Catalyst pellets, diffusion coefficients

Diffusivity and tortuosity affect resistance to diffusion caused by collision with other molecules (bulk diffusion) or by collision with the walls of the pore (Knudsen diffusion). Actual diffusivity in common porous catalysts is intermediate between the two types. Measurements and correlations of diffusivities of both types are Known. Diffusion is expressed per unit cross section and unit thickness of the pellet. Diffusion rate through the pellet then depends on the porosity d and a tortuosity faclor 1 that accounts for increased resistance of crooked and varied-diameter pores. Effective diffusion coefficient is D ff = Empirical porosities range from 0.3 to 0.7, tortuosities from 2 to 7. In the absence of other information, Satterfield Heterogeneous Catalysis in Practice, McGraw-HiU, 1991) recommends taking d = 0.5 and T = 4. In this area, clearly, precision is not a feature. 

Diffusion effects can be expected in reactions that are very rapid. A great deal of effort has been made to shorten the diffusion path, which increases the efficiency of the catalysts. Pellets are made with all the active ingredients concentrated on a thin peripheral shell and monoliths are made with very thin washcoats containing the noble metals. In order to convert 90% of the CO from the inlet stream at a residence time of no more than 0.01 sec, one needs a first-order kinetic rate constant of about 230 sec-1. When the catalytic activity is distributed uniformly through a porous pellet of 0.15 cm radius with a diffusion coefficient of 0.01 cm2/sec, one obtains a Thiele modulus y> = 22.7. This would yield an effectiveness factor of 0.132 for a spherical geometry, and an apparent kinetic rate constant of 30.3 sec-1 (106). 

Provided that the catalyst is active enough, there will be sufficient conversion of the pollutant gases through the pellet bed and the screen bed. The Sherwood number of CO is almost equal to the Nusselt number, and 2.6% of the inlet CO will not be converted in the monolith. The diffusion coefficient of benzene is somewhat smaller, and 10% of the inlet benzene is not converted in the monolith, no matter how active is the catalyst. This mass transfer limitation can be easily avoided by forcing the streams to change flow direction at the cost of some increased pressure drop. These calculations are comparable with the data in Fig. 22, taken from Carlson 112). 

Fig. 3.3.4 Variation of the tortuosity x inside the catalyst pellets during coking and regeneration, obtained by measuring the self-diffusion coefficient of n-heptane at room temperature.

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. 

Barrer (19) has developed another widely used nonsteady-state technique for measuring effective diffusivities in porous catalysts. In this approach, an apparatus configuration similar to the steady-state apparatus is used. One side of the pellet is first evacuated and then the increase in the downstream pressure is recorded as a function of time, the upstream pressure being held constant. The pressure drop across the pellet during the experiment is also held relatively constant. There is a time lag before a steady-state flux develops, and effective diffusion coefficients can be determined from either the transient or steady-state data. For the transient analysis, one must allow for accumulation or depletion of material by adsorption if this occurs. 

Consider the spherical catalyst pellet of radius R shown in Figure 12.4. The effective diffusivity approach presumes that diffusion of all types can be represented in terms of Fick s first law and an overall effective diffusion coefficient that can be taken as a constant. That is, the appropriate flux representation is... 

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). 

Pore diffusion can be increased by choosing a catalyst with the proper geometry, in particular the pellet size and pore structure. Catalyst size is obvious (r if pore diffusion limited for the same total surface area). The diameter of pores can have a marked influence on r) because the diffusion coefficient of the reactant Da witl be a function of dp if molecular flow in the pore dominates. Porous catalysts are frequently designed to have different distributions of pore diameters, sometimes with macropores to promote diffusion into the core of the catalyst and micropores to provide a high total area. 

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. 

The pellets of the commercial catalyst were crushed to grain size from 0.5 to 1 mm. A calculation on the basis of the measurements of the effective diffusion coefficient showed that the reaction proceeded in the kinetic region. Bed density of the catalyst was 1.23 g/cm3, specific surface after kinetic experiments was 36 m2/g. In the temperature range of 150-225°C reaction (342) is practically irreversible. The experiments proved (348) to be valid thus, the kinetics on low- and high-temperature catalysts is the same. 

Figure 5 shows the simulation of the reaction kinetic model for VO-TPP hydro-demetallisation at the reference temperature using a Be the network with coordination 6. The metal deposition profiles are shown as a function of pellet radius and time in case of zero concentration of the intermediates at the edge of the pellet. Computer simulations were ended when pore plugging occurred. It is observed that for the bulk diffusion coefficient of this reacting system the metal deposition maximum occurs at the centre of the catalyst pellet, indicating that the deposition process is reaction rate-determined. The reactants and intermediates can reach the centre of the pellet easily due to the absence of diffusion limitations. 

Figure 6 shows the influence of the bulk diffusion coefficient, Dhi on the metal deposition profiles. Obviously, by decreasing the diffusivity the metal deposition process becomes more diffusion rate-determined. With decreasing diffusivity the transport of reactant and intermediates is decreased resulting in a less deep penetration into the catalyst pellet. Therefore, the metal deposition maximum is shifted further to the exterior of the catalyst pellet. 

S/m and fl/ x = Biot numbers for mass and heat transfer 4 and 4 x = Thiele modulus Le = Lewis number A0 i = dimensionless adiabatic temperature rise t) = effectiveness factor kg =mass transfer coefficient (ms-1) Rp = radius of catalyst pellet (m) Da = effective diffusion coefficient (ms-2) r =rate of reaction (molm-3s-1) C —concentration of reactant (molm-3) ... 

Obviously the effectiveness factor, r, depends upon the effective diffusion coefficient, DeA, and kinetic parameters such as a first-order rate coefficient, kVipi as well as on the shape of the catalyst pellet. 

The question remains as to when the various diffusion effects really influence the conversion rate in fluid-solid reactions. Many criteria have been developed in the past for the determination of the absence of diffusion resistance. In using the many criteria no more information is required than the diffusion coefficient DA for fluid phase diffusion and for internal diffusion in a porous pellet, the heat of reaction and the physical properties of the gas and the solid or catalyst, together with an experimental value of the observed global reaction rate (R ) per unit volume or weight of solid or catalyst. For the time being the following criteria are recommended. Note that intraparticle criteria are discussed in much greater detail in Chapter 6. 

The effective diffusion coefficient may become a function of the reactant concentration and thus it will depend upon the place inside the catalyst pellet. 

The effective diffusion coefficients, which up till now have been assumed constant, become a function of the concentration and, therefore, due to the internal concentration gradients, the effective diffusion coefficients will depend on the distance inside the catalyst pellet. 

Because of volume changes due to the reaction, pressure gradients may occur inside the catalyst pellet. This can give rise to two effects. First, it influences the effective diffusion coefficients, since the gas-phase diffusion coefficients depend on pressure. Second, the pressure gradients affect the concentrations (or more accurately, chemical activities), which determine the reaction rate. Hence pressure gradients must directly influence the effectiveness factor. 

In this equation ep is the porosity of the catalyst pellet and yp the tortuosity of the catalyst pores as discussed in Chapter 3 (the rest of the symbols are as defined before). From this formula it follows that the effective diffusion coefficient depends on both the gas composition and the pressure. Since we know the pressure as a function of the concentration, Equation 7.74 provides the effective diffusion coefficient as a function of the concentration. If we define... 

Differential equation 7.120 describes the concentration profile in an isotropic ring-shaped catalyst pellet with an effective diffusion coefficient DeAH, a height H, and an inner radius... 

Rather than accounting for anisotropy by modifying both the characteristic dimension and effective diffusion coefficient, this is achieved by modifying the effective diffusion coefficient only. Thus, for anisotropic catalyst pellets a modified effective diffusion coefficient D is defined, which accounts for the anisotropy. Hence, for anisotropic catalyst pellets and simple reactions the Aris numbers can be calculated from... 

Comparison of Equations 7.126 and 7.127 with 7.124 and 7.125 shows that for ring-shaped catalyst pellets the modified effective diffusion coefficient Dj follows from... 

From this discussion it follows that all conclusions drawn for isotropic catalyst pellets hold for anisotropic catalyst pellets as well. In fact, with a single catalyst geometry, it is not possible to distinguish between isotropic and anisotropic pellets. The effect of anisotropy is lumped in with the effective diffusion coefficient. If the catalyst pellet is isotropic, then from Equation 7.128 it follows that we measure the effective diffusion coefficient D / = D J = D. For anisotropic pellets, we measure for pellets with a large height (0 large) and D H for flat pellets (0 small). For intermediate values of 0 the value of DeA is between De/iR and D H. 

It is assumed that component A is the component not in excess, thus /3g < 1, )8C < 1,... and that the numbersPb, 0. a do not depend on the temperature. They do not depend on the gas composition inside the catalyst pellet either, since the gas diluted. Notice the superscript + of the effective diffusion coefficient, which denotes that the catalyst pellet may be anisotropic. 

A reaction of the order Vi is carried out in a spherical catalyst pellet. For the given surface temperature and concentration the product ksCAjmii equals 0.2 s 1. The effective diffusion coefficient has been determined as 2 x Iff7 m2 s"1. The diameter of the sphere is 6 mm. Fur-... 

Assume the catalyst pellet to be anisotropic. The radial effective diffusion coefficient is De R = 5x10 m2 s 1 the longitudinal effective diffusion coefficient equals DeAH = 1.8xl0 6 m2 s 1. Then, if the effective diffusion coefficient is determined from measurements of the effectiveness factor, this gives... 

In these equations JA is the mole flux of A in moles of A per second per square metre flowing through surface area of the catalyst pores. This is not the same as the mole flux in moles of A per second per square metre flowing through surface area of the catalyst pellet. This is elaborated in Appendix E. The term DAP is the Maxwell diffusion coefficient of A in a binary mixture with P and DM is the Knudsen diffusion coefficient of A inside the catalyst pores. ka is the mole fraction of A. J, Dpdp Dn, kp etc. are similarly defined. VA is a factor that accounts for viscous flow inside the pores. If VA is much smaller than one, viscous flow can be neglected. We will neglect viscous flow for all components and substitute... 

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