Effectiveness factor as a function of Thiele modulus

Figure 10.11. Effectiveness factor >) as a function of Thiele modulus 4> (4> = A./, for platelet, 4> = Xrc for...

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

Figure 4a and 4b. Catalyst effectiveness factor as a function of thiele modulus... 

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. 

For exothermic processes the reactions cause a temperature rise inside the particle. This usually leads to increased values of the rate constants. This increase of the rate constants can sometimes overcompensate for the lower concentrations (compared to those in the bulk fluid) caused by the diffusion limitations in the particle. As a result, the reaction rate becomes higher than the reaction rate that is obtained with the concentrations in the bulk phase and temperature. Consequently, the effectiveness factor exceeds 1 This effect is particularly emphasized at small values of the Thiele modulus. The catalyst effectiveness factors as a function of Thiele modulus at different values of the Prater numbers are illustrated in Figure 9.15. 

Figure 9.15. The catalyst effectiveness factors as a function of Thiele modulus at different values of the Prater numbers.

Substituting the effectiveness factor as a function of Thiele modulus, the factor is inversely proportional to Thiele modulus in the presence of strong diffusion effects. Therefore, from Equation 18.29, we obtain ... 

Figure 2.10 Internal effectiveness factor as a function of Thiele modulus for porous particles of various shapes. (Source Levenspiel [31]. Reproduced with permission of John Wiley Sons.)...

Fig. 10.15 The catalyst effectiveness factors as a function of Thiele modulus at different values of the Prater numbers. (Experimental data from P.B. Welsz, J.S. Hicks, Chem. Eng. Scl. 17 (1962) 265-275. Copyright 1962 Elsevier).

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

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... 

Figure 18.5 The effectiveness factor as a function of the parameter mL or Mj, called the Thiele modulus, prepared from Aris (1957) and Thiele (1939).

In Fig. 13, typical curves for the effectiveness factor as a function of the Thiele modulus arc given for a first order, irreversible reaction in a spherical catalyst pellet. These curves have been obtained numerically by Weisz and Hicks [110], for the case of negligible intcr-... 

When the effective reaction rate is controlled by pore diffusion, then the asymptotic solution of the catalyst effectiveness factor as a function of the generalized Thiele modulus can be utilized (cq 108). This (approximate) relationship has been derived in Section 6.2.3.1. It is valid for arbitrary order of reaction and arbitrary pellet shape. 

Figure 12 shows the effectiveness factor as a function of the Wheeler-Weisz modulus for different reaction orders, indicating that criterion (33) holds for the generalized Thiele modulus. Due to the definition of L it is fairly independent of the catalyst geometry. 

Figure 13. Internal effectiveness factor as a function of the Thiele modulus for nonisothermal reactions at different values for the Prater number and y, = 10 (numerical solutions for a first order reaction).

Several formulae have been given for the calculation of the effectiveness factor as a function of one of the Aris numbers An or An, or as a function of a Thiele modulus. These formulae can become very complex and, for most kinetic expressions and catalyst geometries, it is impossible to derive analytical solutions for the effectiveness factor, so... 

A plol of the effectiveness factor as a function of the Thiele modulus is shown in Figure 12-5. Figure 12-5a shows "q as a function of (j> for a spherical catalyst pellet for reactions of zero-, first-, and second-order. Figiue 12-5b corresponds to a first-order reaction occurring in three differently shaped pellets of voliune Vp and external siuface area A,. When volume change accompanies a reaction, the corrections shown in Figure 12-6 apply to the effectiveness factor for a first-order reaction. 

Figure 7.4 Effectiveness factor as a function of the Weisz (observable) modulus ( nd also the Thiele modulus ).

Figure 19.10 Triphase catalysis effectiveness factor as a function of dimensionless time for various values of the Thiele modulus (Desikan and Doraiswamy, 1995). Solid lines represent irreversible reactions K —xx) and broken lines reversible reactions (AT = 0.1). Note T]i=T of Equation 19.24...

The effectiveness factor is plotted as a function of Thiele modulus (logarithmic scale) in Figure 18.3. This curve is similar to that mentioned in several books. 

Figure 4.5.20 shows the effectiveness factor as a function of the Thiele modulus for a slab, a sphere, and a cyclinder. It is apparent that all curves can be described with acceptable accuracy by the exact solution for a slab [ ypore = tanh(0/0) ijpom = 1/0 for 0> 2]. Thus, Figures 4.5.21 and 4.5.22 derived fora flat plate can be used to a good approximation for any particle geometry (with Vp/Ap,ex as characteristic length for 0). 

Here the attention is focused on the behavior of the overall effectiveness factor as a function of an overall Thiele modulus. For a given particle size,... 

Fig. 10.13 Effectiveness factor as a function of generalized Thiele modulus. (From J. Hong, W.C. Hecker, T. H. Fletcher, Improving the accuracy of predicting effectiveness factors for mth order and Langmuir rate equations In spherical coordinates, Energy Fuels 14 (2000) 663-670. Copyright 2000 American Chemical Society).

To solve the design equation, we must know how rj and —depend on temperature and composition. For an nth-order reaction, the effectiveness factor is a function of a single dimensionless variable, the Thiele modulus, as given by Eqns. (9-13) and (9-13a). The Thiele modulus depends on temperature because the defining equations contain k, Keq, DA,eS, and aU of which are temperature dependent. The Thiele modulus may also depend... 

Figure S.3S. Effectiveness factor e plotted as a function of the Thiele diffusion modulus Og. The effective factor is well approximated by 3/Og for Og > 10.

Numerical integration yields the effective rate of reaction, as a function of the concentration with the modulus (Figure 11.21). The same results can also be represented in the form of graphics of effectiveness factor against this Thiele... 

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. 

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).

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