Diffusion coefficient, porous catalyst

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. 

Internal diffusion in porous catalysts, if dominant, also reduces the observed activity of the biocatalyst. The decisive coefficient for mass transfer is the effective diffusion coefficient De((, which is defined in Eq. (5.56), where D0is the diffusion coefficient in solution, e the porosity of the carrier, and t the tortuosity factor. 

The general approach for modelling catalyst deactivation is schematically organised in Figure 2. The central part are the mass balances of reactants, intermediates, and metal deposits. In these mass balances, coefficients are present to describe reaction kinetics (reaction rate constant), mass transfer (diffusion coefficient), and catalyst porous texture (accessible porosity and effective transport properties). The mass balances together with the initial and boundary conditions define the catalyst deactivation model. The boundary conditions are determined by the axial position in the reactor. Simulations result in metal deposition profiles in catalyst pellets and catalyst life-time predictions. 

This equation can be used to modify the effective diffusion coefficient an catalyst active surface area under flooded conditions. Another important parameter is irreducible liquid saturation, also called the immobile saturation (Ji ), which represents the amount of isolated trapped water in the pores of the PM. That is, even when a high flow rate of gas is introduced into the porous media, some fraction of liquid will remain (unless evaporated) primarily due to discontinuity or isolation with the rest of the pores. The irreducible fraction does not represent the fraetion of liquid in the porous media which cannot be removed from the fuel cell media. In fact, removal from drag forces is not possible, but removal from evaporation is. 

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. 

Pore diffusion limitation was studied on a very porous catalyst at the operating conditions of a commercial reactor. The aim of the experiments was to measure the effective diffiisivity in the porous catalyst and the mass transfer coefficient at operating conditions. Few experimental results were published before 1970, but some important mathematical analyses had already been presented. Publications of Clements and Schnelle (1963) and Turner (1967) gave some advice. 

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). 

Temperature gradients within the porous catalyst could not be very large, due to the low concentration of combustibles in the exhaust gas. Assuming a concentration of 5% CO, a diffusion coefficient in the porous structure of 0.01 cms/sec, and a thermal conductivity of 4 X 10-4 caI/sec°C cm, one can calculate a Prater temperature of 1.0°C—the maximum possible temperature gradient in the porous structure (107). The simultaneous heat and mass diffusion is not likely to lead to multiple steady states and instability, since the value of the 0 parameter in the Weisz and Hicks theory would be much less than 0.02 (108). 

Checking the absence of internal mass transfer limitations is a more difficult task. A procedure that can be applied in the case of catalyst electrode films is the measurement of the open circuit potential of the catalyst relative to a reference electrode under fixed gas phase atmosphere (e.g. oxygen in helium) and for different thickness of the catalyst film. Changing of the catalyst potential above a certain thickness of the catalyst film implies the onset of the appearance of internal mass transfer limitations. Such checking procedures applied in previous electrochemical promotion studies allow one to safely assume that porous catalyst films (porosity above 20-30%) with thickness not exceeding 10pm are not expected to exhibit internal mass transfer limitations. The absence of internal mass transfer limitations can also be checked by application of the Weisz-Prater criterion (see, for example ref. 33), provided that one has reliable values for the diffusion coefficient within the catalyst film. 

Internal resistance relates to the diffusion of the molecules from the external surface of the catalyst into the pore volume where the major part of the catalyst s surface is found. To determine the diffusion coefficients inside a porous space is not an easy task since they depend not only on the molecules diffusivity but also on the pore shape. In addition, surface diffusion should be taken into account. Data on protein migration obtained by confocal microscopy [8] definitely demonstrate that surface migration of the molecules is possible, even though the mechanism is not yet well understood. All the above-mentioned effects are combined in a definition of the so-called effective diffusivity [7]. 

The species diffusivity, varies in different subregions of a PEFC depending on the specific physical phase of component k. In flow channels and porous electrodes, species k exists in the gaseous phase and thus the diffusion coefficient corresponds with that in gas, whereas species k is dissolved in the membrane phase within the catalyst layers and the membrane and thus assumes the value corresponding to dissolved species, usually a few orders of magnitude lower than that in gas. The diffusive transport in gas can be described by molecular diffusion and Knudsen diffusion. The latter mechanism occurs when the pore size becomes comparable to the mean free path of gas, so that molecule-to-wall collision takes place instead of molecule-to-molecule collision in ordinary diffusion. The Knudsen diffusion coefficient can be computed according to the kinetic theory of gases as follows... 

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. 

In chemical heterogeneous catalysis, it is common to use highly porous catalysts that come in particles of millimeter to centimeter size to increase the effective catalyst surface. In practical electrocatalysis, in particular applying electrocatalysis in fuel cells, it is also usual to use highly porous— although accounting for the low diffusion coefficients in liquid electrolytes compared to gases, 10 5 cm2/sec vs 1 cm2/sec, much smaller—catalyst particles. 

One should take into account the specific features of gas diffusion in porous solids when measuring effective diffusion coefficients in the pores of catalysts. The measurements are usually carried out with a flat membrane of the porous material. The membrane is washed on one side by one gas and on the other side by another gas, the pressure on both sides being kept... 

An important problem in catalysis is to predict diffusion and reaction rates in porous catalysts when the reaction rate can depend on concentration in a non-linear way.6 The heterogeneous system is modeled as a solid material with pores through which the reactants and products diffuse. We assume for diffusion that all the microscopic details of the porous medium are lumped together into the effective diffusion coefficient De for reactant. 

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

Hore importantly, the response curves are noticeably affected where one or both of the components is adsorbable, even at low tracer concentrations. The interpretation of data is then much more complex and requires analysis using the non-isobaric model. Figs 7 and 8 show how adsorption of influences the fluxes observed for He (the tracer), despite the fact that it is the non-adsorbable component. The role played by the induced pressure gradient, in association with the concentration profiles, can be clearly seen. It is notable that the greatest sensitivity is exhibited for small values of the adsorption coefficient, which is often the case with many common porous solids used as catalyst supports. This suggests that routine determination of effective diffusion coefficients will require considerable checks for consistency and emphasizes the need for using the Wicke-Kallenbach cell in conjunction with permeability measurements. 

Table 4 contains a collection of diffusion coefficients determined experimentally for a variety of adsorbate systems. It shows that the values may vary considerably, which is of course due to the specific bonding of the adsorbate to the surface under consideration. Surface diffusion plays a vital role in surface chemical reactions because it is one factor that determines the rates of the reactions. Those reactions with diffusion as the rate-determining step are called diffusion-limited reactions. The above-mentioned photoelectron emission microscope is an interesting tool to effectively study diffusion processes under reaction conditions [158], In the world of real catalysts, diffusion may be vital because the porous structure of the catalyst particle may impose stringent conditions on molecular diffusivities, which in turn leads to massive consequences for reaction yields. 

The effective-scale models have been most often used in the description of transport and reaction processes within the porous structure of catalysts. Such models are based on the introduction of an effective diffusion coefficient De, that is used in the analogy to the Fick s law for the description of diffusion... 

Equation (74) were solved within the section of the reconstructed porous catalyst, represented by the 3D matrix. Here x, y, and z are the spatial coordinates in the porous catalyst, cy denotes the molar concentration of the kth component, Dff is the effective diffusivity of the /cth component, vkj is the stoichiometric coefficient of the component k in the /th reaction step, and r, is the reaction rate of the y th reaction step. 

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. 

Because of the large pores of the catalysts and the dilution with steam in a wa-ter butene ratio of 12, constant molecular diffusion coefficients of 9.69 xlO 5 mV1 (butene/water) and 9.95 xlO"5 m2 s1 (butadiene/water) at a temperature of 933 K can be assumed The porous structure of the catalyst is represented by a sjyp value of 0.1. The concentration profiles are given by Equations 8.22 and 8.23 for plate geometry. The results for R = 10 mm and R = 2 mm are shown in Figure 8.11 with = 0. 

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. 

There are theoretically sound correlations available for estimating effective diffusion coefficients in porous catalyst pellets (cf., 22). It has been shown that for most gas-solid... 

It has been demonstrated that the combined application of various NMR techniques for observing molecular rotations and migrations on different time scales can contribute to a deeper understanding of the elementary steps of molecular diffusion in zeolite catalysts. The NMR results (self-diffusion coefficients, anisotropic diffiisivities, jump lengths, and residence times) can be correlated with corresponding neutron scattering data and sorption kinetics as well as molecular dynamics calculations, thus giving a comprehensive picture of molecular motions in porous solids. 

It is further assumed that B is not present in the feed oil, i.c. Cno =0 and that the reactions are taking place in spherical particles of porous material and are controlled by effective diffusion coefficients Da = Da = D. (The actual catalyst particles used in the experiments were short cylinders). 

Many catalysts are pellets or extrusions formed from porous particles. Such catalysts usually give a bimodal or wide pore size distribution, and measurements of effective diffusion coefficients based on mean pore radius can be very misleading. Smith etal, have developed and tested experimentally some models which... 

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