Hostname: page-component-5c6d5d7d68-xq9c7 Total loading time: 0 Render date: 2024-08-31T20:04:01.320Z Has data issue: false hasContentIssue false
  • English
  • Français

An analytically predictive model for moderately rarefied gas flow

Published online by Cambridge University Press:  05 April 2012

Thomas Veltzke  and
Jorg Thöming
Thomas Veltzke*
Affiliation:
Center for Environmental Research and Sustainable Technology (UFT), University of Bremen, Leobener Strasse, 28359 Bremen, Germany
Jorg Thöming
Affiliation:
Center for Environmental Research and Sustainable Technology (UFT), University of Bremen, Leobener Strasse, 28359 Bremen, Germany
*
†Email address for correspondence: tveltzke@uni-bremen.de
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Mathematical Foundations: General fluid mechanics Rarefied Gas Flow: Kinetic theory Micro-/Nano-fluid dynamics: Microfluidics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 698 , 10 May 2012 , pp. 406 - 422
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Agrawal, A. & Prabhu, S. V. 2008 Survey on measurement of tangential momentum accommodation coefficient. J. Vac. Sci. Technol. A 26, 634–645.Google Scholar
2. Arkilic, E. B. 1994 Gaseous flow in micron-sized channels. Master’s thesis, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
3. Arkilic, E. B. 1997 Measurement of the mass flow and tangential momentum accommodation coefficient in silicon micromachined channels. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
4. Arkilic, E. B., Breuer, K. S. & Schmidt, M. A. 2001 Mass flow and tangential momentum accommodation in silicon micromachined channels. J. Fluid Mech. 437, 29–43.CrossRefGoogle Scholar
5. Arkilic, E. B., Schmidt, M. A. & Breuer, K. S. 1997a Gaseous slip flow in long microchannels. J. Microelectromech. Syst. 6, 167–178.CrossRefGoogle Scholar
6. Arkilic, E. B., Schmidt, M. A. & Breuer, K. S. 1997b TMAC measurement in silicon micromachined channels. In Proceedings of 20th Symposium on Rarefied Gas Dynamics, Bejiing. Peking University Press.Google Scholar
7. Atkins, P. & De Paula, J. 2006 Physical Chemistry, 8th edn. Oxford University Press.Google Scholar
8. Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2007 Transport Phenomena, 2nd edn. John Wiley and Sons.Google Scholar
9. Brenner, H. 2011 Beyond the no-slip boundary condition. Phys. Rev. E 84, 046309.Google Scholar
10. Cao, B.-Y., Chen, M. & Guo, Z.-Y. 2006 Effect of surface roughness on gas flow in microchannels by molecular dynamics simulation. Intl J. Engng Sci. 44, 927–937.Google Scholar
11. Cao, B.-Y., Sun, J. & Guo, Z.-Y. 2009 Molecular momentum transport at fluid-solid interfaces in MEMS/NEMS: A review. Intl J. Mol. Sci. 10, 4638–4706.CrossRefGoogle ScholarPubMed
12. Colin, S. 2005 Rarefaction and compressibility effects on steady and transient gas flows in microchannels. Microfluid Nanofluid 1, 268–279.Google Scholar
13. Colin, S., Lalonde, P. & Caen, R. 2004 Validation of a second-order slip flow model in rectangular microchannels. Heat Transfer Engng 25, 23–30.CrossRefGoogle Scholar
14. Cunningham, R. E. & Williams, R. J. J. 1980 Diffusion in Gases an Porous Media. Plenum.Google Scholar
15. Cussler, E. L. 2009 Diffusion-Mass Transfer in Fluid Systems, 3rd edn. Cambridge University Press.CrossRefGoogle Scholar
16. Dadzie, S. K. & Reese, J. M. 2010 A volume-based hydrodynamic approach to sound wave propagation in a monatomic gas. Phys. Fluids 22, 016103.Google Scholar
17. Dadzie, S. K., Reese, J. M. & McInnes, C. R. 2008 A continuum model of gas flows with localized density variations. Physica A 387, 6079–6094.Google Scholar
18. Dongari, N., Dadzie, S. K., Zhang, Y. & Reese, J. M. 2011 Isothermal micro-channel gas flow using a hydrodynamic model with dissipative mass flux. In Rarefied Gas Dynamics: 27th International Symposium. 1333, pp. 718–724. AIP.Google Scholar
19. Ewart, T. P., Perrier, P., Graur, I. A. & Meolans, J. G. 2006 Mass flow rate measurements in gas micro flows. Exp. Fluids 41, 487–498.CrossRefGoogle Scholar
20. Ewart, T. P., Perrier, P., Graur, I. A. & Meolans, J. G. 2007a Mass flow rate measurements in a microchannel, from hydrodynamic to near free molecular regimes. J. Fluid Mech. 584, 337–356.Google Scholar
21. Ewart, T. P., Perrier, P., Graur, I. A. & Meolans, J. G. 2007b Tangential momemtum accommodation in microtube. Microfluid Nanofluid 3, 689–695.Google Scholar
22. Fick, A. 1855 Ueber diffusion. Ann. Phys. 170, 59–86.CrossRefGoogle Scholar
23. Gad-el-Hak, M. 1999 The fluid mechanics of microdevices – The Freeman scholar lecture. Trans. ASME: J. Fluids Engng 121, 5–33.Google Scholar
24. Geankoplis, J. G. 1972 Mass Transport Phenomena. Holt, Rinehart and Winston.Google Scholar
25. Graur, I. A., Perrier, P., Ghozlani, W. & Meolans, J. G. 2009 Measurements of tangential momentum accommodation coefficient for various gases in plane microchannel. Phys. Fluids 21, 102004.CrossRefGoogle Scholar
26. Hsieh, S.-S., Tsai, H.-H., Lin, C.-Y, Huang, C.-F. & Chien, C.-M. 2004 Gas flow in a long microchannel. Intl J. Heat Mass Transfer 47, 3877–3887.Google Scholar
27. Jang, J., Zhao, Y., Wereley, S. T. & Gui, L. 2002 Mass flow measurement of gases in deep-RIE microchannels. In Proceedings of IMECE2002 ASME International Mechanical Engineering Congress & Exposition, New Orleans, Louisiana, pp. 17–22. ASME.Google Scholar
28. Ji, Y., Yuan, K. & Chung, J. N. 2006 Numerical simulation of wall roughness on gaseous flow and heat transfer in a microchannel. Intl J. Heat Mass Transfer 49, 1329–1339.CrossRefGoogle Scholar
29. Kaerger, J. & Ruthven, D. M. 1992 Diffusion in Zeolites and Other Microporous Solids. Wiley-Interscience.Google Scholar
30. Kennard, E. H. 1938 The Kinetic Theory of Gases. McGraw-Hill.Google Scholar
31. Knudsen, M. 1909 Die Gesetze der Molekularstroemung und der inneren Reibungsstroemung der Gase durch Roehren. Ann. Phys. 333, 75–130.Google Scholar
32. Lalonde, P. 2001 Etude experimentale d’ ecoulements gazeux dans les microsystemes et fluides. PhD thesis, Institut National des Sciences Appliquees, Tolouse, France.Google Scholar
33. Lund, L. M. & Berman, A. S. 1966 Flow and self-diffusion of gases in capillaries. Part I. J. Appl. Phys. 37, 2489–2495.CrossRefGoogle Scholar
34. Malek, K. & Coppens, M.-O. 2003 Knudsen self- and Fickian diffusion in rough nanoporous media. J. Chem. Phys. 119, 2801–2811.CrossRefGoogle Scholar
35. Maurer, J., Tabeling, P., Joseph, P. & Willaime, H. 2003 Second-order slip laws in microchannels for helium and nitrogen. Phys. Fluids 15, 2613–2621.CrossRefGoogle Scholar
36. Maxwell, J. C. 1879 On stresses in rarified gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
37. Myong, R. S. 2004 Gaseous slip models based on the Langmuir adsorption isotherm. Phys. Fluids 16, 104–117.Google Scholar
38. Panton, R. L. 1996 Incompressible Flow, 2nd edn. Wiley-Interscience.Google Scholar
39. Perrier, P., Graur, I. A., Meolans, T. & Ewart, J. G. 2011 Mass flow rate measurements in microtubes: From hydrodynamic to near free molecular regime. Phys. Fluids 23, 042004.Google Scholar
40. Pitakarnnop, J., Varoutis, S., Valougeorgis, D., Geoffroy, S., Baldas, L. & Colin, S. 2010 A novel experimental setup for gas microflows. Microfluid Nanofluid 8, 57–72.Google Scholar
41. Pollard, W. G. & Present, R. D. 1948 On gaseous self-diffusion in long capillary tubes. Phys. Rev. 73, 762–774.Google Scholar
42. Sherman, F. S. 1990 Viscous Flow. McGraw-Hill.Google Scholar
43. Shih, J., Ho, C., Liu, J. & Tai, Y. 1996 Monoatomic and polyatomic gas flow through uniform microchannels. In Application of Microfabrication to Fluid Mechanics, ASME Winter Annual meeting, Atlanta, GA, pp. 197–203. ASME.Google Scholar
44. Steel, R. G. & Torrie, J. H. 1980 Principles and Procedures of Statistics: A Biometrical Approach, 2nd edn. McGraw-Hill.Google Scholar
45. Stevanovic, N. D. 2007 A new analytical solution of microchannel gas flow. J. Micromech. Microengng 17, 1695–1702.Google Scholar
46. Sun, H. & Faghri, M. 2003 Effect of surface roughness on nitrogen flow in a microchannel using the direct simulation Monte Carlo method. Numer. Heat Transfer A-Appl. 43, 1–8.CrossRefGoogle Scholar
47. Sun, J. & Li, Z.-X. 2010 Two-dimensional molecular dynamic simulations on accommodation coefficients in nanochannels with various wall configurations. Comput. Fluids 39, 1345–1352.Google Scholar
48. Sutherland, W. 1895 The viscosity of mixed gases. Phil. Mag. 40, 421–431.CrossRefGoogle Scholar
49. Tang, G. H., Li, Z., He, Y. L. & Tao, W. Q. 2007 Experimental study of compressibility, roughness and rarefaction influences on microchannel flow. Intl J. Heat Mass Transfer 50, 2282–2295.CrossRefGoogle Scholar
50. Turner, S. E., Lam, L. C., Faghri, M. & Gregory, O. J. 2004 Experimental investigation of gas flow in microchannels. J. Heat Transfer 126, 753–763.Google Scholar
51. Yamaguchi, H., Hanawa, T., Yamamoto, O., Matsuda, Y., Egami, Y. & Niimi, T. 2011 Experimental measurement on tangential momentum accommodation coefficient in a single microtube. Microfluid Nanofluid 11, 57–64.CrossRefGoogle Scholar
52. Zohar, Y., Lee, S. Y. K., Lee, W. Y., Jiang, L. & Tong, P. 2002 Subsonic gas flow in a straight uniform microchannel. J. Fluid Mech. 472, 125–151.CrossRefGoogle Scholar