Showing papers on "Knudsen number published in 1973"

Intrinsic nonstoichiometry in the lead zirconate‐lead titanate system determined by Knudsen effusion

Abstract: The range of the intrinsic nonstoichiometry in lead zirconate‐titanate at 1100 °C has been established independently by two gravimetric techniques. A modified Knudsen effusion experiment allowed for an extended interpretation of the vapor pressure data. A single experiment was used to determine the width of any Pb1−x □x (TiyZr1−y)O3−xφx single‐phase region, the equilibrium PbO vapor pressure, activity data as a function of both temperature and composition, and the exact location of the stoichiometric composition within the single‐phase region. The ``vapor phase equilibration'' (VPE) method was used as a supplemental measurement of the width of most PZT single‐phase regions. Results obtained by the two methods for the extent of nonstoichiometry were in close agreement (xmax = 0.10 at y = 1.0; xmin = 0.016 at y = 0.40; T = 1100°C). The Knudsen effusion experiment established that for all the compositions studied, the stoichiometric composition, Pb(TiyZr1−y)O3, was coincident with the (liquid + PZT) phase bo...

Variational Methods Applied to Problems of Diffusion and Reaction

Abstract: 1. Introduction and Preliminaries.- 1.1. General Survey.- 1.2. Phenomenological Descriptions of Diffusion and Reaction..- 1.3. Correlation Functions for Random Suspensions...- 1.4. Mean Free Path Statistics.- 1.5. Void Point-Surface Statistics.- 1.6. Variational Principles Applied to the Diffusion Equation...- 1.7. Notation.- 2. Diffusion Through a Porous Medium.- 2.1. Introduction.- 2.2. Diffusion Through an Isotropic Porous Medium...- 2.3. Variational Formulation for De.- 2.4. Bounds on De for an Isotropic Suspension.- 2.5. Best Possible Bounds on De.- 2.6. Bounds on De for a Random Bed of Solid Spheres...- 2.7. Knudsen Diffusion Through a Porous Medium.- 2.8. Variational Formulation for Knudsen Diffusion...- 2.9. Upper Bounds on the Knudsen Permeability.- 3. Diffusion Limited Reaction.- 3.1. Introduction.- 3.2. Diffusion Limited Precipitation.- 3.3. The Spherical Cell Approximation.- 3.4. Variational Principle and Upper Bound on the Precipitation Rate.- 3.5. Diffusion Controlled Quenching.- 3.6. Upper Bound on ke.- 3.7. Lower Bound on ke.- 3.8. Random Quencher Particles.- 4. Heterogeneous Catalysis.- 4.1. Introduction.- 4.2. Variational Principles for Heterogeneous Catalysis: the Homogeneous Model.- 4.3. Application to First Order Kinetics in a Slab.- 4.4. Application to Langmuir-Hinshelwood Kinetics in a Slab..- 4.5. Effects of Pellet Shape on the Effectiveness Factor...- 4.6. Variational Principles for Heterogeneous Catalysis: the Discrete Model.- 4.7. Application to Linear Kinetics in a Slab: Discrete Model.- 4.8. Analysis of Experimental Data.- 4.9. Non-monotonic Kinetics.- Notation.- Author Index.

Poiseuille flow at arbitrary Knudsen numbers and tangential momentum accommodation

Abstract: A Poiseuille-flow problem in a cylindrical capillary in the whole range of Knudsen numbers with incomplete tangential momentum accommodation of molecules incident on the wall has been worked out. The linear non-homogeneous integral equation for the macroscopic gas velocity flow has been solved by the Bubnov-Galerkin method. For a limited range of Knudsen numbers, generally known results have been obtained.An experimental investigation of the rare gases helium, neon and argon in the range of Knudsen numbers 103−10−3 has been made on packets consisting of 10 and 100 glass capillaries with molten walls. Comparison of theoretical and experimental data enables us to define both slip constants and tangential momentum accommodation coefficients. In the free-molecule flow regime the accommodation coefficients are 0·935, 0·929 and 0·975 for helium, neon and argon, respectively. In the viscous slip-flow regime these coefficients are equal to 0·895, 0·865 and 0·919, respectively. This difference in the tangential momentum accommodation coefficients is, most probably, due to the variable density of adsorbed molecules coating the capillary wall. Gas viscosity coefficients which coincide with those of Kestin within 0.5% have also been calculated. Argon was used as the calibrating gas.

Photophoretic force in the free molecule and transition regimes

Abstract: The photophoretic force of a black body spherical particle is analysed in the free molecule regime. The analysis is more exact than previous theories and provides a check on Hidy and Brock's [J. Geophys. Res. 72, 455 (1967)] approximate result. A transition regime solution for an insulated particle is obtained by the direct simulation Monte Carlo method. This shows that the photophoretic force for a particle of fixed size and composition under constant illumination intensity has a maximum value at a Knudsen number of the order of unity. Schmitt's [Vac. Tech. 10, 238 (1961)] experiment results are discussed briefly.

Poiseuille flow and thermal creep of a rarefied gas between parallel plates

Abstract: A numerical solution of the Boltzmann equation with the Bhatnagar‐Gross‐Krook model is obtained for Poiseuille flow and thermal creep of a rarefied gas between two parallel plates. The numerical results of Poiseuille flow are in fair agreement wth the experimental data of Dong and predict the Knudsen minimum in volume flow rate. The present results of the thermal creep approach a thermal transpiration formula analyzed on the basis of the elementary kinetic theory of gases as the density of the gas decreases and agree well with Maxwell's equation continuum theory in the range of low Knudsen number. The effect of the thermal creep on the velocity profile of Poiseuille flow is very large, especially at the Knudsen layer near the plate.

Heat transfer between parallel plates with arbitrary surface accommodation

Abstract: Elementary solutions of the coupled pair of integrodifferential equations arising from the decomposition of the linearized BGK equation in the kinetic theory of gases are used to solve the problem of heat transfer between parallel plates with arbitrary surface accommodation. A coupled pair of Fredholm equations is derived, and rapidly convergent iterative solutions are constructed. These solutions are then used to obtain accurate values of the heat flux between the plates and the temperature and density profiles, for various values of the accommodation coefficient and inverse Knudsen number. Numerical results for the heat flux are presented and compared to existing variational solutions. Also, explicit results for the temperature and density profiles are given.

Knudsen effusion study of silica

Abstract: Knudsen effusion experiments with SiO2(β-cristobalite) on a thermomicrobalance have been carried out over the temperature range 1823–1983 K. Interpretation of the effusion data is based on the following reactions: (a) SiO2(s)→ SiO(g)+½ O2(g), and (b) O2(g)→ 2O(g). Third law calculations give the standard Gibbs free energy of the reaction (a) as ΔGT= 182 050–58.29 T± 2500 cal mol–1 in the measured temperature range, the heat of formation of SiO(g) as ΔH°298=–27.9 ± 3.5 kcal mol–1, and the dissociation energy of SiO(g) as D°0= 8.45 ± 0.20 eV. Determination of the vaporization coefficient of SiO2(s) gives αv=(2.2 ± 0.8)× 10–2 in the experimental temperature range.

Reaction with mole changes in porous catalysts in the molecular, transition, and Knudsen regimes

Abstract: Equations are derived and solved numerically for the intrapellet pressure and composition profiles in porous catalysts. The extended dusty-gas model is used to describe the transport for a zero-order, irreversible reaction with mole changes. Results agree with other models in the purely Knudsen and molecular regimes. In the transition region, however, the effectiveness factor is a function of five dimensionless parameters. The calculations predict a significant pressure gradient in the Knudsen regime and transition region but an insignificant gradient in the molecular regime for reactions with mole changes. Since many industrial catalysts operate in the transition region of transport, these calculations are highly desirable in reactor design.

Some remarks about dispersion and absorption of sound in monoatomic rarefied gases

Abstract: In the present paper we consider sound propagation in rarefied monoatomic gases from the point of view of a previously proposed modified Navier-Stokes equation. The agreement between theory and experiment is rather good for sound dispersion and absorption in the region where the ratio between the mean free path and the wavelength lies within the limits 0.1 and 1. We also discuss the asymptotic behaviour for high-frequency values of the sound dispersion and absorption and suggest that further measurements in the Knudsen regime of sound dispersion (neglecting absorption) would be useful to test the Validity of sound wave as well as free-molecule propagation theory.

Asymptotic theory of the Boltzmann equation at large Knudsen number

Abstract: A study of an asymptotic theory of hyperthermal flow past a blunt object for large Knudsen number is presented. The greatest demands on such a theory are present for flow past two‐dimensional bodies. This is because the large lateral extent of the emitted molecules in free molecular flow gives rise to a singularity in the integrated collision frequency of the incoming free stream molecules. This singularity is removed by accounting for the effects of collisions upon the emitted molecules which augments the usual geometric free molecular decay. The method of matched asymptotic expansions is applied to regions near the body and far from the body (one mean free path and greater). For two‐dimensional bodies, the surprising result is obtained that the first correction to free molecular flow [Kn−1 ln(Kn)−1] cannot be obtained without explicitly taking into account the collisional effects on the emitted particles in the far region. For three‐dimensional bodies, although there is no problem in the determination of the leading term (Kn−1), the evaluation of the next order term [Kn−2 ln(Kn)−1 involves precisely the same difficulties as the leading term for two‐dimensional bodies.

Drag coefficients and the generalized Boltzmann equation

Abstract: There is a close relation between the Knudsen number dependence of the drag coefficient for a macroscopic body in a gas stream, and the density dependence of the transport coefficients for moderately dense gases in both two and three dimensions.

Equations for Thermal Transpiration

Abstract: A formalism for analytically obtaining an expression for the thermal transpiration pressure ratio R is presented. An experimental parameter σ, which is associated with the type of molecule-solid surface collisions, is introduced. A completely diffuse scattering and a completely specular scattering from a solid surface correspond to σ = 0 and σ = 1, respectively. A known distribution function is used to derive a practical formula for R in the case of long tubes and very low pressures. Quantitative results obtained from this formula indicate that deviations from completely diffuse scattering of molecules from solid surfaces give rise to an anomalous Knudsen limit.

Comparative study of accommodation coefficients of helium and neon on clean tungsten under transition, temperature jump, and free molecule conditions

Abstract: Measurements of heat conduction by neon and helium between concentric cylinders under free molecule, transition, and temperature jump conditions have been made with particular care to insure that a well defined clean surface state of the inner (tungsten) cylindrical surface was maintained. Several kinetic theoretical treatments applicable to the various regimes have been utilized to calculate the thermal accommodation coefficients of these gases. Comparison of these accommodation coefficient values obtained by analysis of the data in the transition and temperature jump regimes by the Smoluchowski‐Kennard temperature jump method, the Langmuir‐Wachman mean free path method, and the Lees‐Liu moment method with the accommodation coefficient values obtained on the same filament by the well established low pressure Knudsen method has been made, and the values are in remarkable agreement.

A kinetic theory, near-continuum model for membrane transport.

Abstract: Continuum flow theory (Poiseuille flow, Fick diffusion) is of dubious validity for small pores. For gases, kinetic theory permits an extension of continuum theory to Knudsen near-continuum slip-flow, by means of a change only in boundary conditions. The range of validity of near-continuum gas flow theory is for channel widths larger than a small number of mean free paths. Here a comparable kinetic theory extension of continuum theory has been developed for liquids. A “saltation,” near rolling, boundary condition is shown to be applicable. The range of validity of this near continuum liquid flow theory is for channels wider than about five molecular diameters (i.e., 15 A for water). Predictions of the theory are tested against the well calibrated data of Beck and Schultz (1972) as well as data from biological membranes. The results are compared with the continuum theory used by Renkin (1954) and others.

Brownian coagulation of aerosols at low Knudsen number

Abstract: Dibutyl phthalate aerosols of narrow size distribution have been prepared in a falling-film generator using nitrogen rather than helium as the carrier gas. The Knudsen number in nitrogen is considerably lower so that the Cunningham correction is much less. A large number of coagulation experiments give average coagulation times in excellent agreement with theory. However, the spread of the values is much greater than obtained earlier in helium. The spread of the results reported here may be due to the lower thermal conductivity of nitrogen.

Rarefied gas flows based on variational principle

Abstract: A variational principle has been utilized to study Couette flow, and Kramers' velocity slip problem with specular-diffuse reflection. The method leads to extremely satisfactory analytic results for the velocity slip coefficient and also for the variation of shear stress with inverse Knudsen number. This leads us to the conclusion that the present variational principle even with extremely simple trial functions, essentially suggested by continuum flow theory, is a useful means of computing macroscopic quantities of physical interest in rarefied gas dynamics.

Impact of a freely expanding gas on a wall

Abstract: A study is made of the flow which results when an initially uniform gas expands into a vacuum through a complete one‐dimensional rarefaction wave, and then strikes either a solid wall or a similar opposite facing rarefaction. A Knudsen number K n is defined by the ratio of the undisturbed mean free path to the distance between the center of the rarefaction and the wall. The direct simulation Monte Carlo method has been used to obtain numerical solutions over a range of K n from 10−4 to 103. Results for the wall pressure profile were also obtained from experiment and from approximate continuum calculations. The pressure profile on the wall or the line of symmetry was found to be insensitive to assumptions about the initial highly rarefied stage of the reflection. On the other hand, when considering the temperature history of the gas adjacent to the wall, a proper molecular analysis of the initial stage of reflection was required for all values of the Knudsen number.

Cylindrical Poiseuille flow and thermal creep of a rarefied gas

Abstract: The rarefied gas flow in a circular tube whose wall is nonuniform in temperature is analyzed using the Maxwell moment method, with special reference to thermal creep. The results obtained are in fair agreement with both Sone and Yamamoto's and the calculations worked out at the low Knudsen number by the elementary kinetic theory of gases.

The Axial Flow Molecular Pump : 4th Reprot, Performance of a Rotor with a Single Blade Row in the Transition Flow Regime

Abstract: As a forevacuum pump of an axial flow molecular pump, a diffusion pump is commonly used. In this case, the pressure in the forevacuum side of the molecular pump is kept below 10-4 mmHg, and therefore a free molecule flow takes place in all the blade rows. Occasionally, only a rotary pump is used as the forevacuum pump. In this case, the pressure in the forevacuum side is 10-3∼10-2 mmHg and collisions between gas molecules occur as frequently as between gas molecules and blade plates of some rows placed near the forevacuum side. This state is called the transition flow regime. The performance of a rotating blade row in this regime is studied theoretically and experimentally. The theoretical performance agrees with the experimental results when the Knudsen number is larger than 5 to 1, but the difference between the theoretical and experimental results increases as the Knudsen number decreases.

Relaxation of an alkali-metal halide aerosol in a shock wave

Abstract: The interaction of a shock wave with an alkali-metal halide aerosol has been studied over a temperature range of 900–3600 K. The particle velocity relaxation and evaporation of aerosols of known size distribution were monitored with a light transmission technique.Times for complete velocity relaxation were found to be comparable with total evaporation times. At temperatures above 1550 K, the total evaporation times were independent of the gas temperatures. At temperatures below 1550 K, an activation phenomenon was observed in the evaporation times.Results were analysed using theoretical expressions for particle drag, heat transfer and particle evaporation. Expressions for the transition region in Knudsen number gave the best agreement with experiment.

Mechanisms of fibrous filtration

Abstract: This thesis is concerned with the removal of particulate material from gases using fibrous filters. described under three headings:- (a) The Collision Efficiency The particle fibre collision efficiency is calculated by computing trajectories in the Davies and Kuwabara flow fields. Electrostatic and gravitational field forces are taken into account. The influence of fibre Knudsen number and Reynolds number on inertial interception is predicted. A model is described which takes into account a log normal distribution of fibre spacing in a filter. It is used to predict the pressure drop across a random fibre mat and its mean efficiency of inertial interception both of which are a factor of two or three less than predicted by the simple Kuwabara model. (b) Particle Retention Mechanisms' It is shown that bounce is the only significant mechanism responsible for particle non-adhesion in fibrous filters. An equilibrium model is used to predict the critical particle size above which adhesion fails. The behaviour of filters in the low adhesion region is examined by measuring the collection efficiency of model filters using narrow sized fractions of dust. The efficiency is a decreasing function of particle size and velocity, trends which agree with the equilibrium model. (c) Non-stationary Filtration The behaviour of filters under load is examined experimentally. The efficiency may either increase or decrease initially with loading, the characteristics depending on the same factors which influence the single fibre efficiency.

Monte Carlo Simulation of Ion Collection by a Rocket-Borne Mass Spectrometer for Collisionless and Transitional Flowfields.

Abstract: : Monte Carlo calculations of ion collection by a rocket-borne mass spectrometer have been completed for the collisionless case for the following range of parameters: Debye number 0.01 to 1000, speed ratio 0.1 to 3.1 and nondimensional potential -10 to -200. Calculations made in the transition regime show that minimum altitude at which the free molecule flow assumption is valid is approximately 120 km. A calculation at 70 km corresponding to a Knudsen number 0.007 indicates that the flowfield can be characterized as near-continuum but one in which the rarefaction effects are still important. The study demonstrates that the Monte Carlo technique is an efficient method to determine ion collection over the complete range of the flight parameters: Knudsen number, Debye number, plate potential, and speed ratio. (Modified author abstract)