THE difficulty of reconciling line spectra with the kinetic theory of gases, has been referred to by Prof. Fitzgerald (NATURE, January 3, p. 221). The following considerations show that it is possible under certain suppositions to have a number of spectral rays with a very restricted number of degrees of freedom. Most of us, I believe, now accept a definite atomic charge of electricity, and if each charge is imagined to be capable of moving along the surface of an atom, it would represent two degrees of freedom. If a molecule is capable of sending out a homogeneous vibration, it means that there must be a definite position of equilibrium of the “electron.” If there are several such positions, the vibrations may take place in several periods. Any one molecule may perform for a certain time a simple periodic oscillation about one position of equilibrium, and owing to some impact the electron may be knocked over into a new position. The vibrations under these circumstances would not be quite homogeneous, but if the electron oscillates about any one position sufficiently long to perform a few thousand oscillations, we should hardly notice the want of homogeneity. Each electron at a given time would only send out vibrations which in our instruments would appear as homogeneous. Each molecule could thus successively give rise to a number of spectral rays, and at any one time the electron in the different molecules would, by the laws of probability, be distributed over all possible positions of equilibrium, so that we should always see all the vibrations which any one molecule of the gas is capable of sending out. The probability of an electron oscillating about one of its positions of equilibrium need not be the same in all cases. Hence a line may be weak not because the vibration has a smaller amplitude, but because fewer molecules give rise to it. The fact that the vibrations of a gas are not quite homogeneous, is borne out by experiment. If impacts become more frequent by increased pressure, we should expect from the above views that the time during which an electron performs a certain oscillation is shortened; hence the line should widen, which is the case. I have spoken, for the sake of simplicity, as if an electron vibrating about one position of equilibrium could only do so in one period. If the forces called into play, by a displacement, depend on the direction of the displacement, there would be two possible frequencies. If the surface is nearly symmetrical, we should have double lines.

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The Kinetic Theory of Gases

Accurate modeling of gas microflow is crucial for the microfluidic devices in MEMS. Gas microflows through these devices are often in the slip and transition flow regimes, characterized by the Knudsen number of the order of 10−2~100. An increasing number of researchers now dedicate great attention to the developments in the modeling of non-equilibrium boundary conditions in the gas microflows, concentrating on the slip model. In this review, we present various slip models obtained from different theoretical, computational and experimental studies for gas microflows. Correct descriptions of the Knudsen layer effect are of critical importance in modeling and designing of gas microflow systems and in predicting their performances. Theoretical descriptions of the gas-surface interaction and gas-surface molecular interaction models are introduced to describe the boundary conditions. Various methods and techniques for determination of the slip coefficients are reviewed. The review presents the considerable success in the implementation of various slip boundary conditions to extend the Navier–Stokes (N–S) equations into the slip and transition flow regimes. Comparisons of different values and formulations of the first- and second-order slip coefficients and models reveal the discrepancies arising from different definitions in the first-order slip coefficient and various approaches to determine the second-order slip coefficient. In addition, no consensus has been reached on the correct and generalized form of higher-order slip expression. The influences of specific effects, such as effective mean free path of the gas molecules and viscosity, surface roughness, gas composition and tangential momentum accommodation coefficient, on the hybrid slip models for gas microflows are analyzed and discussed. It shows that although the various hybrid slip models are proposed from different viewpoints, they can contribute to N–S equations for capturing the high Knudsen number effects in the slip and transition flow regimes. Future studies are also discussed for improving the understanding of gas microflows and enabling us to exactly predict and actively control gas slip.

A review on slip models for gas microflows

A Unified Model for Gas Transfer in Nanopores of Shale-Gas Reservoirs: Coupling Pore Diffusion and Surface Diffusion

Apparent gas permeability in an organic-rich shale reservoir

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

Nonempirical Apparent Permeability of Shale

In microducts deviation from continuum flow behaviour of a gas increases with rarefaction. When using Navier–Stokes equations to calculate a flow under slightly and moderately rarefied conditions, slip boundary conditions are used which in turn refer to the tangential momentum accommodation coefficient (TMAC). Here we demonstrate that, in the so-called slip and transition regime, the flow in microducts can be reliably described by a consistently non-empirical model without considering the TMAC. We obtain this equation by superposition of convective transport and Fickian diffusion using two-dimensional solutions of Navier–Stokes equations and a description for the Knudsen diffusion coefficient as derived from kinetic theory respectively. For a wide variety of measurement series found in the literature the calculation predicts the data accurately. Surprisingly only size of the duct, temperature, gas properties and inlet and outlet pressure are necessary to calculate the resulting mass flow by means of a single algebraic equation. From this, and taking the discrepancies of the TMAC concerning surface roughness and nature of the gases into account, we could conclude that neither the diffusive proportions nor the total mass flow rates are influenced by surface topology and chemistry at Knudsen numbers below unity. Compared to the tube geometry, the model slightly underestimates the flow rate in rectangular channels when rarefaction increases. Likewise, the dimensionless mass flow rate and the diffusive proportion of the total flow are distinctly higher in a tube. Thus the cross-sectional geometry has a significant influence on the transport mechanisms under rarefied conditions.

An analytically predictive model for moderately rarefied gas flow

Moderately rarefied gas flows are clearly distinguished from viscous flow in the continuum regime and from free molecular flow at high rarefaction. Being of relevance for various technical applications, the understanding of such flow processes is crucial for considerable enhancement in micro electromechanical systems (MEMS) and vacuum techniques. In this work, we focus on the isothermal rarefied gas flow through long channels with longitudinally varying cross section. We apply two approaches, an analytical one and a numerical one that is based on the solution of the linearized S-model, both allowing us to predict the mass flow rate in diverging and converging flow directions for arbitrary pressure gradients. Both approaches are validated by CO2, N2 and Ar permeation experiments on tapered microchannels manufactured by means of micromilling. The local Knudsen numbers ranged from 0.0471 to 0.2263. All the numerical and analytical results are in good agreement to the experimental data and show that the mass flow rate is significantly higher when the duct is perfused in converging direction. The understanding of the physical phenomenon of this gas flow diode effect might pave the way for novel components in MEMS such as static one-way valves.

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The gas flow diode effect: theoretical and experimental analysis of moderately rarefied gas flows through a microchannel with varying cross section

Moderately rarefied gas flows are clearly distinguished from viscous flow in the continuum regime and from molecular diffusion at high rarefaction. They are an intermediate of the two border cases referred to as slip flow and transition regime flow. Here, we present a new pencil-and-paper approach for modeling flows in these regimes by a superposition of convection and Fickian diffusion. It allows us to predict mass flows for helium, argon, nitrogen, and carbon dioxide in microducts with parallel walls and with slightly varying cross section. The model was validated by measurement series taken from literature and by own permeation experiments on tapered microchannels. Analytical investigation of the approach showed that the diffusive flow is proportional to the cross-sectional area at the channel entrance. Hence, the mass flow in a tapered channel is unequal in both directions when diffusion dominates due to increased rarefaction. In contrary to the common Maxwellian slip approach the superposition model ...

Thomas Veltzke

Papers

An analytically predictive model for moderately rarefied gas flow

The gas flow diode effect: theoretical and experimental analysis of moderately rarefied gas flows through a microchannel with varying cross section

The contribution of diffusion to gas microflow: An experimental study

Molecular dynamics simulations on scattering of single Ar, N2, and CO2 molecules on realistic surfaces

Thermodynamic interpretation and prediction of CO2 solubility in imidazolium ionic liquids based on regular solution theory