Thiele modulus, discussion

The concentration dependence of z/l vs. c/c0 is plotted in Figure 11.14a. It can be seen that from a Thiele modulus cp > 3 the educt does not reach the internal part of the pore. The inner part of the pore system is useless for catalysis. This is especially relevant if expensive metals serve as active components on a porous carrier, which are then wasted. There are chances to master this diffusion limitation, which will be discussed later in detail. Another important variable is the efficiency factor tj. The efficiency factor r is defined as the quotient of the speed of reaction rs to the maximal possible speed of reaction rsmax. r is related to q> as the quotient of the hyperbolic tangent of the Thiele modulus qy. 

The above discussion shows that the progress of deactivation may occur in different ways depending on the type of decay reaction occurring and on the value of the pore diffusion factor. For parallel and series poisoning, the Thiele modulus for the main reaction is the pertinent pore diffusion parameter. For side-by-side reactions, the Thiele modulus for the deactivation is the prime parameter. 

It may also be useful to note that for simple Michaelis-Menten kinetics the Thiele modulus we have been discussing is given, in conventional notation, by... 

In the case of distillate hydrotreating the simulation models are simply described even for the complicated reaction scheme employed in the reaction model as discussed in the later section, because the catalyst deactivation is not necessarily predicted for local sites in the system. And the assumption confirmed in the previous discussion that the relationship between the Thiele-modulus and effectiveness factor is approximately represented by that of a first order reaction for any reaction order makes the simulation model simpler and easier to develop. 

This is a rather nasty problem to solve numerically, because boundary conditions over the whole range of x are assigned at t = 0 and t = 1. A perturbation expansion around l/P = 0 yields, as expected, the CSTR at the zero-order level at all higher orders, one has a nested series of second-order linear nonhomogeneous differential equations that can be solved analytically if the lower order solution is available. The whole problem thus reduces to the solution of Eq. (133), which has been discussed before. This is, of course, the high-diffusivity limit that corresponds to a small Thiele modulus in the porous catalyst problem. 

As discussed earlier, the effectiveness factor is simply the inverse of the Thiele modulus for the case of severe diffusional limitations (Figure 6.3.9.) Thus, the observed rate under strong diffusional limitations can be written as ... 

We now introduce dimensionless variables

catalytic reactions, the Thiele modulus. Let... 

Closure After completing this chapter, the reader should be able to derive differential equations describing diffusion and reaction, discuss the meaning of the effectiveness factor and its relationship to the Thiele modulus, and identify the regions of mass transfer control and reaction rate control. The reader should be able to apply the Weisz-Prater and Mears criteria to identify gradients and diffusion limitations. These principles should be able to be applied to catalyst particles as well as biomaierial tissue engineering. The reader should be able to apply the overall effectiveness factor to a packed bed reactor to calculate the conversion at the exit of the reactor. The reader should be able to describe the reaction and transport steps in slurry reactors, trickle bed reactors, fluidized-besd reactors, and CVD boat reactors and to make calculations for each reactor. 

The discussion above leaves the impression that there in a separate Thiele modulus and corresponding effectiveness factor for every specific form of reaction kinetics. Unfortunately, this is indeed so, and for most complex kinetic laws the mathematics is not tractable to direct analytical solution. Is it reasonable to see if we can pose the problem in more general form to get around this If so, how can this be done ... 

This is not a theoretical dependence of on temperature but is useful for the following discussions. At strong influence of internal diffusion on the reaction rate, the effectiveness factor was found to be inversely proportional to the Thiele modulus (e.g.. Equation 2.176). Accordingly, the effective rate constant is given by ... 

Figure 1 shows the dependence effectiveness factor tj = roi/ri on Thiele modulus for the selected values of parameter p and for = 2, v = 2 (sphere), yt = 7, yi = 12, 5 = 1 (endothermic reactions) and xa(1) = 1. One can find that for Peffectiveness factor rj assumes in the certain ranges of Thiele modulus values much higher than unity. This means that in the cases discussed the internal diffusion, in contrast to the classical isothermal or endothermic catalytic reactions, may considerably increase the rate of the heterogeneous autocatalytic reactions. 

Note that the length scale in the definition of the Thiele mod-tdus in Equation 8.46 is now as given in Equation 8.42a. The extension of this result to nonlinear kinetics and nonuniform coating geometries was also discussed in Lopes et al. [94]. Reaction rate expressions other than first order can be treated using the concept of generalized Thiele modulus O, where the previously defined parameter is modified so that its introduction in an expression such as... 

Before going on to discuss the importance of the effectiveness factor for reactor design let us briefly emphasize the point that we need in the previous treatment to know the Thiele modulus (and hence the kinetic constant k) in order to calculate n. In practice... 

Catalyst pellets are on occasion cast in the form of hollow cylinders (Raschig Rings). Discuss the advantages and drawbacks of this geometry. How would you define the dimension needed to define the Thiele modulus ... 

In many cases in the Hterature, when experimental data are reported for the influence of the reaction rate or TOP on the cluster size the impact of mass transfer, internal diffosion is not discussed in particular. Intuitively, it can be anticipated that when the experimental observations are influenced by mass transfer, the reaction kinetics and thus cluster-size dependence should be less prominent. For the two-step sequence it is, however, worth to consider such dependence in a quantitative way. As discussed in the previous section, for the case of Langmuir adsorption and subsequent transformation of the adsorbed species, the plots for the catalyst effectiveness factor dependence of the Thiele modulus are available when the rate expression is given by v = kKC/ + KC) and the Thiele modulus rp is rp = L kKCs/ D 1 + KQ)), where Dg is the effective diffosion coefficient, Q is the concentration on the surface and L is the grain characteristic length (ratio of volume to surface area). When 3, the effectiveness factor is inversely proportional to the Thiele modulus and thus to the particle size. 

In discussions of mass transfer, this ratio is called the second Damkohler number and given the symbol Dm. In discussions of catalysis, it is called the square of the Thiele modulus and given the symbol In either case, it is central to the description of coupled... 

As discussed above, the transport properties of porous catalyst particles of ca 3 to 100 pm are extremely important for the selectivity of catalytic reactions in which the desired initial products are liable to further reaction to undesired material. The ratio of the rate of catalytic reaction to that of transport within the pore system of catalyst particles is represented by Thiele s modulus [1], which is proportional to the pore length and to the square root of the diameter of the pores. Accordingly reducing the size of the catalyst particles is more elfective than increasing the diameter of the pores. 

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