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Abstract A new explicit semi-empirical model was proposed for predicting the steady state gas flow rate through short tubes under rarefied condition. The experimental data produced by Fujimoto and Usami and the DSMC data provided by Varoutis et al. were utilized to obtain the model coefficients. The model was further validated by the published experimental data. The proposed model was capable to make predictions while meeting all the criteria of the rarefied gas flow in the following ranges of the variables: 0< P r < 0.7, 0< δ < 4000 and 0< ω < 13. The model was capable to predict the reduced flow rate of rarefied systems, not only in the free molecular region and the hydrodynamic region, but also in the transition region, hence covering all the Knudsen number domain within the utilized data. It was also found that by increasing the pressure ratio and decreasing the length to diameter ratio, the transition region shifted to the higher values of the rarefaction parameter. The proposed model is useful in the vacuum gas community for engineering purposes by offering a practical closed form expression for the DSMC data to predict the rarefied gas flow rate. Highlights • By increasing P r transition region was not fixed and it shifted to higher values of δ. • The proposed model can meet almost all the criteria of a rarefied gas flow system. • An explicit semi-empirical model was proposed for short tubes gas flow rate. [ABSTRACT FROM AUTHOR]

Abstract The conductive heat transfer through a rarefied binary gas mixture of helium and argon is experimentally investigated as a function of a molar fraction. The heat flux is measured from the free-molecular to near free-molecular flow regimes, and it is analyzed both in dimensional and dimensionless forms to clarify the effect from the gas-surface interaction. Then, the thermal accommodation coefficient is obtained by assuming a gas mixture as a single "virtual" species gas. The measured thermal accommodation coefficients for single component gases of helium and argon show good agreement with our previous results and data in literature. The thermal accommodation coefficient for a binary gas mixture is compared with the theoretical value derived in the free-molecular flow regime as a superposition of an independent heat transfer by each gas component. The measured results are well explained by the theory. However, there is a little discrepancy between them, suggesting variation in the thermal accommodation coefficient with a molar fraction in flow regimes other than the free-molecular flow regime. Highlights • The conductive heat transfer through a rarefied binary gas mixture of helium and argon is measured. • The dimensionless heat flux monotonically decrease with increasing the molar fraction of helium. • The thermal accommodation coefficient is well described by a superposition of components in the free-molecular flow regime. • The dimensionless heat fluxes divided by the measured accommodation coefficients agree well with numerical results. [ABSTRACT FROM AUTHOR]

Abstract Fokker-Planck kinetic models have been devised as an approximation of the Boltzmann collision operator. Cubic Fokker-Planck-DSMC hybrid method is employed to simulate the diatomic gas flow through a thin slit and a thin orifice. Pressure driven nitrogen expansion gas flows with two different pressure ratios are investigated at Knudsen number 0.001. The DSMC method is physically accurate for all flow regime; however it is computationally expensive in high density or near continuum regions. The Fokker-Planck-DSMC hybrid scheme employs DSMC in rarefied regions and Fokker-Planck method in near continuum flow regions for an efficient and accurate solution. Numerical procedures of the cubic Fokker-Planck method are implemented within the framework of an existing DSMC-solver, SPARTA. The Fokker-Planck-DSMC hybrid solution reproduces pure DSMC solution with improved computational efficiency up to a factor of five for vacuum flow through a thin orifice. In addition, breakdown of translational equilibrium is investigated. Domain criterion of FP-DSMC is safely smaller than Bird's breakdown criterion. Highlights • Rarefied diatomic gas flow through a thin slit and a thin orifice. • Access of accuracy and efficiency of cubic FP-DSMC for diatomic gas flow. • Investigation of breakdown of translational equilibrium on cubic FP-DSMC hybrid. • Comparison of diatomic and monoatomic flow through a thin slit and a thin orifice. [ABSTRACT FROM AUTHOR]

Burnett equations were originally derived in 1935 by Burnett by employing the Chapman-Enskog expansion to Classical Boltzmann equation to second order in Knudsen number Kn . Since then several variants of these equations have been proposed in the literature; these variants have differing physical and numerical properties. In this papers, we considered four such variants which are known in the literature as ‘the Original Burnett (OB) equations, the Conventional Burnett (CB)'equations, ‘the Augmented Burnett (AB)’ equations and the recently formulated by the authors ‘the Simplified Conventional (SCB) equations.’ One of the most important issues in obtaining numerical solutions of the Burnett equations is their stability under small perturbations. In this paper, we perform the linearized stability (known as the Bobylev Stability) analysis of three-dimensional Burnett equations for all the four variants (OB, CB, AB, and SCB) as far as the authors are aware for the first time in the literature on this subject. By introducing small perturbations in the steady state flow field, the trajectory curve and the variation in attenuation coefficient with wave frequency of the characteristic equation are obtained for all four variants of Burnett equations to determine their stability. The results show that the 3-D Augmented Burnett (AB) equations and the Simplified Conventional Burnett (SCB) equations are unconditionally stable under small wavelength perturbations. However, the Original Burnett (OB) and the Conventional Burnett (CB) equations are unstable when the Knudsen number becomes greater than a critical value and the stability condition worsens in 3-D when compared to the stability condition for 1-D and 2-D equations. The critical Knudsen number for 3-D OB and CB equations is 0.061 and 0.287 respectively. It should be noted that although both AB and SCB equations are unconditionally stable, SCB equations are much simpler to use numerically compared to AB equations without compromising accuracy. [ABSTRACT FROM AUTHOR]

A backward facing micro-step is a building block for many microfluidic devices. Due to micron sized characteristic dimensions, the gas flow in such a geometry is rarefied in nature. Such rarefied gas flows are widely solved using the Direct Simulation Monte Carlo (DSMC) technique. Flow separation, circulation and re-attachment are some of the basic characteristics of step flows. The objective of this study is to analyze the effect of rarefaction on the flow properties and the separation of the flow. The range of selected Knudsen number (Kn) covers the slip and transition regime from a value of 0.0311–13.25. The pressure ratios employed are 3 and 5. It is observed that the slip velocity continuously increases while the centre-line velocity first decreases, then remains constant and finally increases with increase in Kn. At the step, separation of the flow is seen for Kn < 0.1325 while no such separation is observed in the range of Kn from 0.198 to 13.25. The corresponding Re for these ranges are 6.43 to 0.67 and 0.392 to 0.012 respectively. The re-attachment length decreases with increase in Kn whereas it increases with increase in Re. A stronger pressure force and a weaker diffusion effect leads to flow separation in the slip regime whereas stronger diffusion and weaker pressure force lead to an absence of flow separation in the transition regime. Finally, this work presents for the first time the existence of the Knudsen minimum for such a backward step geometry. [ABSTRACT FROM AUTHOR]

The problem about the unsteady outflow of a mixture of gases into a vacuum, periodically interrupted by bodies moving toward the jet, was studied. This problem was considered as a possible gas separation technique. The unsteady flow of the gas mixture was simulated using the event-driven molecular dynamics (EDMD) approach under the following conditions: (1) moving boundaries, (2) a wide range of Knudsen number, (3) a wide range of Mach number, (4) violation of equilibrium between rotational and translational degrees of freedom. The problem of periodically interrupted flow was solved as close to reality as possible, taking into account the finite gap between the stator and the moving runner blades. The dependences of the molar fraction and flow rate of the resulting mixture on the system parameters were obtained. As a result, a number of practically important recommendations for the design of such systems have been formulated. • Periodically interrupted outflow of a mixture of gases into a vacuum was studied as a possible gas separation technique. • Interruption of the outflow allows increasing the molar fraction growth by 2.5 times. • It is better to use short slots, as they allow high flow rates at high molar fraction of the product. • The maximum molar fraction growth is achieved at values of the rarefaction parameter δ = 5 − 10. [ABSTRACT FROM AUTHOR]

The heat transfer in buoyancy driven rarefied gas in a conjugate cavity has been investigated in this paper. The governing equations along with the slip flow and temperature jump boundary conditions are solved using finite-volume technique. The simulations are carried by varying cavity aspect ratios, conductivity ratios and cavity tilt angles. The Nusselt number values are calculated for different Knudsen, Rayleigh and Prandtl numbers. It was found that Kn number is the most influential parameter on the heat transfer; as it increases the heat transfer in the cavity decreases. The effect of all other investigated factors was found and documented; however, it is found that, in general, at high Kn Numbers their effect on the heat transfer is marginal. It is also found that cavity walls with good conducting materials will have minor effect on heat transfer while insulation material will have moderate effect on heat transfer. Two correlations for average Nusselt number are proposed: one for horizontal cavities; N u = 0.149 k r 0.0101 R a 0.135 K n − 0.393 Pr 0.178 and the second correlation equation introduces the effect of the cavity tilt angle along with Ra and Kn; Nu = 0.158 Ra 0.130 , Kn −0.396 , cos( θ ) 0.216 . [ABSTRACT FROM AUTHOR]

A model gas separator composed of two Knudsen pumps is fabricated. The only energy source of the gas separator is the temperature difference up to 100 K. The temperature difference induces the thermal transpiration flow of the rarefied gas in the micro-pores of the mixed cellulose ester (MCE) membrane in the Knudsen pump. The gas flow and the resultant pressure variation are used to separate the feed mixture of helium and neon gases into two gases having different gas compositions. The fabricated model gas separator, which has 27 cm 2 total membrane area, induces 5–6% difference of mole percentage between two product gases under the flow rate 5 sccm of the feed gas. The operation of the gas separator is continuous since the present device as well as the Knudsen pump requires no moving parts. [ABSTRACT FROM AUTHOR]

The paper presents some results of computing of a gas mixture flows in Knudsen pumps. The simulations are carried out by numerically solving the Boltzmann kinetic equation by the conservative projection method. Interactions of gas molecule are modeled by Lennard-Jones and the newest ab initio potentials. The values of the pumping rate, gas mixture separation, and the graphics of gas flow parameters inside the pumps are given. The results obtained with different molecular models are compared and the influence of the molecular potentials on macroscopic flow parameters is shown. The simulations are performed in plane and 3D geometries. [ABSTRACT FROM AUTHOR]

Rarefied thermally-induced flows generate Knudsen thermal force on structures immersed in a gas and are significant for vacuum microbalances as well as for microsystem sensors and actuators. We study experimentally and numerically the influence of transverse thermal gradients on the Knudsen thermal force. A Peltier device provided the thermal gradient in a beam located in close proximity of a large reaction plate, whose displacement has been sensed using a torsional balance. The pressure in the vacuum chamber varied between 1 and 100 Pa giving a range of Knudsen numbers from the slip to nearly free molecular regimes, based on the gap between the beam and the movable plate. Switching polarity of the Peltier heater produces reversal of the Knudsen thermal force, from repulsive to attractive. The numerical solution of Ellipsoidal Statistical Bhatnagar–Gross–Krook (ESBGK) kinetic model of the Boltzmann equation agrees well with experimental data. The obtained non-dimensional Knudsen force coefficient, dependent on the Knudsen number, temperature ratio and the gap height to beam width aspect ratio, quantifies the magnitude of the force for a wide range of generic conditions and could be used for design and analysis of sensors and actuators exploiting this rarefied flow effect. [ABSTRACT FROM AUTHOR]

Steady flows of rarefied single gases through long channels with double trapezoidal cross-section shapes are investigated numerically. The flow is between two large reservoirs having the gas at different pressures. The channels have constant cross-sections along the axial direction. The gas is modeled by the linearized Bhatnagar–Gross–Krook kinetic equation. The diffuse reflection boundary condition is used at the channel walls. The solution of the problem is divided into two stages. At a particular cross-section, the flow is driven by the local pressure gradient. First, the local problem for an arbitrary driving term is solved by using the discrete velocity method. This solution yields the dimensionless flow rate and the velocity profile for a wide range of the gaseous rarefaction. Second, the global flow behavior, i.e. the flow rate and the distribution of the pressure, is deduced for global pressure driven flows on the basis of the conservation of mass. The numerical solution of the kinetic equation is based on the discretization of the spatial and velocity spaces. The spatial space is represented on a rectangular grid. The walls of the channels are aligned parallel to the grid lines or along the diagonal of the grid. Such a choice provides a straightforward calculation of the spatial derivatives. In the interior part of the domain and near the channel walls, second- and first-order finite difference forms are used, respectively. The velocity space is represented by a Gauss–Legendre quadrature. The resulting discrete equations are solved in an iterative manner. The dimensionless flow rates are calculated and tabulated for particular cross-sections in a wide range of the gaseous rarefaction. The flow rate function exhibits the Knudsen minimum. The results are compared to the corresponding ones with other cross-sections. Typical velocity profiles are also shown. Finally, representative results are delivered for global pressure driven flows. [ABSTRACT FROM AUTHOR]

The paper is devoted to the analysis of the time-dependent rarefied gas flow into vacuum from a circular pipe closed at one end. The problem is studied numerically on the basis of the S-model kinetic equation. Limiting solutions corresponding to the one-dimensional free-molecular and hydrodynamic flow regimes are also considered. The results demonstrate the dependence of flow pattern and evacuation time on the Knudsen number. [ABSTRACT FROM AUTHOR]

Several designs of Knudsen pumps, the principle of which is based on a thermally driven flow generated along a microchannel, have been proposed in the literature. The fabrication of efficient prototypes, however, is generally limited by the difficulty to control the temperature distribution along the walls. An alternative possibility, which only requires isothermal hot or cold walls, is investigated. The pumping element consists of two facing isothermal ratchet surfaces with different temperatures. The asymmetric saw-tooth like surfaces lead to a rectified Knudsen flow along the walls. This flow is numerically simulated in the slip flow regime with Navier–Stokes equations and appropriate first-order velocity slip, including thermal creep and wall curvature effects, as well as temperature jump, boundary conditions. The influence of the geometrical parameters is investigated, among which the ratchet angle and the alignment of the ratchet pattern. The influence of the Knudsen number and of the temperature difference is analyzed as well and guidelines are drawn for designing a prototype. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of a rarefied gas flow through a composite circular pipe into vacuum. The pipe is made of two cylindrical sections of different diameters. Two cases are studied: wider pipe followed by narrower pipe (converging configuration) and the narrower pipe followed by the wider (diverging configuration). The analysis is based on the direct numerical solution of the Boltzmann kinetic equation with the S-model collision integral. The results are presented for two length to radius ratios and a large range of Knudsen numbers. The main computed characteristic is the mass flow rate through the pipe. The dependence of the flow field on pipe's geometry and Knudsen number is established. Formation of special features of the flow, such as recirculation zones and a Mach disk, is studied. [ABSTRACT FROM AUTHOR]

The Holweck pump is a device widely used in many pumping systems. It can be a self-standing device or part of a multi-stage system, usually combined with a turbo-molecular pump (TMP). In the present work, an existing methodology [Sharipov et al., 2006] for the simulation of a Holweck pump with channels of constant cross sections, is extended and becomes more general by including pumps with tapered channels. The implemented procedure is based on the kinetic approach, is applicable in the whole range of the Knudsen number and is divided into two steps. First the flow through a tapered channel of a T-shape cross section is decomposed in four simple two-dimensional flows and a database for the dimensionless flow rate in terms of the reference gas rarefaction and the channel geometrical data is built based on linear kinetic theory. Then, using the mass conservation principal and the kinetic database, an ordinary differential equation is solved to yield the pressure evolution along the pump and the pump characteristic quantities of practical interest. Comparison of simulation results with existing experimental data has shown good agreement, proving the validity of the method and its capability of simulating the flow in a Holweck pump with small computational effort. In addition, the great advantage of the method for fast and easy-to-go parametric and sensitivity analysis concerning various operational and geometrical characteristics of the pump is demonstrated. [ABSTRACT FROM AUTHOR]

The spherical expansion of mon- and diatomic gases in a spherically symmetric gravitational field is studied in kinetic simulations. The simulations are performed with the direct simulation Monte Carlo (DSMC) method in the range of the source Knudsen number, the ratio of the mean free path of gas molecules to the source radius, from 0.01 to 0.001, and in the range of the source Jeans parameter, the ratio of the characteristic gravitational binding energy of a gas molecule to its thermal energy, from 0 to 6. The exit boundary position R 1 of the computational domain was varied from 10 to 800 times the source radii R 0 in order to quantify the effect of the size of the computational domain and kinetic escape boundary conditions on the flow structure and escape rates. It is shown that the flow structure and escape rates for both mon- and diatomic gases are highly sensitive to the position of the exit boundary, when the source Knudsen number is sufficiently small and the source Jeans parameter corresponds to the transition from blow-off to Jean-like thermal escape. Outside the transitional range, the flow structures and escape rates are only marginally affected by the position of the exit boundary. In the case of a diatomic gas, the calculated flow structures converge with increasing size of the computational domain. In the case of a monatomic gas, the flow structures calculated for the transitional range of the Jeans parameter do not exhibit convergence with respect to increasing size of the computational domain in the considered range of R 1 / R 0 . It is shown that such flows remain subsonic and retain local equilibrium at arbitrary large distances from the source. The solution of the kinetic problem for a given molecular model in these conditions is uniquely defined by three parameters including the source Jeans parameter and Knudsen number and an additional parameter that constrains the flow far downstream. This additional parameter can be chosen in the form of the imposed number escape rate. [ABSTRACT FROM AUTHOR]

A hybrid solver, dynamically coupling the direct numerical solution of the kinetic equation and hydrodynamic Navier–Stokes equations is applied for numerical investigation of the gas flow through a slit at pressure ratio of 0.1, 0.5 and 0.9, including flow into vacuum and Knudsen number from slip to transitional regime. The coupling is achieved by matching half fluxes at the interface of kinetic and Navier–Stokes sub-domains, which are decomposed based on the local Knudsen number and macroparameters gradients. The solver efficiency is increased via MPI (Message Passing Interface) parallelization. The improvement in accuracy over Navier–Stokes equations and the computational efficiency of the proposed solver are assessed via comparison with pure kinetic solution in terms of mass flow and local parameters. The capability of hybrid solver to apply for vacuum science problems is demonstrated. [ABSTRACT FROM AUTHOR]

The thermophoretic force on a sphere with arbitrary thermal conductivity immersed in a rarefied gas subject to a uniform temperature gradient is calculated in the framework of kinetic theory of gases in a range of the Knudsen number which covers the free molecular, transition and continuum regimes of the gas flow around the sphere. The condition of continuity of the radial heat flux at the gas–solid interface is used to calculate the thermophoretic force as a superposition of the thermophoretic force on a sphere with an uniform temperature and the radiometric force on a sphere due to its thermal polarization. Both solutions were previously obtained from the Shakhov model to the linearized Boltzmann equation and the Cercignani–Lampis model of gas–surface interaction. To analyze the influence of the gas and particle thermal conductivities on the thermophoretic force, spheres of PolyStyrene Latex (PSL), glass and nickel in helium and argon gases are considered. The force is calculated for several values of tangential momentum accommodation coefficient and normal energy accommodation coefficient as introduced in the Cercignani–Lampis scattering kernel. A comparison with experimental data available in the literature allowed to extract the values of the accommodation coefficients for helium interaction with the particle surface. • Thermophoretic force from the S-model for the linearized Boltzmann equation. • Cercignani–Lampis scattering kernel as the boundary condition. • Analysis of the influence of the accommodation coefficients on the force. • Comparison with experimental data for PolyStyrene Latex, glass and nickel spheres in helium. [ABSTRACT FROM AUTHOR]

A binary gas mixture conductance measurement apparatus with an independent gas distribution system was established, the constant volume method to measure the conductance of the mixed gas was used. Also, the conductance of pure gas Ar, He, N 2 and binary gas mixtures Ar–He, N 2 –He, Ar–N 2 through a short tube under different gas distribution chamber pressures and different molar fractions were measured. Meanwhile, the equivalent single gas method was used to calculate the conductance of binary gas mixtures. To verify the accuracy of this method, the relative error between the calculated and the measurement was carried out. The result shows that the equivalent single gas method is suitable for a small Knudsen number (1 / K n ≥ 10), the molar fraction of lighter species is low and a small difference in the molecular mass of the components in the mixed gas. The uncertainty of conductance measurement is better than 3.1%. • A binary mixed gas conductance measurement apparatus with an independent gas distribution system was established. • The conductance of binary mixed gas at different G-D chamber pressures and different molar fractions was measured. • The equivalent single gas method was used to calculate the conductance of binary gas mixture. [ABSTRACT FROM AUTHOR]

Cercignani-Lampis-Lord (CLL) is a boundary condition used in the DSMC method, due to the use of tangential and normal momentum accommodation coefficients (TMAC and NMAC) extractable by Molecular Dynamics (MD) simulations or experimental platforms, the accuracy of this boundary condition is by up to 15% relative to diffuse model (Mehta et al., 2014) [ 1 ]. These coefficients in slip, transitional, and free molecular flow regimes takes a value other than the unit. Since the temperature and velocity of the molecules after their scattering from the surface in the CLL model, drag coefficient, slip velocity, shear stress, and jumping temperature is completely dependent on the values of these coefficients, it is very important to determine these coefficients for different gases and surfaces. In this paper, first the DGEBA-TETA/carbon composite was simulated with MD method using an efficient algorithm and then using this method shown that in the interaction of heavy gases such as uranium, tellurium and sulfur hexafluoride with a composite surface, TMAC and NMAC were strongly dependent on Knudsen number and temperature of gases, velocity and temperature of composite, and the dependency of the TMAC was less than the NMAC for different conditions of gas characteristics. In these cases, tellurium hexafluoride had the greatest impact. • Momentum and energy exchange occurs in the interaction of gas molecules and the surface. • The exact amount of this information exchange can be determined by molecular dynamics simulation. • The values of momentum accommodation coefficients of a composite surface with heavy gases were extracted. • An efficient algorithm was used to molecular dynamics simulation of DGEBA-TETA/Carbon composite. • Variations of the gas and composite conditions, cause the scattering on the surface deviates from the diffuse model. [ABSTRACT FROM AUTHOR]

In this article, a new hybrid approach combining Fokker Planck (FP) and Direct Simulation Monte Carlo (DSMC) is introduced. The FP approach, which is an approximation of the Boltzmann equation, has been recently introduced for modeling rarefied gas flows. Compared to standard molecular methods like DSMC, it has a lower computational cost. However, the FP model may provide erroneous results in modeling some flow features, including vortex region and shock waves. The DSMC approach is accurate enough; yet, it has a high computational cost, especially at low Knudsen/low Mach number conditions. The purpose of this article is to suggest an optimized hybrid algorithm to benefit from high-speed modeling and sufficient accuracy simultaneously. We show a modified form of the gradient length Knudsen number, i.e., Kn GL /Kn, must be considered to define the range of accuracy of the FP method. We used Kn GL /Kn as a parameter for switching between DSMC and FP. The cavity flow and the nozzle flow are considered for the investigation of the proposed hybrid algorithm. The dependence of the accuracy and computational cost on Kn GL /Kn is reported. The results demonstrate that the optimized Kn GL /Kn is between 3 and 4, which gives suitable computational cost and acceptable accuracy. • Report on the evaluation of a hybrid Fokker-Plank-DSMC approach to treat micro-cavity and micro-nozzle flows. • Using a novel criterion for switching between Fokker-Plank and DSMC solvers. • A thorough study of the numerical efficiency of the hybrid scheme at various Kn and back pressure magnitudes. [ABSTRACT FROM AUTHOR]

A detailed study on the performance and accuracy of the Fokker Planck (FP) approach in treating shear driven flows over a wide range of Knudsen numbers and Mach numbers at subsonic and supersonic regimes is considered. One-dimensional Couette flow and the two-dimensional cavity problem are considered. The FP method is evaluated in the Couette flow at a subsonic Mach number of 0.16 (U w = 50 m/s) and at the supersonic Mach number of 3.1 (U w = 1000 m/s), where Knudsen numbers range from 0.005 to 1. Correspondingly, the cavity flow is investigated at a wall Mach number of 0.31 (U w = 100 m/s) and wall Mach number of 0.93 (U w = 300 m/s) at Knudsen numbers ranging from 0.05 to 20. Interestingly, the results show that by increasing the wall velocity and Knudsen numbers, the accuracy of the FP approach increases in treating the cavity flow. In addition to the standard Knudsen number, we show that gradient length Knudsen number, Kn GL , should be considered to determine the range of accuracy of the FP scheme. The latter depends on the strength of the center vortex of the cavity diminishing at higher rarefied conditions. The results demonstrate that the computational efficiency of the FP approach enhances at higher lid velocity. • First report on the evaluation of the Fokker-Plank scheme at high Knudsen shear-driven flows. • A thorough study of the numerical efficiency of the FP scheme at various Kn and Mach numbers. • Presenting a novel parameter in detecting the border of the accuracy of the FP scheme. [ABSTRACT FROM AUTHOR]