Thiele modulus first-order reaction

Thiele modulus in a cylindrical pellet for a first-order reaction. Thiele modulus in a flat plate for a first-order reaction. 

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... 

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 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.

Fig. 5.4-14. Effectiveness factor versus Thiele modulus for 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... 

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. 

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... 

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 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 ... 

Equation (5.59) gives the solution in the limiting case of a first-order reaction, where Thiele modulus L is the characteristic length, which is half the width of a flat plate, r/2 of a long... 

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.

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 ... 

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. 

Effect of Thiele modulus on the normalized concentration profiles in a spherical catalyst particle with first-order reaction. The external surface of the particle is located at /Rp = 1. 

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. 

A plol of the effectiveness factor as a function of the Thiele modulus is shown in Figure 12-5. Figure 12-5a shows "q as a function of (j> for a spherical catalyst pellet for reactions of zero-, first-, and second-order. Figiue 12-5b corresponds to a first-order reaction occurring in three differently shaped pellets of voliune Vp and external siuface area A,. When volume change accompanies a reaction, the corrections shown in Figure 12-6 apply to the effectiveness factor for a first-order reaction. 

To express the overall rate of reaction in terms of the Thiele modulus, we rearrange Equation (12-28) and use the rate law for a first-order reaction in Equation (12-29)... 

The simplest kinetics to assume would be a one-step reaction from an organometallic compound to metallic deposit, i.e. the A to B first order reaction. This is essentially the approach used by Sato et al [1], which led to a straight forward definition of the Thiele modulus and the pattern of metal deposition as the fantiliar... 

The effectiveness factor can be written as a function of a dimensionless independent variable called the Thiele modulus, which for a first-order reaction is defined below together with the corresponding effectiveness factor derived by integration of the corresponding diffusion-reaction equation (7-97) tiveness factor is ti = - (7-104) 0... 

To find the effectiveness under poisoned conditions, this form of the Thiele modulus is substituted into the appropriate relation for effectiveness. For example, for a first-order reaction in slab geometry, the effectiveness factor is... 

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