Diffusion coefficient Knudsen

Note the use of a script for the binary pair mutual diffusion coefficient, as distinct from the Roman D already used to represent Knudsen diffusion coefficients. This convention will be adhered to throughout. 

Let us now turn attention to situations in which the flux equations can be replaced by simpler limiting forms. Consider first the limiting case of dilute solutions where one species, present in considerable excess, is regarded as a solvent and the remaining species as solutes. This is the simplest Limiting case, since it does not involve any examination of the relative behavior of the permeability and the bulk and Knudsen diffusion coefficients. 

The limiting cases of greatest interest correspond to conditions in which the mean free path lengths are large and small, respectively, compared with the pore diameters. Recall from the discussion in Chapter 3 that the effective Knudsen diffusion coefficients are proportional to pore diameter and independent of pressure, while the effective bulk diffusion coefficients are independent of pore diameter and inversely proportional to pressure. 

The Knudsen diffusion coefficients are given by equations (2.11) in which K. is independent of pressure and proportional to pore diameter a, so that we can write... 

It may seem curious that Knudsen diffusion coefficients still appear in equations (5.18) and (5.19), which supposedly give the flux relations at the limit of bulk diffusion control. However, inspection reveals that only ratios of these coefficients are effectively present, and from equation (2,11) it follows that... 

The first case corresponds to a situation in which all Knudsen diffusion coefficients are equal, and all binary pair bulk diffusion coefficients are equal ... 

It ls not surprising chat such a relation should hold at the Limit of Knudsen diffusion, since Che Knudsen diffusion coefficients are themselves inversely proportional to the square roots of molecular weights, but the pore diameters in Graham s stucco plugs were certainly many times larger chan the gaseous mean free path lengths at the experimental conditions. 

Che Knudsen diffusion coefficient is independent of composition and pressure,... 

Knudsen diffusion coefficient for the test gas in a micropore. represents the total void fraction and c that part of of the void fraction... 

In Chase equaclons Che symbols Z u, A, b and id are noc defined quite as In equations (12.39), since Che bulk diffusion coefficient muse now be replaced by a Knudsen diffusion coefficient. Thus... 

The Knudsen diffusion coefficient may be evaluated from equation 12.2.4 if the catalyst property values are used to estimate the average pore radius. From equation C of Illustration 6.2,... 

Substitution of the numerical values for the bulk and Knudsen diffusion coefficients into equation 12.2.8 gives... 

It may be assumed that the accumulation of hydrogen within the pellet is negligible and that it may be treated as being in a quasi-steady-state condition. The finite difference form of Fick s first law may be used to determine the flow rate of hydrogen through the pellet. The diffusion constant appearing in this equation may be considered as an effective Knudsen diffusion coefficient. 

In Figure 3, we have presented the experimentally obtained reciprocal values of (Di )t.ff of oxygen in a sample of the nano-porous hydrophobic material as a function of the total pressure P of gas mixture (02-N2) when the oxygen concentration in the mixture is 21%. From the intercept of the straight line with the ordinate the value of the Knudsen diffusion coefficient can be also determined. It must be underlined that the value of Knudsen diffusion coefficient obtained by these diffusion measurements (2,86.10"2 cm2/s) is in very good coincidence with the value obtained by the gas permeability measurements. 

Reid, Sherwood and Prausnitz [11] provide a wide variety of models for calculation of molecular diffusion. Dr is the Knudsen diffusion coefficient. It has been given in several articles as 9700r(T/MW). Once we have both diffusion coefficients we can obtain an expression for the macro-pore diffusion coefficient 1/D = 1/Dk -i-1/Dm- We next obtain the pore diffusivity by inclusion of the tortuosity Dp = D/t, and finally the local molar flux J in the macro-pores is described by the famiUar relationship J = —e D dcjdz. Thus flux in the macro-pores of the adsorbent product is related to the term CpD/r. This last quantity may be thought of as the effective macro-pore diffusivity. The resistance to mass transfer that develops due to macropore diffusion has a length dependence of R]. 

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... 

In the PEFC system, the mean pore radii of catalyst layers are of the order of 0.1 pm. The Knudsen diffusion coefficients at 80 °C for O2 and H2O through the catalyst layer are thus estimated to be 0.32 and 0.43 cm /s, respectively. These values are comparable to the respective ordinary diffusion coefficients, indicating that Knudsen diffusion further restricts the rates of oxygen and water transport through the cathode catalyst layer in PEFCs and should be taken into account. 

The parameters D and Dk > whether for macro (denoted by subscript m) or for micro (denoted by subscript ju) regions, are normal bulk and Knudsen diffusion coefficients, respectively, and can be estimated from kinetic theory, provided the mean radii of the diffusion channels are known. Mean radii, of course, are obtainable from pore volume and surface area measurements, as pointed out in Sect. 3.1. For a bidisperse system, two peaks (corresponding to macro and micro) would be expected in a differential pore size distribution curve and this therefore provides the necessary information. Macro and micro voidages can also be determined experimentally. 

The Knudsen diffusion coefficient Dk corresponds to the movement of gaseous solutes in small pores and can be estimated by using the following equation (Perry and Green, 1999) ... 

The region of flow where collisions of molecules with the container walls are more frequent than intermolecular gaseous collisions was the subject of detailed study by Knudsen(8) early in the twentieth century. From geometrical considerations it may be shown(9) that, for the case of a capillary of circular cross-section and radius r, the proportionality factor is Snr3/3. This results in a Knudsen diffusion coefficient ... 

Effective diffusivity in Knudsen regime Effective diffusivity in molecular regime Knudsen diffusion coefficient Diffusion coefficient for forced flow Effective diffusivity based on concentration expressed as Y Dispersion coefficient in longitudinal direction based on concentration expressed as Y Radial dispersion coefficient based on concentration expressed as Y Tube diameter Particle diameter... 

Axial, film, and macropore Maxwell and Knudsen diffusion coefficients are estimated based on relatively standard formulations ( 6, 7, 8, 9), using estimates of physical properties shown in Table I to compute the diffusivity values for the systems studied. The effect of errors in the estimated values of these properties will be discussed later. 

The first hypothesis seems unlikely to be true in view of the rather wide variation in the ratio of carbon dioxide s kinetic diameter to the diameter of the intracrystalline pores (about 0.87, 0.77 and 0.39 for 4A, 5A and 13X, respectively (1J2)). The alternative hypothesis, however, (additional dif-fusional modes through the macropore spaces) could be interpreted in terms of transport along the crystal surfaces comprising the "walls" of the macropore spaces. This surface diffusion would act in an additive manner to the effective Maxwell-Knudsen diffusion coefficient, thus reducing the overall resistance to mass transfer within the macropores. 

Here, the length L in (7.38) has been replaced by porous layer thickness d and the surface area Aeff. The effective diffusion coefficient D0,eff characterizes the transport through porous medium and includes both regular diffusion and the Knudsen diffusion coefficient >o,Kn, which has a different temperature dependence from diffusion in bulk. 

The Knudsen diffusion coefficient may be computed according to the kinetic theory of gases ... 

To use this formula, the assumption has been made that the fuel consists of a binary mixture of hydrogen and water, while the cathodic gas is a binary mixture of oxygen and nitrogen. The diffusion coefficient for binary mixtures D y eff is estimated by the equation proposed by Hirschfelder, Bird and Spotz [12], and the Knudsen diffusion coefficient for species i is given by free molecule flow theory [11], Finally, combining Equations (6.15-6.18) the anodic and the cathodic concentration overvoltages are given by (see also Equations (A3.20) and (A3.21)) ... 

In the porous medium, diffusion is affected by the porosity and tortuosity of the medium itself therefore Knudsen diffusion is computed as well as the ordinary diffusion. Eventually, an effective diffusion coefficient is calculated that depends on the ordinary and Knudsen diffusion coefficients and on the ratio between porosity and tortuosity of the medium (Equation (3.58)). 

In Equation (9.6), x is the direction of flux, nt [mol m-3 s 1 ] is the total molar density, X [1] is the mole fraction, Nd [mol m-2 s 1] is the mole flux due to molecular diffusion, D k [m2 s 1] is the effective Knudsen diffusion coefficient, D [m2 s 1] is the effective bimolecular diffusion coefficient (D = Aye/r), e is the porosity of the electrode, r is the tortuosity of the electrode, and J is the total number of gas species. Here, a subscript denotes the index value to a specific specie. The first term on the right of Equation (9.6) accounts for Knudsen diffusion, and the following term accounts for multicomponent bulk molecular diffusion. Further, to account for the porous media, along with induced convection, the Dusty Gas Model is required (Mason and Malinauskas, 1983 Warren, 1969). This model modifies Equation (9.6) as ... 

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