Knudsen Diffusion Gases

Knudsen Diffusion (Gases) Suppose that a gas is diffusing in a straight, round pore with a radius r that is much larger than molecular dimensions. The molecules in the pore will be in random thermal motion and will collide with other gas molecules and with the walls of the pore. The mean free path is defined as the average distance that a molecule travels before it collides with another molecule. The mean free path can be predicted approximately from kinetic theory, for a pure gas. 

In Eqn. (9-17), Am is the mean free path, d is the equivalent diameter of the molecule, Aa is Avogadro s number, and Ca is the molar concentration of the pure gas. The product CA a is the number density (molecules/volume) of the gas. For most gases at atmospheric conditions, Am is between 10 and 100 nm (100 A A 1000 A). 

If Am collisions of molecules with the walls of the pore will be much more frequent than collisions with other molecules. Diffusion will take place through molecule-wall interactions rather than through molecule-molecule interactions, as depicted below. 

This type of diffusion is known as Knudsen diffusion. In the Knudsen diffusion regime, the equation for the flux is 

In this equation, Na is the flux of A in the z-direction, i.e., along the length of the pore. In the Knudsen regime, the flux of any component is purely diffusive, exactly as assumed in Eqn. (9-3). 

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The Knudsen diffusion gas separation layer can be modified by e.g. sol-gel, cvd, or crystallisation techniques to enhance the selectivity, but this decreases the permeation. Silica is the material mainly used for modification. However, data on reproducibility and stability are still scarce. The large scale use of high selective inorganic membranes and these membranes at high temperatures, up to at least 600°C, will probably last another 5-10 years. On a laboratory scale (maximum membrane surface area of about 50 cm ) these high selective membranes are now available, although stability can be a problem in certain atmospheres. 

Mesoporous 2-50 Knudsen diffusion Gas separation, ultrafiltration, nanofiltration... 

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 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... 

For gas-phase diffusion in small pores at lowpressure, the molecular mean free path may be larger than the pore diameter, giving rise to Knudsen diffusion. Satterfield (Ma.s.s Tran.sfer in Heterogeneous Catalysis, MIT, Cambridge, MA, 1970, p. 43), gives the following expression for the pore dimisivity ... 

In the discussion so far, the fluid has been considered to be a continuum, and distances on the molecular scale have, in effect, been regarded as small compared with the dimensions of the containing vessel, and thus only a small proportion of the molecules collides directly with the walls. As the pressure of a gas is reduced, however, the mean free path may increase to such an extent that it becomes comparable with the dimensions of the vessel, and a significant proportion of the molecules may then collide direcdy with the walls rather than with other molecules. Similarly, if the linear dimensions of the system are reduced, as for instance when diffusion is occurring in the small pores of a catalyst particle (Section 10.7), the effects of collision with the walls of the pores may be important even at moderate pressures. Where the main resistance to diffusion arises from collisions of molecules with the walls, the process is referred to Knudsen diffusion, with a Knudsen diffusivily which is proportional to the product where I is a linear dimension of the containing vessel. 

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. 

In bulk diffusion, the predominant interaction of molecules is with other molecules in the fluid phase. This is the ordinary kind of diffusion, and the corresponding diffusivity is denoted as a- At low gas densities in small-diameter pores, the mean free path of molecules may become comparable to the pore diameter. Then, the predominant interaction is with the walls of the pore, and diffusion within a pore is governed by the Knudsen diffusivity, K-This diffusivity is predicted by the kinetic theory of gases to be... 

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. 

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. 

It can be concluded that the predominating mode of gas transport in the investigated nano-porous hydrophobic material is Knudsen diffusion, so that the diffusion is the main mechanism of gas transport in electrochemical systems based on such material and operating with gaseous reactants. 

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. 

With anodic oxidation very controlled and narrow pore size distributions can be obtained. These membranes mounted in a small module may be suitable for ultrafiltration, gas separation with Knudsen diffusion and in biological applications. At present one of the main disadvantages is that the layer has to be supported by a separate layer to produce the complete membrane/support structure. Thus, presently applications are limited to laboratory-scale separations since large surface area modules of such membranes are unavailable. 

There are few studies in literature reporting pure gas permeabilities as well as separation factors of mixtures. Vuren et al. (1987) reported Knudsen diffusion behavior of pure gases for y-alumina membranes with a mean pore radius of 1.2 nm. Separation experiments with a 1 1 H2/N2 mixture showed, that the theoretical Knudsen separation factor [of 3.7, Equation 6.4)j for this mixture could be obtained (Keizer et al. 1988 see also Figure 6.2). In Figure 6.2, the effect of process parameters is also demonstrated. The separation factor is a function of the pressure ratio over the membrane, which is the ratio of the pressure on the permeate-side to that on the feed-side. For pressure ratios approaching unity, which means the pressure on both sides of the... 

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). 

Beside the partial pressure differences between the various gas components, the total pressure on both sides of the membrane is also important. Though mean total pressure does not directly affect the permeation rate in the case of a Knudsen diffusion mechanism, it governs the gas flow through... 

From an order-of-magnitude analysis, when the mean-free path of a molecule is less than 0.01 times the pore radius, bulk diffusion dominates, and when it is greater than 10 times the pore radius, Knudsen diffusion dominates. This means that Knudsen diffusion is significant when the pore radius is less than about 0.5 fim. For reference, a typical carbon gas-diffusion layer has pores between 0.5 and 20 /rm22 229 in radius, and a microporous layer contains pores between 0.05 and 2 Thus, while Knudsen... 

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... 

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