Journal of Engineering Mathematics

, Volume 89, Issue 1, pp 13–25 | Cite as

Analysis of heat transfer and entropy generation for a low-Peclet-number microtube flow using a second-order slip model: an extended-Graetz problem

Article

Abstract

The classical Graetz problem, which is the problem of the hydrodynamically developed, thermally developing laminar flow of an incompressible fluid inside a tube neglecting axial conduction and viscous dissipation, is one of the fundamental problems of internal-flow studies. This study is an extension of the Graetz problem to include the rarefaction effect, viscous dissipation term and axial conduction with a constant wall temperature thermal boundary condition. The energy equation is solved to determine the temperature field analytically using general eigenfunction expansion with a fully developed velocity profile. To analyze the low-Peclet-number nature of the flow, the flow domain is extended from \(-\infty \) to \(+\infty \). To model the rarefaction effect, a second-order slip model is implemented. The temperature distribution, local Nusselt number, and local entropy generation are determined in terms of confluent hypergeometric functions. This kind of theoretical study is important for a fundamental understanding of the convective heat transfer characteristics of flows at the microscale and for the optimum design of thermal systems, which includes convective heat transfer at the microscale, especially operating at low Reynolds numbers.

Keywords

Extended-Graetz problem Low Pe Microtube Second-order slip model 

References

  1. 1.
    Graetz L (1883) Uben die wärmeleitungsfähigkeit von flüssigkeiten (On the thermal conductivity of liquids, part 1). Ann Phys Chem 18:79–94Google Scholar
  2. 2.
    Graetz L (1885) Uben die wärmeleitungsfähigkeit von flüssigkeiten (On the thermal conductivity of liquids, part 2). Ann Phys Chem 25:337–357ADSCrossRefGoogle Scholar
  3. 3.
    Nusselt W (1910) Die abhängigkeit der wärmebergangszahl von der rohrlänge (The dependence of the heat transfer coefficient on the tube length). VDI Z 54:1154–1158Google Scholar
  4. 4.
    Shah RK, London AL (1978) Laminar flow forced convection in ducts: a source book for compact heat exchanger analytical data. Academic Press, New York, pp 78–138Google Scholar
  5. 5.
    Lahjomri J, Quabarra A, Alemany A (2002) 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 45:1127–1148CrossRefMATHGoogle Scholar
  6. 6.
    Lahjomri J, Zniber K, Quabarra A, Alemany A (2003) Heat transfer by laminar hartmanns flow in thermal entrance region with uniform wall heat ux: the Graetz problem extended. Energy Convers Manag 44:11–34CrossRefGoogle Scholar
  7. 7.
    Dutta P, Horiuchi K, Yin HM (2006) Thermal characteristics of mixed electroosmotic and pressure-driven microflows. Comput Math Appl 52:651–670CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Horiuchi K, Dutta P, Hossain A (2006) Joule-heating effects in mixed electroosmotic and pressure-driven microflows under constant wall heat flux. J Eng Math 54:159–180CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Barron RF, Wang XM, Warrington RO, Ameel TA (1997) The Graetz problem extended to slip flow. Int J Heat Mass Transf 40:1817–1823CrossRefMATHGoogle Scholar
  10. 10.
    Ameel TA, Barron RF, Wang XM, Warrington RO (1997) Laminar forced convection in a circular tube with constant heat flux and slip flow. Microscale Thermophys Eng 1:303–320CrossRefGoogle Scholar
  11. 11.
    Larrode FE, Housiadas C, Drossinos Y (2000) Slip-flow heat transfer in circular tubes. Int J Heat Mass Transf 43:2669–2680CrossRefMATHGoogle Scholar
  12. 12.
    Ameel TA, Barron RF, Wang XM, Warrington RO (2001) Heat transfer in microtubes with viscous dissipation. Int J Heat Mass Transf 44:2395–2403CrossRefGoogle Scholar
  13. 13.
    Tunc G, Bayazitoglu Y (2002) Convection at the entrance of micropipes with sudden wall temperature change. In ASME IMECE 2002, November 17–22, New Orleans, LousianaGoogle Scholar
  14. 14.
    Mikhailov MD, Cotta RM, Kakac S (2005) Steady state and periodic heat transfer in micro conduits. In: Kakac S, Vasiliev L, Bayazitoglu Y, Yener Y (eds) Microscale heat transfer-fundamentals and applications in biological systems and MEMS. Kluwer Academic Publisher, DordrechtGoogle Scholar
  15. 15.
    Jeong HE, Jeong JT (2006) Extended Graetz problem including streamwise conduction and viscous dissipation in microchannels. Int J Heat Mass Transf 49:2151–2157CrossRefMATHGoogle Scholar
  16. 16.
    Aydin O, Avcı M (2006) Analysis of micro-Graetz problem in a microtube. Nanoscale Microscale Thermophys Eng 10(4):345–358CrossRefGoogle Scholar
  17. 17.
    Çetin B, Yuncu H, Kakaç S (2006) Gaseous flow in microconduits with viscous dissipation. Int J Transp Phenom 8:297–315Google Scholar
  18. 18.
    Haji-Sheikh A, Beck JV, Amos DE (2008) Axial heat conductioin effects in the entrance region of parallel plate ducts. Int J Heat Mass Transf 51:5811–5822CrossRefMATHGoogle Scholar
  19. 19.
    Çetin B, Yazıcıog̃lu AG, Kakaç S (2009) Slip-flow heat transfer in microtubes with axial conduction and viscous dissipation—an extended Graetz problem. Int J Therm Sci 48:1673–1678CrossRefGoogle Scholar
  20. 20.
    Yaman M, Khudiyev T, Ozgur E, Kanik M, Aktas O, Ozgur EO, Deniz H, Korkut E, Bayindir M (2011) Arrays of indefinitely long uniform nanowires and nanotubes. Nat Mater 10(7):494–501ADSCrossRefGoogle Scholar
  21. 21.
    Karniadakis GE, Beskok A, Aluru N (2005) Microflows and nanoflows: fundamentals and simulations. Springer, New York, pp 51–74, 167–172Google Scholar
  22. 22.
    Çetin B (2013) Effect of thermal creep on heat transfer for a 2D microchannel flow: an analytical approach. ASME J Heat Transf 135(101007):1–8Google Scholar
  23. 23.
    Weng HC, Chen C-K (2008) A challenge in Navier–Stokes based continuum modeling: Maxwell–Burnett slip law. Phys Fluids 20:106101ADSCrossRefGoogle Scholar
  24. 24.
    Deen WM (1998) Analysis of transport phenomena. Oxford University Press, Oxford, pp 391–392Google Scholar
  25. 25.
    Ash RL, Heinbockel JH (1970) Note on heat transfer in laminar fully developed pipe flow with axial conduction. Math Phys 21:266–269Google Scholar
  26. 26.
    Hsu CJ (1971) An exact analysis of low Peclet number thermal entry region heat transfer in transversely non-uniform velocity field. AIChE J 17:732–740CrossRefGoogle Scholar
  27. 27.
    Davis EJ (1971) Exact solutions for a class of heat and mass transfer problems. Can J Chem Eng 51:562–572CrossRefGoogle Scholar
  28. 28.
    Taitel V, Tamir A (1972) Application of the integral method to flows with axial conduction. Int J Heat Mass Transf 15:733–740CrossRefMATHGoogle Scholar
  29. 29.
    Taitel V, Bentwich M, Tamir A (1973) Effects of upstream and downstream boundary conditions on heat(mass) transfer with axial diffussion. Int J Heat Mass Transf 16:359–369CrossRefMATHGoogle Scholar
  30. 30.
    Papoutsakis E, Ramkrishna D, Lim HC (1980) The extended Graetz problem with dirichlet wall boundary condition. Appl Sci Res 36:13–34MATHMathSciNetGoogle Scholar
  31. 31.
    Papoutsakis E, Ramkrishna D, Lim HC (1980) The extended Graetz problem with prescribed wall flux. AIChE J 26:779–787CrossRefMathSciNetGoogle Scholar
  32. 32.
    Acrivos A (1980) The extended Graetz problem at low Peclet numbers. Appl Sci Res 36:35–40MATHMathSciNetGoogle Scholar
  33. 33.
    Vick B, Ozisik MN, Bayazitoglu Y (1980) Method of analysis of low Peclet number thermal entry region problems with axial conduction. Lett Heat Mass Transf 7:235–248ADSCrossRefGoogle Scholar
  34. 34.
    Vick B, Ozisik MN (1981) An exact analysis of low Peclet number heat transfer in laminar flow with axial conduction. Lett Heat Mass Transf 8:1–10ADSCrossRefGoogle Scholar
  35. 35.
    Lahjomri J, Qubarra A (1999) Analytical solution of the Graetz problem with axial conduction. J Heat Transf 121:1078–1083CrossRefGoogle Scholar
  36. 36.
    Çetin B, Yazıcıog̃lu AG, Kakaç S (2008) Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation. Int Comm Heat Mass Transf 35:535–544CrossRefGoogle Scholar
  37. 37.
    Bejan A (1996) Entropy generation minimization: the method of thermodynamic optimization of finite-size systems and finite-time processes. CRC Press, Boca Raton, pp 71–74MATHGoogle Scholar
  38. 38.
    Eckert EGR, Jr Drake RM (1972) Analysis of heat and mass transfer. McGraw-Hill, New YorkMATHGoogle Scholar
  39. 39.
    Kays WM, Crawford M, Weigand B (2005) Convective heat and mass transfer. McGraw Hill, New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Mechanical Engineering Department, Microfluidics & Lab-on-a-chip Research Groupİhsan Doğramacı  Bilkent UniversityAnkaraTurkey

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