Knudsen Flow Research Articles

A modified Maxwell-Stefan model for transport through inert membranes: the binary friction model

This paper focuses mainly on the development of a model for permeation through inert membranes, as encountered in many cases in ultrafiltration and in gas permeation through inert porous plugs. The ultrafiltration model is made up of a boundary layer transport model and a porous membrane model in series, which are connected by an equilibrium relation. The boundary layer model is developed with the Vieth approximation for turbulent diffusivity. For the internal membrane transport, a modification of the Maxwell-Stefan-Lightfoot equation is derived (the binary friction model), which in a natural way includes both interspecies (diffusive) and species-wall forces. Application for the partial separation of PEG-3400 from aqueous solution shows that membrane friction coefficients can simply be estimated from membrane resistance measurements and mixture viscosity data. The only adjustable parameter to be determined is the distribution coefficient between the free solution and the membrane pores. The differences between the Lightfoot approach and the dusty gas model (DGM) are shown to stem from errors in the drivations of the latter, thus invalidating the dusty gas approach in the normal region in which viscous friction effects become important. For gases, the binary friction model is developed to include Knudsen and viscous wall friction terms as well as intermolecular diffusion. It is shown to give excellent coverage of the He-Ar diffusion data of Evans et al. ( J. Appl. Phys., 33 (1962) 2682; 34 (1963) 2020), with wall friction coefficients derived directly from Knudsen coefficients and gas viscosity data. The apparent success of the DGM in describing the same phenomena is shown to be caused by the relatively small importance of the wall friction forces at elevated pressures, and by the correct transition to Knudsen flow at low pressures. In addition, it is shown that diffusive slip phenomena in capillaries can be described well by the binary friction model.

Models of the structure and evolution of comet P/Wirtanen

Model calculations are presented for the nucleus of comet P/Wirtanen, the main target of the upcoming Rosetta mission. A porous, spherical nucleus, made of ice and dust in equal mass fractions is considered, assuming a few percent of CO and CO 2 occluded in the amorphous ice. The main effects included in the code are: crystallization of amorphous ice and release of occluded gases, condensation, sublimation and (Knudsen) flow of gases through the pores, changing pore sizes, flow of dust grains of different radi. Evolution is followed through several orbits, for different initial parameter combinations. It is found that a stratified structure develops for the outer layer of the nucleus: a highly porous dust mantle, a denser layer of crystalline ice and dust below it, followed by layers of amorphous ice and dust including frozen CO 2, and deeper down, frozen CO. The production rate of H 2O varies strongly with heliocentric distance; at perihelion the calculated rate is very close to that inferred from observations. The CO production rate is maintained nearly constant throughout the orbit and that of CO 2 varies moderately. Outbursts are obtained for some models at large heliocentric distances.

Transport resistances during pervaporation through a composite membrane: experiments and model calculations

For an investigation of transport resistances of a GFT PVA-PAN Pervap 1000 type membrane the pervaporation flux of pure water was measured as a function of downstream pressure, with the membrane oriented in normal and in reverse position ( =PVA layer facing the feed or the permeate). Assuming equilibrium across all interfaces, it was possible to break up the total resistance into (a) a support layer resistance part, and (b) an active layer resistance part. (a) Supprot layer resistance part=resistance of the support excluding its upstream skin: A model of liquid water transported by capillary forces through small pores, and of vapor by Poiseuille and Knudsen flow and eventually by surface diffusion along pore walls is presented for describing the 3 transport regimes observed experimentally. The support layer was found to consume up to 80% of the total resistance. (b) Active layer resistance part=resistances of the PVA layer and the slightly porous PAN skin: 3 models describing the influence of the PAN skin are discussed and judged in detail. Then the skin resistance is estimated and used to derive the permeability of water in PVA as a function of water activity or ‘equivalent partial pressure’ EPP in the polymer which is then compared to vapor permeation, vapor sorption and diffusivity data published by Heintz and Lichtenthaler. Strong discrepancies are found and discussed in terms of possible interface resistances, of free and bound water and of cluster formation which would mean that in cases like this diffusivities can be derived from stationary state pervaporation but not from non-stationary state kinetic sorption experiments.

Pore size distributions in microporous membranes. A critical analysis of the bubble point extended method

An extended bubble point method has been used to examine the porous morphology of several track-etched microporous polycarbonate membranes with nominal pore sizes ranging from 0.1 to 5.0 μm. The technique has been carefully analyzed and corrected to take into account the diverse non-ideal factors in flow along with the prevalence of Knudsen flow over the Hagen-Poiseuille one in the smaller pores. The pore size distributions were obtained in terms of flow or pore numbers. Bulk porosity and pore densities were in good agreement with nominal and SEM values for each filter. Performancewise, measured water permeabilities agreed well with those obtained from pore density distributions, assuming Knudsen air flow through the narrowest pores.

Preparation of silica-based microporous inorganic gas separation membranes

Fully inorganic silica-based membranes have been prepared using sol-gel procedures starting from sodium silicate solutions. An asymmetric membrane structure is obtained by multiple deposition of colloidal silica sols onto commercial ceramic supports. The microporous structure of the silica which forms the active top layer remains stable up to 873 K and is characterised by a mean pore diameter of 1.6 nm, a narrow pore size distribution and a porosity of 35%. The continuous silica top layer of the membrane was analysed by SEM and checked by gas permeation measurements. The permeabilities of CH 4, N 2 and CO 2 were measured for membranes deposited onto zirconia supports as a function of temperature. The preliminary experimental results indicate reproducible and relatively high fluxes as well as a transport mechanism which basically follows Knudsen flow with contributions from surface and microporous diffusion.

Modelling endothermic reactions in a compound membrane reactor

This paper models the performance of a membrane reactor. The membrane, a composite alumina-based one, is packed with a catalyst and allows low molecular weight gases to diffuse through it at a faster rate than gases with a higher molecular weight. This allows a greater conversion to be achieved in one pass through the reactor. The reaction that is specifically considered in this paper is the dehydrogenation of methyl-cyclohexane to toluene with the production of hydrogen. This latter species is preferentially removed by the membrane. Data for the performance of the membrane have been estimated from previous experiments using single gases and the mechanisms considered are Knudsen and bulk flow. Surface flow is not considered in the model as it is possibly not important as the endothermic reaction is carried out at a high temperature. A standard kinetic model is also incorporated in the calculations. The correlations of maximum effective length of membrane reactors and maximum percentage conversion as functions of the feed velocity and the membrane diameter are demonstrated in this paper. This paper also considers the behaviour of a compound reactor in which the first section is a straightforward ‘plug flow’ reactor where the catalyst is confined in an impermeable tube with the same internal diameter as the membrane. This is followed by a section containing the membrane. The reason for considering this configuration is to avoid unnecessary leakage of methyl-cyclohexane feed in the initial stages of the reaction. This innovation leads to predicted increases in the overall conversion of the process.

Knudsen flow and the thermomechanical effect in a Fermi liquid

We show that conventional kinetic theory, based on Fermi liquid theory, disagrees with the recent measurements by Greenwood et al. of the thermomechanical effect in normal liquid3He. The theory predicts that, in the Knudsen or long-free-path regime, dP/dT across a porous plug should be 1/3 of the entropy per unit volume ps. Experimentally, the ratio decreases from ∼4 to ∼1 between 2 mK and 20 mK. The theoretical result does not depend on whether the quasiparticles exchange energy with the walls or whether the scattering is partly specular. We consider possible reasons for the discrepancy between theory and experiment. One alternative approach, which treats the3He in the porous plug as a mesoscopic system, gives the same result as conventional kinetic theory. The relation between the thermomechanical effect in3He and thermoelectricity in metals is also discussed.

Transient modelling of a packed tower: Mass and heat transfer with reaction

A packed tower is modelled transiently for fluid-solid reactions. Heat and mass transfer are considered, with allowance for structural changes. The Knudsen, molecular and bulk flow terms are accounted for in the packed tower and within the pellets of the packing. Homogeneous, heterogeneous and catalytic reactions are handled. Conduction, convection and radiation model of heat transfer are considered. The thermophysical, structural and transport properties are included as variables. Three levels of space are used in the system: packed tower, macroporous pellets, and microporous zone (grain). The solution in these spaces includes the distributions of concentrations, temperature, pressure and solids conversion as well as all other parameters at any given time. The three zones are linked together via boundary values.

Preparation and separation properties of silicalite composite membranes

Silicalite-alumina composite membranes were prepared by an in situ zeolite synthesis method using an alumina membrane tube with a 5-nm pore diameter, γ-alumina layer as a substrate. Single gas permeances of H 2, Ar, n-C 4H 10, i-C 4H 10, and SF 6 were measured and mixtures of H 2 i- C 4H 10 and H 2 SF 6 were separated to characterize the silicalite membrane. These measurements were made from 300 to 737 K, and are compared to an alumina membrane without a silicalite layer. Permeances were lower in the silicalite membrane (a factor of 8 for Ar at 298 K). Permeances for the alumina membrane decreased as the temperature increased, and separation selectivities were lower than values expected for Knudsen diffusion. Transport through the alumina membrane was by Knudsen flow and surface diffusion. The silicalite membrane showed dramatically different behavior, and transport appeared to be controlled by molecular size and adsorption properties. Permeances of all components studied were activated in the silicalite membrane, and activation energies ranged from 8.5 to 16.2 kJ/mol. The ratio of single gas permeances was as high as 136 for H 2 to SF 6 and 1100 for H 2 to i-C 4H 10 at 298 K. Separation selectivities at elevated temperatures were significantly above Knudsen diffusion selectivity for the silicalite membrane and were larger than ratios of pure gas permeances at the same temperature. The largest permeance ratio for the separation of mixtures was 12.8 for H 2 SF 6 at 583 K. Separation selectivities for both membranes were higher when a pressure drop was maintained across the membrane than when an inert sweep gas was used because of counter diffusion of the sweep gas.

Maximum gas flow in circular ducts at various pressures: an approximate approach

An approximate solution is given for the prediction of the gas flow rate depending on the pressure drop for compressible laminar flow in circular ducts. The pressure and pressure drop range from high to low values, i.e. both Newtonian flow and Knudsen flow as well as the transition regime have to be modelled. The resulting equations are easy to handle and may be sufficient to serve engineering purposes. A numerical example displaying how fast a vessel can be evacuated is also included in the paper.

Hydrodynamic electron flow in high-mobility wires.

Hydrodynamic electron flow is experimentally observed in the differential resistance of electrostatically defined wires in the two-dimensional electron gas in (Al,Ga)As heterostructures. In these experiments current heating is used to induce a controlled increase in the number of electron-electron collisions in the wire. The interplay between the partly diffusive wire-boundary scattering and the electron-electron scattering leads first to an increase and then to a decrease of the resistance of the wire with increasing current. These effects are the electronic analog of Knudsen and Poiseuille flow in gas transport, respectively. The electron flow is studied theoretically through a Boltzmann transport equation, which includes impurity, electron-electron, and boundary scattering. A solution is obtained for arbitrary scattering parameters. By calculation of flow profiles inside the wire it is demonstrated how normal flow evolves into Poiseuille flow. The boundary-scattering parameters for the gate-defined wires can be deduced from the magnitude of the Knudsen effect. Good agreement between experiment and theory is obtained.

Degradation mechanism of impregnated cathodes

AbstractTo better understand the degradation mechanism of impregnated cathodes, chemical compositions of impregnated cathodes after extended usage were analyzed using a newly devised integral method. The results indicate that cathode degradation is caused by a decrease in Ba supply rate. This decrease is caused by an inward drift of the reaction front as the reaction between an impregnant and a tungsten substrate proceeds. It is also confirmed that the Ba supply rate is controlled by the Knudsen flow of Ba atoms.To compare cathode lives economically, a new diode tube has been designed with an anode that has a taper hole. The diode tube allowed economical measurement of cathode performance (which is almost identical to that of a traveling wave tube (TWT)). The testing of diode tubes confirms that it is possible to estimate the lives of the cathodes with various compositions by the method based on ΔT90. The methodology has been applied to identify the cathode with the longest life from four impregnated cathodes employed in satellite applications.

Gas permeation of porous organic/inorganic hybrid membranes

Selective gas permeation of porous organic/inorganic hybrid membranes via sol-gel route and its thermal stability are described. Separation performance of the hybrid membrane was improved compared with porous membranes governed by the Knudsen flow, and gas permeability was still much higher than that through nonporous membranes. Additionally, it was shown that these membranes were applicable at higher temperatures than organic membranes. SEM observation demonstrated that the thin membrane was crack-free. Nitrogen physisorption isotherms showed the pore size was in the range of nanometers. Gas permeability through this membrane including phenyl group was in the range of 10−8 [cc(STP) cm/(cm2 s cmHg)] at 25°C. The ratios of O2/N2 and CO2/N2 were 1.5 and 6.0, respectively, showing the permeation was not governed by the Knudsen flow. The permeability decreased as the temperature increased. Furthermore, the specific affinity between gas molecules and surface was observed not only in the permeation data of the hybrid membranes but in the physisorption data. These results suggested that the gas permeation through the hybrid membrane was governed by the surface flow mechanism. Thermal analysis indicated that these functional groups were still stable at higher temperatures. The phenyl group especially remained undamaged even at 400°C.

Quantum transport in porous media: Inelastic scattering of 4He atoms and its temperature dependence.

In porous media characterized by pore sizes in the range 5--20 \AA{}, fluid transport characteristics at low temperatures may deviate significantly from the classical behavior, due to the interference effect arising from the quantum wave nature of the fluid atoms. An important consideration in this regard is the size and temperature dependence of the inelastic scattering length ${\mathit{l}}_{\mathrm{in}}$ for the fluid atom, which directly governs the magnitude of the quantum deviation. For a sample of size smaller than ${\mathit{l}}_{\mathrm{in}}$ the transport of the fluid atoms should be described by the Schr\"odinger equation confined in the pores. We intend to address this problem in two separate papers. In this paper we present the results of calculations for the inelastic scattering length in the case of a $^{4}\mathrm{He}$ atom confined in a cylindrical pore. Inside the pore, a monolayer of physisorbed $^{4}\mathrm{He}$ atoms is assumed to line the pore wall. Outside the pore, the solid material is assumed to be elastic. The thermal excitation of elastic waves results in the distortion of the cylindrical pore surface, which in turn causes transitions between the quantum eigenstates of $^{4}\mathrm{He}$ inside the pore. By choosing some reasonable parameters of a porous medium, we have obtained the inelastic scattering rate (length) as a function of temperature for two pore radii. Our results show that the inelastic scattering length is on the order of 1 \ensuremath{\mu}m even at 50 K. This suggests that the quantum interference effect could be important at low temperatures, and that a deviation from Knudsen flow may be expected. The explicit calculation of quantum transport in this regime is planned to be the subject of a second paper.

Observation of Knudsen and Gurzhi transport regimes in a two-dimensional wire

We have observed electronic Knudsen and Poiseuille flow in a current heating experiment on electrostatically defined wires in (Al,Ga)As heterostructures. Current heating induces an increase in the number of electron-electron collisions in the wire, leading first to an increase (Knudsen regime) and subsequently to a decrease (due to Poiseuille electron flow, and known as the Gurzhi effect) of the resistance of the wire.

A generalized solution—diffusion model of the pervaporation process through composite membranes Part II. Concentration polarization, coupled diffusion and the influence of the porous support layer

The central part of the model is based on the solution—diffusion process which describes fluxes of low molecular weight components in the dense active layer. using the Maxwell-Stefan theory, coupled equations for the diffusive fluxes through the dense layer are derived. The concentrations on the phase boundaries inside the membrane are determined by solubility equilibria with the adjacent phases of the feed and permeate phase, respectively. They are calculated using the UNIQUAC model as described in Part I of this series. Fluxes through the porous sublayer are calculated by the general kinetic gas flow model which includes Knudsen flow and laminar flow as special cases. The solution—diffusion model and the pore flow model are combined resulting in a resistance-in-series model for fluxes through the dense layer and the support layer. The generalized model is applied to the pervaporation of six aqueous/organic mixtures through the standard poly(vinyl alcohol)/poly(acrylonitrile) (PVA/PAN) composite membrane. It turns out that the strong non-ideal solubility behavior of the liquid mixture components in PVA and diffusive coupling effects play the dominant roles in the pervaporation process. More simplified models which do not account for these physico-chemical effects fail to describe fluxes and separation diagrams in PVA/PAN membranes. Introducing the influence of the porous support layer leads to almost complete agreement with experimental data even at elevated permeate pressures. The influence of concentration polarization has also been studied in some detail and is found to be of minor importance compared to the other contributions.

Gas Permeation Characteristics of Zirconia-Silica Composite Membranes at High Temperature.

Permeabilities for pure gases of H2, He, CH4, NH3, H2O, N2, N2, O2 and CO2 were measured at trans-membrane pressures up to 0.2 MPa and 303, 423, 523 and 773 K for two types (with small pores and large pores) of composite zirconia-silica membranes coated on the surface of porous ceramic tubes (0.5μm). The composition of the metal-alkoxides solution used successfully in the coating process was Zr (OC3H7) 4 : 3.09, Si (OC2H5) 4 : 7.20, Y (CH3COO) 3.4H2O : 0.21, i-PrOH : 89.50 in molar %.At low temperatures, with large-and small-pore membranes, the permeation mechanism for H2O, NH3 and CO2 is surface diffusion, and for other gases it is Knudsen flow.At higher temperature, with a large-pore membrane, the permeation mechanism for H2O, NH3, CO, and CH4 is Knudsen flow, and for other gases it is a combination of Knudsen flow and activated diffusion. With a small-pore membrane, for H2O it is Knudsen flow, and for all other gases it is activated diffusion.

Characterization of gas transport through micromachined submicron channels in silicon

Gas transport through narrow channels etched in a silicon surface is characterized by mass-spectrometric measurements. The channel structures consist of 33 parallel 100 nm deep and 5 cm long channels, i.e. with a L d ratio of 500,000. which appears to be among the largest experimental ratios reported on so far. Some unexpected findings regarding for example diffusion time lag and its dependence on molecular mass and absolute pressure are reported. Furthermore, the possibility to study wall induced reactions is demonstrated by the water formation reaction on channel walls modified with a thin layer of platinum. The observations are discussed in relation to models for Knudsen flow through narrow capillaries. The main purpose is, however, to point out the special features of micromachined channels for the study of gas transport and chemical reactions controlled by the properties of the channel walls.

Modeling of evacuated pulse micro-reactors

The system for temporal analysis of products consists of a micro-reactor, a high-speed gas pulsing valve, and a computer-controlled real-time quadrupole mass spectrometer. A pulse of reactant is introduced at the entrance of the micro-reactor and evacuated through the catalyst bed creating a mass spectrometer response. Quantification of kinetic parameters from such measurements requires a well-defined flow mechanism in the micro-reactor and an appropriate model of the system. Experimental methods for establishing Knudsen flow in the catalyst bed are outlined. The governing differential equation for the packed-bed micro-reactor is established, and the suitability of various boundary conditions is discussed. The effect of the inlet section on the response is illustrated and quantified. A simple boundary condition of zero exit concentration is justified. The developed model provides the foundation for quantification of results obtained in transient response experimental systems that utilize evacuated packed-bed micro-reactors.

Electrical and electrochemical properties of TiO 2-SiO 2 porous glass ceramics and their application

TiO 2-SiO 2 porous glass ceramics (TPGC) prepared by phase separation and crystallization of a glass have a pore structure which allows gas permeation by Knudsen flow. The TPGC skeleton is covered by many surface hydroxyl groups and has an electrical conductivity of the order of 10 −7 S m −1 at room temperature, which is advantageous for use in a humidity sensor. A sensor constructed from the TPGC worked well, and had a linear relation between the logarithm of impedance and the square root of humidity. It was confirmed by cyclic voltammetry of hexacyanoferrate(II) that the TPGC electrode has a surface area about 6.5 times as large as that of a Pt electrode. A cyclic voltammogram of the TPGC electrode showed anodic currents due to oxygen generation or desorption of hydrogen and cathodic currents due to hydrogen adsorption or leading to hydrogen generation in the potential range −0.02 V to + 1.7 V. The electrode showed a response to irradiation with light similar to that of a TiO 2 semiconductor. When the electrode was illuminated with a xenon lamp, an unstable photoinduced current was observed.