Heat Transfer in the Slip Flow with Axial Heat Conduction in a Microchannel with Walls Having a Constant Temperature

The problem on thermally developed laminar slip gas flow with axial heat conduction in a microchannel (a micropipe or a parallel-plate microchannel) with walls having a constant temperature was solved analytically with the use of the self-adjoint formalism method involving the rearrangement of the energy equation for this flow into a system of two partial differential equations of the first order. The temperature distribution and Nusselt numbers in such a flow have been determined on the assumption that it is hydrodynamically fully developed in the region of the thermal entrance of a microchannel. The data obtained show that the heat transfer in this flow is substantially dependent on the axial heat conduction in it and its rarefaction.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    J. M. Ha and G. P. Peterson, The heat transport capacity of micro heat pipe, J. Heat Transf.,120, No. 4, 1064–1071 (1998).

    Google Scholar 

  2. 2.

    G. Karniadakis, A. Beskok, and N. Aluru, Microflows and Nanoflows: Fundamentals and Simulation, Springer, New York (2005), Part 2, p. 60, Part 10, p. 400.

  3. 3.

    B. Weigand, Analytical Methods for Heat Transfer and Fluid Flow Problems, Springer, Berlin–Heidelberg (2015).

  4. 4.

    E. Papoutsakis, D. Ramkrishna, and H. C. Lim, The extended Graetz problem with Dirichlet wall boundary conditions, Appl. Sci. Res.,36, No. 1, 13–34 (1980).

    MathSciNet  MATH  Google Scholar 

  5. 5.

    E. Papoutsakis, D. Ramkrishna, and H. C. Lim, The extended Graetz problem with prescribed wall flux, AIChE J., 26, No. 5, 779–787 (1980).

    MathSciNet  MATH  Google Scholar 

  6. 6.

    E. Papoutsakis and D. Ramkrishna, Heat transfer in a capillary flow emerging from a reservoir, ASME J. Heat Transf., 103, No. 3, 429–435 (1981).

    Google Scholar 

  7. 7.

    B. Weigand and D. Lauffer, The extended Graetz problem with piecewise constant wall temperature for pipe and channel flows, Int. J. Heat Mass Transf., 47, No. 24, 5303–5312 (2004).

    MATH  Google Scholar 

  8. 8.

    J. Lahjomri, K. Zniber, A. Oubarra, and A. Alemany, Heat transfer by Hartmann flow in thermal entrance region with uniform heat flux: The Graetz problem extended, Energy Convers. Manage., 44, No. 1, 11–34 (2003).

    Google Scholar 

  9. 9.

    F. T. Akyildiz and H. Bellout, Graetz problem extended for Dipolar fluid, Int. J. Heat Mass Transf., 47, Nos. 12–13, 2747–2753 (2004).

  10. 10.

    E. M. Sparrow and S. H. Lin, Laminar heat transfer in tubes under slip-flow conditions, ASME J. Heat Transf., 84, No. 4, 363–369 (1962).

    Google Scholar 

  11. 11.

    R. F. Barron, X. M. Wang, T. A. Ameel, and R. O. Warrington, The Graetz problem extended to slip-flow, Int. J. Heat Mass Transf., 40, No. 8, 1817–1823 (1997).

    MATH  Google Scholar 

  12. 12.

    F. E. Larrode, C. Housiadas, and Y. Drossinos, Slip-flow heat transfer in circular tubes, Int. J. Heat Mass Transf., 43, No. 15, 2669–2680 (2000).

    MATH  Google Scholar 

  13. 13.

    G. Tunc and Y. Bayazitoglu, Heat transfer in microtubes with viscous dissipation, Int. J. Heat Mass Transf., 44, No. 13, 2395–2403 (2001).

    MATH  Google Scholar 

  14. 14.

    H. E. Jeong and J. T. Jeong, Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel, Int. J. Heat Mass Transf., 49, Nos. 13–14, 2151–2157 (2006).

  15. 15.

    B. Cetin, A. G. Yazicioglu, and S. Kakac, Slip-flow heat transfer in microtubes with axial conduction and viscous dissipation: An extended Graetz problem, Int. J. Therm. Sci., 48, No. 9, 1673–1678 (2009).

    Google Scholar 

  16. 16.

    A. K. Satapathy, Slip flow heat transfer in an infinite microtube with axial conduction, Int. J. Therm. Sci., 49, No. 1, 153–160 (2010).

    Google Scholar 

  17. 17.

    Y. Haddout, A. Oubarra, and J. Lahjomri, Convection forcée d’un écoulement glissant dans une microconduite chauffée avec une température uniforme à la paroi: influence de la conduction axiale, Rev. Méc. Appl. Théor., 2, No. 4, 359–376 (2011).

    Google Scholar 

  18. 18.

    B. Çetin and S. Zeinali, Analysis of heat transfer and entropy generation for a low-Peclet-number microtube flow using a second-order slip model: An extended-Graetz problem, J. Eng. Math., 89, No. 1, 13–25 (2014).

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Y. Haddout and J. Lahjomri, The extended Graetz problem for a gaseous slip flow in micropipe and parallel-plate microchannel with heating section of finite length: Effects of axial conduction, viscous dissipation and pressure work, Int. J. Heat Mass Transf., 80, 673–687 (2015).

    Google Scholar 

  20. 20.

    G. Tunc and Y. Bayazitoglu, Heat transfer in rectangular microchannels, Int. J. Heat Mass Transf., 45, No. 4, 765–773 (2002).

    MATH  Google Scholar 

  21. 21.

    B. Cetin, H. Yuncu, and S. Kakac, Gaseous flow in microconduits with viscous dissipation, Int. J. Transp. Phenom., 8, No. 4, 297–315 (2006).

    Google Scholar 

  22. 22.

    B. Cetin, A. G. Yazicioglu, and S. Kakac, Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation, Int. Commun. Heat Mass Transf., 35, No. 5, 535–544 (2008).

    Google Scholar 

  23. 23.

    V. J. Rij, T. Ameel, and T. Harman, The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer, Int. J. Therm. Sci., 48, No. 2, 271–281 (2009).

    Google Scholar 

  24. 24.

    A. Aziz and N. Niedbalski, Thermally developing microtube gas flow with axial conduction and viscous dissipation, Int. J. Therm. Sci., 50, No. 3, 332–340 (2011).

    Google Scholar 

  25. 25.

    Y. Haddout, E. Essaghir, A. Oubarra, and J. Lahjomri, Convective heat transfer for a gaseous slip flow in micropipe and parallel-plate microchannel with uniform wall heat flux: Effect of axial heat conduction, Indian J. Phys., 92, No. 6, 741–755 (2018).

    Google Scholar 

  26. 26.

    H. C. Ku and D. Hatziavramidis, Chebyshev expansion methods for the solution of the extended Graetz problem, J. Comput. Phys., 56, No. 3, 495–512 (1984).

    MathSciNet  MATH  Google Scholar 

  27. 27.

    J. Lahjomri, A. Oubarra, and A. Alemany, Heat transfer by laminar Hartmann flow in thermal entrance region with a step change in wall temperatures: The Graetz problem extended, Int. J. Heat Mass Transf., 45, No. 5, 1127–1148 (2002).

    MATH  Google Scholar 

  28. 28.

    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publ., Inc., New York (1972).

  29. 29.

    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77, 2nd ed., Cambridge Univ. Press (1992).

  30. 30.

    M. H. Holmes, Introduction to Numerical Methods in Differential Equations, Springer, Business Media, LLC (2007).

  31. 31.

    S. P. Yu and T. A. Ameel, Slip-flow heat transfer in rectangular microchannels, Int. J. Heat Mass Transf., 44, No. 22, 4225–4235 (2001).

    MATH  Google Scholar 

  32. 32.

    S. P. Yu and T. A. Ameel, A universal entrance number for internal slip flow, Int. Commun. Heat Mass Transf., 28, No. 7, 905–910 (2001).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Y. Haddout.

Additional information

Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 93, No. 3, pp. 625–636, May–June, 2020.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Haddout, Y., Oubarra, A. & Lahjomri, J. Heat Transfer in the Slip Flow with Axial Heat Conduction in a Microchannel with Walls Having a Constant Temperature. J Eng Phys Thermophy 93, 605–616 (2020). https://doi.org/10.1007/s10891-020-02158-9

Download citation

Keywords

  • heat transfer
  • forced convection
  • slip flow
  • axial heat conduction
  • rarefaction
  • self-adjoint formalism