Showing papers on "Knudsen number published in 1987"

Analysis of Ultra-thin Gas Film Lubrication Based on the Linearized Boltzmann Equation : Influence of Accommodation Coefficient

Abstract: A generalize Reynolds-type lubrication equation valid for both arbitrary Knudsen numbers, defined as the ratio of the molecular mean free path to the film thickness, and arbitrary accommodation coefficients at boundaries, is derived form a linearized Boltzmann equation. Numerical analyses of lubrication characteristics through the equation for high Knudsen numbers reveal that, if the accommodation coefficient is less than 1, that is, if not all the molecules reflect diffusely as is the case with real gases, load carrying capacities are smaller than those for diffuse reflection.

Steady gas flows past bodies at small Knudsen numbers

Abstract: Steady gas flows at small Knudsen numbers around arbitrary bodies (Asymptotic behavior for small Knudsen numbers of the solution of time-independent boundary value problems of the Boltzmann equation over a general domain) are considered when the Reynolds number of the system is of the order of unity. The generalized slip flow theory developed for the Boltzmann-Krook-Welander equation is extended for the standard Boltzmann equation. From the result, the effect of gas rarefaction on the flow (the relation between Boltzmann and hydrodynamic systems) is clarified, and several features of the force on a closed body in the gas are derived.

Kinetic theory of binary gas mixtures with large mass disparity

Abstract: The Boltzmann equations for a binary mixture of gases are considered in the asymptotic limit when their molecular weight ratio and the light gas Knudsen number are small quantities. A first mass‐ratio expansion reduces the cross‐collision operator of the light gas Boltzmann equation to a Lorentz form, uncoupling its kinetic behavior from that of the heavy gas. The light gas distribution function is then determined to first order in the Knudsen number, independently of the degree of nonequilibrium characterizing the heavy gas, whose influence is felt only through its hydrodynamic quantities. All transport coefficients arising are determined variationally for arbitrary interaction potentials using Sonine polynomial expansions as trial functions. A remarkable feature of this analysis is that it yields binary transport information (i.e., diffusion and thermal diffusion coefficients) from considering only the Boltzmann equation for the light gas. A second mass expansion reduces the cross‐collision operator of the heavy gas equation to a Fokker–Planck form. The corresponding coefficients involve integrals over the light gas distribution function determined previously and are evaluated explicitly in terms of the hydrodynamic quantities and transport coefficients of the light gas. The heavy gas distribution function can be determined by solving a Fokker–Planck equation at dilutions large enough to make heavy–heavy collisions negligible, or by a new Knudsen number expansion when the molar fraction of the heavy gas is of order 1. In this latter case, the heavy gas kinetic behavior is independent of the light gas, being characterized by the same transport coefficients of the pure heavy gas. The problem is then reduced to a set of two‐fluid hydrodynamic equations.

Temperature field induced around a sphere in a uniform flow of a rarefied gas

Abstract: The temperature variation induced around a sphere with a uniform temperature in a slow uniform flow of a rarefied gas with the same temperature as the sphere is analyzed on the basis of the linearized Boltzmann–Krook–Welander equation with diffuse or specular reflection as the boundary condition. The analytic expressions of the temperature field are obtained for small and large Knudsen numbers.

Monte Carlo Simulation of Rarefied Flow Along a Flat Plate

Abstract: The direct simulation Monte Carlo technique is applied to a diverging flowfield along a flat plate. The model accounts for a plate of finite thickness with a nonzero leading edge bevel angle, and permits the specification of surface accommodation coefficients ranging from diffuse to specular reflection. The computed density, velocity, and temperature profiles at several axial stations along the plate are compared with experimental data. These comparisons indicate good agreement when the leading edge bevel is included in the calculation, and when the surface accommodation coefficients are each assigned a value of 0.8. Results are also presented that indicate the sensitivity of the computed skin friction to the ratio of the computational cell size to molecular mean free path. The results indicate that the skin friction is insensitive to the cell size if the local Knudsen number based on the cell size is maintained larger than unity everywhere in the flowfield. The results also indicate that the skin friction is underpredicted toward the plate leading edge and overpredicted at downstream axial positions when the cell size exceeds the molecular mean free path.

Motion of a spherical particle in a rarefied gas. Part 1. A liquid particle in its saturated vapour

Abstract: The results of a theoretical investigation of the low-speed motion of a volatile spherical particle in its saturated vapour at arbitrary values of the Knudsen number, evaporation coefficient and tangential momentum accommodation coefficient are presented. The problem is solved on the basis of the Bhatnagar, Gross & Krook (1954) linearized kinetic equation with the model collision integral. The variational method developed for the solution of the axisymmetric problems (Cercignani & Pagani 1968) is employed to calculate the drag. Phase changes on the particle surface have been shown to reduce the drag by an order depending on the Knudsen number. Incomplete accommodation of molecules colliding with the particle also results in the reduction of the drag over the whole range of Knudsen numbers.

Calibration of a Fenn‐type nozzle beam source

Abstract: Calibration of a Fenn‐type nozzle beam source and the limitations due to background attenuation, skimmer interference, and condensation are discussed. The nozzle flow rate Nn is calculated, and the peaking factor κ is determined from both radial pressure surveys and effusive‐to‐supersonic transition measurements. Stage pressure measurements verify both Nn and κ. These quantities specify the ideal beam flux in the absence of attenuation, interference, or condensation. Background attentuation depends on the effective scattering cross section, which can be quite large for finely collimated beams. Serious skimmer interference occurs below a critical value of the skimmer Knudsen number and depends on individual skimmer details. Condensation is observed and found to be predictable according to the known scaling laws. A calculation of absolute beam fluxes is presented. Nonideal behavior of the speed ratio and average particle velocity are also examined. Data are given for the gases H2, He, Ne, and Ar.

High Knudsen Number Molecular Rarefaction Effects in Gas-Lubricated Slider Bearings for Computer Flying Heads

Abstract: Effets de rarefaction moleculaire avec nombre de Knudsen eleve dans une glissiere a gaz (tete oscillante d'ordinateur)

Dynamics of Particle—Shock Interactions: Part I: Similitude

Abstract: The equation of particle motion is normalized for either relaxation behind a shock front or particle impaction from supersonic streams. Algebraic expressions are introduced for the drag coefficient in both cases to account for non-Stokesian drag and free molecular flow. Expressions are developed for the particle Knudsen, Reynolds, Mach, and Stokes numbers and for the ratio of the local gas-to-particle density and maximum particle drag. It is demonstrated that knowledge of the particle stagnation Knudsen number, local gas Mach number, and specific heat ratio are sufficient to predict the particle Knudsen, Reynolds, and Stokes numbers and the maximum drag behind the shock front. Graphs are also provided to estimate these important particle parameters.

Two‐fluid Chapman–Enskog theory for binary gas mixtures

Abstract: As a result of the increasing inefficiency in the transfer of energy in collisions between species with a decreasing ratio of molecular masses, the Knudsen number range of validity of the Chapman–Enskog (CE) theory for binary gas mixtures decreases linearly with the molecular mass ratio. To remedy the situation, a two‐fluid CE theory uniformly valid in the molecular mass ratio is constructed here. The analysis extends previous two‐fluid theories to allow for arbitrary potentials of intermolecular interaction and arbitrary mass ratios. The treatment differs from the CE formulation in that the mean velocities and temperatures of the two gases are not required to be identical to lowest order. To first order, the streaming terms of the Boltzmann equation are thus computed in terms of the derivatives of the two‐fluid hydrodynamic quantities, rather than the mean mixture properties as in the CE theory. As a result, associated with the nonconservation of momentum and energy for each species alone, two new ‘‘driving forces’’ appear in the first‐order integral equations. The amount of momentum and energy transferred per unit time between the species appear in the theory as free constants, which allow satisfying the constraint that all hydrodynamic information be contained within the lowest‐order two‐fluid Maxwellians. Simultaneously, this constraint fixes the rate of momentum and energy interchange in terms of the two‐fluid hydrodynamic quantities and their gradients. The driving force d12 of the CE theory is directly related to the rate of interspecies momentum transfer, and the corresponding CE functions D1 and D2 appear here unmodified.But the physical interpretation of d12 is very different in the two pictures. On the CE side there is only one momentum equation, while d12 provides constitutive information fixing the diffusion flux (velocity differences) in the mass conservation equation. Here, the similar constitutive information associated to d12 is used to couple two different momentum equations. Although the CE theory captures some of the two‐velocity aspects of the problem, no CE analog exists with the functions E1 and E2 associated here with temperature differences, which now require solving new integral equations. Finally, the presence of two velocities and two temperatures leads to four coefficients of viscosity and of thermal conductivity for the two stress tensors and heat flux vectors. Also, two thermal diffusion factors enter now into the expression for d12. Although all these new coefficients arise as portions of the overall CE transport coefficients, their independent optimal determination requires new developments. The corresponding variational formulation is presented here and used to first order to obtain explicit expressions for all two‐fluid transport coefficients by means of Sonine polynomials as trial functions.

Chapter 2: Long Mean Free Paths in Quantum Fluids

Abstract: Publisher Summary This chapter discusses a number of phenomena, in which boundary scattering plays an important role in the behavior of quantum liquids and solids. The phenomena are generally characterized by the mean free path being comparable to other characteristic lengths. To illustrate the limitations of hydrodynamics and to introduce the concepts of fluid slip and Knudsen flow, an elementary example, the motion of a spherical object through a viscous medium is considered. The chapter introduces the slip approximation and applies it to a variety of flow problems of experimental interest. The slip length is calculated microscopically, for both classical and quantum systems, starting from a Boltzmann equation for the particle distribution function. Flow at arbitrary Knudsen numbers is presented in the chapter and frequencies comparable to or greater than the microscopic relaxation rates are considered. The role of different types of boundary scattering is discussed and the comparison with experiment is carried out in the chapter. The main emphasis is on the quantum fluids 3He and 4He as well as mixtures of 3He and 4He.

Knudsen-effect errors with transient line-source measurement in fluids

Abstract: This paper decribes the Knudsen-effect errors of the transient line-source method used for accurate measurements of the thermal conductivity and thermal diffusivity of fluids. The analysis demonstrates that the instrument can be used with a good accuracy (>0.5%) to lower densities than previously thought. The principal errors are illustrated by measurements on propane in the temperature range 250–300 K at densities less than 9 kg · m−3.

Determination of the Dissociation Energy of the Cr2 Molecule.

Abstract: A Knudsen cell-mass spectrometer system has been developed which allows the determination of extremely small partial pressures.

Method for increasing quantity of compressed gaseous body

Abstract: PURPOSE:To increase the quantity of a compressed gaseous body with a simple construction and without need for using a rotary part and increasing power by introducing the gaseous body into said compressed gaseous body from a low pressure side, using a Knudsen diffusion effect CONSTITUTION:A gaseous body 2 which is compressed by a compressor and reached a high temp, is discharged from an ejector nozzle 3, and fed into an ejector horn pipe 5 The ejector nozzle 3 and the inlet of the ejector horn pipe 5 are covered by a Knudsen film 1 This Knudsen film 1 is made of a resin film, a ceramic, etc, and makes the gaseous body pass through it and diffused from a low temp and low pressure side to a high temp and high pressure side, when there is difference in temp between its inside and outside And, by using the diffusion effect of this Knudsen film 1, the gaseous body is introduced from the outside of the Knudsen film 1 into its inside, and sucked into the ejector horn pipe 5, to increase the quantity of the compressed gaseous body

The Influence of a Plane Boundary on the Thermophoretic Force in the Knudsen Regime

Abstract: The effect of a plane diffusely reflecting wall on the thermophoretic force acting on a particle in the Knudsen regime is investigated. A general expression for the force on a particle in a nonuniform gas is given and then the velocity distribution function for the temperature jump problem is inserted. This results in an expression for the thermophoretic force that is a function of the distance of the particle from the wall. At large distances from the wall, the classic results of Waldmann and Schmitt are obtained, but near the wall there is a small reduction of about 1% in this value. The wall correction is independent of the gas-particle interaction, at least for specular, diffuse, Lambert, and backward scattering. The temperature jump problem has been solved using a collision cross section that is independent of velocity. This model, with diffuse reflection at the wall, predicts that the effect of the wall on the thermophoretic force becomes negligible at distances greater than a few mean free paths from the surface.

Accurate measurement of gas lubricated slider bearing separation using visible laser interferometry.

Abstract: Highly accurate (minimum measuring separation 0.025 μm, accuracy 0.002μm), wide frequency range (dc to more than 100kHz) measurement of the separation of gas lubricated slider bearings is presented using visible laser interferometry. The new technology for realizing this measurement includes ; 1) elimination of undesirable interference patterns by adopting optics that focus the laser beam into a small cavity, 2) elimination of laser tube resonance noise, 3) automatic compensation for variations in the optical parameters. Experiments adopting this measurement are performed in an ambient helium atmosphere and the results are compared with numerical solutions based on the modified Reynolds equation. The experimental and numerical results have excellent agreement within the range of separation down to 0.025μm, and with Knudsen numbers as high as 8. As a result, it is confirmed that the modified Reynolds equation can be used to predict slider bearing characteristics accurately within the experimental range of separation and Knudsen numbers.

An Analysis and a Consideration of Molecular Flow in a Long Conduit Pipe

Abstract: The old problem on the molecular flow conductance in a long conduit pipe with a uniform cross sections was analyzed by using a concept of gas diffusion process in a uniform medium. This problem was first treated by Knudsen giving a famous approximate expression for the molecular flow conductance and, after a year, Smoluchowski has given a rigorous expression by summing up motions of individual molecules. In this report, a trial was made to understand molecular flow in a conduit pipe as one of the diffusion processes. It turned out that the well known expression of diffusion coefficient (D=1/3·υλ) did not apply to the problem. A careful consideration has given a more rigorous expression of diffusion coefficient, i.e.D=1/2·υ (l cos θ) 2/lwhere l and l2 are the average and average square of free paths of gas molecules and θ is the angle between l and molecular density gradient. By using the above expression, one can treat the molecular flow as an diffusion process and can obtain the regorous expression of molecular flow conductance.

Theory of thermomagnetic effects in the transition regime

Abstract: A simple model is presented that describes the complex behavior of the magnetic field dependence of the thermal conductivity of a polyatomic gas as a function of the Knudsen number. The model treats the solid surface as one component of a gas mixture, and a first‐order Chapman–Enskog solution is adequate to account for the thermomagnetic effects in single gases. The results provide an interpolation scheme between the continuum and Knudsen regimes, in the form of scaling rules in which the Knudsen number appears in the coefficients and arguments of the functions that describe the limiting cases. Good agreement is obtained with the available experimental data, and leads to the conclusion that different surface accommodation coefficients are needed for the translational and internal energies of the gas molecules. This conclusion is consistent with analysis of independent field‐free conductivity data, and with independent vibrating‐surface measurements.

Boundary kinetic effects in the problem of slow flow over an evaporating heat-conducting spherical particle

Abstract: It is shown that for sufficiently large values of the thermal conductivity of the condensed phaseλ' as compared with the thermal conductivity of the vaporλ (λ/λ' ∼ Kn) the effects associated with the presence of a Knudsen layer on the evaporating surface must be taken into account in order to obtain a solution of the problem of a spherical particle in a slow (Re∞, ≪ 1) continuum (Knudsen number Kn ≪ 1) flow of its own vapor. The drag is calculated for various types of boundary conditions on the particle surface.

Magnetic field effects on a molecular flow at Knudsen numbers of order unity for CO and CH4

Abstract: Magnetically induced changes in the flow rate of CO and CH 4 are investigated experimentally in the intermediate pressure regime (Knudsen numbers between 0.1 and 10) for the three main field orientations. From the results a qualitative understanding of the role of the various angular momentum polarizations involved can be obtained.

Vapour pressure and thermodynamics of the indium determined by the Knudsen effusion method

Abstract: Mesure par une methode de perte de masse entre 1096 K et 1390 K de la pression de vapeur de l'indium liquide. Obtention d'une equation dependante de la temperature

Knudsen Flow and Effective Shear Viscosity of Liquid3He

Abstract: We present new torsional oscillator data on the shear viscosity of liquid 3He at low temperatures. They are analyzed with a theory of Poiseuille flow, which is also applicable in regimes of temperature and pressure where the Knudsen number, characterizing the flow of the thermal excitations, may become large.

Cohesive Energies of Transition Metal Silicides and Phosphides.

Abstract: : Vaporization studies have been carried out by mass loss Knudsen effusion on the Ni-P and chromium-phosphorus systems and by Knudsen effusion mass spectrometry on the V-Si, Cr-Si and Ti-Si systems. The experimental data were used to calculate enthalpies of formation of the intermediate phases in these systems. The stabilities of the 1:1 compounds, expressed as enthalpies of atomization, have been compared with each other and with data on silicides, phosphides, and sulfides from the literature to assess factors for inclusion in a phenomenological model for stability. These factors include a) the number and type of bonding electrons, b) effective nuclear charges, c) valence state energies, d) the degree of partial electron transfer with a resulting Made lung energy, and e) contributions from Lewis acid-base interactions involving back-donation of electrons. Keywords: Transition metal silicides, Transition metal phosphides, Mass loss knudsen effusion, Knudsen effusion mass spectrometry, Vaporization behavior, Congruent vaporization, Thermodynamic activities, Phase equilibria, Free energy of formation, Enthalpy, Formation, Stability.

Mass transfer in a plane finite pore on a broad interval of Knudsen numbers with allowance for condensation

Abstract: The flow field and the flow rate of a gas in a two-dimensional plane pore of finite length are found numerically over a broad interval of Knudsen numbers with allowance for condensation on the walls.

The asymptotic properties of analytical solutions of the non-linear Boltzmann equation

Abstract: The Cauchy problem for the non-linear Boltzmann equation is considered The initial distribution is assumed to be fairly close to an equilibrium one and analytical with respect to the spatial variable For small Knudsen numbers an approximate solution is constructed which differs from the well-known locally Maxwellian solution in its correction, which guarantees the uniform asymptotic accuracy in a fixed closed segment of time which includes the initial layer