Diffusion intrapellet

In the case of nonequimolal cpunterdiffusion, equation 12.2.6 suffers from the serious disadvantage that the combined diffusivity is a function of the gas composition in the pore. This functional dependence carries over to the effective diffusivity in porous catalysts (see below), and makes it difficult to integrate the combined diffusion and transport equations. As Smith (12) points out, the variation of 2C with composition (YA) is not usually strong, and it has been an almost universal practice to use a composition independent form of Q)c (12.2.8) in assessing the importance of intrapellet diffusion. In fact, the concept of a single effective diffusivity loses its engineering utility if the dependence on composition must be retained. 

In this paper we will first describe a fast-response infrared reactor system which is capable of operating at high temperatures and pressures. We will discuss the reactor cell, the feed system which allows concentration step changes or cycling, and the modifications necessary for converting a commercial infrared spectrophotometer to a high-speed instrument. This modified infrared spectroscopic reactor system was then used to study the dynamics of CO adsorption and desorption over a Pt-alumina catalyst at 723 K (450°C). The measured step responses were analyzed using a transient model which accounts for the kinetics of CO adsorption and desorption, extra- and intrapellet diffusion resistances, surface accumulation of CO, and the dynamics of the infrared cell. Finally, we will briefly discuss some of the transient response (i.e., step and cycled) characteristics of the catalyst under reaction conditions (i.e.,... 

As will be shown later, the surface coverages of CO vary with distance into the pellet during CO adsorption and desorption, as a result of intrapellet diffusion resistances. However, the infrared beam monitors the entire pellet, and thus the resulting absorption band reflects the average surface concentration of CO across the pellet s depth. Therefore, for the purpose of direct comparison between theory and experiment, the integral-averaged CO coverage in the pellet... 

Parametric sensitivity analysis showed that for nonreactive systems, the adsorption equilibrium assumption can be safely invoked for transient CO adsorption and desorption, and that intrapellet diffusion resistances have a strong influence on the time scale of the transients (they tend to slow down the responses). The latter observation has important implications in the analysis of transient adsorption and desorption over supported catalysts that is, the results of transient chemisorption studies should be viewed with caution, if the effects of intrapellet diffusion resistances are not properly accounted for. 

R. Madon and E. Iglesia, Hydrogen and CO intrapellet diffusion effects in ruthenium-catalyzed hydrocarbon synthesis, J. Catal., 1994, 149, 428 137. 

Maximum intrapellet temperatures, 25 305 effect of diffusion collision integral on, 25 301-303... 

Hydrodemetallation reactions are revealed to be diffusion limited by examination of metal deposition profiles in catalysts obtained from commercial hydroprocessing reactors. Intrapellet radial metal profiles measured by scanning electron x-ray microanalysis show that vanadium tends to be deposited in sharp, U-shaped profiles (Inoguchi et al, 1971 Oxenrei-ter etal., 1972 Sato et al., 1971 Todo et al., 1971) whereas nickel has been observed in both U-shaped (Inoguchi et al., 1971 Todo et al., 1971) and... 

We previously proposed that intrapellet (pore) diffusion within liquid-filled catalyst pores decreases the rate of a-olefin removal. This increases the residence time and the fugacity of a-olefins within catalyst pellets and increases the probability that they will readsorb onto FT chain growth sites and initiate new chains. This occurs even for small catalyst particles ("0.1 mm pellet diameter) at normal FT conditions. Larger a-olefins remain longer within catalyst particles because diffusivity decreases markedly with increasing molecular size (carbon number). As a result, readsorption rates increase with increasing carbon number. 

The effective diffusivity Dn decreases rapidly as carbon number increases. The readsorption rate constant kr n depends on the intrinsic chemistry of the catalytic site and on experimental conditions but not on chain size. The rest of the equation contains only structural catalyst properties pellet size (L), porosity (e), active site density (0), and pore radius (Rp). High values of the Damkohler number lead to transport-enhanced a-olefin readsorption and chain initiation. The structural parameters in the Damkohler number account for two phenomena that control the extent of an intrapellet secondary reaction the intrapellet residence time of a-olefins and the number of readsorption sites (0) that they encounter as they diffuse through a catalyst particle. For example, high site densities can compensate for low catalyst surface areas, small pellets, and large pores by increasing the probability of readsorption even at short residence times. This is the case, for example, for unsupported Ru, Co, and Fe powders. 

Reactants and products must diffuse through high-molecular-weight liquid hydrocarbons during FT synthesis. The liquid phase may be confined to the mesoporous structure within catalyst pellets or extend to the outer surface and the interstitial spaces between pellets, depending on the reactor design and hydrodynamic properties. In packed-bed reactors, the characteristic diffusion distance equals the radius of the pellets plus the thickness of any liquid boundary layer surrounding them. Intrapellet diffusion usually becomes... 

The effects of diffusional restrictions on the activity and selectivity of FT synthesis processes have been widely studied (32,52,56-60). Intrapellet diffusion limitations are common in packed-bed reactors because heat transfer and pressure-drop considerations require the use of relatively large particles. Bubble columns typically use much smaller pellets, and FT synthesis rates and selectivity are more likely to be influenced by the rate of mass transfer across the gas-liquid interface as a gas bubble traverses the reactor (59,61,62). 

Higher intrapellet residence times increase the contribution of chain initiation by a-olefins to chain growth pathways. This intrapellet delay, caused by the slow diffusion of large hydrocarbons, leads to non-Flory carbon number distributions and to increasingly paraffinic long hydrocarbon chains during FT synthesis. But intrapellet residence time also depends on the effective diameter and on the physical structure (porosity and tortuosity) of the support pellets. The severity of transport restrictions and the probability that a-olefins initiate a surface chain as they diffuse out of a pellet also de-... 

The probability of readsorption increases as the intrinsic readsorption reactivity of a-olefins (k,) increases and as their effective residence time within catalyst pores and bed interstices increases. The Thiele modulus [Eq. (15)] contains a parameter that contains only structural properties of the support material ( <>, pellet radius Fp, pore radius 4>, porosity) and the density of Ru or Co sites (0m) on the support surface. A similar dimensional analysis of Eqs. (l9)-(24), which describe reactant transport during FT synthesis, shows that a similar structural parameter governs intrapellet concentration gradients of CO and H2 [Eq. (25)]. In this case, the first term in the Thiele modulus (i/>co) reflects the reactive and diffusive properties of CO and H2 and the second term ( ) accounts for the effect of catalyst structure on reactant transport limitations. Not surprisingly, this second term is... 

The high asymptotic value of C5+ selectivity at large values of occurs on pellets that restrict the removal of reactive a-olefins and allow many readsorption events in the time required for intrapellet olefin removal by diffusion. Yet, transport restrictions within these pellets must not significantly hinder the rate of arrival of CO and H2 reactants to the active sites. Carbon number distributions also obey Flory kinetics for high values of because even the smaller olefins significantly react within a catalyst pellet. 

Fig. 27. Effect of diffusion-enhanced a-olefin cracking catalytic function on carbon number distribution (simulations experimental/model parameters as in Fig. 15, 10% CO conversion). (A) FT synthesis without cracking function (B) with intrapellet cracking function, jS = 1.2 (C) with extra pellet cracking function, jS = 1.2. (a) Carbon selectivity vs. carbon number (b) Flory plots.

Diffusive and convective transport processes introduce flexibility in the design of catalyst pellets and in the control of FT synthesis selectivity. Transport restrictions lead to the observed effects of pellet size, site density, bed residence time, and hydrocarbon chain size on chain growth probability and olefin content. The restricted removal of reactive olefins also allows the introduction of other intrapellet catalytic functions that convert olefins to other valuable products by exploiting high intrapellet olefin fugacities. Our proposed model also describes the catalytic behavior of more complex Fe-... 

It is evident from the foregoing discussion that the effective diffusivity cannot be predicted accurately for use under reaction conditions unless surface diffusion is negligible and a valid model for the pore structure is available. The prediction of an effective thermal conductivity is even more difficult. Hence sizable errors are frequent in predicting the global rate from the rate equation for the chemical step on the interior catalyst surface. This is not to imply that for certain special cases accuracy is not possible (see Sec. 11-10). It does mean that heavy reliance must be placed on experimental measurements for effective diffusivities and thermal conductivities. Note also from some of the examples and data mentioned later that intrapellet resistances can greatly affect the rate. Hence the problem is significant. 

Surface migration is pertinent to a study of intrapellet mass transfer if its contribution is significant with respect to diffusion in the pore space. When multimolecular-layer adsorption occurs, surface diffusion has been explained as a flow of the outer layers as a condensed phase. However, surface transport of interest in relation to reaction occurs in the monomolecular layer. It is more appropriate to consider, as proposed by deBoer," that such transport is an activated process, dependent on surface characteristics as well as those of the adsorbed molecules. Imagine that a molecule in the gas phase strikes the pore wall and is adsorbed. Then two alternatives are possible desorption into the gas (Knudsen diffusion) or movement to an adjacent active site on the pore wall (surface diffusion). If desorption occurs,... 

For such large intraparticle diffusion has a large effect on the rate. Practically these conditions mean that diffusion into the pellet is relatively slow, so that reaction occurs before the reactant has diffused far into the pellet. In fact, an alternate definition for is the fraction of the whole surface that is as active as the external surface. If 1, Eq. (11-43) shows that the rate for the whole pellet is the same as the rate if all the surface were available to reactant at concentration Q i.e., the rate at the center is the same as the rate at the outer surface—all the surface is fully effective. In this special case the concentration profile shown in Fig. 11-6 would be horizontal, with C = Cj. In contrast, if 77 1, only the surface near the outer periphery of the pellet is effective the concentration drops from Q to nearly zero in a narrow region near r, e, or high Ar. The latter factor shows that low effectiveness factors are more likely with a very active catalyst. Thus the more effective the active catalyst, the more likely it is that intrapellet diffusion resistance will reduce the rate per pellet. 

Should intrapellet diffusion resistance be considered in evaluating the global rate That is, is r significantly less than unity ... 

Suppose that the rate is measured t a given bulk concentration of reactant. Suppose also that either the external resistance is negligible, or the surface concentration Q has been evaluated from the bulk value by the methods discussed in Chap. 10. Weisz has provided a criterion for deciding, from these measurements and whether intrapellet diffusion may be disregarded. The basic premise is that if then I j is not much less... 

The -in. pellets are the largest for which intrapellet diffusion has a negligible effect on the rate. 

Example 11-7 illustrates one of the problems in scale-up of catalytic reactors. The results showed that for all but -in. pellets intrapellet diffusion significantly reduced the global rate of reaction. If this reduction were not considered, erroneous design could result. For example, suppose the laboratory kinetic studies to determine a rate equation were made with f-in. pellets. Then suppose it was decide tojise f-ih. pellets in the commercial reactor to reduce the pressure drop through the bed. If the rate equation were used for the -in. pellets without modification, the rate would be erroneously high. At the conditions of part b) of Example 11-7 the correct would be only 0.68/0.93, or 73% of the rate measured with -in. pellets. 

The rate measurement for the small particles determines the rate of the chemical reaction at the catalyst site, i.e., without intrapellet-diffusion resistance. Then the rate constant k can be calculated directly. Thus it is... 

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