Thiele modulus, nonisothermal

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

The reactor feed mixture was "prepared so as to contain less than 17% ethylene (remainder hydrogen) so that the change in total moles within the catalyst pore structure would be small. This reduced the variation in total pressure and its effect on the reaction rate, so as to permit comparison of experiment results with theoretical predictions [e.g., those of Weisz and Hicks (61)]. Since the numerical solutions to the nonisothermal catalyst problem also presumed first-order kinetics, they determined the Thiele modulus by forcing the observed rate to fit this form even though they recognized that a Hougen-Watson type rate expression would have been more appropriate. Hence their Thiele modulus was defined as... 

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).

Figure 9.4. Effectiveness factor as function of modified Thiele modulus under nonisothermal conditions (from Weisz and Hicks [27]).

Another approach, which requires a bit of foreknowledge concerning the magnitude of the Thiele modulus likely to be encountered, employs the direct analysis of Liu given in equations (7-28) and (7-33). For example, if we have a first-order nonisothermal reaction system that fits the restrictions accompanying equation (7-33), then... 

In these equations, (j> is the Thiele modulus at any point within the catalyst, 0, is the modulus at the surface, and the parameters a, and are, as in any fluid-solid (catalytic) reaction (Chapter 7), the additional parameters necessary to characterize nonisothermal operation. 

Simulation results are plotted as RX concentration versus time in Figure 19.12 for both isothermal and nonisothermal situations (it is only necessary to ignore the heat transfer groups to simulate the isothermal condition). Clearly, there is some effect at higher reaction times, but it is not significant. From the values of the Thiele modulus ((j> = 0.33), we can assume that the reaction is kinetically controlled. The low value of 0 also justifies the small effect of nonisothermicity. 

In the results reported by Weisz and Hicks [39], the nonisothermal t] was described in terms of three dimensionless parameters, that is, two new independent parameters were introduced in addition to the Thiele modulus ... 

Figure 9 -Effectiveness factor for first order irreversible reaction,convection and diffusion in nonisothermal slab catalysts as a function of the Thiele modulus (Y=20 6=0.1 X =0)...

A typical, unsealed plot of versus the nonisothermal Thiele modulus is shown in Figure 9.10. Two additional parameters that contain the thermal factors make their appearance here the Arrhenius number EJRT which contains the important activation energy E and the dimensionless parameter P, which reflects the effect due to the heat of reaction and the transport resistances. For p = 0 (i.e., for a vanishing heat of reaction or infinite thermal conductivity), the effectiveness factor reduces to that of the isothermal case. P > 0 denotes an exothermic reaction, and here the rise in temperature in the interior of the pellet is seen to have a significant impact on E which may rise above unity and reach values as high as 100. This means that the overall reaction rate in the pellet is up to 100 times faster than would be the case at the prevailing surface conditions. This is due to the strong exponential dependence of reaction rate on temperature, as expressed by the Arrhenius relation... 

The intraparticle deactivation of nonisothermal pellets has been analyzed by Ray (21) using a pore mouth poisoning, slab geometry model. Heat flux at the boundary of the active and inactive portions of the particle (Figure 2a) was computed via Bischoff s ( ) asymptotic solution for large Thiele modulus. An effective diffusional modulus for this case can be defined as ... 

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

---