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This work aims to verify whether the continuum mechanics assumption holds for the numerical simulation of a typical sample delivery system in serial femtosecond crystallography (SFX). Knudsen numbers were calculated based on the numerical simulation results of helium flow through the gas-focused liquid sheet nozzle into the vacuum chamber, representing the upper limit of Knudsen number for such systems. The analysed flow is considered steady, compressible, and laminar. The numerical results are mesh-independent, with a Grid Convergence Index significantly lower than 1% for global and local analysis. This study is based on an improved definition of the numerical Knudsen number: a combination of the cell Knudsen number and the physical Knudsen number. In the analysis, no-slip boundary and low-pressure boundary slip conditions are compared. No significant differences are observed. This study justifies using computational fluid dynamics (CFD) analysis for SFX sample delivery systems based on the assumption of continuum mechanics. [ABSTRACT FROM AUTHOR]

A gas pressure distribution calculation method for viscous laminar and intermediate flows is developed. By introducing the pseudo-conductance for viscous laminar flow, linearized equations describing pressure distribution are obtained. In intermediate flow, the pressure distribution is iteratively obtained. The validity of the method is confirmed by demonstrating a pressure in a long-tube flow path with various flow rates. The result smoothly connects theoretical curves of molecular flow and viscous laminar flow between two regions. Then, the calculation method is applied to an actual device and is compared to the measurement. The comparison between the calculation and measurement shows the qualitative agreement and reveals the requirement for a more detailed implementation of the actual conditions. [ABSTRACT FROM AUTHOR]

Multiple flow mechanisms that coexist in nanoscale porous media are responsible for deviations from the linear Klinkenberg equation. The use of mathematical models in the literature has obvious limitations in evaluating this flow phenomenon because the viscosity/diffusion coefficients of nanoscale porous media are more accurate only in the limited Knudsen number region. By introducing, the concept of an effective molecular mean free path, this paper proposes single models of viscosity/diffusion for the full Knudsen number range to replace the combination model in the literature. On this basis, a new apparent permeability model is developed with multiple coexisting mechanisms for the full Knudsen number range, and the effectiveness of the proposed model is verified by using published data. The discontinuous problem of the combination model of the viscosity/diffusion coefficient in the literature for the full Knudsen number range is solved using the new viscosity and diffusion coefficient models. The new apparent permeability model accurately predicts the absolute permeability and explains the phenomenon of deviation from the linear Klinkenberg equation. This paper further discusses the influence of different mechanisms on the permeability. The rarefaction effect weakens the diffusion ability in porous media but increases the contribution of Darcy flow to permeability. The viscous flow increment, absolute permeability and slippage effect were the most important flow mechanisms in nanopores. Article Highlights: We present the equation of the gas viscosity/diffusion coefficient in nanopores for the full Knudsen number range. We present a new apparent permeability model with multiple coexisting mechanisms for the full Knudsen number range. Our model predicts the absolute permeability of nanoscale porous media. [ABSTRACT FROM AUTHOR]

Gas catalytic reaction in porous materials is integral to the chemical industry. A novel reactive direct simulation Monte Carlo model is proposed to investigate the hydrogen gas reaction in three dimensional porous fibrous materials with flow state across from rarefied to continuum regimes. Complicated processes involving mass, gas flow, heat transfer, radiation effect, and evolving process of the solid skeleton within chemical reactions can be considered simultaneously. The effects of catalyst coverage and gas ratio on heat and mass transfer in the evolved porous fibrous materials are investigated. Results show that the porous fibrous skeleton evolution results in an increase in temperature compared to the scenarios neglecting skeleton evolution across rarefied to continuum regimes. The temperature increases with an increase in catalyst coverage and inlet hydrogen content. Both chemical reaction heat induced by gas molecules and skeleton evolution rate are contributed to the temperature rise within porous fibrous materials. A formula is derived to accurately and efficiently predict the effect of various factors on the temperature in porous fibrous materials. Above findings can improve the understanding of the real chemical reaction process in porous materials. • Reactive DSMC model under full regimes in evolved porous media is developed. • Skeleton evolution results in an increase in temperature with Kn decreasing. • Chemical reaction heat and skeleton evolution both increase temperature. • Prediction formula is constructed to predict factor effect on temperature. [ABSTRACT FROM AUTHOR]

The influence of the deposition of nanoparticles of an aerosol, moving in a continuous reactor at a constant velocity, on the channel walls of the reactor and of the coagulation of these particles as a result of their Brownian diffusion on the nanoaerosol parameters was investigated. A simple one-dimensional model, adequately defining the diffusion, coagulation, and deposition of nanoaerosol particles in a wide range of change in its Knudsen number, has been constructed. It is shown analytically that, as the distance from the inlet of the reactor increases, the volume fraction of particles in the nanoaerosol decreases by the exponential law, and the radius of the clusters formed in the nanoaerosol as a result of the coagulation of its particles increases and tends to a limiting (maximum) value. A nonlinear integro-differential equation for the radius of such clusters has been obtained, and, from it, compact formulas for approximate calculation of its limiting radius have been derived. The characteristic distributions of the radii of clusters and of their concentrations along the reactor channel, calculated by the numerical method, are presented. It was established that, if this channel has a fairly large length, the clusters moving in it increase to their limiting size. The influence of the determining parameters of the flow of a nanoaerosol at the inlet to the continuous reactor on the distribution of its dispersion characteristics along the reactor channel, on the limiting size of the clysters in it, and on the characteristic length of the channel, at which the processes of deposition and coagulation of nanoaerosol particles are completed, is discussed. A comparison of the results of calculations of the parameters of a nanoaerosol moving in a continuous reactor by the one- and two-dimensional models has shown that the simple one-dimensional model defines the behavior of these parameters quite adequately. [ABSTRACT FROM AUTHOR]

The article presents the results of an analytical study of the flow dynamics in a microdiffuser. An expression for the velocity profile is obtained with account for slip effects. The effect of the Knudsen number on the velocity profiles is shown. The effect of the microdiffuser angle on the flow velocity profile and the friction coefficient is analyzed. The opening angle of the microdiffuser was determined, at which slip effects do not affect the velocity profile. [ABSTRACT FROM AUTHOR]

The interaction of rarefied gases with functionalized surfaces is of great importance in technical applications such as gas separation membranes and catalysis. To investigate the influence of functionalization and rarefaction on gas flow rate in a defined geometry, pressure-driven gas flow experiments with helium and carbon dioxide through plain and alkyl-functionalized microchannels are performed. The experiments cover Knudsen numbers from 0.01 to 200 and therefore the slip flow regime up to free molecular flow. To minimize the experimental uncertainty which is prevalent in micro flow experiments, a methodology is developed to make optimal use of the measurement data. The results are compared to an analysis-based hydraulic closure model (ACM) predicting rarefied gas flow in straight channels and to numerical solutions of the linearized S-model and BGK kinetic equations. The experimental data shows that if there is a difference between plain and functionalized channels, it is likely obscured by experimental uncertainty. This stands in contrast to previous measurements in smaller geometries and demonstrates that the surface-to-volume ratio of 0.4 μ m - 1 seems to be too small for the functionalization to have a strong influence and highlights the importance of geometric scale for surface effects. These results also shed light on the molecular reflection characteristics described by the TMAC. [ABSTRACT FROM AUTHOR]

In the present work, the gas flows through diverging channels driven by small, moderate, and large pressure drops are studied, considering a wide range of the gas rarefaction from free molecular limit through transition flow regime up to early slip regime. The analysis is performed using the Shakhov kinetic model, and applying the deterministic DVM method. The complete 4D flow problem is considered by including the upstream and downstream reservoirs. A strong effect of the channel geometry on the flow pattern is shown, with the distributions of the macroscopic quantities differing qualitatively and quantitatively from the straight channel flows. The mass flow rate data set from the complete solution is compared with the corresponding set obtained from the approximate kinetic methodology, which is based on the fully developed mass flow rate data available in the literature. In addition, the use of the end-effect approach significantly improves the applicability range of the approximate kinetic methodology. The influence of the wall temperature on the flow characteristics is also studied and is found to be strong in less-rarefied cases, with the mass flow rate in these cases being a decreasing function of the temperature wall. Overall, the present analysis is expected to be useful in the development and optimization of technological devices in vacuum and aerospace technologies. [ABSTRACT FROM AUTHOR]

The objective of this investigation is to assess the effect of obstacles on numerical heat transfer and fluid flow momentum in a rectangular microchannel (MC). Two distinct configurations were studied: one without obstacles and the other with alternating obstacles placed on the upper and lower walls. The research utilized the thermal lattice Boltzmann method (LBM), which solves the energy and momentum equations of fluids with the BGK approximation, implemented in a Python coding environment. Temperature jump and slip velocity conditions were utilized in the simulation for the MC and extended to all obstacle boundaries. The study aims to analyze the rarefaction effect, with Knudsen numbers (Kn) of 0.012, 0.02, and 0.05. The outcomes indicate that rarefaction has a significant impact on the velocity and temperature distribution. The presence of nine obstacles led to slower fluid movement inside the microchannel MC, resulting in faster cooling at the outlet. In MCs with obstacles, the rarefaction effect plays a crucial role in decreasing the Nusselt number (Nu) and skin friction coefficient (Cf). Furthermore, the study demonstrated that the obstacles played a crucial role in boosting fluid flow and heat transfer in the MC. The findings suggest that the examined configurations could have potential applications as cooling technologies in micro-electro-mechanical systems and microdevice applications. [ABSTRACT FROM AUTHOR]

By way of mathematical and numerical simulation of an nonisothermic flow of a gas with nanoparticles in a tubular continuous-flow reactor with an nonuniform temperature field, the influence of the diffusion (Brownian diffusion and thermodiffusion) of nanoparticles in this flow and their diffusion deposition on the walls of the reactor on the parameters of the gas-suspension flow in it was investigated. The problem on the two-dimensional flow of a nanogas suspension with a nonuniform transverse velocity profile in such a reactor was solved. The longitudinal and transverse distributions of the concentration of nanoparticles in this suspension and its temperature in the reactor were obtained by the numerical finite difference method. The influence of the initial concentration of nanoparticles in the gas flow at the inlet cross section of the reactor and the Knudsen number of the nanoparticles (their sizes in fact) on the distributions of the characteristics of the disperse-mixture flow in the reactor was determined. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the tangential momentum accommodation coefficient α v (TMAC) on an assumed planar, compressible, and slightly rarefied nitrogen gas flow through a microchannel was numerically investigated. Given the small values of Knudsen numbers (0. 04 < K n < 0. 07) and the low Mach number ( M a < 0. 3), the 2 D governing equations were solved using the finite element method with first-order slip boundary conditions using the COMSOL Multiphysics software. The obtained results were presented and discussed. Those results are in good agreement compared to analytical solutions and to the DSMC results obtained by the literature, which confirms the validity of the present model. A comparison with experimental data provided from the literature for pressure distribution of airflow through a long microchannel was also presented and comparable results were found. It has been proved that COMSOL is faster in resolving such problems compared to DSMC method. It has been shown that decreasing the tangential momentum accommodation coefficient α v values increases the gas velocity and the velocity jump near walls, while the pressure gradient is reduced for the given α v values. It has been also shown that for tending to zero values of α v , vortices in the inlet chamber become more intense. [ABSTRACT FROM AUTHOR]

The idea of transferring vital drugs through nanotubes in the human body has attracted the attention of many researchers in the field of nanomedicine. Accordingly, the main objective of this article is to investigate the forced vibration response of two hybrid smart carbon nanotubes connected using springs and conveying a nanofluid, with each one including either a piezoelectric or electrorheological fluid layer. To this aim, a slip boundary condition along with a Knudsen number is employed to address the vibration behavior of smart hybrid sandwich nanotubes acted upon by a moving sinusoidal load, while equations are derived with the aid of the Timoshenko beam model and nonlocal strain gradient theory. The former theory is complemented with hardening and softening material effects which can greatly enhance the precision of the results. Furthermore, Hamilton's principle is implemented to obtain the equations of motion. Regarding the time response of the structure, a combination of modal analysis and the Laplace transform is applied. The accuracy of the developed procedure is verified through a set of comparisons of static deflection (as a result of a point load) and vibration frequencies. In addition, the impact of a number of parameters ranging from nanoparticle velocity to material length scale is investigated. [ABSTRACT FROM AUTHOR]

We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann equation in a rigorous manner in Sobolev spaces. To this end, we formally show that, in a diffusive scaling, the hydrodynamical limit of the kinetic system is the Fick model supplemented with a closure relation and we give explicit formulae for the macroscopic diffusion coefficients from the Boltzmann collision operator. Then, we provide a perturbative Cauchy theory in Sobolev spaces for the constructed Fick system, which turns out to be a dilated parabolic equation. We finally prove the stability of the system in the Boltzmann equation, ensuring a rigorous derivation between the two models. [ABSTRACT FROM AUTHOR]

The gas Couette flow for a cylindrical geometry of bounding surfaces that move in the longitudinal direction relative to their symmetry axis is considered. For a monoatomic gas, the relation between the shear stress and the energy flux transferred to the longitudinal-flow surface (Reynolds analogy) is studied. In the case of continuous medium and free molecular flow regimes, simple explicit analytical expressions for the Reynolds analogy coefficient are obtained, which depend only on the Eckert number and are independent of the ratio of the cylinder radii. The transitional regime for various values of the Knudsen number is studied using the direct simulation Monte Carlo (DSMC) method. It is shown that, in this case, the Reynolds analogy coefficient at a fixed ratio of the radii and the Knudsen number depends on the relative velocity and temperatures of the surfaces mainly through the Eckert number. A relationship between the linear energy fluxes transferred to the cylindrical surfaces is found. [ABSTRACT FROM AUTHOR]

We are proposing a model mathematical description of droplet evaporation using the kinetic approach. We have obtained the basic equation of the theory by using the law of conserving the full power of the vapor–liquid system, which has not been done before. We have found the range of droplet sizes at which it is stable. We have given a comparison of the obtained results with the known traditional ones. We have given numerical estimates for the critical size of the fine-dispersed phase up to the value of which ordinary evaporation takes place (that is for Knudsen number K n = l R , inequality K n ≪ 1 must be fulfilled, where l − is the free path of the molecule and R − is the droplet radius). We have given the optimal droplet size which is the most effective from the point of view of technical use in extinguishing flammable oil transformers. [ABSTRACT FROM AUTHOR]

Anthracite coal of Zhijin Block was taken as the coal samples and NaClO as the oxidants to investigate enhancement of multi-gas transport process in coalbed methane reservoir by oxidation treatment. Industrial microscope observation, SEM observation, low-temperature N 2 sorption measurement and NMR test were used to comprehensively characterize the multiscale pore structure of coal. The change of macromolecular structure of coal before and after oxidation treatment was tested by 13C NMR test, and the force between coal matrix and gas molecules was estimated. Based on the pore size distribution of coal obtained by NMR analysis, the Knudsen number was calculated. Oxidation treatment effectively promoted the desorption of methane in coal, increased the content of free methane in coal. It also led to the CBM gas well entering the stable production period earlier, and shortens the gas breakthrough time of the gas well. Oxidation treatment improved micropores to larger pores by dissolving the coal matrix, and the porosity and permeability of coal were increased. Oxidation treatment led to the widening of pore diameter, and the effect of Knudsen diffusion on the core seepage of the matrix was gradually enhanced. Macroscopically, the gas slippage effect was more significant and the permeability was enhanced. The mechanism of oxidation treatment on multi-gas transport process was mainly reflected in two aspects: increasing the content of free methane and enhancing the gas supply capacity of coal. • The adsorption/desorption characteristics of coal before and after oxidation treatment were analyzed. • The force between coal matrix and gas molecules before and after oxidation treatment was estimated. • The variation of multiscale pore structure of coal before and after oxidation treatment was comprehensively characterized. • The mechanism of oxidation treatment enhances the multi-gas transport process in CBM reservoir was revealed. [ABSTRACT FROM AUTHOR]

The size of micro- and nanofluidic devices accounts for their operation in modes that differ significantly from those for the corresponding macroscopic counterparts. Deep understanding of gas-dynamic processes occurring in micro- and nanofluidic systems opens new opportunities for the practical use of molecular transport at the micro- and nanoscale. Models and simulation methods with high reliability are described. The article also outlines the important flow parameters which must be considered in the first place to correctly simulate gas-dynamic processes in micro- and nanofluidic systems. The review will be useful as a reference for researchers interested in implementing preliminary analysis in the development and optimization of micro- and nanofluid devices. [ABSTRACT FROM AUTHOR]

A microchannel with a 90° bend is one of the essential components of a complex microchannel in microdevices. Therefore, a fundamental understanding of flow through a microchannel with bends is required. This paper investigates gaseous flow experimentally and numerically through a microchannel with bends in the slip regime. A microdevice with 20 microchannels of dimensions 393.96 Ă— 1.03 Ă— 25 200 µ m3 with double 90° bends has been fabricated using standard silicon microfabrication processes. These channels have a common inlet and outlet plenum, and nitrogen gas is used as the working fluid. The mass flow rate through the microchannel is measured for various pressure ratios ranging between 1.30 and 1.55 at different outlet Knudsen numbers in the slip regime. Three-dimensional simulations are performed for rarefied isothermal flow using Ansys fluent 18.1 with slip boundary conditions on the top and bottom walls. The slip effect is quantified by comparing the measured data of the mass flow rates with the corresponding no-slip boundary conditions. To the authors’ best knowledge, experimental data in the slip regime covering the Knudsen number range 0.0662â€"0.0805 have not been reported previously. The gas flow was modelled numerically using the first-order model with an appropriate slip coefficient (C 1 = 1.15), and the equivalent straight length per bend in the slip regime was also calculated. [ABSTRACT FROM AUTHOR]

In this study, high-vacuum flow was analyzed using the direct simulation Monte Carlo (DSMC) method, and various forces acting on fine particles in a high-vacuum flow field were studied. The DSMC method is a Lagrangian method that models the flow as particles and analyzes the collisions and behaviors of each particle, which costs a large computing resource. To validate DSMC method, computational results of a Poiseuille flow in microchannel are compared with analytical results. In addition, the force acting on the particles in the high-vacuum rarefied gas region was verified using the outputs of previous studies. Through this numerical analysis, it is possible to analyze about regions that are difficult to proceed with experiments. As a result, the drag forces according to the Knudsen number which indicates the ratio of vacuum and the particle size, it was confirmed that the drag force can be predicted through the empirical formula of previous studies. [ABSTRACT FROM AUTHOR]

Underground salt cavern storage has become the preferred medium for storing gas energy and strategic substances. Salt caverns are suitable for storing small molecular gases due to the low porosity and permeability of salt rocks. This paper comprehensively analyzes the physical properties of four gases - hydrogen, helium, methane, and carbon dioxide - under subsurface temperature and pressure conditions. It categorizes the flow regimes of these gases in salt rocks under geological pressure conditions based on the Knudsen number. In a typical 1000 m salt cavern, the predominant permeation flow regime of four gases in the surrounding rock is Klinkenberg flow, with helium potentially undergoing transitional flow at low operation pressures. A 3D numerical model is established for an actual salt cavern to compare the permeation and leakage characteristics of these gases within salt rock formations. Results indicate that, under identical operation conditions, the permeation range of the gases decreases in the following order: hydrogen > methane > helium > carbon dioxide. Under the cyclic operation pressures, the cumulative leakage amount of hydrogen, helium, and methane increases over time, while that of carbon dioxide initially rises and then decreases. This behavior is attributed to the fact that the reverse-permeation rate of carbon dioxide at low pressures exceeds its permeation rate at high pressures. Over 30 years cyclic operation, the leakage ratios of the gases are as follows: hydrogen (13.29 %), methane (9.34 %), helium (7.47 %), and carbon dioxide (0.93 %), with hydrogen exhibiting the highest and carbon dioxide the lowest leakage ratios. Larger permeability results in a larger permeation range, while larger porosity leads to a smaller permeation range. When the permeability of salt layer is greater than 1e-20 m2, the permeation range of hydrogen significantly increases. Gas leakage ratios increase with permeability nonlinearly and increase with porosity linearly. The impact of salt layer permeability on leakage ratios is greater than that of porosity. This study provides crucial guidance for the selection of geological formations for storing hydrogen, helium, methane, and carbon dioxide in salt caverns, as well as the investigation of gas permeation characteristics in salt layers. [ABSTRACT FROM AUTHOR]

• Micromechanical modelling of the thermal conductivity of gas-solid mixture medium with imperfect interface. • Eshelby's ellipsoidal inhomogeneity problem. • Green's function technique and integral equation, • Ellipsoidal harmonics method for the thermal conduction inside gas-solid mixture medium. • Analytical expressions of local mean intensity and effective thermal conductivity. The present paper deals with the micromechanical modeling of the thermal conductivity of two-phase gas-solid mixtures with ellipsoidal inhomogeneities. In some structural morphologies, when the characteristic size of the gas-filled inhomogeneity is comparable to the mean path of the gas molecule, the classical heat conduction theory is no longer applicable to the regions very close to the solid interfaces. The present work assumes a sliding flow regime and a Knudsen number such that: 0 < K n ≪ 1. K n = λ / l where λ is the mean path of the gas molecule and l is the size of the inhomogeneity. In this situation, the thermal behavior of the gas can be described by the classical thermal conductivity with modified boundary conditions, considering discontinuities in the local heat flux and temperature fields across gas-solid interfaces. Within this framework, the solution of two problems involving a gaseous ellipsoidal inhomogeneity immersed in a solid matrix, and conversely a solid ellipsoidal inhomogeneity suspended in a gas, is developed. For the first problem, the proposed micromechanical approach is based on the use of Green's function technique and the integral equation established in the general case of anisotropic thermal conductivity per phase. For the second issue, the solution is obtained directly by solving the Fourier equation describing heat transfer in the heterogeneous medium. Then, the Mori–Tanaka (MT) method and the Differential (DS) Scheme are employed to express the effective thermal conductivity of the gas-solid mixture media. [ABSTRACT FROM AUTHOR]

In this study, pressure-driven flow through a slit-type obstacle with various length (L) and height (H) placed in between two parallel plates was investigated by Event Driven Molecular Dynamics (EDMD) simulation. Mach number, temperature and pressure distributions were obtained along the channel in the transition regime. The change in these macroscopic properties and flow rate were examined for different cases created by changing Knudsen number (Kn) of the gas, the geometry of the slit and the outlet/inlet pressure ratio of the flow. Collision of gas molecules with plates and the obstacle were modeled with diffuse reflection boundary condition. The flow rate showed a sudden change in the transition regime and significant differences in the molecular regime depending on the pressure ratio. Except for the Kn, H and L dimensions were found to be effective in Mach disc formation. Pressure drops at the exit of the slit were shaped differently in normalized pressure profiles depending on Kn, H and L dimensions. In addition, the structure of the vortices formed at the entrance and exit of the slit varies depending on Kn. Some of the results obtained were confirmed to be consistent with the similar studies in the literature. [ABSTRACT FROM AUTHOR]

This paper deals with a mathematical model that represents non-Newtonian bile flow through a calculus duct under the influence of heat transfer with wall slip conditions. The peristaltic flow of bile is characterized by a generalized Carreau's model. An ordinate transformation is initiated to map the cosine geometry of the stone into a rectangular grid. It allows evaluation velocity, pressure distribution, flow rate, and reflux occurrence conditions, adopting the perturbation technique, the analytical solution obtained under various parameters, such as Knudsen number, amplitude ratio, Grashof number, Weissenberg number, and power index. These mathematical expressions are analyzed by plotting the graph in MATLAB R2018b software and observed that the axial velocity and the pressure gradient are strongly affected by heat and wall slip parameters. [ABSTRACT FROM AUTHOR]

In this paper, we examine a one-dimensional system of equations for a discrete gas model (the Godunov–Sultangazin system). The Godunov–Sultangazin system is the Boltzmann kinetic equation for a model one-dimensional gas consisting of three groups of particles. In this model, the momentum is preserved whereas the energy is not. We prove the existence of a unique global solution to the Cauchy problem for a perturbation of the equilibrium state with periodic initial data. For the first time, we find the rate of stabilization to the equilibrium state (exponential stabilization). [ABSTRACT FROM AUTHOR]

Gas flow in complex porous media can vary across rarefied to continuum regimes depending on the characteristic size of porous media. To study the gas reactive transport process in three dimensional porous fibrous media across rarefied to continuum regimes, a comprehensive reactive direct simulation Monte Carlo model coupling with heat conduction and radiation model as well as reaction process is proposed. The results calculated by the proposed model are verified to fit well with those obtained using analytical model and experimental data. The "inlet" effect promoting the skeleton temperature is found. There are existing competitions between the heat conduction, reaction rate and convection, radiation process. The temperature of porous fibrous media skeleton increases with the catalyst coverage, and decreases with increase of Knudsen number. When the proportion of hydrogen increases, the increase in the temperature of the porous fibrous media skeleton is initially rapid and then gradual. These studies help to deeply understand the chemical reaction mechanism in porous media across rarefied to continuum regimes. • Reactive DSMC model coupling with heat conduction and radiation models is developed. • The "inlet" effect promoting the skeleton temperature is found. • Relationship among heat conduction, radiation, reaction rate and convection is identified. • Comprehensive model helps to understand chemical reaction mechanism in porous media. [ABSTRACT FROM AUTHOR]

There are wide applications for flow in a microporous medium. In this study, a computational analysis of airflow through a porous microannulus constructed by circular‐ based Sierpinski has been performed in a slip flow regime, where several parameters played an important role in the flow characteristics. These parameters are the Knudsen number, the average friction factor, radius ratio, and porosity. The impacts of these parameters on permeability and the gas flow characteristics are examined and analyzed thoroughly. The ranges of the investigated parameters are as follows (0.001≤Kn≤0.1 and the porosity range is 0.75≤ε≤0.95). The results showed that porosity has a significant impact on the velocity distribution and Darcy number. The Knudsen number has also a direct effect on the velocity distribution, while it has a positive logarithmic proportionality with a dimensionless permeability but the radius ratio does have a neglected effect on the Darcy number. Moreover, the effect of the average friction factor has an inverse proportional relationship to the Darcy number. [ABSTRACT FROM AUTHOR]

In this study, a computational fluid analysis of a slip flow developing thermally and hydrodynamically in a triangular microchannel has been considered. The wall of the channel is preserved at a fixed temperature. Temperature jump and velocity slip boundary conditions are imposed on the surfaces. The microchannel is constructed from seven triangular‐shaped channels and the angle of the microchannel (θ) is varied from 0° to 40°. Simulations are carried out for 1 ≤≤ ≤ ≤ Re Kn 8, 0 0.1, and the air is used as a working fluid with variable physical properties. The simulation results show that the triangular microchannel can improve heat transfer as compared with plain microchannels. An optimal θ of 20° is found to provide the best heat transfer at moderate frictional losses. Finally, correlations for the average Nusselt number and the average friction coefficient among all parameters are proposed as follows: Nu Re Kn- = 4.015 (sin ) θ 0.0255 -0.1 -0.0479, C Ref- = 3.71 -1.218 Kn (sin ) θ -0.296 -0.495. [ABSTRACT FROM AUTHOR]

Rarefied gas flow over a backward-facing step (BFS) is often encountered in separating flows prevalent in aerodynamic flows, engine flows, condensers, space vehicles, heat transfer systems, and microflows. Direct Simulation Monte Carlo (DSMC) is a powerful tool to investigate such flows. The purpose of this research is to assess the impact of Mach number and wall temperature on the flow and surface properties in the transitional flow regime. The Mach numbers considered are 5, 10, 25, 30, and the ratio of the temperature of the wall to that of freestream considered are 1, 2, 4, 8. The Reynolds number for the cases studied is 8.6, 17.2, 43, and 51.7, respectively. Typically the flow properties near the wall are found to increase with both Mach number and wall temperature owing to compressibility and viscous dissipation effects. The variation in flow properties is more sensitive to Mach number than the wall temperature. The surface properties are found to decrease with Mach number and increase with wall temperature. Moreover, in the wake of the step, the vortex's recirculation length is reasonably independent of both free stream Mach number and wall temperature, whereas it decreases with Knudsen number. [ABSTRACT FROM AUTHOR]

With the help of the classical Boltzmann kinetic equation, a correction to the Navier–Stokes equation has been found in the form of a biharmonic term in the Laplace operator. It is shown that upon taking this term into account, corresponding corrections in the Knudsen number appear in the Stokes formula. [ABSTRACT FROM AUTHOR]

In unconventional reservoir rocks, pore anisotropy and gas high Knudsen number (Kn) effect are prominent, while gas slippage factor is a crucial parameter to evaluate their apparent permeability. To analyze the correlation of gas slippage factor with pore anisotropy of porous media and Kn, two-dimensional bundle models and anisotropic porous media with same characteristic length were skillfully constructed in this work. A multi-relaxation-time Lattice Boltzmann model combining diffusive reflection boundary condition and Bosanquet-type viscosity model was applied to simulate gas high-Kn flow (Kn = 0.05–0.53) in them. The results showed that Kn and pore-scale anisotropy jointly determine gas slippage factor of anisotropic porous media, which has nothing to do with porosity, specific surface area, and intrinsic permeability in nature. Pore-scale anisotropy leads to the distinct nonlinear changes of gas slippage factor with Kn. When pore-scale anisotropy factor is between 5.37 and 14.58, gas slippage factor of porous media is positively correlated with Kn. But as pore-scale anisotropy factor is in a range from 1.0 to 5.37, gas slippage factor decreases with an increase of Kn. In addition, gas slippage factor of porous media increases with an increase of pore-scale anisotropy as Kn is in a range of 0.18 to 0.53. This work further improves the understanding of gas slippage factor and gas high-Kn effect in anisotropic porous media. [ABSTRACT FROM AUTHOR]

A Forward-Facing Step (FFS) can be approximated as a simplified geometry for modeling the irregularities around space vehicles, the flow around which is often rarefied. These irregularities lead to flow separation, circulation, and re-attachment. This study aims to examine one of the fundamental surface irregularities that can have a significant effect on the design of space vehicles. In this article, Direct Simulation Monte Carlo (DSMC), an effective method for studying rarefied flows, is used to investigate the role of rarefaction on the non-reacting hypersonic rarefied gas flow over a forward-facing step. A comprehensive analysis of the effects of Knudsen number on the on flow-field properties, i.e., velocity, pressure, temperature, density, and surface properties, i.e., pressure coefficient, skin friction coefficient, and heat transfer coefficient is performed. The selected Knudsen number range from 0.05 to 21.33, covering the various rarefaction regimes. The Mach number of the free-stream flow employed was 25. The sections in the vicinity and downstream of the step are examined in-depth, describing the predominant nonequilibrium aspects. The normalized flow properties increased with Knudsen number and are primarily affected by the interplay of compressibility and viscous dissipation effects. Flow recirculation occurred in the slip and transition regime, whereas it was absent in the free-molecular regime; moreover, the recirculation lengths decreased with rarefaction. The surface properties were found to increase with Knudsen number with peak magnitudes located on the upstream and frontal face of the step. • Rarefied hypersonic gas flow over a forward-facing step is investigated using DSMC. • Effects of Knudsen number in all rarefaction regimes is analyzed. • Flow and surface properties increase with rarefaction. • Recirculation lengths decrease with rarefaction. • Results are primarily influenced by the interplay of compressibility and viscous dissipation effects. [ABSTRACT FROM AUTHOR]

Aim of this study is to investigate the properties of mono-atomic gas flow through the porous medium by using Event-Driven Molecular Dynamics (EDMD) simulation in the transition regime. The molecules and the solid particles forming the porous structure were modelled as hard spheres hence molecule trajectories, collision partners, interaction times and post-collision velocities were calculated deterministically. The porous medium is formed of spherical particles suspended in the middle of the channel and these particles are distributed into the channel in a regular cubic array. Collisions of gas molecules with porous medium were provided by means of the specular reflection boundary condition. A negative pressure boundary condition was applied to the inlet and outlet of the porous media to ensure gas flow. Porosity, solid sphere diameter and Knudsen number (Kn) were initially input to the simulation for different Cases. Thus, the effects of these parameters on mass flow rate, dynamic viscosity, tortuosity and permeability were calculated by EDMD simulation. The results were compared with the literature and were found to be consistent. [ABSTRACT FROM AUTHOR]

Under near-vacuum conditions, the fluid frictional dissipation or approximately the inverse of the quality factor of a microcantilever once the intrinsic dissipation can be neglected is proportional to the low pressure. We shall investigate the dynamic behavior of micro-electromechanical systems (MEMS) devices via the calculation of the quality factor or frictional damping forces resulting from surrounding gases. Here, we illustrated some specific examples relevant to the computation of the quality factor or dynamical friction for an oscillating microcantilever in air via measurements of the paper of Okada et al. (Okada, H., Itoh, T., and Suga, T., 2008, Wafer Level Sealing Characterization Method Using Si Micro Cantilevers,Â" Sens. Actuators A, 147(2), pp. 359-364) considering the quality factors of the CM (a label for a microcantilever: 500 × 90 × 5 μm³ Si microcantilever (the measured resonance frequency: 23.7 kHz) and the paper of Kara et al. (Kara, V., Yakhot, V., and Ekinci, K. L., 2017, Generalized Knudsen Number for Unsteady Fluid Flow, Phys. Rev. Lett., 118(7), p. 074505) in rarefied gases regime. We present the corrected quality factor or dynamical friction over the whole range of the Knudsen number considering the CM part by Okada et al. Our new plot considering the quality factor which is proportional to the inverse of the dissipative friction parameter per unit length, pressure as well as the Knudsen number over the whole range should be useful to researchers in this field. [ABSTRACT FROM AUTHOR]

In the present work, the effect of an oriented low magnetic field on near‐continuum gaseous slip flow inside a two‐dimensional rectangular microchannel has been studied using first‐order boundary conditions. The flow was assumed to be compressible, laminar, and steady. The governing equations were solved analytically to obtain the solutions of velocity, temperature, and the pressure of the flow. The influence of different parameters such as Knudsen number, aspect ratio, Hartmann number, and pressure ratio were studied and analyzed. It was found that the electric and magnetic field with an inclined angle had significant effects on the flow properties. The results showed that the velocity increases and the temperature decreases as the inclination angle of the magnetic field decreases. The velocity increases as the Knudsen number, pressure ratio, and aspect ratio increase, while it decreases with increasing of the Hartmann number. The temperature decreases with increasing of the Knudsen number, pressure ratio, and aspect ratio, while the temperature increases as the Hartmann number increases. The results of the present study were validated with published results in the literature. [ABSTRACT FROM AUTHOR]

In this study, the influence of slip flow and temperature jump on the entropy generation rate are investigated in rectangular microducts. The Knudsen numbers are considered in the range between 0.001 and 0.1, and the aspect ratio lies between 0 and 1. The dimensionless governing equations are solved numerically using Chebyshev spectral collocation method, and the dimensionless velocity and temperature gradients are employed in the entropy generation model. The influences of the dimensionless numbers including Bejan number and irreversibility distribution ratio on the entropy generation rates are investigated and discussed through surface plots and contour diagrams. It is demonstrated that the minimum entropy generation rate exists corresponding to an optimal aspect ratio for each dimensionless number. This minimum entropy generation rate depends upon the nature of dimensionless numbers. [ABSTRACT FROM AUTHOR]

We consider the inverse problem of reconstructing the optical parameters of the radiative transfer equation (RTE) from boundary measurements in the diffusion limit. In the diffusive regime (the Knudsen number Kn ≪ 1), the forward problem for the stationary RTE is well approximated by an elliptic equation. However, the connection between the inverse problem for the RTE and the inverse problem for the elliptic equation has not been fully developed. This problem is particularly interesting because the former one is mildly ill-posed, with a Lipschitz type stability estimate, while the latter is well known to be severely ill-posed with a logarithmic type stability estimate. In this paper, we derive stability estimates for the inverse problem for RTE and examine its dependence on Kn. We show that the stability is Lipschitz in all regimes, but the coefficient deteriorates as e 1/eKn, making the inverse problem of RTE severely ill-posed when Kn is small. In this way we connect the two inverse problems. Numerical results agree with the analysis of worsening stability as the Knudsen number gets smaller. [ABSTRACT FROM AUTHOR]

The flow instability of particle-laden jet has been widely studied for large Stokes numbers. However, there is little attention on the case with small Stoke number, which often occurs in practical applications with nanoparticle-laden fluid. In this paper, the instability of nanofluids in round jet is studied numerically for S t ≤ 0. 0 1. The results show that the law of nanofluids instability is quite similar to regular particle instability for axisymmetric azimuthal mode n = 0. However, for asymmetric azimuthal mode n = 1 , the regular pattern of instability is quite complex and different compared to common particle instability. The variations of wave amplification with wave number for different jet parameter B , Reynolds number Re, particle mass loading Z , Knudsen number Kn, Stokes number St and the azimuthal modes n are given. The flow usually gets more unstable as Knudsen number Kn increases, but the varying law gets inverse at high Reynolds number and at n = 1. The flow gets more unstable as Stokes number St increases at n = 0 but gets more stable at n = 1. The decreases in wave number stimulate the flow instability at n = 1 which shows distinct difference for the case at n = 0. Some unusual results of the effect of B, Re, Z on the flow instability are also discussed. [ABSTRACT FROM AUTHOR]

The thermal transpiration effect has great potential applications for the hydrogen energy. In this paper, the thermal transpiration effect and the hydrogen flow behaviors are studied in the microchannel with the semicircular obstacles. Firstly, the slip boundary model is used in the simulation of the flow performance in the microchannel. The validity of the model at different Kn is verified by comparing with some previous work. Then, the hydrogen flow characteristics of the thermal transpiration effect with the semicircular obstacle are investigated. The result shows that as the size of the semicircular obstacle increases, the hydrogen flow path of the thermal transpiration effect becomes longer, and the temperature gradient decreases. As the characteristic length of the hydrogen flow decreases, there is an obviously negative influence on the thermal transpiration flow. A deeper analysis shows that the thermal driven flow and the pressure driven flow will produce y -component velocity, which leads to a backflow under the effect of semicircles, and the semicircular obstacles make the Knudsen layer spread to the channel center. • The hydrogen flow behaviors in microchannel with semicircular obstacle. • Semicircular obstacles affect the pressure increase of the hydrogen Knudsen compressor. • Semicircular obstacles affect the state of the hydrogen flow through the channel. • Semicircular obstacles in the microchannel affect the Knudsen layer. [ABSTRACT FROM AUTHOR]

Failure analysis revealed that capacitor short-circuit failures were related to a MIM metal missing/peeling defect. During subsequent investigation a correlation between MIM top metal defects and Knudsen flow conditions within the metal deposition system was revealed. MIM surface condition prior to and after top metal deposition was extensively studied. MIM surface characterization showed large size particles formed on the MIM feature after the wafer experienced Knudsen flow. Experiments show such defects could also be produced after multiple oxygen plasma cleans and long staging time of exposed MIM features. These same changes of the MIM surface condition occurred at an accelerated rate under Knudsen flow conditions in the metal deposition system. Inline AOI results showed that when the Knudsen number was at or greater than ~0.2 during vacuum load lock pumping, MIM metal missing/peeling defects were formed, affecting up to 20% of die. [ABSTRACT FROM AUTHOR]

The paper presents results of a study of forced convection in a vertical flat and circular microchannels incorporating porous medium under slip boundary conditions of the first and second orders. The problem was solved analytically and compared with numerical simulations based on the lattice Boltzmann method. Effects of porosity and slip velocity on velocity and temperature profiles were investigated. Behavior of the normalized Nusselt number as a function of the Knudsen and Prandtl numbers, as well as the parameter M characterizing porosity of the medium in the microchannel, was also elucidated. Computations indicated that a decrease in porosity (an increase in the parameter M) causes a decrease in the velocity and temperature jumps on the wall, which contributes to the increase in the Nusselt number. Considering the effect of the second-order slip boundary conditions leads to the decrease in the velocity jump on the wall, when the coefficient A2 (this coefficient considers the second-order slip boundary conditions) changes from negative to positive values. The heat transfer rate at high Prandtl numbers increases with the increasing Knudsen number, because of the improved thermal interaction of the flow with the channel wall. Given the second-order boundary conditions, the effect of parameter A2 was not observed at small Prandtl numbers (Pr ≤ 1). For A2 < 0, the second-order boundary conditions cause an increase in the normalized Nusselt number, whereas for A2 > 0 the normalized Nusselt number decreases in comparison with the case of A2 = 0 (the first-order boundary conditions). The results by the analytical solution and the lattice Boltzmann method are in a good agreement with each other for A2 ≥ 0, with the differences ≤ 1%. For A2 < 0, the differences between two models are more noticeable. For A2 ≥ 0, predictions by the analytical model lie higher than the numerical results, whereas for A2 < 0, numerical simulations exceed the analytical solution. [ABSTRACT FROM AUTHOR]

To characterize the microflow over a larger range of Knudsen numbers, an improved kinetic equation considering the volume diffusion effect for nonideal gases was presented based on Klimontovich's kinetic equation and Enskog equation-based lattice Boltzmann Bhatnagar-Gross-Krook (LBGK) model. Then, with the modified effective viscosity and the second-order slip boundary condition, a series of numerical simulations of gas flows with different mean Knudsen numbers were carried out based on the proposed model. Compared with the solutions of Navier-Stokes equations, Navier-Stokes equations with different slip boundary conditions, bivelocity hydrodynetics, and experimental data, we found that the present model can be valid up to a Knudsen number of 30. It is also shown that the present model furnishes a better solution in the transitional flow regime (0.1

• Rarefied gas flow in a microchannel with backward facing step is simulated by LBM. • Mach number, pressure, and mass flow rate are studied in slip and transient regimes. • Results obtained by LBM have a good agreement with the results of DSMC method. • Mach numbers in transient regime are lower than Mach numbers of slip one. • Slip regime has lower values of Knudsen number as compared with transient one. In this paper, the rarefied gas flow in a microchannel with backward facing step is simulated by Lattice Boltzmann method. A model with two relaxation times in slip and transient flow regimes is used. The slip velocity in the walls of the microchannel is improved by using the wall operation function, known as power-law analytical function. The advantage of this model in comparison with a single relaxation time model is the accuracy of the slip velocity and its advantage over a model with a multiple relaxation times is the low cost and running time along with simplicity of programing. The boundary conditions used in the inlet and outlet sections of this geometry are Zou-He condition and the bounce back and Specular Reflection boundary condition is considered on the walls. The profiles of Mach number, pressure, and mass flow rate are studied in the slip and transient flow regimes at Reynolds numbers of 6.43 and 0.052, respectively. The effects of Knudsen number on the velocity and vortex formation are also discussed. The results are also compared with the results obtained by the Direct Simulation Monte Carlo method and a good agreement between two results is observed, which shows the accuracy of the model. [ABSTRACT FROM AUTHOR]

In this article, rarefied gas flow was investigated and analyzed by the Fokker-Planck approach in different Knudsen numbers and Mach numbers at subsonic and supersonic regimes. The presented Fokker-Planck approach is used to solve the rarefied gas flows in different sheardriven micro/nano geometries like one-dimensional Couette flow and the two-dimensional cavity problem. Boltzmann’s equation, and especially statistical technique of the Direct Simulation Monte Carlo (DSMC), are precise tools for simulating non-equilibrium flows. However, as the Knudsen number becomes small, the computational costs of the DSMC are greatly increased. In order to cope with this challenge, the Fokker-Planck approximation of the Boltzmann equation is considered in this article. The developed code replaces the molecular collisions in DSMC with a set of continuous stochastic differential equations. In this study, the Fokker-Planck method was evaluated in the Couette flow in the subsonic Mach number of 0.16 (wall velocity was 50 m/s) and in the supersonic Mach number of 3.1 (wall velocity was 1000 m/s), where Knudsen numbers range from 0.005-0.3. Also, the cavity flow with a wall Mach number of 0.93 (wall velocity was 300 m/s) in Knudsen numbers ranging from 0.05-20 was investigated. The results show that by increasing speed and Knudsen numbers, the accuracy of Fokker-Planck increases. In addition, despite using larger number of simulator particles, the rapid convergence and lower computational costs relative to other methods are the features of this method. [ABSTRACT FROM AUTHOR]

Highlights • New LBM study to consider free & force convections in microflow under heat flux. • Use suitable models for slip velocity and heat flux boundary conditions in LBM. • Gravity should be considered at Kn < 0.01 in air microflow imposed constant heat flux. Abstract Mixed convection heat transfer of air in a 2-D microchannel is investigated numerically by using lattice Boltzmann method. The effects of buoyancy forces on slip velocity and temperature profiles are presented while the microchannel side walls are under a constant heat flux boundary condition. Three states are considered as no gravity, Gr = 100 and Gr = 500. At each state, the value of Knudsen number is chosen as Kn = 0.005, Kn = 0.01 and Kn = 0.02 respectively; while Reynolds number and Prandtl number are kept fixed at Re = 1 and Pr = 0.7. Density-momentum and internal energy distribution functions are used in order to simulate the hydrodynamic and thermal domains in LBM approach. Develop the ability of LBM to simulate the constant heat flux boundary condition along the microchannel walls in the presence of slip velocity and buoyancy forces is proved for the first time at present work. The new and interesting results are achieved such as generating a rotational cell through the fluid flow due to buoyancy forces which leads to see the negative slip velocity at these areas. [ABSTRACT FROM AUTHOR]

Abstract In this study, the lattice Boltzmann method is utilized to simulate gas flow at micro/nano pores of tight and shale gas reservoirs. The flow regime domain for this work is in the Knudsen range of slip flow, i.e., 0.01 < Kn < 0.1. The Langmuir slip model incorporated in the lattice Boltzmann model is introduced using slip regime boundary condition, which is a combination of bounce-back and specular reflections. Appropriate parameters, such as reflection factor, relaxation time and Knudsen number, are considered to accurately predict the slip velocity on the pore walls. A pore-throat model is developed to observe changes in flow characteristics and slip velocity when gradual contraction happens along the flow channel (throat), followed by steady expansion to the original pore size. Although the literature shows various studies on the lattice Boltzmann simulation of gas flow under micro/nano-confinement, realistic pore-throat structures of shale have received limited attention to date. In fact, the shape, size and diameter of the pore-throat system and its effect on the gaseous flow are of crucial importance in micro/nano gas flow due to the dynamic behavior of the Knudsen number at different length scales. The throat to pore ratio is defined as the ratio of the diameter of pore throat, D2, to that of the pore body, D1. It is found that the gas flow behavior depended on the Knudsen number and the D2/D1 ratio. In addition, both the velocity profile and slip velocity are investigated at different Knudsen numbers (within the slip flow range) and D2/D1 ratios. At Kn numbers of 0.01, 0.05 and 0.1, and decreasing D2/D1 ratios from 0.4 to 0.1, the slip velocity adjacent to the wall increased by 57, 46 and 31%, respectively. It is also revealed that decreases in throat size and increases in Kn value enhance the slip effect within the throat. When the D2/D1 ratio decreases from 0.4 to 0.1 and Kn rises from 0.01 to 0.1, the ratio of maximum velocity to slip velocity alters from 8.11 to 1.2. This shows that at the highest Kn value and lowest throat to pore ratio, the slip effect is at the most significant level. Graphical abstract Image 1 Highlights • Gas flow in micro/nano pore-throat structure is modeled. • New LBM model is compared with previous data/models. • Slip velocity and maximum velocity is studied at different Kn values. • Effect of different pore characteristics on slip velocity is studied. • Effect of pore contradiction is investigated in slip flow regime. [ABSTRACT FROM AUTHOR]

Comsol software was used to investigate the effect of temperature jump and velocity slip on the hydrodynamic and thermal behavior of MHD flows under free convection conditions between parallel vertical plates and along a vertical plate. Further, the continuum model of fluid was used with Knudssen (Kn) number regime 0.001

The system of hyperbolic type differential equations is considered for non–Fourier heat transfer in thin films and wires. Structures of the harmonic solutions for telegrapher equation and for Guyer–Krumhansl equation are identified and compared with each other. The exact harmonic solution is obtained; the influence of the phonon heat transfer is explored. The maximum principle violation is demonstrated. Frequency dependence of the solution for Guyer–Krumhansl equation is shown. The exact analytical solution is derived for the inhomogeneous ballistic–diffusion set of differential equations, governing heat propagation in thin films. The contribution of the ballistic, wave and diffusive heat transfer components as well as the influence of the initial conditions and of the Knudsen number on heat conduction in thin films and wires are investigated. The cases of identical and different from each other Knudsen numbers for ballistic and diffusive components are explored. The examples of exact solutions for the heat transfer in each case are demonstrated. [ABSTRACT FROM AUTHOR]

The main factors/mechanisms that influence coal permeability are effective stress, swelling, shrinkage, deformation and gas slippage. After an extended period of gas production, coal can have a rebound phenomenon where permeability increases with increasing effective stress. This rebound can have a significant impact on gas recovery during the late stages of a reservoir life cycle. This paper aims to characterise coal permeability by combining laboratory measurements with a simple gas slippage model that explains the rebound phenomenon. Gas and Klinkenberg corrected permeabilities of coal are measured at (1) constant confining pressure and (2) constant effective stress. We estimate the length scales relevant to gas flow using mercury intrusion, a permeability slip model, and the kinetic theory of gases, which allows us to estimate the Knudsen number for gas flow. Results show a linear relationship between slip length and the mean free path of gas for all of the tested mean pore pressures. This result suggests that a first order slip boundary condition is sufficient to explain the momentum exchange at the gas/solid boundary during flow under normal reservoir conditions. A correlation between Knudsen number and increased permeability is developed, which further demonstrates that slippage cannot be neglected in coals when Knudsen number is greater than 0.1. Overall, we present a simple model that explains permeability rebound in coal by considering only gas slippage. We do not discredit the mechanism of coal shrinkage, which could also influence coal permeability. We confirm that gas slippage should be considered in coal permeability models. [ABSTRACT FROM AUTHOR]