Showing papers on "Knudsen number published in 1975"

Drag on a small sphere moving through a gas

Abstract: An approximate theoretical solution is presented for the drag force on a small sphere moving through a gas. The result is based on a moment solution to the Boltzmann equation which places no restriction on the Knudsen number. The equation which is obtained for the drag force reduces to the exact solution in the case of both large and small Knudsen number and gives excellent agreement with experimental data.

Unipolar charging of small aerosol particles

Abstract: Calculations are presented of the unipolar charging rate of small aerosol particles in the transition and free molecule regimes of Knudsen number. Numerical results for the charging rate in the transition regime are obtained from a constant collision frequency model of the Boltzmann equation. These results confirm experimental observations that the unipolar charging rates for larger particle charges are only weakly pressure dependent even when the Knudsen number is relatively small. Also, the modification of the near free molecular unipolar charging rate by ion-neutral molecule collisions, is found to depend importantly and in a complex manner on particle size and total particle charge when the image force is included in the ion-particle interaction.

Brownian coagulation of aerosols in the transition regime

Abstract: Earlier experimental studies of Brownian coagulation of aerosols have been extended into the transition regime, ie, Knudsen number values 08–16 This was done by working with the same range of particle size as earlier, but at reduced pressure A number of modifications were made in the experimental technique including the use of diethylhexylsebacate instead of dibutylphthalate in order to avoid the possibility of loss to the walls by evaporation The rate of coagulation at Kn = 02 agreed closely with that predicted by Smoluchowski's coagulation constant for the continuum regime, as modified by the Stokes-Cunningham correction, with a coalescence efficiency of unity The rate at higher Knudsen numbers (Kn = 08-16) was somewhat lower (about 20%) than predicted by Fuchs' formula for interpolation between the continuum and free-molecule regimes

Thermal force on a sphere

Abstract: The thermal force on a sphere of insulating material centered between parallel plates has been measured in the three monatomic gases He, Ne, and Ar and the two diatomic gases N2 and HD over a broad range of mean free paths. These measurements are on a finite system, an apparatus where the sphere radius and the plate spacing are of the same order of magnitude. Behavior near the observed maximum of the thermal force is stressed. The translational heat flux characterizes the dependence of the thermal force on the gas. Comparison with existing theories is made for the three basic Knudsen regimes, and the data are compared to previous measurements of the thermal force on aerosols.

High-order hydrodynamics and boundary conditions. Application to the thermal force

Abstract: For higher-order constitutive laws boundary conditions are derived by a thermodynamic method due to Waldmann. The application to the thermal force yields an expression which can account for the experimental data over a wide range of Knudsen numbers and all ratios of the heat conductivities of the aerosol particle and the gas.

High‐temperature thermodynamic properties of MnS by the dynamic Knudsen–torsion effusion method

Abstract: The vaporization and thermodynamics of MnS were investigated from 1536–1693 K by Knudsen effusion and dynamic‐torsion effusion. The principal vaporization reaction was confirmed to be MnS(s) = Mn(g)+(1/2)S2(g). Equations representing vapor pressures as functions of temperature were derived. Third‐law ΔH °m (298 K) of the principal vaporization reaction was 563.7±0.3 kJ from torsion effusion and 562.8±0.2 kJ from Knudsen effusion. A vapor analysis from the combined Knudsen and torsion effusion results revealed the average molecular weight of effusing vapor was 62.5±8.3 at 1600 K. Results were compared with those from previous investigations and with the enthalpy calculated from a thermodynamic cycle. Efficacy of the dynamic torsion effusion method was tested by observing the torsion effusion pendulum in three ways, at different parts of its oscillation, and comparing the results. The method gave results in excellent agreement with Knudsen effusion results and was effective in detecting and analyzing extran...

Solution by an iterative method of stationary problems of the kinetic theory of gases at moderate and low knudsen numbers

Abstract: THE INVESTIGATION is based on a model kinetic equation. An iterative method based on the use of the conservation differential equations is proposed for the numerical solution of stationary problems at moderate and low knudsen numbers. The problems of heat conduction, Couette flows and shock wave structure are solved for the whole range of Knudsen numbers as examples of the use of the method. The method is most efficient for low Knudsen numbers.

Incompressible viscous flow near the leading edge of a flat plate admitting slip

Abstract: The shear stress at the leading edge, calculated on basis of the Navier-Stokes equations and the no-slip boundary condition, approaches infinity. However, taking into account the mean free path of the molecules, which implies admitting a certain slip, the shear stress becomes inversely proportional to the square root of the Knudsen numberk ifk→0.k is defined as the ratio between the mean free path and the viscous length. The new boundary condition modifies the shear stress only within the Knudsen region of which the size is of the order of 3 to 4 times the mean free path.

Monte Carlo simulation of ion collection by a rocket‐borne mass spectrometer

Abstract: Monte Carlo calculations of ion collection by a rocket-borne mass spectrometer have been completed for the collisionless and transition flow regimes. In the collisionless case, computations have been completed for the following range of parameters: Debye numbers of 0.01 to 1000, speed ratios of 0.1 to 3.1, and nondimensional potentials of −10 to −200. For a speed ratio of 1.5 and a potential of −200 the ion flux coefficient at the stagnation point decreases from approximately 40 at a Debye number of 1000 to 3 at a Debye number of 0.01. The stagnation flux varied from 11 at a speed ratio of 3.1 to 260 at a speed ratio of 0.1 for a Debye number of 1000 and a potential of −200. The effect of varying the plate potential was found to be a linear variation of flux coefficient with collecting potential. Calculations made in the transition regime show that the minimum altitude at which the free molecule flow assumption is valid is approximately 120 km. A calculation at 70 km corresponding to a Knudsen number of 0.007 indicates that the flow field can be characterized as being near continuum but is such that rarefaction effects are still important. An investigation of the effects of collision cross section shows that the ion flux for an ion-neutral collision cross section 7 times larger than the neutral cross section is about half the flux for equal cross sections. Comparison of the calculations with flight data shows that the numerical results are higher by a factor of 3 or 4 than the measured values.

Propagation of acoustic modes in the transitional ionosphere

Abstract: Continuum theory is unsatisfactory for describing wave propagation in the transitional ionosphere, since the mean free path is of the same order as the characteristic length (i.e., Knudsen number in this region is of the order of unity). Therefore a transition model in which isotropic electrons behave as a fluid while the ions are governed by the kinetic equation modified by ion-electron collision effects is proposed to study acoustic wave propagation in the upper atmosphere in the altitude region from 500 to 2000 km. The results show that the dissipation of acoustic waves by electrons is through the viscous and thermal conduction effects, and the dissipation of acoustic waves by ions is through Landau damping. Comparisons of the characteristics of acoustic mode propagation and wave dissipation mechanisms in the collision-dominated, transitional from collisional to collisionless, and collisionless media are also discussed.

Kinetic theory of the heat transfer near a cylinder

Abstract: The heat flow on a cylinder immersed in an infinite rarefied gas is well defined in the free molecular limit To first order in the inverse Knudsen number, the correction to this free molecular value diverges logarithmically, for a reason quite similar to that explaining the divergence of the virial expansion of transport coefficients of two−dimensional gases When one accounts for the most divergent terms of an expansion of this flux in powers of the inverse Knudsen number, this logarithmic divergence does not disappear, due to hydrodynamical phenomena taking place at a distance from the cylinder that is much larger than the mean free path This situation is somewhat similar to that encountered in studying the divergent transport coefficients of two−dimensional gases It is discussed from the point of view of a possible experiment

Derivation of equations of motion of a slightly rarefied gas around highly heated bodies from boltzmann's equation

Abstract: The motion of a slightly rarefied gas (K ≪ 1, where K is the Knudsen number) around highly heated bodies is examined. On the assumption that the characteristic macroscopic velocity of gas motion generated during contact with a highly heated body is on the order of or much greater than the velocity of the impinging stream, the corresponding hydrodynamic equations are derived from Boltzmann's equation by Hubert's method [1]. A qualitative study is made of the region of applicability of the equations obtained. A class of flows of a continuous medium in which the characteristic change in enthalpy is much larger than the characteristic kinetic energy was studied in [2]. The Navier-Stokes equations with boundary conditions of adhesion proved to be inadequate for a description of these flows since it was already necessary in the first basic approximation to take into account part of the Barnett terms and slippage. The authors of [2] suggest using simplified Barnett equations with the condition of creep, with the Barnett terms being on the same order as the inertial and Navier-Stokes terms. On the other hand, it is known that the Barnett equations are derived on the assumption that the additional terms are small in comparison with the Navier-Stokes and Eulerian terms. This makes it desirable to obtain equations describing this class of flows directly from Boltzmann's equation.

Improved upper bounds on Knudsen permeabilities through a bed of spheres

Abstract: In a previous publication, a variational upper bound on the permeability of a porous medium to Knudsen flow was presented. The bounds were calculated for a model pore structure generated from randomly placed, freely overlapping solid spheres. In this paper, the improvement of these bounds is discussed.

Longitudinal rarefied gas flow over a plate

Abstract: Results are given in the present article from a numerical solution of the two-dimensional problem of longiCazdinal supersonic flow of a rarefied gas over a plate of infinite span and finite length, The problem is solved for a kinetic equation [1] approximating the Boltzmann equation [2, 3]. A precise numerical method (method of characteristics) Chat the author developed and applied earlier to the problem of transverse flow over a plate [4] is used in the present study. The results of the calculations are compared with the existing experimental and theoretical work of other authors. The agreement withthe datais satisfactory, The first attempt at a theoretical solution of the problem in the kinetic context was undertaken in [5], There the authors solved the problem of a semiinfinite plate for the Krook equation by an approximate numerical method, using the formal analogy with one-dimensional nonsteady-state problems. The problem was stated and solved as a parabolic problem with initial data in a certain cross section in front of the leading edge. The results of calculations were given for a Mach number M = 1.5. The same ntm~.erical method was later applied to an ellipsoidal model [6] for M = 1.5 and 5.5. For M = 5.5, however, t~e results of the calculations are given only with respect to the pressure coefficient. Some preliminary (in the author's own words) results of a numerical solution of the problem of longitudinal flow over a plate of finite length for the Krook equation have been published in [7]. By contrast with [5, 6], the problem was stated at the outset as a boundary-value problem in both variables. A new difference scheme was used. However, the details of the numerical algorithm are not entirely clear from the description given in I7]. An example is given of the flow field calculations for M = 1.5 and a Knudsen number K = 0.5. A complete description of the results has never, to the author's knowledge, been published. For large Mach numbers the problem of longitudinal flow past a plate of finite length has been solved by the Monte Carlo method [8, 9]. The authors of [8] cite the approximateness of the method used in [5, 6] and the departure of their results from the calculations carried out in [10] by the method of [5] for hypersonic flow. In [8] the calculated and experimental density profiles were compared and found to be .in good agreement. The results of a determination of the dependence of the friction and heat-transfer co~fficients on the Mach number, Knudsen number, and temperature ratio are not given. It is important to point out the large statistical scatter of the results obtained by the Monte Carlo method. For example, the scatter of the data about the average for the pressure on the surface of a plate at M = 24 is about 15%. The main series of calculations in the present study is carried out for a velocity ratio U = 4.6, which corresponds to M = 5.5 in a diatomic gas or to M -~ 5 in a monatomic gas. At these values of M all the effects of hypersonic flow are fully established. The viscosity- coefficient is assumed to have a power-law dependence on the temperature, with power exponent 0.816, and the Prsxldtl number a = ~. The results 'of the calculations are qualitatively consistent with the data of [8] and with the results of density profile measurements, but discrepancies also exist in the results, particularly for cold plates. The analytical dependences of the friction coefficient concur with the experimental data. For comparison, calculations are carried out for M = 1.5 in a monatomic gas. The hat,are of the flow for M = 1.5 differs significantly from the case M = 5. The flow field calculations give results different from [5, 6] as well as from [7], but the behavior of the friction coefficient is close to that obtained in [5]. The heat-transfer coefficient does not match the results of [5].

The Knudsen Boundary Layer in Low Density Gas Flow Past a Crystal

Abstract: The concept of the boundary layer has proved to be a very useful tool in fluid mechanics since its invention in 1904. It has been the subject of considerable analytical and experimental research, and Sydney Goldstein and his students and associates have played a crucial role in that development.We examine the concept of the boundary layer for a rarefied gas near a wall. In that situation, we find a hierarchy of phenomena connected to each other by boundary layer-type relationships. The first of these is the descent from the Liouville equation to the Boltzmann equation which describes that set approximately in terms of a single particle distribution function. The next is the descent from the Boltzmann equation to the Navier–Stokes equations with suitable slip conditions for the outside flow and a Knudsen layer adjacent to a solid boundary (or a specified initial state).The first part of the detailed work reported in this paper concerns the structure of that Knudsen layer in the gas and the conditions which...

Vaporization of cadmium oxide by Knudsen effusion

Abstract: The vaporization of CdO in alumina by Knudsen effusion cells was investigated over the temperature range 1014-1188$sup 0$K. The equilibrium constant for the reaction CdO(s) = Cd(g) + $sup 1$/$sub 2$O$sub 2$(g) was found to be log K = 10.68 - 19,084/T, and the second-law $delta$H$sup 0$$sub 2$$sub 9$$sub 8$ for this reaction calculated by the sigma method was found to be 89.2 +- 3.0 kcal, closely agreeing with the mean third-law $delta$H$sup 0$$sub 2$$sub 9$$sub 8$ of 88.9 + 1.0 kcal. It was determined that the vaporization of CdO is congruent with a sublimation coefficient close to unity. (auth)

The radiative knudsen layer

Abstract: The analogy between the kinetic theory of gases and photon gas dynamics (see [1], for instance) means that many qualitative results obtained in kinetic theory can be extended to photon gas dynamics. This includes the behavior of a radiating optically thick gas near solid surfaces. It is known that in an optically thick gas the radiative heat conduction approximation is inapplicable at distances of the order of the mean free path of the radiation from the surface of a body [1, 2]. The behavior of a photon gas near a surface can be predicted on the basis of the complete radiative transfer equation. In the present paper the problem of such wall layers in an optically thick gas (which have much in common with the well-known Knudsen layers) is solved in the approximation of a gray gas in local thermodynamic equilibrium. For simplicity only the steady-state case is considered. Expressions for the temperature jumps at the surface in a wide range of gas temperatures are obtained.

Thermal force on a disk in rarefied gas

Abstract: A theory for thermal force on a finite disk is developed by revising an earlier theory for an infinitely large plate. Coupling between force and temperature gradient is provided as before by constitutive equations relating momentum and heat flux to the second derivative of temperature. Viscosity and thermal conductivity coefficients depend on Knudsen number so as to span the transition regime. The theory includes surface accommodation effects and is in satisfactory agreement with recent experimental data.

The assignment of ion currents detected in the mass spectrum of a molecular beam

Abstract: A technique for assigning ion currents in the mass spectra of molecular beams is described with reference to the particular case of a molecular beam generated from a Knudsen effusion oven containing InP. Areas where the technique would find application are suggested.