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Thermal Analysis of Non-linear Convective–Radiative Hyperbolic Lumped Systems with Simultaneous Variation of Temperature-Dependent Specific Heat and Surface Emissivity by MsDTM and BPES

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Abstract

With the advent of temperatures near absolute zero, it is often claimed that at very low temperatures the effect of thermal wave propagation must be included by the hyperbolic heat conduction equation (HHCE). In this paper the non-linear convective–radiative HHCE is investigated. Opposite to common numerical analyses, analytical expressions are obtained for the temperature variations by the multi-step differential transformation method. Some conclusions about alteration of the specific heat of the material, temperature steeping, and Vernotte number have been formulated.

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Abbreviations

\(A\) :

Dimensionless parameter describing variation of the thermal conductivity

\(B\) :

Dimensionless parameter describing variation of the surface emissivity

\(c\) :

Specific heat (J\(\,\cdot \,\)kg\(^{-1}\cdot \, \)K\(^{-1}\))

\(c_\mathrm{a} \) :

Specific heat at the temperature \(T_\mathrm{a}\) (J\(\,\cdot \, \)kg\(^{-1}\cdot \, \)K\(^{-1}\))

\(D\) :

Domain

\(E\) :

Surface emissivity

\(E_\mathrm{s} \) :

Surface emissivity at the temperature \(T_\mathrm{s} \)

\(Fo\) :

Fourier number

\(H\) :

Constant

\(h\) :

Heat transfer coefficient (W\(\cdot \, \)m\(^{-2}\cdot \, \)K\(^{-1}\))

\(Nr\) :

Dimensionless radiation–conduction parameter

\(S\) :

Surface area (m\(^{2}\))

\(T\) :

Temperature (K)

\(t\) :

Temporal coordinate (s)

\(T_\mathrm{a} \) :

Convection sink temperature (K)

\(T_\mathrm{i} \) :

Initial temperature (K)

\(T_\mathrm{s} \) :

Radiation sink temperature (K)

\(V\) :

Volume (m\(^{3}\))

\(Ve\) :

Vernotte number

\(X(k)\) :

Transformed analytical function

\(x(t)\) :

Original analytical function

\(\alpha \) :

Measure of thermal conductivity variation with temperature (K\(^{-1}\))

\(\beta \) :

Measure of surface emissivity variation with temperature (K\(^{-1}\))

\(\rho \) :

Mass density (kg\(\cdot \, \)m\(^{-3}\))

\(\sigma \) :

Stefan–Boltzmann constant (W\(\cdot \, \)m\(^{-2}\cdot \, \)K\(^{-4}\))

\(\tau \) :

Thermal relaxation time (s)

\(\theta \) :

Dimensionless temperature

\(\theta _\mathrm{a} \) :

Dimensionless convection sink temperature

\(\theta _\mathrm{s} \) :

Dimensionless radiation sink temperature

References

  1. C. Cattaneo, C. R. Acad. Sci. 247, 431 (1958)

    MathSciNet  Google Scholar 

  2. P. Vernotte, C.R. Acad. Sci. 246, 3154 (1958)

    MathSciNet  Google Scholar 

  3. W. Shen, S. Han, Heat Mass Transfer 39, 499 (2003)

    ADS  Google Scholar 

  4. H.E. Jackson, C.T. Walker, Phys. Rev. B: Condens. Matter 3, 1428 (1971)

    Article  ADS  Google Scholar 

  5. V. Narayanamurti, R.C. Dynes, Phys. Rev. Lett. 28, 1461 (1972)

    Article  ADS  Google Scholar 

  6. J. Zhou, Y. Zhang, J.K. Chen, Int. J. Therm. Sci. 48, 1477 (2009)

    Article  Google Scholar 

  7. J. Zhou, J.K. Chen, Y. Zhang, Comput. Biol. Med. 39, 286 (2009)

    Article  Google Scholar 

  8. F. Xu, K.A. Seffen, T.J. Liu, Int. J. Heat Mass Transf. 51, 2237 (2008)

    Article  MATH  Google Scholar 

  9. C. Korner, H.W. Bergmann, Appl. Phys. A 67, 397 (1998)

    Article  ADS  Google Scholar 

  10. M. Al-Nimr, M. Naji, Microscale Thermophys. Eng. 4, 231 (2000)

    Article  Google Scholar 

  11. S. Galovic, D. Kostoski, G. Stamboliev, E. Suljovrujic, Radiat. Phys. Chem. 67, 459 (2003)

    Article  ADS  Google Scholar 

  12. D. Jou, J. Casas-Vazquez, G. Lebon, Rep. Prog. Phys. 51, 1105 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  13. J.A. López Molina, M.J. Rivera, M. Trujillo, E.J. Berjano, Med. Phys. 36, 1112 (2009)

    Article  Google Scholar 

  14. M. Lewandowska, L. Malinowski, Int. Commun. Heat Mass Transf. 33, 61 (2006)

    Article  Google Scholar 

  15. A. Moosaie, Forsch. Ingenieurwes. 71, 163 (2007)

    Article  Google Scholar 

  16. A. Moosaie, Int. Commun. Heat Mass Transf. 35, 103 (2008)

    Article  Google Scholar 

  17. D.W. Tang, N. Araki, Int. J. Heat Mass Transf. 39, 1585 (1996)

    Article  MATH  Google Scholar 

  18. D. Zhang, L. Li, Z. Li, L. Guan, X. Tan, Physica B 364, 285 (2005)

    Article  ADS  Google Scholar 

  19. A. Saleh, M. Al-Nimr, Int. Commun. Heat Mass Transf. 35, 204 (2008)

    Article  Google Scholar 

  20. F.M. Jiang, A.C.M. Sousa, J. Thermophys. Heat Transf. 19, 595 (2005)

    Article  Google Scholar 

  21. M. Torabi, S. Saedodin, J. Thermophys. Heat Transf. 25, 239 (2011)

    Article  Google Scholar 

  22. H.T. Chen, J.Y. Lin, Int. J. Heat Mass Transf. 36, 2891 (1992)

    Google Scholar 

  23. C.Y. Yang, J. Thermophys. Heat Transf. 19, 217 (2005)

    Article  Google Scholar 

  24. J. Zhou, Y. Zhang, J.K. Chen, Numer. Heat Transf. Part A 54, 1 (2008)

    Article  ADS  Google Scholar 

  25. C.H. Huang, C.Y. Lin, J. Thermophys. Heat Transf. 22, 766 (2008)

    Article  Google Scholar 

  26. C.Y. Yang, Appl. Math. Modell. 33, 2907 (2009)

    Article  MATH  Google Scholar 

  27. M.H. Babaei, Z. Chen, J. Thermophys. Heat Transf. 24, 325 (2010)

    Article  Google Scholar 

  28. J.K. Zhou, Differential Transform and its Applications for Electrical Circuits (Huarjung University Press, Wuhan, 1986)

    Google Scholar 

  29. C.W. Bert, ASME J. Heat Transf. 124, 208 (2002)

    Article  Google Scholar 

  30. H.P. Chu, C.Y. Lo, Numer. Heat Transf. Part A 53, 295 (2008)

    Article  Google Scholar 

  31. C.Y. Lo, B.Y. Chen, Numer. Heat Transf. Part B 55, 219 (2009)

    Article  ADS  Google Scholar 

  32. H. Yaghoobi, M. Torabi, Int. Commun. Heat Mass Transf. 38, 815 (2011)

    Article  Google Scholar 

  33. S. Kim, C.-H. Huang, J. Phys. D: Appl. Phys. 40, 2979 (2007)

    Article  ADS  Google Scholar 

  34. P. Barry, A. Hennessy, J. Integer. Seq. 13, 1 (2010)

    Google Scholar 

  35. A. Belhadj, J. Bessrour, M. Bouhafs, L. Barrallier, J. Therm. Anal. Calorim. 97, 911 (2009)

    Article  Google Scholar 

  36. M. Agida, A.S. Kumar, El. J. Theor. Phys. 7, 319 (2010)

    Google Scholar 

  37. O.B. Awojoyogbe, K. Boubaker, Curr. Appl. Phys. 9, 278 (2008)

    Article  ADS  Google Scholar 

  38. A. Belhadj, O. Onyango, N. Rozibaeva, J. Thermophys. Heat Transf. 23, 639 (2009)

    Article  Google Scholar 

  39. S. Fridjine, M. Amlouk, Mod. Phys. Lett. B 23, 2179 (2009)

    Article  ADS  MATH  Google Scholar 

  40. S. Fridjine, K. Boubaker, M. Amlouk, Can. J. Phys. 87, 653 (2009)

    Article  ADS  Google Scholar 

  41. J. Ghanouchi, H. Labiadh, K. Boubaker, Int. J. Heat Technol. 26, 49 (2008)

    Google Scholar 

  42. A. Chaouachi, K. Boubaker, M. Amlouk, H. Bouzouita, Eur. Phys. J. Appl. Phys. 37, 105 (2007)

    Article  ADS  Google Scholar 

  43. S. Fridjine, M. Amlouk, Mod. Phys. Lett. B 23, 2179 (2009)

    Article  ADS  MATH  Google Scholar 

  44. T. Ghrib, K. Boubaker, M. Bouhafs, Mod. Phys. Lett. B 22, 2893 (2008)

    Article  ADS  MATH  Google Scholar 

  45. N. Guezmir, T. Ben Nasrallah, K. Boubaker, M. Amlouk, S. Belgacem, J. Alloys Compd. 481, 543 (2009)

    Article  Google Scholar 

  46. A.S. Kumar, J. Franklin Inst. 347, 1755 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  47. O.D. Oyodum, O.B. Awojoyogbe, M. Dada, J. Magnuson, Eur. Phys. J. Appl. Phys. 46, 21201 (2009)

    Article  Google Scholar 

  48. A. Milgram, J. Theor. Biol. 271, 157 (2011)

    Article  Google Scholar 

  49. S. Slama, J. Bessrour, M. Bouhafs, K.B. Ben Mahmoud, Numer. Heat Transf. Part A 55, 401 (2009)

    Article  ADS  Google Scholar 

  50. S. Slama, K. Boubaker, J. Bessrour, M. Bouhafs, Thermochim. Acta 482, 8 (2009)

    Article  Google Scholar 

  51. T.G. Zhao, Y.X. Wang, K. Ben Mahmoud, Int. J. Math. Comp. 1, 13 (2008)

    MathSciNet  Google Scholar 

  52. S. Tabatabaei, T. Zhao, O. Awojoyogbe, F. Moses, Heat Mass Transf. 45, 1247 (2009)

    Article  ADS  Google Scholar 

  53. S. Slama, M. Bouhafs, K.B. Ben Mahmoud, Int. J. Heat Technol. 26, 141 (2008)

    Google Scholar 

  54. A. Yildirim, S.T. Mohyud-Din, D.H. Zhang, Comp. Math. Appl. 59, 2473 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Young Researchers Club, Central Tehran Branch, Islamic Azad University, Tehran, Iran

    Mohsen Torabi & Hessameddin Yaghoobi

  2. École Supérieure de Sciences et Techniques de Tunis (ESSTT), Université de Tunis, 63 Rue Sidi Jabeur, 5100 , Mahdia, Tunisia

    Karem Boubaker

Authors
  1. Mohsen Torabi
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  2. Hessameddin Yaghoobi
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  3. Karem Boubaker
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Correspondence to Karem Boubaker.

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Torabi, M., Yaghoobi, H. & Boubaker, K. Thermal Analysis of Non-linear Convective–Radiative Hyperbolic Lumped Systems with Simultaneous Variation of Temperature-Dependent Specific Heat and Surface Emissivity by MsDTM and BPES. Int J Thermophys 34, 122–138 (2013). https://doi.org/10.1007/s10765-012-1388-5

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