Showing papers on "Knudsen number published in 1972"
TL;DR: In this article, the authors explored the nature of certain special classes of solutions to the equation of coagulation for the case of Knudsen aerosols and showed that self-preserving solutions may exist.
246 citations
TL;DR: In this paper, a new kind of mechanism which induces a flow around a solid body in a slightly rarefied gas is proposed, and the behavior of gas around a sphere with a constant temperature which is placed in an infinite expanse of gas at rest with a uniform temperature gradient is investigated.
Abstract: A new kind of mechanism which induces a flow around a solid body in a slightly rarefied gas is proposed In order to demonstrate the flow induced by this mechanism the behavior of gas around a sphere with a constant temperature which is placed in an infinite expanse of gas at rest with a uniform temperature gradient is investigated on the basis of the asymptotic theory for a slightly rarefied gas A flow with magnitude of the order of the Knudsen number squared is induced from the hotter to the colder region The sphere is subject to a force in the direction of the given temperature gradient
93 citations
TL;DR: In this article, the Orowan, Petch, and Knudsen equations were examined against 46 sets of strength-vs-grain-size data, and the results showed that the Orowsan-Petch treatment does not remove systematic variations of strength as a function of grain size.
Abstract: The Orowan, Petch, and Knudsen equations were examined against 46 sets of strength-vs-grain-size data. For the 30 sets which are most discriminating, represented by 229 averaged-data points, the variances of the Orowan-Petch and Knudsen treatments are ∼4.5 and ∼17.1 kpsi2, respectively. Statistical considerations give preference to the Orowan-Petch treatment at a high confidence level in these 30 cases, showing that the Knudsen equation does not remove systematic variations of strength as a function of grain size. The Orowan-Petch treatment also appears to provide a sounder basis for extrapolation. The identification of the Orowan and Petch equations with postulated physical models permits interpretation of data in terms of probable underlying causes. Several examples are discussed and in most cases agree at least qualitatively with present understanding. A satisfactory physical model for Petch behavior in “nonyielding” ceramics is needed.
64 citations
TL;DR: In this article, the effect of the thermal creep coefficient upon the evaluation of rotational collision numbers from thermal transpiration measurements is discussed, and experimental data for He, Ne, Ar, Kr, and N2 agree best with the recent theoretical work of Loyalka and indicate little or no dependence on gas type.
Abstract: The development of a pressure gradient in a gas‐filled capillary due to the application of a temperature gradient is known as thermal transpiration. In the near‐continuum limit (small Knudsen number) the mechanism for the transpiration effect is a creeping motion of the gas in a thin layer adjacent to the surface. Measurements performed in the near‐continuum limit are compared with a number of theoretical calculations for the thermally induced creep velocity. The experimental data for He, Ne, Ar, Kr, and N2 agree best with the recent theoretical work of Loyalka and indicate little or no dependence on gas type, in contrast with another recent theory which indicates a marked dependence on the thermal accommodation coefficients. In addition, the effect of the thermal creep coefficient upon the evaluation of rotational collision numbers from thermal transpiration measurements is discussed.
53 citations
TL;DR: In this article, a theoretical analysis for the thermal force exerted on a spherical particle between two flat plates is presented, where the Knudsen number KL, based on the plate spacing, is nonzero.
Abstract: A theoretical analysis is presented for the thermal force exerted on a spherical particle between two flat plates. The case is considered where the Knudsen number KL, based on the plate spacing, is nonzero. The analysis is based on the assumption that the particle radius is much smaller than the plate spacing and the result is valid for all values of KL from zero to infinity. A moment solution to the Boltzmann equation coupled with a two‐stream Chapman‐Enskog distribution function is employed, and the collision integral is evaluated on the basis of Maxwell molecules.
31 citations
TL;DR: In this article, the effects of overheat ratio, Knudsen number, Peclet number, length to diameter ratio, Grashof number and yaw angle are discussed.
24 citations
TL;DR: In this article, a variational principle was used to solve the problem of heat transfer between two parallel plates with arbitrary accommodation coefficients on the plates for all ranges of Knudsen number with a simple trial function.
Abstract: A kinetic model equation for a polyatomic gas is used to investigate heat transfer between two parallel plates. In particular, a general kinetic boundary‐value problem is studied for the case of arbitrary accommodation coefficients on the plates. This problem is solved for all ranges of Knudsen number using a variational principle with a simple trial function. The dependence of heat transfer on Knudsen number, the internal energy, the Eucken factor and the ratio of translational collision time to internal collision time for a polyatomic gas has been obtained. Moreover, density and temperature profiles have been calculated. Comparison is made with measurements on nitrogen, and it is found that the solution is in very good agreement with experiment for heat transfer, and in qualitative agreement with experimental density profiles.
18 citations
TL;DR: The Couette problem is the simplest problem of steady shear flow of rarefied gas in a region bounded by solid surfaces, and it has been solved by the Monte Carlo method.
Abstract: The Couette problem is the simplest problem of steady shear flow of rarefied gas in a region bounded by solid surfaces. This problem has been examined in the linear formulation by many authors, using either the linearized Krook equation or the moment methods (see [1]). It has recently been solved by the Monte Carlo method [2].
17 citations
15 citations
TL;DR: In this article, a kinetic theory for the mass flux to a liquid droplet surrounded by its pure vapor is presented, where the coefficients of the first three terms of this expansion are derived by solving the Boltzmann equation using a modified Knudsen number iteration procedure.
Abstract: A kinetic theory is presented for the mass flux to a liquid droplet surrounded by its pure vapor. When the mass flux Γ is expanded in terms of a parameter α which is the ratio of the droplet size to the mean free path (inverse Knudsen number), one obtains a series of the form Γ=Γ(0)+Γ(1)α +Γ(2)α2 lnα +···. The coefficients of the first three terms of this expansion are derived by solving the Boltzmann equation using a modified Knudsen number iteration procedure. It is shown that the coefficients are determined by integrals associated with sequences of successive collisions among a number of vapor molecules and the droplet. These collision integrals bear a close similarity to the collision integrals derived earlier from the generalized Boltzmann equation for the density dependence of the transport properties of gases.
13 citations
TL;DR: In this article, the problem of strong recondensation between two infinite parallel plates over a wide range of Knudsen numbers for a one-component and a two-component gas, on the basis of the model Boltzmann kinetic equation, was studied.
Abstract: As is known, surface phenomena such as evaporation, absorption, and reflection of molecules from the surface of a body depend strongly on its temperature [1–5]. This leads to the establishment of a flow of a substance between two surfaces maintained at different temperatures (“recondensation”). The phenomenon of recondensation was studied in kinetic theory comparatively long ago. However, up to the present, only the case of small mass flows in a onecomponent gas has been investigated completely [3,4]. Meanwhile it is clear that by the creation of appropriate conditions we can obtain considerable flows of the recondensing substance, so that the mass-transfer rate will be of the order of the molecular thermal velocity. Such a numerical solution of the problem with strong mass flows along the normal to the surface for small Knudsen numbers for a model Boltzmann kinetic equation was obtained in [7]. In this study we numerically solve the problem of strong recondensation between two infinite parallel plates over a wide range of Knudsen numbers for a one-component and a two-component gas, on the basis of the model Boltzmann kinetic equation [6] for a one-component gas and the model Boltzmann kinetic equation for a binary mixture in the form assumed by Hamel [8], for a ratio of the plate temperatures equal to ten. We also investigate the effect of the relative plate motion on the recondensation flow.
TL;DR: In this article, precise measurements of the pressure dependence of the flow of gases through annuli of various radius ratios are presented. But the results are limited to the free molecules and near free molecules.
Abstract: Results of precise measurements of the pressure dependence of the flow of gases through annuli of various radius ratio are presented. The data cover a range of Knudsen numbers for about 0.01 to 450. The depth of the Knudsen minimum is found to vary markedly with radius ratio and increases as the radius ratio increases. (Radius ratio is defined as the ratio of inside to outside radii). About 50% of the total change in the depth of the Kundsen minimum occurring between flat plates (radius ratio = 1.0) and capillaries (radius ratio = 0) is effected by a 13% change in radius ratio from 1.0 to 0.8711. Previously observed deviations from classical kinetic theory are verified; and, in particular, in the free‐molecule and near free‐molecule regimes, the flows are found to be lower than those predicted based on diffuse scattering of gas molecules from the annulus wall. The data are generally found to be in good agreement with flows predicted by the BGK and variational techniques presented by Berman and Maegley [Phys. Fluids 15, 772 (1972)].
TL;DR: In this paper, a phenomenological description of the relation between pressure drop, gas velocity and other gas and filter variables for a fibrous filter medium is presented, which extends to viscous and inertial flow, compressible and incompressible flow, and to conditions of slip and slip-free flow.
TL;DR: In this paper, the authors determined the transmission probability of 15 orifices with length to diameter ratios from 0.02 to 9.53, with a pressure of CsCl vapor from 3.7 × 10−3 to 1.826 torr corresponding to values of the Knudsen number from 17.6 to 0.051.
Abstract: Orifice transmission probabilities (defined as the ratio of the number of molecules leaving an orifice into vacuum to the number of molecules entering the orifice from a gas container according to the kinetic theory of gases) have been determined for 15 orifices with length to diameter ratios from 0.02 to 9.53, with a pressure of CsCl vapor from 3.7 × 10−3 to 1.826 torr corresponding to values of the Knudsen number (ratio of mean free path to orifice diameter) from 17.6 to 0.051. Total effusion measurements have been made using both a multiple cell effusion method and a vacuum balance apparatus. The data agree well with previous experimentally determined values, with Clausing orifice transmission probabilities for large K calculated by DeMarcus, and with values calculated from equations derived by Wahlbeck. It is now possible to calculate vapor pressures from transition region data provided the molecular diameter is known.
30 May 1972
TL;DR: In this paper, the third-law heat of formation was derived with the thermodynamic functions of Strauss and Thiele as was the second-law value, and the entropy values were also determined.
Abstract: : Partial pressures of C3(g) have been measured with a high-resolution mass spectrometer and a Knudsen effusion cell in the temperature range from 2300 to 2800K. The third-law heat of formation was derived with the thermodynamic functions of Strauss and Thiele as was the second-law value. The entropy values were also determined.
01 Jul 1972
TL;DR: In this paper, the authors presented atomization, drop size, and penetration data for cross stream water injection at conditions simulating high altitude reentry (low Weber number, high static temperature, high Knudsen number, and low static pressure).
Abstract: Atomization, drop size, and penetration data are presented for cross stream water injection at conditions simulating high altitude reentry (low Weber number, high static temperature, high Knudsen number, and low static pressure). These results are applied to the RAM C-1 and C-3 flights. Two primary breakup modes are considered, vapor pressure or flashing and aerodynamic atomization. Results are given for breakup boundaries and mean drop size for each of these atomization mechanisms. Both standard and flight orifice geometries are investigated. The data were obtained in both a static environment and in conventional aerodynamic facilities at Mach numbers of 4.5 and 8. The high temperature aspects of reentry were simulated in a Mach 5.5 cyanogen-oxygen tunnel with total temperature of 4500 K.
TL;DR: In this paper, a theory of the thermomagnetic torque at pressures so high that Knudsen effects can be neglected is presented, based on the kinetic theory of Levi, McCourt, and Beenakker.
Abstract: This paper presents a theory of the thermomagnetic torque which is applicable at pressures so high that Knudsen effects can be neglected. Kinetic theory techniques are used to derive momentum and energy boundary conditions appropriate to transport in the presence of a magnetic field. It is found that thermal creep generated by the field‐induced annular flow of heat contributes substantially to the torque. Calculations based upon a combination of the kinetic theory of Levi, McCourt, and Beenakker and Senftleben—Beenakker experimental data are found to be in reasonable agreement with measurements of torque.
TL;DR: In this article, a model for the transport of either heat, momentum, or mass in the transition regime is proposed based on third-order constitutive equations for the fluxes.
Abstract: A description of transport of either heat, momentum, or mass in the transition regime is based upon third‐order constitutive equations for the fluxes. The one‐dimensional steady‐state equation which governs the dependent variable P* (either dimensionless temperature, velocity, or concentration) as a function of distance x* is the linearized fourth‐order differential equation N2(d4P*/dx*4) + (d2P*/dx*2)=0, where N is proportional to the Knudsen number. A solution is proposed based on reasonable boundary conditions for parallel plate geometry. Expressions are derived for the profile of P*, the slip or jump of P* at a wall, the effective transport coefficient, and the flux. The expressions have the proper limits for the continuum and free molecule regimes, and compare well with other theories for the transition regime and with subsonic experimental data.
TL;DR: In this article, the problem of nearly free-molecular flow of a gas from one reservoir to another through a two-dimensional channel is studied, where the ratio of the equilibrium pressures in the reservoirs is finite, whereas the equilibrium temperatures are the same.
Abstract: The problem of nearly free‐molecular flow of a gas from one reservoir to another through a two‐dimensional channel is studied. The width of the channel is much smaller than the length which is of the order of the mean free path of the gas. The ratio of the equilibrium pressures in the reservoirs is finite, whereas the equilibrium temperatures are the same. The fundamental solution of the linear Boltzmann equation is used for the evaluation of total mass flow rate. The result is presented in the form of an asymptotic series, of which the first‐order terms are of the order of α ln α and α, where α is the inverse Knudsen number. Computations shows that the first‐order terms, which represent intermolecular collisions in the counter flows, have a net negative value. The total mass flow rapidly decreases as the length of the channel increases.
TL;DR: The revised theory of thermal transpiration by Wu has been applied to the case of a modified Absolute Manometer for any temperature ratios and arbitrary accommodation coefficients on the boundaries of the three plates as discussed by the authors.
Abstract: The revised theory of thermal transpiration by Wu has been applied to the case of a modified Absolute Manometer for any temperature ratios and arbitrary accommodation coefficients on the boundaries of the three plates. According to the Lockenvitz’s approximate theory the pressure on the central vane is independent of its temperature. This paper explicitly shows that the pressure on the central vane is dependent on its temperature. A comparison of both theories has been discussed extensively for arbitrary temperature on the central plate.
TL;DR: In this paper, it was shown that the effect of temperature and concentration on the velocity of a mixture of gases can be modeled as a tensor of the first derivatives with respect to T along the coordinates.
Abstract: In the general case, changes in the temperature of a gas bring about motion of the gas, i.e., thermostrcss convection [1, 2]. It is shown below that in mixtures of gases there is an analogous phenomenon, i.e., concentration-stress convection, resulting from concentration gradients; examples of the latter are given. Some of the results of [1, 2] are also correlated. In [1, 2] investigations were made of the basic properties of a class of slow flows, “nonrarefied” (Knudsen number K→0) of a monoatomic gas with Reynolds numbers R∼1 and with relative temperature drops in the flow θ=T*−1Δ T≲1, which, with adhesive boundary conditions, cannot be described by the Navier-Stokes equations (T* is the characteristic temperature). Under these circumstances, it is necessary to take account of some of the Barnett terms of the momentum equation, due to temperature gradients, and to slipping of the temperature. In the case of slow movements of a mixture of gases, there are analogous effects, due to concentration gradients, if the relative concentration drops Nα=yα*−1Δyα are of the order of magnitude of unity [2]. Here, yα-nα/n; nα is the number of particles of the α-th component of the mixture in unit volume, n=Σnα. Maxwell [3] was the first to investigate the question of the stresses in rarefied gases, due to nonhomogeneity of the temperature. In this case, he neglected terms of the tensor of the temperature stresses containing products of the first derivatives with respect to T along the coordinates. Then, the Barnett temperature terms entering into the momentum equation have a “gradient” form and can be combined with the pressure. Therefore, Maxwell drew the conclusion that temperature stresses do not bring about motion, and that motion can arise only as the result of slipping of the temperature; his investigation reduced to a consideration of the redistribution of the pressure due to temperature stresses in a quiescent gas. After this, the above question was touched upon in the book [4], in which the following note is made: “for velocity gradients on the order of 1 sec−1 and values of [∂2T/∂xi∂xj] on the order of 1 deg/cm2, it is impossible to completely neglect the temperature stresses in comparison with the ordinary viscous stresses, even at ordinary pressures, and they can play a role in experiments aimed at determination of the viscosity, in which inequality of the temperatures is allowed.” As far as is known to the authors, this exhausts the existing literature discussion of the effect of Barnett stresses (both temperature and concentration) on gas flows with K→0. In concluding our review, we note still another phenomenon, for a description of which, with K≪1, we must bring in the Barnett approximation: in the case of slow flows of a mixture of gases, some of the Barnett (due to velocity gradients) terms in the expression for the diffusional velocity are of the same order as the usual barodifferential term of this expression [4, 5]. There is demonstrated below the “symmetry” of some of the flow properties of a one-component gas in a mixture of gases, in the presence, respectively, of Barnett temperature and concentration stresses. The article considers conditions with the absence of convection due to the corresponding Barnett stresses. It gives examples of concentration-stress flows and discusses the properties of flows due to weak temperature and diffusion slipping.
TL;DR: In this article, the problem of heat transfer between two infinite parallel plates is investigated on the basis of equations obtained by averaging the Boltzmann kinetic equation with respect to the transverse velocity.
Abstract: The problem of heat transfer between two infinite parallel plates is investigated on the basis of equations obtained by averaging the Boltzmann kinetic equation with respect to the transverse velocity. A numerical solution of the problem is accomplished for a temperature ratio between the plates of T0/T1=1/4 and for various Knudsen numbers.
TL;DR: In this article, the authors proposed a number based on probe length path length between horns (microwave interferometer) or probe length Mach number particle mass particle density Reynolds number and Schmidt number for probe length dimensionless mobility ion Schmidt number fluid to probe temperature ratio freestream velocity angle of attack phase difference in radians.
Abstract: width of waveguide (interferometer) electron charge current to probe dimensionless probe current I/neu^n^ sin# ion mobility Knudsen number based on probe length path length between horns (microwave interferometer) or probe length Mach number particle mass particle density Reynolds number based on probe length dimensionless mobility ion Schmidt number fluid to probe temperature ratio freestream velocity angle of attack phase difference in radians (microwave interferometer) dielectric constant conical probe half-angle Debye length over probe length free space wavelength (microwave interferometer)
TL;DR: In this article, the acceleration of a shock wave in an ideal gas of decreasing density has been investigated with empirical inclusion of real-gas effects for strong shocks in hydrogen, and experimental results suggest that previous shock acceleration models are valid only for a limited range of the Knudsen number in finite geometries.
Abstract: The acceleration of a shock wave in an ideal gas of decreasing density has previously been studied. The problem is reconsidered here with empirical inclusion of real-gas effects for strong shocks in hydrogen. Experimental results suggest that previous shock acceleration models are valid only for a limited range of the Knudsen number in finite geometries and that for large final-state Knudsen numbers a free-expansion model best describes the experimental results.
01 Jan 1972
TL;DR: In this paper, it is shown that Knudsen forces can be used for measuring low gas pressures and buoyancy forces for measuring higher gas pressures simultaneously and from the results conclusions are drawn on the possibility of designing a wide range manometer.
Abstract: It is wellknown that Knudsen forces can be used for measuring low gas pressures and that buoyancy forces can be used for measuring higher gas pressures. An experiment is described where these two forces were studied simultaneously and from the results conclusions are drawn on the possibility of designing a wide range manometer.
01 Mar 1972
TL;DR: In this paper, the authors measured drag, heat transfer and number flux on sharp cones in the near free, molecule flow regime, and the results were compared with available Monte Carlo calculations.
Abstract: Drag, heat transfer, and number flux were measured on sharp cones in the near free, molecule flow regime, and the results were compared with available Monte Carlo calculations. In general, the calculations predicted the magnitude of the data; however, the heat transfer and drag increased with increasing Knudsen number at a faster rate than predicted. Also the drag coefficients measured for the slender cones at high Knudsen number were higher than predicted for free molecule flow. These disagreements between theory and experiment could possibly be attributed to the simplicity of the surface interaction laws assumed in the theory. Reynolds analogy factors obtained from the experimental measurements agreed with free-molecule values and also with that obtained by the Monte Carlo technique.
TL;DR: In this paper, the filtration of rarefied gas through porous bodies of various structures, porosities, and geometries was studied under various flow modes with the Knudsen number ranging from 10−4to 2.5.
Abstract: The filtration of a rarefied gas through porous bodies of various structures, porosities, and geometries was studied under various flow modes with the Knudsen number ranging from 10−4to 2.5.
TL;DR: In this article, an elementary stochasti treatment of thermal transpiration is given for the case of an imperfect gas in the transition region between the Knudsen and continuum limits.
Abstract: By working in terms of the “dusty-gas” model, in which a capillary or porous solid is treate formally as a conglomerate of statistically-motionless giant molecules, an elementary stochasti treatment of thermal transpiration is given. Specific attention is drawn to the case of an imperfect gas in the transition region between the Knudsen and continuum limits.
01 Jan 1972
TL;DR: Free molecule flow heat transfer occurs when the predominant mechanism of molecular transport is by direct molecule-wall collisions rather than through a sequence of molecule-molecule collisions as mentioned in this paper. And it is generally accepted that a Knudsen number of three or more is sufficient to achieve this condition.
Abstract: Free molecule flow heat transfer occurs when the predominant mechanism of molecular transport is by direct molecule-wall collisions rather than through a sequence of molecule-molecule collisions. It is generally accepted that a Knudsen number of three or more is sufficient to achieve this condition.
TL;DR: In this paper, the conductance of channels for rarefied molecular flow is estimated by optical analogue experiments, based on the Lambert's and Knudsen's laws of scattering.
Abstract: The scattering of photons from perfectly diffusing surfaces and the scattering of molecules from surfaces covered with adsorbates follow similar laws, namely Lambert's and Knudsen's. Hence estimates of the conductance of channels for rarefied molecular flow may be simply obtained by optical analogue experiments.