Catalysts pore tortuosity

In the special case of an ideal single catalyst pore, we have to take into account that diffusion is quicker than in a porous particle, where the tortuous nature of the pores has to be considered. Hence, the tortuosity r has to be regarded. Furthermore, the mass-related surface area AmBEX is used to calculate the surface-related rate constant based on the experimentally determined mass-related rate constant. Finally, the gas phase concentrations of the kinetic approach (Equation 12.14) were replaced by the liquid phase concentrations via the Henry coefficient. This yields the following differential equation ... 

If data are available on the catalyst pore- structure, a geometrical model can be applied to calculate the effective diffusivity and the tortuosity factor. Wakao and Smith [36] applied a successful model to calculate the effective diffusivity using the concept of the random pore model. According to this, they established that ... 

The effectiveness factors and n, defined as the ratios of the actual reaction rates at time 0 to the maximum reaction rates on a clean catalyst, are obtained nEmerically from equations [4] -[9]. An explicit finite difference method was used to solve the partial differential equations without further simplifications. Densities, porosities and clean catalyst pore diameters were measured experimentally. The maximum coke content is assumed to be that which fills the pore completely. The tortuosity is taken as 2.3, as discussed by Satterfield et al. (14). 

Coppens and Froment (1995a, b) employed a fractal pore model of supported catalyst and derived expressions for the pore tortuosity and accessible pore surface area. In the domain of mass transport limitation, the fractal catalyst is more active than a catalyst of smooth uniform pores having similar average properties. Because the Knudsen diffusivity increases with molecular size and decreases with molecular mass, the gas diffusivities of individual species in... 

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... 

Here, is the experimental mean rate of reaction per unit volume of catalyst, L is a characteristic length of the porous photocatalyst (i.e., the film thickness), t is the pore tortuosity (taken as three), D is the diffusion coefficient of the pollutant in air, Cg is the mean concentration at the external surface, and e is the catalyst grain porosity (0.5 for Degussa s P25). Such a treatment was performed by Doucet et al. (2006) while taking D of the pollutants to be approximately 10 m s. The estimated Weisz modulus ranged between 10 and 10, depending on the type of pollutant, that is, some three to five orders of magnitude smaller than the value of unity, which is often taken as a criterion for internal mass transport limitation. 

Catalyst particles size -35 -i-48 Tyler mesh were used in all tests. Porosity was measured using a mercury porosimeter. A 0.1356 pm pore mean diameter was determined. The Satterfield and Sherwood (7) methodology was used to verify that reaction occurs without any diSusional limitation (internal or external). The effective diffusivity was estimated from the porosity measurements and binary diffusion coefficient and pore tortuosity pubhshed in the hterature, leading to an estimated value of 10 for the generahzed Thiele Modvdus based on the reaction rate. The efi ectiveness factor was then considered as 1.0. 

Experimentally, it is difficult to separate the effects of F and z on the effective diffusivity. Often, empirical values of the product Ft are reported as the tortuosity. This is particularly true in the literature of transport/reac-tion in porous catalyst pellets. For the large variety of catalysts, the tortuosity —equal to /h(SK in Equation 4-42—ranges from 1 to 10 [103]. Although it has been difficult to correlate these tortuosity values with experimentally determined pore structure parameters, the tortuosity almost always decreases with increasing porosity. 

Coking of the catalyst pore structure can also lead to modifications of the tortuosity factor. Ren et al. [63] studied diffusion of heptane in coked alumina catalyst samples using NMR methods. They found that tortuosity of an alumina catalyst increased from 2.4 when fresh to 3.2 after 16 wt. % coke had been deposited on its surface. Wood and Gladden [17] also observed increases in tortuosity experienced by pentane and heptane probe molecules of 19 and 57%, respectively, in a coked hydroprocessing catalyst compared with a fresh catalyst sample. 

For an actual catalyst, pore system is very complex, because the sizes and shapes of pores are different from each other and intercross with each other. In addition, their walls may not be smooth.These factors can influence the diffusion and hydrodynamics of gas, and a tortuosity factor, expressed by 6, which is introduced to account for or check their effects. The value of 6 is determined experimentally. The effective diffusion coefficient of component A is defined as ... 

Satterfield and Cadle [38] determined the tortuosity factors for 17 commercially manufactured, pelleted catalysts and catalyst supports using the parallel-pore model. Except for two materials that had been calcined at very high temperatures, all tortuosity factors fell between 3 and 7. For about half the catalysts, the tortuosity factor was about 4, regardless of macroporosity or composition. 

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. 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

X. H. Ren, M. Bertmer, H. Kuhn, S. Stapf, D. E. Demco, B. Blumich, C. Kem, A. Jess 2002, ( H, 13C and 129Xe NMR study of changing pore size and tortuosity during deactivation and decoking of a naphtha reforming catalyst), NATO Sci. Ser. ITMath., Phys. Chem. 7b, 603. 

The measured value of k Sg is 0.716 cm3/(sec-g catalyst) and the ratio of this value to k ltTueSg should be equal to our assumed value for the effectiveness factor, if our assumption was correct. The actual ratio is 0.175, which is at variance with the assumed value. Hence we pick a new value of rj and repeat the procedure until agreement is obtained. This iterative approach produces an effectiveness factor of 0.238, which corresponds to a differs from the experimental value (0.17) and that calculated by the cylindrical pore model (0.61). In the above calculations, an experimental value of eff was not available and this circumstance is largely responsible for the discrepancy. If the combined diffusivity determined in Illustration 12.1 is converted to an effective diffusivity using equation 12.2.9, the value used above corresponds to a tortuosity factor of 2.6. If we had employed Q)c from Illustration 12.1 and a tortuosity factor of unity to calculate eff, we would have determined that rj = 0.65, which is consistent with the value obtained from the straight cylindrical pore model in Illustration 12.2. 

The catalyst activity depends not only on the chemical composition but also on the diffusion properties of the catalyst material and on the size and shape of the catalyst pellets because transport limitations through the gas boundary layer around the pellets and through the porous material reduce the overall reaction rate. The influence of gas film restrictions, which depends on the pellet size and gas velocity, is usually low in sulphuric acid converters. The effective diffusivity in the catalyst depends on the porosity, the pore size distribution, and the tortuosity of the pore system. It may be improved in the design of the carrier by e.g. increasing the porosity or the pore size, but usually such improvements will also lead to a reduction of mechanical strength. The effect of transport restrictions is normally expressed as an effectiveness factor q defined as the ratio between observed reaction rate for a catalyst pellet and the intrinsic reaction rate, i.e. the hypothetical reaction rate if bulk or surface conditions (temperature, pressure, concentrations) prevailed throughout the pellet [11], For particles with the same intrinsic reaction rate and the same pore system, the surface effectiveness factor only depends on an equivalent particle diameter given by... 

For typical catalyst layers impregnated with ionomer, sizes of hydrated ionomer domains that form during self-organization are of the order of 10 nm. The random distribution and tortuosity of ionomer domains and pores in catalyst layers require more complex approaches to account properly for bulk water transport and interfacial vaporization exchange. A useful approach for studying vaporization exchange in catalyst layers could be to exploit the analogy to electrical random resistor networks of... 

One example of this type of reactor is in the synthesis of catalyst powders and pellets by growing porous soHd oxides from supersaturated solution. Here the growth conditions control the porosity and pore diameter and tortuosity, factors that we have seen are crucial in designing optimal catalysts for packed bed, fluidized bed, or slurry reactors. 

The pure compound rate constants were measured with 20-28 mesh catalyst particles and reflect intrinsic rates (—i.e., rates free from diffusion effects). Estimated pore diffusion thresholds are shown for 1/8-inch and 1/16-inch catalyst sizes. These curves show the approximate reaction rate constants above which pore diffusion effects may be observed for these two catalyst sizes. These thresholds were calculated using pore diffusion theory for first-order reactions (18). Effective diffusivities were estimated using the Wilke-Chang correlation (19) and applying a tortuosity of 4.0. The pure compound data were obtained by G. E. Langlois and co-workers in our laboratories. Product yields and suggested reaction mechanisms for hydrocracking many of these compounds have been published elsewhere (20-25). 

A naphtha is desulfurized by reducing its thiophene content with hydrogen at 660 K and 30 atm.The reaction is apparently first order with k = 0.3 cc thiophene/(g catalyst)(sec). The catalyst particle diameter is 0.35 cm, true density 2.65 g/cc, specific surface 180 m2/g, porosity 40%, In an experiment with very fine particles, conversion was 90%, but with the particles in question it was 70%. Find the effectiveness of the catalyst and the tortuosity of the pores. 

The following results refer to a bed 0.91 m deep containing spherical catalyst pellets of diameter 1.52 mm, with porosity 0.4 due to pores of diameter 75 A and tortuosity factor 3.5. 

Early efforts to model catalyst deactivation either utilized simplified models of the catalyst s porous structure, such as a bundle of nonintersecting parallel pores, or pseudo-homogeneous descriptions in terms of effective diffusivities and tortuosity... 

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