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Because of the existence of multiscale pores from nano- to macroscale, a multimechanistic shale gas flow process involving the Darcy and Knudsen flows occurs during gas shale well depletion. The respective contribution of the Darcy and Knudsen flows to the permeability is constantly changing with pressure evolution. In this study, laboratory measurements of shale permeability with CO2 injections were carried out under hydrostatic conditions, using the transient pulse-decay method. The “U”-shape permeability curve resulted in both positive and negative effective stress coefficients (Biot’s coefficient) χ . A permeability turning point was thus created to partition permeability curves into the Darcy and Knudsen sections. The Knudsen effect was proven to be significant at low pressure/late time in the laboratory. Effective stress and sorption-induced deformation have been found to govern the Darcy permeability evolution under the tested experimental conditions. Thus, negative effective stress coefficients, together with the positive ones, should be applied to a nonmonotonic pressure-permeability evolution to explain the concurrent effect of the Darcy flow and Knudsen flow at different pore pressures.

A comprehensive methodology has been developed for treating Gas Liquid Displacement (GLD) porometry data with a flow model called Weber model (WM) describing mixed Poiseuille-Knudsen flow regime. The model has been applied in two options: i) considering that the gas viscosity in porometry experiments is the same as that available in reference books for Poiseuille flow regime and ii) equating the expression for Darcy coefficient in gas flow to that obtained in additional liquid permeability experiments and thus leaving the gas viscosity to be an adaptable parameter. In the analysis of GLD porometry data for a range of different microfiltration membranes, it is found that with the WM in both options identical relative pore-number distribution is estimated; and this distribution satisfactorily reproduces both dry and wet flow data from the GLD experiments. The absolute pore-number distributions obtained by the two options are quite similar, but differ in the absolute value of the pore numbers. The pore-number distribution obtained by the second option describes the liquid permeability well, while the first option fails. The WM as a method of GLD porometry data treatment is quite similar to the earlier introduced variable viscosity Poiseuille model (VVPM), and the variable viscosity from the latter model appears to be a combined effect of an uncertainty about actual gas viscosity and the contribution of Knudsen flow. It is concluded that a standard test method for determining pore-size distribution by GLD porometry must include prediction or description of liquid permeability of the membrane. Then, any acceptable gas flow model with adjusted Darcy coefficient obtained from liquid permeability experiment will be suitable for advanced GLD porometry data treatment, beyond the methods typically implemented in gas flow-based porometers, currently used in academia and industry.

RATIONALE Knudsen effusion mass spectrometry (KEMS) shows improved performance with the “restricted collimation” method of Chatillon and colleagues, which consists of two apertures between the Knudsen cell orifice and the ionizer. These apertures define the shape and position of the molecular beam independently of the sample and effusion orifice and as a result reduces background and improves sampling from the Knudsen cell. Modeling of the molecular beam in restricted collimation allows optimization of the apertures’ diameters and spacing. METHODS Knudsen flow is easily simulated with a Monte Carlo method. In this study a Visual Basic for Excel (VBA) code is developed to simulate the molecular beam originating from a vaporizing condensed phase in a Knudsen cell and passing through the cell orifice and the two apertures. RESULTS The code is able to calculate the transmission coefficient through the cell orifice, through the cell orifice and the first aperture, and through the cell orifice and first and second apertures. Also calculated are the angular distributions of the effusate density emerging from the cell and average number of collisions with the orifice walls. CONCLUSION This code allows the geometry (aperture spacing and diameters) of the sampling system to be optimized for maximum transmission. The calculated effusate distributions and low average number of orifice wall collisions illustrated the advantages of restricted collimation. Calculated transmission factors are also compared to literature values calculated via the analytical method of Chatillon and colleagues.

International audience; This study proposes new approaches for measuring the gas permeability and the accessible porosity of porous media. Two techniques are used: the usual permeameter, of the Cembureau type (measurement under pressure), and a new technique named a "double-cell" permeameter, based on a vacuum technique. Theoretical and experimental results point out that the apparent permeability measured in vacuum is proportional to the permeability measured under pressure. For a given pressure, the theoretical expression of the coefficients of proportionality leads to a quasi-constant value for a very large range of concrete permeability. A new equation is also proposed to evaluate the accessible porosity from the Time to Reach Steady State (TRSS) recorded during permeability tests. Concordance between the porosity accessible to gas obtained in this way and the porosity measured by the usual technique of hydrostatic weighing is discussed.

The streaming instability, as an example of instabilities driven by particle feedback on a gas flow, has been proven to have a major role in controlling the formation of planetesimals. These instabilities in protoplanetary disks occur at the transition from being gas-dominated to being dust and ice particle dominated. Here, we present experiments to approach this situation in the laboratory for particles in the Knudsen flow regime. In these experiments, we observe a particle cloud trapped for about 30 s in a rotating system under Earth’s gravity. For average dust-to-gas ratios up to 0.08, particles behave like individual test particles. Their sedimentation speed is identical to that of a single free-falling particle, even in locally denser regions. However, for higher dust-to-gas ratios, the motion of particles becomes sensitive to clumping. Particles in locally denser regions now sediment faster. Their sedimentation speed then depends linearly on the overall dust-to-gas ratio. This clearly shows a transition from tracerlike behavior to collective behavior. Beyond these findings, these types of experiments can now be used as a gauge to test particle feedback models in astrophysical hydrocodes, which are currently used for numerical simulations of streaming instabilities.

Monte Carlo analysis of rarefied-gas heat transfer between parallel plates under various flow conditions

We report on the results of analysis of numerical scenarios for motion of a dust cloud interacting with a Knudsen gas flow. Two types of evolution of the dust formation are observed: regular, in which the characteristic points of the cloud are attracted to attractors, or the development of instability in a bounded region.

The performance of standard Lattice Boltzmann, LB, is confined to the micro scale flows with a Knudsen number, Kn , less than 0.1. In the previous works we proposed a modified LB which is able to extend the ability of LBM to simulate wide range of Knudsen flow regimes. To study the ability of this model for porous media, in the present work we investigate nanoscale gaseous flows through porous media confined between parallel plates by means of LB, and molecular dynamics, MD, simulations. The comparison between modified LB and MD shows a very good agreement at sufficiently low solid fractions. In addition, flow characteristics through various porous structures are studied via the MD simulation and the effect of various parameters such as driving force, porosity and Knudsen number on the velocity and temperature distributions is investigated. Our results demonstrate the good ability of MDS for investigating different flow phenomena such as permittivity and Darcian flow in porous media.

Micro-bends are frequently encountered in micro-electro-mechanical systems as a basic unit of complex geometry. It is essential for a deep understanding of the rarefied gas flow through bent channels. In this paper, a two-dimensional pressure-driven gas flow in a micro-channel with two bends is investigated by solving the Bhatnagar-Gross-Krook kinetic equation via the discrete velocity method in the slip and transition flow regimes. The results show that the mass flow rate (MFR) through the bent channel is slightly higher than that in the straight channel in the slip flow regime but drops significantly as the Knudsen number increases further. It is demonstrated that the increase in MFR is not due to the rarefaction effect but due to the increase in cross section of the bent corners. As the rarefaction effect becomes more prominent, the low-velocity zones at the corners expand and the gas flow is “squeezed” into the inner corner. The narrowed flow section is similar to the throttling effect caused by the valve, and both the changes in MFRs and the pressure distribution also confirm this effect. The classical Knudsen minimum changes due to this “rarefaction throttling effect.” The Knudsen number at which the minimum MFR occurs gradually increases with the bend angle and finally disappears in the transition flow regime. In addition, the onset of rarefaction throttling effect shifts to a smaller Knudsen number with a lower tangential momentum accommodation coefficient.Micro-bends are frequently encountered in micro-electro-mechanical systems as a basic unit of complex geometry. It is essential for a deep understanding of the rarefied gas flow through bent channels. In this paper, a two-dimensional pressure-driven gas flow in a micro-channel with two bends is investigated by solving the Bhatnagar-Gross-Krook kinetic equation via the discrete velocity method in the slip and transition flow regimes. The results show that the mass flow rate (MFR) through the bent channel is slightly higher than that in the straight channel in the slip flow regime but drops significantly as the Knudsen number increases further. It is demonstrated that the increase in MFR is not due to the rarefaction effect but due to the increase in cross section of the bent corners. As the rarefaction effect becomes more prominent, the low-velocity zones at the corners expand and the gas flow is “squeezed” into the inner corner. The narrowed flow section is similar to the throttling effect caused by the v...

We reconstructed pore structures of three porous solids that differ from each other in morphology and topology of pore space. To achieve this, we used a stochastic method based on simulated annealing and X-ray computed microtomography. Simulated annealing was constrained by the following microstructural descriptors sampled along the principal and diagonal directions: the two-point probability function for the void phase and the lineal-path functions for both void and solid phases. The stochastic method also assumed the isotropic pore structures in accordance with a recent paper (Capek et al. in Transp Porous Media 88(1): 87–106 (2011)). With the exception of the solid with the widest pores, we made tomographic volume images in high and low resolution, which enabled us to study the effect of resolution on microstructural descriptors and effective transport properties. A comparison of the two-point probability function and the lineal-path function sampled in the principal directions revealed that the pore structures derived from the tomographic volume images were slightly anisotropic, in opposition to the assumption of the stochastic method. Besides the anisotropy, other microstructural descriptors including the pore-size function and the total fraction of percolating cells indicated that the morphological and topological characteristics of the pore structures depended on the reconstruction method and its parameters. Particularly, the pore structures reproduced using the stochastic method contained wider pores than those obtained using X-ray tomography. Deviations between the pore structures derived from low- and high-resolution tomographic volume images were also observed and imputed to partial volume artefacts. Then, viscous flow of incompressible liquid, ordinary diffusion, Knudsen flow and self-diffusion of water in the reconstructed pore spaces were simulated. As counterparts, experimental data were measured by means of permeation and Wicke–Kallenbach cells and pulsed field gradient NMR. Deviations between the simulated quantities on the one hand and experimental data on the other hand were generally acceptable, which corroborated the pore-space models. As expected, the predictions based on the tomographic models of pore space were more successful than those derived from the stochastic models. The stationary effective transport properties, i.e. the effective permeability, the effective pore size and the geometric factor, were sensitive to a bias in long-range pore connectivity. Furthermore, the time-dependent effective diffusivity was found to be especially sensitive to relatively small morphological deviations between the real and reconstructed pore structures. It is concluded that the combined predictions of the effective permeability, the effective pore size, the geometric factor and time-dependent effective self-diffusivity of water are needed for the reliable evaluation of pore-space reconstruction.

Economic exploration for shale gas is highly dependent on the complexity of the fracture network caused by hydraulic fracturing technology, so it is necessary to accurately assess the effect of the fracture network on gas flow behavior and productivity. Having multiple scales is a remarkable characteristic of gas flow in fractured shale gas reservoirs, which involves multiple flow regimes, including the gas desorption from shale matrix, Knudsen flow in nanoscale pores, Darcy flow in general porous media, Darcy flow in the fracture networks and fluid exchange between the matrix system and the fracture system. The extended finite element method (XFEM) has been integrated with the dual-permeability method (DPM) to investigate the multi-scale problem. Some previous studies have adopted the same solution scheme, but usually considered gas flow in the natural fracture network and in the hydraulic fractures as belonging to the same scale, and in addition, the coverage area of the stimulated reservoir volume (SRV) was underestimated. Based on the XFEM–DPM, this paper subsumes flow in the micro-mecro fractures and macro-hydraulic fractures under two scales because of their different effects on shale gas flow and then presents a new multi-scale extended finite element model to study the multi-scale flow problem in shale gas reservoirs. Moreover, the Lagrangian multiplier method is integrated to introduce the internal well boundary conditions into the XFEM, so the arbitrariness and the asymmetry of the complex fracture network can be taken into account easily to reflect the real flow mechanisms in fractured shale. Case studies indicate that the improved extended finite element model constructed in this paper is effective, especially for complicated asymmetrical physical condition problems.

A porous medium transport model with a distributed sublimation front is developed for low pressure freeze drying of beef by radiant surface heating and volumetric microwave heating. The model incorporates the importance of Knudsen flow in porous materials during low pressure freeze drying. This effort is part of fundamental physics-based framework building for simulating food and biomaterial processes involving rapid evaporation/sublimation. Temperature, pressure and ice saturation histories were computed. Drying rates and spatial temperature profiles showed excellent agreement with literature experimental data. Sublimation front width, a novel result, is seen to increase as ice saturation decreases, justifying the importance of this distributed sublimation formulation in contrast with the sharp sublimation front commonly employed in literature. The insulation effect of the gas fraction in the pores is observed by the slow movement of the sublimation front in ‘thick’ samples. Effects of porosity, initial ice saturation and microwave heating are illustrated.

In this study, nonlinear vibration and stability of a fluid-conveying nanotube (FCNT), elastically coupled to a smart piezoelectric polymeric beam (PPB) is investigated based on nonlocal elasticity theory, Euler–Bernoulli beam model and energy approach. In order to obtain an active instability control of FCNT, the PPB is longitudinally polarized as an actuator while in the absence of an imposed electric field it is also possible to be used as an alarm biosensor. Simulating the above smart coupled nanobeam system alike the double nanobeam systems (which are relatively developed by other authors) leads to obtain nonlinear differential equations of motion. The linear natural and damping frequencies are achieved by ignoring all the system nonlinearities which are then considered to obtain nonlinear frequencies using an iterative method. The effects of geometric nonlinearity, small scale parameter, coupled medium constants, Knudsen number, temperature change, aspect ratio and external applied voltage on critical flow velocity are studied in details. It is concluded that applying an electric voltage on PPB will increase the stability of FCNT. It is hoped that this research will provide a new approach to smart instability control of FCNTs which is no yet reported.

A model capable of predicting transport properties based solely on the pore size distribution of the material and the fluid characteristics has been developed previously by the authors. This model combines a Darcy description of the fluid flow at the macroscale with a Poiseuille/Knudsen flow at the micro-scale. A stochastic approach is then used with regards to pore network generation, in a manner consistent with mercury intrusion porosimetry (MIP). The present paper extends this model to multi-phase flow in an attempt to model partially saturated porous media. Kelvin's law defines which pore sizes are invaded with liquid and rules for accounting contributions of the pore network to liquid and vapour flows are introduced. The extended model provides data that are consistent with the Van Genuchten approach to liquid and gas relative permeabilities, compares well with experimental data and provides information of the relative permeability to vapour and liquid water upon damage. © ASCE.

This work presents the study of isothermal rarefied gas flows in converging microchannels. Experiments are carried out on microchannels of three different converging angles (4 degrees , 8 degrees , and 12 degrees). Numerical investigation is carried out using commercial software to study the local behaviour of the flow parameters. The simulations show a sudden drop in the fluid temperature at the exit of the microchannel. Knudsen minimum, which was experimentally observed for the first time recently in diverging microchannels, is also noted here in the case of flow in converging cross section. It is interesting to note that, at the location of Knudsen minimum, the Knudsen number and the value of the minimum mass flow rate are same for both converging and diverging cross sections, for all the angles tested. This result implies the absence of any flow preference at high Knudsen numbers when the flow is subjected to converging and diverging orientations of the microchannel. Published by AIP Publishing.

The rarefied effect of gas flow in microchannel is significant and cannot be well described by traditional hydrodynamic models. It has been know that discrete Boltzmann model (DBM) has the potential to investigate flows in a relatively wider range of Knudsen number because of its intrinsic kinetic nature inherited from Boltzmann equation. It is crucial to have a proper kinetic boundary condition for DBM to capture the velocity slip and the flow characteristics in the Knudsen layer. In this paper, we present a DBM combined with Maxwell-type boundary condition model for slip flow. The tangential momentum accommodation coefficient is introduced to implement a gas-surface interaction model. Both the velocity slip and the Knudsen layer under various Knudsen numbers and accommodation coefficients can be well described. Two kinds of slip flows, including Couette flow and Poiseuille flow, are simulated to verify the model. To dynamically compare results from different models, the relation between the definition of Knudsen number in hard sphere model and that in BGK model is clarified., Comment: 9 pages,9 figures. Comm. Theor. Phys.(Accepted)

The purpose of this work is to evaluate the recently derived Onsager-Burnett (OBurnett) equations [N. Singh, R.S. Jadhav, and A. Agrawal, "Derivation of stable Burnett equations for rarefied gas flows," Phys. Rev. E 96, 013106 (2017)] for force-driven compressible plane Poiseuille flow. This classical internal flowproblem depicts several non-equilibrium phenomena, for instance, non-constant pressure profile in the transverse direction and tangential heat flux, which are not captured by the classical Navier-Stokes-Fourier equations. The results of OBurnett equations for conserved and non-conserved variables are validated against the existing direct simulation Monte Carlo (DSMC) and molecular dynamics (MD) simulation results. These results suggest that the OBurnett equations are able to predict most of the variables well with respect to DSMC and MD simulation results. We find that the OBurnett equations predict a strictly monotonic pressure profile, in contrast to the bimodal profile predicted by the DSMC results and the conventional Burnett equations, but in agreement with the molecular dynamics simulation results. The equations also recover the non-zero tangential heat flux but fail to capture the peculiar temperature dip at the center, owing to its second order accuracy. These results suggest that the evaluated equations are accurate in predicting the non-equilibrium phenomena observed in the rarefied gas flows for the case considered. Published by AIP Publishing.

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.

Summary Physics of fluid flow in shale reservoirs cannot be predicted from standard flow or mass-transfer models because of the presence of nanopores, ranging in size from one to hundreds of nanometers, in shales. Conventional continuum-flow equations, such as Darcy's law, greatly underestimate the fluid-flow rate when applied to nanopore-bearing shale reservoirs. As a result of the existence of nanopores in shales, the molecular mean free path becomes comparable with the characteristic geometric scale, and we hypothesize that under this condition, Knudsen diffusion, in addition to correction for the slip boundary condition, becomes the dominant mechanism. Recently, a few models have been developed that use various empirical parameters to account for these modifications (Javadpour 2009; Civan 2010; Darabi et al. 2012). This paper aims to provide a different approach to modeling apparent permeability in shale reservoirs. The proposed model is analytical, free of any empirical coefficients, and has been derived without invoking the assumption of slip flow at the pore wall. Our model of apparent permeability represented by a single analytical equation, depends only on pore size, pore geometry, temperature, gas properties, and average reservoir pressure. The proposed model is valid for Knudsen numbers less than unity and it stands up under the complete operating conditions of a shale reservoir. Our model reasonably predicts results as reported by other models. Finally, the model shows that pore-surface roughness and mineralogy have a negligible influence on gas-flow rate, whereas pore geometry and pore size play a significant role in the proportion of diffusion in total flow rate. Our study shows that a combination of Darcy flow and Knudsen flow—ignoring the Klinkenberg effect—can describe gas flow for a range of Knudsen flow applicable to a shale-gas system.

Microfractures are commonly observed in coal reservoirs. During the coalbed gas production process, stress sensitivity and coalbed gas viscosity changes are significant factors that affect gas transport. By using research methods based on desorption theory and elastic-plastic mechanics, a coal confined microfracture gas transport model that considers dynamic microfracture width variations and gas effective viscosity is established in this paper. This model comprehensively fuses the Knudsen diffusion model, the slip flow model, the surface diffusion model, and the cubic grid model. The reliability of this model is verified via molecular simulations, and the influence factors of gas transport capacity in confined microfractures of coal reservoirs are then discussed in detail. The results demonstrate the following findings. (1) The analyzed flows are well simulated in coal confined microfractures by the model established in this paper, which considers stress sensitivity and coalbed methane viscosity change. (2) Under low formation pressure (less than 5 MPa), the effective gas viscosity rapidly decreases with the decrease in formation pressure, and the negative contribution of gas viscosity change to the microfracture permeability of coal is large. The smaller the initial fracture width of coal, the more obvious this negative effect. (3) When the initial fracture width is fairly small (near 1 nm), Knudsen flow and surface diffusion greatly contribute to the microfracture permeability of coal. However, the larger the initial microfracture width, the smaller the contribution of the two flow regimes to the microfracture permeability, and the permeability is mainly provided by slip flow. (4) Under given conditions, the microfracture permeability of coal is positively related to rock mechanical parameters (Young's modulus and Poisson's ratio) and negatively related to fracture compressibility. Under low formation pressures (less than 15 MPa), the microfracture permeability of coal is positively related to gas desorption. When the formation pressure exceeds 15 MPa, the influence of gas desorption performance on permeability is nearly constant.

This work investigates the species separation in the rarefied flow of the argon-helium mixture through convergent-divergent micronozzles. Imposing a molecular mass ratio in the order of 10, the flow of this mixture can lead to the formation of serious nonhomogeneous phenomena such as the species separation. This study is performed in the ranges of 2.0–4.0 for the geometrical expansion ratio, 200–400 K for the wall temperature, and 0.003–1.454 for the inlet Knudsen number. The effects of these parameters are examined on the separative performances of micronozzle. The direct simulation Monte Carlo method is selected as the solution method because it can provide reliable solutions in the current rarefied flow regime study. The current study reveals two important separative effects in the mixture flow through micronozzles. The first effect is the lateral species separation, which results in the enrichment of heavier species near the centerline. The second effect is the streamwise separation, which leads to the enrichment of one species, mostly the lighter one, as the mixture passes through the micronozzle. The current results show that increasing the expansion ratio will enhance the lateral separation monotonically. However, there are specific wall temperature and Knudsen values, which can result in optimum lateral separative effects. In addition, it is observed that the expansion ratio has little effect on the streamwise separation. However, increasing either the wall temperature or the Knudsen number will enhance the streamwise separation, albeit with a limiting value at very high Knudsen numbers.This work investigates the species separation in the rarefied flow of the argon-helium mixture through convergent-divergent micronozzles. Imposing a molecular mass ratio in the order of 10, the flow of this mixture can lead to the formation of serious nonhomogeneous phenomena such as the species separation. This study is performed in the ranges of 2.0–4.0 for the geometrical expansion ratio, 200–400 K for the wall temperature, and 0.003–1.454 for the inlet Knudsen number. The effects of these parameters are examined on the separative performances of micronozzle. The direct simulation Monte Carlo method is selected as the solution method because it can provide reliable solutions in the current rarefied flow regime study. The current study reveals two important separative effects in the mixture flow through micronozzles. The first effect is the lateral species separation, which results in the enrichment of heavier species near the centerline. The second effect is the streamwise separation, which leads to th...

The hydrodynamic mix of fusion fuel and inert pusher can simultaneously generate smaller fuel pockets and finer pusher layers that separate them. Smaller fuel pockets have greater local Knudsen numbers, which tend to exacerbate the Knudsen layer reactivity reduction. A thinner pusher layer separating the neighboring fuel pockets, on the other hand, can enable the diffusive tunneling of Gamow fuel ions through the pusher layer and hence alleviate the Knudsen layer reactivity degradation. Here, the diffusive tunneling phenomenon describes a random walk process by which the Gamow fuel ions from one fuel pocket can traverse the inert pusher layer to join a neighboring fuel pocket without losing much of their energy. This is made possible by the much slower collisional slowing down rate compared with the pitch angle scattering rate of light fuel ions with heavier pusher ions. In an isobaric target mixture where fuel and pusher segments can have distinct temperatures, due to their different compressibilities, the temperature effect on the critical pusher layer areal density below which diffusive tunneling can occur, which is a property of the hydrodynamic mix, is understood by computing the ion charge state distribution using a collisional radiative model. This information is fed into the collisionality evaluation, enabling a parametric scan of the diffusive tunneling physics in terms of the target pressure, fuel, and pusher temperatures. It is found that when the gold pusher layer has a temperature above 1 keV, the variation of the pusher temperature has little effect on the critical areal mass density below which diffusive tunneling can occur. If the pusher layer is 1 keV or below, the critical areal mass density rises sharply, indicating that for a stronger fuel-pusher temperature disparity, the onset of diffusive tunneling will be at an earlier stage of the hydrodynamic mix when the fuel-pusher mixing structures are of less reduced size.The hydrodynamic mix of fusion fuel and inert pusher can simultaneously generate smaller fuel pockets and finer pusher layers that separate them. Smaller fuel pockets have greater local Knudsen numbers, which tend to exacerbate the Knudsen layer reactivity reduction. A thinner pusher layer separating the neighboring fuel pockets, on the other hand, can enable the diffusive tunneling of Gamow fuel ions through the pusher layer and hence alleviate the Knudsen layer reactivity degradation. Here, the diffusive tunneling phenomenon describes a random walk process by which the Gamow fuel ions from one fuel pocket can traverse the inert pusher layer to join a neighboring fuel pocket without losing much of their energy. This is made possible by the much slower collisional slowing down rate compared with the pitch angle scattering rate of light fuel ions with heavier pusher ions. In an isobaric target mixture where fuel and pusher segments can have distinct temperatures, due to their different compressibilities, t...

A two-dimensional comprehensive model was developed to predict the transport of water in the nanoporous membrane contactors. The considered membrane distillation device was a counter-current flat-sheet membrane contactor for production of pure water from saline water. The developed model formulates the fundamental transport equations of heat, mass, and momentum in the membrane contactor. The computational fluid dynamics techniques were applied for numerical simulation of model equations. The simulation results were compared with experimental data obtained from literature and showed great agreement with the measured values. A combination of the Knudsen flow and Poiseuille flow was used in the model for estimation of diffusion inside the membrane pores and increased the accuracy of the model. Simulation results revealed that, in the regions adjacent to the membrane wall, the temperature difference is significant. This could be attributed to the fact that a temperature boundary layer is formed near the membr...

Electronic hydrodynamic flow—making electrons flow like a fluid—has been observed [Also see Reports by Bandurin et al. , Crossno et al. and Moll et al. ]

Study of rarefied gas flow in converging and diverging cross sections is crucial to the development of micro-nozzles and micro-thrusters. In other practical cases too, a microchannel may not always be straight and may include diverging and converging sections in the flow path. In this context, isothermal rarefied gas flow in microchannels of longitudinally varying cross section is studied experimentally in this work. The primary objective is to investigate the existence of Knudsen minimum in microchannels of varying cross sections. The effect of geometrical cross section and fluid properties on the Knudsen minimum are also investigated by performing experiments on three divergence angles (4 degrees, 8 degrees, and 12 degrees) and three different gases (argon, nitrogen, and oxygen) to prove the robustness of the result. The Knudsen minimum, which is one of the characteristic features of rarefied flows, is experimentally observed for the first time in a microchannel of varying cross section. The position of the Knudsen minimum (at Kn approximate to 1) is seen to depend only weakly on the divergence angle and fluid properties. (C) 2016 AIP Publishing LLC.

International audience; It is well known that the roughness of the wall has an effect on microscale gas flows. This effect can be shown for large Knudsen numbers by using a numerical solution of the Boltzmann equation. However, when the wall is rough at a nanometric scale, it is necessary to use a very small mesh size which is much too expansive. An alternative approach is to incorporate the roughness effect in the scattering kernel of the boundary condition, such as the Maxwell-like kernel introduced by the authors in a previous paper. Here, we explain how this boundary condition can be implemented in a Discrete Velocity approximation of the Boltzmann equation. Moreover, the influence of the roughness is shown by computing the structure scattering pattern of mono-energetic beams of the incident gas molecules. The effect of the angle of incidence of these molecules, of their mass, and of the morphology of the wall is investigated and discussed in a simplified two-dimensional configuration. The effect of the azimuthal angle of the incident beams is shown for a three-dimensional configuration. Finally, the case of non-elastic scattering is considered. All these results suggest that our approach is a promising way to incorporate enough physics of gas-surface interaction, at a reasonable computing cost, to improve kinetic simulations of micro and nano-flows.

Energy and exergy mechanisms of laminar-transitional micropipe flows are computationally investigated by solving the variable fluid property continuity, Navier–Stokes and energy equations. Analyses are carried for wide ranges of Reynolds number (Re = 10–2,000), micropipe diameter (d = 0.50–1.00 mm), non-dimensional surface roughness (e* = 0.001–0.01) and wall heat flux ( $$ {\text{q}}^{\prime \prime } $$ = 1,000–2,000 W/m2) conditions. Computations revealed that friction coefficient (Cf) elevates with higher e* and Re and with lower d, where the rise of e* from 0.001 to 0.01 induced the Cf to increase by 0.7 → 0.9% (d = 1.00 → 0.50 mm), 3.4 → 4.2%, 6.6 → 8.1%, 9.6 → 11.9% and 12.4 → 15.2% for Re = 100, 500, 1,000, 1,500 and 2,000, respectively. Earlier transition exposed with stronger micro-structure and surface roughness at the descriptive transitional Reynolds numbers of Re tra = 1,656 → 769 (e* = 0.001 → 0.01), 1,491 → 699 and 1,272 → 611 at d = 1.00, 0.75 and 0.50 mm; the corresponding shape factor (H) and intermittency (γ) data appear in the narrow ranges of H = 3.135–3.142 and γ = 0.132–0.135. At higher Re and lower d, e* is determined to become more influential on the heat transfer rates, such that the Nue*=0.01/Nue*=0.001 ratio attains the values of 1.002 → 1.023 (d = 1.00 → 0.50 mm), 1.012 → 1.039, 1.025 → 1.056 and 1.046 → 1.082 at Re = 100, 500, 1,000 and 2,000. As e* comes out to cause minor variations in the cross-sectional thermal entropy generation rates $$ \left( {{\text{S}}_{{\Updelta {\text{T}}}}^{\prime } } \right) $$ , $$ {\text{q}}^{\prime \prime } $$ is confirmed to augment $$ {\text{S}}_{{\Updelta {\text{T}}}}^{\prime } $$ , where the impact becomes more pronounced at higher Re and d. Frictional entropy generation values $$ \left( {{\text{S}}_{{\Updelta {\text{P}}}}^{\prime } } \right) $$ are found to be motivated by lower d, higher Re and e*, such that the $$ {\text{S}}_{{\Updelta {\text{P}}_{{{\text{d}} = 0.50{\text{mm}}}} }}^{\prime } /{\text{S}}_{{\Updelta {\text{P}}_{{{\text{d}} = 1.00{\text{mm}}}} }}^{\prime } $$ ratio is computed as 4.0011 → 4.0014 (e* = 0.001 → 0.01), 4.002 → 4.007, 4.006 → 4.027 and 4.023 → 4.102 at Re = 100, 500, 1,000 and 2,000. As the role of $$ {\text{q}}^{\prime \prime } $$ on total entropy generation $$ ({\text{S}}^{\prime } ) $$ turns out to be more remarkable at higher d and lower Re, the task of e* becomes more sensible at higher Re.

This paper addresses diffusion of air within a cellular metal. A Knudsen diffusion simulation is performed on both simplified model structures and, for the first time, computed tomography data of real materials. In the first part, the three deflection models (mirror reflection, random deflection and random cosine deflection) are compared. It is found that the Knudsen diffusivities of the model structures exhibit different values for each deflection model. The second part is a Knudsen diffusion simulation on computed tomography data of M-Pore® aluminium foam from which an estimate of its Knudsen diffusivity is obtained. Four sub-geometries of the same specimen are converted into voxel calculation models. All calculations show high Knudsen diffusivities with average values of 1.99 m 2 /s (random cosine deflection) and 2.09 m 2 /s (random deflection). Plots of the Knudsen diffusivity versus porosity show an increase of diffusivity with porosity.

The resistance to gas accumulation in constant volume systems has been evaluated by means of a position-depended time lag. The greater the absolute value of the time lag, the greater the error in membrane diffusivity determined using the constant volume system. The time lag, and thus the error in the diffusion coefficient, depends on the position within the receiver of a constant volume system, and regardless of the actual configuration of the receiver, there is always at least one zero time lag position within the receiver. In this paper we have developed an analytical model that allows assessment of the position-dependent time lag in any configuration of the receiver, including multi-tank receivers commonly used in membrane testing laboratories. The model allows optimization of the position of the pressure transducer(s) in the existing receivers, as well as, to understand the effect of important design parameters such as the diameter of tubes, the length of tubes forming direct connection between a membrane cell and the accumulation tank(s), the distribution of the volume between the tanks, and the position of tanks relative to each other and to the membrane cell. The model was verified experimentally using an existing two-tank receiver equipped with three pressure transducers.

This paper investigates the thickness distribution of substrate coating of electron-beam physical vapor deposition (EBPVD) using a novel spatial-distribution-function, which is different from the empirical modified law. The Knudsen cosine law and empirical modified law were usually used for predicting coating thickness in EBPVD. However, these two laws obtained from experiments and geometric concepts cannot provide the elucidation for physical point of view and all-inclusive spatial-distribution-functions. Therefore, instead of the spatial-distribution-functions based on the Knudsen cosine law for low evaporation rates and empirical modified law for the higher evaporation rates, the spatial distribution function based on the mass diffusion theory is proposed to analyze the coating thickness of EB-induced physical vapor deposition. The coating thicknesses predicted by these spatial-distribution-functions are compared with the data measured by the published papers. The results reveal that the coating thicknesses predicted by the spatial distribution function of the diffusion theory are more consistent with the measured data than these obtained from the spatial distribution function of the empirical modified law. The spatial distribution function proposed by this study is validated by correctly predicting the measured coating thickness for EBPVD of titanium and aluminum.

Poiseuille flow and the thermal transpiration of a rarefied gas through a long straight pipe with a rectangular cross section are analyzed numerically based on the linearized Boltzmann equation for a hard sphere molecular gas. A deterministic numerical analysis based on the finite difference is carried out, where the collision integral is evaluated with the aid of the numerical kernel method [Y. Sone et al., Phys. Fluids A 1, 363 (1989)]. The numerical solution, the mass flow rate as well as the macroscopic variables, is presented and discussed for a wide range of the Knudsen number and various aspect ratios of the cross section.

Normal stress effects are investigated on tube flow of a single-component non-Newtonian fluid under a constant pressure gradient in a constant temperature field. The generalized hydrodynamic equations are employed, which are consistent with the laws of thermodynamics. In the cylindrical tube flow configuration, the solutions of generalized hydrodynamic equations are exactly solvable and the flow velocity is obtained in a simple one-dimensional integral quadrature. Unlike the case of flow in the absence of normal stresses, the flow develops an anomaly in that the flow in the boundary layer becomes stagnant and the thickness of such a stagnant velocity boundary layer depends on the pressure gradient, the aspect ratio of the radius to the length of the tube, and the pressure (or density and temperature) at the entrance of the tube. The volume flow rate formula through the tube is derived for the flow. It generalizes the Knudsen flow rate formula to the case of a non-Newtonian stress tensor in the presence of normal stress differences. It also reduces to the Navier–Stokes theory formula in the low shear rate limit near equilibrium.

In this study, the novel method to predict the performance of a multi-stage Knudsen pump is proposed. The performance prediction method is carried out in two steps numerically with the assistance of a simple experimental result. In the first step, the performance of a single-stage Knudsen pump was measured experimentally under various pressure conditions, and the relationship of the mass flow rate was obtained with respect to the average pressure between the inlet and outlet of the pump and the pressure difference between them. In the second step, the performance of a multi-stage pump was analyzed by a one-dimensional model derived from the mass conservation law. The performances predicted by the 1D-model of 1-stage, 2-stage, 3-stage, and 4-stage pumps were validated by the experimental results for the corresponding number of stages. It was concluded that the proposed prediction method works properly.

A computational study of the integrated effects of surface roughness, heat flux, and Reynolds number on the 1st and 2nd law characteristics of laminar-transitional flow in a micropipe is presented. Analyses are carried by solving the variable fluid property continuity, Navier-Stokes, and energy equations for the surface roughness, heat flux, and Reynolds number ranges of 1-50 m, 5-100 W/m2, and 1-2000, respectively. Computations put forward that surface roughness not only accelerates transition to lower Reynolds number but also augments heat transfer rates, such that the transitional Reynolds numbers and intermittency values are evaluated as 1650, 575, and 450 and 0.132, 0.117, and 0.136 for the surface roughness cases of 1, 20, and 50 m, respectively. Thermocritical Reynolds numbers are identified by determining the viscous dissipation rates, which characterize the heating/cooling behavior and the related Reynolds number range. Surface roughness comes out to have no role on entropy generation at low Reynolds numbers; moreover, entropy generation is found to be inversely proportional with mean temperature variation, where the trends become almost asymptotic at the lower limit of the investigated Reynolds number range. Being independent of surface roughness, heat flux, and Reynolds number, radial irreversibility distribution ratio is determined to be negligible at the pipe centerline, indicating that the frictional entropy is minor and the major portion of the total entropy generation is thermal based.

A rarefied gas flow through a thin slit into vacuum is studied on the basis of the direct simulation Monte Carlo method. The mass flow rate and flow field are calculated over the whole range of the gas rarefaction from the free-molecular regime to the viscous one. A comparison to other results on the same problem available in literature is performed. An interpolating formula for the reduced flow rate is obtained.

This article describes the derivation of the efficiency for the Knudsen pump for two cases. First, by neglecting the thermal losses to calculate the maximum efficiency, and second, by including the thermal losses along the length of the channel. The Knudsen pump is a thermally driven pump with no moving parts and which functions on the principle of thermal transpiration. Diffusion is the main mode of mass transport, as opposed to bulk fluid motion. The pump functions when the gases are in a rarefied state. The efficiency without thermal losses is shown to asymptotically approach the value of the ideal gas constant divided by the specific heat capacity, while the efficiency with thermal losses is given for rectangular and circular cross sections and is shown to be independent of length.

A two-dimensional mathematical model was theoretically developed to predict the temperature polarization profile of direct contact membrane distillation (DCMD) processes. A concurrent flat-plate device was designed to verify the theoretical prediction of pure water productivity on saline water desalination. The numerical results from the temperature polarization profile were obtained using the finite difference technique to reduce the two-dimensional partial differential equations into an ordinary differential equations system. The resultant simultaneous linear equations system was solved with the fourth-order Runge-Kutta method. The results show theoretical prediction agreement with the measured values from the experimental runs. A combination of the Knudsen flow and Poiseuille flow models in the present mathematical formulation for membrane coefficient estimation was used to establish theoretical agreement. The influence of the inlet saline water temperature and volumetric flow rate on the pure water productivity as well as the hydraulic dissipated energy are also delineated.

A computational and experimental study has been performed for the investigation of fully developed rarefied gas flows through channels of circular, orthogonal, triangular, and trapezoidal cross sections. The theoretical-computational approach is based on the solution of the Bhatnagar-Gross-Krook kinetic equation subject to Maxwell diffuse-specular boundary conditions by the discrete velocity method. The experimental work has been performed at the vacuum facility “TRANSFLOW,” at Forschungszentrum Karlsruhe and it is based on measuring, for assigned flow rates, the corresponding pressure differences. The computed and measured mass flow rates and conductance are in all cases in very good agreement. In addition, in order to obtain some insight in the flow characteristics, the reference Knudsen, Reynolds, and Mach numbers characterizing the flow at each experimental run have been estimated. Also, the pressure distribution along the channel for several typical cases is presented. Both computational and experimental results cover the whole range of the Knudsen number.

The Knudsen compressor, based on the rarefied flow phenomenon of thermal creep (or thermal transpiration), is an unconventional nano/micro/mesoscale gas pump or compressor with no moving parts. Recent experimental studies of single micro/mesoscale stages have indicated that, at low pressures (les 100 Pa), stage performance was decreased in qualitatively somewhat understood and quantitatively not-well-understood ways. The investigation reported in this paper was undertaken to establish a detailed picture of the thermally driven flow's behavior for a range of representative Knudsen compressor stage configurations and flow conditions. The objectives of the study were to identify, quantify, and ultimately understand the sources of negative impacts on the performance of single stages using the direct simulation Monte Carlo technique. The findings suggest that once the Knudsen number in the cooling region is less than 0.1, the maximum pressure gain from a single stage of a Knudsen compressor can be carried forward to the next stage with relatively minor performance losses. Simultaneously, for Knudsen numbers below 0.1, the working gas' temperature can be reduced to within a few percent of its initial temperature entering the stage. What was found relates to a necessary characteristic of successful Knudsen compressors, with the same occurrences being anticipated for thermal-creep-driven gas transport channels in other nano/microscale devices.

Rarefied gas flow plays an important role in the design and performance analysis of micro-electro-mechanical systems (MEMS) under high-vacuum conditions. The rarefaction can be evaluated by the Knudsen number (Kn), which is the ratio of the molecular mean free path length and the characteristic length. In micro systems, the rarefied gas flow usually stays in the slip- and transition-flow regions (10−3 < Kn < 10), and may even go into the free molecular flow region (Kn > 10). As a result, conventional design tools based on continuum Navier-Stokes equation solvers are not applicable to analyzing rarefaction phenomena in MEMS under vacuum conditions. In this paper, we investigate the rarefied gas flow by using the lattice Boltzmann method (LBM), which is suitable for mesoscopic fluid simulation. The gas pressure determines the mean free path length and Kn, which further influences the relaxation time in the collision procedure of LBM. Here, we focus on the problem of squeezed film damping caused by an oscillating rigid object in a cavity. We propose an improved LBM with an immersed boundary approach, where an adjustable force term is used to quantify the interaction between the moving object and adjacent fluid, and further determines the slip velocity. With the proposed approach, the rarefied gas flow in MEMS with squeezed film damping is characterized. Different factors that affect the damping coefficient, such as pressure of gas and frequency of oscillation, are investigated in our simulation studies.

Constant volume systems are commonly used for the determination of the diffusion, permeability and solubility coefficients of gases in porous and nonporous media. In this paper we present important considerations for the design of an outflow receiver of a constant volume system to minimize the possible resistance to gas transport downstream from the tested medium. The receiver considered in this paper consists of main tube, a tank, and a tube connecting the main tube with the tank. The resistance of the receiver is quantified using the concepts of the asymptotic solution and the time lag by assuming that gas accumulation in the receiver occurs in the Knudsen flow regime. The effects of the volume of the tank, the position at which the tank is connected to the main tube, the length of the connecting tube, as well as, the position of the pressure sensor in the receiver are discussed.

The problem of predicting deposition rates and film thickness variation is relevant to many high-vacuum physical vapor deposition (PVD) processes. Analytical methods for modeling the molecular flow fail when the geometry is more complicated than simple tubular or planar sources. Monte Carlo methods, which have traditionally been used for modeling PVD processes in more complicated geometries, being probabilistic in nature, entail long computation times, and thus render geometry optimization for deposition uniformity a difficult task. Free molecular flow is governed by the same line-of-sight considerations as thermal radiation. Though the existence of an analogy between the two was recognized by Knudsen (1909, Ann. Phys., 4(28), pp. 75–130) during his early experiments, it has not been exploited toward mainstream analysis of deposition processes. With the availability of commercial finite element software having advanced geometry modelers and built-in cavity radiation solvers, the analysis of diffuse thermal radiation problems has become considerably simplified. Hence, it is proposed to use the geometry modeling and radiation analysis capabilities of commercial finite element software toward analyzing and optimizing high-vacuum deposition processes by applying the radiation-molecular flow analogy. In this paper, we lay down this analogy and use the commercial finite element software ABAQUS for predicting radiation flux profiles from planar as well as tube sources. These profiles are compared to corresponding deposition profiles presented in thin-film literature. In order to test the ability of the analogy in predicting absolute values of molecular flow rates, ABAQUS was also employed for calculating the radiative flux through a long tube. The predictions are compared to Knudsen’s analytical formula for free molecular flow through long tubes. Finally, in order to see the efficacy of using the analogy in modeling the film thickness variation in a complex source-substrate configuration, an experiment was conducted where chromium films were deposited on an asymmetric arrangement of glass slides in a high-vacuum PVD chamber. The thickness of the deposited films was measured and the source-substrate configuration was simulated in ABAQUS. The variation of radiation fluxes from the simulation was compared to variation of the measured film thicknesses across the slides. The close agreement between the predictions and experimental data establishes the feasibility of using commercial finite element software for analyzing high vacuum deposition processes.

Gaseous flow through a microchannel is treated numerically and analytically in order to assess pressure and mass flow losses due to a constriction of a finite length. Numerical modeling of two-dimensional (2-D) microchannel flow in the slip and transitional regimes is carried out using the direct simulation Monte Carlo (DSMC) method. The prediction of pressure losses and mass flow based on a simple analytic model for constriction microchannel flow are found to be in excellent agreement with DSMC simulations. Constriction has a dramatic effect on pressure loss and mass flow rate for the considered cases. The DSMC results indicate that the flow in the constriction microchannel separates in the transition section, but the separation does not significantly impact pressure distributions.

Compact models for micromechanical squeezed-film dampers with gap sizes comparable to the surface dimensions are presented. Two different models considering both the border flow and non-uniform pressure distribution effects are first derived for small squeeze numbers. In the first 'surface extension' model the border effects are considered simply by calculating the damping with extended surface dimensions, and in the second 'border flow channel' model an additional short fictitious flow channel is placed at the damper borders. Utilizing a large amount of two-dimensional (2D) FEM simulation results by varying the damper dimensions, mainly the ratio a/h between the surface length and the air gap height, surface elongations are extracted using both elongation models. Both linear and torsional modes of motion are considered at the continuum flow regime. These results show that the 'surface extension' model is superior, since the extracted elongation Δa is almost constant (Δa = 1.3h), leading to a very simple model. Next, the rare gas effects are included in the 'surface extension' model in the slip flow regime (Knudsen number 0 4 in the linear motion and for a/h > 10 in the torsional motion. The model assumes incompressible flow and thus the maximum frequency where the models are valid is limited. In typical MEMS topologies where the elongations must be considered, this means that the models are valid below frequencies of 500 kHz. To also model rectangular 2D squeezed-film dampers, these elongations are applied directly in the surface length and width used in the compact models. Comparison with three-dimensional (3D) FEM simulations shows that the new model gives excellent results, and it extends the validity range of existing compact models. The maximum relative error of the models is smaller than 10% for a/h > 16 in the linear motion and for a/h > 16 in the torsional motion. The new surface extension model is useful in simulating both the circuit level and the system level behavior of gas-damped microelectromechanical devices with aspect ratios greater than 2 in the time and frequency domains.