Effectiveness factors nonisothermal reactions

FIGURE 10.3 Nonisothermal effectiveness factors for first-order reactions in spherical pellets. (Adapted from Weisz, P. B. and Hicks, J. S., Chem. Eng. Sci., 17, 265 (1962).)... 

ILLUSTRATION 12.4 EFFECTIVENESS FACTOR DETERMINATION FOR A NONISOTHERMAL CATALYST PELLET-EXOTHERMIC REACTION... 

Figure 13. Internal effectiveness factor as a function of the Thiele modulus for nonisothermal reactions at different values for the Prater number and y, = 10 (numerical solutions for a first order reaction).

The above suggests that the discussion of the Aris numbers for simple reactions also holds for nonisothermal pellets. For example, effectiveness factors larger than one are found if the number An, becomes negative. According to Equation 7.14 this is the case if... 

Therefore, Equations 8.48 and 8.49 can be combined into one equation for concentration only. The effectiveness factor for the case considered can be calculated with the technique described in Chapter 7. It is important to stress that the effectiveness factor changes along the reactor because parameters of the reaction rate expression, Equation 6.18, e and a depend on the surface concentration and temperature. The calculated modified effectiveness factors for nonisothermal first-order reaction at different conversions = (1- CAJCa) are shown in Figure 8.9 versus the ratio of inner and outer diameters of the hollow cylinder. The parameters chosen for the calculation are ... 

The above discussion of effectiveness factors is valid only for isothermal conditions. When a reaction is exothermic and nonisothermal, the effectiveness factor can be significantly greater than 1 as shown in Figiue 12-7. Values of t greater than 1 occur because the external smface temperature of the pellet is less than the temperature inside the pellet where the exothermic reaction is taking place. Therefore, the rate of reaction inside the pellet is greater than the rate at the surface. Thus, because the effectiveness factor is the ratio of the actual reaction rate to the rate at smface conditions, the effectiveness factor... 

Fig. 6.10 Effectiveness factors for nonisothermal first order reaction in the slab. [Adapted from Petersen (see references) with permission.]...

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

In the previous examples, we have exploited the idea of an effectiveness factor to reduce fixed-bed reactor models to the same form as plug-flow reactor models. This approach is useful and solves several important cases, but this approach is also limited and can take us only So far. In the general case, we must contend with multiple reactions that are not first order, nonconstant thermochemical properties, and nonisothermal behavior in the pellet and the fluid. For these cases, we have no alternative but to solve numerically for the temperature and species concentrations profiles in both the pellet and the bed. As a final example, we compute the numerical solution to a problem of this type. 

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... 

Since the major thermal resistances in nonisothermal reaction systems are encountered in the boundary layer, while the major mass transfer resistances occur within the particle, we can entertain some simplification of the overall effectiveness factor problem we have been considering. This simplified model envisions interphase temperature gradients and intraphase concentration gradients only. For this case... 

Figure 3.7.a-I Effectiveness factor with first-order reaction in a spherical nonisothermal catalyst pellet from Weisz and Hicks [112]). 

ILLUSTRATION 12.4 Effectiveness Factor Determination for a Nonisothermal Catalyst Pellet Employed to Effect an Exothermic Reaction... 

On the other hand, the effectiveness factor varies significantly with temperature. Therefore, for nonisothermal conditions, especially in the exothermic reactions, it will vary since the temperature varies within the pores or particles due to the temperature gradient caused by the chemical reaction. It is then necessary to construct an... 

Figure 2.29 Effectiveness factor as function of the Thiele modulus. Nonisothermal sphere, first order reaction [30]. (Adapted with permission from Elsevier.)...

For non-isothermal kinetics, we adapt the result derived by Tavera [100] originally for Dirichlet conditions, and as expected b will be a function of kinetic descriptors (e.g., order of reaction w), nonisothermal parameters (y and P), and surface conditions (c j, and T, y). The dependence on the latter quantities (generally unknown) leads to an implicit calculation of the effectiveness factor. Substituting Equation 3.42 into 3.54a,... 

Theoretical work has been done on the effectiveness factor for nonisothermal particles. Fig. 3.13.1-1 shows the results of the computations by Weisz and Hicks [1962] for y = EIRT = 20. For > 0.1, that is, for sufficiently exothermic reactions, the effectiveness factor can exceed the value of 1. In such a case the temperature rise, which increases the value of the rate constant, would more than offset the decrease in reactant concentration Cas, so that fA averaged over the particle exceeds that at surface conditions. The converse is true for endothermic reactions. 

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