for overcoming this phenomenon are presented, two more additional terms shall be introduced the Thiele modulus and the effectiveness factor. 

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... 

Then the classical Thiele modulus ( ) and the effectiveness factor (t/), expressing the extent of internal diffusion limitations in the catalytic washcoat layer of thickness 8, can be calculated according to (cf. Aris, 1975 Froment and Bischoff, 1979, 1990)... 

The Thiele modulus tfi for the case of a spherical catalyst particle of radius R (cm), in which a first-order catalytic reaction occurs at every point within the particles, is given as... 

The definition of the electrochemical Thiele modulus [Eq. (9b)J characterizing the degree of electrocatalyst utilization is a prerequisite for properly tailoring the micromorphology of porous electrocatalytic electrode coatings and fuel cell electrodes, as it allows matching of the coating or catalyst particle dimension to the catalytic activity of the material ... 

In this section we have presented and solved the BVPs associated with the diffusion and reaction that take place in the pores of a porous catalyst pellet. The results were expressed graphically in terms of the effectiveness factor rj versus the Thiele modulus d> for two cases One with negligible external mass and heat transfer resistances, i.e., when Sh and Nu —> oo, and another with finite Sh and Nu values. This problem is very important in the design of fixed-bed catalytic reactors. The sample results presented here have shown that for exothermal reactions multiple steady states may occur over a range of Thiele moduli d>. Efficient numerical techniques have been presented as MATLAB programs that solve singular two-point boundary value problems. 

Construct the diagrams of the effectiveness factor and the desired yield versus the Thiele modulus of the reactant A for a first-order consecutive exothermic catalytic reaction... 

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. 

Both studies show that at relatively low temperatures, i.e., during ignition of the catalyst, the rate-limiting step shifts from chemical kinetics to diffusion in the washcoat. This is clear from Fig. 7, computed using a one-dimensional model by Nakhjavan et al. [54]. Figure 7A shows the Thiele modulus and Fig. 7B an external diffusion limiting factor F versus dimensionless axial position in the reactor at various times on-stream for the catalytic combustion of propene in monolith reactors. The time is defined as the time after injection of the fuel in a preheated air flow. 

As stated before, the volume of catalyst per unit volume of reactor space, which is to be termed the fractional catalyst volume (equal to 1 minus the voidage) and the degree of utilization of this catalytic material are important factors in a high-pressure, relatively slow catalytic process, such as the hydroprocessing of oils. The effectiveness factor of catalyst particles of arbitrary shape can be correlated with a generalized Thiele modulus 4>gen, defined by... 

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. 

Contrary to the afore mentioned Authors, Chien considering various polyethylene and polypropylene catalytic systems, on the basis of the Thiele modulus criteria (Q 5 for a > 5 and Q 1 for a 1) recently concluded that Ziegler-Natta catalysis is not controlled by diffusion phenomena while Phillips catalysis is. 

In the case of heterogeneous high yield catalysts, Thiele modulus criterion would predict an appreciable broadening of MWD in comparison to traditional catalysts (higher values). This broadening could be also due to a notable enlargement of the polymer particle and, therefore, to an increase in the Thiele modulus in proportion to catalytic yield... 

We now introduce dimensionless variables

catalytic reactions, the Thiele modulus. Let... 

It is well established that the smallest crystals are the most effective as catalysts as long as the catalytic reaction proceeds in the intercrystalline void volume [1,8]. Increased crystal size will result in an increase in pore length and thus in the Thiele modulus. This will result in a reduced effectiveness factor, viz. a reduced actual rate of reaction. 

In order to have significant electrochemical promotion of metal-support interaction promotion of a catalytic reaction, t]p must be at least 0.2. Equation (66) and the corresponding Figure 44 show the range of Op and J values that allow for this to happen the Thiele modulus Op must be smaller than 5. This means small film thickness or catalyst particle size, small kinetic constant k for promoter destruction, and finite surface diffusivity, Dg, of the promoter. Also the dimensionless current J must be larger than 2, and this again dictates a small k value for promoter destruction, a finite current for electrochemical promotion, and a fast catalytic rate, r, for metal-support interactions. 

Sousa et al [5.76, 5.77] modeled a CMR utilizing a dense catalytic polymeric membrane for an equilibrium limited elementary gas phase reaction of the type ttaA +abB acC +adD. The model considers well-stirred retentate and permeate sides, isothermal operation, Fickian transport across the membrane with constant diffusivities, and a linear sorption equilibrium between the bulk and membrane phases. The conversion enhancement over the thermodynamic equilibrium value corresponding to equimolar feed conditions is studied for three different cases An > 0, An = 0, and An < 0, where An = (ac + ad) -(aa + ab). Souza et al [5.76, 5.77] conclude that the conversion can be significantly enhanced, when the diffusion coefficients of the products are higher than those of the reactants and/or the sorption coefficients are lower, the degree of enhancement affected strongly by An and the Thiele modulus. They report that performance of a dense polymeric membrane CMR depends on both the sorption and diffusion coefficients but in a different way, so the study of such a reactor should not be based on overall component permeabilities. 

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. 

Exercise 7.6 Thiele modulus and first-order kinetics A catalytic reaction that is first order in the concentration of A... 

Recall now that Uq = k/oX and that a L) is the characteristic parameter (similar to the Thiele modulus for catalytic reactions), indicating the magnitude of diffusional effects. Here we wish to define the values of this parameter for F > 0.95 and < 0.05 corresponding to the kinetic regimes indicated in the problem statement. If we let cosh(noT) = M, then... 

If the catalyst particles are not completely wetted by the liquid phase and the pores consequently not completely filled with liquid phase (static holdup gives some indication of whether this is the case or not), the situation is considerably more complex. In addition to being a function of the Thiele modulus, the catalytic effectiveness will now depend on the fraction of external wetting, rjcs, and the fraction of pore volume filled with liquid, rji. Dudokovic [M.P. Dudokovic, Amer. Inst. Chem. Eng. Jl., 23, 940 (1977)] proposed a reasonable approach that accounts for all three factors. If the reaction proceeds only on the catalyst surface effectively wetted by the liquid phase and components of the reaction mixture are nonvolatile, then one can in principle modify the definition of the Thiele modulus to... 

Catalytical Activity and Transport Limitation Catalytical lltficiency and Thiele Modulus- As is well known in heterogeneous catalysis, the relevance of transport limitations in such a reaction system can be evaluated if the Thiele modulus is known (23), ). For a set of assumptions (28), the Thiele modulus for s erical particles and a zero-order reaction can... 

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