Thiele modulus zero-order reaction

Figure 12.7 adapted from Satterfield (40) contains a plot of the effectiveness factor for a zero-order reaction versus the Thiele modulus... 

The modulus defined by eqn. (10) then has the advantage that the asymptotes to t (0) are approximately coincident for a variety of particle shapes and reaction orders, with the specific exception of a zero-order reaction (n = 0), for which t = 1 when 0 < 1 and 77 = 1/0 when 0 > 1. The curve of 77 as a function of 0 is thus quite general for practical catalyst pellets. Figure 2 illustrates the form of For 0 > 3, it is found that 77 = 1/0 to an accuracy within 0.5%, while the approximation is within 3.5% for 0 > 2. The errors involved in using the generalised curve to estimate 77 are probably no greater than the errors perpetrated by estimating values of parameters in the Thiele modulus. 

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 ... 

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. 

The latter depends on reactant concentration, as in conventional kinetics 267, 402), as well as on the catalyst potential (412). Figure 33 shows the drastic decrease of the local concentration along a pore for a zero-order reaction as the Thiele modulus increases. Similar results apply to other order reactions but with concentration approaching zero only asymptotically. If external mass transport is fast compared to pore diffusion,... 

Recognizing the second tenii is just the ratio of a reaction rale to a diffusion rate for a zero order reaction, we call this ratio the Thiele modulus, We divide and multiply by two to facilitate the integration ... 

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... 

In the case of a zero-order reaction (r = feof) fhe Thiele modulus jS has been modified to suit the interpretation for biofilm kinetics. It is then the reciprocal of the normal Thiele modulus for a zero-order reaction,... 

In Figure 6a we show E as a function of X with the Thiele modulus as a parameter,for first order irreversible reactions.Similar plots can be calculated for zero order reactions as shown in Figure 6b. 

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... 

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... 

Figure 23. Relative selectivity A. 2,i of a Type II reaction (first and zero order) plotted against the observable modulus

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. 

The concentration history appears to be a function of three dimensionless parameters, a modified Thiele modulus, y, the mass Biot number, gj, and the dimensionless feed concentration, /3. The set of non-linear equations is uncoupled by introducing an effectiveness factor, r, and numerically solved. In order to reduce computer time, the effectiveness factor has been conveniently expressed as a weighted sum of its value for the zero and first order reaction rate. Different regime conditions are depicted in terms of a dimensionless parameter,... 

The last parameter is the reciprocal of saturation factor 0. Consequently, low values of p,(fi 0) correspond to a first-order kinetics, whereas high values (j3 -> oo) approximate the actual reaction rate to zero-order kinetics. The square of the Thiele modulus has a precise physical meaning, since it compares a first-order reaction rate with a pure diffusion rate. 

This difference is maximized, if CA.center becomes zero. Figure 4.7.1 shows the influence of the Thiele modulus on the concentration of reactant A in the center of a spherical particle (first-order reaction), calculated by the equation in Table 4.5.5 [note that sinh(ax)/x= a for Oj ... 

Utilize the data given in Problem 4.5, assume the benzene hydrogenation reaction is zero order in benzene and first order in H2, and then calculate the Thiele modulus for the largest and smallest catalyst particles assuming all the Pd is in the mesopores. What is the Thiele modulus for the largest (500 p) particles if all the Pd were dispersed in only the micropores Should there be any concern about pore diffusion limitations for any of the three possibilities Why ... 

Ramachandran and Smith (7) assumed the surface area per unit volume of the pore, given by tCq/2, is equal to that of the solid particles from which the experimental data are obtained. Furthermore the effective length of the pore L is chosen so that the Thiele modulus at zero time given by eq. (19) is equal to that characterizing reaction and diffusion in a sphere of radius R with first order kinetics. It then follows that... 

In the case of internal diffusion, analytical solutions can be used direcdy only for simple kinetics, such as first or zero orders thus, approximations should be applied, since many reactions are not of zero or first order. Alternatively, utilization of the generalized diagrams of Aris [7] is possible (Fig. 10.19). Such diagrams relate for various kinetics effectiveness factors and the generafized Thiele modulus expressed by Eq. (10.121). 

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