Effectiveness factor plot first-order reaction

Plot of effectiveness factor versus Thiele modulus for first-order reaction. 

Effectiveness factor plot for spherical catalyst particles based on effective diffusivities (first-order reaction). 

A plot of the effectiveness factor from cq 53 against the Weisz modulus 1ppn from cq 58 gives the curve depicted in Fig. 8 for a first order reaction (flat plate). On the basis of this diagram, the effectiveness factor can be determined easily once the effective reaction rate and the effective diffusivity arc known. 

From now on the two-parameter model is used because it is almost as accurate as the three-parameter model and it gives a better insight. For example, the curves which were drawn by Weisz and Hicks [2] for different values of a and s (Figure 6.4) reduce to one. This is illustrated in Figure 7.1 where the effectiveness factor is plotted versus An0 for several values of C, and for a first-order reaction occurring in a slab. Notice that all the curves in Figure 7.1 coincide in the low ij region, since ij is plotted versus An0. The formulae used for Ana now follow. 

Figure 12-5 (a) Effectiveness factor plot for nth-order kinetics spherical catalyst particles (from Mass Transfer in Heterogeneous Catalysis, hy C. N. Satterfield, 1970 reprint edition Robert E. Krieger Publishing Co., 1981 reprinted by permission of the author), (b) First-order reaction in different pellet geometries (from R. Aris, Introduction to the Analysis of Chemical Reactors, 1965, p. 131 reprinted by permission of Prentice-Hall, Englewood Cliffs, NJ)... 

Catalyst deactivation in large-pore slab catalysts, where intrapaiticle convection, diffusion and first order reaction are the competing processes, is analyzed by uniform and shell-progressive models. Analytical solutions arc provid as well as plots of effectiveness factors as a function of model parameters as a basis for steady-state reactor design. 

A plot of the effectiveness factor as a function of the Thiele modulus is shown in Figure 12-5. Figure l2-3(a) shows t) as a function of the Thiele modulus < )j for a spherical catalyst pellet for reactions of zero, first, and second order. Figure 12-5(b) corresponds to a first-order reaction occurring in three differently shaped pellets of volume Vp and external surface area Ap, and the Thiele modulus for a first-order reaction, < >], is defined difierently for each shape. When volume change accompanies a reaction (i.e., 0) the corrections shown in Figure 12-6 apply to the effectiveness factor for a first-order reaction. 

Figure 12.2 Plot of effectiveness factor versus Thiele modulus for the first-order reaction hj = LyJlk l/ 7T) ).

Figure 17.2 Overall effectiveness factor plots for a first-order reaction (Chaudhari and...

The 4>s versus if plot has the same shape as that of (pi versus ij for a flat plate but is shifted by a factor of 3 on the same log-log plot, if Equations 2.60 and 2.63 are used as basis. Aris (29,38] has shown that, for a first-order reaction in various particle geometries, the plots between the Thiele modulus q> and the isothermal internal effectiveness factor ij become identical at high and low values of

size parameter in the Thiele modulus is defined on a common basis. The characteristic size parameter L is, therefore, defined as the ratio of the particle volume to the external surface area available for reactant penetration, which enables its use for any arbitrary particle shape ... 

A variety of concave pyridines 3 (Table 1) and open-chain analogues have been tested in the addition of ethanol to diphenylketene (59a). Pseudo-first-order rate constants in dichloromethane have been determined photometrically at 25 °C by recording the disappearance of the ketene absorption [47]. In comparison to the uncatalyzed addition of ethanol to the ketene 59a, accelerations of 3 to 25(X) were found under the reaction conditions chosen. Two factors determine the effectiveness of a catalyst basicity and sterical shielding. Using a Bronsted plot, these two influences could be separated from one another. Figure 4 shows a Bronsted plot for some selected concave pyridines 3 and pyridine itself (50). 

Presumably less nucleophilically assisted solvolyses could show higher a-deuterium isotope effects, and there is a linear relationship between the magnitude of nucleophilic solvent assistance (Table 2) and the a-deuterium isotope effect for solvolyses of 2-propyl sulpho-nates (Fig. 7). Another measure of nucleophilic assistance is the ratio k2 (OH )/, where k2 is the second-order rate constant for nucleophilic attack by OH and kx is the first-order rate constant for reaction with the solvent water, and a linear correlation was obtained by plotting the ratio versus the experimentally observed isotope effects for methyl and trideuteriomethyl sulphonates, chlorides, bromides and iodides (Hartman and Robertson, 1960). Using fractionation factors the latter correlation may also be explained by a leaving group effect on initial state vibrational frequencies (Hartshorn and Shiner, 1972), but there seems to be no sound evidence to support the view that Sn2 reactions must give a-deuterium isotope effects of 1-06 or less. 

This is illustrated in Figure 7.4 where the effectiveness factor is plotted versus the low ij Aris number An0 for a bimolecular reaction with (1,1) kinetics, and for several values of/ . P lies between 0 and 1, calculations were made with a numerical method. Again all curves coincide in the low tj region, because rj is plotted versus An0. For p = 0, the excess of component B is very large and the reaction becomes first order in component A. For p = 1, A and B match stoichiometrically and the reaction becomes pseudosecond order in component A (and B for that matter). Hence the rj-An0 graphs for simple first- and second-order reactions are the boundaries when varying p. 

M. Calvin and 11. W. Alter, J. Chem. Phys., 19, 768 (1952), have attempted to investigate cis-trans isomerizations in the liquid phase. They found experimental difficulties in side reactions, and while they could represent their data by means of first-order plots, they did not investigate any catalytic effects of the presemte of free radicals. Their frequency factors fell in the range 10 sec to 10 sec , for wliich they found no reasonable explanation. 

Figure 4.32. Plots of effectiveness factor of reaction f/r,ext versus modulus in case of external transport limitation as a function of substrate concentration s/Xg. The rate-determining steps (rds) are indicated with their range of validity by dotted lines, together with the limiting first-order effectiveness factor at low s values. (From Horvath and Engasser, 1974.)...

The results indicate that the use of Eq. (3.3.33), which is exact for first-order kinetics, for other reaction orders may introduce a serious error. This is in contrast to the result for the heterogeneous catalysis in a porous pellet or on a solid surface, in which the difference between the effectiveness factors for different reaction orders is much smaller, when plotted against the Thiele modulus generalized for different reaction order [25, 26]. 

The team has brainstormed the factors and developed a prioritized list of the key ones. For each factor, ranges were established that were well outside of the anticipated operating range in order to maximize the effect size (but not so large as to change the fundamental reaction chemistry or overly stress the process facilities). Table 5 summarizes the final list of factors and their ranges that are to be included in the first DOE. Time, of course, will also be a (nested) factor, but, as previously discussed, its levels will be determined after the whole plot design is established. 

---