Advertisement

Sādhanā

, 43:105 | Cite as

Analysis of extended micro-Graetz problem in a microtube

  • Mete Avci
  • Orhan Aydin
Article
  • 39 Downloads

Abstract

In this numerical study, hydrodynamically developed but thermally developing forced convection in a microtube subjected to a step change in the wall heat flux is analysed using a finite-volume method. The slip velocity and temperature jump conditions at the wall and the axial conduction in the fluid are included in the analysis. The combined effects of the Peclet number and the Knudsen number on the local Nusselt numbers as well as on the wall and bulk temperatures are determined in the continuum and slip flow regimes (0 ≤ Kn ≤ 0.1). In the entrance region, large reductions are observed in the Nusselt number with decreasing Peclet number or increasing Knudsen number. The results also show that the thermal length increases with decreasing Peclet number.

Keywords

Microtube axial conduction Knudsen number 

Nomenclature

D

diameter of the microtube (m)

F

tangential momentum accommodation coefficient

Ft

thermal accommodation coefficient

k

thermal conductivity (W/m K)

Kn

Knudsen number

L

half-length of the microtube (m)

L*

dimensionless half-length of the microtube

Nu

local Nusselt number

Pe

Peclet number

Pr

Prandtl number

\( q^{\prime\prime}_{w} \)

wall heat flux (W/m2)

r

radial coordinate (m)

R

dimensionless radial coordinate

T

temperature (K)

u

velocity (m/s)

x

axial direction (m)

X

dimensionless axial coordinate

Greek symbols

\( \gamma \)

specific heat ratio

\( \lambda \)

molecular mean free path (m)

υ

kinematic viscosity (m2/s)

\( \theta \)

dimensionless temperature, Eq. (3)

\( \theta_{b} \)

dimensionless fluid bulk temperature, Eq. (11)

\( \theta_{s - w} \)

dimensionless temperature jump between the fluid and wall, Eq. (9)

\( \theta_{w} \)

dimensionless wall temperature, Eq. (14)

Subscripts

s

fluid properties at the wall

w

wall

References

  1. 1.
    Palm B 2001 Heat transfer in microchannels. Microsc. Thermophys. Eng. 5: 155–175CrossRefGoogle Scholar
  2. 2.
    Hennecke D K 1968 Heat transfer by Hagen–Poiseuille flow in the thermal development region with axial conduction. Warme Stoffübertrag. 1: 177–184CrossRefGoogle Scholar
  3. 3.
    Hsu C J 1970 Theoretical solutions for low-Peclet-number thermal-entry-region heat transfer in laminar flow through concentric annuli. Int. J. Heat Mass Transfer 13: 1907–1924CrossRefMATHGoogle Scholar
  4. 4.
    Hsu C J 1971 An exact analysis of low Peclet number thermal entry region heat transfer in transversely non-uniform velocity fields. AIChE J. 17: 732–740CrossRefGoogle Scholar
  5. 5.
    Jones A S 1972 Laminar forced convection at low Peclet number. Bull. Aust. Math. Soc. 6: 83–94CrossRefMATHGoogle Scholar
  6. 6.
    Verhoff F H and Fisher D P 1973 A numerical solution of the Graetz problem with axial conduction included. J. Heat Transfer 95: 132–134CrossRefGoogle Scholar
  7. 7.
    Papoutsakis E, Ramkrishna D and Lim H C 1980 The extended Graetz problem with prescribed wall flux. AlChE J. 26: 779–787MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Vick B, Ozisik M N and Bayazitoglu Y 1980 A method of analysis of Peclet number thermal entry region problems with axial conduction. Lett. Heat Mass Transfer 7: 235–248CrossRefGoogle Scholar
  9. 9.
    Ebadian M A and Zhang H Y 1989 An exact solution of extended Graetz problem with axial conduction. Int. J. Heat Mass Transfer 32: 1709–1717CrossRefGoogle Scholar
  10. 10.
    Ebadian M A and Zhang H Y 1990 Effects of heat generation and axial heat conduction in laminar flow inside a circular pipe with a step change in wall temperature. Int. Commun. Heat Mass Transfer 17: 621–635CrossRefGoogle Scholar
  11. 11.
    Bilir S 1995 Laminar flow heat transfer in pipes including two-dimensional wall and fluid axial conduction. Int. J. Heat Mass Transfer 38: 1619–1625CrossRefMATHGoogle Scholar
  12. 12.
    Lahjomri J and Oubarra A 1999 Analytical solution of the Graetz problem with axial conduction. J. Heat Transfer 121: 1078–1083CrossRefGoogle Scholar
  13. 13.
    Weigand B and Lauffer D 2004 The extended Graetz problem with piecewise constant wall temperature for pipe and channel flows. Int. J. Heat Mass Transfer 47: 5303–5312CrossRefMATHGoogle Scholar
  14. 14.
    Jeong H E and Jeong J T 2006 Extended Graetz problem including streamwise conduction and viscous dissipation in microchannels. Int. J. Heat Mass Transfer 49: 2151–2157CrossRefMATHGoogle Scholar
  15. 15.
    Myong R S, Lockerby D A and Reese J M 2006 The effect of gaseous slip on microscale heat transfer: an extended Graetz problem. Int. J. Heat Mass Transfer 49: 2502–2513CrossRefMATHGoogle Scholar
  16. 16.
    Dutta P, Horiuchi K and Yin H M 2006 Thermal characteristics of mixed electroosmotic and pressure-driven microflows. Comput. Math. Appl. 52: 651–670MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Cetin B, Yazicioglu A G and Kakac S 2008 Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation. Int. Commun. Heat Mass Transfer 35: 535–544CrossRefGoogle Scholar
  18. 18.
    Cetin B, Yazicioglu A G and Kakac 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
  19. 19.
    Aziz A and Niedbalski A 2011 Thermally developing microtube gas flow with axial conduction and viscous dissipation. Int. J. Therm. Sci. 50: 332–340CrossRefGoogle Scholar
  20. 20.
    Mecili M and Mezaache E H 2013 Slug flow-heat transfer in parallel plate microchannel including slip effects and axial conduction. Energy Proc. 36: 268–277CrossRefGoogle Scholar
  21. 21.
    Cole K D, Cetin B and Brettmann L 2014 Microchannel heat transfer with slip flow and wall effects. J. Thermophys. Heat Transfer 28: 455–462CrossRefGoogle Scholar
  22. 22.
    Haddout Y and Lahjomri J 2015 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 Transfer 80: 673–687CrossRefGoogle Scholar
  23. 23.
    Balaj M, Roohi E, Akhlaghi H and Myong R S 2014 Investigation of convective heat transfer through constant wall heat flux micro/nano channels using DSMC. Int. J. Heat Mass Transfer 71: 633–638CrossRefGoogle Scholar
  24. 24.
    Balaj M, Roohi E and Akhlaghi H 2015 Effects of shear work on non-equilibrium heat transfer characteristics of rarefied gas flows through micro/nanochannels. Int. J. Heat Mass Transfer 83: 69–74CrossRefGoogle Scholar
  25. 25.
    Aydin O and Avci M 2006 Heat and flow characteristics of gases in micropipes. Int. J. Heat Mass Transfer 49: 1723–1730CrossRefMATHGoogle Scholar
  26. 26.
    Aydin O and Avci M 2006 Analysis of micro-Graetz problem in a microtube. Nanosc. Microsc. Thermophys. Eng. 10: 345–358CrossRefGoogle Scholar
  27. 27.
    Aydin O and Avci M 2006 Thermally developing flow in microchannels. J. Thermophys. Heat Transfer 20: 628–631CrossRefGoogle Scholar
  28. 28.
    Aydin O and Avci M 2007 Analysis of laminar heat transfer in micro-Poiseuille flow. Int. J. Therm. Sci. 46: 30–37CrossRefGoogle Scholar
  29. 29.
    Avci M and Aydin O 2008 Laminar forced convection slip-flow in a micro-annulus between two concentric cylinders. Int. J. Heat Mass Transfer 51: 3460–3467CrossRefMATHGoogle Scholar
  30. 30.
    Aydin O and Avci M 2015 Laminar forced convective slip flow in a microduct with a sinusoidally varying heat flux in axial direction. Int. J. Heat Mass Transfer 89: 606–612CrossRefGoogle Scholar
  31. 31.
    Patankar S V 1980 Numerical heat transfer and fluid flow. New York: McGraw HillMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringKaradeniz Technical UniversityTrabzonTurkey

Personalised recommendations