Thiele modulus normalization

Then the Thiele modulus normalized for shape is, for first-order kinetics ... 

Table 8.2 Thiele modulus ( ) normalized with respect to shape and asymptotic value of tj...

Thiele modulus normalized with respect to shape, equation 8.5-17... 

Hence, the concentration in the particle depends only on the Thiele modulus, s and a normalized radius r/R as illustrated in Fig. 5.34. 

Figure 5.34. Normalized concentration profiles in a porous sphere for different values of the Thiele modulus. Note that if the latter is large, only a small part of the catalyst near the surface contributes to conversion.

The results for spherical and cylindrical shapes are approximately in accordance with those shown in Figure 8.11, and in the limit of large, become the same, if the Thiele modulus is normalized in terms of a common effective diflusion-path-length parameter, Le, defined 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... 

A normalization for the Thiele modulus (letter with correction in November issue). Ind. Eng. Chem. Fund. 4, 227-229 (1965). 

As a rule, hydrogen ion is involved not only in the pH-dependency of the reaction term (Thiele modulus) but also as the actively participating species involved in the acid-base equilibrium of all the substrates, reaction intermediates, products, and even the gel matrix. Furthermore, enzymatic reactions are always carried out in the presence of the mobile buffer. By mobile we mean a weak acid or a weak base that can move in and out of the reaction layer, as opposed to the fixed buffer represented by the gel (and by the protein) itself. Thus, we have to include the normalized diffusion-reaction equations for hydrogen ion and for the buffer. 

Thiele modulus. It can be increased by the enzyme loading but, obviously, only up to a point. The normal dynamic range is approximately between 10 4 and 10-1 M. 

Figure 4. Normalized concentration profiles of reactant A versus the pellet radius, calculated from eq 49 for different values of the Thiele modulus (isothermal, first order, irreversible reaction in a sphere).

However, whereas effectiveness factors above unity under nonisothcrmal conditions can be explained quite easily, the observation of multiple steady states is a new and unexpected feature. These arise at small values of provided the reaction is substantially exothermic and, additionally, has a high activation energy. This means that, for a single value of the Thiele modulus, several possible solutions for the steady state overall effectiveness factor may exist (operating points), usually up to three. The middle operating point is normally unstable. Whenever the temperature and/or the... 

Eifect of Thiele modulus on the normalized concentration profiles in a catalyst pore with first-order surface reaction. 

An early normalization of the Thiele modulus for an isothermal pellet and arbitrary kinetics was given by R. B. Bird, W. E. Stewart, and E. N. Lightfoot on pages 335-41 of their Notes on Transport Phenomena, the precursor to their well-known Transport Phenomena (New York John Wiley Sons, Inc., 1960). Slightly more general forms—all of them equivalent—have been given independently and almost simultaneously in ... 

The relevant dimensionless parameters are a modified Thiele modulus, the normalized adiabatic temperature rise 0), and the Arrhenius number — a IRT. Plots for a first-order reaction in a spherical particle are shown in Figure 9.4 (next page). For highly exothermic reactions and large values of (3 the rate can be multivalued at modified Thiele moduli around 0.5, with two stable and one unstable steady states. At which state the particle performs depends on its prior history. 

Figure 7.19 Effectiveness factor versus normalized Thiele modulus for a first-order reaction in a non isothermal spherical pellet.

Effectiveness factor versus normalized Thiele modulus... 

Thiele moduli. These also appear in the smaller area below the single-file profile curve for /c = 1.27 x 10 in Fig. 5, in comparison to the area under the profile resulting from the conditions of normal diffusion with the same Thiele modulus. By comparing single-file diffusion with ordinary diffusion the prevailing effect is clearly the dramatic enhancement of rintra and hence of the Thiele modulus, leading to a correspondingly dramatic reduction of the effectiveness factor. 

In the case of a zero-order reaction (r = feof) fhe Thiele modulus jS has been modified to suit the interpretation for biofilm kinetics. It is then the reciprocal of the normal Thiele modulus for a zero-order reaction,... 

The concentration in pores depends on the electrode potential, and parameter B relates to the normalized Thiele modulus o [444], which characterizes the mass transfer in pores. Examples of concentration gradients obtained at different potentials — are presented in Eig. 9.30. As the potential becomes more negative. 

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