International Journal of Thermophysics

, Volume 27, Issue 2, pp 596–613 | Cite as

Analysis for Microscopic Hyperbolic Two-Step Heat Transfer Problems


This work analyzes the heat transfer problems in thin metal films using the microscopic hyperbolic two-step model. It is necessary in dealing with such problems to solve a set of the coupled energy equations or an equation containing higher-order mixed derivatives in both time and space. This present numerical scheme eliminates the coupling between energy equations with the Laplace transform technique and leads to a second-order governing differential equation in the transform domain. Afterward, the transformed second-order governing differential equation is discretized by the control volume scheme. To demonstrate the efficiency and accuracy of the present numerical scheme, a comparison between the present numerical results and the analytical solution is made. Theoretical insight into the hyperbolic two-step heat conduction is provided. Results show that the thermal propagation velocity is finite and is independent of the coupling factor and the volumetric heat capacity ratio between electrons and the lattice.


ballistic behavior high-rate heating hybrid numerical scheme hyperbolic two-step model 



volumetric heat capacity


dimensionless heating source, defined as \(E=\frac{S\tau }{C_{\rm e} T_{\rm r} } \widetilde {E}\)

\(\widetilde {E}\)

Laplace transform of E


coupling factor


thermal conductivity

distance between neighboring nodes


parameter, defined as \(N=\frac{G\tau }{C_{\rm e}}\)


dimensionless heat flux, defined as \(Q=\frac{q}{T_{\rm r} \sqrt {kC_{\rm e}/\tau}}\)

\(\widetilde {Q}\)

Laplace transform of Q


heat flux


parameter, defined as \(R_{\rm C} =\frac{C_{\rm l}}{C_{\rm e}}\)


heating source


Laplace transform parameter




initial temperature


reference temperature




propagation speed of thermal signal, defined as \(V=\sqrt {k \mathord C_{\rm e} \tau } \)


space coordinate

Greek Symbols



parameter, defined in Eq. (9)


dimensionless space coordinate, defined as \(\eta\, =\,\frac{x}{\sqrt {k\tau /C_{\rm e}} }\)


dimensionless temperature, defined as \(\theta =\frac{T-T_{\rm 0}}{T_{\rm r} }\)

\(\widetilde {\theta }\)

Laplace transform of θ


dimensionless time, defined as \(\xi =\frac{t}{\tau }\)


relaxation time






node number




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fujimoto J.G., Liu J.M., and Ippen E.P. (1984). Phys. Rev 53:416Google Scholar
  2. 2.
    Brorson S.D., Fujimoto J.G., and Ippen E.P. (1987). Phys. Rev. Lett 59:1962CrossRefADSGoogle Scholar
  3. 3.
    Elsayed-Ali H.E. (1991). Phys. Rev. B43:4488CrossRefADSGoogle Scholar
  4. 4.
    Kaganov M.I., Lifshitz I.M., and Tanatarov L.V. (1957). Sov Phys JETP 4:173MATHGoogle Scholar
  5. 5.
    Anisimov S.I., Kapeliovich B.L., and Perel’man T.L. (1974). Sov. Phys. JETP 39:375ADSGoogle Scholar
  6. 6.
    Elsayed-Ali H.E., Norris T.B., Pessot M.A., and Mourou G.A. (1987). Phys. Rev. Lett 58:1212CrossRefADSGoogle Scholar
  7. 7.
    Corkum P.B., Brunel F., and Sherman N.K. (1987). Phys. Rev. Lett. 61:2886CrossRefADSGoogle Scholar
  8. 8.
    Qiu T.Q., and Tien C.L. (1992). Int. J. Heat Mass Transfer 35:719CrossRefADSGoogle Scholar
  9. 9.
    Qiu T.Q., and Tien C.L. (1994). Int. J. Heat Mass Transfer 37:2789CrossRefGoogle Scholar
  10. 10.
    Kiwan S. and Al-Nimr M.A. (2000). Jpn. J. Appl. Phys 39:4245CrossRefADSGoogle Scholar
  11. 11.
    Al-Nimr M.A., Hader M., and Naji M. (2003). Int. J. Heat Mass Transfer 46:333MATHCrossRefGoogle Scholar
  12. 12.
    Qiu T.Q., and Tien C.L. (1993). J. Heat Transfer 115:835CrossRefGoogle Scholar
  13. 13.
    Tzou D.Y. (1995). Int. J. Heat Mass Transfer 38:3231CrossRefGoogle Scholar
  14. 14.
    Al-Nimr M.A., and Arpaci V.S. (2000). Int. J. Heat Mass Transfer 43:2021MATHCrossRefGoogle Scholar
  15. 15.
    Al-Nimr M.A.,, Haddad O.M., and Arpaci V.S. (1999). Heat Mass Transfer 35:459CrossRefADSGoogle Scholar
  16. 16.
    Naji M., Al-Nimr M.A.,, and Hader M. (2003). Int J Thermophys. 24:545CrossRefGoogle Scholar
  17. 17.
    Al-Nimr M.A., and Alkam M.K. (2003). Int. J. Thermophys. 24:577CrossRefGoogle Scholar
  18. 18.
    Tzou D.Y. (1995). ASME J. Heat Transfer 117:8CrossRefGoogle Scholar
  19. 19.
    Tzou D.Y. (1997). Macro- to Microscale Heat Transfer: The Lagging Behavior. Taylor & Francis, Washington, DC, Chap. 10.Google Scholar
  20. 20.
    Tzou D.Y. (2003). ASME J. Dynamic Systems, Measurement, and Control 125:563CrossRefGoogle Scholar
  21. 21.
    Shih T.M. (1984). Numerical Heat Transfer. Hemisphere, New York, p. 55.MATHGoogle Scholar
  22. 22.
    Chen H.T., and Liu K.C. (2003). Int. J. Heat Mass Transfer 46:2809MATHCrossRefGoogle Scholar
  23. 23.
    Chen H.T., and Liu K.C. (2003). Comput. Phys. Commun 150:31CrossRefADSGoogle Scholar
  24. 24.
    Chen H.T., and Liu K.C. (2003). Int. J. Numer. Methods Eng. 57:637MATHCrossRefGoogle Scholar
  25. 25.
    Honig G., and Hirdes U. (1984). J. Comput. Appl. Math. 10:113MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Hays-Stang K.J., Haji-Sheikh A. (1999). Int. J. Heat Mass Transfer 42:455MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringFar East CollegeTaiwanRepublic of China

Personalised recommendations