Diffusion within catalyst pellet

Fig. 20. Spin density, and water diffusion images for a 2.2-inm-diameter, spherical silica catalyst support pellet. In-plane pixel resolution was 45 pm x 45 pm image slice thickness was 0.3 mm. (a) Spin-density map lighter shades indicate higher liquid content, (b) map (150 00 ms) lighter shades indicate longer values of Ti. (c) Diffusivity map ((0-1.5) x 10 m s ) lighter shades indicate higher values of water diffusivity within the pellet.

Based on the studies on the KD306-type sulfur-resisting methanation catalyst, the non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key-components have been established, which were solved using an orthogonal collocation method. The simulation values of the effectiveness factors for the methanation reaction ch4 and the shift reaction fco2are in fair agreement with the experimental values, which indicates that both models are able to predict intraparticle transport and reaction processes within catalyst pellets. 

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. 

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... 

In Section IV,H, we described how selectivity depends on the transport-limited arrival of reactants and removal of reactive olefins within catalyst pellets. Specifically, we showed how FT synthesis selectivity can be described accurately and entirely by a structural parameter regardless of whether olefin removal [Eq. (15)] or reactant arrival [Eq. (25)] is the controlling diffusive resistance. The preceding two sections described how transport-limited removal of olefins from catalyst pellets can enhance the rate of secondary reactions. Here, we show how such transport limitations can be controlled by varying the liquid composition within catalyst pellets. In the discussion that follows, we refer to the experimental and simulation results of Fig. 20, where we showed how Cs+ selectivity depends on the value of the structural parameter ( ). 

Cs+ selectivity can be controlled by varying the density of sites within catalyst pellets and the diameter of these pellets. The density of sites determines the reactant requirements and the pellet size controls the required path length of diffusing molecules. Such modifications affect the value of x, causing the performance of these catalysts to move along the curve in Fig. 20. The shape of the selectivity curve, however, depends only on the intrinsic readsorption rate constant (/Sr) and on the kinetic dependence of chain growth pathways on reactant pressure. 

Catalysts are normally used in the form of pellets, except in fluidized-bed reactors where powders are used. Thus problems of resistance to diffusion within the pellets are common. These have been quite extensively studied and many texts written (see in particular Aris, 1975). We present some basic equations here and outline methods for a quick evaluation of these effects. 

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). 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

When consecutive reactions take place within a porous catalyst, the concentrations of A and V within the pellet will be significantly different from those prevailing at the external surface. The intermediate V molecules formed within the pore structure have a high probability of reacting further before they can diffuse out of the pore. 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

Figure 7-6 Different size scales in a packed bed catalytic reactor. We must consider the position z in the bed, the flow around catalyst pellets, diffusion within pores of pellets, and adsorption and reaction on reaction sites. These span distance scales from meters to Angstroms.

In the preceding example we assumed that reaction occurred on the external surface so we did not have to be concerned with diffusion within the catalyst pellet. Now we consider the effect of pore diffusion on the overall rate. We have to do considerable mathematical manipulation to find the proper expressions to handle this, and before we begin, it is worthwhile to consider where we are going. As before, we want the rate as a function of bulk concentration Ca >i and we need to know the rate coefficient for various approximations (Figure 7-1 1). 

Some simulation results for trilobic particles (citral hydrogenation) are provided by Fig. 2. As the figure reveals, the process is heavily diffusion-limited, not only by hydrogen diffusion but also that of the organic educts and products. The effectiviness factor is typically within the range 0.03-1. In case of lower stirrer rates, the role of external diffusion limitation becomes more profound. Furthermore, the quasi-stationary concentration fronts move inside the catalyst pellet, as the catalyst deactivation proceeds. 

The structure-transport relationship characteristic of the catalyst pellet is shown by comparison of Figs 20a-c the spatial heterogeneity in the values of the molecular diffusion coefficient is much more consistent with the heterogeneity in the intensity shown in the Ti map than that of the spin-density map. Thus, we conclude that it is the spatial variation of local pore size that has the dominant influence on molecular transport within the pellet. 

Thiele(I4>, who predicted how in-pore diffusion would influence chemical reaction rates, employed a geometric model with isotropic properties. Both the effective diffusivity and the effective thermal conductivity are independent of position for such a model. Although idealised geometric shapes are used to depict the situation within a particle such models, as we shall see later, are quite good approximations to practical catalyst pellets. 

The manner in which Ni and V sulfide deposits accumulate on individual catalyst pellets depends on the kinetics of the HDM reactions as influenced by catalyst properties, feed characteristics, and operating conditions. The dynamic course of deactivation of catalytic reactor beds is also determined by the kinetics of the HDM reaction. The lifetime and activity of a reactor bed are directly related to the details of the metal deposit distribution within individual pellets. This section will review deactivation behavior of reactor beds in light of our understanding of the reaction and diffusion phenomena occurring in independent catalyst pellets. Unfortunately, this is an area of research which remains mostly proprietary with too little information published. What has been published is generally lacking in detail for the same reason. 

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