Skip to main content
SpringerLink
Log in

Numerical investigation of the effect of second order slip flow conditions on interfacial heat transfer in micro pipes

  • Published:
Sādhanā Aims and scope Submit manuscript

Abstract

Heat transfer that occurs in micro scale devices has a very important place among the engineering applications that cooling or heating. This heat transfer mechanism in devices having dimensions at micron level is a completely different problem in the macro level analysis. Therefore, in the calculations made, the flow events and heat transfer in micron scale pipes are calculated by using more realistic expressions. For this reason, in this study heat transfer in a circular micro pipe with wall and fluid conjugation for laminar rarefied gas flow in transient regime is investigated under the second order slip boundary conditions at the interface. Patankar’s control volume method is used here to solve the problem numerically. This analysis includes of axial conduction, viscous dissipation and rarefaction effects which are indispensable in micro-flow structure. From the results, it is seen that the values that are indicating heat transfer are excessively affected by wall thickness, viscous heating and gas rarefaction especially in transient regime.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8

Similar content being viewed by others

Abbreviations

\( a \) :

discretization constants

\( b \) :

residual term

\( Br \) :

Brinkman number [\( = \frac{{\mu_{f} u_{m}^{2} }}{{k\left( {T_{1} - T_{0} } \right)}} \)]

\( c_{p} \) :

specific heat [J/kgK]

\( d \) :

wall thickness [m]

\( D \) :

hydraulic diameter [m]

\( k \) :

thermal conductivity [W/mK]

\( Kn \) :

Knudsen number [=λ/D]

\( Nu \) :

Nusselt Number [\( = \frac{hD}{k} \)]

\( Pe \) :

Peclet number [=\( \text{Re} \Pr = \tfrac{{2r_{wi} u_{m} \rho_{f} c_{pf} }}{{k_{f} }} \)]

\( Pr \) :

Prandtl number [=µcp/k]

\( Po \) :

Poiseuille number

\( q \) :

heat flux [W/m2]

R :

radial coordinate [m]

Re :

Reynolds number [=umD/ν]

\( t \) :

time [s]

\( T \) :

temperature [K]

\( T_{s} \) :

slip fluid temperature [K]

\( u \) :

axial velocity [m/s]

\( u_{s} \) :

slip velocity [m/s]

\( v \) :

radial velocity [m/s]

\( x \) :

axial coordinate [m]

\( \alpha \) :

thermal diffusivity [m2/s]

γ :

specific heat ratio

λ :

molecular mean free path [m]

\( \delta r \) :

radial grid difference [m]

\( \delta x \) :

axial grid difference [m]

\( \kappa \) :

kappa [\( = \tfrac{2\gamma }{\gamma + 1}\frac{1}{\Pr } \)]

\( \Delta r \) :

radial step [m]

\( \Delta t \) :

time step [s]

\( \Delta x \) :

axial step [m]

\( \rho \) :

density [kg/m3]

σm :

tangential momentum accommodation coefficient

σT :

thermal accommodation coefficient

\( \mu \) :

dynamic viscosity [Pa.s]

\( b \) :

bulk

\( f \) :

fluid

\( i \) :

inner

\( i,j \) :

at nodal point i, j

\( m \) :

mean

\( w \) :

wall

\( ' \) :

dimensionless quantity

0 :

at previous time step

References

  1. Patankar S 1980 Numerical Heat Transfer and Fluid Flow. CRC Press, Boca Raton

    MATH  Google Scholar 

  2. Kandlikar S G, Garimella S Li D, Colin S and King M R 2006 Heat Transfer and Fluid Flow in Minichannels and Microchannels. Oxford: Elsevier

    Google Scholar 

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

    Article  Google Scholar 

  4. Barletta A and Di Schio E R 2000 Periodic forced convection with axial heat conduction in a circular duct. Int. J. Heat Mass Transf. 43(16): 2949–2960

    Article  Google Scholar 

  5. Çetin B, Yazicioglu A G and Kakac S 2008 Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation. Int. Commun. Heat Mass Transf. 35(5): 535–544

    Article  Google Scholar 

  6. Rosa P, Karayiannis T G and Collins M W 2009 Single-phase heat transfer in microchannels: the importance of scaling effects. Appl. Therm. Eng. 29: 3447–3468

    Article  Google Scholar 

  7. Zhu X and Liao Q 2006 Heat transfer for laminar slip flow in a microchannel of arbitrary cross section with complex thermal boundary conditions. Appl. Therm. Eng. 26: 1246–1256

    Article  Google Scholar 

  8. Hong C and Asako Y 2008 Heat transfer characteristics of gaseous flows in microtube with constant heat flux. Appl. Therm. Eng. 28: 1375–1385

    Article  Google Scholar 

  9. Nam T P and Le Roohi E 2015 A new form of the second-order temperature jump boundary condition for the low-speed nanoscale and hypersonic rarefied gas flow simulations. Int. J. Therm. Sci. 98: 51–59

    Article  Google Scholar 

  10. Kushwaha H M and Sahu S K 2014 Analysis of gaseous flow in a micro pipe with second order velocity slip and temperature jump boundary conditions. Heat Mass Transfer 50: 1649–1659

    Article  Google Scholar 

  11. Dongari N and Agrawal A 2007 Analytical solution of gaseous slip in long microchannels. Heat Mass Transfer 50: 3411–3421

    Article  Google Scholar 

  12. Xiao N, Elsnab J and Ameel T 2009 Microtube gas flows with second order slip flow and temperature jump boundary conditions. Int. J. Therm. Sci. 48: 243–251

    Article  Google Scholar 

  13. Aubert C and Colin S 2001 High order boundary conditions for gaseous flows in rectangular microducts. Microscale Thermophys. Eng. 5: 41–54

    Article  Google Scholar 

  14. Cercignani C and Lorenzani S 2010 Variational derivation of second order coefficients on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids 22: 062004

    Article  Google Scholar 

  15. Duan Z 2012 Second-order gaseous slip flow models in long circular and noncircular microchannels and nanochannels. Micro Nanofluidics 12: 805–820

    Article  Google Scholar 

  16. Lelea D and Cioabla A E 2010 The viscous dissipation effect on heat transfer and fluid flow in micro-tubes. Int. Commun. Heat Mass Transf. 37(9): 1208–1214

    Article  Google Scholar 

  17. Rahimi M and Mehryar R 2012 Numerical study of axial heat conduction effects on the local Nusselt number at the entrance and ending regions of a circular microchannel. Int. J. Therm. Sci. 59: 87–94

    Article  Google Scholar 

  18. Aziz A and Niedbalski N 2011 Thermally developing microtube gas flow with axial conduction and viscous dissipation. Int. J. Therm. Sci. 50(3): 332–340

    Article  Google Scholar 

  19. Barron R F, Wang X, Warrington R O and Ameel T 1996 Evaluation of the eigenvalues for the Graetz problem in slip-flow. Int. Commun. Heat Mass Transf. 23(4): 563–574

    Article  Google Scholar 

  20. Barron R F, Wang X, Ameel T A and Warrington RO 1997 The Graetz problem extended to slip-flow. Int. J. Heat Mass Transf. 40(8): 1817–1823

    Article  Google Scholar 

  21. Kabar Y, Bessaïh R and Rebay M 2013 Conjugate heat transfer with rarefaction in parallel plates microchannel. Superlattices Microstruct. 60: 370–388

    Article  Google Scholar 

  22. Bilir Ş 1995 Laminar flow heat transfer in pipes including two-dimensional wall and fluid axial conduction. Int. J. Heat Mass Transf. 38(9): 1619–1625

    Article  Google Scholar 

  23. Bilir Ş 2002 Transient conjugated heat transfer in pipes involving two-dimensional wall and axial fluid conduction. Int. J. Heat Mass Transf. 45(8): 1781–1788

    Article  Google Scholar 

  24. Bilir Ş and Ateş A 2003 Transient conjugated heat transfer in thick walled pipes with convective boundary conditions. Int. J. Heat Mass Transf. 46(14): 2701–2709

    Article  Google Scholar 

  25. Altun A H, Bilir Ş Ateş A 2016 Transient conjugated heat transfer in thermally developing laminar flow in thick walled pipes and minipipes with time periodically varying wall temperature boundary condition. Int. J. Heat Mass Transf. 92: 643–657

    Article  Google Scholar 

  26. Wei-Mon Y 1993 Transient conjugated heat transfer in channel flows with convection from yhe ambient. Int. J. Heat Mass Transf. 36(5): 1295–1301

    Article  Google Scholar 

  27. Lın T F and Kuo J C 1988 Transient conjugated heat transfer in fully developed laminar pipe flows. Int. J. Heat Mass Transf. 31(5): 1093–1102

    Article  Google Scholar 

  28. Knupp D C, Naveira-Cotta C P and Cotta R M 2012 Theoretical analysis of conjugated heat transfer with a single domain formulation and integral transforms. Int. Commun. Heat Mass Transf. 39: 355–362

    Article  Google Scholar 

  29. Knupp D C, Cotta R M, Naveira-Cotta C P and Kakaç S 2015 Transient conjugated heat transfer in microchannels: Integral transforms with single domain formulation. Int. J. Therm. Sci. 88:248–257

    Article  Google Scholar 

  30. Lelea D 2007 The conjugate heat transfer of the partially heated microchannels. Heat Mass Trans. Heat Mass Transf. 44: 33–41

    Article  Google Scholar 

  31. Şen S and Darici S 2017 Transient conjugate heat transfer in a circular microchannel involving rarefaction, viscous dissipation and axial conduction effects. App. Therm. Eng. 111:855–862

    Article  Google Scholar 

  32. Yener Y, Kakac S, Avelino M and Okutucu T 2005 Single phase forced convection in microchannels-state-of-the-art-review. Microscale Heat Transfer-Fundamentals and Applications. The Netherlands: Kluwer Academic Publishers 1–24

    Google Scholar 

  33. Rebay M, Kakaç S and Cotta R M 2016 Microscale and Nanoscale Heat Transfer: Analysis, Design and Application. CRC Press, Boca Rataon 391–405

  34. Karniadakis G, Beskok A and Aluru N 2006 Microflows and Nanoflows: Fundamentals and Simulation. Springer Science & Business Media, Berlin

  35. Bilir Ş 1992 Numerical solution of Graetz problem with axial conduction. Num. Heat Transf. 21(4): 493–500

    Article  Google Scholar 

  36. Faghri M and Sparrow E 1980 Simultaneous wall and fluid axial conduction in laminar pipe-flow heat transfer. J. Heat Transf. 102(1): 58–63

    Article  Google Scholar 

Download references

Acknowledgements

The Authour is thankful to Selçuk University for the support.

Author information

Authors and Affiliations

  1. Department of Airplane Airframe and Engine Maintenance, School of Civil Aviation, Selçuk University, 42250, Konya, Turkey

    Soner Şen

Authors
  1. Soner Şen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Soner Şen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Şen, S. Numerical investigation of the effect of second order slip flow conditions on interfacial heat transfer in micro pipes. Sādhanā 44, 159 (2019). https://doi.org/10.1007/s12046-019-1137-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12046-019-1137-6

Keywords

Search

Navigation