Thiele factor

For the case of a first order reaction in a spherical particle a relationship between an efficiency factor, n, and a dimensionless number, the so called Thiele factor,, can be shown to be given by ... 

The overall characteristics of the Arrhenius plot of flow velocity half life are shown in figure 10. The plot is linear with a slope corresponding to the deactivation activation enthalpy of the free enzyme at low temperatures (low Thiele factors). The half life of the immobilized enzyme activity is equal to that of the free enzyme. In the intermediary range of Thiele factors or temperatures the slope slowly decreases then increases again to the slope corresponding to the deactivation activation enthalpy of the free enzyme. At hi Thiele factors or high temperatures the slope again corresponds to the deactivation activation enthalpy but the activity half life of the immobilized enzyme is twice that of the free enzyme. 

The Thiele factor to be substituted in equation (8) is given as a function of time by the following equation ... 

With the aid of the equation for the relationship between Thiele factor and time, equation (20), equation (28) can now be used to calculate productivity as a function of time at various operating temperatures. (In evaluating (28) it has to be borne in mind that k, which was expressed inl/day, has to be transformed to 1/s before substitution in equation (28)). 

The solution was first obtained independently by Wertheim [32] and Thiele [33] using Laplace transfonns. Subsequently, Baxter [34] obtained the same solutions by a Wiener-Hopf factorization teclmique. This method has been generalized to charged hard spheres. 

Table II.1 which depends on the pellet size, so the familiar plot of effectiveness factor versus Thiele modulus shows how t varies with pellet radius. A slightly more interesting case arises if it is desired to exhibit the variation of the effectiveness factor with pressure as the mechanism of diffusion changes from Knudsen streaming to bulk diffusion control [66,...

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). 

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 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. 

The mass transport influence is easy to diagnose experimentally. One measures the rate at various values of the Thiele modulus the modulus is easily changed by variation of R, the particle size. Cmshing and sieving the particles provide catalyst samples for the experiments. If the rate is independent of the particle size, the effectiveness factor is unity for all of them. If the rate is inversely proportional to particle size, the effectiveness factor is less than unity and

experimental points allow triangulation on the curve of Figure 10 and estimation of Tj and ( ). It is also possible to estimate the effective diffusion coefficient and thereby to estimate Tj and ( ) from a single measurement of the rate (48). 

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). 

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

The relationship between effectiveness factor p and Thiele modulus < >l may be calculated for several other regular shapes of particles, where again the characteristic dimension of the particle is defined as the ratio of its volume to its surface area. It is found that... 

The solution of this equation is in the form of a Bessel function 32. Again, the characteristic length of the cylinder may be defined as the ratio of its volume to its surface area in this case, L = rcJ2. It may be seen in Figure 10.13 that, when the effectiveness factor rj is plotted against the normalised Thiele modulus, the curve for the cylinder lies between the curves for the slab and the sphere. Furthermore, for these three particles, the effectiveness factor is not critically dependent on shape. 

Estimate the Thiele modulus and the effectiveness factor for a reactor in which the catalyst particles are ... 

Figure 11.13. Dependence of promotional effectiveness factor, r]p, on Thiele modulus 0>p and dimensionless current J.23...

The quantitative solution to the problem is given in section 11.3. The effectiveness factor T)P (< 1) which expresses the extent to which the promoting ion is fully utilized (qP=l) depends on three dimensionless parameters n, J and P n is the dimensionless dipole moment of the promoting ion, J is a dimensionless current and P, a promotional Thiele modulus, is proportional to the film thickness, L. 

II. The promotional effectiveness factor, t]p, (Chapter 11) must be significant, larger than, at least, 0.1. This requires small promotional Thiele modulus, Op, and significant dimensionless current, J, values. This implies thin (low L) catalyst films and slow kinetics of promoter destmction (low k values, Chapter 11). 

Other peptides that have been linked to the actions of ethanol are corticotropin-releasing factor, urocortin, leptin, cholecystokinin, melanocortins, and galanin (for reviews, see Cowen et al. 2004 Egli 2003 Thiele et al. 2003). 

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.

Various simple equations have been available. However, each equation has its own disadvantages. For example, the equation by Wijnggaarden et al.[3] showing excellent estimates of effectiveness factors for various problems fails to provides appropriate estimates for a large Thiele modulus when f (l) is negative[2]. It is well-known that... 

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.

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 ... 

In order for diffusional limitations to be negligible, the effectiveness factor must be close to 1, i.e. nearly complete catalyst utilization, which requires that the Thiele modulus is suffieiently small (< ca. 0.5), see Figure 3.32. Therefore, the surface-over-volume ratio must be as large as possible (particle size as small as possible) from a diffusion (and heat-transfer) point of view. There are many different catalyst shapes that have different SA/V ratios for a given size. 

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

Fig. 5.4-14. Effectiveness factor versus Thiele modulus for first-order reaction.

---