Diffusion catalyst pores

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). 

Mass transport selectivity is Ulustrated by a process for disproportionation of toluene catalyzed by HZSM-5 (86). The desired product is -xylene the other isomers are less valuable. The ortho and meta isomers are bulkier than the para isomer and diffuse less readily in the zeoHte pores. This transport restriction favors their conversion to the desired product in the catalyst pores the desired para isomer is formed in excess of the equUibrium concentration. Xylene isomerization is another reaction catalyzed by HZSM-5, and the catalyst is preferred because of restricted transition state selectivity (86). An undesired side reaction, the xylene disproportionation to give toluene and trimethylbenzenes, is suppressed because it is bimolecular and the bulky transition state caimot readily form. 

Fig. 9. Catalyst pore and reaction. The CO diffuses into a precious metal site D reacts with O2 and leaves as CO2.

Note that this parameter has the same form as the Thiele number which occurs in the theory of diffusion/reac tion in catalyst pores. 

Other cases, neglecting heat effects would cause serious errors. In such cases the mathematical treatment requires the simultaneous solution of the diffusion and heat conductivity equations for the catalyst pores. 

For the first assumption, the value of Kw for the shift appears to be too high. It must be this high because it is necessary to make C02 appear while both C02 and CO are being consumed rapidly by methanation. The data may be tested to see if the indicated rate appears unreasonable from the standpoint of mass transfer to the gross catalyst surface. Regardless of the rate of diffusion in catalyst pores or the surface reaction rate, it is unlikely that the reaction can proceed more rapidly than material can reach the gross pill surface unless the reaction is a homogeneous one that is catalyzed by free radicals strewn from the catalyst into the gas stream. 

Step 3. Transport within a catalyst pore is usually modeled as a one-dimensional diffusion process. The pore is assumed to be straight and to have length The concentration inside the pore is ai =ai(l,r,z) where I is the position inside the pore measured from the external surface of the catalyst particle. See Figure 10.2. There is no convection inside the pore, and the diameter of the pore is assumed to be so small that there are no concentration gradients in the radial direction. The governing equation is an ODE. 

Fig. 2 shows the liquid product distributions over catalysts. Main product over ferrierite is C5 hydrocarbon, while products were distributed over mainly C,-C, over HZSM-5. Table 4 shows the effect of mixing ratio on product distribution. While HZSM-5/PP ratio does not affect product distribution, higher amount of gas is obtained with increasing ferrierite/PP ratio. This is ascribed to the increased possibility of polypropylene diffusion into pore as the amount of ferrierite is increased. 

Thus, considering diffusion in pores leads to very similar results to those we obtained when describing diffusion in catalyst particles. 

For catalytic reactions carried out in the presence of a heterogeneous catalyst, the observed reaction rate could be determined by the rate of mass transfer from the bulk of the reaction mixture and the outer surface of the catalyst particles or the rate of diffusion of reactants within the catalyst pores. Consider a simple first order reaction its rate must be related to the concentration of species S at the outer surface of the catalyst as follows ... 

Transfer of products from the interior catalyst pores to the gross external surface of the catalyst by ordinary molecular diffusion and/or Knudsen diffusion. 

Scanning electron microscopy and other experimental methods indicate that the void spaces in a typical catalyst particle are not uniform in size, shape, or length. Moreover, they are often highly interconnected. Because of the complexities of most common pore structures, detailed mathematical descriptions of the void structure are not available. Moreover, because of other uncertainties involved in the design of catalytic reactors, the use of elaborate quantitative models of catalyst pore structures is not warranted. What is required, however, is a model that allows one to take into account the rates of diffusion of reactant and product species through the void spaces. Many of the models in common use simulate the void regions as cylindrical pores for such models a knowledge of the distribution of pore radii and the volumes associated therewith is required. 

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. 

Ordinary or bulk diffusion is primarily responsible for molecular transport when the mean free path of a molecule is small compared with the diameter of the pore. At 1 atm the mean free path of typical gaseous species is of the order of 10 5 cm or 103 A. In pores larger than 1CT4 cm the mean free path is much smaller than the pore dimension, and collisions with other gas phase molecules will occur much more often than collisions with the pore walls. Under these circumstances the effective diffusivity will be independent of the pore diameter and, within a given catalyst pore, ordinary bulk diffusion coefficients may be used in Fick s first law to evaluate the rate of mass transfer and the concentration profile in the pore. In industrial practice there are three general classes of reaction conditions for which the bulk value of the diffusion coefficient is appropriate. For all catalysts these include liquid phase reactions... 

If the dominant mode of transport within the catalyst pores is ordinary molecular diffusion, the analysis becomes somewhat more complex. The ordinary molecular diffusivity is inversely proportional to the pressure so that in this case... 

Now consider the other extreme condition where diffusion is rapid relative to chemical reaction [i.e., hT( 1 — a) is small]. In this situation the effectiveness factor will approach unity for both the poisoned and unpoisoned reactions, and we must retain the hyperbolic tangent terms in equation 12.3.124 to properly evaluate Curve C in Figure 12.11 is calculated for a value of hT = 5. It is apparent that in this instance the activity decline is not nearly as sharp at low values of a as it was at the other extreme, but it is obviously more than a linear effect. The reason for this result is that the regions of the catalyst pore exposed to the highest reactant concentrations do not contribute proportionately to the overall reaction rate because they have suffered a disproportionate loss of activity when pore-mouth poisoning takes place. 

Carbon deposits or heavy hydrocarbons (>C100) may block the catalyst pores causing diffusion problems.39... 

Taking these effects into account, internal pore diffusion was modeled on the basis of a wax-filled cylindrical single catalyst pore by using experimental data. The modeling was accomplished by a three-dimensional finite element method as well as by a respective differential-algebraic system. Since the Fischer-Tropsch synthesis is a rather complex reaction, an evaluation of pore diffusion limitations... 

The most difficult problem to solve in the design of a Fischer-Tropsch reactor is its very high exothermicity combined with a high sensitivity of product selectivity to temperature. On an industrial scale, multitubular and bubble column reactors have been widely accepted for this highly exothermic reaction.6 In case of a fixed bed reactor, it is desirable that the catalyst particles are in the millimeter size range to avoid excessive pressure drops. During Fischer-Tropsch synthesis the catalyst pores are filled with liquid FT products (mainly waxes) that may result in a fundamental decrease of the reaction rate caused by pore diffusion processes. Post et al. showed that for catalyst particle diameters in excess of only about 1 mm, the catalyst activity is seriously limited by intraparticle diffusion in both iron and cobalt catalysts.1... 

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 ... 

The modeling of the internal pore diffusion of a wax-filled cylindrical single catalyst pore was accomplished by the software Comsol Multiphysics (from Comsol AB, Stockholm, Sweden) as well as by Presto Kinetics (from CiT, Rastede, Germany). Both are numerical differential equation solvers and are based on a three-dimensional finite element method. Presto Kinetics displays the results in the form of diagrams. Comsol Multiphysics, instead, provides a three-dimensional solution of the problem. 

The data derived from modeling at different conversion degrees (X = 5, 40, and 80%) were also compared to the results obtained from the calculation of the classical Thiele modulus. The calculated (by the Thiele modulus) and modeled (by Presto Kinetics) effectiveness factors showed comparable values. Hence, the usage of simulation software is not required to get a first impression of the diffusion limitations in a Fischer-Tropsch catalyst pore. Nevertheless, modeling represents a valuable tool to better understand conditions within a catalyst pore. 

A large number of analytical solutions of these equations appear in the literature. Mostly, however, they deal only with first order reactions. All others require solution by numerical or other approximate means. In this book, solutions of two examples are carried along analytically part way in P7.02.06 and P7.02.07. Section 7.4 considers flow through an external film, while Section 7.5 deals with diffusion and reaction in catalyst pores under steady state conditions. 

Mass transfer effects are very important for the selectivity in the Fischer-Tropsch synthesis. Even though the reactants are in the gas phase, the catalyst pores will be filled with liquid products. Diffusion in the liquid phase is about 3 orders of magnitude slower than in the gas phase and even slow reactions may become diffusion limited. Diffusion limitations may occur through limitation on the arrival of CO to the active points or through the limited removal of reactive products.8,9... 

As shown, there are numerous reports in the literature describing qualitatively the influence of water on the rate and the selectivity of the FTS, but there are fewer quantitative descriptions and mechanistic explanations. The FTS provides a complex reaction environment, where catalyst pores are wax-filled, and water is the main reaction product. This means that the reactants must be dissolved and diffuse through the liquid to reach the active site, and the products (including water) must be transported in the opposite direction. Although the reaction in general is slow,... 

For reasons of simplicity, the Thiele modulus will be defined and calculated for a catalyst plate with pore access at both ends of the plate and not at the bottom or top. Note that for most cases in real-life applications the assumptions have to be modified using polar coordinates for the calculations. The Thiele modulus q> is therefore defined as the product of the length of the catalyst pore, /, and the square root of the quotient of the constant of the speed of the reaction, k. divided by the effective diffusion coefficient DeS ... 

Surface area is by no means the only physical property which determines the extent of adsorption and catalytic reaction. Equally important is the catalyst pore structure which, although contributing to the total surface area, is more conveniently regarded as a separate factor. This is because the distribution of pore sizes in a given catalyst preparation may be such that some of the internal surface area is completely inaccessible to large reactant molecules and may also restrict the rate of conversion to products by impeding the diffusion of both reactants and products throughout the porous medium. 

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.

Figure 7-13 Plots of effectiveness factor 17 versus Thiele modulus

The reactant A and the product B diffuse into and out of a cylindrical catalyst pore with length L and radius r. The material balance for reactant A at steady state for a differential length dr of the catalyst pore is written as diffusion flux in - diffusion flux out - disappearance by reaction = 0... 

Reactant selectivity occurs when some of the molecules in a reaction mixture can enter the pores and react in the catalyst pores. However, the molecules that are too large to diffuse through the pores cannot react. 

Product selectivity occurs when some of the products formed in the catalyst pore are too bulky to diffuse out, being converted to less bulky molecules (e.g., by equilibration or cracking). The large product molecules, which cannot diffuse out, may eventually deactivate the catalytic sites by blocking the pores. 

Mobil ZSM-5 zeolite catalysts can be modified to reduce the effective pore and channel dimensions. These modified zeolites allow discrimination between molecules of slightly different dimensions. Because of this shape-selective action, p-ethyltoluene is able to diffuse out of the catalyst pores at a rate about three orders of magnitude greater than the two regioisomers. As a result, p-ethyltoluene is formed with very high (97%) selectivity.333... 

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