Knudsen diffusion, in porous

For Knudsen diffusion in porous solids of porosity e and tortuosity t,... 

Knudsen diffusion in porous catalysts with a fractal internal surface. Fractals, Dutch Antilles, Curacao. 3 (4), 807-820. 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

Feng, Kostrov and Stewart (1974) reported multicomponent diffusion data for gaseous mixtures of helium (He), nitrogen (N2) and methane (CH4) through an extruded platinum-alumina catalyst as functions of pressure (1 to 70 atm), temperature (300 to 390 K), and terminal compositions. The experiments were designed to test several models of diffusion in porous media over the range between Knudsen and continuum diffusion in a commercial catalyst (Sinclair-Engelhard RD-150) with a wide pore-size distribution. 

Although Eq. (11-2) is the proper one to use for regions where both Knudsen and bulk diffusion are important, it has a serious disadvantage the combined diffusivity i) is a function of gas composition in the pore. This dependency on composition carries over to the effective diffusivity in porous catalysts (see Sec. 11-2) and makes it difficult later to integrate the combined diffusion and transport equations. The variation of D with y is not usually strong (see Example 11-3). Therefore it has been almost universal, in assessing the importance of intrapellet resistances, to use a composition-independent form for Z), for example, Eq. (11-4). In fact, the concept of a single effective diffusivity loses its value if the composition dependency must be retained. 

Effective diffusivities in porous catalysts are usually measured under conditions where the pressure is maintained constant by external means. The experimental method is discussed in Sec. 11-2 it is mentioned here because under this condition, and for a binary counterdiffusing system, the ratio is the same regardless of the extent of Knudsen and bulk... 

Example 1.22 Combined Molecular and Knudsen Diffusion in a Porous Solid... 

Modeling multicomponent gaseous diffusion in porous media depends upon the value of the Knudsen number. For very large pores, corresponding to very small values of Kn, the Maxwell-Stefan equations (1-34) can be used with effective binary dif-fusivities given by... 

Dependence of the diffusion coefficient on the degree of bead saturation is also known for porous adsorbents with a rigid structure, such as activated carbons [14, 15]. In this case, two processes - Knudsen diffusion (diffusion in porous materials) and migration of sorbate molecules along the pore... 

The steady state Wicke-Kallabach method is usually conducted with a binary system with an aim of determining the binary diffusivity and Knudsen diffusivity in the porous medium. In this binary system, one gas (A) is flowing into and out of one chamber, and the other gas (B) is flowing into and out of the other chamber. 

In actual diffusion in porous solids the pores are not straight and cylindrical but are irregular. Hence, the equations for diffusion in pores must be modified somewhat for actual porous solids. The problem is further complicated by the fact that the pore diameters vary and the Knudsen diffusivity is a function of pore diameter. 

Gas diffusion in porous solid. In this type a gas phase is present on both sides of the membrane, which is a microporous solid. The rates of molecular diffusion of the various gas molecules depend on the pore sizes and the molecular weights. This type of diffusion in the molecular, transition, and Knudsen regions was discussed in detail in Section 7.6. 

Instead of the molecular diffusion coefficient in a single pore, we have to use an effecttive diffusion coefficient DA.efrthat considers the porosity of the porous particle and the tortuous nature of pores and pore constrictions. These two aspects (details in Section 3.2.2.3) lead to a value of DA.eff that is a factor of about 10 smaller than the molecular diffusivity. For Knudsen diffusion in narrow pores, this deviation is even higher. 

The reactant gas species transport to reaction sites through the porous electrodes based on the concept of gas diffusion in porous media. In porous media, the diffusion mechanism can be of three different types ordinary diffusion, Knudsen diffusion, and surface diffusion. If the pores are much larger than the mean free path length, then the molecules collide with each other more frequently than with the pore walls, and ordinary diffusion is assumed to be the dominant diffusion mechanism. Knudsen diffusion is encoimtered in smaller pores or at lower pressure or density. In this case, molecules collide more frequently with the walls than with other gas molecules. The Knudsen diffusion coefficient given is based on kinetic theory as... 

For Knudsen diffusion in a porous medium, an average pore radius may be used for r (in cm) in Eq. 14.7. It is instructive in this regard to compare the relative magnitude of molecular and Knudsen diffusivities with the help of an approximate expression for the former based on the simple kinetic theory, given by ... 

In an operating SOFCs, gases (air or fuel) diffuse through the porous electrodes to reach electrode/electrolyte interfaces. The gas diffusion rate must be fast enough to minimize the diffusion loss. Three primary processes are involved with gas diffusion in porous electrodes ordinary diffusion, Knudsen diffusion, and surface diffusion. The surface diffusion is negligible at elevated temperatures (e.g., 600-800°C). The question remains as to whether ordinary diffusion or Knudsen diffusion is the limiting step of the diffusion process. The answer resides in the pore... 

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

The above estimates of pressure variations suggest that their magni-tude as a percentage of the absolute pressure may not be very large except near the limit of Knudsen diffusion. But in porous catalysts, as we have seen, the diffusion processes to be modeled often lie in the Intermediate range between Knudsen streaming and bulk diffusion control. It is therefore tempting to try to simplify the flux equations in such a way as to... 

For the same reaction in a pellet of finely porous structure, where Knudsen diffusion controls, the appropriate dynamical equations sre (12.20) and (12.21) if we once more adopt approximations which are a consequence of Che large size of K. These again have a dimensionless form, which may be written... 

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. 

The ratio of the overall rate of reaction to that which would be achieved in the absence of a mass transfer resistance is referred to as the effectiveness factor rj. SCOTT and Dullion(29) describe an apparatus incorporating a diffusion cell in which the effective diffusivity De of a gas in a porous medium may be measured. This approach allows for the combined effects of molecular and Knudsen diffusion, and takes into account the effect of the complex structure of the porous solid, and the influence of tortuosity which affects the path length to be traversed by the molecules. 

Classical commercial ceramic porous materials, as those obtained via sol-gel processes, generally have adequate permeabilities but could present some drawbacks They indeed have a limited thermal stability and are generally not permselective enough their pores are in the mesoporous range and maximum separation factors correspond to Knudsen diffusion mechanisms. 

Thus a zero-order reaction appears to be 1/2 order and a second-order reaction appears to be 3/2 order when dealing with a fast reaction taking place in porous catalyst pellets. First-order reactions do not appear to undergo a shift in reaction order in going from high to low effectiveness factors. These statements presume that the combined diffusivity lies in the Knudsen range, so that this parameter is pressure independent. 

Villet and Wilhelm Ind. Eng. Chem., 53 (837), 1961] have studied the Knudsen diffusion of hydrogen in porous silica-alumina cracking catalyst pellets. They used apparatus of the type depicted in Figure 12P.1. 

In Figure 2 we presented the permeability coefficient K of oxygen as a function of the mean gas pressure experimentally obtained for a sample of porous material from acetylene black modified with 35% PTFE. The experimental linear dependence is obtained. The intercept with the abscissa corresponds to the Knudsen term DiK. The value obtained is 2,89.1 O 2 cm2/s. The slope of the straight line is small, so that the ratio K,/ Dik at mean gas pressure 1 atm. is small ( 0.1) which means that the gas flow is predominantly achieved by Knudsen diffusion and the viscous flow is quite negligible. At normal conditions (1 atm, 25°C) the mean free path of the air molecules (X a 100 nm) is greater than the mean pore radii in the hydrophobic material (r 20 nm), so that the condition (X r) for the Knudsen-diffusion mechanism of gas transport is fulfilled. 

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