Thiele modulus for first-order reaction

Fig. 5.4-14. Effectiveness factor versus Thiele modulus for first-order reaction.

Terminal velocity, 405 Thiele modulus for first-order reaction, 97 general, 108 generalized, 118 Thomas method, 302 Titration method, 27 Transformation ratio, 84 Transport criteria for intrinsic kinetics, 74, 138... 

In the past, a number of attempts have been made to generalize the definition of the Thiele modulus. Aris [6] noticed that all the Thiele moduli for first-order reactions were of the form ... 

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] ... 

Thiele number for first order reaction or specified with subscripts. Thiele modulus /g for first-order reaction. 

For first order reaction in slab geometry, evaluate the ratio of effectiveness with uniform poisoning, 7)un> and pore mouth poisoning, T)pm, in terms of fractional poisoning and the Thiele modulus. 

Even though the governing phenomena of coupled reaction and mass transfer in porous media are principally known since the days of Thiele (1) and Frank-Kamenetskii (2), they are still not frequently used in the modeling of complex organic systems, involving sequences of parallel and consecutive reactions. Simple ad hoc methods, such as evaluation of Thiele modulus and Biot number for first-order reactions are not sufficient for such a network comprising slow and rapid steps with non-linear reaction kinetics. 

The function expressing the Thiele modulus in terms of kinetic parameters and the catalyst properties depends on the intrinsic reaction rate. For first-order reactions, the modulus is... 

The Flatta number is the ratio of the reaction in the liquid surface/mass transfer into the bulk phase or a modified Thiele modulus for GL systems to correct the mass transfer for chemical reaction. Because the ratio involves the reaction rate, the actual form of the Hatta number depends on the reaction kinetics. For first order reactions. 

Figure 2.1 Dependence of the effectiveness factor on the Thiele modulus for a first-order irreversible reaction. Steady-state diffusion and reaction, slab model, and isothermal conditions are assumed.

The internal effectiveness factor is a function of the generalized Thiele modulus (see for instance Krishna and Sie (1994), Trambouze et al. (1988), and Fogler (1986). For a first-order reaction ... 

Figure 3.32. Interna effectiveness factor as a function of the generalized Thiele modulus for a first order reaction.

For comparison reasons, the results derived from the simulation were additionally calculated by means of the Thiele modulus (Equation 12.12), i.e., for a simple first-order reaction. The reaction rate used in the model is more complex (see Equation 12.14) thus, the surface-related rate constant kA in Equation 12.12 is replaced by... 

The asymptotic solution ( - large) for tj is [2/(n + l)]1/2/, of which the result given by 8.5-14c is a special case for a first-order reaction. The general result can thus be used to normalize the Thiele modulus for order so that the results for strong pore-diffusion resistance all fall on the same limiting straight line of slope - 1 in Figure 8.11. The normalized Thiele modulus for this purpose is... 

This approach is analytically correct for isothermal reactors and first-order rate laws, since concentration does not appear in the expression for the Thiele modulus. For other (nonlinear) rate laws, concentration changes along the reactor affect the Thiele modulus, and hence produce changes in the local effectiveness factor, even if the reaction is isothermal. Problem 21-15 uses an average effectiveness factor as an approximation. 

The Thiele modulus for the cylinder and sphere differ from that for the slab. However, if, for a first-order reaction, this modulus is redefined in... 

A certain spherical porous catalyst with a pellet diameter of 1/8 in. has a Thiele modulus of 0.5 for a first-order reaction and gives 90% conversion in a packed bed reactor. It is proposed to... 

The Thiele modulus for spherical particles, and first-order reaction ... 

In assessing whether a reactor is influenced by intraparticle mass transfer effects WeiSZ and Prater 24 developed a criterion for isothermal reactions based upon the observation that the effectiveness factor approaches unity when the generalised Thiele modulus is of the order of unity. It has been showneffectiveness factor for all catalyst geometries and reaction orders (except zero order) tends to unity when the generalised Thiele modulus falls below a value of one. Since tj is about unity when 0 < ll for zero-order reactions, a quite general criterion for diffusion control of simple isothermal reactions not affected by product inhibition is < 1. Since the Thiele modulus (see equation 3.19) contains the specific rate constant for chemical reaction, which is often unknown, a more useful criterion is obtained by substituting l v/CAm (for a first-order reaction) for k to give ... 

FIG. 19-15 Effectiveness factors versus Thiele modulus for a first-order reaction in spheres under adiabatic conditions. [Weisz and Hicks, Chem. Eng. Sci., 17 265 (1962).]... 

Fig. 7.10. Reactant concentration profiles wthin a catalyst slab for an irreversible first-order reaction, at different values of the Thiele modulus.

Hie Aris numbers An0 and An, are much alike. This is illustrated in Table 6.4 where the formulae for An0 and An, are given for arbitrary kinetics, for n-th order kinetics and for first-order kinetics. In practice, reaction kinetics do not differ too much from first-order kinetics, and hence the values of An0 and An, will remain very close to each other (as also the geometry factor T is close to one). In that case both Aris numbers will be roughly equal to the square power of the shape-generalized Thiele modulus of Aris [6]. 

In the case of distillate hydrotreating the simulation models are simply described even for the complicated reaction scheme employed in the reaction model as discussed in the later section, because the catalyst deactivation is not necessarily predicted for local sites in the system. And the assumption confirmed in the previous discussion that the relationship between the Thiele-modulus and effectiveness factor is approximately represented by that of a first order reaction for any reaction order makes the simulation model simpler and easier to develop. 

The relationship between the Thiele-modulus and effectiveness factor represented here for a first order reaction can also be applied to other reaction orders for approximations in practical use. This makes the model simpler and easier to use and develop quantitatively. 

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 ... 

Since the equations are nonlinear, a numerical solution method is required. Weisz and Hicks calculated the effectiveness factor for a first-order reaction in a spherical catalyst pellet as a function of the Thiele modulus for various values of the Prater number [P. B. Weisz and J. S. Hicks, Chem. Eng. Sci., 17 (1962) 265]. Figure 6.3.12 summarizes the results for an Arrhenius number equal to 30. Since the Arrhenius number is directly proportional to the activation energy, a higher value of y corresponds to a greater sensitivity to temperature. The most important conclusion to draw from Figure 6.3.12 is that effectiveness factors for exothermic reactions (positive values of j8) can exceed unity, depending on the characteristics of the pellet and the reaction. In the narrow range of the Thiele modulus between about 0.1 and 1, three different values of the effectiveness factor can be found (but only two represent stable steady states). The ultimate reaction rate that is achieved in the 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. 

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