Molecular Mean Free Path Research Articles

Multiscale simulation of molecular gas flows by the general synthetic iterative scheme

The in-depth knowledge of rarefied gas dynamics is crucial to address challenges in a wide range of engineering problems, where gas flows are usually multiscale, i.e., covering a wide range of Knudsen numbers. As the traditional Navier–Stokes equations fail, gas kinetic equations are required to model the flows. So far, very few numerical methods are designed to efficiently solve the multiscale gas dynamics and reveal the role of internal degrees of freedom of gas molecules. In this work, a general synthetic iterative scheme (GSIS) is proposed to find steady-state solutions of the gas kinetic equations for molecular gas flows accurately and efficiently, where the gas kinetic equations are solved together with the macroscopic synthetic equations that expedite solutions towards the steady state. In the macroscopic synthetic equations, while the momentum equation is the same as that used in the GSIS for monatomic gas, two energy equations are introduced here for polyatomic gases: one is for the translational energy and the other for the internal energy; these equations are derived exactly from the gas kinetic equations hence no approximation is made in final solutions. The Fourier stability analysis is performed to show that the GSIS permits fast convergence to steady-state solutions in the entire flow regime; meanwhile the asymptotic analysis shows that the GSIS recovers the Navier–Stokes equations when the Knudsen number is small, even on the spatial grid with cell size much larger than the molecular mean free path. With all these unique features, several challenging numerical examples are given to show that the proposed GSIS is a promising tool to simulate multiscale molecular gas flows and investigate the effects of internal degrees of freedom.

Coupled confined phase behavior and transport of methane in slit nanopores

The proximity of the order of magnitude molecular mean free path and pore sizes in nanopores leads to remarkable interactions between molecules and walls. In such systems, the thermodynamic property and the transport behavior of fluids deviate from those at bulk conditions. Molecular dynamics (MD) simulation may be used to investigate the effects of confinement on fluid physics in nano-size pores. However, translating these subscale observations into a larger scale of interest remains a challenging task. In this work, we propose a modified extension of Peng-Robinson equation of state (PR-EOS) motivated and guided by MD simulation results. The shift of critical properties, i.e., pressure and temperature, are evaluated independently. A temperature dependent parameter is introduced to account for the existence of capillary pressure in the two-phase region. This formulation is capable of capturing the shift in critical properties as well as density phase diagrams under various confinement scenarios. We incorporate the proposed EOS in the lattice Boltzmann method (LBM) to study coupled confined phase behavior and transport of methane in nano-size slit pores. Adsorptive strength is determined such that the resulting density ratio matches that from MD simulation. Results indicate that the use of the proposed EOS leads to smaller fluid velocities compared to PR-EOS. This effect is due to stronger interactions between fluid particles under confinement. Also, results show that transport characteristics are impacted by the pressure of the system: in systems with a relatively low pressure, transport seems to be dominated by Knudsen diffusion; however, at a higher pressure, the contribution from viscous flow increases as the pore widens while the influence of Knudsen and surface diffusion diminishes.

On the motion of slightly rarefied gas induced by a discontinuous surface temperature

The motion of a slightly rarefied gas in a long straight two-dimensional channel caused by a discontinuous surface temperature is investigated on the basis of kinetic theory with a special interest in the fluid-dynamic description. More precisely, the channel is longitudinally divided into two parts and each part is kept at a uniform temperature different from each other, so that the surface temperature of the whole channel has a jump discontinuity at the junction. Under the assumption that the amount of jump in the surface temperature is small, the steady behaviour of the gas induced in the channel is studied on the basis of the linearized Boltzmann equation and the diffuse reflection boundary condition in the case where the Knudsen number, defined by the ratio of the molecular mean free path and the width of the channel, is small. Using a matched asymptotic expansion method combined with Sone’s asymptotics, a Stokes system describing the overall macroscopic behaviour of the gas inside the channel is derived, with a new feature of the ‘slip boundary condition’ for the flow velocity due to the jump discontinuity in the surface temperature of the channel. This condition takes the form of a diverging singularity with source and sink located at the point of discontinuity, with a multiplicative factor determined through the analysis of a spatially two-dimensional Knudsen-layer (or a Knudsen-zone) problem. Some numerical demonstrations based on the Bhatnagar–Gross–Krook equation are also presented.

Nanoconfined Fluids: What Can We Expect from Them?

Nanoconfined fluids (NCFs), which are confined in nanospaces, exhibit distinctive nanoscale effects, including surface effects, small-size effects, quantum effects, and others. The continuous medium hypothesis in fluid mechanics is not valid in this context because of the comparable characteristic length of spaces and molecular mean free path, and accordingly, the classical continuum theories developed for the bulk fluids usually cannot describe the mass and energy transport of NCFs. In this Perspective, we summarize the nanoscale effects on the thermodynamics, mass transport, flow dynamics, heat transfer, phase change, and energy transport of NCFs and highlight the related representative works. The applications of NCFs in the fields of membrane separation, oil and gas production, energy harvesting and storage, and biological engineering are especially indicated. Currently, the theoretical description framework of NCFs is still missing, and it is expected that this framework can be established by adopting the classical continuum theories with the consideration of nanoscale effects.

A mean free path approach to the micro/nanochannel gas flows

We investigate the gas flows near to solid surfaces in terms of the local spatial variation in the molecular mean free path (MFP). Molecular dynamics (MD) is the appropriate scientific tool for obtaining molecularly-accurate dynamic information in micro and nano-scale gas flows, and has been used to evaluate the molecular mean free path of gases. In the calibration procedure, the viscosity of a gas in the homogeneous case can be recovered in our MD simulations and reach good agreement with the theoretical prediction and data from NIST. In surface-bounded gas flows, if the collisions between gas molecules and walls are counted, a spatially-varying mean free path is presented, and for the first time we have observed that the distribution of the free paths deviates from the exponential one and spikes appear in their distributions at larger Kn, i.e. in the transition flow regime. Based on elementary kinetic theory, the effective viscosity of the gas derived from the mean free path has been incorporated into the framework of the continuum-fluid dynamics equations, and micro-Couette flows are performed to demonstrate this potential application.

Fast Convergence and Asymptotic Preserving of the General Synthetic Iterative Scheme

Recently the general synthetic iteration scheme (GSIS) was proposed for the Boltzmann equation [W. Su et al., J. Comput. Phys., 407 (2020), 109245], where various numerical simulations have shown that (i) the steady-state solution can be found within dozens of iterations at any Knudsen number $K$, and (ii) the solution is accurate even when the spatial cell size in the bulk region is much larger than the molecular mean free path, i.e., the Navier--Stokes solutions are recovered at coarse grids. The first property indicates that the error decay rate between two consecutive iterations decreases to zero along with $K$, while the second one implies that the GSIS asymptotically preserves the Navier--Stokes limit when $K$ approaches zero. This paper is first dedicated to the rigorous proof of both properties. Second, several numerically challenging cases (especially the two-dimensional thermal edge flow) are used to further demonstrate the accuracy and efficiency of GSIS.

Development of a nanoscale hot-wire probe for supersonic flow applications

A new nanoscale thermal anemometry probe (NSTAP) was designed and fabricated to measure mass flux in supersonic flows. This sensor was evaluated in the Trisonic Wind Tunnel Munich (TWM) at both subsonic and supersonic speeds. Subsonic compressible flow tests were performed to confirm the new sensor’s repeatability and to compare its behaviour to measurements from a conventional cylindrical hot-wire, while supersonic tests were performed to investigate the nature of the convective heat transfer from the nanoscale sensor at those conditions. For the range of mass fluxes tested in the supersonic regime, a linear relationship between the Nusselt number and the Reynolds number fit the data well. A linear relationship has previously been noticed at length scales close to the molecular mean free path of the flow and has been attributed to the free-molecule flow regime, where the Knudsen number is on the order of unity.

Gas permeability in rarefied flow conditions for characterization of mineral membrane support

Abstract Gas Permeability Measurement Technique (GPMT) has the advantage of being a non-destructive method, which is efficient in characterizing filtration membranes. Ceramic filtration membranes consist of successive layers of micro (support) to nano size (skin) pores. When gas flows through such a small scale structure, the molecular mean free path becomes comparable to the pore size. The Slip flow model, validated to describe the gas transport properties under rarefied flow conditions in a microchannel, is extended to porous media. The porous structure is modeled as a cluster of several identical cylindrical channels. By measuring the pressure drop Δ P at several different mean pressures, the pore radius and the porosity on square tortuosity ratio ϵ ∕ τ 2 of the porous model structure that have the same flow property were estimated.

Effective mean free path and viscosity of confined gases

The molecular mean free path (MFP) of gases in confined geometries is numerically evaluated by means of the direct simulation Monte Carlo method and molecular dynamics simulations. Our results show that if calculations take into account not only intermolecular interactions between gas molecules but also collisions between gas molecules and wall atoms, then a space-dependent MFP is obtained. The latter, in turn, permits one to define an effective viscosity of confined gases that also varies spatially. Both the gas MFP and viscosity variation in surface-confined systems have been questioned in the past. In this work, we demonstrate that this effective viscosity derived from our MFP calculations is consistent with those deduced from the linear-response relationship between the shear stress and strain rate using independent nonequilibrium Couette-style simulations as well as the equilibrium Green-Kubo predictions.

An extended gas-kinetic scheme for shock structure calculations

In super/hyper-sonic flows in near space and low speed flows around MEMS, the gas distribution function deviates from the equilibrium, because the molecular mean free path is comparable with the characteristic length of the flow in certain part of the flow flied. The non-equilibrium distribution function makes the constitutive relations of stress and heat flux deviate from the linear form used in NS equation. Methods for rarefied flow (DSMC and DVM) and unified methods (UGKS, GKUA and DUGKS) are able to simulate non-equilibrium flows, while, for flows in near continuum regimes, their computational costs are much larger than that of the continuum NS method. For the flows in near space and around MEMS, the rarefaction and non-equilibrium effects are not too strong as these in free molecular flows. Therefore, an Extend Gas-Kinetic Scheme (EGKS) is proposed in the present article for near continuum flows and weak non-equilibrium flows where certain amount of non-equilibrium exists. As an extension of the Gas-Kinetic Scheme (GKS), by using a perturbation of the distribution function that has the maximum entropy under the constrains of conservation variables, stress and heat flux, the nonlinear constitutive relations can be directly introduced into the EGKS. There are many nonlinear constitutive relations so far, whose numerical predictions are conducted in their own frameworks. There is a huge demand for a unified and physical framework in order to investigate their validity and accuracy. Since all these constitutive relations are derived from the gas-kinetic theory, a GKS framework is chosen in consideration of physical accommodation. Burnett, SCB, RCE and NCCR nonlinear constitutive relations are used in the EGKS along with linear Stokes-Fourier relations. Using the shock structure cases in a wide range of Mach number from weak non-equilibrium to high non-equilibrium (Mach number from 1.2 to 20), the EGKS scheme and the nonlinear constitutive relations are verified. The density, temperature, stress and heat flux profiles along with shock thickness and density-temperature separation, are systematically examined. Since the accuracy of the EGKS depends on the model of stress and heat flux used, the EGKS with nonlinear constitutive relations gives much better predictions than that with linear constitutive relations in high Mach number cases. Benefit from the present framework, the RCE and NCCR constitutive relations are also modified using the concept and property of the entropy generation of collision term, the modified constitutive relations provide good predictions in high Mach number shock structure calculations.

Photon attenuation coefficients of oxide dispersion strengthened steels by Geant4, XCOM and experimental data

The total mass attenuation coefficients (μ/ρ) of oxide dispersion strengthened (ODS) steels have been investigated by using Geant4 simulation toolkit at energies ranging from 0.01 MeV to 100 GeV. Moreover, Geant4 is applied to further study on the partial mass attenuation coefficient of three main interactions including photoelectric effect, compton scattering, gamma conversion between incident photon and ODS steels. Then molecular cross-section (σm) and mean free path (λ) are obtained by the total mass attenuation coefficients. The results obtained by Geant4 are compared with those obtained by experiments and XCOM program and these results agree with each other very well. It is concluded that Geant4 Monte Carlo toolkit is a powerful tool in studying interaction of photons in material and energies. The shielding parameters of ODS steels presented in this paper can be very useful for the potential nuclear application and other industries in future.

Characterization of a Knudsen force based vacuum sensor for N2[sbnd]H2O gas mixtures

Radiometric phenomena can arise in gas flows in which the local molecular mean free path is in the same order as the temperature gradient length scale. The forces exerted on immersed bodies by such rarefied flows are known as Knudsen (or thermophoretic) forces and can play a major role in nonisothermal microflows. In this work, the direct simulation Monte Carlo (DSMC) method is used to numerically characterize the force and thermal mechanisms on a second generation Microscale In-Plane Knudsen Radiometric Actuator (MIKRA) designed for gas pressure and composition sensing. First, a brief verification analysis of the present DSMC solver in reproducing the thermal conductivity for the gas species of interest – nitrogen and water vapor – is carried out. The geometric impact of the gap between MIKRA beams is then investigated for pure N2 environments. In addition, the beam heat transfer coefficients for N2H2O binary mixtures are extracted from the DSMC data and a dimensionless correlation is proposed for predicting the corresponding Knudsen forces. These results are intended to assist in the operation and design of the MIKRA device.

Meteor echo height ceiling effect and mesospheric temperature estimation from meteor radar observations

Abstract. The mesospheric temperature estimation from meteor height distribution is reevaluated by using the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) and the King Sejong Station (KSS) meteor radar observations. It is found that the experimentally determined proportionality constant between the full width at half maximum (FWHM) of the meteor height distribution and temperature is in remarkable agreement with theoretical value derived from the physics-based equation and it is nearly time-invariant for the entire observation period of 2012–2016. Furthermore, for the first time we found that the FWHM provides the best estimate of temperature at slightly lower height than the meteor peak height (MPH) by about 2–3 km. This is related to the asymmetric distribution of meteor echoes around MPH, which is known to be caused by the meteor echo height ceiling effect (MHC). At higher altitude above MPH, the meteor detection rate is greatly reduced due to the MHC, and the cutoff height for this reduction follows a fixed molecular mean free path of the background atmosphere. This result indicates that the meteor height distribution can be used to estimate the mesospheric temperature, even under the asymmetric meteor echo distribution caused by the MHC at high altitude.

Gas flow regimes judgement in nanoporous media by digital core analysis

Abstract A method to judge shale gas flow regimes based on digital core analysis is proposed in this work. Firstly, three-dimensional shale digital cores in an anonymous shale formation in the Sichuan Basin are reconstructed by a Markov Chain Monte Carlo (MCMC) algorithm based on two-dimensional Scanning Electron Microscope (SEM) images. Then a voxel-based method is proposed to calculate the characteristic length of the three-dimensional shale digital core. The Knudsen number for three-dimensional shale digital cores is calculated by the ratio of the molecular mean free path to the characteristic length and is used to judge the flow regimes under different reservoir conditions. The results indicate that shale gas flow regimes are mainly located at the slip flow and transition flow region. Furthermore, adsorption has no obvious influence on the free gas flow regimes. Because adsorption only exists in organic pores, three-dimensional inorganic pores and organic pores in the Haynesville shale formation are reconstructed by a MCMC algorithm based on two-dimensional SEM images. The characteristic lengths of the three-dimensional inorganic pores and three-dimensional organic pores are both calculated and gas flow regimes in organic pores and inorganic pores are judged.

The Effects of Intermolecular Energy Transfers on a Closed Power Cycle’s Efficiency

The efficiency of the power cycle described in this paper is affected by the transfer of energy between molecular potential and translational during the expansion and contraction of its working fluid. How the thermodynamic interrelationships between the temperature, volume changes, and changes in the molecular mean free paths determine this energy shift and their affects on cycle efficiency is discussed.

New phenomenological model of multicomponent gas diffusion

A new phenomenological model is proposed for diffusion of multicomponent gas mixtures. Obvious general structural relations between the mass diffusion fluxes and the gradients of gas mixture components are obtained within the framework of the model in the case of the multicomponent diffusion coefficients expressed in terms of the “intrinsic” diffusion coefficients of individualmixture components. The molecular mean free path model can be used to estimate the latter.

A wall grid scale criterion for hypersonic aerodynamic heating calculation

The theory of molecular motion was used to study the wall cell Reynolds number method, and it found that the height of the first grid point off wall is related to the molecular mean free path near wall. A wall grid scale criterion based on the molecular mean free path at the stagnation point was proposed by considering both the flow parameters and the wall temperature boundaries. The criterion was verified through numerical calculations and comparative analyses of the wall heat flux in different hypersonic flow fields. Research results draw the following conclusions: The optimal height of the first grid point off wall is a molecular mean free path. The influence of height of the first grid point off wall on aerodynamic heat is much greater than that of aerodynamic force. The criterion is not only effective for the wall heat flux predictions in air conditions, but also useful to that of non air environments. With the weakening of aerodynamic heating along the flow direction, the effect of wall grid scale on the wall heat flux gradually decreases.

Preperetions of bi-layer and multi-layer graphene on copper substrates by atmospheric pressure chemical vapor deposition and their mechanisms

Chemical vapor deposition is widely utilized to synthesize graphene with controlled properties for many applications. And it is one of the most important methods for the preparation of graphene with high quality in large area. Cu substrate is most commonly used for the preparation of graphene in chemical vapor deposition. As is well known, the properties of graphene are greatly affected by the number of layers. However, the syntheses and mechanisms of bi-layer and multi-layer graphene on Cu substrates are still under debate. And how to make a breakthrough in realizing the controllable syntheses of bi-layer and multi-layer graphene on Cu substrates has become a direction for many researchers. In this work, we report bi-layer and multi-layer graphene on Cu substrates prepared by atmospheric pressure chemical vapor deposition. Firstly, the Cu foil is placed on the quartz slides of the tube furnace and heated to 1000℃ with a rate of 15℃/min. After reaching 1000℃, the Cu foilis annealed for 2 h in a gas mixture of hydrogen (20 sccm) and argon (380 sccm). After that, the graphene growth is carried out at 1000℃ under an 80 sccm gas mixture of argon and ethanol. Then, the samples are cooled down to the room temperature with a rate of 100℃/min in a protection gas of hydrogen and argon, and then taken out of the furnace. The graphene is prepared on the Cu foils and finally transferred onto the SiO2/Si substrates. The quality and number of layers of the as-produced graphene are assessed by field emission scanning electron microscopy (SEM), transmission electron microscopy (TEM), Raman spectroscopy, and optical microscopy. By tuning the graphene growth, the monolayer, bi-layer and multi-layer graphene with higer quality and better continuity are obtained. And the growth times of monolayer, bi-layer, and four-layers graphene are respectively 25, 40, and 60 s. And wefind that the graphene layer will be increased in the process of insulation. The growth mechanisms of bi-layer and multi-layer graphene on copper substrates by atmospheric pressure chemical vapor deposition are also discussed. There will be some indiffusible carbon atoms or radicals near the copper foil surface due to the small molecular diffusion mean free path under normal pressure. We suggeste that the bi-layer and multi-layer graphene grown on copper substrates by atmospheric pressure chemical vapor deposition is dominated by van der Waals epitaxial mechanism. This work provides a reference for improving the quality of chemical vapor deposition monolayer, bi-layer and multi-layer graphene.

Separation of Hydrogen from Carbon Dioxide through Porous Ceramics.

The gas permeability of α-alumina, yttria-stabilized zirconia (YSZ), and silicon carbide porous ceramics toward H2, CO2, and H2–CO2 mixtures were investigated at room temperature. The permeation of H2 and CO2 single gases occurred above a critical pressure gradient, which was smaller for H2 gas than for CO2 gas. When the Knudsen number (λ/r ratio, λ: molecular mean free path, r: pore radius) of a single gas was larger than unity, Knudsen flow became the dominant gas transportation process. The H2 fraction for the mixed gas of (20%–80%) H2–(80%–20%) CO2 through porous Al2O3, YSZ, and SiC approached unity with decreasing pressure gradient. The high fraction of H2 gas was closely related to the difference in the critical pressure gradient values of H2 and CO2 single gas, the inlet mixed gas composition, and the gas flow mechanism of the mixed gas. Moisture in the atmosphere adsorbed easily on the porous ceramics and affected the critical pressure gradient, leading to the increased selectivity of H2 gas.

Effect of various surface conditions on nanochannel flows past permeable walls

In this paper, effects of surface conditions on the gas flow in nanochannels with permeable walls have been investigated by molecular dynamics simulations. The hydrodynamic characteristics of the gas flow in nanochannels, including the density distributions, slip length and boundary friction coefficient, have been significantly influenced by the molecular interactions between gas molecules and wall atoms. The density layering phenomena are observed at the fluid–wall interface under a strong fluid–wall interaction (FWI). Particularly, there is a peak of the gas density on the permeable surface where the gas density is increased and about 3 times larger than the bulk one under the strong FWI. It indicates a strong non-continuum density distribution on the permeable surface and the step down in density from nanopores to the bulk region. On the other hand, the nanoscale vortices are produced in the nanopores. Moreover, the mass flux of gas flow in nanopores is reduced, and the hydrodynamic boundary has been shifted above the permeable surface further under the weak FWI. Slip characteristics on the permeable surface under various conditions are explored. It has been found that the slip length on the permeable surface may vary as a logarithmic function with the molecular mean free path. Apparently, the skin friction on the permeable surface is affected by the velocity slip. These results are significant in the understanding of nanoscale hydrodynamics.