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Heat transfer in a binary gas mixture between two parallel plates: an application of linear extended thermodynamics

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Abstract

In this paper, we present a heat transfer problem in a binary inert mixture of ideal gases at rest between two parallel plates. For the description of the phenomenon, we have considered the field equations of a linearized extended thermodynamics theory with 13 moments, under the assumption of a common temperature for both the constituents. The solution of the system of equations presents non-controllable boundary values, for which a fluctuation principle is applied. We found that, unlike classical thermodynamics, the 13-moment field equations predict thermal diffusion effects and exhibit solutions with boundary layers. Moreover, the results are qualitatively similar to those obtained from the kinetic theory.

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References

  1. Bassanini P., Cercignani C., Pagani C.D.: Comparison of kinetic therory analyses of linearized heat transfer between parallel plates. Int. J. Heat Mass Transf. 10, 447–460 (1967)

    Article  Google Scholar 

  2. Ohwada T., Aoki K., Sone Y.: Heat transfer and temperature distribution in a rarefied gas between two parallel plates with different temperatures: Numerical analysis of the Boltzmann equation for hard sphere molecule. In: Muntz, E.P., Weaver, D.P., Campbell, D.H. (eds) Rarefied Gas Dynamics:Theoretical and Computational Techniques, pp. 70–81. AIAA, Washington, DC (1989)

    Google Scholar 

  3. Ohwada T.: Heat flow and temperature and density distribution in a rarefied gas between parallel plates with different temperatures. Finite-difference analysis of the nonlinear Boltzmann equation for hard-sphere molecules. Phys. Fluids 8, 2153–2160 (1996)

    Article  MATH  Google Scholar 

  4. Barbera E., Müller I., Reitebuch D., Zhao N.: Determination of the boundary conditions in extended thermodynamics via fluctuation theory. Cont. Mech. Thermodyn. 16, 411–425 (2004)

    MATH  Google Scholar 

  5. Struchtrup, H.: Heat transfer in the transition regime: solution of boundary value problems for Grad’s moment equations via kinetic schemes. Phys. Rev. E. 65, 041204-1-16 (2002)

    Google Scholar 

  6. Müller I., Ruggeri T.: Stationary heat conduction in radially symmetric situations—an application of extended thermodynamics. J. Non Newtonian Fluid Mech. 119, 139–143 (2004)

    Article  MATH  Google Scholar 

  7. Barbera E., Müller I.: Inherent frame dependence of thermodynamic fields in a gas. Acta Mech. 184, 205–216 (2006)

    Article  MATH  Google Scholar 

  8. Barbera E., Müller I.: Secondary heat flow between confocal ellipses—an application of extended thermodynamics. J. Non Newtonian Fluid Mech. 153, 149–156 (2008)

    Article  Google Scholar 

  9. Barbera E., Brini F.: On stationary heat conduction in 3D symmetric domains—an application of extended thermodynamics. Acta Mech. 215, 241–260 (2010)

    Article  MATH  Google Scholar 

  10. Barbera, E., Brini, F.: Linearized extended thermodynamics for stationary heat conduction in a gas at rest confined to the gap between two tori. In: Greco, A.M., Rionero, S., Ruggeri, T. (eds.) Proceedings 15th Conference on Waves and Stability in Continuous Media (WASCOM 09), World Scientific, Singapore, pp. 15–20 (2010)

  11. Yeh B.T., Frohn A.: Heat conduction between parallel flat plates in binary gas mixtures. Phys. Fluids 16, 330–332 (1973)

    Article  Google Scholar 

  12. Braun D., Frohn A.: Heat transfer in simple monatomic gases and in binary mixtures of monatomic gases. Int. J. Heat Mass Transf. 19, 1329–1335 (1976)

    Article  Google Scholar 

  13. Kosuge, S., Aoki, K., Takata, S.: Heat transfer in a gas mixture between two parallel plates: finite-difference analysis of the Boltzmann equation. In: Bartel, T.J., Gallis, M.A. Rarefied Gas Dynamics, pp. 289–296. AIP, Melville (2001)

  14. Garcia, R.D.M., Siewert, C.E.: Heat transfer between two parallel plates: an approach based on the linearized Boltzmann equation for a binary mixture of rigid-sphere gases. Phys. Fluids, 19, 027102-1-7 (2007)

    Google Scholar 

  15. Ruggeri T., Lou J.: Heat conduction in multi-temperature mixtures of fluids: the role of the average temperature. Phys. Lett. A 373, 3052–3055 (2009)

    Article  Google Scholar 

  16. Heckl M., Müller I.: Frame dependence, entropy, entropy flux, and wave speed in mixtures of gases. Acta Mech. 50, 71–95 (1983)

    Article  MATH  Google Scholar 

  17. Fick A.:Über Diffusion. Poggendorf’s Annalen Phys. Chem. 94, 59–86 (1855)

    Article  Google Scholar 

  18. Enskog D.: Über eine Verall gemeinerung der zweiten Maxwellschen Theorie der Gase. Physik. Zs. Leipzig 12, 56–60 (1911)

    Google Scholar 

  19. Enskog D.: Bemerkungen zu einer Fundamentalgleichung in der kinetischen Gastheorie. Physik. Zs. Leipzig 12, 533–539 (1911)

    Google Scholar 

  20. Chapman S.: On the kinetic theory of gas. Part II: a composite monoatomic gas: diffusion, viscosity and thermal conduction. Phil. Trans. R. Soc. Lond. A 217, 115–197 (1918)

    Article  MATH  Google Scholar 

  21. Chapman S., Dootson F.W.: A note on thermal diffusion. Phil. Mag. 33, 248–256 (1917)

    Google Scholar 

  22. Ibbs T. L.: Thermal diffusion mesurements. Proc. R. Soc. Lond. A 107, 470–486 (1925)

    Article  Google Scholar 

  23. Powell R.W.: Heat. Rep. Progr. Phys. 5, 164–181 (1938)

    Article  Google Scholar 

  24. Clusius K., Dickel G.: Neues Verfahren zur Gasentmischung und Isotopentrennung. Natruwiss 26, 546 (1938)

    Article  Google Scholar 

  25. Clusius K., Dickel G.: Isolierung des leichten Chlorisotops mit dem Atomgewicht 34,979 im Trennrohr. Naturwiss. 27, 487 (1939)

    Article  Google Scholar 

  26. Furry W.H., Jones R.C., Onsager L.: On the theory of isotope separation by thermal diffusion. Phys. Rev. 55, 1083–1095 (1939)

    Article  MATH  Google Scholar 

  27. Leaf B., Wall F.T.: Separation of gas mixtures by thermal diffusion. J. Phys. Chem. 46, 820–826 (1942)

    Article  Google Scholar 

  28. Wall F.T., Holley C.E.: Separation by thermal diffusion of mixtures of gases having the same molecular weight. J. Chem. Phys. 8, 949–953 (1940)

    Article  Google Scholar 

  29. Grew K.E.: Diffusion in mixtures of the inert gases. Proc. R. Soc. Lond. A 189, 402–414 (1947)

    Article  Google Scholar 

  30. de Groot S.R., Mazur P.: Non-equilibrium Thermodynamics. North-Holland Publishing Company, Amsterdam (1962)

    Google Scholar 

  31. Müller I.: A thermodynamic theory of mixture of fluids. Arch. Rat. Mech. Anal. 28, 1–39 (1968)

    Article  MATH  Google Scholar 

  32. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases. 3rd edition Cambridge University Press (1970)

  33. Raines A.A.: Heat transfer in a gas mixture between parallel plates. Computational Mathematics and Mathematical Physics 48, 306–313 (2008)

    Article  MathSciNet  Google Scholar 

  34. Sharipov F., Cumin L.M.G., Kalempa D.: Heat flux between parallel plates through a binary gaseous mixture over the whole range of the Knudsen number. Phys. A 378, 183–193 (2007)

    Article  Google Scholar 

  35. Truesdell C.: Sulle basi della termodinamica I. Atti Accad. Naz. Lincei: Rend. Sc. fis. mat. nat. 22, 33–38 (1957)

    MathSciNet  Google Scholar 

  36. Truesdell C.: Sulle basi della termodinamica II. Atti Accad. Naz. Lincei: Rend. Sci. Fis. Mat. Nat 22, 158–166 (1957)

    MathSciNet  Google Scholar 

  37. Truesdell C., Toupin R.A.: The Classical field Theories. In: Flügge, S. (ed.) Handbuch der Physik. Prinzipen der klassischen Mechanik und Feldtheorie. III/1, Springer, Berlin (1960)

    Google Scholar 

  38. Truesdell C.: Mechanical basis of diffusion. J. Chem. Phys. 37, 2336–2344 (1962)

    Article  Google Scholar 

  39. Truesdell C.: Sulle basi della Termodinamica delle Miscele. Atti Accad. Naz. Lincei: Rend. Sci. Fis. Mat. Nat. 44, 381–383 (1968)

    Google Scholar 

  40. Müller I.: A new approach to thermodynamics of simple mixtures. Zeit. Naturforschung 28a, 1801–1813 (1973)

    Google Scholar 

  41. Müller I.: Thermodynamics. Pitman, London, New York (1985)

    MATH  Google Scholar 

  42. Müller I., Müller W.H.: Fundamentals of Thermodynamics and Applications: With Historical Annotations and Many Citations from Avogadro to Zermelo. Springer, Berlin, Heidelberg (2009)

    MATH  Google Scholar 

  43. Kremer G.M.: Extended thermodynamics of mixtures of ideal gases. Int. J. Eng. Sci. 25, 95–115 (1987)

    Article  MATH  Google Scholar 

  44. Rajagopal K.R., Tao L.: Mechanics of Mixtures. Series on Advances in Mathematics for Applied Sciences, vol. 35. World Scientific, Singapore (1995)

    Google Scholar 

  45. Müller I., Ruggeri T.: Rational Extended Rational Thermodynamics. Springer, New York (1998)

    MATH  Google Scholar 

  46. Grad H.: On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2, 331–407 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  47. Grad H.: Principles of the Kinetic Theory of Gases. Handbuch der Physik XII, pp. 205–294. Springer, Heidelberg (1958)

    Google Scholar 

  48. Bhatnagar P.L., Gross E.P., Krook M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys.Rev. 94, 511–525 (1954)

    Article  MATH  Google Scholar 

  49. Struchtrup H., Weiss W.: Maximum of the local entropy production becomes minimal in stationary processes. Phys. Rev. Lett. 80, 5048–5051 (1998)

    Article  Google Scholar 

  50. Brini F., Ruggeri T.: Entropy Principle for the Moment Systems of degree alpha associated to the Boltzmann Equation. Critical Derivatives and Non Controllable Boundary Data. Cont. Mech. Thermodyn. 14, 165–189 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  51. Thatcher T., Zheng Y., Struchtrup H.: Boundary conditions for Grad’s 13 moment equations. Prog. Comput. Fluid Dyn. 8, 69–83 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  52. Barbera, E.: Boundary conditions in extended thermodynamics and applications. In: Wang, Y., Hutter, K. (eds.) Proceedings “4XIVth international Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2004)”6, published in Trends in Applications of Mathematics to Mechanics, Shaker Verlag, Aachen, pp. 31–40 (2005)

  53. Barbera, E.: An application of extended thermodynamics and fluctuation principle to stationary heat conduction in radial symmetry. In: Manganaro, N., Monaco, R., Rionero, S. (eds.) Proceedings “14th Conference on Waves and Stability in Continuous Media (WASCOM 07)”. World Scientific, Singapore, pp. 31–37 (2008)

  54. Barbera E., Brini F.: Heat transfer in gas mixtures: advantages of an extended thermodynamics approach. Phys. Lett. A 375, 827–831 (2011)

    Article  Google Scholar 

  55. Barbera E., Valenti G.: Kawashima condition and acceleration waves for binary nonreacting mixtures. Acta Mech. 187, 203–217 (2006)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Di.S.I.A., Università degli Studi di Messina, c/da di Dio, S. Agata, 98166, Messina, Italy

    Elvira Barbera

  2. Dipartimento di Matematica, C.I.R.A.M., Università degli Studi di Bologna, via Saragozza, 8, 40123, Bologna, Italy

    Francesca Brini

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  1. Elvira Barbera
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  2. Francesca Brini
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Correspondence to Elvira Barbera.

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Barbera, E., Brini, F. Heat transfer in a binary gas mixture between two parallel plates: an application of linear extended thermodynamics. Acta Mech 220, 87–105 (2011). https://doi.org/10.1007/s00707-011-0465-3

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  • DOI: https://doi.org/10.1007/s00707-011-0465-3

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