Effectiveness factor greater than unity

Reactions in porous catalyst pellets are Invariably accompanied by thermal effects associated with the heat of reaction. Particularly In the case of exothermic reactions these may have a marked influence on the solutions, and hence on the effectiveness factor, leading to effectiveness factors greater than unity and, In certain circumstances, multiple steady state solutions with given boundary conditions [78]. These phenomena have attracted a great deal of interest and attention in recent years, and an excellent account of our present state of knowledge has been given by Arls [45]. 

Operative. For the non isothermal case, effectiveness factors greater than unity are possible. Weisz and Hicks have considered this problem in some detail and constructed a number of graphs for various heats of reaction and activation energies. When a reaction is limited by pore diffusion, the reaction rate is proportional to yjky. If the temperature effects can be expressed as a simple Arrhenius relationship = A txp —E/RT), then the measured activation energy E will be about half the true activation energy. Very low values of the activation energy, i.e, 1-2 kcal. mole are only observed when mass transfer to the external catalyst surface is limiting the rate. 

One of the most interesting features is that for > 0 (exothermic), there are regions where > 1. This behavior is based on the physical reasoning that with sufficient temperature rise caused by heat transfer limitations, the increase in the rate constant, Ic,., more than offsets the decrease in reactant concentration, C so that the internal rate is actually larger than that at surface conditions of C/ and T, , leading to an effectiveness factor greater than unity. The converse is, of course, true for endothermic reactions. 

For reactions between ions of unlike sign, solvation effects give an increase in entropy on activation. Consideration of internal structure gives a decrease in A S and the two effects could balance, or partially balance, each other. The electrostatic modifications would, however predict an increase in A S, indicating p factors greater than unity. Since the other two effects are in opposite directions to each other, the electrostatic modifications must dominate, with this being the major factor influencing the A factors. 

The reactant concentration C will be greater than zero throughout the entire length of the pore provided that h0 < y/2. In this case the effectiveness factor will be unity because the reaction rate is independent of concentration. For values of h0 > yfl, equation 12.3.44 would call for negative values of the reactant concentration at large values of x/L, a situation that is clearly impossible. Hence the boundary conditions on equation 12.3.43 must be changed so that both the reactant concentration and its gradient become zero at a point in the pore that we label with a coordinate xc. In this situation, the concentration profile becomes... 

Recall that k T) is a strong function of temperature. The effectiveness factor for an exothermic reaction can be le than, equal to, or greater than unity, depending on how k T) increases relative to F(Q) within the particle. Thus, there are cases where the increase in k T) can be much larger than the decrease in F Ca), for example,... 

By convention, the magnitude of an equilibrium isotope effect is expressed as a fractionation factor (a). If the product is enriched in the heavy isotope relative to the reactant, a is greater than unity. For reaction (5.3), a has the form... 

An important factor in commercial operation is the relative amounts of alkene produced, relative to alkanes. Alkene/alkane ratios for the Cl to C5 range are presented in Fig. 3 for n-hexadecane and for 1% and 10% additions of quinoline and phenanthrene to the n-hexadecane feedstock. In all cases the ratio was greater than unity, with 1% addition of additives having relatively little effect on this ratio. However, at 10% addition, phenanthrene enhanced this ratio, whilst quinoline showed a corresponding decrease. Thus, although these additives diminished the individual yields of the gaseous components, with a marked reduction in the case of quinoline, small concentrations had little effect on the alkene/alkane ratio. 

In practice, (f) can be calculated by inserting experimental copolymerization rates into Eq. (7.64). The values of (j> thus obtained are frequently greater than unity, and these deviations are ascribed to polar effects that favor cross-termination over homotermination. However, this is not always unambiguous, since the apparent cross-termination factor may vary with monomer feed composition in a given system [25,26]. It is clear also that termination reactions are at least partially diffusion controlled [27,28]. A dependence of segmental diffusivity on the structure of macroradicals is to be expected and dependence of diffusion controlled termination on copolymer composition seems reasonable. It is therefore plausible that the value of the overall termination rate constant ku in copolymerizations should be functions of fractions F and Fi) of the comonomers incorporated in the copolymer. An empirical expression for ku has thus been proposed [27] ... 

Effectiveness factor versus Thiele modulus in a spherical pellet reaction orders greater than unity.. 378... 

For the single-reaction, nonisothermal problem, we solved the so-called Weisz-Hicks problem, and determined the temperature and concentration profiles within the pellet. We showed the effectiveness factor can be greater than unity for this case. Multiple steady-state solutions also are possible for this problem, but for realistic values of the... 

Figures 7.5 and 7.6 display the effectiveness factor versus Thiele modulus relationship given in Equation 7.33. The log-log scale in Figure 7.6 is particularly useful, and we see the two asymptotic limits of Equation 733. At small p 1, and at large q l/4>. Figure 7.6 shows that the asymptote q = I/O is an excellent approximation for the spherical pellet for O > 10. For large values of the Thiele modulus, the rate of reaction is much greater than the rate of diffusion, the effectiveness factor is much less than unity, and we say the pellet is diffusion limited. Conversely, when the diffusion rate is much larger than the reaction rate, the effectiveness factor is near unity, and we say the pellet is reaction limited.

In other words, reactants exist everywhere within the pores of the catalyst when the chemical reaction rate is slow enough relative to intrapellet diffusion, and the intrapellet Damkohler number is less than, or equal to, its critical value. These conditions lead to an effectiveness factor of unity for zerofli-order kinetics. When the intrapellet Damkohler number is greater than Acnticai, the central core of the catalyst is reactant starved because criticai is between 0 and 1, and the effectiveness factor decreases below unity because only the outer shell of the pellet is used to convert reactants to products. In fact, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler number for zeroth-order kinetics exhibits an abrupt change in slope when A = Acriticai- Critical spatial coordinates and critical intrapeUet Damkohler numbers are not required to analyze homogeneous diffusion and chemical reaction problems in catalytic pellets when the reaction order is different from zeroth-order. When the molar density appears explicitly in the rate law for nth-order chemical kinetics (i.e., n > 0), the rate of reaction antomaticaUy becomes extremely small when the reactants vanish. Furthermore, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler nnmber does not exhibit an abrupt change in slope when the rate of reaction is different from zeroth-order. 

The factor x (Equation 8.14) depends on two key ratios GmlEf, which is typically 0.01-0.02 and Rlr, which is not much greater than unity. Figure 8.5 indicates that the length correction factor becomes significant for values of ax less than 10. In practice the corresponding aspect ratio for effective reinforcement is usually greater than 100. 

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