Permeation Knudsen diffusion

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... 

Du 1986). This reflects the importance of smaU pores in order to apply effectively capillary condensation as a separation mechanism. Uhlhom (1990) demonstrated the effect of multilayer diffusion of propylene through a modified y-alumina membrane at 0°C. The separation factor for the N2/CjHg mixture was 27, where propylene is the preferentially permeating component, while the permeability increased to 7 times the Knudsen diffusion permeability. Although this mechanism appears to be very effective because of a high separation factor and a high permeability, it is limited by the obvious need for a condensable component. This in turn restricts the applicability range, due to limits set by temperature and pressure, needed for formation of multilayers or capillary condensation. 

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... 

In order to predict correctly the fluxes of multicomponent mixtures in porous membranes, simplified models based solely on Fields law should be used with care [28]. Often, combinations of several mechanisms control the fluxes, and more sophisticated models are required. A well-known example is the Dusty Gas Model which takes into account contributions of molecular diffusion, Knudsen diffusion, and permeation [29]. This model describes the coupled fluxes of N gaseous components, Ji, as a function of the pressure and total pressure gradients with the following equation ... 

The constitutive equations of transport in porous media comprise both physical properties of components and pairs of components and simplifying assumptions about the geometrical characteristics of the porous medium. Two advanced effective-scale (i.e., space-averaged) models are commonly applied for description of combined bulk diffusion, Knudsen diffusion and permeation transport of multicomponent gas mixtures—Mean Transport-Pore Model (MTPM)—and Dusty Gas Model (DGM) cf. Mason and Malinauskas (1983), Schneider and Gelbin (1984), and Krishna and Wesseling (1997). The molar flux intensity of the z th component A) is the sum of the diffusion Nc- and permeation N contributions,... 

For the tortuous and irregular capillaries of porous media, it has been reported theoretically and experimentally that a minimum in the permeability of adsorbates at low pressures is not expected to appear. In our study of n-hexane in activated carbon, however, a minimum was consistently observed for n-hexane at a relative pressure of about 0.03, while benzene and CCI4 show a monotonically increasing behavior of the permeability versus pressure. Such an observation suggests that the existence of the minimum depends on the properties of permeating vapors as well as the porous medium. In this paper a permeation model is presented to describe the minimum with an introduction of a collision-reflection factor. Surface diffusion permeability is found to increase sharply at very low pressure, then decrease modestly with an increase in pressure. As a result, the appearance of a minimum in permeability was found to be controlled by the interplay between Knudsen diffusion and surface diffusion for each adsorbate at low pressures. 

Wu et al. (1993] have developed a mathematical model based on Knudsen diffusion and intermolecular momentum transfer. Their model applies the permeability values of single components (i.e., pure gases) to determine two parameters related to the morphology of the microporous membranes and the reflection behavior of the gas molecules. The parameters are then used in the model to predict the separation performance. The model predicts that the permeability of carbon monoxide deviates substantially from that based on Knudsen diffusion alone. Their model calculations are able to explain the low gas separation efficiency. Under the transport regimes considered in their study, the feed side pressure and pressure ratio (permeate to feed pressures) are found to exert stronger influences on the separation factor than other factors. A low feed side pressure and a tow pressure ratio provide a maximum separation efficiency. 

Finally, it is important to notice the effect of the support in the pervaporation flux, analyzed by Bruijn et al. [117] who proposed a model and evaluated the contribution of the support layer to the overall resistance for mass transfer in the selected literature data. They found that in many cases, the support is limiting the flux the permeation mechanism through the support corresponds to a Knudsen diffusion mechanism, which makes improvements in the porosity, tortuosity, pore diameter, and thickness necessary for an increase in the pervaporation flux. 

The gas is applied as a mixture to the retentate (high pressure) side of the membrane, the components of the mixture diffuse with different rates through the membrane under the action of a total pressure gradient and are removed at the permeate side by a sweep gas or by vacuum suction. Because the only segregative mechanisms in mesopores are Knudsen diffusion and surface diffusion/capillary condensation (see Table 9.1), viscous flow and continuum (bulk gas) diffusion should be absent in the separation layer. Only the transition state between Knudsen diffusion and continuum diffusion is allowed to some extent, but is not preferred because the selectivity is decreased. Nevertheless, continuum diffusion and viscous flow usually occur in the macroscopic pores of the support of the separation layer in asymmetric systems (see Fig. 9.2) and this can affect the separation factor. Furthermore the experimental set-up as shown in Fig. 9.11 can be used vmder isobaric conditions (only partial pressure differences are present) for the measurement of diffusivities in gas mixtures in so-called Wicke-Callenbach types of measurement. 

According to Eq. (9.9a) the viscous flow increases with and with P, while the Knudsen diffusion increases with r and is independent of pressure. This means that the contribution of the viscous flow to the total permeation increases with r and p. 

Mathematically HkH accoimts for the porosity (e) and the tortuosity (x) for gas permeation dominated by Knudsen diffusion (see Eq. (9.16)). P[-] is used to correct for behaviour deviating from the ideal Knudsen behaviour, e.g., due to reflection conditions deviating from elastic specular collisions with the pore wall. 

It should be noted that surface diffusion of CO is possible, but is probably negligible because the permeation is decreased (with respect to expectations based on Knudsen diffusion) instead of increased (if surface diffusion is important). 

For He, H2, O2 and N2 a linear dependence of the transmembrane flux on the pressure gradient across the membrane was observed. So the permeation is constant and independent of pressure as expected for Knudsen diffusion and sorbed gases in the Henry regime (and accordingly to the sum of both mechanisms). 

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. 

In the feasibility studies relatively simple models have been used because these concern mostly a rough estimate of the possibilities. For high selective and microporous types of membranes, permeation through the membrane is assumed to occur only via diffusion which obeys Fick s law. In the case of the Knudsen diffusion membranes the contribution of the non-separating viscous flow through the membrane is also accounted for. 

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