Knudsen Flow Research Articles

Permeability of Green Ceramic Tapes as a Function of Binder Loading

The permeability of green ceramic tapes was determined as a function of binder content for binder removed by air oxidation. The tapes were comprised of barium titanate as the dielectric, and polyvinyl butyral and dioctyl phthalate were the main components of the binder mixture. The flow in porous media through the tapes was analyzed in terms of models for describing Knudsen, slip, and Poiseuille flow mechanisms. The characteristic pore size was determined to be 1–2 μm, and thus Poiseuille flow was the dominant transport mechanism contributing to the flux. The permeability was then determined from Darcy's law for flow in porous media. The permeability was also determined from microstructural attributes in terms of the specific surface, the pore fraction, and a term to account for tortuosity and constrictions.

Nano Filter from Sintered Rice Husk Silica Membrane

A nano filter showing the Knudsen flow was demonstrated by a modification of a membrane constructed from rice husk silica. The membrane was prepared by pressing and sintering micron sized rice husk silica with 4 nm pores. The membrane showed a permeability of 5.2 x 10(-8) mol m(-1) sec(-1) Pa(-1) for H2 and ratios of gas permeability 2.1 and 3.2 for k(H2)/k(CH4) and k(H2)/k(CO2), respectively. When the membrane was treated by filtration of approximately 100 nm sized rice husk silica particles, the permeability decreased to 4.9 x 10(-8) mol m(-1) sec(-1) Pa(-1) and the ratios increased to 2.2 and 3.4. In the case of the membrane after treatments with the dispersion and chemical deposition of tetraethylorthosilicate (TEOS), the corresponding permeability and ratios of the membrane were 1.8 x 10(-8) mol m(-1) sec(-1) Pa(-1), and 2.9 and 4.5, respectively. From the change of the ratio of gas permeability for the membrane with modifications, it is suggested that approximately 100 nm sized rice husk silica particles pack the large pores among the micron sized rice husk silica particles while the chemical deposition of tetraethylorthosilicate (TEOS) reveals the gas flow through 4 nm pores in the rice husk silica by blocking large pores.

Mass Diffusion in a Hydrophobic Membrane Humidification/Dehumidification Process: the Effects of Membrane Characteristics

Humidification and dehumidification are common processes in air conditioning industry. In this study, a new technique‐the hydrophobic membrane based air humidification and dehumidification are investigated. The effects of membrane characteristics on moisture exchange effectiveness of the system are studied. A two‐dimensional transient model which takes into account the combined mechanisms of Knudsen flow and ordinary diffusion in membrane pores is proposed and validated. Four membranes, 2 hydrophobic treated Nylon and 2 PVDF, are used in the test to validate the model. Then the effects of variations of membrane properties on the system performance and membrane to total resistance ratio, are evaluated. A dimensionless Number of Transfer Units (NTU) can be summarized to govern the moisture exchange performance. Following, a correlation has been obtained to reflect the relations between the moisture exchange effectiveness and the Number of Transfer Units.

General solution for the time lag of a single-tank receiver in the Knudsen flow regime and its implications for the receiver's configuration

Constant volume systems are commonly used for the determination of the diffusion, permeability and solubility coefficients of gases in porous and nonporous media. In this paper we present important considerations for the design of an outflow receiver of a constant volume system to minimize the possible resistance to gas transport downstream from the tested medium. The receiver considered in this paper consists of main tube, a tank, and a tube connecting the main tube with the tank. The resistance of the receiver is quantified using the concepts of the asymptotic solution and the time lag by assuming that gas accumulation in the receiver occurs in the Knudsen flow regime. The effects of the volume of the tank, the position at which the tank is connected to the main tube, the length of the connecting tube, as well as, the position of the pressure sensor in the receiver are discussed.

Liquid Expulsion Permporometry – a Tool for Obtaining the Distribution of Flow‐Through Pores

AbstractThe permporometry technique, common in the fields of thin filters, membranes etc., was extended to porous solids with far larger thicknesses. The basic idea of permporometry is the controlled expulsion of a suitable liquid from pores of different sizes by increasing the gas pressure difference across the tested porous solid. The gas flow rate through liquid‐free pores then corresponds to the amount of pores with different sizes. Two modes of measurement were suggested and tested. The contribution of the Knudsen flow mechanism was taken into account in addition to the usually anticipated viscous gas flow mechanism.It was confirmed that both measurement modes give consistent pore size distributions (PSD) of flow‐through pores. Permporometry PSDs were compared with results from mercury porosimetry and pseudo‐steady state permeation. It appears that permporometry should be considered as a standard textural characterization technique suitable for the prediction and simulation of gas flow through porous media in all processes where gas transport takes place.

The heterogeneous decomposition of ozone on atmospheric mineral dust surrogates at ambient temperature

AbstractThe rate of uptake of ozone on various mineral dust surrogates, expressed as uptake coefficient γ, has been studied employing a Knudsen flow reactor. Experiments were performed at T = 298 ± 2 K on substrates of kaolinite, CaCO3, natural limestone, Saharan dust, and Arizona test dust. Initially, the uptake coefficients have been calculated on the basis of the geometric surface area of the powder samples. Both initial and steady‐state uptake coefficients γ0 and γss were found very similar for all the examined substrates. In addition, additional uptake experiments on marble sample have shown that γ0 and γss may be overestimated between a factor of 50 and 100, respectively. Based on these considerations, we proposed initial and steady‐state uptake values of the order of 10−4 and 10−5, respectively. On kaolinite, the uptake coefficient decreased with increasing O3 residence time τg thus indicating a complex mechanism. In contrast, γ decreased and became independent of τg at long residence time after long exposure to O3. For all uptake experiments the disappearance of O3 was accompanied by the formation of O2. The different mineral dust surrogates may be more accurately distinguished by their time‐dependent O2 yield r(t) rather than the magnitude of γ. The heterogeneous reaction of O3 on mineral dust has been found to be noncatalytic and of limited importance in the atmosphere. © 2006 Wiley Periodicals, Inc. Int J Chem Kinet 38: 407–419, 2006

Non-equilibrium Molecular Dynamics Simulation on Pure Gas Permeability Through Carbon Membranes

The permeation of various pure gas (H 2, He, Ne, CH 4 and Ar) through carbon membranes is investigated using a dual control volume grand canonical molecular dynamics method. A two-dimensional slit pore is employed instead of the one-dimensional pore. Compared with the experiments, simulation results show that the improvement of pore model is very necessary. The effects of membrane thickness, pore width and temperature on gas permeance and ideal separation factor are also discussed. Results show that gas permeates through membrane according to Knudsen diffusion in large pore, while Knudsen diffusion is accompanied by molecular sieving in small pore. Moreover, methane is easily adsorbed on the membrane surface due to strong attractive interactions of membrane and shows higher permeance than that of Knudsen flow. In addition, it is noted that when membrane thickness is thin enough the permeance of gas does not decrease with the increase of membrane thickness due to the strong adsorption until membrane resistance becomes dominant.

Phonon Knudsen flow in nanostructured semiconductor systems

We determine the size effect on the lattice thermal conductivity of nanoscale wire and multilayer structures formed in and by some typical semiconductor materials, using the Boltzmann transport equation and focusing on the Knudsen-flow effect. For both types of nanostructured systems we find that the phonon transport is reduced significantly below the bulk value by boundary scattering off interface defects and/or interface modes. The Knudsen-flow effects are important for almost all types of semiconductor nanostructures but we find them most pronounced in Si and SiC systems due to the very large phonon mean free paths. We apply and test our wire thermal-transport results to recent measurements on Si nanowires. We further investigate and predict size effects in typical multilayered SiC nanostructures, for example, a doped-SiC∕SiC∕SiO2 layered structure that could define the transport channel in a nanosize transistor. Here the phonon-interface scattering produces a heterostructure thermal conductivity smaller than what is predicted in a traditional heat-transport calculation, suggesting a breakdown of the traditional Fourier analysis even at room temperatures. Finally, we show that the effective thermal transport in a SiC∕SiO2 heterostructure is sensitive to the oxide depth and could thus be used as an in situ probe of the SiC oxidation progress.

A Galerkin Method for the Simulation of the Transient 2-D/2-D and 3-D/3-D Linear Boltzmann Equation

Many production steps used in the manufacturing of integrated circuits involve the deposition of material from the gas phase onto wafers. Models for these processes should account for gaseous transport in a range of flow regimes, from continuum flow to free molecular or Knudsen flow, and for chemical reactions at the wafer surface. We develop a kinetic transport and reaction model whose mathematical representation is a system of transient linear Boltzmann equations. In addition to time, a deterministic numerical solution of this system of kinetic equations requires the discretization of both position and velocity spaces, each two-dimensional for 2-D/2-D or each three-dimensional for 3-D/3-D simulations. Discretizing the velocity space by a spectral Galerkin method approximates each Boltzmann equation by a system of transient linear hyperbolic conservation laws. The classical choice of basis functions based on Hermite polynomials leads to dense coefficient matrices in this system. We use a collocation basis instead that directly yields diagonal coefficient matrices, allowing for more convenient simulations in higher dimensions. The systems of conservation laws are solved using the discontinuous Galerkin finite element method. First, we simulate chemical vapor deposition in both two and three dimensions in typical micron scale features as application example. Second, stability and convergence of the numerical method are demonstrated numerically in two and three dimensions. Third, we present parallel performance results which indicate that the implementation of the method possesses very good scalability on a distributed-memory cluster with a high-performance Myrinet interconnect.

Knudsen Diffusion and Viscous Flow Dusty-Gas Coefficients for Pelletised Zeolites from Kinetic Uptake Experiments

A simple volumetric uptake apparatus was used to determine uptake data of N2 on a sample of LiLSX zeolite for two different particle sizes, two temperatures, and a variety of different dosing pressure levels. Using a mass and energy conservation model for the dosing and sample volumes and the Dusty Gas Model + viscous flow for the mass transfer description at the pellet level, the Knudsen and viscous flow structural parameters were derived. Our analysis gave structural coefficients C K = 0.0827 ± 0.018 and C v = 0.0608 ± 0.026 which gave good agreement across all of the experimental runs conducted for both particle sizes and all pressure ranges. From these, tortuosity coefficients for Knudsen and viscous flow were derived and gave τ K = e P,macro/C K = 3.7 ± 0.8 and τ v = e P,macro/C v = 5.1 ± 2.2 respectively. These are in good agreement with reported values. The apparatus and procedure is not very sensitive to the viscous flow coefficient but is sensitive to the Knudsen coefficient. All other parameters of the model were measured or determined by calibration experiments. This study suggests that the apparatus may be useful for determination of some of the fundamental structural coefficients employed in the Dusty Gas Model.

Glass membranes for purification of aggressive gases: Part II. Adsorption measurements and diffusion coefficient estimations

In the article Part I of this work durability and separation properties for several types of glass membranes in aggressive gas environment have been evaluated. A surface modified glass membrane (modified with (heptadecafluoro-1,1,2,2-tetrahydrodecyl) dimethyl chlorosilane, Pf-C10), proved to be the best choice with respect to stability, permeability and selectivity. For a better understanding of the gas separation taking place according to the governing mechanism, selective surface flow, the sorption and diffusion coefficients were investigated more closely. This is reported in the current paper for the gases Cl 2, HCl, R22 (CHF 2Cl), He, H 2, N 2, CO, O 2, Xe, SF 6 and CO 2. Temperature and pressure range focused on were 1–4 bar and 30–45 °C, respectively, as these ranges were judged to be most interesting with respect to possible changes in the transport through the Pt-C10 surface modified glass membrane. The degree of selective surface flow, SSF, relative to the Knudsen flow is also discussed in this work. When plotting the sorption coefficient versus degree of SSF, there seems to be two distinct patterns: The Cl 2, HCl and R22 (all containing chlorine and have high critical temperatures) are described by exceptionally high sorption coefficients, while the other gases are best described by an exponential fit. The heat of adsorption for Cl 2 and HCl was found to be comparable to the heat of condensation at the same pressure. It was assumed that the adsorption of these gases corresponds to the proposed “sliding liquid layer” flow described in the literature, while the exponential behaviour of the other gases correspond to the “site to site hopping” or 2D gas flow.

Analytical solution for the effective time lag of a membrane in a permeate tube collector in which Knudsen flow regime exists

Gas permeation in a simplified constant volume system, in which the outflow volume consists of a cylindrical collector tube, initially at high vacuum, has been modeled. Assuming a Knudsen flow regime in the permeate collector the analytical solution for the effective time lag of a membrane has been obtained by using the concept of linear asymptotes and Laplace transformation of the governing partial differential equation. The solution indicates that the effective time lag is different from the actual time lag of the membrane; the magnitude and the sign of the difference between the two time lags depends on the location where the pressure rise in the permeate tube collector is monitored, as well as, on the length and the Knudsen diffusion coefficient in the tube.

Interpretation of the Transport Properties of α-Al2O3 Supports in Terms of Pore Structure Characteristics

The absolute permeability, k, and Knudsen diffusion coefficient, Dkn, of mesoporous alumina (α-Al2O3) supports are measured by fitting transient and steady-state measurements of the flow rate versus the pressure drop across a disk-shaped sample to analytical solutions of the relevant macroscopic equations. The microscopic properties of the pore structure of the alumina disks, such as the pore and throat radius distributions and network accessibility functions, are determined by employing Hg intrusion/retraction data. To this scope, the estimates of an optimization algorithm, based on analytical percolation models of Hg intrusion/retraction, are coupled with results of simulators of the drainage/imbibition in pore-and-throat networks. The permeability and Knudsen diffusion coefficient are computed by solving the viscous flow and Knudsen flow problems separately in cubic networks of constricted unit cells. The numerically predicted values of k and Dkn may differ by ±30−40% from the experimentally measured o...

Dip-coating technique for the manufacture of alumina microfilters using PVA and Na-CMC as binders: a comparative study

α-Alumina microfiltration flat-membranes have been prepared using the dip-coating procedure. The effect of slip concentration (<30 wt.%), coating time and binder type (PVA and Na-CMC) on the thickness, porosity, roughness and effective pore diameter of the membrane layers has been investigated. A new method for the characterization of the membrane layer based on gas permeation has been presented and the pertaining equations derived. The method is applicable if permeability measurements embrace both Knudsen and Poiseuille flow regimes.

The heterogeneous interaction of HOCl with solid KBr substrates: The catalytic role of adsorbed halogens

The heterogeneous reactivity of HOCl on solid KBr at ambient temperature has been studied using a Knudsen flow reactor. On solid KBr steady-state uptake experiments reveal the formation of Br- and Cl-containing reaction products formed in secondary reactions such as Br(2), BrCl, HOBr, BrOCl, Cl(2) and Cl(2)O with the latter two predominating in the late stages of the reaction. The uptake coefficient gamma spanning a range between 0.15 and 1 x 10(-3) and product yields of HOCl strongly depend on the nature of the solid sample, whether grain, ground grain or thin sprayed film, as well as on sample processing such as pumping and/or heating. Furthermore, the presence of adsorbed halogen species such as Br(2)(a) are crucial for the kinetics of the reaction of HOCl with solid KBr substrates. The presence of surface-adsorbed water (SAW) leads to deactivation of KBr whereas mechanical stress such as grinding leads to the formation of surface defects that become reaction centers. Desorption of SAW at T > 620 K induces high reactivity of the KBr sample at ambient temperature. A reaction mechanism encompassing all significant observations including unusual autocatalytic activity is given as there is no direct reaction of HOCl with solid KBr. It stresses the importance of adsorbed Br-containing species such as Br(2)(a) and HBr(a) that initiate the heterogeneous chemistry of HOCl on solid KBr in the presence of SAW. The role of surface acidity and SAW for the extent of reaction is emphasized.

Inertial effects and diffusion

Fick’s law of diffusion is equivalent to an equation of motion for an ideal substance in which the thermodynamic driving force is assumed to be balanced by a drag force proportional to drift velocity. Neglect of the inertial force required by Newton’s second law leads to anomalous short-time predictions such as infinite speeds of propagation. Resolution of the propagation anomaly for chemical diffusion is considered for three applications: Knudsen flow in porous materials, interstitial diffusion in elastic solids, and interstitial diffusion in viscoelastic solids. The transients associated with inertial effects for the first two cases occur at very short-times and are typically of little practical significance. Diffusion in viscoelastic solids appears as a possible exception based on observations of linear transport kinetics in ordinary timescales. The constitutive equations for the Maxwell model lead to the prediction that solute-induced dilatations propagate according to the telegraph equation, a widely proposed correction for avoiding the propagation anomaly in standard diffusion theory. The telegraph equation is not universally valid, however. In the case of Knudsen flow, for example, inertial effects give rise to an additional term related to material convection.

Generalized hydrodynamics and microflows.

In this paper, a mathematical model within the framework of generalized hydrodynamics is developed for the description of flows in microsystems where the Knudsen number is large and the aspect ratio [(width)/(length)] is not so small. The model is based on a set of empirical generalized hydrodynamic equations, which are fashioned from the steady-state generalized hydrodynamic equations derived from the Boltzmann equation in a manner consistent with the laws of thermodynamics. The constitutive equations used for the model are highly nonlinear, unlike the Newtonian law of viscosity and the Fourier law of heat conduction, but they are thermodynamically consistent. In the absence of heat conduction, the model yields exact solutions for the velocity components and a nonlinear differential equation for the pressure distribution in the rectangular microchannel. The Langmuir adsorption model for surface-gas interaction is used for boundary conditions for the velocity. The calculated flow rate exhibits a Knudsen minimum with respect to the Knudsen number. The longitudinal velocity profile is also non-Poiseuille. The differential equation for pressure distribution is also solved approximately in order to obtain an analytic formula for the flow rate, which exhibits a Knudsen minimum. The formula, although approximate, provides considerable insights into the Knudsen flow phenomena in microchannels.

Pressure Driven3He Flow Through Submicron Sized Apertures

The flow of liquid 3He through different weak link structures was investigated at temperatures of 1 mK to 1 K and at pressures ≈0 bar by measuring the characteristic time τ which was needed for pressure equilibration between two volumes connected by the weak link. Two types of weak links were used, one with a regular structure of 65×65 holes of 150 nm diameter in a 50 nm thick SiN membrane, and another with 105 holes of 15 nm diameter statistically distributed over a 14 nm thick nitrocellulose film. For the flow through the 150 nm holes the expected transition from Knudsen flow at low T to viscous flow at high T was observed. To the contrary, the normal fluid 3He-flow through the 15 nm structure does not show any temperature dependence. The relaxation time τ dropped by orders of magnitude for both types of weak links below the superfluid transition.

Dynamics of flow and diffusion of adsorbing gases in Al2O3 and Pd‐Al2O3 pellets

AbstractA nonisobaric pulse‐response technique was proposed for the simultaneous analysis of flow and diffusion of adsorbing tracers in porous solids. Relative contributions of viscous flow, Knudsen flow, surface flow, and effective diffusion of adsorbing (hydrogen and carbon dioxide) and inert (helium) tracers in alumina and Pd‐alumina pellets were evaluated. Hydrogen permeability was shown to increase by Pd impregnation into alumina, due to the contribution of surface flow, especially at low temperatures. The tortuosity factor, corresponding to the viscous and Knudsen flow, was found to be less than the tortuosity factors evaluated from effective diffusivities obtained from isobaric pulse‐response experiments.

The accommodation of the translational and rotational energy of a gas in a Knudsen flow past a thin wire

The paper presents the results obtained in determining the accommodation coefficients for the translational and rotational energy of gas molecules in a Knudsen flow past a thin wire. The method used was based on numerically solving the complete heat balance equation for a wire probe. The accommodation coef- ficients were determined for H 2 , N 2 , CH 4 , and CO 2 on a gilded tungsten surface. For hydrogen with a quenched rotational energy, a negative accommodation coefficient of rotational energy was obtained due to the conversion of the rotational energy of incident molecules into the translational energy of reflected molecules. © 2003 MAIK "Nauka/Interperiodica". PLASMA, GASES