Knudsen phase, diffusion

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 connection with multiphase diffusion another poorly understood topic should be mentioned—namely, the diffusion through porous media. This topic is of importance in connection with the drying of solids, the diffusion in catalyst pellets, and the recovery of petroleum. It is quite common to use Fick s laws to describe diffusion through porous media fJ14). However, the mass transfer is possibly taking place partly by gaseous diffusion and partially by liquid-phase diffusion along the surface of the capillary tubes if the pores are sufficiently small, Knudsen gas flow may prevail (W7, Bl). 

The combined effects of Knudsen, DK, and molecular (fluid-phase) diffusion Dm are commonly estimated from the expression ... 

Fig. 8.1 Oxidation of a carbon substrate beneath a protective film containing cracks. The maximum rate represents the oxidation rate beneath the crack and is based on the cross-sectional area of the crack. The average rate depends on the crack size and crack distribution and is shown here for two selected examples. For the 10 nm crack, oxygen transport occurs via Knudsen diffusion, and the oxidation rate is essentially independent of the total pressure. For the 1 pim crack, oxygen transport occurs via normal gas-phase diffusion, and the oxidation rate varies inversely with the total pressure the results shown here are for a total pressure of 10 bars.2...

Hindered diffusion, the primary transport mechanism in porous solids, can be qualitatively described as a series of hops by the analyte, via gas-phase diffusion, from one surface site to the next. Thus, hindered diffusion is composed of two main components a pure diffusion-related term, often Fickian in nature, associated with movement of the analyte in the gas phase and a term describing the noninstantaneous equilibration between gas-phase analyte and the solid surface at each point where the analyte touches down (adsorbs). In extended porous solids (e.g., a chromatographic column tightly packed with porous beads), transport is often more complex, requiring the consideration of such factors as eddy diffusion and Knudsen effusion. This is important if there is a significant pressure drop along the path of the analyte [109]. Finally, the presence of any external fields (thermal, electric, etc.) must be considered as well. 

Ceramic membrane is the nanoporous membrane which has the comparatively higher permeability and lower separation fector. And in the case of mixed gases, separation mechanism is mainly concerned with the permeate velocity. The velocity properties of gas flow in nanoporous membranes depend on the ratio of the number of molecule-molecule collisions to that of the molecule-wall collision. The Knudsen number Kn Xydp is characteristic parameter defining different permeate mechanisms. The value of the mean free path depends on the length of the gas molecule and the characteristic pore diameter. The diffusion of inert and adsorbable gases through porous membrane is concerned with the contributions of gas phase diffusion and sur u e diffusion. 

In this model also the decrease of the pore radius due to the formation of an adsorbed layer is incorporated. Flow 1 in Fig. 9.9 is the case of combined Knudsen molecular diffusion in the gas phase and multilayer (surface) flow in the adsorbed phase. In case 2, capillary condensation takes place at the upstream end of the pore (high pressure Pi) but not at the downstream end (P2), and in case 3 the entire capillary is filled with condensate. The crucial point in cases 3 and 4 is that the liquid meniscus with a curved surface not only reduces the vapour pressure (Kelvin equation) but also causes a hydrostatic pressure difference across the meniscus and so causes a capillary suction pressure Pc equal to... 

The equations and plots presented in the foregoing sections largely pertain to the diffusion of a single component followed by reaction. There are several other situations of industrial importance on which considerable information is available. They include biomolecular reactions in which the diffusion-reaction problem must be extended to two molecular species, reactions in the liquid phase, reactions in zeolites, reactions in immobilized catalysts, and extension to complex reactions (see Aris, 1975 Doraiswamy, 2001). Several factors influence the effectiveness factor, such as pore shape and constriction, particle size distribution, micro-macro pore structure, flow regime (bulk or Knudsen), transverse diffusion, gross external surface area of catalyst (as distinct from the total pore area), and volume change upon reaction. Table 11.8 lists the major effects of all these situations and factors. 

The type of diffusion in the pore, bulk or Knudsen, depends on whether the diffusing species collide more often with each other or with the pore wall surface. Liquid-phase diffusion is described by liquid-liquid... 

The temperature dependence of the effective intrapellet diffusion coefficient conforms to the assumption that ordinary molecular diffusion provides the dominant resistance to mass transfer in the pores, relative to Knudsen diffusion. This is valid when the pore diameter is larger than 1 tim. Gas-phase diffusivities are approximately proportional to the three-halves power of absolute temperature. Hence,... 

Fig. 6 Temperature dependence of the coefficient of long-range self-diffusion of ethane measured by PFG NMR in a bed of crystallites of zeolite Na-X points) and comparison with the theoretical estimate (line). The theoretical estimate is based on the sketched models of prevailing Knudsen diffusion (low temperatures, molecular trajectories consist of straight lines connecting the points of surface encounters) and gas-phase diffusion (high temperatures, mutual collisions of the molecules lead to Brownian-type trajectories in the intercrystalline space). From [92] with permission...

The Knudsen cell reactor is a versatile tool in the study of heterogeneous atmospheric chemistry. The rate of interaction with the surface is measured relative to a physical process, escape through an aperture, which is straightforward to calibrate. This puts the gas-surface rate measurement on an absolute basis. In addition, the low operating pressure means that complicated corrections for gas-phase diffusion do not need to be made. 

It can be shown that the intrinsic growth or evaporation rate associated with a given organic volatility is given by vdC where the characteristic velocity is 226 nm h V(pg m ) [75]. This is modified by three important terms - the mass accommodation coefficient, a, the surface-energy (Kelvin) term for particles smaller than 50 nm or so, and the Fuchs term for gas-phase diffusion limitations in the boundary layer around a particle for particles larger than 50 nm or so (with Knudsen... 

At the catalyst-electrolyte surface we have gas-phase diffusion, and there can also be additional surface diffusion. In surface diffusion, gas molecules physically or chemically absorb onto a solid surface. If it is physical absorption, the species are highly mobUe. If it is chemisorption and the molecule is more strongly bonded to the specific site, species are not directly mobile but can move via a hopping mechanism. Surface diffusion rates can be measured by direct measurement of the flux of a nonreacting gas across the material surface. The difference between the measured diffusion and predicted Knudsen diffusion is calculated to be the surface diffusion component. Values of the surface diffusion coefficient (Ds) are 10 cm /s in solids and liquids, but these vary widely since surface interaction is involved. Also, Ds is a strong function of temperature and surface concentration. Surface diffusion adds to the overall diffusion but is typically less than one-half of the Knudsen component and so has been mostly neglected in fuel cell analysis. 

A quantitative answer to above questions may be given through the theoretical modeling of non-isobaric, non-isothermal single component gas phase adsorption. External heat and mass transfer, intrapai ticle mass transport through Knudsen diffusion, Fickian diffusion, sorbed phase diffusion and viscous flow as well as intraparticle heat conduction are accounted for. Fig. 1 presents the underlying assumption on the combination of the different mass transport mechanisms in the pore system. It is shown elsewhere that the assumption of instantaneous... 

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... 

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. 

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). 

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