Spherical catalysts first-order kinetics

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)... 

Diffusion effects can be expected in reactions that are very rapid. A great deal of effort has been made to shorten the diffusion path, which increases the efficiency of the catalysts. Pellets are made with all the active ingredients concentrated on a thin peripheral shell and monoliths are made with very thin washcoats containing the noble metals. In order to convert 90% of the CO from the inlet stream at a residence time of no more than 0.01 sec, one needs a first-order kinetic rate constant of about 230 sec-1. When the catalytic activity is distributed uniformly through a porous pellet of 0.15 cm radius with a diffusion coefficient of 0.01 cm2/sec, one obtains a Thiele modulus y> = 22.7. This would yield an effectiveness factor of 0.132 for a spherical geometry, and an apparent kinetic rate constant of 30.3 sec-1 (106). 

Effectiveness factor ratios for first-order kinetics on spherical catalyst pellets. 

Find the effectiveness factor for a spherical catalyst particle with first-order kinetics. 

Figure 12-6 Effectiveness factor ratios for first-order kinetics on spherical catalyst pellets for various values of the Thiele modulus, <1), for a sphere. [From V, W. Weekman and R. L. Goring, J. Catal, 4, 260 (1965).]...

It is advantageous to linearize the rate law, given by equations (22-38), because analytical solutions are available for diffusion and chemical reaction within porous catalysts of all geometries when the kinetics are first-order. Consequently, one calculates the effectiveness factor in spherical pellets rather easily after linearization is performed. The best value of the pseudo-first-order kinetic rate constant for irreversible reactions that achieve 100% conversion is... 

Consider the case of a nonisothermal reaction A B occurring in the interior of a spherical catalyst pellet of radius R (Figure 6.4). We wish to compute the effect of internal heat and mass transfer resistance upon the reaction rate and the concentration and temperature profiles within the pellet. If Z)a is the effective binary diffusivity of A within the pellet, and we have first-order kinetics, the concentration profile CA(f) is governed by the mole balance... 

It is rather straightforward to employ numerical methods and demonstrate that the effectiveness factor approaches unity in the reaction-rate-controlled regime, where A approaches zero. Analytical proof of this claim for first-order irreversible chemical kinetics in spherical catalysts requires algebraic manipulation of equation (20-57) and three applications of rHopital s rule to verify this universal trend for isothermal conditions in catalytic pellets of any shape. 

Effectiveness factors for diffusion and zeroth-order chemical kinetics in spherical catalysts, described by equations (20-61) and (20-62), are illustrated in Figure 20-2 and compared with the results for diffusion and first-order irreversible chemical kinetics in the same catalyst geometry, given by... 

The analysis in this section focuses on the appropriate dimensionless numbers that are required to analyze convection, axial dispersion and first-order irreversible chemical reaction in a packed catalytic tubular reactor. The catalytic pellets are spherical. Hence, an analytical solution for the effectiveness factor is employed, based on first-order irreversible chemical kinetics in catalysts with spherical symmetry. It is assumed that the catalytic pores are larger than 1 p.m (i.e., > 10 A) and that the operating pressure is at least 1 atm. Under these conditions, ordinary molecular diffusion provides the dominant resistance to mass transfer within the pores because the Knudsen diffusivity,... 

The objective of this problem is to calculate reactant conversion in the exit stream of a packed catalytic tubular reactor. The chemical kinetics are irreversible and first-order. The reactor is packed with catalysts that are spherically symmetric. The following data are available. Be careful with units, because the kinetic rate constant and the volumetric flow rate are given in minutes, whereas the net intrapeiiet diffusivity is given in seconds. 

For first-order reactions and moderate values of less than the value of i] for a slab with the same volume/surface ratio. As shown in Table 4.2, the maximum difference is about 14%. This small difference means that solutions for complex kinetic models that were obtained for the flat-slab case can be used to get approximate effectiveness factors for spherical catalysts. 

The molecular weight differences between lignin and its model compounds also complicate the use of model compound kinetics in a predictive simulation. The mobility of a high-molecular weight polymer would be much less than that of smaller model substrates (14). As for catalyst decay, a simple model was used to probe transport issues. For a first order, irreversible reaction in an isothermal, spherical catalyst pellet with equimolar counterdiffusion, the catalyst effectiveness factor and Thiele modulus provide the relevant information as... 

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...

Reaction rates inside a spherical isothermal porous catalyst particle depend on die rate of diffusion and the kinetics. When tiie reaction is fast compared to the diffusion, e.g. at higher temperatures, the reactants will be consumed near the surface and not aU the catalyst will be effectively used. The ratio of the actual reaction rate tiiroughout the particle to the reaction rate without any diffusion limitation is a good measure of how effectively the catalyst is used. This is called the effectivenesss factor, ri. Consider calculating the effectiveness factor for a spherical catalyst particle for a first-order reaction at the conditions specified below. The governing equation and boimdary conditions are given in the dimensionless form, and a first-order reaction is assumed ... 

The kinetics of the wear in a spherical aluminosilicate catalyst is shown in Figure 7.57a. This figure shows the relative percentage of mass removed as a function of time, m(f) = [Mq - M(t)]/Afo, where Mq is the initial material mass and M(t) is the residual mass after milling time, t. The three curves labeled 1,2, and 3 represent the abrasion process as a reaction of the zeroth, first, and second order, respectively. Curve 2, representing the exponential dependence m(f) = 1 - exp(-t/x), yields the best fit of the experimental data. The time constant, t, may serve as a measure of the catalyst s resistance to abrasion. 

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