Thiele modulus internal

For large values of the Thiele modulus, internal diffusion limits the rate of reaction... 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

Typically, the him resistance is important only when the internal pore resistance is also important. If the Thiele modulus is small, the him resistance will usually be negligible. This idea is explored in Problems 10.11 and 10.12. 

Most of the actual reactions involve a three-phase process gas, liquid, and solid catalysts are present. Internal and external mass transfer limitations in porous catalyst layers play a central role in three-phase processes. The governing phenomena are well known since the days of Thiele [43] and Frank-Kamenetskii [44], but transport phenomena coupled to chemical reactions are not frequently used for complex organic systems, but simple - often too simple - tests based on the use of first-order Thiele modulus and Biot number are used. Instead, complete numerical simulations are preferable to reveal the role of mass and heat transfer at the phase boundaries and inside the porous catalyst particles. 

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 ... 

Problem P7.06.02 reproduces one result from the literature. There it is apparent that in some ranges of the parameters, effectiveness can be much greater than unity, and also that at low values of Thiele modulus several steady states are possible. When both external and internal adiabatic diffusion occur, moreover, other studies find that half a dozen or more steady states can exist. Those kinds of findings involve much computer work. A book by Aris (Mathematical Theory of Diffusion and Reaction in Permeable Catalysts,... 

The concentration dependence of z/l vs. c/c0 is plotted in Figure 11.14a. It can be seen that from a Thiele modulus cp > 3 the educt does not reach the internal part of the pore. The inner part of the pore system is useless for catalysis. This is especially relevant if expensive metals serve as active components on a porous carrier, which are then wasted. There are chances to master this diffusion limitation, which will be discussed later in detail. Another important variable is the efficiency factor tj. The efficiency factor r is defined as the quotient of the speed of reaction rs to the maximal possible speed of reaction rsmax. r is related to q> as the quotient of the hyperbolic tangent of the Thiele modulus qy. 

Calculation of the internal effectiveness factor for spherical pellets and fust order reaction The Thiele modulus is... 

This study employed conventional diffusion-reaction theory, showing that with diffusion-limited reactions the internal effectiveness factor of a heterogeneous catalyst is inversely related to the Thiele modulus. Using a standard definition of the Thiele modulus [100], the observed reaction rate of an immobilized-enzyme reaction will vary with the square root of the immobilized-enzyme concentration in a diffusion-limited system. In this case, a plot of the reaction rate versus the enzyme loading in the catalyst formulation will be nonlinear. 

Then the classical Thiele modulus ( ) and the effectiveness factor (t/), expressing the extent of internal diffusion limitations in the catalytic washcoat layer of thickness 8, can be calculated according to (cf. Aris, 1975 Froment and Bischoff, 1979, 1990)... 

Agrawal (1980) computed the deposit concentration in the microspheres as a function of position in the pellet and time. Initially, the deposit concentrates in the microspheres near the outer region of the pellet. With the outer region deactivated, deposit formation progresses inward because of access to the inner microspheres through the unobstructed macropores. The profiles in the microspheres are more or less uniform because of the low Thiele modulus. In the case of the unimodal catalyst without macropores, the buildup of deposit in the outer region of the pellet would seal the outer pores and prevent access to internal sites. 

Internal diffusional limitations are possible any time that a porous immobilized enzymatic preparation is used. Bernard et al. (1992) studied internal diffusional limitations in the esterification of myristic acid with ethanol, catalyzed by immobilized lipase from Mucor miehei (Lipozyme). No internal mass diffusion would exist if there was no change in the initial velocity of the reaction while the enzyme particle size was changed. Bernard found this was not the case, however, and the initial velocity decreased with increasing particle size. This corresponds to an efficiency of reaction decrease from 0.6 to 0.36 for a particle size increase from 180 pm to 480 pm. Using the Thiele modulus, they also determined that for a reaction efficiency of 90% a particle size of 30 pm would be necessary. While Bernard et al. found that their system was limited by internal diffusion, Steytler et al. (1991) found that when they investigated the effect of different sizes of glass bead, 1 mm and 3 mm, no change in reaction rate was observed. 

In Table 7 the effectiveness and corresponding Thiele modulus for the different support materials is given. The particle size for the ASA, SiC>2 and HT supports was taken equal to the sieve fraction. This is a worst-case scenario, since it is far more likely that the particles in the sieve fraction are constructed of several crystallites which contain the relevant pores and Pt particles. Between those crystallites, the pore radii will be very large compared to the pore radius in the support material. Even in this worst case scenario, the effectiveness is still high, close to unity, for all catalysts. This demonstrates that the observed reaction kinetics reflect the intrinsic catalyst properties, since internal diffusion limitations are absent. 

The overall process can be affected by pore diffusion and external mass transfer. Molecular diffusion coefficients DPB may be calculated by Aspen Plus. Effective pore diffusion may be estimated by the relation DP = Dpb( j,/tp) = 0.1 DPE, in which ep is the particle porosity and rp the tortuosity. Furthermore, the Thiele modulus and internal effectiveness can be calculated as ... 

The value 1.693 suggests that the internal diffusion should only slightly affect the catalyst effectiveness. From a Thiele-modulus calculation, the effectiveness should be at least 90%. On the other hand, because of rapid acid consumption on down-... 

When the Thiele modulus is small no internal concentration profile exists. In case of large values, due to the existence of a concentration profile, the catalyst is not effectively used, and an effectiveness factor is defined as... 

Figure 13. Internal effectiveness factor as a function of the Thiele modulus for nonisothermal reactions at different values for the Prater number and y, = 10 (numerical solutions for a first order reaction).

Figure 7.10 shows the internal concentration profiles calculated according to Eqn. 7.101 for different values of the Thiele modulus. Knowing the exact internal concentration profile, Eqn. 7.101, allows us to calculate the molar flux through the external surface of the slab from Eqn. 7.80) ... 

We have demonstrated for the first time that we could apply the theory of generalized Thiele modulus to an enzymatic reaction both in n-hexane and SC CO2. The comparison between the two reaction media is not so clear in n-hexane the real reaction velocity is higher than that obtained in SC C02. Nevertheless, the Thiele modulus values indicates a limitation due to the internal mass transfer rate g 1. Thus we observed, in the hexane case, a diffusional control, while in SC C02 an intermediate rate between the reactional and diffusional rates was apparent. It therefore, seems that SC CO2 should be the solvent of choice in reactions catalyzed by immobilized enzymes, since it reduces problems with internal mass transfer. An other advantage is that the value of the inhibition constant is 43 mM in n-hexane and 120 mM in SC CO2 [14], so SC CO2 should be more convenient if we have to work with higher ethanol concentration. The economic feasibility of an industrial scale lipase catalyzed reaction on C02 may depend upon possible costs for high-pressure equipment. 

To check for internal mass transfer limitation, it is possible to use the nondimensional Weisz modulus, Thiele modulus (Levenspiel, 1998) ... 

If the catalyst is dispersed throughout the pellet, then internal diffusion of the species within the pores of the pellet, along with simultaneous reaction(s) must be accounted for if the prevailing Thiele modulus > 1. This aspect gives rise to the effectiveness factor" problem, to which a significant amount of effort, summarized by Aris ( ), has been devoted in the literature. It is important to realize that if the catalyst pellet effectiveness factor is different from unity, then the packed-bed reactor model must be a heterogeneous model it cannot be a pseudohomogeneous model. 

It is desired to find the particle size that gives an internal effectiveness factor equal to 0.70. For any geometry, the Thiele modulus is determined from ... 

When the Thiele modulus is large, internal diffusion usually limits the overall rate of reaction when cj) is small, the surface reaction is usually rate-limiting. If for the reaction... 

It is well known that, under the assumption that the exchange between the product molecules in the interior of the zeolite crystallites and the reactant molecules in the intercrystalline space is limited by intracrystalline diffusion, the effectiveness of reactions catalyzed by the internal surface of zeolite crystallites is a function of a single parameter, the Thiele modulus [3. It is defined by the relation... 

Figure 8.19 shows the dimensionless internal concentration profiles calculated using Eqn. (8.66) for different values of the Thiele modulus (C = C/C z - z/L). 

When 1, strong diffusion limitations occur. Indeed, large values of the Wheeler-Weisz modulus can only correspond to large values of the generalized Thiele modulus, and hence, to strong internal diffusion limitations. 

In view of the calculated values of < )c and ( )ni (Table 2) the limitation to internal oxygen diffusion under the experimental conditions is even lower in the zeolite erystal than in the catalyst particle. The Thiele modulus in zeolite crystals for a severe deactivation state corresponding to activity = 0.20 [7] is < )c = 3.09 10 3 (Sample 5), for an initial coke content of 1.9 wt% (approximately 40 wt% of the coke content needed for blockage of the internal zeolite channels). This low value of Thiele modulus is evidence that oxygen-coke contact is not limited by internal diffusion in the deactivated eatalyst. 

A measure of the absence of internal (pore diffusion) mass transfer limitations is provided by the internal effectiveness factor, t, which is defined as the ratio of the actual overall rate of reaction to the rate that would be observed if the entire interior surface were exposed to the reactant concentration and temperature existing at the exterior of the catalyst pellet. A value of 1 for rj implies that all of the sites are being utilized to their potential, while a value below, say, 0.5, signals that mass transfer is limiting performance. The value of rj can be related to that of the Thiele modulus, 4>, which is an important dimensionless parameter that roughly expresses a ratio of surface reaction rate to diffusion rate. For the specific case of an nth order irreversible reaction occurring in a porous sphere,... 

Certain rough criteria can be made for the importance of internal diffusion in terms of the general Thiele modulus, h. Consider first the case of the first order reaction in the slab. It is clear from the left-hand curve of Fig. 6.7 that if h is less than I the effect of diffusion limitation is not serious, whereas... 

We observe that as the particle diameter becomes very small. decreases, so that the effectiveness factor approaches 1 and the reaction is surface-reaction-limited. On the other hand, when the Thiele modulus < ) is large (- 30), the internal effectiveness factor v) is small (i.e., < 1), and the reaction is diffu-... 

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