Effectiveness factor plot, first-order

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

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)... 

Figure 12.7 Effectiveness factor plots for sphere and slab geometries for first- and zero-order kinetics. The Thiele modulus is given by = RyJfor spherical catalysts or by...

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

Eq. (12.14) is recovered. The presence of traps lowers ihe mobility as expected. The essential message of Figures 12-17 and 12-18 is that, to a first order approximation, Eq. (12.14) maintains the icmperalurc dependency of the mobility if one replaces the disorder parameter by an effective disorder parameter ocJj or, equivalently, an effective width of the DOS that depends on both the concentration and the depth of the traps. Deviations from the behavior predicted by Eq. (12.14) become important for ,>0.3 eV, notably at lower temperatures. It is noteworthy, though, that the T- oo intercepts of p(7), if plotted as In p versus 7 2, vary by no more than a factor of 2 upon varying trap depth and concentration. 

Currently, benzene alkylation to produce ethylbenzene and cumene is routinely carried out using zeohtes. We performed a study comparing a zeohte Y embedded in TUD-1 to a commercial zeolite Y for ethylbenzene synthesis. Two different particle diameters (0.3 and 1.3 mm) were used for each catalyst. In Figure 41.7, the first-order rate constants were plotted versus particle diameter, which is analogous to a linear plot of effectiveness factor versus Thiele modulus. In this way, the rate constants were fitted for both catalysts. 

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

Curve B of Figure 12.3 [adopted from Wheeler (38)] represents the dependence of the effectiveness factor on Thiele modulus for second-order kinetics. Values of r for first-order and zero-order kinetics in straight cylindrical pores are shown as curves A and C, respectively. Each curve is plotted versus its appropriate modulus. 

Plots of effectiveness factors versus corresponding Thiele moduli for zero-, first-, and second-order kinetics based on straight cylindrical pore model. For large hr, values of r are as follows ... 

Figure 11.19 Plots of the external effectiveness factor as a function of the substrate modulus Da for different values of the dimensionless bulk substrate concentration is the limiting first-order effectiveness factor attained at sufficiently low concentrations. Adapted from C.Horvath and J.M.Engasser. Biotechnol.Bioeng., 16, 909 (1974).

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). 

The rates of movement of foreign compound into and out of the central compartment are characterized by rate constants kab and kei (Fig. 3.23). When a compound is administered intravenously, the absorption is effectively instantaneous and is not a factor. The situation after a single, intravenous dose, with distribution into one compartment, is the most simple to analyze kinetically, as only distribution and elimination are involved. With a rapidly distributed compound then, this may be simplified further to a consideration of just elimination. When the plasma (blood) concentration is plotted against time, the profile normally encountered is an exponential decline (Fig. 3.24). This is because the rate of removal is proportional to the concentration remaining it is a first-order process, and so a constant fraction of the compound is excreted at any given time. When the plasma concentration is plotted on a logio scale, the profile will be a straight line for this simple, one compartment model (Fig. 3.25). The equation for this line is... 

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. 

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. 

Equations 6.57 and 6.58 are illustrated in Figure 6.15 where A according to Equation 6.56 is plotted versus 17 for first-order kinetics in a slab. The effectiveness factor 17 and the approximation rj were calculated from Tables 6.1 and 6.2 and Equations 6.57 and... 

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. 

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. 

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. 

The scenarios in series A concern initiation by purposely added initiator RX. Figures 1-8 (A-D) and 10 (A-D) show the diagnostic plots for the ideal case (instantaneous initiation, scenario Al, dotted lines) together with the effect of the various complicating factors, i.e., instantaneous initiation plus zero order chain transfer to monomer (scenario A2a), instantaneous initiation plus first order chain transfer to monomer (scenario A2b), instantaneous initiation plus both... 

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