Acta Applicandae Mathematicae

, Volume 132, Issue 1, pp 41–50 | Cite as

Heat Transfer Problem in a Van Der Waals Gas

Article
First Online:
Received:
Accepted:

Abstract

This paper is devoted to the study of a heat transfer problem in a van der Waals gas. We found that the extended thermodynamics theory for such a fluid predicts the non-vanishing dynamic pressure and the shear stress tensor, even in the simplest stationary planar case. However, the order of magnitude of the non-equilibrium variables is very small and, consequently, their experimental observation seems to be quite difficult.

A comparison between the results derived from the classical Navier–Stokes Fourier theory and those from extended thermodynamics is also presented.

Keywords

Heat transfer Van der Waals gas Extended thermodynamics 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

This paper was supported by GNFM-INdAM, by University of Bologna Farb Project 2012 “Termodinamica Estesa dei Processi di Non Equilibrio dalla Macro- alla Nano-Scala” and by Japan Society of Promotion of Science No. 25390150.

References

  1. 1.
    Müller, I.: Thermodynamics. Pitman, London (1985) MATHGoogle Scholar
  2. 2.
    Müller, I., Müller, W.H.: Fundamentals of Thermodynamics and Applications: With Historical Annotations and Many Citations from Avogadro to Zermelo. Springer, Berlin (2009) Google Scholar
  3. 3.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer, New York (1998) CrossRefMATHGoogle Scholar
  4. 4.
    Müller, I., Ruggeri, T.: Stationary heat conduction in radially symmetric situations—an application of extended thermodynamics. J. Non-Newton. Fluid Mech. 119, 139–143 (2004) CrossRefMATHGoogle Scholar
  5. 5.
    Barbera, E., Müller, I.: Secondary heat flow between confocal ellipses—an application of extended thermodynamics. J. Non-Newton. Fluid Mech. 153, 149–156 (2008) CrossRefMATHGoogle Scholar
  6. 6.
    Barbera, E., Brini, F.: On stationary heat conduction in 3D symmetric domains: an application of extended thermodynamics. Acta Mech. 215, 241–260 (2010) CrossRefMATHGoogle Scholar
  7. 7.
    Barbera, E., Brini, F., Valenti, G.: Some non-linear effects of stationary heat conduction in 3D domains through extended thermodynamics. Europhys. Lett. 98, 54004 (2012), 6 pp. CrossRefGoogle Scholar
  8. 8.
    Barbera, E., Brini, F.: Heat transfer in gas mixtures: advantages of an extended thermodynamics approach. Phys. Lett. A 375(4), 827–831 (2011) CrossRefGoogle Scholar
  9. 9.
    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) CrossRefMATHGoogle Scholar
  10. 10.
    Barbera, E., Brini, F.: Heat transfer in multi-component gas mixtures described by extended thermodynamics. Meccanica 47(3), 655–666 (2012) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Barbera, E., Brini, F.: An extended thermodynamics description of stationary heat transfer in binary gas mixtures confined in radial symmetric bounded domains. Contin. Mech. Thermodyn. 24, 313–331 (2012) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Engholm, H. Jr., Kremer, G.: Thermodynamics of a diatomic gas with rotational and vibrational degrees of freedom. Int. J. Eng. Sci. 32(8), 1241–1252 (1994) CrossRefMATHGoogle Scholar
  13. 13.
    Kremer, G.M.: Extended thermodynamics and statistical mechanics of a polyatomic ideal gas. J. Non-Equilib. Thermodyn. 14, 363–374 (1989) MATHGoogle Scholar
  14. 14.
    Kremer, G.M.: Extended thermodynamics of molecular ideal gases. Contin. Mech. Thermodyn. 1, 21–45 (1989) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Kremer, G.M.: Extended thermodynamics of non-ideal gases. Physica 144A, 156–178 (1987) CrossRefGoogle Scholar
  16. 16.
    Carrisi, M.C., Mele, M.A., Pennisi, S.: On some remarkable properties of an extended thermodynamic model for dense gases and macromolecular fluids. Proc. R. Soc. A 466, 1645–1666 (2010) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Carrisi, M.C., Montisci, S., Pennisi, S.: Entropy principle and Galilean relativity for dense gases, the general solution without approximations. Entropy 15, 1035–1056 (2013) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Contin. Mech. Thermodyn. 24, 271–292 (2012) CrossRefMATHGoogle Scholar
  19. 19.
    Arima, T., Sugiyama, M.: Characteristic features of extended thermodynamics of dense gases. Atti R. Accad. Pelorit. 90(1), 1–15 (2012) Google Scholar
  20. 20.
    Pavić, M., Ruggeri, T., Simić, S.: Maximum entropy principle for rarefied polyatomic gases. Physica A 392(6), 1302–1317 (2013) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Dispersion relation for sound in rarefied polyatomic gases based on extended thermodynamics. Contin. Mech. Thermodyn. 25(6), 727–737 (2013) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Thermodynamic theory of the shock wave structure in a rarefied polyatomic gas: beyond the Bethe–Teller theory. Phys. Rev. E 89, 013025 (2014) CrossRefGoogle Scholar
  23. 23.
    Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Effect of dynamic pressure on the shock wave structure in a rarefied polyatomic gas. Phys. Fluids 26, 016103 (2014) CrossRefGoogle Scholar
  24. 24.
    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of real gases with dynamic pressure: an extension of Meixner’s theory. Phys. Lett. A 376, 2799–2803 (2012) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: On the six-field model of fluidsbased on extended thermodynamics. Meccanica (2014). doi: 10.1007/s11012-014-9886-0 MathSciNetGoogle Scholar
  26. 26.
    Meixner, J.: Absorption und dispersion des schalles in gasen mit chemisch reagierenden und anregbaren komponenten. I. Ann. Phys. 43, 470–487 (1943) CrossRefMathSciNetGoogle Scholar
  27. 27.
    Meixner, J.: Allgemeine theorie der schallabsorption in gasen und flussigkeiten unter berucksichtigung der transporterscheinungen. Acustica 2, 101–109 (1952) MathSciNetGoogle Scholar
  28. 28.
    Arima, T., Barbera, E., Brini, F., Sugiyama, M.: The role of the dynamic pressure in stationary heat conduction of a rarefied polyatomic gas. Phys. Lett. A, submitted Google Scholar
  29. 29.
    Prangsma, G.J., Alberga, A.H., Beenakker, J.J.M.: Ultrasonic determination of the volume viscosity of N2, CO, CH4 and CD4 between 77 and 300 K. Physica 64, 278–288 (1973) CrossRefGoogle Scholar
  30. 30.
    Barbera, E., Müller, I.: Heat conduction in a non-inertial frame. In: Podio-Guidugli, B. (ed.) Rational Continua, Classical and New, pp. 1–10. Springer, Milano (2002) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Elvira Barbera
    • 1
  • Francesca Brini
    • 2
  • Masaru Sugiyama
    • 3
  1. 1.Department of Mathematics and Computer ScienceUniversity of MessinaMessinaItaly
  2. 2.Department of Mathematics and Research Center of Applied Mathematics (CIRAM)University of BolognaBolognaItaly
  3. 3.Graduate School of EngineeringNagoya Institute of TechnologyNagoyaJapan

Personalised recommendations