Thiele modulus, isothermal

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). 

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.

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. 

Figure 8.11 Effectiveness factor (tj) as a function of Thiele modulus (< >) for an isothermal particle three regions indicated ...

This approach is analytically correct for isothermal reactors and first-order rate laws, since concentration does not appear in the expression for the Thiele modulus. For other (nonlinear) rate laws, concentration changes along the reactor affect the Thiele modulus, and hence produce changes in the local effectiveness factor, even if the reaction is isothermal. Problem 21-15 uses an average effectiveness factor as an approximation. 

For simple power law rate equations the effectiveness can be expressed in terms of the Thiele modulus, Eq 7.28. In those cases restriction is to irreversible, isothermal reactions without volume change. Other cases can be solved, but then the Thiele modulus alone is not sufficient for a correlation. 

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

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... 

The affect of diffusion on catalyst selectivity in porous catalysts operating under non-isothermal conditions has been examined by a number of workers. The mathematical problem has been comprehensively stated in a paper [21] which also takes into account the affect of surface diffusion on selectivity. For consecutive first-order exothermic reactions, the selectivity increases with an increase in Thiele modulus when the parameter A (the difference between the activation energy for reaction... 

A great deal of attention has been devoted to this topic because of the interesting and often solvable mathematical problems that it presents. Results of such calculations for isothermal zero-, first-, and second-order reactions in uniform cylindrical pores are summarized in Figure 17.6. The abscissa is a modified Thiele modulus whose basic definition is... 

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 ... 

Fig. 3.9. Selectivity as a function of the Thiele modulus for non-isothermal conditions...

Calculate the isothermal effectiveness factor rj for the porous catalyst pellet in problem 1 as a function of the Thiele modulus d> for the first reaction A —> B utilizing the fact that the rate constant of the second reaction B —> C is half the rate constant of A —> B, the pellet is isothermal, and the external mass transfer resistance is negligible. 

Also compute the yield and selectivity rj of the desired product B and the range of values from 0.01 to 10, where is the isothermal Thiele modulus. 

Figure 4. Normalized concentration profiles of reactant A versus the pellet radius, calculated from eq 49 for different values of the Thiele modulus (isothermal, first order, irreversible reaction in a sphere).

Figure 5. Effectiveness factor rj as a function of the Thiele modulus for different pellet shapes. Influence of intraparticle diffusion on the effective reaction rate (isothermal, first order, irreversible reaction).

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. 

This type of reaction has been investigated by Smith and Amundsen [95] and Carberry [14]. Without derivation it may be stated that under isothermal conditions the same solution for the effectiveness factor is obtained as in the case of an irreversible reaction if a modified Thiele modulus rev is introduced [91] ... 

The dependence of the modified effectiveness factor on t for a first-order isothermal irreversible reaction (RA = kACA) is presented in Figure 8.8 for different values of Thiele modulus = Ru kAl De A. All curves in this case are characterized by a gentle maximum, which becomes more pronounced when the ratio of the cylinder height to its external radius A = HIRh increases. The value of i providing the maximum also depends on A with increasing A the maximum position shifts towards higher values of t. The maximum relative increase of the modified effectiveness factor... 

The isothermal, reversible, first-order reaction A = B occurs in a flat plate catalyst pellet. Plot the dimensionless concentration of A (Ca/C s) as a function of distance into the pellet for various values of the Thiele modulus and the equilibrium constant. To simplify the solution, let Cas = 0.9(Ca + Cg) for all cases. 

For calculation of the Thiele modulus in the zeolite crystals, c )c, eq 5 was applied with the following parameters zeolite crystal radius, Rp = 3.1 10 cm temperature, 550°C density of the zeolite crystals, Pc = 1.73 10 g 1 molecular weight of coke. Me = 300 [4] molecular weight of oxygen = 16. The calculation of values of porosity and mean pore radius were carried out from CO2 adsorption-desorption isotherm for Samples 1-4 (time on stream 2 h) e = 0.29... 

The Thiele modulus for the mesoporous structure of the eatalyst, ( ) i, was calculated using the following parameters particle size, Rp = 0.0137 cm mean pore radius, rpore.ave = 20 10 cm catalyst porosity, e = 0.52 catalyst density, pg = 1210 g 1. N2 adsorption-desorption isotherms were used for measurement. The calculated value of effective diffusivity coefficient in the mesoporous structure of the catalyst is Dg = 9.71 10 2 cm min . This value is not affeeted by coke deposition. 

An early normalization of the Thiele modulus for an isothermal pellet and arbitrary kinetics was given by R. B. Bird, W. E. Stewart, and E. N. Lightfoot on pages 335-41 of their Notes on Transport Phenomena, the precursor to their well-known Transport Phenomena (New York John Wiley Sons, Inc., 1960). Slightly more general forms—all of them equivalent—have been given independently and almost simultaneously in ... 

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. 

Fir . 13. Isothermal effectiveness factor, tj, inside catalyst particles as function of the generalized Thiele modulus gt.n. 

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