Free Molecule Flow Research Articles

Effect of inlet guide vanes and sharp blades on the performance of a turbomolecular pump

The direct simulation Monte Carlo method is applied to a two-dimensional representation of the flow through a turbomolecular pump. It is shown that a conventional pump with a rotor as the first stage produces a significant disturbance in the vacuum chamber upstream of the pump. This disturbance can be prevented, without any adverse effect on the performance of the pump, by the addition of inlet guide vanes, similar to a stator, upstream of the rotor. In addition, it is shown that the compression ratio across the early stages that are in near free-molecule flow is significantly improved if the blades have sharp leading edges. The calculations required runs of just several hours on an ordinary PC.

The impact force acting on a flat plate exposed normally to a rarefied plasma plume issuing from an annular or circular nozzle

With the indirect thrust measurement of electric thrusters working at a low vacuum chamber pressure as the research background, this paper analyses the impact force acting on a flat plate exposed normally to a rarefied plasma plume issuing from a thruster with an annular or circular exit section for the free-molecule flow regime (at large Knudsen numbers). The constraint relation proposed by Cai and Boyd (2007 J. Spacecr. Rockets 44 619, 1326) about the velocity components of gas particles leaving a location on the nozzle exit section and arriving at a given spatial point outside the nozzle has been employed here to derive the analytical expressions for calculating the impact force. Sample calculation results show that if the flat plate is sufficiently large, the impact force acting on the flat plate calculated for the case without accounting for gas particle reflection at the plate surface agrees well with the axial momentum flux calculated at the thruster exit or the theoretical thrust force of the studied thruster, while accounting for the contribution of gas particles reflected from the plate surface to the impact force production may significantly increase the calculated impact force acting on the flat plate. For a Hall-effect thruster in which the thrust force is dominantly produced by the ions with high directional kinetic energy and the ions are not directly reflected from the plate surface, the contribution to the impact force production of atom species and of gas particles reflected from the plate surface is negligibly small and thus the measured axial impact force acting on a sufficiently large plate can well represent the thrust force of the thruster. On the other hand, if the contribution of the gas particles reflected from the plate surface to the impact force production cannot be neglected (e.g. for the electric thrusters with comparatively low thruster exit temperatures), appreciable error would appear in the indirect thrust measurement.

Numerical study of a single blade row in turbomolecular pump

Research on turbomolecular pumps (TMPs) based on the analysis of single blade row has been carried out by a number of investigators Kruger CH, Ph.D. Thesis, Massachusetts Institute of Technology; 1960; Katsimichas S, Goddard AJH, Lewington R, Oliveira de CRE. J Vac Sci Technol A 1995;13:2954; Schneider TN, Katsimichas S, de Oliveira CRE, Goddard AJH. J Vac Sci Technol A 1998;16:175; De Simon M. Vacuum 1990;41:2021; Chu JG, Hua ZY. J Vac Sci Technol 1982;20:1101; Amoli A, Ebrahimi R, Hosseinalipour SM. Vacuum 2004;72:427; Sawada T. Bull Jpn Soc Mech Eng 1973;16:993; Joong-Sik Heo, Hwang Young-Kyu. Vacuum 2001;56:133; Sawada T. Bull Jpn Soc Mech Eng 1979;22:362; Amoli A, Hosseinalipour SM. Vacuum 2004;75:361; Sheng Wang, Hisashi Ninokata. Prog Nucl Energy 2005;47:664, but those calculations were either limited to free-molecule flow, or failed to obtain good results in the transition regime. Furthermore, they were usually focused on pumping performance and did not report detailed flow field characteristics. In this study, a three-dimensional direct simulation Monte Carlo (DSMC) code adopting an accurate intermolecular collision model, i.e., a generalized soft sphere (GSS) model, has been developed to simulate the single blade row of a one-stage TMP from free-molecule flow to transition flow without any geometrical simplification. Molecular velocities and position equations are deduced under a rotating frame with consideration for the Coriolis and centrifugal acceleration, and the number of sample molecules is tested for sensitivity. The validity of this study and its accuracy are verified through comparison with previous DSMC results and experimental data, and the deterioration of the maximum pressure ratio in the transition flow regime and the influence of the intermolecular collision model on the maximum pressure ratio calculation are analyzed with the aid of transmission probability. Pumping performance under differing geometrical parameters (clearance between the blade tip wall and the housing wall, and the spacing–chord ratio) is investigated, and detailed flow field characteristics, including the number of molecule–surface collisions, temperature and density distributions, particle traces of single gas components and gas mixtures are all obtained under zero pumping speed.

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.

Analysis of fluid‐structure interaction in low pressure MEMS by Integral Equations

AbstractThe evaluation of gas dissipation occurring in inertial polysilicon MEMS is addressed focusing the attention on the free–molecule flow. In this regime, which is very often of interest for industrial applications, collisions between molecules can be neglected and the momentum transfer to the moving shuttle can be easily computed. Since the surfaces of silicon MEMS are generally very rough, a complete diffusion model is adopted to describe the wall–molecule interaction. A Boundary Integral Equation approach is proposed and it is shown that the introduction of the key assumption of small perturbations is crucial in the development of a robust and fast numerical tool. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On a deterministic approach for the evaluation of gas damping in inertial MEMS in the free-molecule regime

The evaluation of gas dissipation occurring in inertial polysilicon MEMS is addressed focusing the attention on the free-molecule flow. In this regime, which is very often of interest for industrial applications, collisions between molecules can be neglected and the momentum transfer to the moving shuttle can be easily computed. Since the surfaces of silicon MEMS are generally very rough, a complete diffusion model is adopted to describe the wall–molecule interaction. A Boundary Integral Equation approach is proposed and compared to a classical Test Particle Monte Carlo method, pointing out the clear advantages of the former for the applications at hand. It is shown that the introduction of the key assumptions of small perturbations and quasi-static response is crucial in the development of a robust and fast numerical tool. Several examples are presented validating the formulation with analytical and experimental results.

A BEM technique for free-molecule flows in high frequency MEMS resonators

A boundary element approach is proposed for the analysis of free-molecule flows in order to evaluate mechanical dissipation in micro-systems working at low pressure. The formulation is first presented for the general case of large perturbations and then adapted to the typical working conditions of micro-electro-mechanical-systems (MEMS). Since polysilicon micro-systems are addressed, the high surface roughness justifies the adoption of a diffuse reflection model at surfaces. The final linear boundary integral methods (BIE), which represents a flow balance of molecules on the surface, is implemented for medium scale 3D problems and validated against both analytical and experimental results with very promising results.

Influence of surface phenomena on free-molecule gas flow in fine channels

The paper deals with a theoretical study of the joint effect of the character of gas molecule scattering by channel (capillary) walls and surface diffusion of adsorbed molecules on the channel permeability in the free-molecule gas flow regime. The effect of the surface diffusion on radiation-induced mass transfer in channels is discussed.

Continuum models of rarefied gas flows in problems of hypersonic aerothermodynamics

The limits of applicability of continuum flow models in the problem of the hypersonic rarefied gas flow over blunt bodies are determined by an asymptotic analysis of the Navier–Stokes equations, the numerical solution of the viscous shock layer equations and the numerical and asymptotic solution of the thin viscous shock layer equations for low Reynolds numbers. It is shown that the thin viscous shock layer model gives correct values of the skin friction coefficient and the heat transfer coefficient in the transitional to free-molecule flow regime. The asymptotic solutions, the numerical solutions obtained within the framework of different continuum models, and the results of a calculation by Direct Simulation Monte Carlo method are compared.

Modelling of multi-component gas flows in capillaries and porous solids

The paper presents a derivation of the governing equations for multi-component convective-diffusive flow in capillaries and porous solids starting from a well-defined model and clear assumptions. The solution for the continuum regime is discussed in detail including a derivation of the diffusion slip boundary condition based on an improved momentum transfer theory. The Stefan–Maxwell species momentum equations are also re-examined and important distinctions made between the local and tube-averaged equations. An equation for the pressure gradient is derived and some examples of binary flows in capillaries are discussed. The theory for free-molecule flow is standard but the equations are recast into a form identical to the continuum equations which suggests an obvious method of interpolation for flow at arbitrary Knudsen number. There are no problems concerning viscous terms which have marred other derivations. The extension to flow in porous bodies is achieved by introducing a porosity–tortuosity factor but, unlike other treatments, this parameter is not absorbed into the gas diffusivities and flow permeability. It can then be eliminated from all but one of the equations and, with appropriate boundary conditions, the flux ratios can be obtained in terms of a mean pore radius only. The porosity–tortuosity parameter simply controls the absolute flux level and is best interpreted as a length scale-factor. The theory is applied with success to the prediction of some experimental data for helium–argon counter-diffusion and it is shown that, contrary to common belief, the mean pore radius is well-defined by flux ratio measurements if these are made with non-zero pressure differences.

Monte Carlo simulation of hydrodynamic drag and thermophoresis of fractal aggregates of spheres in the free-molecule flow regime

The hydrodynamic drag and thermophoretic properties of fractal-like aggregates of spheres in the free-molecular flow regime are examined using a Monte Carlo method. A modification of the sampling procedure that is used to assign velocities to the computational molecules is developed, and allows for a relatively fast and accurate determination of the particle transport properties for small-Mach number (i.e., slow-moving) conditions. Aggregate targets in the simulation are generated using an algorithm which mimics cluster–cluster aggregation, and simulations are performed on aggregates with a range of fractal dimensions and structure factors. The results indicate that for fractal dimensions in the range of D f = 1.7 –2.0, the hydrodynamic radius of the aggregate scales as a m ∼ aN S 0.47 , in which a is the sphere radius and N S is the number of spheres. Results also show that the thermophoretic velocity of a randomly-oriented aggregate will be slightly larger than that of the isolated spheres. The enhancement in velocity increases with N S , yet for relatively large-scale aggregates with N S = 3000 the increase is only around a factor of 1.08.

Supersonic dusty-gas flows with Knudsen effect in interphase momentum exchange

On the basis of the two-continuum model of dilute gas-solid suspensions, the dynamic behavior of inertial particles in supersonic dusty-gas flows past a blunt body is studied for moderate Reynolds numbers, when the Knudsen effect in the interphase momentum exchange is significant. The limits of the inertial particle deposition regime in the space of governing parameters are found numerically under the assumption of the slip and free-molecule flow regimes around particles. As a model problem, the flow structure is obtained for a supersonic dusty-gas point-source flow colliding with a hypersonic flow of pure gas. The calculations performed using the full Lagrangian approach for the near-symmetry-axis region and the free-molecular flow regime around the particles reveal a multi-layer structure of the dispersed-phase density with a sharp accumulation of the particles in some thin regions between the bow and termination shock waves.

Modeling for Actual Surface Roughness at Rarefied Gas Flow between Two Flat Plates

Rarefied gas flows in narrow clearances are affected by the surface roughness of the wall. For the purpose of clarifying the effects of angle distributions of surface roughness on the rarefied gas flow through passages, surface roughness is modeled by small inclines, which have various slope angles and azimuth angles depending on the angle distributions of actual surface roughness. Molecules are assumed to scatter from inclines according to the cosine law. Conductance of rarefied gas flows through passages composed of two flat plates is calculated using the Monte Carlo method. The calculated results agree well with the experimental results in near free molecule flow region.

希薄気体流における流路の表面形状と下流側圧力の影響−真空用微小流量測定システムの製作−

The aim of this paper is to clarify the effects of surface roughness and downstream pressure on the rarefied gas flow through passages. Surface roughness is modeled by two-dimensional triangular roughness. In the calculation, molecules are assumed to scatter from slope of triangular roughness according to the cosine law. Conductance of rarefied gas flows through passages is calculated using the Monte Carlo method. In the experiment, a flow measurement system for rarefied gas is manufactured to measure flow rate through passages. Experiments and calculations are conducted in the near free molecule flow region. Results of this study are as follows; The manufactured flow measurement system can measure flow rates below 1 × 10 -4 [Pa·m3/sec] accurately. The conductance of rarefied gas flows is affected by the downstream pressure even if pressure difference is equal in two flows. The effect of surface roughness does not change with the downstream pressure of passages.

Flow in a Liquid Sphere Moving with a Hypersonic Velocity in a Rarefied Gas: An Analytic Solution of Linearized Equations

The flow in a liquid sphere moving in a rarefied gas with a hypersonic velocity is obtained analytically. The flow in the rarefied gas is approximated as free molecule flow. We assume that the non-linear terms of the hydrodynamical equations as well as the surface deformation are sufficiently small to employ linearized equations. The solution shows that an eddy is formed owing to the tangential momentum flux of gas molecules on the surface of the sphere. This solution is relevant to models describing the formation of cosmic spherules and chondrules in meteorites, where solid dust particles melt due to gas frictional heating.

Gas dynamic calibration of a nano-Newton thrust stand

The ability to measure extremely low thrust levels with unusual precision is becoming more critical as attempts are made to characterize the performance of emerging micropropulsion systems. Many new attitude control concepts for nanospacecraft involve the production of thrust below 1 μN. A simple, but uniquely successful thrust stand has been developed and used to measure thrust levels as low as 86.2 nano-Newtons (nN) with an estimated accuracy of ±11%. Thrust levels in the range of 712 (nN) to 1 μN have been measured with an estimated accuracy of ±2%. Thrust is measured from an underexpanded orifice operating in the free molecule flow regime with helium, argon, and nitrogen propellants. The thrust stand is calibrated using results from direct simulation Monte Carlo numerical models and analytical solutions for free molecule orifice flow. The accuracy of the gas dynamic calibration technique, using free molecule orifice flow, has also been investigated. It is shown that thrust stand calibration using high Knudsen number helium flow can be accurate to within a few percent in the 80 to 1 μN thrust range for thin walled orifices when the stagnation pressure is accurately measured. The thrust stand and calibration technique exhibit significant improvement for accurate, low thrust measurements compared to currently published results.

Drag Model for Free Molecule Flow

A new drag model for convex surfaces engaged in free molecule flow has been developed under the condition of surface adsorption equilibrium using the hard cube model of atom scattering and the surface fractional coverage involved in gas adsorption. The new differential surface element drag and lift equations are only valid when the mass ratio of gas molecule to surface molecule is less than unity

Drift velocity of C(60)(+) in gases follows rarefied gas dynamics.

The drift velocities of C(60)(+) in He, Ne, Ar, and Kr were found to be estimated with high accuracy using the drag coefficient of a solid body in free molecule flow. That is, massive C(60)(+) behaves in gases as if it were a large classical body.

Effects of surface roughness on gas flow conductance in channels estimated by conical roughness model

Passages with rough walls allow for smaller conductance for rarefied gas flow than passages with smooth walls. For the purpose of clarifying the effects of surface roughness on rarefied gas flow through channels, surface roughness is modeled by circular cones with the same base radius. The cones are located in a statistical manner consistent with the measured angle distributions of actual surface roughness. Molecules are assumed to scatter on the cone surfaces according to the cosine law. The calculated results predict well the measured reduction of free-molecule and near-free molecule flow conductance through the passages composed of two flat plates.

Measurement of axial pressure distribution on a rotor of a helical grooved molecular drag pump

The theory in regard to the pumping performance of a helical grooved molecular drag pump proposed by the authors has been employed successfully as a design and evaluation tool. However, some researchers presented results different from the authors’ using the direct simulation Monte Carlo method. Their method included the inlet effect and the secondary flow in grooves, both of which were neglected in the authors’ theory. This study was planned to examine the validity of the authors’ theory precisely. A helical grooved cylindrical rotor was located concentrically in a smooth sleeve. In order to obtain the pressure distribution on the rotor, several taps were installed on the sleeve along the axial direction. The taps led to one side of a differential pressure gauge via valves and the inlet side of the rotor was connected to the other side of the gauge. The pressure was measured by switching the valves in turn. The measurements show that (a) the pressure in the free molecule flow regime increases exponentially from the inlet to outlet, (b) the pressure increases almost linearly from the inlet to outlet in the slip and viscous flow regime, and (c) the inlet effect becomes remarkable in the turbulent flow regime. The validity of the theory has been proved from the free molecule to viscous flow regimes, but the theory needs modification so as to include the inlet effect in the turbulent flow regime.