微尺度通道内稀薄气体高阶努森数渗透率修正模型
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摘要
气体在微纳米尺度通道内流动时会产生稀薄效应,应用经典理论很难准确预测气体的真实流量,亟需建立精度更高、更具普适性的渗透率修正模型来描述稀薄气体的流动行为。为此,首先采用R26矩方法对平板微通道中的气体流动进行数值模拟,并与直接模拟蒙特卡洛法(DSMC法)、R13矩方法的模拟结果进行对比,然后基于R26矩方法的模拟结果建立平板微通道与圆管微通道内气体渗透率修正模型,运用所建立的模型描述微尺度通道内稀薄气体的流动行为,计算不同努森数下气体渗透率修正系数,并与Tang等模型预测结果、实验数据及线性Boltzmann方程解进行对比分析。研究结果表明:①采用R26矩方法描述气体稀薄效应,其模拟结果与DSMC法计算结果吻合情况良好,并且计算结果的精度高于R13矩方法 ;②采用平板微通道高阶努森数气体渗透率修正模型计算的气体渗透率修正系数与实验数据、线性Boltzmann方程解吻合良好;③采用圆管微通道高阶努森数气体渗透率修正模型计算的气体渗透率修正系数与线性Boltzmann方程解吻合良好。结论认为,所建立的高阶努森数气体渗透率修正模型预测精度高且具有普适性,可用于描述微纳米尺度通道内气体的稀薄效应。
Rarefaction effect appears when gas flows in micro-or nano-scale channels, so it is difficult to accurately predict the real gas flow rate by using the classical theory. To solve this problem, it is necessary to establish a more accurate and universal permeability correction model to describe the flowing behavior of rarefied gas. In this paper, the gas flow in a parallel microchannel was numerically simulated using R26 moment method, and the simulation results were compared with those of the direct simulation Monte Carlo method(DSMC method) and R13 moment method. Then, a gas permeability correction model for parallel microchannels and circular microtubes was established based on the simulation results of the R26 moment method, and used to describe the flowing behavior of rarefied gas in micro-scale channels. Finally, the gas permeability correction coefficient for different Knudsen numbers was calculated and compared with the prediction results of the Tang model, the experimental data and the solution of linearized Boltzmann equation. And the following research results were obtained. First, when the R26 moment method is used to describe the rarefaction effect of gas, its result is accordant with the calculation result of the DSMC method, and its calculation accuracy is higher than that of R13 moment method. Second, the gas permeability correction coefficient which is calculated by using the higher-order Knudsen's gas permeability correction model for parallel microchannels is in accordance with the experimental data and the solution of linearized Boltzmann equation. Third, the gas permeability correction coefficient which is calculated by using the higher-order Knudsen's gas permeability correction model for circular microtubes is accordant with the solution of linearized Boltzmann equation. In conclusion, this higher-order Knudsen's gas permeability correction model is advantageous with high prediction precision and universality, and it can be used to describe the rarefaction effect of gas in micro/nano-scale channels.
Rarefaction effect appears when gas flows in micro-or nano-scale channels, so it is difficult to accurately predict the real gas flow rate by using the classical theory. To solve this problem, it is necessary to establish a more accurate and universal permeability correction model to describe the flowing behavior of rarefied gas. In this paper, the gas flow in a parallel microchannel was numerically simulated using R26 moment method, and the simulation results were compared with those of the direct simulation Monte Carlo method(DSMC method) and R13 moment method. Then, a gas permeability correction model for parallel microchannels and circular microtubes was established based on the simulation results of the R26 moment method, and used to describe the flowing behavior of rarefied gas in micro-scale channels. Finally, the gas permeability correction coefficient for different Knudsen numbers was calculated and compared with the prediction results of the Tang model, the experimental data and the solution of linearized Boltzmann equation. And the following research results were obtained. First, when the R26 moment method is used to describe the rarefaction effect of gas, its result is accordant with the calculation result of the DSMC method, and its calculation accuracy is higher than that of R13 moment method. Second, the gas permeability correction coefficient which is calculated by using the higher-order Knudsen's gas permeability correction model for parallel microchannels is in accordance with the experimental data and the solution of linearized Boltzmann equation. Third, the gas permeability correction coefficient which is calculated by using the higher-order Knudsen's gas permeability correction model for circular microtubes is accordant with the solution of linearized Boltzmann equation. In conclusion, this higher-order Knudsen's gas permeability correction model is advantageous with high prediction precision and universality, and it can be used to describe the rarefaction effect of gas in micro/nano-scale channels.
引文
[1]蒋裕强,董大忠,漆麟,沈妍斐,蒋婵,何溥为.页岩气储层的基本特征及其评价[J].天然气工业,2010,30(10):7-12.Jiang Yuqiang,Dong Dazhong,Qi Lin,Shen Yanfei,Jiang Chan&He Puwei.Basic features and evaluation of shale gas reservoirs[J].Natural Gas Industry,2010,30(10):7-12.
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[7]Knudsen M.Die Gesetze der Molekularstr?mung und der inneren Reibungsstr?mung der Gasedurch R?hren[J].Annalen der Physik,1909,333(1):75-130.
[8]Beskok A&Karniadakis GE.Report:A model for flows in channels,pipes,and ducts at micro and nano scales[J].Microscale Thermophysical Engineering,1999,3(1):43-77.
[9]Tang GH,Tao WQ&He YL.Gas slippage effect on microscale porous flow using the lattice Boltzmann method[J].Physical Review E,2005,72(5):056301.
[10]Schamberg R.The fundamental differential equations and the boundary conditions for high speed slip-flow,and their application to several specific problems[D].California:California Institute of Technology,1947.
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[12]Sreekanth AK.Slip flow through long circular tubes[C]//paper presented at the 6th International Symposium on Rarefied Gas Dynamics,1969,New York City,New York,USA.
[13]Mitsuya Y.Modified Reynolds equation for ultra-thin film gas lubrication using 1.5-order slip-flow model and considering surface accommodation coefficient[J].Journal of Tribology,1993,115(2):289-294.
[14]Beskok A,Karniadakis GE&Trimmer W.Rarefaction and compressibility effects in gas microflows[J].Journal of Fluids Engineering,1996,118(3):448-456.
[15]Wu L&Bogy D.A generalized compressible Reynolds lubrication equation with bounded contact pressure[J].Physics of Fluids,2001,13(8):2237-2244.
[16]Kim SH&Pitsch H.Analytic solution for a higher-order lattice Boltzmann method:Slip velocity and Knudsen layer[J].Physical Review E,2008,78(1):016702.
[17]Grad H.Note on N-dimensional hermite polynomials[J].Communications on Pure and Applied Mathematics,1949,2(4):325-330.
[18]Struchtrup H.Macroscopic transport equations for rarefied gas flows[M].New York:Springer,2005.
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[20]Gu XJ&Emerson D.A high-order moment approach for capturing non-equilibrium phenomena in the transition regime[J].Journal of Fluid Mechanics,2009,636:177-216.
[21]Karlin IV,Gorban AN,Dukek G&Nonnenmacher TF.Dynamic correction to moment approximations[J].Physical Review E,1998,57(2):1668-1672.
[22]Tang GH,Zhai GX,Tao WQ,Gu XJ&Emerson DR.Extended thermodynamic approach for non-equilibrium gas flow[J].Communications in Computational Physics,2013,13(5):1330-1356.
[23]Alves MA,Oliveira PJ&Pinho FT.A convergent and universally bounded interpolation scheme for the treatment of advection[J].International Journal for Numerical Methods in Fluids,2003,41(1):47-75.
[24]Patankar S.Numerical heat transfer and fluid flow[M].Boca Raton:CRC Press,1980.
[25]Struchtrup H.Linear kinetic heat transfer:Moment equations,boundary conditions,and Knudsen layers[J].Physica A:Statistical Mechanics and its Applications,2008,387(8/9):1750-1766.
[26]Zheng Q,Xu J,Yang B&Yu BM.A fractal model for gaseous leak rates through contact surfaces under non-isothermal condition[J].Applied Thermal Engineering,2013,52(1):54-61.
[27]秦丰华,孙德军,尹协远.微管道气体流动的蒙特卡洛直接模拟[J].计算物理,2001,18(6):507-510.Qin Fenghua,Sun Dejun&Yin Xieyuan.DSMC for gas flows in a microchannel[J].Chinese Journal of Computational Physics,2001,18(6):507-510.
[28]沈青.稀薄气体动力学[M].北京:国防工业出版社,2003.Shen Qing.Rarefied gas dynamics[M].Beijing:National Defence Industry Press,2003.
[29]Wang M&Li ZX.Simulations for gas flows in microgeometries using the direct simulation Monte Carlo method[J].International Journal of Heat and Fluid Flow,2004,25(6):975-985.
[30]Kazemi M&Takbiri-Borujeni A.An analytical model for shale gas permeability[J].International Journal of Coal Geology,2015,146:188-197.
[31]Young JB.Calculation of Knudsen layers and jump conditions using the linearised G13 and R13 moment methods[J].International Journal of Heat and Mass Transfer,2011,54(13/14):2902-2912.
[32]Zhu GY,Liu L,Yang ZM,Liu XG,Guo YG&Cui YT.Experiment and mathematical model of gasflow in low permeability porous media[C]//paper presented at the 5th International Conference on Fluid Mechanics,15-19 August 2007,Shanghai,China.
[33]Maurer J,Tabeling P,Joseph P&Willaime H.Second-order slip laws in microchannels for helium and nitrogen[J].Physics of Fluids,2003,15(9):2613-2621.
[34]Aubert C&Colin S.Second-order effects in gas flows through microchannels[J].Microscale Thermophysical Engineering,2001,5(1):41.
[35]Ohwada T,Sone Y&Aoki K.Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules[J].Physics of Fluids A,1989,1(12):2042-2049.
[36]Ewart T,Perrier P,Graur IA&Meolans JG.Mass flow rate measurements in a microchannel:From hydrodynamic to free molecular regimes[J].Journal of Fluid Mechanics,2007,584:337-356.
[37]Yamaguchi H,Hanawa T,Yamamoto O,MatsudaY,Egami Y&Niimi T.Experimental measurement on tangential momentum accommodation coefficient in a single microtube[J].Microfluidics and Nanofluidics,2011,11(1):57-64.
[38]Zhang WM,Meng G&Wei XY.A review on slip models for gas microflows[J].Microfluidics and Nanofluidics,2012,13(6):845-882.
[39]Loyalka SK&Hamoodi SA.Poiseuille flow of a rarefied gas in a cylindrical tube:Solution of linearized Boltzmann equation[J].Physics of Fluids A,1990,2(11):2061-2065.
[2]郭小哲,李景,张欣.页岩气藏压裂水平井物质平衡模型的建立[J].西南石油大学学报(自然科学版),2017,39(2):132-138.Guo Xiaozhe,Li Jing&Zhang Xin.Study on the establishment of material balance model for fractured horizontal well in shale gas reservoir[J].Journal of Southwest Petroleum University(Science&Technology Edition),2017,39(2):132-138.
[3]Xue H,Fan Q&Shu C.Prediction of micro-channel flows using direct simulation Monte Carlo[J].Probabilistic Engineering Mechanics,2000,15(2):213-219.
[4]刘杰,张永利,胡志明,李英杰,杨新乐.页岩气储层纳米级孔隙中气体的质量传输机理及流态实验[J].天然气工业,2018,38(12):87-95.Liu Jie,Zhang Yongli,Hu Zhiming,Li Yingjie&Yang Xinle.An experimental study on the mass transfer mechanism and the flow regime of gas in nano-scale pores of shale gas reservoirs[J].Natural Gas Industry,2018,38(12):87-95.
[5]陆家亮,赵素平,孙玉平,唐红君.中国天然气产量峰值研究及建议[J].天然气工业,2018,38(1):1-9.Lu Jialiang,Zhao Suping,Sun Yuping&Tang Hongjun.Natural gas production peaks in China:Research and strategic proposals[J].Natural Gas Industry,2018,38(1):1-9.
[6]Maxwell JC.On Stresses in rarified gases arising from inequalities of temperature[J].Philosophical Transactions of the Royal Society of London,1879,170:231-256.
[7]Knudsen M.Die Gesetze der Molekularstr?mung und der inneren Reibungsstr?mung der Gasedurch R?hren[J].Annalen der Physik,1909,333(1):75-130.
[8]Beskok A&Karniadakis GE.Report:A model for flows in channels,pipes,and ducts at micro and nano scales[J].Microscale Thermophysical Engineering,1999,3(1):43-77.
[9]Tang GH,Tao WQ&He YL.Gas slippage effect on microscale porous flow using the lattice Boltzmann method[J].Physical Review E,2005,72(5):056301.
[10]Schamberg R.The fundamental differential equations and the boundary conditions for high speed slip-flow,and their application to several specific problems[D].California:California Institute of Technology,1947.
[11]Cercignani C.Higher order slip according to the linearized Boltzmann equation[R].Berkeley:California University,1964.
[12]Sreekanth AK.Slip flow through long circular tubes[C]//paper presented at the 6th International Symposium on Rarefied Gas Dynamics,1969,New York City,New York,USA.
[13]Mitsuya Y.Modified Reynolds equation for ultra-thin film gas lubrication using 1.5-order slip-flow model and considering surface accommodation coefficient[J].Journal of Tribology,1993,115(2):289-294.
[14]Beskok A,Karniadakis GE&Trimmer W.Rarefaction and compressibility effects in gas microflows[J].Journal of Fluids Engineering,1996,118(3):448-456.
[15]Wu L&Bogy D.A generalized compressible Reynolds lubrication equation with bounded contact pressure[J].Physics of Fluids,2001,13(8):2237-2244.
[16]Kim SH&Pitsch H.Analytic solution for a higher-order lattice Boltzmann method:Slip velocity and Knudsen layer[J].Physical Review E,2008,78(1):016702.
[17]Grad H.Note on N-dimensional hermite polynomials[J].Communications on Pure and Applied Mathematics,1949,2(4):325-330.
[18]Struchtrup H.Macroscopic transport equations for rarefied gas flows[M].New York:Springer,2005.
[19]Grad H.On the kinetic theory of rarefied gases[J].Communications on pure and applied mathematics,1949,2(4):331-407.
[20]Gu XJ&Emerson D.A high-order moment approach for capturing non-equilibrium phenomena in the transition regime[J].Journal of Fluid Mechanics,2009,636:177-216.
[21]Karlin IV,Gorban AN,Dukek G&Nonnenmacher TF.Dynamic correction to moment approximations[J].Physical Review E,1998,57(2):1668-1672.
[22]Tang GH,Zhai GX,Tao WQ,Gu XJ&Emerson DR.Extended thermodynamic approach for non-equilibrium gas flow[J].Communications in Computational Physics,2013,13(5):1330-1356.
[23]Alves MA,Oliveira PJ&Pinho FT.A convergent and universally bounded interpolation scheme for the treatment of advection[J].International Journal for Numerical Methods in Fluids,2003,41(1):47-75.
[24]Patankar S.Numerical heat transfer and fluid flow[M].Boca Raton:CRC Press,1980.
[25]Struchtrup H.Linear kinetic heat transfer:Moment equations,boundary conditions,and Knudsen layers[J].Physica A:Statistical Mechanics and its Applications,2008,387(8/9):1750-1766.
[26]Zheng Q,Xu J,Yang B&Yu BM.A fractal model for gaseous leak rates through contact surfaces under non-isothermal condition[J].Applied Thermal Engineering,2013,52(1):54-61.
[27]秦丰华,孙德军,尹协远.微管道气体流动的蒙特卡洛直接模拟[J].计算物理,2001,18(6):507-510.Qin Fenghua,Sun Dejun&Yin Xieyuan.DSMC for gas flows in a microchannel[J].Chinese Journal of Computational Physics,2001,18(6):507-510.
[28]沈青.稀薄气体动力学[M].北京:国防工业出版社,2003.Shen Qing.Rarefied gas dynamics[M].Beijing:National Defence Industry Press,2003.
[29]Wang M&Li ZX.Simulations for gas flows in microgeometries using the direct simulation Monte Carlo method[J].International Journal of Heat and Fluid Flow,2004,25(6):975-985.
[30]Kazemi M&Takbiri-Borujeni A.An analytical model for shale gas permeability[J].International Journal of Coal Geology,2015,146:188-197.
[31]Young JB.Calculation of Knudsen layers and jump conditions using the linearised G13 and R13 moment methods[J].International Journal of Heat and Mass Transfer,2011,54(13/14):2902-2912.
[32]Zhu GY,Liu L,Yang ZM,Liu XG,Guo YG&Cui YT.Experiment and mathematical model of gasflow in low permeability porous media[C]//paper presented at the 5th International Conference on Fluid Mechanics,15-19 August 2007,Shanghai,China.
[33]Maurer J,Tabeling P,Joseph P&Willaime H.Second-order slip laws in microchannels for helium and nitrogen[J].Physics of Fluids,2003,15(9):2613-2621.
[34]Aubert C&Colin S.Second-order effects in gas flows through microchannels[J].Microscale Thermophysical Engineering,2001,5(1):41.
[35]Ohwada T,Sone Y&Aoki K.Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules[J].Physics of Fluids A,1989,1(12):2042-2049.
[36]Ewart T,Perrier P,Graur IA&Meolans JG.Mass flow rate measurements in a microchannel:From hydrodynamic to free molecular regimes[J].Journal of Fluid Mechanics,2007,584:337-356.
[37]Yamaguchi H,Hanawa T,Yamamoto O,MatsudaY,Egami Y&Niimi T.Experimental measurement on tangential momentum accommodation coefficient in a single microtube[J].Microfluidics and Nanofluidics,2011,11(1):57-64.
[38]Zhang WM,Meng G&Wei XY.A review on slip models for gas microflows[J].Microfluidics and Nanofluidics,2012,13(6):845-882.
[39]Loyalka SK&Hamoodi SA.Poiseuille flow of a rarefied gas in a cylindrical tube:Solution of linearized Boltzmann equation[J].Physics of Fluids A,1990,2(11):2061-2065.