Diffusion within the catalyst pellet

The most difficult step to include within the scheme of physical and chemical steps is that of diffusion within the catalyst pellet. To show what kind of 

Its curved wall, but not its flat end, is capable of catalyzing the reaction —  

B so that the rate of disappearance of A is k X (the local concentration of A) in moles per unit time per unit catalyst area. We let the blank end be x = 0, the 

Since there is no flux at the unreactive end x = 0, we have the boundary condition 

Equation (6.4.1) should be one of the better known equations, but if it is not, recall that it is linear, of the second order, and amenable to the solution outlined in Sec. 5.1. The solution is 

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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... 

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). 

The ratio of timescales for internal diffusion within the catalyst pellet and for external mass transfer is of the order of the mass Biot number given in (3.43e). Since for many practical conditions Bim is reasonably high (10 -10 as estimated in Ref. [72]), it is expected that the external mass transfer dominates over the internal one. If we extend this consideration to the reactor scale, then a very useful pseudo-homogeneous model [51] is obtained. The scaling condition that expresses fast fluid-solid mass transfer compared with other processes (X) in Equation 3.22 is... 

The modeling equations contain two key dimensionless parameters a Thiele modulus ( o) and a Peclet number (Peo) they characterize reaction/diffusion processes within the catalyst pellets and convection effects within the interpellet voids, respectively (81). Accordingly, both quantities depend on catalyst properties and reactor conditions. 

In our discussion of surface reactions in Chapter 11 we assumed that each point in the interior of the entire catalyst surface was accessible to the same reactant concentration. However, where the reactants diffuse into the pores within the catalyst pellet, the concentration at the pore mouth will be higher than that inside the pore, and we see that the entire catalytic surface is not accessible to the same concentration. To account for variations in concentration throughout the pellet, we introduce a parameter known as the effectiveness factor. In this chapter we will develop models for diffusion and reaction in two-phase systems, which include catalyst pellets and CVD reactors. The types of reactors discussed in this chapter will include packed beds, bubbling fluidized beds, slurry reactors, and trickle beds. After studying this chapter you will be able to describe diffusion and reaction in two- and three-phase systems, determine when internal pore diffusion limits the overall rate of reaction, describe how to go about eliminating this limitation, and develop models for systems in which both diffusion and reaction play a role (e.g., CVD). 

We w ilJ discuss each of the steps shown in Figure 10-6 and Table 10-2. As mentioned earlier, this chapter focuses on Steps 3. 4. and 5 (the adsorption, surface reaction and desorption steps by assuming Steps 1, 2, 6, and 7 are very rapid. Consequently to understand when this assumption is valid, we shall give a quick overview of Steps 1, 2. 6. and 7. Steps I and 2 involve diffusion of the reactants to and within the catalyst pellet. While these steps are covered in detail in Chapters 11 and 12. it is worthwhile to give a brief description of these two mass transfer steps to better understand the entire sequence of steps. 

As discussed in chapter 5, diffusion through catalyst pores represents a resistance to mass and heat transfer, which gives rise to concentration and temperature gradients within the catalyst pellet. This causes the rate of reaction in the solid phase to be different from that if the bulk phase conditions prevail inside the particle, and the rate of reaction should be integrated along the radius of the pellet to get the actual rate of reaction. 

An important embellishment to the foregoing treatment of packed-bed reactors is to allow for temperature and concentration gradients within the catalyst pellets. Intrapellet diffusion of heat and mass is governed by differential equations that are about as complex as those governing the bulk properties of the bed. See Section... 

In a fixed-bed reactor, if coke is formed primarily from the reactants, there will be a gradient in catalyst activity, with the highest coke content and the lowest activity in the first part of the bed. If a reaction product is the main coke precursor, the coke level will be highest and the activity low toward the end of the bed. Additional complexity arises if the coke-deposition reaction is diffusion limited, which leads to a gradient of coke content within the catalyst pellet. Many theoretical studies have been made showing the shape of profiles of coke content and catalytic activity in catalyst beds subject to fouling, but there are few comparisons of theory and experiment and no way to predict the fouling rate for a new system. 

Eor strong diffusion resistance within the catalyst pellet, Equation 2.190 can be simplified to ... 

Let rA be the rate of transfer of A into the catalyst pellet had the resistance to internal pore diffusion been negligible. If the resistance to internal pore diffusion is neglected, then the concentration of A at all the active sites within the pores inside the catalyst pellet will be the same as the concentration of A at the surface of the catalyst pellet, that is, Q = Qs for all values ofx,-Lspecific reaction rate of A is the same at all locations within the catalyst pellet and is equal to (-ri) = RCas- So,... 

Diffusion and Reaction in a Single Cylindrical Pore within the Catalyst Pellet... 

If a catalyst pellet (of any shape) has well-structured pores that are of imiform diameter d and length L and the pores are uniformly distributed throughout the volume of the pellet, then the overall rate equation can be derived by accounting for the rate of diffusion and rate of reaction in one single pore within the catalyst pellet. Consider a cylindrical pore of diameter d and length L (Figure 4.24) in a catalyst pellet in contact with a gas stream containing reactant A at concentration Ag- AS is the concentration of A in the gas at the pore mouth on the outer surface of the catalyst pellet. 

The kinetics discussed so far are intrinsic kinetics that are valid in the absence of difihisional or heat transfer resistances. Bulk phase eoneentrations (partial pressures) and temperatures are not present at the active sites within the catalyst pellets, since the pore structure offers very significant resistance to diffusion, and mild resistance to heat transfer exists between the bulk phase and the pellet surface. Little resistance to heat transfer exists within the pellets (Reference 4, page 69). 

Ethane is dehydrogenated on a porous platinum catalyst. Here, the ethane can diffuse through the catalyst pores at a rate similar to diffusion in the surrounding gas. The reaction takes place on pore walls within the catalyst pellet. This reaction is usually treated as homogeneous, even though the chemistry is similar to the previous example. 

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.

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. 

Clearly by working with typical spatial resolutions of approximately 30-50 pm, individual pores within the material are not resolved. However, a wealth of information can be obtained even at this lower resolution (53,54,55). Typical data are shown in Fig. 20, which includes images or maps of spin density, nuclear spin-lattice relaxation time (Ti), and self-diffusivity of water within a porous catalyst support pellet. In-plane spatial resolution is 45 pm x 45 pm, and the image slice thickness is 0.3 mm. The spin-density map is a quantitative measure of the amount of water present within the porous pellet (i.e., it is a spatially resolved map of void volume). Estimates of overall pellet void volume obtained from the MR data agree to within 5% with those obtained by gravimetric analysis. 

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.

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. 

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