Zeroth-order kinetics effectiveness factors

Figure 6.11 holds for a slab. Similar figures can be obtained for other catalyst geometries. This is illustrated in Figure 6.13 where the effectiveness factor is plotted versus 8 for zeroth-order kinetics in an infinite slab, infinite cylinder and a sphere. Figure 6.13 has been constructed on the basis of the formulae given in Table 6.6. Hence, the discussion that follows is not restricted to a slab, but holds for any arbitrary catalyst geometry. 

Figure 6.13 Effectiveness factor r versus Thiele modulus 8 of Equation 6.48 for zeroth-order kinetics in an infinitely long slab, an infinitely long cylinder and a sphere.

Table 6.6 Effectiveness factor tj as a function of the Thiele modulus 8 (given by Equation 6.48) for zeroth-order kinetics an infinite slab, an infinite cylinder and sphere...

In other words, reactants exist everywhere within the pores of the catalyst when the chemical reaction rate is slow enough relative to intrapellet diffusion, and the intrapellet Damkohler number is less than, or equal to, its critical value. These conditions lead to an effectiveness factor of unity for zerofli-order kinetics. When the intrapellet Damkohler number is greater than Acnticai, the central core of the catalyst is reactant starved because criticai is between 0 and 1, and the effectiveness factor decreases below unity because only the outer shell of the pellet is used to convert reactants to products. In fact, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler number for zeroth-order kinetics exhibits an abrupt change in slope when A = Acriticai- Critical spatial coordinates and critical intrapeUet Damkohler numbers are not required to analyze homogeneous diffusion and chemical reaction problems in catalytic pellets when the reaction order is different from zeroth-order. When the molar density appears explicitly in the rate law for nth-order chemical kinetics (i.e., n > 0), the rate of reaction antomaticaUy becomes extremely small when the reactants vanish. Furthermore, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler nnmber does not exhibit an abrupt change in slope when the rate of reaction is different from zeroth-order. 

Equations (20-48) require knowledge of the dimensionless molar density profile to calculate the molar flux of reactant A into the pellet via Pick s law. At first glance, equations (20-47) allow one to calculate the effectiveness factor for zeroth-order kinetics via trivial integration that does not require knowledge of the molar density profile, because n = 0. Hence,... 

Obviously, the molar density profile is required to calculate the effectiveness factor for zeroth-order kinetics when A > Acnticai because ijciiticai = /(A) is defined by I a = 0. 

The kinetic effect of increased pressure is also in agreement with the proposed mechanism. A pressure of 2000 atm increased the first-order rates of nitration of toluene in acetic acid at 20 °C and in nitromethane at 0 °C by a factor of about 2, and increased the rates of the zeroth-order nitrations of p-dichlorobenzene in nitromethane at 0 °C and of chlorobenzene and benzene in acetic acid at 0 °C by a factor of about 559. The products of the equilibrium (21a) have a smaller volume than the reactants and hence an increase in pressure speeds up the rate by increasing the formation of H2NO. Likewise, the heterolysis of the nitric acidium ion in equilibrium (22) and the reaction of the nitronium ion with the aromatic are processes both of which have a volume decrease, consequently the first-order reactions are also speeded up and to a greater extent than the zeroth-order reactions. 

If pore diffusion is unimportant, i.e., if the effectiveness factor r is equal to 1, then with constant conversion policy both CSTR and PFR yield half-lives identical to Eq. (5.74) with arbitrary kinetics. With constant flow rate policy, the measured half-life is identical to that obtained through Eq. (5.74) only if the enzyme is saturated, i.e., [S] fCM, and the reaction is zeroth order. 

The effects of added species. The rate of nitration of benzene, according to a rate law kinetically of the first order in the concentration of aromatic, was reduced by sodium nitrate, a concentration of io 3 mol l-1 of the latter retarding nitration by a factor of about 4.llc>28 Lithium nitrate anticatalysed the nitration and acetoxylation of o-xylene in solutions of acetyl nitrate in acetic anhydride. The presence of 6 x io-4 mol 1 1 of nitrate reduced the rate by a factor of 4, and modified the kinetic form of the nitration from a zeroth-order dependence on the concentration of aromatic towards a first-order dependence. However, the ratio of acetoxylation to nitration remained constant.146 Small concentrations of sodium nitrate similarly depressed the rate of nitration of anisole and again modified the reaction away from zeroth to first-order dependence on the concentration of the aromatic.116... 

Figure 6.5 Effectiveness factor t] versus the zeroth Aris number An, for first-order kinetics in an infinite slab, infinite cylinder and sphere.

Hence, it is not possible to redefine the characteristic length such that the critical value of the intrapellet Damkohler number is the same for all catalyst geometries when the kinetics can be described by a zeroth-order rate law. However, if the characteristic length scale is chosen to be V cataiyst/ extemai, then the effectiveness factor is approximately A for any catalyst shape and rate law in the diffusion-limited regime (A oo). This claim is based on the fact that reactants don t penetrate very deeply into the catalytic pores at large intrapellet Damkohler numbers and the mass transfer/chemical reaction problem is well described by a boundary layer solution in a very thin region near the external surface. Curvature is not important when reactants exist only in a thin shell near T] = I, and consequently, a locally flat description of the problem is appropriate for any geometry. These comments apply equally well to other types of kinetic rate laws. 

The mass balance with homogeneous one-dimensional diffusion and irreversible nth-order chemical reaction provides basic information for the spatial dependence of reactant molar density within a catalytic pellet. Since this problem is based on one isolated pellet, the molar density profile can be obtained for any type of chemical kinetics. Of course, analytical solutions are available only when the rate law conforms to simple zeroth- or first-order kinetics. Numerical techniques are required to solve the mass balance when the kinetics are more complex. The rationale for developing a correlation between the effectiveness factor and intrapellet Damkohler number is based on the fact that the reactor design engineer does not want to consider details of the interplay between diffusion and chemical reaction in each catalytic pellet when these pellets are packed in a large-scale reactor. The strategy is formulated as follows ... 

If the kinetics are not zeroth-order, then these integral expressions are more tedions to nse than the ones developed earlier in this chapter based on mass transfer across the external snrface of the catalyst. The preferred expressions for the effectiveness factor are summarized below for nth-order irreversible chemical kinetics when the rate law is only a function of the molar density of one reactant ... 

Problem. Consider zeroth-order chemical kinetics in pellets with rectangular, cylindrical and spherical symmetry. Dimensionless molar density profiles have been developed in Chapter 16 for each catalyst geometry. Calculate the effectiveness factor when the intrapellet Damkohler number is greater than its critical value by invoking mass transfer of reactant A into the pellet across the external surface. Compare your answers with those given by equations (20-50). 

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... 

Obtain an analytical expression for the effectiveness factor (i.e., E vs. tjcriticai) in Spherical catalysts when the chemical kinetics are zeroth-order and the intrapeUet Damkohler number is greater than its critical value. Use the definition of the effectiveness factor that is based on mass transfer via diffusion across the external surface of the catalyst. 

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