Showing papers on "Knudsen number published in 1968"

Flow of Rarefied Gas through a Circular Pipe

Abstract: The asymptotic behavior at small Knudsen number for two problems involving the flow of rarefied gas through a circular pipe, Poiseuille flow and thermal creep is treated using the Boltzmann‐Krook equation. By a matching procedure the outer solution for the continuum regime and the solution for the Knudsen layer are determined simultaneously. Actual calculations have been carried out up to second order. A short discussion on second‐order slip condition is also given.

Hypersonic viscous interaction on slender cones.

Abstract: Hypersonic surface measurements and flowfield properties about slender cones in hypersonic viscous interaction regime

Nonlinear Rarefied Couette Flow with Heat Transfer

Abstract: In order to calibrate the new, general discrete ordinate method proposed by Huang against the known standard solutions, the proposed method is applied to the problem of nonlinear Couette flow with heat transfer whose numerically exact solutions are available. It is shown that the proposed method yields excellent results over a large range of Knudsen numbers (from free molecular to near‐continuum flow regime). In contrast to other methods, an important advantage of the proposed method is the calculation of the local molecular velocity distribution function which is useful in the direct comparison with molecular beam experimental results.

Heat Conduction in a Rarefied Gas between Concentric Cylinders

Abstract: Heat conduction through a rarefied gas contained between concentric cylinders is studied analytically. Emphasis is placed on the connection with the experimental determination of thermal accommodation coefficients by the low‐pressure method. The linearized BGK‐model equation is used, based on small relative temperature difference. Two coupled linear integral equations for the perturbed temperature and density are then formulated. An asymptotic analysis is carried out for the case of mean free path much greater than the inner but of the same order as the outer cylinder radius, which corresponds to typical operating conditions for experiments. The lowest‐order deviation of the heat flux from its free‐molecular value is of O[ln(Kn)/Kn], the same as that obtained by Knudsen iteration. Numerical calculations are performed to find this term. It is found that for a mean free path 20 or more times the inner cylinder radius the relative error introduced by using the free‐molecular value for the heat flux is less t...

On the solution of the Boltzmann equation for an unsteady cylindrically symmetric expansion of a monatomic gas into a vacuum

Abstract: The problem of an unsteady axisymmetric expansion of a monatomic gas into a vacuum is considered in the limit of small source Knudsen number. It is shown that a solution of the Boltzmann equation for Maxwell molecules valid for large time can be constructed, which matches with the known equilibrium solution for an inviscid expansion of a fixed mass of gas into a vacuum provided that the region near the zero density front is excluded. This solution is formally the same as that obtained for the similar problem of steady spherical expansion into a vacuum—the variations along each particle path of the unsteady flow being the same as that in the steady flow.Near the front, the expansion procedure breaks down and the equations require a different scaling. A modified form of the Boltzmann equation is obtained which leads to a corresponding set of moment equations. Unfortunately, the set of moment equations is no longer closed and no essential simplification has been made.

Study of Some of the Parameters Affecting Knudsen Effusion. V. Free‐Path Considerations in Small Knudsen Cells

Abstract: Monte Carlo methods have been used to calculate distributions describing the behavior of particles in a typical, small Kundsen cell. For pressure conditions corresponding to free‐molecule flow, three distributions were obtained: (1) the wall collisions per particle, (2) the path lengths of the particles between collisions, and (3) the total path lengths of the particles. Knowledge of these distributions provides considerable insight into the behavior of particles at the beginning of the transition region. In particular, the relationship between gaseous collisions and the Knudsen number is clarified. Also, it is shown that the population of free paths depends on the geometry of the cell but is independent of the condensation coefficient αc. The lifetime of a given particle, however, is strongly dependent on αc. The analysis, which can be applied to any vapor species, is described for gold.

A finite difference analysis of developing slip flow

Abstract: A finite difference analysis is presented for slip flow in the entrance region of the parallel plate channel and the plain tube. Results are given for Knudsen numbers equal to 0, 0.01, 0.03, 0.05, and 0.1. The present results for the pressure distribution in the entrance region do not favour any particular one of the linear solutions to the problem which have been given previously. There is, however, very good agreement between the finite difference prediction of the developing velocity profiles and that of a certain linear solution.

Theory of Thermal Transpiration in a Knudsen Gas

Abstract: The theory of thermal transpiration has been revised in a rigorous treatment. This phenomenon, discovered by Newmann, Feddersen, and Reynolds, has been treated by Maxwell and Knudsen from kinetic theory considerations. In the present theory, it has been shown that the Maxwell and Knudsen theories do not give a detailed, microscopic description and are only valid for a near‐Maxwellian distribution function. However, the revised and exact theory contains a set of new invariants which deviate from the classical theory by a geometrical factor I. This factor (isotropy) is a measure of the isotropy of the velocity distribution function in the thermal transpiration system. The new invariants in the Knudsen regime are pI / T1 / 2 and n(T)1 / 2I. In general, I is not always equal to unity or a constant value in a thermal transpiration system.

Effusion. II. Angular Number Distributions of Gaseous Cadmium from a Right-Circular Cylindrical Orifice into Vacuum

Abstract: The angular number distributions of Cd(g) atoms effusing from a Pyrex right‐circular cylindrical orifice have been studied by use of a condensation target. The profile of the deposit on the condensation target was analyzed by an interferometric means. Only qualitative agreement with the Clausing prediction for the angular number distribution could be obtained. A high‐pressure limit for molecular flow was found to be 3.9 < K (Knudsen number; ratio of mean free path to orifice diameter) <4.6. At higher pressures in the transition region, the probability of effusion in the forward direction decreases with increasing pressure.

Some of the Parameters Affecting Knudsen Effusion. IV. Monte Carlo Calculations of Effusion Probabilities and Flux Gradients for Knudsen Cells

Abstract: Data are presented for effusion probabilities and cell‐wall flux gradients calculated dynamically by means of Monte Carlo computer studies. Typical effusion cell dimensions were programmed for both knife‐edged and Clausing orifices; all calculations apply to a flat sample. The influence of the size and the length‐to‐radius ratio of the orifice and the effects of different condensation coefficients are described. An analytical expression relating the equilibrium sample flux to the flux at steady state has been derived. It is shown that for a given cell design, the ratio of these fluxes is constant and independent of the nature of the vaporizing species. Because the Monte Carlo method requires no mathematical approximations or simplifying assumptions, results analogous to real cell behavior are to be expected. Comparison of these results with the treatment derived by Motzfeldt shows that his formula is applicable and quite accurate over a wide range of experimentally encountered conditions.

New boundary conditions in the transition régime

Abstract: Starting from the Boltzmann equation, new boundary conditions are derived to be matched with the Navier—Stokes equations, that are supposed to hold in the main body of a gas. The idea upon which this method is based goes back to Maxwell and Langmuir. Since the distribution function is supposed to be completely determined by the Navier—Stokes equations, this new set of boundary conditions extends in some sense the validity of the macroscopic equations to the transition and free molecular regimes. In fact, it is shown that the free molecular and slip flow regimes are correctly described by this method; the latter is also supposed to give a reasonable approximation for the complete range of Knudsen numbers. The new procedure is applied to different problems such as plane Couette flow, plane and cylindrical Poiseuile flow, heat transfer between parallel plates and concentric cylinders. Results are obtained and compared with the exact numerical solutions for the above-mentioned problems.

The diffusion force in the transition region of knudsen number

Abstract: The free molecule theory of the diffusion Force is reviewed briefly. The theory of the diffusion Force in the transition region is developed for binary mixtures for equimolar counterdiffusion and for diffusion of one component of the mixture through the second stagnant component. The transition region theory of the diffusion Force appears to agree with experimental results for mixtures the components of which do not differ greatly in molecular weight.

Evaluation of Knudsen Measurements of the Sublimation of Manganese(II) Selenide

Abstract: MnSe sublimation Knudsen measurements at various temperatures and pressures noting congruent vaporization

Kinetic Description of Cylindrical Heat Conduction in a Polyatomic Gas

Abstract: A kinetic description has been obtained for a polyatomic gas between concentric cylinders maintained at different temperatures. Knudsen iteration on a polyatomic kinetic model equation has been employed to obtain first collision corrections to the free molecular distribution function. It is assumed that the temperature difference between the cylinders is small. The heat transfer vector obtained from this theory has been compared with the results of Lees' moment method to obtain an estimate of the range of Knudsen number for which the theory should be valid.

Effusion from Knudsen Cells with Conical Channels

Abstract: Effusion from conical channels in Knudsen cells, including both vapor‐transport and surface‐diffusion contributions, is analyzed. A computer solution is obtained for the resulting integro‐differential equations describing flow. The effects on effusion of various combinations of channel geometries and surface properties are examined. The results indicate that the relative contribution of surface diffusion increases with increasing conical angle, with decreasing channel radius, with decreasing channel‐length‐to‐radius ratio, and with increasing surface diffusivity.

Effect of Gas–Surface Interaction on Sound Propagation in Highly Rarefied Gases

Abstract: Previous theoretical investigation of sound propagation in rarefied gases is clarified and extended with emphasis on the role played by gas–surface interaction effects. The problem is formulated for the case of one‐dimensional motion generated by a piston undergoing small amplitude sinusoidal oscillation where the Knudsen number based on the distance from the piston is large. The resulting dispersive properties such as phase velocity and attenuation rate are in qualitative agreement with experimental data and previous theoretical predictions. However, the dependence of dispersion on the molecule–surface interaction is found to be quantitatively different from that suggested by earlier investigators.

A kinetic theory of catalysis and mass transfer in a cylinder

Abstract: The diffusion and mass transfer of a trace component in a background gas in a cylinder, is treated according to a relaxation transport equation The system is assumed isothermal and at rest An expression for an effective axial diffusivity is found which goes to the appropriate high and low density limits Numerically, this diffusivity is similar to one half of the harmonic mean of the high density diffusivities, which is often used as an empirical extrapolating formula The radial mass transfer problem results in an integral equation which is solved numerically for all Knudsen numbers The case of catalytic reaction on the walls is considered in detail Rarefaction effects become important under conditions of fast reaction The problem of sublimation or evaporation from a cylinder wall can be treated by the same equations

Variational Principles for the Heat Flux in a Rarefied Gas between Concentric Cylinders

Abstract: In studying the heat conduction in a rarefied monatomic gas between concentric cylinders two coupled linear integral equations are obtained using the linearized BGK‐model equation for the molecular velocity distribution function. These integral equations are transformed into a symmetric form. Variational principles are then formulated for the heat flux which hold for all Knudsen numbers.

Nearly free molecular heat transfer from a sphere

Abstract: Consider a sphere immersed in a rarefied monatomic gas with zero mean flow. The distribution function of the molecules at infinity is chosen to be a Maxwellian. The boundary condition at the body is diffuse reflection with perfect accommodation to the surface temperature. The microscopic flow of particles about the sphere is modeled kinetically by the Boltzmann equation with the Krook collision term. Appropriate normalizations in the near and far fields lead to a perturbation solution of the problem, expanded in terms of the ratio of body diameter to mean free path (inverse Knudsen number). The distribution function is found directly in each region, and intermediate matching is demonstrated. The heat transfer from the sphere is then calculated as an integral over this distribution function in the inner region. Final results indicate that the heat transfer may at first increase over its free flow value before falling to the continuum level.

Rarefied Gas Flow through Passages with Static Boundaries

Abstract: A resume of the current state-of-knowledge concerning rarefied gas flow through passages with static boundaries is presented. A brief development of the theoretical gas flow models which have been used for long tubes and annuli is included. These theoretical models have been developed to cover the flow regime extending from laminar continuum flow to free molecular flow. The analysis required to use these models is simple and straightforward. Experimental data are presented for the flow of argon through a long tube and for the flow of argon and xenon through annuli. The experimental values of mass flow rate per unit pressure drop as a function of Knudsen number are graphically presented for a long tube and for two annuli. Also presented on the same figures are the values predicted by the theoretical models. The agreement between theory and experiment for the complete regime between continuum and free molecular flow is very good. Presented at the ASLE Annual Meeting in Cleveland, May 6–9, 1968.

Simulation of non-continuum free jet plume impingement.

Abstract: Noncontinuum free jet plume impingement simulation, effect of total temperature to model surface temperature ratio, Knudsen number and total enthalpy