Knudsen Transport

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 

In which (p is the volume fraction of porosity, r is the tortuosity of the pore and the M is the molecular weight of the species / (either 1 or 2). The separation factor can 

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Just as one considers two regions of flow for homogeneous media, so one may have molecular or Knudsen transport for heterogeneous media. 

The choice of small holes in a thin plate assures that molecules do not collide with each other during their transport through the hole. Thus, the Knudsen transport is due to the collision of molecules with the pore wall (periphery of the hole), and as a result the movement of different molecules are independent of each other. The flux of species from one side having the molecular density n and vacuum at the other side is given by the following equation ... 

As a model we have used the Mean Transport-Pore Model (MTPM) [6] which assumes that the decisive part of the gas transport takes place in transport-pores that are visualized as cylindrical capillaries. The transport-pore radii are distributed around the mean value (first model parameter). The width of this distribution is characterized hy the mean value of the squared transport-pore radii, (second model parameter). The third model parameter is the ratio of porosity, y/i, and tortuosity of transport-pores, qt, q/= Pore diffusion is described by the Maxwell-Stefan diffusion equation extended to account for Knudsen transport [6]. For gas permeation the simplified form of Weber equation [8-10] is used. 

The effective Knudsen diffusion coefficients are proportional to the pore diameter and independent of pressure, while the bulk diffusivities are independent of pore diameter and inversely proportional to pressure [62]. However, both bulk difiusiv-ities and Knudsen diffusivities are dependent on the temperature. Relations for the Knudsen transport coefficient and binary diffusivities are represented in Table 2.1. 

As an indication of a successful preparation, it is commonly accepted that the membrane shows molecular sieving which is different from the Knudsen transport. 

The pressure-independent permeances of (34.7) and (34.9) can be used to calculate the overall permeance for gases in a multilayer sequence with type 1 or Knudsen transport with a simple series resistance expression ... 

Since in this case different gas species move independent of each other for both type 1 as well as Knudsen transport, (34.12) can be used for single gases as weU as mixtures. Because the gas viscous flow permeance (34.10) contains a pressure-dependent term p ), it is not possible to use it in expressions as simple as (34.12). However, for incompressible liquids, an expression similar to (34.12) can also be written for a meso/macroporous... 

We can reduce diffusion by working with polymers with a high glass transition temperature Tg. Above the Tg, the polymer behaves as a viscous liquid, with diffusion coefficients typically around 10 cm /sec. Below the Tg, the polymer forms a glass willing to form small crystals whose growth is inhibited by the polymer chains running between crystals. Diffusion in such crystals is small, like that in metals diffusion between crystals includes Knudsen transport and diffusion in a viscous liquid. 

FIGt 22-48 Transport mechanisms for separation membranes a) Viscous flow, used in UF and MF. No separation achieved in RO, NF, ED, GAS, or PY (h) Knudsen flow used in some gas membranes. Pore diameter < mean free path, (c) Ultramicroporoiis membrane—precise pore diameter used in gas separation, (d) Solution-diffusion used in gas, RO, PY Molecule dissolves in the membrane and diffuses through. Not shown Electro-dialysis membranes and metallic membranes for hydrogen. 

Inspection of Fig. 15.3 reveals that while for jo 0.1 nAcm , the effectiveness factor is expected to be close to 1, for a faster reaction with Jo 1 p,A cm , it will drop to about 0.2. This is the case of internal diffusion limitation, well known in heterogeneous catalysis, when the reagent concentration at the outer surface of the catalyst grains is equal to its volume concentration, but drops sharply inside the pores of the catalyst. In this context, it should be pointed out that when the pore size is decreased below about 50 nm, the predominant mechanism of mass transport is Knudsen diffusion [Malek and Coppens, 2003], with the diffusion coefficient being less than the Pick diffusion coefficient and dependent on the porosity and pore stmcture. Moreover, the discrete distribution of the catalytic particles in the CL may also affect the measured current owing to overlap of diffusion zones around closely positioned particles [Antoine et ah, 1998]. 

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. 

Whether Knudsen or bulk diffusion dominates the mass transport process depends on the relative magnitudes of the two terms in the denominator of equation 12.2.6. The ratio of the two diffusivity parameters is obviously important in establishing these magnitudes. In this regard, it is worth noting that DK is proportional to the pore diameter and independent of pressure whereas DAB is independent of pore size and inversely proportional to the pressure. Consequently, the higher the pressure and the larger the pore, the more likely it is that ordinary bulk diffusion will dominate. 

From the magnitudes of the diffusion coefficients, it is evident that under the conditions cited the majority of the mass transport will occur by Knudsen diffusion. Equation 12.2.9 and the tabulated values of the porosity and tortuosity may be used to determine the effective diffusivity. 

If the effectiveness factor of the catalyst is known to be 0.42, estimate the tortuosity factor of the catalyst assuming that the reaction obeys first-order kinetics and that Knudsen diffusion is the dominant mode of molecular transport. 

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. 

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. 

Hasselbalch, S. G., Knudsen, G. M., Capaldo, B. etal. Blood-brain barrier transport and brain metabolism of glucose during acute hyperglycemia in humans. /. Clin. Endocrinol. Metab. 86 1986-1990, 2001. 

The main emphasis in this chapter is on the use of membranes for separations in liquid systems. As discussed by Koros and Chern(30) and Kesting and Fritzsche(31), gas mixtures may also be separated by membranes and both porous and non-porous membranes may be used. In the former case, Knudsen flow can result in separation, though the effect is relatively small. Much better separation is achieved with non-porous polymer membranes where the transport mechanism is based on sorption and diffusion. As for reverse osmosis and pervaporation, the transport equations for gas permeation through dense polymer membranes are based on Fick s Law, material transport being a function of the partial pressure difference across the membrane. 

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. 

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