Thiele modulus reversible reaction

Figure 4.5.26 shows the radial concentration distribution in a porous spherical particle with diameter 2tp according to Eq. (4.5.115) for two values of the Thiele modulus ( reversible fot the example of a gas phase free of B (cB,g = 0) and Defr/O tp) = 0.05 and K = l. Note that in the case of high values of ( reversible (S>5 in Figure 4.5.26), the external mass transfer determines the effective reaction rate, that is, the equilibrium concentrations are almost reached within the porous particle (for the example of Figure 4.5.26, K<, = 1 and Ca,equilibrium = Cb,equilibrium = 0.5c J, and the concentrations vary strongly in the boundary layer, for example,...

From this expression it can be seen that the modulus rcv transforms to the standard Thiele modulus (eq 27) when the equilibrium constant approaches infinity. Additionally, it is obvious that the effectiveness factor decreases when, at a given value of the forward rate constant k+, the reverse reaction becomes increasingly important (Fig. 18). This holds for all types of reversible reactions [31, 91]. Therefore, the effectiveness factor of a truly reversible reaction might be considerably overestimated if the reaction is treated as irreversible. 

Develop expressions for the Thiele modulus and the concentration profile of A for the following reversible first-order reaction that takes place in a flat plate catalyst pellet ... 

The isothermal, reversible, first-order reaction A = B occurs in a flat plate catalyst pellet. Plot the dimensionless concentration of A (Ca/C s) as a function of distance into the pellet for various values of the Thiele modulus and the equilibrium constant. To simplify the solution, let Cas = 0.9(Ca + Cg) for all cases. 

The results for the other pellet densities are given in the second column of Table 11-6. The Thiele-type modulus is calculated from Eq. (11-56), modified for a reversible reaction by substitutingfor k ... 

Figure 15 shows a typical plot of intraparticle effectiveness factor vs. nondimensionless time for different values of the Thiele modulus. It should be noted that the reversibility of the aqueous phase ion-exchange reaction led to lower effectiveness factors than the simulations by Wang and Yang (1991b) for irreversible reactions. 

Figure 19.10 Triphase catalysis effectiveness factor as a function of dimensionless time for various values of the Thiele modulus (Desikan and Doraiswamy, 1995). Solid lines represent irreversible reactions K —xx) and broken lines reversible reactions (AT = 0.1). Note T]i=T of Equation 19.24...

The Thiele moduli for first-order reactions in various geometries as well as the generalized moduli applicable for other reaction orders all assume that the chemical reaction is irreversible. Studies on the first-order reversible reaction A <-> B have shown that the same q>L-ri function as in the irreversible case can be used, when the Thiele modulus is defined using the characteristic size factor L of Equation 2.65. The reaction equilibrium constant K is used, if the D, value for A and B is more or less the same [13, 29] ... 

Thus, we find that Equations 4.191 and 4.193 are analogous to the corresponding Equations 4.157 and 4.158 derived for an irreversible reaction in the spherical catalyst pellet except that is replaced by Ca and the rate constant fc is replaced by + K/K). So, the equation for Thiele modulus O defined for an irreversible reaction will hold good for a reversible reaction with the rate constant fc in O replaced by ki(l + K/K). Thus, the Thiele modulus O for a reversible reaction in a spherical catalyst pellet is... 

In general, the modified Thiele modulus O for the reversible reaction in a pellet of any regular shape is... 

The rate equation for the reversible reaction is written as where C/le is the equilibrium concentration. The modified Thiele modulus ( ) is... 

Expressions for the effectiveness factor, similar to Eqn. (9-9), can be derived for other particle shapes, other reaction orders, and for reversible as well as irreversible reactions. Fortunately, if the Thiele modulus is redefined somewhat, all of these solutions can be approximated by a single curve of rj versus 0. 

This version of the Thiele modulus looks complicated, primarily because three new parameters ( s, P, and K ) are required to account for reversibility. If the reaction is essentially irreversible K oo), then = 1 and Eqn. (9-13) reduces to... 

Figure 9-7 Effectiveness factor t]) versus Thiele modulus (0) for an nth-order reaction in an isothermal catalyst particle. The modulus is defined by Eqn. (9-13), which applies to both reversible and irreversible reactions. For < 0.10, t 1. For 0 > 10, i) = 1 /0.

Suppose that the forward isomerization reaction is first order in n-C Hu and that the reverse isomerization reaction is first order in i-CsHm. Suppose further that the hydrogenolysis reaction is first order in n-CgHw. If the Thiele modulus is large, will the actual selectivity to i-CeHM be greater than, less than, or the same as the intrinsic selectivity ... 

The rate of the re verse isomerization reaction is not affected by Cn, but it is affected by Q. When the Thiele modulus is high, the concentration of i-CeHw is higher than Ci,s evoywhere iitside the catalyst particle. Therefore, the rate of the reverse isomerization reaction is faster when the Thiele modulus is high [t] < 1) than when the Thiele modulus is low. 

For nth-order reversible reaction, the generalized Thiele modulus is... 

---