Knudsen contribution

From the slope the value of e/t can be calculated, while the intersection with the permeation axis (P = 0) yields the value of e/fT-G) and so 0 can be calculated as well as the Knudsen contribution to the total flow. The pore size should be well defined in these cases and so the pore size distribution should be reasonably sharp. When the total porosity really is representative for all the active pores (thus, e.g., not many "dead ends" should be present), the value of the tortuosity x can then be calculated. Otherwise the parameter e/t is used as a fitting parameter. 

Permeation data of macroporous supports and mesoporous layers for N2 at 20°C and an average pressure p = 1 bar in the tramsition region of viscous to Knudsen flow. The fraction of the viscous flow (b) to the total flow is given by Fr, the remainder is the Knudsen contribution (a)... 

As is shown in Fig. 9.13 for a given pressure ratio, the higher the feed pressure, the lower the separation factor. At all pressures (1-34 atm) the separation factor decreases continuously with P (0.10-0.70). At P = 0.70 all separation factors converge to a value of 1.5. Note that even at the lowest pressure (1 atm) and lowest value of Pj = 0.10 the value of a = 3.20 which is considerably smaller than the ideal value (a = 3.70) as given by Eq. (9.37). So even a small amount of non-Knudsen contribution to the total flow in a pore considerably influences the separation. 

Finally should be stressed that firm conclusions on the magnitude of permeation and separation factors are only possible after appropriate control of the defect level of the membranes (non- micropore/Knudsen contributions to the... 

Even relatively small amoimts of non-Knudsen contributions in the diffusive transport (which hardly affects the permeation) can decrease the separation factor considerably (see Eqs. (9.38) and (9.34)). This implies that to obtain maximum separation factors the support resistance should be as small as possible and vacuum suction is preferred above use of a sweep gas to remove the permeate (from the permeate side). 

So mathematically, what went wrong when we neglect the Knudsen contributions The fact is that the small values of binary diffusivities do not mean that the molecular diffusion term will entirely dominate as the numerator in the first term of eq.(8.8-6) appeared in the form of a difference can also be a small number. 

Catalytic membranes can be prepared by deposition of a catalytically active phase (e.g., a metal) in the inert porous membranes through various techniques commonly used for conventional catalyst preparation - such as wet impregnation, monolayer metal complexation, and ion exchange -followed by heat treatment (activation) [17], The conventional activation steps can also be avoided by direct deposition of solvated metal microclusters [21], The structural characteristics of the porous substrate, and in particular the relative laminar and Knudsen contributions to permeation, have a strong influence on the membrane performance [22], They make use of the membrane structure to optimize the access of disfavored reactants, or to control and rule the residence time and contact of species in the active zone. Furthermore, the porous membrane is required to present a rather homogeneous structure to avoid heterogeneities in the reactant-to-catalyst contact, and also to facilitate CMR operation control. 

The contributions of the individual fluxes were based upon the volume fractions of the respective areas available for diffusion, as would be expected from the simple models given earlier. For low-density pellets only the macropore contribution is significant, and if the macropores are large enough, or the pressure high enough, the Knudsen part of the diffusion in the macropores can be neglected. Under these conditions their expression simplifies to... 

Obviously, there will be a range of pressures or molecular concentrations over which the transition from ordinary molecular diffusion to Knudsen diffusion takes place. Within this region both processes contribute to the mass transport, and it is appropriate to utilize a combined diffusivity (Q)c). For species A the correct form for the combined diffusivity is the following. 

The effusion rate Am/At at identical orifice diameter and temperatures was slowest for the cylindrical hole with long effusion channel and fastest for a cylindrical orifice with very short effusion channel. This means that the back scattering of the vapor molecules is strongly reduced if the ratio L/D between length L of the effusion channel and diameter D of the orifice is small, which is in agreement with theoretical calculations by Clausing66,67. There are three contributions to the Knudsen cell current which originate from the cell interior, from the channel wall, and the cell lid. 

Porous ceramic membrane layers are formed on top of macroporous supports, for enhanced mechanical resistance. The flow through the support may consist of contributions due to both Knudsen-diffusion and convective nonseparative flow. Supports with large pores are preferred due to their low resistance to the flow. Supports with high resistance to the flow decrease the effective pressure drop over the membrane separation layer, thus diminishing the separation efficiency of the membrane (van Vuren et al. 1987). For this reason in a membrane reactor it is more effective to place the reaction (catalytic) zone at the top layer side of the membrane while purging at the support side of the membrane. 

There are two major types of diffusion contributing to mass transport in the monolith washcoat (cf., e.g. Aris, 1975 Froment and Bischoff, 1979, 1990 Poling et al., 2001) volume (molecular) diffusion, Eq. (14), and Knudsen diffusion, Eq. (15), the latter one being dominant in small pores. 

Solid diffusion takes place in pore diameters of about 0.001 pm(10A) (Dg values of order of 10-Jcm-/sec). Since most diffusion types (Volmer, Knudsen, and solid) are orders of magnitude smaller than molecular diffusion, they contribute little to the... 

If compounds with very low odor thresholds and very small concentrations contribute to a material s odor their detection can be very challenging, especially when only applying routine emission measurements like GC—MS. Such compounds will easily be overlooked, for their detection GC—O can often be the only choice, but so far this method is seldom used in material analysis. Instead concentrations determined by emission measurements are compared with published odor thresholds to decide whether a compound might contribute to the odor or not. One problem is that published odor thresholds can differ quite a lot, even by several orders of magnitude (van Gemert, 2003). The value depends on the method and the panel but also on the purity of the compound used for threshold determination (if small impurities of a substance with a low odor threshold were present in a sample the odor threshold determined would have been too low ). Many factors influence odor threshold determination, therefore many published values are questionable and they are hard to rely on. Some authors (Knudsen et al., 1999 Wolkoff, 1999 Wolkoff et al., 2006) assume that many of the odor thresholds reported in the literature are actually much lower, because if they compare concentrations of compounds emitted and measured with odor thresholds published,... 

The relevance of this chemistry is demonstrated by the sensory impressions of linseed-oil based linoleum. Jensen, Wolkoff and Wilkins (1995) studied the autooxidation products emitted from linoleum and identified saturated and unsaturated aldehydes and fatty acids. An odor evaluation of the identified 2-alkenals and the fatty acids as contributing most to odor intensity (Jensen, Wolkoff and Wilkins, 1995), although many odorous products may not be quantified by traditional analytical methods (Knudsen et al., 2007) Knudsen et al. (2007) showed that linseed oil based products exhibited a more negative sensory perception than similar product not containing linseed oil, and that the negative perception persisted for at least a year. 

Macropore diffusion Diffusion in macropores —pores that are large compared with the molecular diameter. Several different mechanisms contribute to macropore diffusion, notably ordinary molecular diffusion in larger macropores at higher pressures or in liquids and Knudsen diffusion in smaller macropores at low pressures. Also referred to as intraparticle diffusion. 

Diffusion in macropores occurs mainly by the combined effects of bulk molecular diffusion (as in the free fluid) and Knudsen flow, with generally smaller contributions from other mechanisms such as surface diffusion and Poiseuille flow. Knudsen flow, which has the characteristics of a diffusive process, occurs because molecules striking the pore wall are instantaneously adsorbed and re-emitted in a random direction. The relative importance of bulk and Knudsen diffusion depends on the relative frequency of molecule-molecule and molecule-wall collisions, which in turn depends on the ratio of the mean free path to pore diameter. Thus Knudsen flow becomes dominant in small pores at low pressures, while in larger pores and at higher pressures diffusion occurs mainly by the molecular mechanism. Since the mechanism of diffusion may well be different at different pressures, one must be cautious about extrapolating from experimental diffusivity data, obtained at low pressures, to the high pressures commonly employed in industrial processes. 

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 ... 

Hydrogen, deuterium, neon, argon, and methane flow through saran charcoal by both Knudsen and surface flow. The latter is effected by the adsorbed molecules sliding from site to site across the surface. This is equivalent to a two-dimensional Knudsen flow where the adsorption site acts as the wall of the three-dimensional case, and a slide across the surface is the same as a flight across the pore. The activation energy for surface diffusion is 75 to 80% of the heat of adsorption. It is possible to calculate theoretically the relative contribution of each mechanism, while comparison with He, which does not adsorb, permits its experimental determination. The efficiency of surface flow is the ratio of the measured to the calculated value this decreases as the size of the molecule increases, being 80% for Ne and 12% for CH4. 

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,... 

A similar model has been applied to the modeling of porous media with condensation in the pores. Capillary condensation in the pores of the catalyst in hydroprocessing reactors operated close to the dew point leads to a decrease of conversion at the particle center owing to the loss of surface area available for vapor-phase reaction, and to the liquid-filled pores that contribute less to the flux of reactants (Wood et al., 2002b). Significant changes in catalyst performance thus occur when reactions are accompanied by capillary condensation. A pore-network model incorporates reaction-diffusion processes and the pore filling by capillary condensation. The multicomponent bulk and Knudsen diffusion of vapors in each pore is represented by the Maxwell-Stefan model. 

The thermal conductivity may be modeled as two conducting paths with transfer of heat between the two. The fluid contribution is particularly complicated since the pore size distribution of many catalysts is such that, at atmospheric pressure, diffusion takes place in the transition region between the Knudsen and bulk modes. Here, the heat flow... 

The effective diffusion coefficients. Die, obtained from the parameter tdjf, include contributions from the Knudsen diffusion mechanism and from the molecular diffusion mechanism. Because of the very low tracer concentrations the Bosanquet formula (3) is applicable. 

One would physically expect that as pressure increases the solid surface may get smoother due to the filling of small pores and cavities with adsorbed molecules, and as a result the reflection time of gas phase molecules from the surface may become shorter. The values of / in Table 1 are close to unity as expected and they are in an increasing order of n-hexane, carbon tetrachloride and benzene. On the other hand, the parameter a for n-hexane is much higher than that of the others. Since the parameter a in Eq. 3 represents how fast the Knudsen diffiisivity increases with pressure, one would expect a substantial contribution of the Knudsen diffusion for n-hexane to the total permeability at very low pressures. Also the parameter is a measure of how fast the activation energy for surface diffusion decreases with adsorbed concentration. As Table 1 indicates, the surface diffusion permeabilities of n-hexane and carbon tetrachloride are expected to increase more sharply than that of benzene. 

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