Knudsen diffusion mechanism

Classical commercial ceramic porous materials, as those obtained via sol-gel processes, generally have adequate permeabilities but could present some drawbacks They indeed have a limited thermal stability and are generally not permselective enough their pores are in the mesoporous range and maximum separation factors correspond to Knudsen diffusion mechanisms. 

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. 

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. 

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... 

Knudsen diffusion Mechanism of diffusion, dominant in smaller macropores at relatively low pressures, when collisions between diffusing molecules and pore walls occur more frequently than collisions between the molecules themselves. 

In this treatment only the ordinary and Knudsen diffusion mechanisms will be considered. Then, mass transport in isothermal, multicomponent gas phase systems is described by the following constitutive equation ... 

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. 

The resistance, Qk, of the Knudsen diffusion mechanism relative to the sum of Knudsen and bulk diffusion resistances was calculated for two pairs tracer-carrier and all porous pellets. Calculated results are summarized in Table 4. The obtained Qk for both pairs of gases for all catalysts are in a good agreement, the largest deviation is less than 12%. It can be seen that the significance of Knudsen diffusion mechanism decreases from ICI pellets (where it has nearly the same significance as bulk diffusion) to G4 pellets (where only about 15-20% of the total diffusion resistance is of the Knudsen type). Figure 7 illustrates the 95% Table 4... 

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. 

To describe the combined bulk and Knudsen diffusion flrrxes the dusty gas model can be used [44] [64] [48] [49]. The dusty gas model basically represents an extension of the Maxwell-Stefan bulk diffusion model where a description of the Knudsen diffusion mechanisms is included. In order to include the Knudsen molecule - wall collision mechanism in the Maxwell-Stefan model originally derived considering bulk gas molecule-molecule collisions only, the wall (medium) molecules are treated as an additional pseudo component in the gas mixture. The pore wall medium is approximated as consisting of giant molecules, called dust, which are uniformly distributed in space and held stationary by an external clamping force. This implies that both the diffusive ffrrx and the concentration gradient with respect to the dust particles vanish. 

On the other hand, the more rigorous Maxwell-Stefan equations and the dusty gas model are seldom used in industrial reaction engineering applications. Nevertheless, the dusty gas model [64] represents a modern attempt to provide a more realistic description of the combined bulk and Knudsen diffusion mechanisms based on the multicomponent Maxwell-Stefan model formulation. Similar extensions of the Maxwell-Stefan model have also been suggested for the surface diffusion of adsorbed molecular pseudo-species, as well as the combined bulk, Knudsen and surface diffusion apparently with limited success [48] [49]. 

Finally, it is important to notice the effect of the support in the pervaporation flux, analyzed by de Bruijn et al. [164] 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. In fact, the researchers of Bussan Nanotech Research Institute Inc. (BNR), Sato et al. [165], designed and patented an appropriate asymmetric ceramic porous support to suppress pressure drop, and in this case, the water flux increased dramatically compared to previous reported results. Wang et al. [166] have clearly shown that the flux of the membranes increased with the porosity of the hollow fiber supports. In spite of the thin 1 pm zeolite layer, prepared by Zhou et al. [167], the flux enhancement compared to layers 10 times thicker [168] was not significant. 

Bulk and Knudsen diffusion mechanisms occur in series and it is always wise to take both mechanisms into account, rather than assuming controlling only [34]. Surface diffusion occurs in parallel to the other mechanisms, and its contribution to the total species flux may be quite significant, especially at elevated pressures [34]. 

Thus for a given pressure gradient across the capillary, the steady state flux under the Knudsen diffusion mechanism is proportional to the pore radius and is inversely proportional to the molecular weight. Viscosity does not affect the Kiiudsen flow as it does not have any meaning at very low pressures when continuum is no longer valid. 

We have addressed in the example 7.4-4 the steady state flux due to the Knudsen diffusion mechanism, but the question which is of significant interest is how long does it take for the system to response from some initial conditions to the final steady state behaviour. This is important to understand the pure diffusion time in a capillary. By pure diffusion time, we mean the diffusion time in the absence of adsorption. In the presence of adsorption, the time to approach equilibrium from some initial state is longer than the pure diffusion time due to the... 

When Knudsen diffusion mechanism controls the mass transfer, the following flux equation is obtained from eq. (7.8-10) ... 

The value of 0.5 for a corresponds to the Knudsen diffusion mechanism, i.e. molecule-wall collision dominates and a = 1.75 corresponds to the molecule-molecule collision. 

The only parameter that could be affected by the total pressure is the pore diffusivity Dp. If the macropore diffusion is controlled purely by the Knudsen diffusion mechanism, the pore diffusivity is and hence it is independent of total pressure, implying that the parameter y is independent of pressure. However, if the macropore diffusion is governed by molecular-molecular collision, then the pore diffusivity is inversely proportional to the total pressure, meaning that the parameter Y increases linearly with the total pressure. This means that the system is moving toward macropore diffusion control as the total pressure increases. 

This time lag (12.2-32) using the pressure response of the supply reservoir or the time lag (eq. 12.2-24) using the pressure response of the receiving reservoir can be used to determine the diffusivity. Thus, the time lag method provides a very convenient if not a straightforward method to determine the diffusion coefficient. The method is not restrictive to the simple Knudsen diffusion mechanism, it is also applicable to other situations, for example... 

Eqn (4.9) describes the Knudsen diffusion mechanism, which is charactaized by a series of collisions of the molecules with the pore walls. Instead, Eqn (4.10) is the mathematical description of a viscous flow that is not capable of high separation performances. 

The high selectivity is compatible with a Knudsen diffusion mechanism, and it is higher than the values obtainable with a simple porous ceramic tube. 

Oxygen transport in the catalyst layer may differ from transport in the GDL due to the nonnegligible impact of its small pores with radii of 0.01 am. In these pores, oxygen transport is controlled by the Knudsen diffusion mechanism (see below). For a detailed discussion of transport mechanisms in the porous CL and GDL, see Hinebaugh et al. (2012), Wang (2004), and Weber and Newman (2004a). 

The transport of gases in mesopores generally occurs by the Knudsen diffusion mechanism if the mean free path of the molecule is larger than the pore diameter . For this flow resume, the permeance can be written as... 

Graaf et al. [51] used the dusty gas model to investigate the relative importance of Knudsen and bulk diffusion for the intra-particle diffusion limitation problem for the methanol synthesis. In their steady-state model, they neglected the pressure gradient and assumed isothermal conditions throughout the spherical particle. Their simulation results indicated that both bulk and Knudsen diffusion mechanisms should be incorporated in the single pellet model to obtain reliable results. 

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