Momentum Accommodation Coefficient Research Articles

Modeling drag coefficients of spheroidal particles in rarefied flow conditions

Transport of particles in flows is often modeled in a combined Eulerian–Lagrangian framework. The flow is evaluated on an Eulerian grid, while particles are modeled as Lagrangian points whose positions and velocities are evolved in time, resulting in particle trajectories embedded in the time-dependent flow field. The method essentially resolves the flow field in complex geometries in detail but uses a closure model for the particle dynamics aimed at including the essential particle–fluid interactions at the cost of detailed small-scale physics. Rarefaction effects are usually included through the phenomenological Cunningham correction on the drag force experienced by the particles. In this Lagrangian point-particle approach, any explicit reference to the finite size and the shape of the particles, and their local orientation in the flow field, is typically ignored. In this work we aim to address this gap by deriving, from fully-resolved Direct Simulation Monte Carlo (DSMC) studies, heuristic or closure models for the drag force acting on prolate and oblate spheroidal particles with different aspect ratios, and a fixed orientation, in uniform ambient rarefied flows covering the transition regime between the continuum and free-molecular limits. These closure models predict the drag in the transition regime for all considered parameter settings (validated with DSMC data). The continuum limit is enforced a priori and we retrieve the free-molecular limit with reasonable accuracy (based on comparisons with literature data). We also include in the models the capability to predict effects related to basic gas-surface interactions via the tangential momentum accommodation coefficient. We furthermore assess the validity of the proposed closure model for particle dynamics in proximity to solid walls. This investigation extends our previous work, which focused on small aspect ratio spheroids with exclusively diffusive gas-surface interactions [see Livi et al. (2022)]. The derived models are obtained for isothermal, subsonic flows relevant for particle contamination control in semiconductor manufacturing.

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

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

On the increased interfacial resistance of hydrogen in carbon nanotube arrays and its effect on gas mixture separation

We outline a surface scattering kernel for rarefied gas flows through ideally ordered nanomaterials, such as high aspect ratio carbon nanotubes. The derived model allows for a comparison of the tangential momentum accommodation coefficient, and, hence, the total effective friction, for different species of gases as a function of the particle diameter. This surface kernel is incorporated with a Fokker–Planck model as an approximation to transport of a rarefied gas through ideally ordered carbon nanotubes. The results of this analysis predict that H2 experiences higher friction in such systems in comparison with larger molecules such as CH4. The results are proposed as a potential explanation of the reduced gas transport of hydrogen gas in nanoporous systems.

Effect of film thickness and temperature on condensation and momentum accommodation at the liquid–vapor methane interphase in contact with a quartz substrate

In this paper, we used the Molecular Dynamics method to simulate the equilibrium vapor–liquid methane in contact with a solid quartz substrate and study the condensation process and momentum exchange at the atomic level. A large potential cutoff radius is used to determine the vapor and liquid densities at good accuracy. By tracking the motions of fluid molecules exchanged between the vapor and liquid phases and carrying out statistical analysis of residence time, penetration depth and especially velocity correlation, the mass and the velocity accommodation coefficients can be determined at the same time. The latter is based on the assumption that the reflected events take place over a short time, near the interface and thus atoms velocities are correlated while the condensation/evaporation events are not. The calculated coefficients are sensitive to the film thickness due to the presence of the quartz substrate and the layering effects. As temperature increases, atoms condense less and reflect more and the reflection is more diffusive due to increasing collision rate. Near the critical temperature, both the condensation and momentum accommodation coefficients vary significantly.

Molecular Dynamics Study of Gas–Surface Interactions on -Cristobalite Surface

In the present work, nonreactive gas–surface interactions between nitrogen molecules and a -cristobalite surface are analyzed using the molecular dynamics framework. A sampling method is employed to perform trajectory calculations, and the tangential momentum accommodation coefficient is computed. The credibility of the reactive force field potential to model cristobalite is investigated, and the effect of the surface and gas temperatures on the tangential momentum accommodation coefficient is studied in detail. The obtained value of the tangential momentum accommodation coefficient (from molecular dynamics analysis) is used as an input parameter in the Maxwell gas–surface interaction model using the direct simulation Monte Carlo method to investigate the surface heat flux on the nose region of a model reentry vehicle. The computed heat-flux results obtained using a molecular-dynamics-derived accommodation coefficient are found to be in excellent agreement with the experimental data.

Impact of surface physisorption on gas scattering dynamics

Engineering flow systems operating under low pressures and/or at the micro/nano scale generally include a physically adsorbed gas layer next to the surface. In this paper, we develop a scattering kernel that accounts for the effect of adsorption, arising from van der Waals interactions, on the dynamics of molecules impinging on solid smooth surfaces. In the limit of low bulk density, surface adsorption becomes negligible and the scattering kernel recovers consistently the Cercignani–Lampis model, which best describes molecular collisions with a clean, smooth surface. In the limit of high bulk density, a dense adsorbed molecular layer forms next to the surface and its presence is picked up by the Maxwell model with complete diffuse reflection, which better captures the multiple collisions suffered by molecules. A weight coefficient based on the Langmuir adsorption isotherm is incorporated into the modelling to handle the transition between these two limiting conditions of low and high densities. The proposed model is validated against high-fidelity molecular dynamics simulations that are performed for a variety of gas–surface combinations and adsorbed molecular layers with different densities. It is shown that the proposed model very well captures the scattering patterns of beams of gas molecules at different velocities impinging on surfaces, as well as momentum and energy accommodation coefficients in the entire range of explored conditions.

Flow-field in a lid driven cavity with slip boundaries: An investigation using lattice Boltzmann modelling

Investigation of the flow behaviour in a lid driven square enclosure has been performed imposing slip boundary conditions at all sidewalls. The slip on the walls is described by prescribing a tangential momentum accommodation coefficient (σ). While the range of Reynolds number (Re) is varied between 20 and 300, the value of σ is taken in the range of 0.01 to 1. Simulations have been carried out using single relaxation time based lattice Boltzmann method (SRT- LBM). To account for the slip, two different methods are used for treatment of boundary conditions, namely tangential momentum accommodation coefficient (TMAC) and modified bounce-back specular reflection (MBSR). While both schemes are comparable in terms of computing requirements, it is observed that the MSBR scheme predicts higher slip velocity on the wall as compared to that of TMAC scheme in lid driven cavity. Results show that the appearance of secondary vortex is affected by the accommodation coefficient. The size of the secondary vortex in the cavity reduces and eventually vanishes as the slip on the wall increases. The threshold accommodation coefficient for disappearance of corner vortex decreases with increasing Reynolds number. With reducing value of σ i.e., as the effect of slip increases, the location of the primary vortex centre shifts towards the centre of the cavity.

Numerical study of the accommodation coefficient influence on rarefied gas flow through a microchannel

In this paper, the influence of the tangential momentum accommodation coefficient αv (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<Kn<0.07) and the low Mach number (Ma<0.3), the 2D 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 αv values increases the gas velocity and the velocity jump near walls, while the pressure gradient is reduced for the given αv values. It has been also shown that for tending to zero values of αv, vortices in the inlet chamber become more intense.

Knudsen Layer Behaviour and Momentum Accommodation from Surface Roughness Modelling

This work analyses the formation of the Knudsen layer in micro/nanoscale flows by linking a rough wall collision model to a continuum flow model via asymptotic matching. Expressions for the accommodation coefficients in terms of the surface characteristics are derived, allowing for boundary layer analysis of rarefied flows without the use of prior determined accommodation coefficients. This derived model, through use of the Lennard–Jones parameters for a nanoscale system, allows for a prediction of the the effective Tangential Momentum Accommodation Coefficient (TMAC) in flows against ordered nanoscale surfaces.

Statistical assessment of tangential momentum accommodation coefficient using internal flow rate model based on rarefied gas conditions

In this paper, we refer previously reported numerical data to design a new analytic equation for Poiseuille flow in a circular channel. Superposition of the continuum, slip, and free-molecular flow equations provides a versatile solution for the entire Knudsen range; however, it is not accurate in the transition regime. Adding correction factors and modifications to the existing flow-rate model enhances the precision and validity for wide flow ranges. Parameters of each equation are fitted for the collected data with Levenberg–Marquardt algorithm and compared in each regime; subsequently, the equation with the best accuracy and simplicity is selected. The newly suggested formula fits well for the referred data by correlating with the dimensionless flow rate, Knudsen number, and tangential momentum accommodation coefficient (TMAC). Furthermore, we calculate the TMAC using previous experimental data, which reflects the reflective interactions between the channel surfaces and gas molecules. Our newly suggested formula demonstrates effectively that the TMAC is significantly affected not only by the channel surface roughness but also by the molecular structure of gas.

A modified Cercignani–Lampis model with independent momentum and thermal accommodation coefficients for gas molecules scattering on surfaces

To accurately predict the aerodynamic forces and heating of hypersonic vehicles, an elaborate gas–surface interaction (GSI) model is essential to describe the rarefied gas effect at the interface. By far, the Cercignani–Lampis (CL) model is the most sophisticated GSI model, but it has not been systematically examined under hypersonic conditions where the gas flow possesses large velocity and temperature differences relative to the boundary surface. In this work, the accuracy of the CL model is first assessed by comparing with molecular dynamics simulation results under a variety of conditions of temperature and velocity differences, surface temperature, and surface roughness. It is found that the tangential scattering kernel of CL model is perfect only when the surface is smooth and the gas kinetic energy is close to the surface kinetic energy. To accurately capture the momentum and temperature of the reflected gas molecules, a modified CL model with independent momentum and thermal accommodation coefficients is developed. A hybrid diffusive and modified CL model is further proposed to account for the adsorption phenomenon. With these modifications, the new GSI model can accurately describe the scattering of gas molecules in a wider range of hypersonic rarefied gas flows.

Gas–surface interactions of a Couette–Poiseuille flow in a rectangular channel

Reduced mass flow rates of a rarefied Couette and Poiseuille flow in a long rectangular channel are calculated in the whole range of the gas rarefaction and a wide range of the width to height ratio. Furthermore, walls may be made of different materials so that different tangential momentum accommodation coefficients (TMACs) may be applied. Analytical solutions are given for the slip regime, where all four surrounding walls may have a different TMAC. Due to a simplified modeling assumption, these solutions can be used to correct the well-known flow rates of a fully diffuse channel for different TMACs in the whole range of the gas rarefaction. If the slip solution and the diffuse solution are known, the procedure can principally be adapted for any channel shape. The results of the analytical model expressions are validated with simulation data of the plane Couette and Poiseuille flow and the Poiseuille flow through a pipe, which are found in the literature. In addition, the analytical solution is compared to results of the Direct Simulation Monte Carlo (DSMC) method of a Couette and a Poiseuille flow in a rectangular channel, which are provided as tabulated data for a variation of the gas rarefaction parameter at different aspect ratios and different combinations of TMACs. The procedure to calculate the mass flow rate of the certain flow as well as the application limits are discussed.

Aerothermodynamics of a sphere in a monatomic gas based onab initiointeratomic potentials over a wide range of gas rarefaction: transonic, supersonic and hypersonic flows

Aerothermodynamic characteristics of a sphere such as drag and energy transfer coefficients are calculated for Mach numbers varying from 1 to 10 over a wide range of the gas rarefaction degree spanning the free molecular, transitional and near hydrodynamic flow regimes. The effects of major factors determining the gas flow are studied using the direct simulation Monte Carlo method. To reveal the effect of gas species, the simulations are performed based onab initiointeratomic potentials for helium, neon, argon and krypton, as well as based on the hard sphere model. The impact of the accommodation coefficients is evaluated by applying the Cercignani–Lampis model of gas–surface interaction. The calculations are carried out for several values of the free stream and sphere temperatures. It is found that the effects of gas species on the drag and energy transfer coefficients are approximately 3 % and 6 %, respectively. The accommodation coefficients in the Cercignani–Lampis model strongly affect all aerothermodynamic characteristics. The drag and energy transfer coefficients calculated for different accommodation coefficients vary within 30 % and 200 %, respectively. It is found that the variation of the tangential momentum and normal energy accommodation coefficients can induce an increase of the drag coefficient compared to the case of diffuse gas–surface interaction. In hypersonic flows, the drag coefficient varies within 30 % when the sphere temperature varies from the free stream temperature to the stagnation temperature. The drag and energy transfer coefficients are found to be non-monotonic functions of the free stream temperature.

Thermophoretic force on a sphere of arbitrary thermal conductivity in a rarefied gas

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

A Comparison between Klinkenberg and Maxwell-Stefan Formulations to Model Tight Condensate Formations

Summary Cyclic solvent (gas) injection is an efficient recovery method for condensate reservoirs. However, in tight, unconventional formations, the added complexity of low permeability results in more physics at play, beyond the widely used Darcy model for conventional reservoirs. In this work, a rigorous mass transfer model is implemented considering the real gas version of the Maxwell-Stefan formulation to evaluate cyclic injection schemes in tight condensate reservoirs. This model is then compared to the more widespread used Klinkenberg formulation, which does not include molecular diffusion. An evaluation is performed to check if a simplified formulation can be used to provide reasonable results in modeling production and enhanced recovery in tight condensate formations. Verification of the implemented equations is performed using experiments (Maxwell-Stefan model) and a commercial reservoir simulator (Klinkenberg model). Furthermore, the cell length used for the numerical studies is selected based on a sensitivity study to evaluate how numerical dispersion impacts recovery factor and liquid saturation for different cell sizes. By comparing the Klinkenberg model with different tangential momentum accommodation coefficient (TMAC) values to the Maxwell-Stefan model during primary production, it is possible to select a value of TMAC that can match closely the recovery values of lighter components when using the Maxwell-Stefan equations. However, for heavier hydrocarbon fractions, difference in recovery is more accentuated owing to increased molecule size (more molecular friction). This results in differences in condensate yield during primary production that may be relevant in a field scale. In the cyclic injection scheme, the importance of accounting for frictional effects between molecules is demonstrated using the Maxwell-Stefan formulation. In this case, molecular diffusion fluxes are influenced by high composition gradients. This results in differences between the Maxwell-Stefan and Klinkenberg models in terms of gas stored and hydrocarbon produced during cyclic injection simulations. Furthermore, a sensitivity study on operational parameters in the cyclic injection stage demonstrated that increasing the length of production cycles may be more beneficial than increasing the length of injection or soaking cycles. For the simulations in this study, the gas is injected above the dewpoint and pressure diffusivity is at least one order of magnitude higher than the other physics present in the process. Therefore, increasing the length of production cycles allows for recovery of heavier hydrocarbon fractions that remain in the gas phase. In this work, it is demonstrated that using a rigorous mass transfer formulation, such as the Maxwell-Stefan equations, can provide more information on a per component basis when evaluating cyclic injection schemes in tight condensate reservoirs.

Methane scattering on porous kerogen surfaces and its impact on mesopore transport in shale

Revealing the scattering behaviour of gas molecules on porous surfaces is essential to develop accurate boundary conditions for kinetic transport models that describe the gas dynamics in shale reservoirs. Here, we use high-fidelity molecular dynamics simulations to resolve the gas–surface interactions between methane molecules and realistic organic kerogen surfaces, and to assess the applicability of the widely used scattering kernels. Our results show that the tight matrix porosities have a negligible effect on the timescale and lengthscale of the scattering process, which can be considered instantaneous in time and local in space. Although reflected velocity distributions reveal that the common Maxwell, Cercignani–Lampis and Yamamoto scattering models fail to fully capture the scattering details of methane on kerogen, especially when the incident molecular speeds are high, the Maxwell model predicts best the reflected angular beam pattern and the overall reflected velocity distribution for rough kerogen surfaces. However, for low-speed impingement, more characteristic of shale applications, all scattering models give similar velocity distributions, which are driven by the high degree of gas–surface accommodation observed. We find that a Maxwell model with a calibrated tangential momentum accommodation coefficient, which approaches unity as the surface roughness increases to ∼2 nm, is enough to reproduce comparable velocity profiles and mass flow rates inside moderately confined kerogen mesopores. Deviations between the Maxwell model and our molecular simulations are only observed for highly rarefied transport problems, but this rarefaction lies beyond the realm of shale reservoir applications. This paper, therefore, reports the first scattering study on porous and rough kerogen surfaces, and demonstrates the applicability of the Maxwell model, which can be readily incorporated into gas kinetic solvers to predict the apparent permeability of shale with mesopore and macropore networks.

Statistical Assessment of Tangential Momentum Accommodation Coefficient Using an Internal Flow Rate Model Based on Rarefied Gas Conditions

Statistical Assessment of Tangential Momentum Accommodation Coefficient Using an Internal Flow Rate Model Based on Rarefied Gas Conditions

Extraction of Tangential Momentum and Normal Energy Accommodation Coefficients by Comparing Variational Solutions of the Boltzmann Equation with Experiments on Thermal Creep Gas Flow in Microchannels

In the present paper, we provide an analytical expression for the first- and second-order thermal slip coefficients, σ1,T and σ2,T, by means of a variational technique that applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator for hard-sphere molecules. The Cercignani-Lampis scattering kernel of the gas-surface interaction has been considered in order to take into account the influence of the accommodation coefficients (αt, αn) on the slip parameters. Comparing our theoretical results with recent experimental data on the mass flow rate and the slip coefficient for five noble gases (helium, neon, argon, krypton, and xenon), we found out that there is a continuous set of values for the pair (αt, αn) which leads to the same thermal slip parameters. To uniquely determine the accommodation coefficients, we took into account a further series of measurements carried out with the same experimental apparatus, where the thermal molecular pressure exponent γ has been also evaluated. Therefore, the new method proposed in the present work for extracting the accommodation coefficients relies on two steps. First of all, since γ mainly depends on αt, we fix the tangential momentum accommodation coefficient in such a way as to obtain a fair agreement between theoretical and experimental results. Then, among the multiple pairs of variational solutions for (αt, αn), giving the same thermal slip coefficients (chosen to closely approximate the measurements), we select the unique pair with the previously determined value of αt. The analysis carried out in the present work confirms that both accommodation coefficients increase by increasing the molecular weight of the considered gases, as already highlighted in the literature.

Experimental measurements of thermal and tangential momentum accommodation coefficients on solid surfaces: Water vapor in comparison with noble gases

In the high Knudsen number flows, like micro gaseous flows and flows in vacuum, the gas-surface interaction plays an important role. Among many gas species, water is a common and useful fluid in wide area of engineering applications, and it is one of the main residual gas components in the high vacuum environment. However, experimental measurements on the gas-surface interaction of water molecules are still limited. In this study, we focus on the accommodation coefficient, which represents the mean behavior of gas molecules on the scattering process from a surface. We measure the thermal accommodation coefficient (TAC) and the tangential momentum accommodation coefficient (TMAC) of water molecules. TAC is measured from the heat flux between two surfaces with different temperatures in vacuum. TMAC is derived through the viscous slip coefficient from the measurement of the mass flow rate through a microtube. To investigate the characteristics of water molecules clearly, the measured accommodation coefficients are compared with those of noble gases, helium, neon, and argon, measured with the same apparatus and procedures. For the accommodation coefficients of noble gases, TAC increases with increasing the molecular weight, while TMAC decreases, which are in good agreement with the previous studies. The TAC and TMAC of water are slightly larger than those estimated from the corresponding molecular weight of noble gases.

Numerical study of supersonic boundary-layer modal stability for a slightly rarefied gas using Navier—Stokes approach

In this paper, a systematic study on the supersonic boundary-layer modal stability for a slightly rarefied gas is conducted by considering velocity slip and temperature jump effects in the Navier–Stokes (NS) equations. The effects of slip boundary on the first- and second-mode instability at different conditions are presented in detail. The laminar flow is obtained by solving the NS equations along with no-slip and slip boundary conditions, which shows that the slip boundary causes the boundary layer becoming thinner and the supersonic region near the wall becoming narrower. The perturbation slip boundary conditions at the wall and their influence on the stability are carefully discussed. The tangential momentum accommodation coefficient and the thermal accommodation coefficient are set equal or unequal for a broad range to study the combined or leading effects of velocity slip and temperature jump, respectively. It is found that velocity slip significantly stabilizes the second-mode disturbances while largely destabilizes the first-mode perturbations. On the contrary, the temperature jump apparently enhances the second-mode instability, while it has little influence on the first mode. When velocity slip and temperature jump are both present, the first mode is more destabilized, while a competitive effect acts on the second mode. Additional results show that the neutral stability curves for the second and third modes as well as the synchronization between fast and slow modes are delayed further downstream due to velocity slip. These findings are shown consistently regardless of the wall cooling for both supersonic and hypersonic flows.