Knudsen diffusion flux

To describe the combined bulk and Knudsen diffusion fluxes the dusty gas model can be used. It is postulated that on a mass basis the dusty gas model can be written in analogy to (2.330) in the following way ... 

In order to calculate bulk and Knudsen diffusion fluxes simultaneously, the dusty gas model is often applied [35-37], where giant dust molecules distributed uniformly in space represent the pore walls. 

Like the molecular diffusion flux the Knudsen diffusion flux (3.4-3) is referred to the total particle surface area, so that ... 

The steady state Knudsen diffusion flux of a gas species i of molecular weight M, through straight circular pores of radius Vp and length S under the condition of total pressure being the same at both ends of the pore is given by... 

The flux expressions for gas transport through porous membranes have been considered in Section 3.I.3.2.4. The steady state Knudsen diffusion flux expression (3.1.115a),... 

S.6 Combined Bulk and Knudsen Diffusion Fluxes The Wilke-Bosanquet and Dusty Gas Models... 

To describe the combined bulk and Knudsen diffusion fluxes in the transition gas transport regime, the simple Wilke-Bosanquet model [11, 70, 151] and the more rigorous dusty gas model [62, 70, 71, 94] can be used either on the mass or molar forms. Knudsen diffusion refers to a gas transport regime where the mean free... 

As a first approach to calculate the combined bulk and Knudsen diffusive flux using a Fickian flux formulation, the Bosanquet formula [11] has been suggested in literature [11,70,151]. In this approach, the Fick form formulation uses a diffusivity computed as the reciprocal additivity of the diffusion coefficients in the ordinary and Knudsen diffusion regimes. 

For multicomponent systems the diffusive flux terms may be written in accordance with the approximate Wilke bulk flux equation (2.450), the approximate Wilke-Bosanquet combined bulk and Knudsen flux for porous media (2.454), the rigorous Maxwell-Stefan bulk flux equations (2.421), and the consistent dusty gas combined bulk and Knudsen diffusion flux for porous media (2.504). The different mass based diffusion flux models are listed in Table 2.3. The corresponding molar based diffusion flux models are listed in Table 2.4. In most simulations, the catalyst pellet is approximated by a porous sphericai pellet with center point symmetry. For such spherical pellets a representative system of pellet model equations, constitutive laws and boundary conditions are listed in Tables 2.5,2.6 and 2.7, respectively. 

High-flux, limited-selectivity gas separations rely on (small) differences in Knudsen diffusion fluxes. Mesoporous 7-alumina membranes have also been reported to exhibit a substantial CO2 flux by surface diffusion on the internal pore surface (Uhlhom et al., 1992). This mechanism might be of use for the separation of CO2 and other polar molecules. 

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. 

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... 

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. 

For an Isothermal pellet with pores sufficiently small that Knudsen diffusion controls, the flux relations are required to take the form (8.1), which can be written... 

In simple cases it is not difficult to estimate the magnitude of the pressure variation within the pellet. Let us restrict attention to a reaction of the form A nB in a pellet of one of the three simple geometries, with uniform external conditions so that the flux relations (11.3) hold. Consider first the case in which all the pores are small and Knudsen diffusion controls, so that the fluxes are given by... 

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... 

In chapter 5 we showed that the dusty gas model flux relations could be solved (fjite easily at the limit of Knudsen diffusion control, when they reduce to the form given in equation (5.25), namely... 

A molecule colliding with the pore wall is reflected in a specular manner so that the direction of the molecule leaving the surface has no correlation with that of the incident molecule. This leads to a Fickian mechanism, known as Knudsen diffusion, in which the flux is proportional to the gradient of concentration of partial pressure. The Knudsen diffusivity is independent of pressure and varies only weaMy with temperature ... 

Mitrovic and Knezic (1979) also prepared ultrafiltration and reverse osmosis membranes by this technique. Their membranes were etched in 5% oxalic acid. The membranes had pores of the order of 100 nm, but only about 1.5 nm in the residual barrier layer (layer AB in Figure 2.15). The pores in the barrier layer were unstable in water and the permeability decreased during the experiments. Complete dehydration of alumina or phase transformation to a-alumina was necessary to stabilize the pore structure. The resulting membranes were found unsuitable for reverse osmosis but suitable for ultrafiltration after removing the barrier layer. Beside reverse osmosis and ultrafiltration measurements, some gas permeability data have also been reported on this type of membranes (Itaya et al. 1984). The water flux through a 50/im thick membrane is about 0.2mL/cm -h with a N2 flow about 6cmVcm -min-bar. The gas transport through the membrane was due to Knudsen diffusion mechanism, which is inversely proportional to the square root of molecular mass. 

Summarizing it can be stated that the separation by gas phase transport (Knudsen diffusion) has a limited selectivity, depending on the molecular masses of the gases. The theoretical separation factor is decreased by effects like concentration-polarization and backdiffusion. However, fluxes through the membrane are high and this separation mechanism can be applied in harsh chemical and thermal environments with currently available membranes (Uhlhorn 1990, Bhave, Gillot and Liu 1989). 

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]. 

Both Knudsen and molecular diffusion can be described adequately for homogeneous media. However, a porous mass of solid usually contains pores of non-uniform cross-section which pursue a very tortuous path through the particle and which may intersect with many other pores. Thus the flux predicted by an equation for normal bulk diffusion (or for Knudsen diffusion) should be multiplied by a geometric factor which takes into account the tortuosity and the fact that the flow will be impeded by that fraction of the total pellet volume which is solid. It is therefore expedient to define an effective diffusivity De in such a way that the flux of material may be thought of as flowing through an equivalent homogeneous medium. We may then write ... 

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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