Effectiveness factors spherical catalyst pellets

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

Effectiveness factor ratios for first-order kinetics on spherical catalyst pellets. 

Using this definition of the Thiele modulus, the reaction rate measurements for finely divided catalyst particles noted below, and the additional property values cited below, determine the effectiveness factor for 0.5 in. spherical catalyst pellets fabricated from these particles. Comment on the reasons for the discrepancy between the calculated value of rj and the ratio of the observed rate for 0.5 in. pellets to that for fine particles. 

This relation is plotted as curve Bin Figure 12.11. Smith (66) has shown that the same limiting forms for are observed using the concept of effective dififusivities and spherical catalyst pellets. Curve B indicates that, for fast reactions on catalyst surfaces where the poisoned sites are uniformly distributed over the pore surface, the apparent activity of the catalyst declines much less rapidly than for the case where catalyst effectiveness factors approach unity. Under these circumstances, the catalyst effectiveness factors are considerably less than unity, and the effects of the portion of the poison adsorbed near the closed end of the pore are not as apparent as in the earlier case for small hr. With poisoning, the Thiele modulus hp decreases, and the reaction merely penetrates deeper into the pore. 

A first-order chemical reaction occurs isothermally in a reactor packed with spherical catalyst pellets of radius R. If there is a resistance to mass transfer from the main fluid stream to the surface of the particle in addition to a resistance within the particle, show that the effectiveness factor for the pellet is given by ... 

Derive an expression for the effectiveness factor of a spherical catalyst pellet in which a first-order isothermal reaction occurs. 

Here we consider a spherical catalyst pellet with negligible intraparticle mass- and negligible heat-transfer resistances. Such a pellet is nonporous with a high thermal conductivity and with external mass and heat transfer resistances only between the surface of the pellet and the bulk fluid. Thus only the external heat- and mass-transfer resistances are considered in developing the pellet equations that calculate the effectiveness factor rj at every point along the length of the reactor. 

Figure 6 presents the effectiveness factors of a spherical catalyst pellet for reaction orders of 1.0, 2.0 ( Figure 6a) and 0.5 (up curve in Figure 6b). A reasonable agreement between the four-term decomposition solutions and the finite difference method is achieved over most of the range of Thiele modulus.

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-... 

So far, we have obtained an explicit expression for the normalized concentration of reactant Ai inside the pellet. Analogously to eq 42, which relates the effectiveness factor of a spherical catalyst pellet to the nor-... 

Figure 6.3.8 illustrates the relationship between the effectiveness factor and the Thiele modulus for a spherical catalyst pellet. 

Since the equations are nonlinear, a numerical solution method is required. Weisz and Hicks calculated the effectiveness factor for a first-order reaction in a spherical catalyst pellet as a function of the Thiele modulus for various values of the Prater number [P. B. Weisz and J. S. Hicks, Chem. Eng. Sci., 17 (1962) 265]. Figure 6.3.12 summarizes the results for an Arrhenius number equal to 30. Since the Arrhenius number is directly proportional to the activation energy, a higher value of y corresponds to a greater sensitivity to temperature. The most important conclusion to draw from Figure 6.3.12 is that effectiveness factors for exothermic reactions (positive values of j8) can exceed unity, depending on the characteristics of the pellet and the reaction. In the narrow range of the Thiele modulus between about 0.1 and 1, three different values of the effectiveness factor can be found (but only two represent stable steady states). The ultimate reaction rate that is achieved in the pellet... 

In this section we will develop the internal effectiveness factor for spherical catalyst pellets. The development of models that treat individual pores and pellets of different shapes is undertaken in the problems at the end of this chapter. We will first look at the internal mass transfer resistance to either the products or reactants that occurs between the external pellet surface and the interior of the pellet. To illustrate the salient principles of this model, we consider the irreversible isomerization... 

Internal effectiveness factor for a first-order reaction in a spherical catalyst pellet... 

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 12-6 Effectiveness factor ratios for first-order kinetics on spherical catalyst pellets for various values of the Thiele modulus, <1), for a sphere. [From V, W. Weekman and R. L. Goring, J. Catal, 4, 260 (1965).]...

Fig. 11-10 Nonisothermal effectiveness factors for first-order reactions in spherical catalyst pellets...

Reactions in series were discussed in Section 2.4.2.1. For reactions occurring in a spherical catalyst pellet with the pore radius R, the concentration of the intermediate product, B, is proportional to the overall effectiveness factor given by... 

The molecular weight differences between lignin and its model compounds also complicate the use of model compound kinetics in a predictive simulation. The mobility of a high-molecular weight polymer would be much less than that of smaller model substrates (14). As for catalyst decay, a simple model was used to probe transport issues. For a first order, irreversible reaction in an isothermal, spherical catalyst pellet with equimolar counterdiffusion, the catalyst effectiveness factor and Thiele modulus provide the relevant information as... 

Figure 12.6 Effectiveness factor plot for spherical catalyst pellets based on the effective diffusivity model for a first-order reaction.

Figure 6.5 Effectiveness factor vs. Thiele modulus for nonisothermal first-order chemical reaction within a spherical catalyst pellet.

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). 

A hydrocarbon is cracked using a silica-alumina catalyst in the form of spherical pellets of mean diameter 2.0 mm. When the reactant concentration is 0.011 kmol/m3, the reaction rate is 8.2 x 10"2 kmol/(m3 catalyst) s. If the reaction is of first-order and the effective diffusivity De is 7.5 x 10 s m2/s, calculate the value of the effectiveness factor r). It may be assumed that the effect of mass transfer resistance in the. fluid external Lo the particles may be neglected. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Effectiveness Factors for Nonisother-mal Catalyst Pellets. Here we indicate how previous effectiveness factor analyses may be extended to situations where the pellet is not isothermal. Consider the case of a spherical... 

For spherical particle geometry, as in the case of a microbial floe, a pellet of mould or a bead of gel-entrapped enzyme, the expression for the effectiveness factor can again be derived by a procedure similar to that used in Chapter 3 for a spherical pellet of conventional catalyst. A material balance for the substrate across an elementary shell of radius r and thickness dr within the pellet will yield ... 

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