Molecular Mean Free Path Research Articles

Multiscale simulation of rarefied gas dynamics via direct intermittent GSIS-DSMC coupling

The general synthetic iterative scheme (GSIS) has proven its efficacy in modeling rarefied gas dynamics, where the steady-state solutions are obtained after dozens of iterations of the Boltzmann equation, with minimal numerical dissipation even using large spatial cells. In this paper, the fast convergence and asymptotic-preserving properties of the GSIS are harnessed to remove the limitations of the direct simulation Monte Carlo (DSMC) method. The GSIS, which leverages high-order constitutive relations derived from DSMC, is applied intermittently, which not only rapidly steers the DSMC towards steady state, but also eliminates the requirement that the cell size must be smaller than the molecular mean free path. Several numerical tests have been conducted to validate the accuracy and efficiency of this hybrid GSIS-DSMC approach.

Evaporation into half-space: Experiments with water at the molecular mean free path scale

Evaporation into half-space: Experiments with water at the molecular mean free path scale

Forces acting on near-wall spherical particles in shear flows of diluted gases

In the present paper, we studied the forces on a spherical particle of radius R moving in the vicinity of the plane wall in a shear flow of free molecular regime. We consider that the distance ratio between the plane wall and the particle (L) and the particle radius (R) is large (e.g., L/R > 5), and the gas molecular mean free path (λ) is much higher than the particle size (λ/R ≫1). An analytical formula for the forces is obtained based on gas kinetic theory and certain simplifying assumptions, and is verified by using Direct Simulation Monte Carlo Method. It is found that the forces acting on the particle can be affected by the momentum accommodation coefficients (σ) of the wall and particle surfaces, the wall/gas temperature ratios (Tw/T), and the velocity gradient (G) of the gas flow. In the cases of specular reflections (σ = 0), the near-wall effect can be neglected. With the increase of the momentum accommodation coefficients, the near-wall effect can be enhanced. When near-wall particles move in the direction parallel to the plane wall, there is a lift force which is perpendicular to the wall due to the near-wall effect and the shear flow. For Tw/T < 1, the lift force for the near-wall particles is in the direction against the wall. While for Tw/T > 1, the force is in the direction away from the plane wall. The findings presented in this paper can provide theoretical guidance for the application of near-wall particles in shear flows.

Transport phenomena in the near-field region of Stefen flow

Stefan flow is a transport phenomenon concerning the movement of one component of a multi-component mixture that is induced by the production or removal of the component at an interface. Fick’s law is often used to describe the transport phenomena in many physical and chemical processes. However, when the system scales down to the order of molecular mean free path, the influence of interface that is not considered in Fick’s law, cannot be neglected. To predict the transport phenomena in the near-field region, the Herz-Knudsen (HK) relation is often adopted, in which the near-field region controlled by the HK relation was assumed as a zero-thickness layer. Our theoretical analysis has shown that this assumption is less physically realistic. In this study, we derive the thickness of the near-field region through the analysis of the theoretical binary diffusion coefficient and corresponding mass flux. Based on the analysis, we propose a modified Fick’s law to describe the far-field and near-field regions separately. The validity of this modified Fick’s law and its difference from Fick’s law coupled with the HK relation are demonstrated by comparing the partial pressure distributions of binary mixtures, predicted by the two versions of Fick’s law and the Direct Simulation Monte Carlo (DSMC) method, due to the deposition of one component on a cold surface. The comparison results show that the modified Fick’s law outperforms the Fick’s law coupled with the HK relation in accuracy at high deposition coefficients for H2O/N2 mixture and in stability within the range of this study. Moreover, the situations when the modified Fick’s law is required are discussed. This study provides a more molecular-level insight into the transport phenomena in the near-field region of Stefen flow.

Impact of fuel properties on the transition of liquid-gas interface dynamics under supercritical pressure

Fuel injection dynamics transition from two-phase theory to single-phase dense fluid dynamics are a crucial phenomenon that markedly affects the mixing process and subsequent combustion under diesel engine-relevant conditions. As the significant effect of fuel properties on the timescale and mechanism of this transition remains elusive yet, this work focuses on the impact of fuel properties on the transition to single-phase dense fluid at supercritical pressure. The detailed thermodynamic structure of non-continuum gas-liquid interfaces at supercritical pressures was constructed through gradient theory, and its evolution was analyzed. The driving mechanism of transition from spray to single-phase dense fluid was discussed. The influences of fuel properties on the timescale and mechanism of breakdown of the two-phase theory were investigated through the comparison of three different alkanes. Theoretical analysis suggests that the transition from spray to single-phase dense fluid is jointly promoted by the thickening interface, the diminished surface tension, and the shorter molecular mean free path. The gas-liquid interface of lighter-alkane/nitrogen binary systems is more inclined to evolve into the continuum length-scale regime. The cause is a combination of two aspects: (1) smaller thermodynamic potential energy density difference ω¯(ϱM)−ω¯s and smaller interfacial tangential pressure across lighter alkanes/nitrogen interfaces lead to broader interfaces and smaller surface tension, (2) the more volatile nature of lighter alkanes results in shorter molecular mean free path. The minimum reduced pressure required for gas-liquid interfaces to reach critical mixing states is lower for lighter and more volatile alkanes at the same reduced temperature. From a time-scale perspective, the non-continuum gas-liquid interfaces evolve into the continuum length scale regime later for heavier alkanes. This suggests that the transition to the diffusive mixing mechanism occurs further downstream for heavier alkane sprays. A conceptual diagram was presented to illustrate the effect of fuel properties on the transition.

A laboratory observation for gases transport in shale nanochannels: Helium, nitrogen, methane, and helium-methane mixture

Mudstones are widely distributed in sedimentary basins with abundant nanopores and has very low fluid mobility. It is widely accepted that gas migration in micro-nanochannels no longer follows Darcy's law and the Hagen-Poiseuille equation and that molecular kinetic features profoundly influence gas migration in micro-nanochannels. Here, we characterize the nanochannel sizes of Longmaxi Shale cores under different overburden pressure and pore pressure conditions. And the apparent permeabilities of individual component gases (helium, nitrogen, and methane) in different size nanochannels are compared. Then the variations for components and methane carbon isotope values of the overflowing helium-methane mixture are analyzed. The results show that the molecular slippage effect contributes significantly to the transport of the three gases, especially at low pore pressures; helium has the largest apparent permeability because of its largest molecular mean free path (or Knudsen number). Methane has the smallest molecular mean free path, and the adsorption effect reduces channel size and increases surface roughness, which inhibits the molecular slip effect and therefore results in methane having the smallest apparent permeability. Helium has a transport advantage in the helium-methane mixture due to its Knudsen number advantage, and the larger Knudsen number of 12CH4 likewise leads to a transport advantage over 13CH4. This study suggests that the molecular slippage effect may be an important molecular dynamics mechanism for the efficient development of shale gas. Pressure gradient-driven transport of helium-bearing gas in nanochannels is an essential mechanism for helium enrichment, and high-grade helium reservoirs should occur in secondary low-pressure reservoirs in the basin.

The ignition of fine iron particles in the Knudsen transition regime

A theoretical model is considered to predict the minimum ambient gas temperature at which fine iron particles can undergo thermal runaway–the ignition temperature. The model accounts for Knudsen transition transport effects, which become significant when the particle size is comparable to, or smaller than, the molecular mean free path of the surrounding gas. Two kinetic models for the high-temperature solid-phase oxidation of iron are analyzed. The first model (parabolic kinetics) considers the inhibiting effect of the iron oxide layers at the particle surface on the kinetic rate of oxidation, and a kinetic rate independent of the gaseous oxidizer concentration. The ignition temperature is solved as a function of particle size and initial oxide layer thickness with an unsteady analysis considering the growth of the oxide layers. In the free-molecular limit (small particles), the thermal insulating effect of transition heat transport can lead to a decrease of ignition temperature with decreasing particle size; however, the presence of the oxide layer slows the reaction kinetics, and its increasing proportion in the small-particle limit can lead to an increase of ignition temperature with decreasing particle size. This effect is observed for sufficiently large initial oxide layer thicknesses. The continuum transport model is shown to predict the ignition temperature of iron particles exceeding an initial diameter of 30 µm to a difference of 3% or less (30 K or less) when compared to the prediction of the transition transport model. The second kinetic model (first-order kinetics) considers a porous, non-hindering oxide layer, and a linear dependence of the kinetic rate of oxidation on the gaseous oxidizer concentration. The ignition temperature is resolved as a function of particle size with the transition and continuum transport models, and the differences between the ignition characteristics predicted by the two kinetic models are identified and discussed.

Theoretical modelling of non-equilibrium reaction–diffusion of rarefied gas on a wall with microscale roughness

Heat and mass transports through a rough surface are among the most fundamental and important phenomena in either natural or engineering problems. In this paper, theoretical modelling and direct simulation Monte Carlo method are employed to study the heterogeneous reaction–diffusion features induced by microscale roughness which is comparable to the molecular mean free path of the ambient gas. A quasi-one-dimensional homogeneous model is proposed, and it consists of an external diffusion region outside the roughness elements and an internal reaction–diffusion region which could be equivalent to a smooth surface with an effective chemical property. The external macroscopic diffusion can be characterized by a non-equilibrium criterion – the Damköhler number. The internal diffusion in micro-cavities must be analysed by considering the rarefied gas effects on the diffusivity, and another non-equilibrium criterion, the Thiele number, is introduced to evaluate the effective boundary condition imposed on the external region. Analytical formulae based on these criteria are derived to predict the equivalent surface reaction–diffusion performance, and the predictions compare well with the numerical results of different types of surface reaction, even on the three-dimensional roughness. This reveals that the roughness could either enhance or weaken the apparent reaction rate depending on the non-equilibrium degree. This study could enrich our understanding of the gas–surface interactions on a rough wall, such as the oxidation, catalysis and energy accommodation, and also preliminarily provides a practical method for evaluation of the aerothermochemical performance of coating materials of hypersonic vehicles.

Thermal Expansion and Adsorption Characteristics of Coal Based on Temperature Change and Its Influence on the Slippage Effect

The slippage effect is an important phenomenon of coalbed methane (CBM) migration. As the exploitation of CBM, the slippage effect becomes more and more obvious. For exploring the evolution of the slippage effect as CBM extraction by heat injection, the thermal expansion test and the temperature increasing desorption test under different stress states, as well as the seepage test under variable temperature conditions, were conducted by operating the high-temperature multifunctional triaxial test system. The effect rules for temperature and pore pressure on the slippage effect were analyzed, and then the evolution mechanism of the slippage factor at the united action of pore pressure and temperature was comprehensively analyzed based on thermal expansion deformation and adsorption characteristics. The results showed that: first, the influence of pore pressure on the slippage effect is mainly caused by the molecular mean free path and effective stress when pore pressure is greater than 0.8 MPa in the range of 30–150 °C. With the pore pressure increase, the slippage effect decreases. Second, the temperature influence on the slippage effect is mostly induced by the variation of cracks in coal due to thermal expansion deformation created by temperature. With the increase in temperature, the slippage factor average increases first from 30 to 60 °C is 19 times and then decreases from 60 to 90 °C is 36%. Finally, at the combined influence of temperature and pore pressure, the slippage effect is mainly influenced by thermal expansion deformation, desorption deformation, and effective stress. At 30–60 °C, the slippage factor is principally affected by thermal expansion deformation. At 60–90 °C, the slippage factor is mostly influenced by desorption deformation and effective stress. The research results provide a foundation for the technology of CBM exploitation by heat injection.

Investigation of Molecular Mean Free Path, Molecular Kinetic Energy, and Molecular Polarity Affecting Knudsen Diffusivity along Pore Channels

The effective purification of corrosive gases at the cathode air stream side is essential for proton exchange membrane fuel cells’ performance in real-world applications. Gas molecular diffusion depth along the pore channel is a sufficient parameter that determines the effectiveness of the porous purification media. The collision between gas molecules and pore surfaces is the crucial determinant of the diffusion depth. An analytical model was developed to predict the gas molecular diffusion depth in the pore channels. Two different crystal sizes of UiO-66 were synthesized to validate against the model result and empirically determine the diffusion depths. The parametric effects of the mean free path, molecular kinetic energy, and molecular polarity on molecular diffusivity were assessed. A smaller molecular mean free path and greater molecular kinetic energy were favorable for larger diffusion depth, owing to the fewer collisions and enhanced bounces after collisions. Greater molecular polarity led to shorter diffusion depth due to the enhanced van der Waals force between molecules and pore surfaces.

Navier-Stokes Equations Do Not Describe the Smallest Scales of Turbulence in Gases.

In turbulent flows, kinetic energy is transferred from the largest scales to progressively smaller scales, until it is ultimately converted into heat. The Navier-Stokes equations are almost universally used to study this process. Here, by comparing with molecular-gas-dynamics simulations, we show that the Navier-Stokes equations do not describe turbulent gas flows in the dissipation range because they neglect thermal fluctuations. We investigate decaying turbulence produced by the Taylor-Green vortex and find that in the dissipation range the molecular-gas-dynamics spectra grow quadratically with wave number due to thermal fluctuations, in agreement with previous predictions, while the Navier-Stokes spectra decay exponentially. Furthermore, the transition to quadratic growth occurs at a length scale much larger than the gas molecular mean free path, namely in a regime that the Navier-Stokes equations are widely believed to describe. In fact, our results suggest that the Navier-Stokes equations are not guaranteed to describe the smallest scales of gas turbulence for any positive Knudsen number.

One-way flow over uniformly heated U-shaped bodies driven by thermal edge effects

The thermal edge flow is a gas flow typically induced near a sharp edge (or a tip) of a uniformly heated (or cooled) flat plate. This flow has potential applicability as a nonmechanical pump or flow controller in microelectromechanical systems (MEMS). However, it has a shortcoming: the thermal edge flows from each edge cancel out, resulting in no net flow. In this study, to circumvent this difficulty, the use of a U-shaped body is proposed and is examined numerically. More specifically, a rarefied gas flow over an array of U-shaped bodies, periodically arranged in a straight channel, is investigated using the direct simulation Monte-Carlo (DSMC) method. The U-shaped bodies are kept at a uniform temperature different from that of the channel wall. Two types of U-shaped bodies are considered, namely, a square-U shape and a round-U shape. It is demonstrated that a steady one-way flow is induced in the channel for both types. The mass flow rate is obtained for a wide range of the Knudsen numbers, i.e., the ratio of the molecular mean free path to the characteristic size of the U-shape body. For the square-U type, the direction of the overall mass flow is in the same direction for the entire range of the Knudsen numbers investigated. For the round-U type, the direction of the total mass flux is reversed when the Knudsen number is moderate or larger. This reversal of the mass flow rate is attributed to a kind of thermal edge flow induced over the curved part of the round-U-shaped body, which overwhelms the thermal edge flow induced near the tip. The force acting on each of the bodies is also investigated.

Inversion of the transverse force on a spinning sphere moving in a rarefied gas

The flow around a spinning sphere moving in a rarefied gas is considered in the following situation: (i) the translational velocity of the sphere is small (i.e. the Mach number is small); (ii) the Knudsen number, the ratio of the molecular mean free path to the sphere radius, is of the order of unity (the case with small Knudsen numbers is also discussed); and (iii) the ratio between the equatorial surface velocity and the translational velocity of the sphere is of the order of unity. The behaviour of the gas, particularly the transverse force acting on the sphere, is investigated through an asymptotic analysis of the Boltzmann equation for small Mach numbers. It is shown that the transverse force is expressed as $\boldsymbol{F}_L = {\rm \pi}\rho a^3 (\boldsymbol{\varOmega} \times \boldsymbol{v}) \bar{h}_L$ , where $\rho$ is the density of the surrounding gas, a is the radius of the sphere, $\boldsymbol {\varOmega }$ is its angular velocity, $\boldsymbol {v}$ is its velocity and $\bar {h}_L$ is a numerical factor that depends on the Knudsen number. Then, $\bar {h}_L$ is obtained numerically based on the Bhatnagar–Gross–Krook model of the Boltzmann equation for a wide range of Knudsen number. It is shown that $\bar {h}_L$ varies with the Knudsen number monotonically from 1 (the continuum limit) to $-\tfrac {2}{3}$ (the free molecular limit), vanishing at an intermediate Knudsen number. The present analysis is intended to clarify the transition of the transverse force, which is previously known to have different signs in the continuum and the free molecular limits.

Non-isothermal rarefied gas flow in microtube with constant wall temperature

In this paper, pressure-driven gas flow through a microtube with constant wall temperature is considered. The ratio of the molecular mean free path and the diameter of the microtube cannot be negligible. Therefore, the gas rarefaction is taken into account. A solution is obtained for subsonic as well as slip and continuum gas flow. Velocity, pressure, and temperature fields are analytically attained by macroscopic approach, using continuity, Navier-Stokes, and energy equations, with the first order boundary conditions for velocity and temperature. Characteristic variables are expressed in the form of perturbation series. The first approximation stands for solution to the continuum flow. The second one reveals the effects of gas rarefaction, inertia, and dissipation. Solutions for compressible and incompressible gas flow are presented and compared with the available results from the literature. A good matching has been achieved. This enables using proposed method for solving other microtube gas flows, which are common in various fields of engineering, biomedicine, pharmacy, etc. The main contribution of this paper is the integral treatment of several important effects such as rarefaction, compressibility, temperature field variability, inertia, and viscous dissipation in the presented solutions. Since the solutions are analytical, they are useful and easily applicable.

A FRACTAL MODEL FOR PREDICTING THE EFFECTIVE THERMAL CONDUCTIVITY OF ROUGHENED POROUS MEDIA WITH MICROSCALE EFFECT

The effective thermal conductivity (ETC) of roughened porous media (RPM) is of interest in a number of applications of heat transfer. In this work, a fractal analytical model for the ETC of RPM with microscale effect is proposed. The proposed fractal model is expressed in terms of relative roughness, the molecular mean free path, porosity, fractal dimensions (pore area fractal dimension and tortuosity fractal dimension), maximum pore diameter, capillary straight length, and the thermal conductivity of the solid matrix and gas. It is observed that the dimensionless ETC of RPM decreases with increasing relative roughness, pore area fractal dimension and tortuosity fractal dimension. Besides, it is found that the dimensionless ETC of RPM increases with thermal conductivity ratio of gas phase over solid phase. In addition, it is found that the dimensionless ETC of RPM is slightly dependent on the relative roughness and tortuosity fractal dimension when [Formula: see text]. The determined dimensionless ETC of RPM is in good agreement with experimental data and existing models reported in the literature. With the proposed fractal model, the physical mechanisms of heat transport through RPM with microscale effect are better elucidated. Every parameter in the fractal analytical model has clear physical meaning, with no empirical constant.

Influence of coal deformation on the Knudsen number of gas flow in coal seams

The effective stress and pore pressure can change the fracture aperture, which in turn can change the methane Knudsen number and transitions between gas flow regimes. A novel Knudsen number (Kn) model for gas flow under coal deformation was established; a revised permeability model was formulated and used to obtain an equation for the Klinkenberg coefficient b. The predictions of the revised permeability model were in good agreement with experimental data, and the parameters used to determine the gas Kn were obtained. An increase in the pore pressure under coal deformation causes the fracture aperture to increase and the gas molecular mean free path to decrease, thereby decreasing the Kn. Thus, the critical pore pressure for the transition between gas flow regimes is reduced. The gas Kn for a deformed fracture is higher than that for a constant fracture aperture. The critical fracture aperture for the transition between gas flow regimes increases under coal deformation. The Klinkenberg coefficient b for methane is a variation value that increases as Kn decreases. The Kn variation is very important for determining the coal permeability and which can be used to resolve problems in CBM recovery.

Acoustic levitation of a rigid nano-sphere at non-continuum conditions

Abstract

A selective volatilization and condensation process for extracting precious metals from noble lead

Noble lead, which is produced by the reduction smelting of lead anode slime, contains copious rare and precious metals. The vacuum distillation procedure for processing noble lead exhibits the advantages of efficiency and simplicity, and has been industrialized in most of China’s lead, copper and precious metal smelting enterprises. However, the condensation behaviors of volatile components in noble lead remain unknown, making effective Pb and Sb removal difficult; thus, the corresponding system equipment fails to meet the production requirements of separating alloys and enriching precious metals. In this paper, the condensation area of volatile components in noble lead is predicted by the molecular mean free path. Then, preliminary verification and experimental research were carried out on the condensation rules of each component to verify and revise the theory. The experimental results indicate that the dominant condensation zones of Pb and Sb are 24–44 mm and 44–59 mm, respectively. Further experimental studies show that Pb and Bi are removed to below 0.8%, and the direct yields of Pb, Bi and Ag reached 91%, 95% and 94%, respectively, with temperature of 1250 °C, holding time for 3 h and pressure of 5 Pa. This research solves the problem of difficult Pb and Sb removal in the conventional vacuum distillation process of noble lead and realizes the efficient recovery of metals, which has practical guiding significance for the technological conditions in the industrial application of this process and the equipment design. This work meets modern cleanliness, efficiency, and simple metallurgical requirements.

Gas Steady-state Diffusion in Fractal Porous Media

Gas diffusion in fractal pores does not follow the classic Fick’s and Knudsen’s laws, so more research on gas diffusion in fractal porous media is needed. Fractal pore models are generated using the random walk method. The gas diffusion governing equations for the fractal pores are derived from the classic kineti theory of gases. The gas diffusion model is used to study the gas diffusion in fractal porous meida and to determine steady-state diffusion coefficient formulas. The results show that the diffusion coefficient is proportional to the mean proe diameter, porosity, and the exponetial function of the fractal dimension in the Knudsen diffusion regime. The diffusion coefficient is not only related to the three pore parameters but is also related to the molecular mean free path in the configurational diffusion regime.

A novel lightweight aerodynamic design for the wings of hypersonic vehicles cruising in the upper atmosphere

Base on the concept of molecular mean free paths in high-altitude rarefied aerodynamics, a novel conceptual lightweight design is proposed and applied to the aerodynamic design of wings of hypersonic vehicles cruising in the upper atmosphere between 120 km to 300 km. The lightweight scheme proposed here is promising in hypersonic vehicles cruising in the upper atmosphere in the sense that it is able to not only reduce the weight of a wing by about 14%, but also save the material by the same amount. A comprehensive aerodynamic analysis indicates that the lightweight aerodynamic design can apply to a wide range of cruising altitudes and Mach numbers. In comparison with the conventional flat-plate wing, the lightweight wing always has smaller drags and larger effective lift-to-drag ratios, proving the feasibility of lightweight design for hypersonic vehicles at high altitudes. Moreover, as the cruising altitude is increased, the lightweight wing has even better lift and lift-to-drag-ratio performances, further demonstrating the excellent promise of lightweight wings for hypersonic vehicles cruising in the upper atmosphere. This lightweight design concept is of great value because the region of the upper atmosphere is relatively unexplored, and hypersonic vehicles cruising in this region can be used for high-resolution earth surveillance, accurate gravity/magnetic field mapping, and global ocean climate exploring.