Thiele modulus, relationship

Figures 7.5 and 7.6 display the effectiveness factor versus Thiele modulus relationship given in Equation 7.33. The log-log scale in Figure 7.6 is particularly useful, and we see the two asymptotic limits of Equation 733. At small p 1, and at large q l/4>. Figure 7.6 shows that the asymptote q = I/O is an excellent approximation for the spherical pellet for O > 10. For large values of the Thiele modulus, the rate of reaction is much greater than the rate of diffusion, the effectiveness factor is much less than unity, and we say the pellet is diffusion limited. Conversely, when the diffusion rate is much larger than the reaction rate, the effectiveness factor is near unity, and we say the pellet is reaction limited.

The relationship between effectiveness factor p and Thiele modulus < >l may be calculated for several other regular shapes of particles, where again the characteristic dimension of the particle is defined as the ratio of its volume to its surface area. It is found that... 

For different geometries and for kinetics of arbitrary order a general expression for the calculation of the Thiele modulus and the relationship between tj and has been developed (Aris, 1957) [Eq. (5.64)]. 

To have a quantitative idea of the problem of intraparticle diffusion, effectiveness factors for the two catalysts were calculated from the observed second order rate constants (based on surface area) using the "triangle method" suggested by Saterfield (4). The effectiveness factors for Monolith and Nalcomo 474 catalysts on Synthoil liquid at 371°C (700 F) were calculated to be 0.94 and 0.216, respectively. In applying the relationship between the "Thiele Modulus," 4>> and the "effectiveness factor," n> the following simplifying assumptions were made ... 

In conclusion, the ADM is a more convenient method to obtain approximate solutions for the models with arbitrary reaction orders, n, so that the relationship between effectiveness c and Thiele modulus can be quantitatively understood in more details for the nonlinear cases. 

The above analysis and Fig. 19-25 provide a theoretical foundation similar to the Thiele-modulus effectiveness factor relationship for fluid-solid systems. However, there are no generalized closed-form expressions of E for the more general case ofa complex reaction network, and its value has to be determined by solving the complete diffusion-reaction equations for known intrinsic mechanism and kinetics, or alternatively estimated experimentally. 

Comparing eqs 56 and 27, and recalling the definition of the effectiveness factor according to cq 40, yields the following simple relationship between the Thiele modulus and the Weisz modulus ... 

When the effective reaction rate is controlled by pore diffusion, then the asymptotic solution of the catalyst effectiveness factor as a function of the generalized Thiele modulus can be utilized (cq 108). This (approximate) relationship has been derived in Section 6.2.3.1. It is valid for arbitrary order of reaction and arbitrary pellet shape. 

Figure 7.11 also shows the relationship between the effectiveness factor and the Thiele modulus for other pellet shapes. In general the linear dimension L of the slab is replaced by ... 

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. 

This relationship is plotted in Figure 6.3.9. The effectiveness factor for a severely diffusion-limited reaction in a catalyst particle is approximated by the inverse of the Thiele modulus. 

Froment and Bischoff (ref. 8) reported a relationship between the Thiele modulus and the reaction rate constant for irreversible 2-nd order reaction. 

Generally as the Thiele modulus of main reaction is smaller than unity, the order of deactivation is equal to unity and, therefore the apparent deactivation rate constant can he extracted from the slope of -In Xg(t) versus t Krishnaswamy and Kittrell (ref. 10) have shown that the relationship between the... 

For more complex reactions, the effect of intraparticle diffusion resistance on rate, selectivity, and yield depends on the particulars of the network. Also, the use of the Thiele modulus-effectiveness factor relationships is not as easily applicable, and numerical solution of the diffusion-reaction equations may be required. 

For simple nth-order irreversible reactions, models for diffusion-reaction lead to the relationship, shown in Fig. 1.6, between the effectiveness factor. 1, and the Thiele modulus... 

Closure After completing this chapter, the reader should be able to derive differential equations describing diffusion and reaction, discuss the meaning of the effectiveness factor and its relationship to the Thiele modulus, and identify the regions of mass transfer control and reaction rate control. The reader should be able to apply the Weisz-Prater and Mears criteria to identify gradients and diffusion limitations. These principles should be able to be applied to catalyst particles as well as biomaierial tissue engineering. The reader should be able to apply the overall effectiveness factor to a packed bed reactor to calculate the conversion at the exit of the reactor. The reader should be able to describe the reaction and transport steps in slurry reactors, trickle bed reactors, fluidized-besd reactors, and CVD boat reactors and to make calculations for each reactor. 

Let us analyze what is the relationship between the Thiele modulus and observable Thiele modulus 0. For that we will set S -S0. At these conditions the observed rate is equal to the rate in the absence of internal diffusion, thus... 

In a more realistic case we may have the reaction first-order in both A and B, where the analysis is complicated by the fact that the Thiele modulus becomes a function of Bj, which is changing with time. This complication is tractable, however, and the final time-conversion relationship is... 

The observation that the slope of the asymptotic solution for ti q>) becomes independent of the particle geometry suggests that the dependence of the effectiveness factor on the Thiele modulus can be described by a generalized relationship, valid... 

In general, the intrinsic kinetic parameters of a catalytic reaction under study are unknown. Therefore, the relationships based on the Thiele modulus cannot be used to estimate the influence of inner mass transfer on the measured overall reaction rate. Observed is the experimentally accessible efficient reaction rate, In... 

The relationship shown in Equation 2.207 suffers from the fact that the Thiele modulus must be specified to estimate the catalyst efficiency. This is, in general, not possible as the intrinsic kinetics is not known. It is, therefore, more convenient to relate the overall effectiveness factor to the Weisz modulus, which is based only on observable parameters. 

The relationship between the isothermal internal effectiveness factor rj and the Thiele modulus q>i for a flat plate is plotted in Figure 2.10 for q>i values ranging from 0.1 to 20. 

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