Diffusivity in a catalyst particle

According to the above definitions, the effectiveness factor for any of the above shapes can adequately describe simultaneous reaction and diffusion in a catalyst particle. The equation for the effectiveness factor in a slab is the simplest in Table 6.3.1 and will be used for all pellet shapes with the appropriate Thiele modulus ... 

Illustration 4.9 Reaction and Diffusion in a Catalyst Particle. The Effectiveness Factor and the Design of Catalyst Pellets... 

Figure 10.5 shows the basic concept of the particle-level MR that gives (i) selective addition of reactants to the reaction zone and (ii) selective removal of products from the reaction zone. In the first case, if the diffusivity of one reactant (A) is much higher than that of the other components (B), the reactant (A) selectively diffuses into a catalyst particle through a membrane. Undesired reactions or the adsorption of poisons on the catalysts can be prevented. In the second case, the reaction has a hmited yield or is selectivity controlled by thermodynamics. The selective removal of the desired product from the catalyst particle gives enhancement of selectivity when the diffusivity of one product (R) is much greater than that of the other products (S). 

A kinetics or reaction model must take into account the various individual processes involved in the overall process. We picture the reaction itself taking place on solid B surface somewhere within the particle, but to arrive at the surface, reactant A must make its way from the bulk-gas phase to the interior of the particle. This suggests the possibility of gas-phase resistances similar to those in a catalyst particle (Figure 8.9) external mass-transfer resistance in the vicinity of the exterior surface of the particle, and interior diffusion resistance through pores of both product formed and unreacted reactant. The situation is illustrated in Figure 9.1 for an isothermal spherical particle of radius A at a particular instant of time, in terms of the general case and two extreme cases. These extreme cases form the bases for relatively simple models, with corresponding concentration profiles for A and B. 

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. 

The classical method of investigation of effects of diffusion on reactions is typically to run a reaction with catalyst particles of various sizes. For zeolites, the resistance of intracrystalline diffusion is normally much larger than that characteristic of molecular diffusion or Knudsen diffusion that could occur in the spaces between the zeolite crystals in a catalyst particle. Thus, the crystal size of the zeolite has to be varied instead of the particle size to determine the effects of diffusion on zeolite-catalyzed reactions. Kinetics of the MTO reaction has been measured with SAPO-34 crystals with identical compositions and sizes of 0.25 and 2.5 pm 89). The methanol conversion was measured as a function of the coke content of the two SAPO-34 crystals in the TEOM reactor. 

This relationship is plotted in Figure 6.3.9. The effectiveness factor for a severely diffusion-limited reaction in a catalyst particle is approximated by the inverse of the Thiele modulus. 

It is obvious from this discussion that control of pore size and distribution is important in developing the optimum catalyst for a specific process. The most common perception of the pores in a catalyst particle is that they are essentially holes of various diameters drilled into the particle at various angles or, perhaps, they resemble various sized tunnels randomly oriented throughout the particle. While such representations may be helplul in discussing the effects of pore diffusion and similar catalytic factors, it does not provide any basis for the development of methods that can produce supports or supported catalysts having specifrc pore characteristics. 

In order to simplify the model, we assumed an effective diffusMty icould be used to describe diffusion in the catalyst particles. We next presented the general mass and energy balances for the catalyst particle. Next we solved a series of reaction-diffusion problems in a single... 

Diffusion of reactants to the external surface is the first step in a solid-catalyzed reaction, and this is followed by simultaneous diffusion and reaction in the pores, as discussed in Chapter 4. In developing the solutions for pore diffusion plus reaction, the surface concentrations of reactants and products are assumed to be known, and in many cases these concentrations are essentially the same as in the bulk fluid. However, for fast reactions, the concentration driving force for external mass transfer may become an appreciable fraction of the bulk concentration, and both external and internal diffusion must be allowed for. There may also be temperature differences to consider these will be discussed later. Typical concentration profiles near and in a catalyst particle are depicted in Figure 5.6. As a simplification, a linear concentration gradient is shown in the boundary layer, though the actual concentration profile is generally curved. 

The stability problem for an exothermic reaction in a catalyst particle is similar to that for a reaction in a CSTR, in that multiple solutions of the heat and mass balance equations are possible. A typical plot of heat generation and removal rates is shown in Figure 5.11. The values of Qg and Qr are in cal/sec, g, and a is the external area in cm /g. The plot differs from the one for a CSTR (Fig. 5.2) in that the highest possible value for Qg is a mass transfer limit corresponding to Cj = 0 and not to complete conversion. The mass transfer limit increases with temperature because of the increase in diffusivity, and the limit also increases with gas velocity. The heat removal... 

The kinetics of the main reaction were studied in a classical differential reactor, on the basis of conversions extrapolated to zero time. Since the rate of the main reaction was diffusion controlled, several catalyst sizes had to be investigated. The conversion and coke profiles in a catalyst particle of industrial size were shown already in Fig, 1, Ex. 5.3xl-l. 

As an alternative to the previous example, we can also solve the problems with inhomogeneous boundary conditions by direct application of the finite integral transform, without the necessity of homogenizing the boundary conditions. To demonstrate this, we consider the following transient diffusion and reaction problem for a catalyst particle of either slab, cylindrical, or spherical shape. The dimensionless mass balance equations in a catalyst particle with a first order... 

The analytical integration of the differential equation for diffusion with chemical reaction in a catalyst particle is achievable just for first-order reactions. A generalized modulus has been proposed for extending the use of the // expression in Equations 2.61 and 2.64a to any type of rate expression [17], at least approximately. For irreversible nth order reactions, the generalized modulus for a sphere becomes dependent on the exterior surface concentration ... 

As an illustration of the application of approximate analytical techniques, we derive the asymptotic forms of the efiectiveness factor and average solid concentration/temperature for a non-isothermal reaction in a catalyst particle with external heat/mass gradients. Adopting the normalization in Equations 3.16-3.20 for a solid porous particle ( o = 0) with characteristic dimension in the direction of diffusion Rp, the mass and heat balances become... 

Luss D, Amundson NR. Uniqueness of the steady state solutions for chemical reactor occurring in a catalyst particle or in a tubular reaction with axial diffusion. Chemical Engineering Science 1967 22 253-266. 

To diedc for the presence of pore diffusion in a catalyst one mu do preliminary runs at a set of reaction conditions, within die range of conditions of interest, idio e these effects can be expected to be most n-ominent. Sudi conditions include high ton-poature, high concentration, and large catalyst particle size. The prdiminary runs are (kne at die most sevoe temperature and concentration conditions diat we expect to mcounto in the study. At this set of omditions we vary die size of catalyst particles used. The results must show no effect of particle size on conversion, over a fiictor of two or diree in particle diameter above the particle size to be used in the study. 

We note in passing that for dissociation of many gases the adsorption of molecules from the gas phase directly upon the surface of catalyst nanoparticles is not necessary. Tsu and Boudart (1961) and others (Henry et al. 1991 Bowker 1996) showed that the molecules can first adsorb onto the oxide support and diffuse to a catalyst particle. This means that the effective capture radius of a catalyst nanoparticle (Pd, Pt, etc.) can be much greater than the nanoparticle s physical radius. As with the spillover zones, the molecule-collection zones (Tsu and Boudart 1961) overlap when the coverage of catalyst nanoparticles exceeds some threshold value, effectively converting the entire surface of the nanostructure into a molecular delivery system for the metal catalyst nanoparticles. This so-called back-spillover effect further increases the likelihood of molecular dissociation and ionosorption on the metal oxide surface. 

However, there is large difference between the conditions in industrial and in laboratory reactor. In addition to the size of catalyst particles for which the surface utilization ratio can be adjusted by Eq. (2.214), the catalyst amount used is much more for industrial reactor than laboratory one the reduction conditions and two-dimension distributions of gas flow and temperature caused by reactor structure are also different, as well as there is catalyst deactivation etc. Therefore, intrinsic rate constant k derived from activity data measured at laboratory cannot be used directly for the design calculation of industrial reactor. In order to simplify calculation, a concept of active coefficient is introduced, since it is too complicated to consider the process factors such as pore diffusion in large catalyst particles, industrial reduction and deactivation and two streams of gas flow and temperature distributions in industrial reactor etc, for which direct engineering applications are more difficult. That means that the intrinsic rate constant k obtained in laboratory or Eq. (2.214) is produced by the active coefficient. The magnitude of the active... 

Diffusion and Reaction in a Catalyst Particle. A Continuum Model... 

DIFFUSION AND REACTION IN A CATALYST PARTICLE. A CONTINUUM MODEL 193... 

The analytical integration of the continuity equation for the diffusing and reacting component A in a catalyst particle is possible only for a first-order reaction. A catalytic reaction generally has a more complicated rate equation, however, as shown in Chapter 2. It is attractive to extend the application of the simple relation (3.6.1-6 or 3.6.1-14) to any type of rate equation, be it only approximately. This is what led Aris [1965a, b], Bischoff [1965], and Petersen [1965a, b] to the introduction of a generalized modulus. 

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