Showing papers on "Knudsen number published in 1999"

Report: a model for flows in channels, pipes, and ducts at micro and nano scales

Abstract: Rarefied gas flows in channels, pipes, and ducts with smooth surfaces are studied in a wide range of Knudsen number (Kn) at low Mach number (M) with the objective of developing simple, physics-based models. Such flows are encountered in microelectromechanical systems (MEMS), in nanotechnology applications, and in low-pressure environments. A new general boundary condition that accounts for the reduced momentum and heat exchange with wall surfaces is proposed and its validity is investigated. It is shown that it is applicable in the entire Knudsen range and is second-order accurate in Kn in the slip flow regime. Based on this boundary condition, a universal scaling for the velocity profile is obtained, which is used to develop a unified model predicting mass flow rate and pressure distribution with reasonable accuracy for channel, pipe, and duct flows in the regime (0 Kn). A rarefaction coefficient is introduced into this two-parameter model to account for the increasingly reduced intermolecular collisions...

Knudsen compressor as a micro- and macroscale vacuum pump without moving parts or fluids

Abstract: Applications of Knudsen compressors as both microscale and macroscale vacuum pumps have been investigated. The study is based on a cascade analysis incorporating available transitional thermal transpiration and Poiseuille flow results for slender channels. It was found that the Knudsen compressor is an attractive possibility for microscale pumps down to a pressure of about 1 mTorr and for macroscale pumps to about 0.1 mTorr. A microscale pump for a micromass spectrometer providing a molecule flow rate of 5×1014 molecules/s results in the following pump characteristics: energy use of 2.4 W, pump volume of 13.9 ml at an inlet pressure of 1 mTorr and an energy use of 28.5 mW, and pump volume of 0.16 ml at an inlet pressure of 10 mTorr. A macroscale pump providing a pumping speed of 103 l/s results in a pump with an energy use of 1786 W, and pump volume of 1695 l at an inlet pressure of 0.1 mTorr. Several Knudsen compressor characteristics such as pressure rise, pumping speed, volume, energy use and mass flow...

Experimental determination of the thermal accommodation and condensation coefficients of water

Abstract: The condensation coefficient of water vapor on liquid water and the thermal accommodation coefficient of air on liquid water are poorly known despite their importance in many applications, such as cloud physics. We have developed a new technique for determining the condensation and thermal accommodation coefficients experimentally. The technique consists of simultaneously measuring the homogeneous nucleation rate of ice and the evaporation rate of liquid water droplets as a function of pressure (droplet Knudsen number). As the Knudsen number increases, surface kinetic processes limit mass and energy fluxes and, as a result, the equilibrium temperature of an evaporating droplet is a function of the condensation and thermal accommodation coefficients. The homogeneous freezing nucleation rate is used as a sensitive measure of the droplet temperature. The nucleation and evaporation rates are determined by observing the scattered light from evaporating water droplets suspended in an electrodynamic levitation s...

Motion of tracer particles in supersonic flows

Abstract: Factors that may act on particle motion in high-speed flow are investigated. The classical expressions of drag coefficient C D for a sphere are reviewed. Then, a drag expression is proposed, extending Cunningham’s method to higher velocities and Knudsen numbers. This law, valid from continuum to free molecule conditions, for Re≲200 and M≲1 (where Re and M are, respectively, the Reynolds and Mach numbers based on relative velocity), is used to compare calculated and experimental values of the drag coefficient, as well as the particle velocities across an oblique shock wave. Calculated results are found to be in agreement with experiments.

Thermodynamically consistent hydrodynamic computational models for high-Knudsen-number gas flows

Abstract: In high-Knudsen-number flows nonequilibrium effects become dominant and the use of Navier–Stokes–Fourier equations becomes questionable since they are based on small deviation from local thermodynamic equilibrium. In this paper new hydrodynamic computational models are proposed for modeling gases in the transition regime. They are based on Eu’s generalized hydrodynamic equations and it turns out that they apply in all Mach numbers and satisfy the second law of thermodynamics to every order of approximation. In order to learn more about the new equations a model equation similar to the Burgers’ equation is studied. From this analysis new insight into constitutive relations of various hydrodynamic equations has been gained. In addition, a convergent iterative method for solving the highly nonlinear constitutive equations is developed. Finally, the shock structure and slip flow problems are computed by using high resolution numerical schemes and issues of extending the one-dimensional solver to multidimensional problems are discussed.

Burnett description for plane Poiseuille flow.

Abstract: Two recent works have shown that at small Knudsen number (K) the pressure and temperature profiles in plane Poiseuille flow exhibit a different qualitative behavior from the profiles obtained by the Navier-Stokes equations. Tij and Santos [J. Stat. Phys. 76, 1399 (1994)] used the Bhatnagar-Gross-Kook model to show that the temperature profile is bimodal and the pressure profile is nonconstant. Malek-Mansour, Baras, and Garcia [Physica A 240, 255 (1997)] qualitatively confirmed these predictions in computer experiments using the direct simulation Monte Carlo method (DSMC). In this paper we compare the DSMC measurements of hydrodynamic variables and non-equilibrium fluxes with numerical solutions of the Burnett equations. Given that they are in better agreement with molecular-dynamics simulations [E. Salomons and M. Mareschal, Phys. Rev. Lett. 69, 269 (1992)] of strong shock waves than Navier-Stokes [F. J. Uribe, R. M. Velasco, and L. S. Garc\'{\i}a-Col\'{\i}n, Phys. Rev. Lett. 81, 2044 (1998)], and that they are second order in Knudsen number suggests that the Burnett equations may provide a better description for large K. We find that for plane Poiseuille flow the Burnett equations do not predict the bimodal temperature profile but do recover many of the other anomalous features (e.g., nonconstant pressure and nonzero parallel heat flux).

Homogenization of Wall-Slip Gas Flow Through Porous Media

Abstract: The permeability of reservoir rocks is most commonly measured with an atmospheric gas. Permeability is greater for a gas than for a liquid. The Klinkenberg equation gives a semi-empirical relation between the liquid and gas permeabilities. In this paper, the wall-slip gas flow problem is homogenized. This problem is described by the steady state, low velocity Navier–Stokes equations for a compressible gas with a small Knudsen number. Darcy's law with a permeability tensor equal to that of liquid flow is shown to be valid to the lowest order. The lowest order wall-slip correction is a local tensorial form of the Klinkenberg equation. The Klinkenberg permeability is a positive tensor. It is in general not symmetric, but may under some conditions, which we specify, be symmetric. Our result reduces to the Klinkenberg equation for constant viscosity gas flow in isotropic media.

Applications of statistical mechanics in subcontinuum fluid dynamics

Abstract: We show results of molecular dynamics studies of uid ows in the Knudsen regime in which the mean free path is comparable to the system size. We elucidate the boundary conditions at the wall{uid interface in such ows and nd scenarios envisioned by Maxwell. We also nd scenarios which do not agree with Maxwell’s hypothesis. We focus primarily on the case of repulsive walls and discuss similarities and dierences in behavior when the wall{uid interaction is attractive or repulsive. Striking many body eects are found on increasing the uid density as one interpolates between the dilute gas and the dense uid regimes. c 1999 Elsevier Science

A discrete-ordinates solution for Poiseuille flow in a plane channel

Abstract: A recently established version of the discrete-ordinates method is used to develop a solution to a class of problems in the theory of rarefied-gas dynamics. In particular, an accurate solution for the flow, described by the Bhatnagar, Gross and Krook model, of a rarefied gas between two parallel plates is developed for a wide range of the Knudsen number.

Two-surface problems of a multicomponent mixture of vapors and noncondensable gases in the continuum limit in the light of kinetic theory

Abstract: The steady behavior of a multicomponent mixture of vapors and noncondensable gases between two parallel plane condensed phases for small Knudsen numbers, especially for the continuum limit (i.e., the limit as the Knudsen number vanishes), is investigated in the light of kinetic theory. By a systematic asymptotic analysis of the Boltzmann equation with kinetic boundary conditions, the flow due to evaporation and condensation on the condensed phases is shown to vanish in the continuum limit, and then the system of fluid-dynamic-type equations and their boundary conditions which describes the behavior in the limit is derived. On the basis of the system, it is shown that the vanishingly weak evaporation and condensation give a finite effect on the behavior of the mixture in the continuum limit. This is an example of the ghost effect discovered recently by Sone and co-workers [e.g., Y. Sone et al., Phys. Fluids 8, 628 and 3403 (1996); Y. Sone, in Rarefied Gas Dynamics, edited by C. Shen (Peking U.P., Beijing, 1997), p. 3]. Finally, for the case of a binary mixture of a vapor and a noncondensable gas, two typical problems, the simultaneous mass and heat transfer and the plane Couette flow, are considered to demonstrate the effect more concretely. The result is also compared with that obtained by the numerical analysis of the Boltzmann equation by the direct simulation Monte Carlo method.

Gaseous flow in microtubes at arbitrary Knudsen numbers

Abstract: New models that describe gas flow behaviour in microtubes are presented. To avoid time-consuming calculations in solving the integral equation which is obtained from the microscopic point of view, the high-order slip-flow boundary condition is utilized to correct the gas flow in such a micron or submicron spacing. The proposed model can be applied to arbitrary Knudsen number conditions under the assumption that the bulk flow velocity is negligible compared with the sonic velocity of the gas. The analytical solution of the pressure distribution for the first-order slip-flow model is obtained. The results show that the first-order slip-flow model is in good agreement with this model. The nonlinear pressure distribution is due to gas compressibility. The dominant mechanism influencing the nonlinear pressure distribution comes from the rarefaction of gas and the inlet pressure. The rarefaction effect increases the pressure drop at the inlet region of the channel and decreases the pressure drop at the exit region of the channel. The decrease of inverse Knudsen number changes the pressure distribution from concave to almost linear and increases the mass flow.

Non-isothermal couette flow of a rarefied gas between two rotating cylinders

Abstract: An accurate numerical solution of the momentum and the heat transfer through a rarefied gas confined between two cylinders rotating with different angular velocities and having different temperatures has been obtained over a wide range of the Knudsen number on the basis of the Bhatnagar, Gross, Krook model equation. The viscous stress tensor, heat flux, and the fields of density, temperature and velocity are found. An analysis of the influence of the angular velocities and the temperature ratio on these quantities is given.

Transport phenomena in sublimation growth of SiC bulk crystals

Abstract: Sublimation growth of SiC bulk crystals in the atmosphere of concentrated multi-component vapor is studied using a specially developed model of transport processes coupled with heterogeneous reactions at the source and the seed surfaces. The convective and multi-component diffusion mechanisms of the gas phase transport, dependence of the pressure level inside the growth chamber on the growth conditions, and kinetic jumps of the species partial pressures at the Knudsen layers on the reactive surfaces are taken into account in the model. The latter effect is described by introduction of novel boundary conditions representing extension of the Hertz–Knudsen relationship for the case of multi-component vapor. The results of calculations are shown to be in a good agreement with the available experimental data.

Sources of error in sorption and density measurements

Abstract: In sorption measurements, volumetric or gravimetric procedures are commonly used to determine the amount adsorbed. At low pressures, thermomolecular flow and pressure differences according to Knudsen's law disturb measurements. In volumetry, calibration of the dead space is required; in gravimetry, the influence of buoyancy has to be taken into account. In both cases, adsorption of the calibrating gas, usually helium, may disturb the measurements [1]. From the calibration measurements, the density of the sample can in principle be calculated. However, it has been observed in many experiments that its value depends on the calibrating gas.

Fokker-Planck Equation, Molecular Friction, and Molecular Dynamics for Brownian Particle Transport near External Solid Surfaces

Abstract: The Fokker–Planck (FP) equation describing the dynamics of a single Brownian particle near a fixed external surface is derived using the multiple-time-scales perturbation method, previously used by Cukier and Deutch and Nienhuis in the absence of any external surfaces, and Piasecki et al. for two Brownian spheres in a hard fluid. The FP equation includes an explicit expression for the (time-independent) particle friction tensor in terms of the force autocorrelation function and equilibrium average force on the particle by the surrounding fluid and in the presence of a fixed external surface, such as an adsorbate. The scaling and perturbation analysis given here also shows that the force autocorrelation function must decay rapidly on the zeroth-order time scale τ0, which physically requires NKn≪1, where NKn is the Knudsen number (ratio of the length scale for fluid intermolecular interactions to the Brownian particle length scale). This restricts the theory given here to liquid systems where NKn≪1. For a specified particle configuration with respect to the external surface, equilibrium canonical molecular dynamics (MD) calculations are conducted, as shown here, in order to obtain numerical values of the friction tensor from the force autocorrelation expression. Molecular dynamics computations of the friction tensor for a single spherical particle in the absence of a fixed external surface are shown to recover Stokes' law for various types of fluid molecule–particle interaction potentials. Analytical studies of the static force correlation function also demonstrate the remarkable principle of force-time parity whereby the particle friction coefficient is nearly independent of the fluid molecule–particle interaction potential. Molecular dynamics computations of the friction tensor for a single spherical particle near a fixed external spherical surface (adsorbate) demonstrate a breakdown in continuum hydrodynamic results at close particle–surface separation distances on the order of several molecular diameters.

Comparison of Burnett and DSMC predictions of pressure distributions and normal stress in one-dimensional, strongly nonisothermal gases

Abstract: The Burnett equations have been shown to provide improved descriptions, relative to the Navier–Stokes equations, of flow structure in high-velocity (i.e., hypersonic) gases. We examine here the accuracy of the Burnett constitutive equation for fluid stress as applied to stationary gases. Specifically, we investigate the effects of “thermal stress” (fluid stress induced by a temperature gradient), as predicted by the Burnett equation, on the pressure distributions and normal stress in a stationary, buoyancy-free, hard-sphere gas for the case of one-dimensional heat transfer. We show, using first-law principles and the Burnett equation, that thermal stress results in a reduction in normal stress in the nonisothermal gas relative to that in the equilibrium state. The normal stress, in turn, can be obtained as an eigenvalue to a second-order ordinary differential equation, representing the Burnett equation, for the pressure distribution in the gas. Simple asymptotic solutions to the Burnett equation are developed, and are used in combination with order-Kn pressure slip relations to formulate pressure boundary conditions at the heated and cooled surfaces. The approximate solutions, as well as exact numerical calculations, are compared with pressure distributions generated from the direct-simulation Monte Carlo (DSMC) method. The Burnett and DSMC predictions of pressure are in good agreement for effective Knudsen numbers (based on the temperature gradient in the gas) less than 0.1. In particular, the Burnett equations can provide a reasonable description of the Knudsen (or rarefaction) layers adjacent to the heated and cooled surfaces that bound the gas, and can also describe the variation in pressure in the bulk gas. In addition, theoretical predictions of the reduction in normal stress correspond well to DSMC-derived values.

Cylindrical Couette flows of a rarefied gas with evaporation and condensation: Reversal and bifurcation of flows

Abstract: A rarefied gas between two coaxial circular cylinders made of the condensed phase of the gas is considered, where each cylinder is kept at a uniform temperature and is rotating at a constant angular velocity around its axis (cylindrical Couette flows of a rarefied gas with evaporation or condensation on the cylinders). The steady behavior of the gas, with special interest in bifurcation of a flow, is studied on the basis of kinetic theory from the continuum to the Knudsen limit. The solution shows profound variety: reversal of direction of evaporation-condensation with variation of the speed of rotation of the cylinders; contrary to the conventional cylindrical Couette flow without evaporation and condensation, bifurcation of a flow in a simple case where the state of the gas is circumferentially and axially uniform.

MP3 Linux Players

Abstract: Is MP3 the wave of the future? Mr. Knudsen describes this new technology and what it will mean to the listener

BGK-Burnett Equations for Flows in the Continuum-Transition Regime

Abstract: To extend the range of applicability of continuum formulations into the continuum-transition regime, an extended set of fluid dynamic equations has been derived. These equations, termed as the Bhatnagar-Gross-Krook (BGK)-Burnett equations, have been derived by taking moments of the Boltzmann equation by using the BGK model for the collision integral, The second-order distribution function that forms the basis of this derivation is formulated by considering the first three terms of the Chapman-Enskog expansion. It is shown that the BGK-Burnett equations have been used to compute the hypersonic shock structure and the hypersonic flow past a blunt body. The results of these computations are compared with the augmented Burnett and Navier-Stokes solutions. The second-order distribution function does not violate Boltzmann's H-theorem; as a consequence the BGK-Burnett equations are entropy consistent for the range of Knudsen numbers for which computations have been performed

Combustion of volatile organic compounds over mixed-regime catalytic membranes

Abstract: Catalytic membranes operating in a mixed permeation regime (i.e., with significant Knudsen and laminar contributions) have been developed. The membranes prepared had wide pores and presented a low pressure drop. After the addition of γ-Al2O3 and Pt, the resulting catalytic membranes were active for the combustion of VOCs. Their performance was compared with that of similar catalytic membranes operating under the Knudsen diffusion regime.

Numerical study of the influence of gravity on the heat conductivity on the basis of kinetic theory

Abstract: The Boltzmann–Krook–Welander (or Bhatnagar–Gross–Krook) model of the Boltzmann equation is solved numerically for the heat transfer problem of a gas enclosed between two parallel, infinite plates kept at different temperatures, in the presence of a constant gravity field normal to the plates. At each point where the direct effect of the boundaries is negligible, a relation among the relevant local quantities (heat flux, temperature gradient, temperature, and density) holds even if the temperature varies over a length scale comparable to the mean free path. The ratio of the actual heat flux to the value predicted by the Fourier law is seen to be determined by the local Knudsen number and the local Froude number which are defined with the local mean free path, local characteristic length, and the magnitude of gravity. It is observed that the gravity produces an enhancement of the effective heat conductivity when the heat flux and the gravity field are parallel, while it produces an inhibition when both vectors are antiparallel. This deviation from the Fourier law, which vanishes in the absence of gravity, increases as the local Knudsen number increases and is more remarkable when the heat flux is parallel to the gravity field rather than otherwise. Comparison of the numerical data with an asymptotic analysis as well as with Pade approximants derived from it is also made.

Thermophoretic Deposition of a Sphere Normal to a Plane Surface

Abstract: An analytical study is presented for the thermophoresis of a sphere constant applied temperature gradient perpendicular to a plane surface. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model with a thermal creep and a hydrodynamic slip at the particle surface. asymptotic formulas for the temperature and velocity fields in the quasisteady situation are obtained by using a method of reflections. The plane surface may be a wall or a free surface. The boundary effect on the thermophoretic motion is found to be weaker than that on the motion driven by a gravitational force. Even so, interaction between the plane and the particle can be very strong when the gap thickness approaches zero. For the motion of a particle normal to a solid plane, the effect of the plane surface is to reduce the thermophoretic velocity of the particle. For case of particle migration toward a free surface owing to thermophoresis, the particle velocity can be either greater or smaller than...

Hydrodynamic interactions between two equal spheres in a highly rarefied gas

Abstract: This paper describes hydrodynamic interactions between two spherical particles having equal radii, a, and translating with velocities U1 and U2 in a highly rarefied gas. The center-to-center distance between the two spheres is aχ. The gas is at rest far from the two particles. The spheres move with speeds that are much smaller than the mean thermal speed of the gas molecules so that the Mach number, M≡max(U1,U2)/c, characterizing the deviation from equilibrium is much less than one. Here c is the mean thermal speed of the gas molecules. Gas molecules are assumed to be diffusively reflected from the particle surfaces. Our analysis is confined to the case where the particle Knudsen number is very large, i.e., Kno≡λo/a→∞, λo being the mean free path of the gas far from the two particles. We first study the free-molecular drag on the two sphere configuration for arbitrary translations of the spheres. For small Mach number, the general time-dependent, nonlinear problem may be approximated by a quasisteady, linear problem in which the spheres are held fixed and molecules reflected from each sphere have a modified Maxwell–Boltzmann distribution of velocities. A standard integral equation formulation based on flux balances at the particle surfaces is then employed to calculate the drag force acting on the spheres. The results obtained can be used as leading estimates for the forces acting on the spheres when Kno≫1 and 2⩽χ≪Kno. We then consider the case where the flow in the vicinity of each sphere is nearly free-molecular, but the flow in the O(aχ) space between the spheres is nearly continuum in nature. In this limit, the flow in the gap between the spheres is studied using the method of reflections. This approach can be used for arbitrary Kno provided Kno≪χ≪Kno M−1. The leading correction to the drag force due to the hydrodynamic interactions between the spheres when Kno≫1 is obtained. In all cases studied, the temperature of the two spheres is assumed to be the same as that of the surrounding gas.

Spectral problems for the discrete velocity models

Abstract: We use the discrete velocity models to study the spectral problems related to the 1D plane wave propagation in monatomic gases which are fundamental in the rarefied gases dynamics and nonequilibrium statistical thermodynamics. The results show that 6- and 8-velocity models can only capture the propagation of diffusion mode (entropy wave) in the intermediate Knudsen number regime. 4-velocity model instead captures the propagation of sound mode quite well after the comparison with the continuum-mechanic results.