Showing papers on "Knudsen number published in 1971"

Heat Transfer in Rarefied Gases

Abstract: Publisher Summary This chapter presents the results of the recent analytical and experimental investigations of heat transfer in rarefied gases. It concentrates on the heat transfer through the electrically neutral nonreacting gases. The effects of ionization, dissociation, etc., that may arise in high-speed high temperature gas flows are not considered. The term “rarefied” means that the molecular mean free path is not small compared to a characteristic dimension . The parameter that describes the degree of rarefaction is the Knudsen number. In defining the Knudsen number, it is important to select the appropriate characteristic mean free path and length. When the Knudsen number is very small, then, in the vicinity of the body, the number of collisions between the molecules is large compared to the number of collisions between the molecules and the body. In this case, the usual continuum concepts are applicable and the Navier-Stokes equations and the Fourier heat conduction law are valid. When the Knudsen number becomes sufficiently large, then the continuum concepts must be modified for calculating the heat transfer. At very high Knudsen numbers where the number of collisions between the molecules and the wall is much larger than the number of collisions between the molecules, the flow is termed “free molecule.”

Pressure characteristics of fibrous aerosol filters

Abstract: Theoretical equations for the pressure drop of high-porosity fibrous aerosol filters for low-speed Newtonian flow are summarized, and the relations between them are discussed. Using these equations the general character of the dependence of the pressure drop on the gas pressure (pressure characteristics) is analyzed over the whole range of Knudsen numbers. The dimensionless equations describing the reduction of the pressure drop with decreasing gas pressure are derived. Analyzing these equations, it is concluded that the pressure-drop reduction is dependent on three factors: the character of flow through a filter at normal pressure (continuum, slip, transient and free molecule flow), the dilution (reduction of the gas pressure), and porosity of the filter. The theory is checked against available experimental data.

Thermal Creep Flow of Rarefied Gas between Two Parallel Plates

Abstract: The thermal creep flow of a rarefied gas between two parallel plates is investigated on the basis of a relaxation model of the Boltzmann equation for all range of the Knudsen number. The velocity profile as well as the volume flow rate is calculated

Effect of gas slip on the pressure drop in a system of parallel cylinders at small reynolds numbers

Abstract: The dependence of pressure drop across a model filter on the Knudsen number has been studied experimentally and theoretically for a simplest fiber filter model—a system of parallel cylinders arranged perpendicular to the flow. It has been shown that the reciprocal value of the drag on the fibers increased linearly with the Knudsen number.

The rate of quasi-steady growth and evaporation of small drops in a gaseous medium

Abstract: The intensity of steady diffusion flux onto a sphere absorbing particles with a known probability is determined at arbitrary Knudsen number. New expression for the concentration jump is given. The accuracy of the well known δ-method developed byN. A. Fuchs is estimated. The formula is derived for the rate of slow condensation growth (or evaporation) of drops of any size in a vapour-gas mixture at low vapour content. The results of different authors are compared.

Pipe flow of suspensions in turbulent fluid

Abstract: The general case of a fully developed pipe flow of a suspension in a turbulent fluid with electrically charged particles or with significant gravity effect, or both, and for any inclination of the pipe with the direction of gravity, is formulated. Parameters defining the state of motion are: pipe flow Reynolds number, Froude number, electro diffusion number, diffusion response number, momentum transfer number and particle Knudsen number. Comparison with experimental results is made for both gas-solid and liquid-solid suspensions. It is shown that the gravity effect becomes significant in the case of large pipe diameters and large particle concentrations.

Velocity Distribution Measurements in a Near-Free Molecule Orifice Flow

Abstract: Measurements have been made of the velocity distribution of the flow of cesium gas from an orifice in the near‐free molecule regime. The apparatus design is based on the time‐of‐flight method using a single disk chopper and a surface ionization gauge. Measured velocity distributions are compared with the corresponding Maxwellian distribution adjusted to have the same peak amplitude. Although the two distributions agree closely at high Knudsen numbers (i.e., Kn>10), the comparison at smaller Knudsen numbers indicates that the measured distribution is deficient in low‐speed molecules. This deficiency increases with decreasing Knudsen number and with decreasing angular position from the centerline of the orifice. The experimental results are compared with the predictions of existing analyses of near‐free molecule flow through an orifice.

Cylindrical Couette Flow Experiments in the Transition Regime

Abstract: Density distributions were measured in rarefied argon contained between two concentric cylinders, the inner one rotating, the outer one stationary. The experiments were performed with a Mach number near unity, based on the surface speed and surface temperature of the rotating cylinder. Particular attention was focused on obtaining data in the slip and transition regimes where the Knudsen number, defined as the ratio of the mean free path to the gap size between the cylinders, varied from 0.04 to 1.07. The density distributions were measured by observing the gas luminescence induced by the passage of a narrow beam of high energy electrons through the gas. In addition to the density measurements, heat transfer and drag measurements were also made in order to estimate the values of the thermal accommodation and the tangential momentum accommodation coefficients. The experimental results were compared to solutions of the Navier‐Stokes and Burnett equations as given by Schamberg and Lin and Street. The results of the Navier‐Stokes and the Burnett equations were found to approximate the density distributions well at Knudsen numbers below ∼0.05 and ∼0.2 , respectively. At higher Knudsen numbers the analytical and experimental results differ considerably.

Gaseous Self‐Diffusion through a Porous Medium

Abstract: The mean free path theory of self‐diffusion of “tagged” molecules is formulated for arbitrary Knudsen number (mean free path to average pore diameter ratio) in terms of two coupled integral equations. A variational upper bound on the net rate of diffusion through a porous slab is derived. An upper bound on the diffusion rate is calculated for a model pore structure generated by randomly placed, freely overlapping, solid spheres.

Atomic and molecular beam scattering from macroscopically rough surfaces

Abstract: The scattering of thermal energy atomic and molecular beams from macroscopically rough surfaces has been found to display scattered flux distributions that are typically skewed towards the incoming, beam i.e., backscattered. Such an observation is consistent with the results reported in the literature describing flow-conductance measurements, which are less than the conductance calculated assuming the Knudsen (diffuse) scattering law. Two types of surfaces were studied in the present work: shot-blasted surfaces and gold-blacked surfaces. Both these surfaces displayed scattered flux distributions of a backscattered nature with only subtle differences noted between the two. The conditions required for backscattering are contrasted with the conditions which appear to be necessary for Knudsen scattering and the possibilities are considered for direct application of these observations to a description of the environment existent in typical vacuum systems.

Brownian diffusion in a nonuniform gas

Abstract: An analysis is made of the effects on the diffusion of Brownian particles whose Knudsen number is large compared to unity, of nonuniformities in the host gas. As examples, in one type of nonuniformity of the host gas, the Chapman-Enskog velocity distribution function for the gas molecules is used; in the other, the host gas is a free-molecule Couette flow. In both cases, a new force on the Brownian particles appears. Two techniques are used (extending Kramers' method and utilizing the Chapman-Enskog method) to transform the new Fokker-Planck equation into generalized Smoluchowski and convective diffusion equations. In these equations, the diffusion coefficient appears as a second-order tensor. Thus, it is demonstrated that Brownian diffusion in a nonuniform gas is anisotropic.

The Effect of Thermal Gas Motion on Microbalance Measurements

Abstract: This paper is a survey of research relevant to disturbances in gravimetric measurements that result from Brownian motion and thermal gas flow (thermomolecular flow, slip flow, and convection). Methods for elimination or minimization of the disturbances are reviewed and discussed. The application of microgravimetric methods for investigating gas flow is mentioned briefly.

Influence of Boundary Conditions on the Thermomagnetic Torque

Abstract: A moment method for solving the Boltzmann equation for a gas of diatomic molecules in an external magnetic field and in cylindrical geometry is presented. It is found that the particular geometry is very important at all pressures while the specific form of the boundary conditions becomes dominant at low pressures. In the continuum region, the torque has the same features as a Burnett effect in being inversely proportional to the pressure while at lower pressures the Knudsen effect on the temperature distribution becomes the major contribution to the pressure dependence. In this case, an ansatz for a two‐sided distribution function is used in solving the heat flow and temperature distribution. A comparison is made with experimental results and other theoretical investigations.

Studies of translational freezing in free expanding jets using molecular beam techniques

Abstract: Experimental values for the perpendicular temperature in freely expanding jets are deduced from beam intensity measurements as a function of the skimmer diameter over a wide range of reduced distances (40 < x/d < 300) for high inverse source Knudsen numbers (500 < Kn" 1 < 6500). The gases used are Ar, N2, CO2. The Tj_ values obtained are well represented by the equation T± ^ (x/d)~~m where in depends on the gas and on the source Knudsen number and is intermediate between the values for isoentropic and molecular flow. Our measurements indicate that the flow is still in the transition region at rather high distances (x/d ^ 300) from the source. The assumptions made in deriving the ideal molecular beam equation (ideal skimming, negligible self-scattering upstream the skimmer, ellipsoidal velocity distribution) are discussed.

Nearly free-molecular slit flow at finite pressure and temperature ratios

Abstract: An analytical study is made of nearly free-molecular flow of a noble gas from one reservoir to another through a two-dimensional slit, with finite pressure and temperature ratios across the slit. The fundamental solution of the linear Boltzmann equation is employed in the study. The total mass flow is calculated to the first-order correction terms, of the order of α ln α and α, where α is the inverse Knudsen number. The coefficients of these terms are in general multiple integrals, but they become explicit functions of the pressure and temperature ratios after the multiple integrations are carried out by using Krook collision model. When the general result is simplified to the isothermal case the first-order correction has a negative value, indicating the reduction of the total mass flow due to intermolecular collisions in the counter flows.

Spherical Expansions into Vacuum: A Higher-Order Analysis

Abstract: This paper deals with a higher order analysis of the steady spherically symmetric expansion of a monatomic gas into a vacuum. The analysis is based on the B.G.K. model of the Boltzmann equation for a gas having a viscosity-temperature dependence of the form Tw. It is assumed that the gas is collision dominated at a reference radius, so that the reference Knudsen number is small. A near-equilibrium solution is obtained correct to 0(^4 -1) where A is inversely proportional to the reference Knudsen number, and is therefore large. The breakdown of this asymptotic inner solution leads to a consideration of the outer non equilibrium region. Solutions in this outer region have been given previously, and this paper describes the perturbations to these solutions. It is shown that in the outer region perturbations of order A~l and A~~^a exist where a. = 3 + 4(1 — co). The 0 (A~ l) terms can be included in the zeroth-order solution by means of a slight straining of the radial coordinate. Equations for the Q(A~^'a) perturbations are derived, and asymptotic solutions given. For Maxwell molecules, i.e., w = 1, these equations can be solved explicitly, and numerical results are presented. It is shown that co = -f- is a special case where the first perturbation to temperature in the outer region is of order log A/A. It is also shown that for co -J, no breakdown of the inner solution occurs so that the equilibrium solution is uniformly valid. For o> = -J, the equilibrium solution is rendered uniformly valid using Lighthill's technique of strained coordinates.

Species Separation by Stagnation of Argon-Helium Mixtures in Supersonic Flow.

Abstract: : Measurements were made of the gas composition in the stagnation region of hemispherical probes immersed in supersonic flows of argon-helium mixtures. Substantial enrichment of heavy species was found at densities high enough that impact pressure measurements indicated no viscous enhancement. As densities (Reynolds numbers) were further decreased, enrichment of heavy species increased. A correlation is presented which describes the enrichment in terms of probe Knudsen number, free stream Mach number and free stream composition. (Author)

Determination of the density of seawater.

Abstract: KREMLING has recently described1 a promising new method for the determination of the density of seawater. The agreement he demonstrates between his results and those based on the equation of state, determined by Knudsen as a function of salinity and temperature, is even better than he suggests. This arises from an error in Kremling's interpretation of the new definition of salinity.

Dynamic upper atmospheric force model on stabilized vehicles for a high-precision trajectory computer program

Abstract: The upper atmosphere model draws heavily on the behavior of the earth's upper atmosphere which exhibits cyclic as well as irregular variations in density profile, temperature, pressure, and composition in unison with solar activities as deduced from the more recent land-based and satellite observations. The lift and drag model is designed specifically for inertially stabilized vehicles of the Mariner class, with possible extension to gravity gradient stabilized vehicles of the GEOS class. The model considers operation in the free molecular flow regimes with large Knudsen numbers. The vehicle is considered a composite structure with basic components having well-defined shapes, each with its own surface characteristics in terms of temperature, reflectivity, and accommodation of free stream molecules. The model takes into account both the calculation of precise aerodynamic force coefficients in terms of expansion of modified Bessel functions in speed ratios and angle of attack, and approximate force coefficients when the speed ratios approach infinity. Other considerations include specular and diffused reflectivity, shielding, and shadow effects.

Mixing of thermal molecular jets produced from knudsen effusion

Abstract: The flow resulting from a mixture of a large number of thermal molecular jets in a region of finite pressure is studied. The theoretical problem is treated by a moment method based on a particular form of the molecular velocity distribution function. The analysis indicates that the resulting one-dimensional flow velocity can attain a slightly supersonic value. The velocity depends on only one parameter (the pressure-porosity parameter) which is naturally introduced into the calculation. In the non-equilibrium region near the plane containing the orifices, the gas experiences a one-dimensional expansion. accompanied by a slight reduction in temperature. The theoretical results concerning this region are confirmed by measurements of the molecular flux near the orifice plane. Paper presented at "3eme Symposium International sur les Jets Moleculaires" 29 June - 2 July 1971. Cannes

Heat transfer from a heated sphere in a rarefied gas at rest

Abstract: We consider the problem of heat transfer from a slightly heated sphere in a resting rarefied gas. We assume that the Krook equation is valid in this case. Two forms of the basic equations are presented, and relations are given which are obtained as a result of calculations of the heat flux and the temperature jump at the sphere surface as a function of a parameter which is inversely proportional to the Knudsen number. The results obtained are compared with results given by the known approximate theories.

Sharp flat plate heat transfer in helium at Mach numbers of 22.8 to 86.8 and in corner flow with air at Mach number of 19

Abstract: Surface heat transfer rates were measured on a sharp flat plate at zero angle of attack in a hypersonic shock tunnel. The density and leading edge Knudsen number were varied to span the continuum to near free molecule regimes. The strong interaction parameter varied from 11 to 16,000 with Knudsen numbers from 0.56 to 17.1 respectively. Local heat transfer rates in the corner flow region produced by the intersection of two perpendicular flat plates with sharp leading edges were determined for various flow densities. The strength of the shock wave from the vertical plate was varied by adjusting the angle of attack from 0 to 5 deg. The unit Reynolds number varied from 1,000 to 17,200 and the Knudsen numbers from 1.6 to 27. The strong interaction parameter varied from 14 to 500.