Effectiveness generalized Thiele modulus

The internal effectiveness factor is a function of the generalized Thiele modulus (see for instance Krishna and Sie (1994), Trambouze et al. (1988), and Fogler (1986). For a first-order reaction ... 

Figure 3.32. Interna effectiveness factor as a function of the generalized Thiele modulus for a first order reaction.

This example illustrates calculation of the rate of a surface reaction from an intrinsic-rate law of the LH type in conjunction with determination of the effectiveness factor (rj) from the generalized Thiele modulus (G) and Figure 8.11 as an approximate representation of the 7]-Q relation. We first determine G, then 17, and finally (—rA)obs. 

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... 

Figure 6. Effectiveness factor rj as a function of the generalized Thiele modulus f for different pellet geometries. Influence of intraparticle diffusion on the effective reaction rate (isothermal, first order, irreversible reaction).

Figure 6 shows the effectiveness factor for any of the three different pellet shapes as a function of the generalized Thiele modulus p. It is obvious that for larger Thiele moduli (i.e. p > 3) all curves can be described with acceptable accuracy by a common asymptote t] — 1 / p. The largest deviation between the solutions for the individual shapes occurs around p x 1. However, even for the extremely different geometries of the flat plate and the sphere, the deviation of the efficiency... 

The above considerations can also be extended in a simple way to different reaction orders, if the modulus p is further modified. In this sense Petersen [85] defined a generalized Thiele modulus 4>pn which takes into account the effects of the pellet shape as well as the influence of the reaction order ... 

In Fig. 7 the effectiveness factor is shown as a function of the generalized Thiele modulus pn for different reaction orders (flat plate). From this figure, it is obvious that, except for the case of a zero order reaction, the curves agree quite well over the entire range of interest. The asymptotic solution t = l/ pn is valid for any reaction order and for values of the modulus pn > 3. 

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. 

Calculations of catalyst effectiveness are readily carried out using the generalized Thiele modulus. For the key component we can write ... 

The problem of Example 1 was reformulated in terms of Equations 16, 17 and 20, and the generalized Thiele modulus was found to be M = 11.40 corresponding to an effectiveness factor of n = 0.0877. Thus the error introduced by the constant diffusivities assumption amounts to less than 5% for this example. Had there been a substantial volume change accompanying the reaction, agreement between the two methods of computation would not have been as good. 

The theory is based upon two newly defined numbers, which we call Aris numbers. This recognizes that, to our knowledge, Aris was the first who substantially contributed to a generalized theory for effectiveness factors, by postulating his shape-generalized Thiele modulus [6]. Aris also wrote a book which gives an excellent survey of all that has been done in this field [31]. 

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 ... 

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... 

Certain rough criteria can be made for the importance of internal diffusion in terms of the general Thiele modulus, h. Consider first the case of the first order reaction in the slab. It is clear from the left-hand curve of Fig. 6.7 that if h is less than I the effect of diffusion limitation is not serious, whereas... 

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. 

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

Figture 16 Effectiveness factor of single-file reaction plotted against the generalized Thiele modulus as defined by Eq. (42). (From Ref. 162.)... 

Using Bischoflf s approximation (1965), implicit expressions can then be derived for the generalized Thiele modulus and overall effectiveness factor E. Based on these, plots can be prepared of E versus (the familiar Thiele modulus for the catalyst) for different values of cta = a[(1 + aM1 )/ w) and Kp [A. A few representative plots are shown in Figure 17.3. 

The effectiveness factor as function of the generalized Thiele modulus is shown in Figure 2.26 for a slab and a sphere. Both curves coincide exactly for -> oo. The maximum deviations are in the order of 10-15%. 

The Weisz-Prater criterion makes use of observable quantities like -Ra)p, the measured global rate (kmol/kg-s) dp, the particle diameter (m) pp, the particle density (kg/m ) Dg, the effective mass diffusivity (m /s) and the surface concentration of reactant (kmol/m ). The intrinsic reaction rate constant ky need not be known in order to use the Weisz-Prater criterion. If external mass transfer effects are eliminated, CAb can be used, and the effective diffusivity can be estimated using catalyst and fluid physical properties. The criterion can be extended to other reaction orders and multiple reactions by using the generalized Thiele modulus, and various functional forms are quoted in the literature [17, 26, 28]. 

To suppress the internal mass transfer resistance in the pores of the solid material, small enough particles should be used. The role of internal mass transfer resistance can be evaluated using the concepts described in Chapter 5, that is, by evaluating the generalized Thiele modulus and the effectiveness factor. 

A property of the effectiveness factor is that the product of the Thiele modulus and effectiveness factor approaches unity as the value of the Thiele modulus approaches infinity. In this asymptotic region of strong diffusion effects, the pellet center concentration approaches zero. Therefore, a generalized Thiele modulus can be defined as ... 

The effective diffusivity can also be determined from reaction conditions provided that the intrinsic kinetics are known. In the region of strong diffusion effects, the product of the effectiveness factor and the generalized Thiele modulus is unity, i.e., riG — 1, where o is given by ... 

Relationship between effectiveness factor and generalized Thiele modulus... 

The value of the generalized Thiele modulus (p will be calculated at the bed inlet and the bed outlet using Eqn. (9-14). The characteristic dimension will be calculated from Eqn. (9-10), and the rate constant will be converted from a weight-of-catalyst basis to a volume-of-catalyst basis using the given particle density. Equation (9-15) or Figure 9-7 can then be used to estimate the effectiveness factor at each position. 

The effectiveness factor t) can be used to estimate the actual rate of reaction, including the effect of internal concentration gradients. For most nth-order reactions, t) can be estimated by calculating the generalized Thiele modulus [Pg.369]

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).

This value of the effectiveness factor is further applied to recalculate the generalized Thiele modulus, followed by another estimation of the corresponding effectiveness factor from the plots. Such an iteration procedure is applied until convergence is reached. 

The catalyst effectiveness factor can be estimated as a function of the Thiele modulus (O). The generalized Thiele modulus for nth-order irreversible reaction is (Froment et al., 2010)... 

Microlevel. The starting point in multiphase reactor selection is the determination of the best particle size (catalyst particles, bubbles, and droplets). The size of catalyst particles should be such that utilization of the catalyst is as high as possible. A measure of catalyst utilization is the effectiveness factor q (see Sections 3.4.1 and 5.4.3) that is inversely related to the Thiele modulus (Eqn. 5.4-78). Generally, the effectiveness factor for Thiele moduli less than 0.5 are sufficiently high, exceeding 0.9. For the reaction under consideration, the particles size should be so small that these limits are met. 

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