Showing papers on "Knudsen number published in 1979"
59 citations
TL;DR: In this article, the condensation rate of trace vapor on a stationary, spherical aerosol particle was studied theoretically under the assumption that flow of the vapor toward the droplet does not disturb the Maxwellian distribution of molecules of the multicomponent bath gas, the stationary Boltzmann equation was solved analytically by the method of Grad.
51 citations
TL;DR: In this paper, an exact solution of the general dynamic equation for aerosol growth by condensation is obtained in dimensionless form for particle growth occurring by gas phase diffusion-, surface reaction-, and volume reaction-controlled processes.
49 citations
TL;DR: In this paper, the asymptotic behavior for small mean free path of steady flow of a gas in contact with its condensed phase of arbitrary shape is investigated on the basis of kinetic theory.
Abstract: The asymptotic behavior for small mean free path of steady flow of a gas in contact with its condensed phase of arbitrary shape is investigated on the basis of kinetic theory. The hydrodynamic equations for the macroscopic quantities, the velocity, temperature, and pressure, of the gas and their boundary conditions on the interface between the gas and its condensed phase are derived up to the order of the Knudsen number squared, together with the Knudsen-layer correction near the interface. In the analysis, the perturbations from a uniform equilibrium state are assumed to be small, but their nonlinear effects are taken into account correctly so that the results obtained can be applied when the Reynolds number of the system is finite. This is an extension of the corresponding linearized theory by the same authors, which is valid only for very small Reynolds number.
45 citations
TL;DR: In this paper, the phenomenological coefficients for one-dimensional mass and energy flows between infinite parallel plates containing polyatomic gases at all degrees of rarefaction were reported and compared to the polyatomic gas results.
Abstract: The phenomenological coefficients for one‐dimensional mass and energy flows between infinite parallel plates containing polyatomic gases at all degrees of rarefaction are reported. The polyatomic gas model equations developed by Hansen and Morse were used in the linearized Wang Chang and Uhlenbeck equation with Maxwell’s diffuse scattering boundary conditions. The solutions reproduce the molecular flow results at large Knudsen numbers and the continuum fluid properties for small Knudsen numbers. The phenomenological coefficients are functions of the Eucken factor, the internal constant volume heat capacity, and the translational Eucken factor. Calculations for the Gross–Jackson and the BGK model are reported and compared to the polyatomic gas results. We find the thermal transpiration effect depends on the translational Eucken factor and is independent of the inelastic collisions. The monatomic gas results can be used to estimate the thermal transpiration effect by ’’scaling’’ the Knudsen numbers used to ...
44 citations
TL;DR: In this article, single spherical latex particles and clusters were observed in a capacitor by means of a microscope in dark field illumination, and the dynamic shape factor of the irregular particles was calculated by noting the time particles took to pass between two marks on the graticule of the microscope.
24 citations
TL;DR: The mean-free-path approach to kinetic theory, initiated by Maxwell, and largely abandoned after the Chapman-Enskog success with Boltzmann's equation, is revised and considerably extended in order to find expressions for the heat flux vector q and pressure tensor p, valid (it is hoped) for all Knudsen numbers, K.
Abstract: The mean-free-path approach to kinetic theory, initiated by Maxwell, and largely abandoned after the Chapman-Enskog success with Boltzmann's equation, is revised and considerably extended in order to find expressions for the heat flux vector q and pressure tensor p, valid (it is hoped) for all Knudsen numbers, K. These expressions (equations (2.24) and (2.26)) are integrals taken over the whole volume of the fluid plus surface integrals taken over the solid boundaries. The one phenomenological element is the mean free path λ, which takes different values according to whether it is mass, momentum or energy that is transported by the molecules. The need for such an approach is evidenced by the existence of critical values of K, above which the Chapman-Enskog expansion in powers of K, truncated after a finite number of terms, fails to yield a solution. For example with the Burnett equations, which are correct to O(K 2), the critical K in a shock wave is only 0·2 based upon the upstream λ.
23 citations
TL;DR: In this article, the authors analyze the mechanisms that cause the motion of objects suspended in inhomogeneously heated gases, and present a methodology for calculating the kinetic coefficients that generalizes the Chapman-Enskog method to the case in which the state of the gas inside the Knudsen layers plays a substantial role.
Abstract: We analyze the mechanisms that cause the motion of objects suspended in inhomogeneously heated gases. We treat two limiting cases: a) a highly rarefied gas in which the mean free path λ of the gas molecules is large in comparison with the characteristic dimension R of the object, and b) a weakly rarefied gas that satisfies the condition λ R. In both cases we assume that the characteristic scale of the temperature inhomogeneities in the gas obeys Lλ. The case of a weakly rarefied gas is of very great interest, primarily from the standpoint of studying the state of a gas near a gas-solid phase boundary in the Knudsen layer. The greater part of the review is devoted to this problem: we treat the methodology of obtaining the boundary conditions for hydrogasdynamics with slip, and present in detail a scheme for calculating the kinetic coefficients that generalizes the Chapman-Enskog method to the case in which the state of the gas inside the Knudsen layers plays a substantial role, and we discuss the problem of the applicability of the thermodynamics of irreversible processes to problems of this type and demonstrate the efficacy of its methods. In addition, we modify the well-known method of half-range expansions on the basis of some physical assumptions and express some ideas on the principles of construction of the system of moment equations in the kinetic theory of gases. We also propose a scheme of experiment for testing the validity of the presented concepts.
21 citations
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TL;DR: In this paper, the parameters relating to the interaction of particles and gas in a dispersed phase flow are presented, and these parameters are essential to the development of numerical and analytic submodels of pulverized coal combustion.
Abstract: The parameters relating to the interaction of particles and gas in a dispersed-phase flow are presented in this chapter. These parameters are essential to the development of numerical and analytic submodels of pulverized-coal combustion.
21 citations
TL;DR: A variational principle based on the mean free path kinetic theory of gases is used to derive an expansion of the effective diffusion coefficient in the Knudsen number for a bed of overlapping, solid spheres.
Abstract: A variational principle based on the mean free path kinetic theory of gases is used to derive an expansion of the effective diffusion coefficient in the Knudsen number for a bed of overlapping, solid spheres. The effects of tortuosity are rigorously and properly considered by explicitly including the flux of diffusing material around obstructions in its path in the trial function. A formula for the ratio of Knudsen to bulk tortuosity in terms of the void fraction is derived. For small void fraction the reciprocal additivity of Knudsen and bulk effective diffusivities is obtained in analytical form.
20 citations
TL;DR: In this article, the vaporization of Hg0.8Cd0.2Te crystals under Knudsen effusion conditions has been investigated by the technique of modulated-beam mass spectrometry.
Abstract: The vaporisation of Hg0.8Cd0.2Te crystals under Knudsen effusion conditions has been investigated by the technique of modulated-beam mass spectrometry. Hg is the only dissociation vapour species detected at temperatures below 200 degrees C. Between 200 degrees C and 450 degrees C both Hg and Te3 was present but Cd was detected only above 450 degrees C. This grossly incongruent vaporisation leads to considerable concentration gradients within the charge crystals which have been observed directly using SEM, EDEX and X-ray diffraction techniques. Crystals heated to 420 degrees C for several hours under Knudsen effusion conditions lose HgTe via the vapour species Hg and Te2 and develop a surface layer of CdTe which, though porous, remains epitaxial. The development of concentration gradients within the charge at lower temperature leads to a diffusion rate limitation of the mercury vapour pressure developed within the Knudsen oven.
TL;DR: In this article, a practical theory of drop growth by condensation is discussed with a view toward its meteorological (mainly cloud physics) applications; the size range contemplated extends to about thirty microns.
TL;DR: In this paper, the boundary condition constitutes a linear relation within this pair, and the Green's relation stemming from the Boltzmann equation is combined with the integral formulation of the boundary value problem.
Abstract: The gas is described by the linearized Boltzmann equation, the wall by hydrodynamics. The rate of interfacial entropy production and an interfacial reciprocity postulate are formulated. Both are rewritten in terms of an interfacial thermodynamical force-flux pair, essentially the sum and the difference of the distribution function and its motion-reversed counterpart, for particles approaching the wall. The boundary condition constitutes a linear relation within this pair. Combination with the Green's relation stemming from the Boltzmann equation, leads to an integral formulation of the boundary value problem. Certain biothonormalized half-range eigenfunctions are introduced. They serve to expand the interfacial force-flux pair and formally to treat Couette flow and heat transfer between parallel plates, for any pressure. Upper and lower bounds for mechanical velocity slip and temperature jump ensue. The Knudsen case as a limit and complete accomodation as a numerical example are considered.
TL;DR: In this article, the transition flow of a gas between two reservoirs through a two-dimensional slot is studied, where the width of the slot is much smaller than the length but finite and the ratio of the equilibrium number densities in the reservoirs is near 1 and the equilibrium temperatures are the same.
Abstract: The transition flow of a gas between two reservoirs through a two-dimensional slot is studied. The width of the slot is much smaller than the length but finite. The ratio of the equilibrium number densities in the reservoirs is near 1 and the equilibrium temperatures are the same. Galerkin's method of solution is applied to the linearized BGK equation and the total mass flow is evaluated as a function of the inverse Knudsen number δ and the length-to-half width ratioL. Limits δ≫ 1, δ≪1 andL≫ 1 are investigated and good agreement with other authors is found. Simple analytical expressions are obtained for these cases.
TL;DR: In this paper, the asymptotic behavior of the steady flow of a rarefied gas over a body with a small accommodation coefficient (of the order of the Knudsen number) is investigated on the basis of the Boltzmann-Krook-Welander equation and the Maxwell type boundary condition.
Abstract: The asymptotic behavior (for small Knudsen number) of the steady flow of a rarefied gas over a body with a small accommodation coefficient (of the order of the Knudsen number) is investigated on the basis of the Boltzmann-Krook-Welander equation and the Maxwell-type boundary condition. It is shown that the present results provide a bridge between the cases of zero accommodation coefficient (specular reflection) and of finite one. As applications of the theory, a sphere with constant temperature in a uniform flow of gas and that in a gas at rest with a uniform temperature gradient are considered.
TL;DR: In this article, the effect of thermomolecular pressure in a cylindrical channel is analyzed, at any values of the Knudsen number, on the basis of the S-model kinetic equation.
Abstract: The effect of thermomolecular pressure in a cylindrical channel is analyzed, at any values of the Knudsen number, on the basis of the S-model kinetic equation.
TL;DR: In this paper, a thermal generator for CaCO3 aerosols was described which produced compact, fairly homogeneous (σg ≅ 1.5) particles with radii from 0.005 to 0.04μm; fewer than 10% of the particles were electrically charged.
TL;DR: Forced flow of a Fermi liquid is studied in this article for a cell geometry consisting of two planes with a separation on the order of mean free path, where an approximate transport equation is used to derive an integral equation for the velocity profile, which is solved numerically.
Abstract: Forced flow of a Fermi liquid is studied for a cell geometry consisting of two planes with a separation on the order of mean free path. An approximate transport equation is used to derive an integral equation for the velocity profile, which is solved numerically. Results for the total flux through the cell, which determines the dissipation, are given as a function of the Knudsen number N (ratio of cell thickness to mean free path). Effects of specular reflection at the boundaries are considered. It is found that the dissipation has a minimum at N approximately equal to 1/2, and behaves linearly for N greater than or equal to 3. Implications for present experimentation are discussed.
TL;DR: In this article, the behavior of a Knudsen gas between two plates at unequal temperatures is discussed and a general formulation for the case of small temperature difference is given for the nonspherical interaction of a polyatomic gas with the solid walls.
Abstract: The behavior of a Knudsen gas between two plates at unequal temperatures is discussed. A general formulation is given for the case of small temperature difference. The use of perturbation theory is then demonstrated and applied to the case of a nonspherical interaction of a polyatomic gas with the solid walls in the presence of a magnetic field.
TL;DR: In this article, the problem of nonisothermal gas motion in a plane channel is considered for arbitrary values of the Knudsen number on the base of an S-model kinetic equation.
Abstract: The problem of nonisothermal gas motion in a plane channel is considered for arbitrary values of the Knudsen number on the base of an S-model kinetic equation.
01 Jan 1979
TL;DR: In this paper, the combined effect of a variable collision frequency and a correct Prandtl number on the structure of a Knudsen layer is investigated and a new collision model having both properties, but simple enough to be amenable to a simple solution, is proposed and used to solve the problem.
Abstract: The combined effect of a variable collision frequency and a correct Prandtl number on the structure of a Knudsen layer is investigated A new collision model having both properties, but simple enough to be amenable to a simple solution, is proposed and used to solve the problem Results turn out to be in exceptionally good agreement with the experimental data of Reynolds, Smolderen and Wendt
TL;DR: In this paper, the authors compared the analytical treatment of Present and de Bethune for the purpose of predicting constraints which are likely to be encountered in the high temperature effusional separation of the equilibrium components of water.
TL;DR: In this paper, the authors used light interferometry using a variable-wavelength pulsed dye laser and a CW HeNe laser as monochromatic sources to measure the bearing clearances in self-acting pivoted narrow-slider gas bearings.
Abstract: : The experimental investigation described here involves the highly accurate measurement of bearing clearances on the order of 10 microinches in self-acting pivoted narrow-slider gas bearings. The experimental measurements are based on light interferometry using a variable-wavelength pulsed dye laser and a CW HeNe laser as monochromatic sources. The light interference in the gas bearing is obtained by flying the slider on a very precise optically flat quartz disk through which the light beam is transmitted. The combined effect of high Knudsen numbers and surface irregularities on the flying height of narrow gas bearings is observed by varying the load on the bearing and the ambient molecular mean free path. The experimentally measured bearing clearances are compared quantitatively with rather accurate theoretical predictions obtained by numerical solution of Reynolds differential equation for compressible fluids with slip boundary conditions. The result of this study indicates that, as clearances in narrow gas bearings get progressively smaller, while the Knudsen number increases to values beyond 0.1, the theoretical model fails to predict the bearing behavior. It is also argued that this failure is because of the weakness of the continuum model. (Author)
TL;DR: In this paper, the plane Couette problem for a Knudsen gas of linear molecules is considered and the rotational degrees of freedom of the molecules are described through the internal angular momentum.
Abstract: The plane Couette problem for a Knudsen gas of linear molecules is considered. The rotational degrees of freedom of the molecules are described through the internal angular momentum. An alignment of the molecules is produced by the orientation dependence of the interaction of the gas molecules with the plates bounding the gas. The presence of a constant magnetic field is taken into account. The tensor describing the influence of the molecular alignment on polarization is evaluated.
TL;DR: In this article, the properties of the thermal conductivity of gases were investigated by the coaxial-cylinder method at low pressures, and it was established that there is a temperature-jump effect, which must be taken into account, especially for light gases at P≈1 atm and room temperature.
Abstract: The properties of the thermal conductivity of gases are investigated by the coaxial-cylinder method at low pressures. It is established that there is a temperature-jump effect, which must be taken into account, especially for light gases at P≈1 atm and room temperature.
TL;DR: In this paper, mass-discrimination effects in stagnation-point ion extraction are treated for a plasma with a wide range of Knudsen number, i.e., when the charged particle's mean free path ⩽ the sampling electrode radius.
TL;DR: In this article, the axially symmetric motion of a gas in a volume confined between an external immobile surface of rotation and a coaxial surface of a rotating sphere is considered, and a solution is obtained by the moment method based on the Boltzmann equation with a collision integral of Maxwellian molecules.
Abstract: The axially symmetric motion of a gas in a volume confined between an external immobile surface of rotation and a coaxial surface of a rotating sphere is considered A solution is obtained by the moment method based on the Boltzmann equation with a collision integral of Maxwellian molecules The gas-velocity distribution and an expression for the friction torque exerted on the sphere are obtained for arbitrary Knudsen numbers and for an arbitrary shape of the outer surface The proportionality of the gas slip velocity over the surface of the sphere to the friction strain is shown The friction torque is investigated for specific shapes of the outer surface The motion of a gas filling the space between concentric spheres, each of which rotates about an arbitrary axis, is treated In the limiting case of small Knudsen numbers the expressions obtained are compared with the corresponding results for a continuous medium
TL;DR: In this article, the authors considered the problem of the flow of a mixture along a cylindrical tube of finite length for given pressure difference Δp between its ends, and gave a consistent asymptotic (with respect to the small parameter ǫ) solution.
Abstract: In a number of experiments (see [1], in which experimental papers are listed), diffusion has been observed in the radial direction in the process of flow of a mixture along tubes at low pressures. The heavier molecules accumulate near the tube axis. The attempt made in [1] to explain this phenomenon by the influence of the Burnett contribution to the diffusion did not lead to success, and the Burnett terms in the radial diffusion velocity indicate a motion of heavy molecules away from the tube axis. In the present paper, a complete analysis is given of this phenomenon. We consider the problem of the flow of a mixture along a cylindrical tube of finite length for given pressure difference Δp between its ends. On the basis of the hydrodynamic equations of the Burnett and super-Burnett approximations, a consistent asymptotic (with respect to the small parameter ɛ) solution is given; ɛ = (Δp/p)R/L is the relative change in the pressure along the tube at a distance of order R (R and L are the radii and length of the tube). Radial diffusion occurs in the quadratic approximation in ɛ. It is shown that the radial diffusion velocity contains new terms not present in [1]; these are due to the inhomogeneity of the temperature and the pressure over the tube section, the expansion of the gas, and the super-Burnett correction to the diffusion velocity. The most important is the thermodiffusion term, which is determined by the hydrodynamic equations of the Navier-Stokes approximation. The remaining terms have order relative to it of Kn2 (Kn = 〈 1 〉/R is the Knudsen number, and 〈 1 〉 is the mean free path of the molecules). The expression obtained for the diffusion velocity agrees in sign with the experiment.