Numerical investigation of gas separation via thermally induced flows in ratchet-like patterned microchannels
TL;DR: In this article, the influence of temperature, Knudsen number, surface accommodation coefficient, and geometric parameters of the microchannel on its gas separation efficiency are studied using the controlled variable method.
About: This article is published in International Journal of Thermal Sciences.The article was published on 2022-02-01. It has received 5 citations till now. The article focuses on the topics: Microchannel & Materials science.
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01 Jan 2002
TL;DR: In this article, the authors discuss the fluid-dynamic type equations derived from the Boltzmann equation as its asymptotic behavior for small mean free path and the boundary conditions that describe the behavior of the gas in the continuum limit.
Abstract: In this series of talks, I will discuss the fluid-dynamic-type equations that is derived from the Boltzmann equation as its the asymptotic behavior for small mean free path. The study of the relation of the two systems describing the behavior of a gas, the fluid-dynamic system and the Boltzmann system, has a long history and many works have been done. The Hilbert expansion and the Chapman–Enskog expansion are well-known among them. The behavior of a gas in the continuum limit, however, is not so simple as is widely discussed by superficial understanding of these solutions. The correct behavior has to be investigated by classifying the physical situations. The results are largely different depending on the situations. There is an important class of problems for which neither the Euler equations nor the Navier–Stokes give the correct answer. In these two expansions themselves, an initialor boundaryvalue problem is not taken into account. We will discuss the fluid-dynamic-type equations together with the boundary conditions that describe the behavior of the gas in the continuum limit by appropriately classifying the physical situations and taking the boundary condition into account. Here the result for the time-independent case is summarized. The time-dependent case will also be mentioned in the talk. The velocity distribution function approaches a Maxwellian fe, whose parameters depend on the position in the gas, in the continuum limit. The fluid-dynamictype equations that determine the macroscopic variables in the limit differ considerably depending on the character of the Maxwellian. The systems are classified by the size of |fe− fe0|/fe0, where fe0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |fe − fe0|/fe0 = O(Kn) (Kn : Knudsen number, i.e., Kn = `/L; ` : the reference mean free path. L : the reference length of the system) : S system (the incompressible Navier–Stokes set with the energy equation modified). (1a) |fe − fe0|/fe0 = o(Kn) : Linear system (the Stokes set). (2) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(Kn) (ξi : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity vi of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) must be retained in the equations (non-Navier–Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(1) : E+VB system (the Euler and viscous boundary-layer sets). E system (Euler set) in the case where the boundary is an interface of the gas and its condensed phase. The fluid-dynamic systems are classified in terms of the macroscopic parameters that appear in the boundary condition. Let Tw and δTw be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified with the nondimensional temperature variation δTw/Tw and Reynolds number Re as shown in Fig. 1. In the region SB, the classical gas dynamics is inapplicable, that is, neither the Euler
501 citations
TL;DR: In this article , a review of the latest advancements and investigations on the gas surface velocity and temperature inequalities, named velocity slip and temperature jump, in rarefied gas flows is presented.
7 citations
TL;DR: In this article , the effects of Knudsen number Kn and equilibrium temperature T 0 on the flow properties of each gas species within the gas mixture, such as velocity, molar fraction, streamlines, temperature, and pressure, are investigated by the Direct simulation Monte Carlo (DSMC) method.
TL;DR: In this paper , the authors used Direct Simulation Monte Carlo (DSMC) method to perform a numerical study of a MEMS-based gas separation device prototype, where the core part of the device is represented by a microchannel with a row of elements-barriers oscillating with high frequency perpendicular to channel axis and periodically blocking the flow.
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TL;DR: In this paper, it was shown that the difference between the maximum and the minimum pressure at a point may be of considerable magnitude when the density of the gas is small enough, and when the inequalities of temperature are produced by small solid bodies at a higher or lower temperature than the vessel containing the gas.
Abstract: 1. In this paper I have followed the method given in my paper “On the Dynamical Theory of Gases” (Phil. Trans., 1867, p. 49). I have shown that when inequalities of temperature exist in a gas, the pressure at a given point is not the same in all directions, and that the difference between the maximum and the minimum pressure at a point may be of considerable magnitude when the density of the gas is small enough, and when the inequalities of temperature are produced by small solid bodies at a higher or lower temperature than the vessel containing the gas. 2. The nature of this stress may be thus defined:— Let the distance from a given point, measured in a given direction, be denoted by h; then the space-variation of the temperature for a point moving along this line will be denoted by dθ/dh, and the spaced variation of this quantity along the same line by d2θ/dh2.
1,319 citations
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12 Aug 2002TL;DR: In this article, the authors discuss the fluid-dynamic type equations derived from the Boltzmann equation as its asymptotic behavior for small mean free path and the boundary conditions that describe the behavior of the gas in the continuum limit.
Abstract: In this series of talks, I will discuss the fluid-dynamic-type equations that is derived from the Boltzmann equation as its the asymptotic behavior for small mean free path. The study of the relation of the two systems describing the behavior of a gas, the fluid-dynamic system and the Boltzmann system, has a long history and many works have been done. The Hilbert expansion and the Chapman–Enskog expansion are well-known among them. The behavior of a gas in the continuum limit, however, is not so simple as is widely discussed by superficial understanding of these solutions. The correct behavior has to be investigated by classifying the physical situations. The results are largely different depending on the situations. There is an important class of problems for which neither the Euler equations nor the Navier–Stokes give the correct answer. In these two expansions themselves, an initialor boundaryvalue problem is not taken into account. We will discuss the fluid-dynamic-type equations together with the boundary conditions that describe the behavior of the gas in the continuum limit by appropriately classifying the physical situations and taking the boundary condition into account. Here the result for the time-independent case is summarized. The time-dependent case will also be mentioned in the talk. The velocity distribution function approaches a Maxwellian fe, whose parameters depend on the position in the gas, in the continuum limit. The fluid-dynamictype equations that determine the macroscopic variables in the limit differ considerably depending on the character of the Maxwellian. The systems are classified by the size of |fe− fe0|/fe0, where fe0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |fe − fe0|/fe0 = O(Kn) (Kn : Knudsen number, i.e., Kn = `/L; ` : the reference mean free path. L : the reference length of the system) : S system (the incompressible Navier–Stokes set with the energy equation modified). (1a) |fe − fe0|/fe0 = o(Kn) : Linear system (the Stokes set). (2) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(Kn) (ξi : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity vi of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) must be retained in the equations (non-Navier–Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(1) : E+VB system (the Euler and viscous boundary-layer sets). E system (Euler set) in the case where the boundary is an interface of the gas and its condensed phase. The fluid-dynamic systems are classified in terms of the macroscopic parameters that appear in the boundary condition. Let Tw and δTw be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified with the nondimensional temperature variation δTw/Tw and Reynolds number Re as shown in Fig. 1. In the region SB, the classical gas dynamics is inapplicable, that is, neither the Euler
501 citations
01 Jan 2002
TL;DR: In this article, the authors discuss the fluid-dynamic type equations derived from the Boltzmann equation as its asymptotic behavior for small mean free path and the boundary conditions that describe the behavior of the gas in the continuum limit.
Abstract: In this series of talks, I will discuss the fluid-dynamic-type equations that is derived from the Boltzmann equation as its the asymptotic behavior for small mean free path. The study of the relation of the two systems describing the behavior of a gas, the fluid-dynamic system and the Boltzmann system, has a long history and many works have been done. The Hilbert expansion and the Chapman–Enskog expansion are well-known among them. The behavior of a gas in the continuum limit, however, is not so simple as is widely discussed by superficial understanding of these solutions. The correct behavior has to be investigated by classifying the physical situations. The results are largely different depending on the situations. There is an important class of problems for which neither the Euler equations nor the Navier–Stokes give the correct answer. In these two expansions themselves, an initialor boundaryvalue problem is not taken into account. We will discuss the fluid-dynamic-type equations together with the boundary conditions that describe the behavior of the gas in the continuum limit by appropriately classifying the physical situations and taking the boundary condition into account. Here the result for the time-independent case is summarized. The time-dependent case will also be mentioned in the talk. The velocity distribution function approaches a Maxwellian fe, whose parameters depend on the position in the gas, in the continuum limit. The fluid-dynamictype equations that determine the macroscopic variables in the limit differ considerably depending on the character of the Maxwellian. The systems are classified by the size of |fe− fe0|/fe0, where fe0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |fe − fe0|/fe0 = O(Kn) (Kn : Knudsen number, i.e., Kn = `/L; ` : the reference mean free path. L : the reference length of the system) : S system (the incompressible Navier–Stokes set with the energy equation modified). (1a) |fe − fe0|/fe0 = o(Kn) : Linear system (the Stokes set). (2) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(Kn) (ξi : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity vi of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) must be retained in the equations (non-Navier–Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(1) : E+VB system (the Euler and viscous boundary-layer sets). E system (Euler set) in the case where the boundary is an interface of the gas and its condensed phase. The fluid-dynamic systems are classified in terms of the macroscopic parameters that appear in the boundary condition. Let Tw and δTw be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified with the nondimensional temperature variation δTw/Tw and Reynolds number Re as shown in Fig. 1. In the region SB, the classical gas dynamics is inapplicable, that is, neither the Euler
501 citations
TL;DR: The results of validation of an open source Direct Simulation Monte Carlo (DSMC) code for general application to rarefied gas flows is presented, showing very good agreement with data provided by both analytical solutions and other contemporary DSMC codes.
251 citations
TL;DR: It is almost certain that the first Monte Carlo simulation of a gas was carried out by William Anderson, the secretary and assistant to Lord Kelvin, and requires the introduction of the Knudsen number Kn as a distinct dimensionless parameter.
Abstract: It is almost certain that the first Monte Carlo simulation of a gas was carried out by William Anderson, the secretary and assistant to Lord Kelvin. As reported by Kelvin (1901), Anderson generated random numbers by shuffling decks of numbered cards and calculated· with "unfailingly faithful perseverance" a total of five thousand molecular impacts with surfaces and three hundred intermolecular collisions. The use of random numbers is the distinguishing feature of a Monte Carlo procedure, and the essentially probabilistic nature of a gas flow at the molecular level makes it an obvious subject for a simulation approach based directly on the physics of the individual molecular interactions. However, prior to the advent. of the digital computer, the approach was effectively ruled out by the enormous number of repetitive arithmetical computations that are required for its application, even to the simplest problem. Typical computer runs of Monte Carlo simulation programs now involve the computation of as many as ten million intermolecular collisions, together with millions of molecule-surface interactions. The molecular or microscopic model of a gas flow must, of course, be viewed against the familiar macroscopic or continuum model. This requires the introduction of the Knudsen number Kn as a distinct dimensionless parameter. The usual definition is
228 citations