Skip to main content
Log in

Conjugate Heat Transfer: Analysis Via Integral Transforms and Eigenvalue Problems

  • Published:
Journal of Engineering Physics and Thermophysics Aims and scope

An integral transform approach to the solution of the problem on conjugate heat transfer, combining the singledomain formulation with the convective eigenfunction expansion basis within the total integral transformation framework, which leads to a nonclassical eigenvalue problem, is presented. The problem on the conjugate heat transfer in the transient two-dimensional incompressible laminar flow of a Newtonian fluid in a parallel-plate channel is considered to illustrate the hybrid numerical-analytical approach. To demonstrate the improvement of the convergence rate achieved with the methodology proposed, a critical comparison against the traditional total integral transformation solution of the diffusive eigenvalue problem is provided, and results are presented and discussed for three representative situations realized with different Peclet numbers: Pe = 1, 10 and 100. A remarkable improvement of the convergence rate, obtained especially with the large Péclet numbers, offers evidence of the validity of the expansion constructed upon the nonclassical eigenvalue problem proposed.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. T. L. Perelman, On conjugated problems of heat transfer, Int. J. Heat Mass Transf., 3, 294–303 (1961).

    Article  Google Scholar 

  2. A. V. Luikov, V. A. Aleksashenko, and A. A. Aleksashenko, Analytical methods of solution of conjugated problems in convective heat transfer, Int. J. Heat Mass Transf., 14, 1047–1056 (1971).

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  5. R. M. Cotta, D. C. Knupp, and C. P. Naveira-Cotta, Analytical Heat and Fluid Flow in Microchannels and Microsystems, Springer, New York (2016).

    Book  Google Scholar 

  6. D. C. Knupp, C. P. Naveira-Cotta, A. Renfer, M. K. Tiwari, R. M. Cotta, and D. Poulikakos, Analysis of conjugated heat transfer in micro-heat exchangers via integral transforms and non-intrusive optical techniques, Int. J. Num. Methods Heat Fluid Flow, 25, No. 6, 1444–1462 (2015).

    Article  Google Scholar 

  7. D. C. Knupp, C. P. Naveira-Cotta, and R. M. Cotta, Theoretical–experimental analysis of conjugated heat transfer in nanocomposite heat spreaders with multiple microchannels, Int. J. Heat Mass Transf., 74, 306–318 (2014).

    Article  Google Scholar 

  8. D. C. Knupp, F. S. Mascouto, L. A. Abreu, C. P. Naveira-Cotta, and R. M. Cotta, Conjugated heat transfer in circular microchannels with slip fl ow and axial diffusion effects, Int. Commun. Heat Mass Transf., 91, 225–233 (2018).

    Article  Google Scholar 

  9. D. C. Knupp, R. M. Cotta, and C. P. Naveira-Cotta, Fluid fl ow and conjugated heat transfer in arbitrarily shaped channels via single domain formulation and integral transforms, Int. J. Heat Mass Transf., 82, 479–489 (2015).

    Article  Google Scholar 

  10. R. M. Cotta, Integral Transforms in Computational Heat and Fluid Flow, CRC Press, Boca Raton, FL (1993).

    MATH  Google Scholar 

  11. R. M. Cotta, C. P. Naveira-Cotta, and D. C. Knupp, Convective eigenvalue problems for convergence enhancement of eigenfunction expansions in convection–diffusion problems, J. Therm. Sci. Eng. Appl., 10, No. 2, 021009 (2018).

  12. C. J. Hsu, An exact mathematical solution for entrance-region laminar heat transfer with axial conduction, Appl. Sci. Res, 17, Nos. 4–5, 359–376 (1967).

  13. C. W. Tan and H. Chia-Jung, Low Péclet number mass transfer in laminar fl ow through circular tubes, Int. J. Heat Mass Transf., 15, No. 11, 2187–2201 (1972).

    Article  Google Scholar 

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

  15. C. W. Tan and M. J. Normandia, Low Peclet number diffusion in narrow rectangular channels, Lett. Heat Mass Transf., 2, No. 3, 259–266 (1975).

    Article  Google Scholar 

  16. B. Vick and M. N. Özişik, An exact analysis of low Peclet number heat transfer in laminar fl ow with axial conduction, Lett. Heat Mass Transf., 8, No. 1, 1–10 (1981).

    Article  Google Scholar 

  17. D. Ramkrishna and N. R. Amundson, A non-self-adjoint problem in heat conduction, J. Heat Transf., 104, No. 1, 185–190 (1982).

    Article  Google Scholar 

  18. M. C. Oliveira, R. Ramos, and R. M. Cotta, On the eigenvalues basic to the analytical solution of convective heat transfer with axial diffusion effects, Commun. Numer. Methods Eng., 11, No. 4, 287–296 (1995).

    Article  Google Scholar 

  19. A. Haji-Sheikh, J. V. Beck, and D. E. Amos, Axial heat conduction effects in the entrance region of parallel plate ducts, Int. J. Heat Mass Transf., 51, Nos. 25–26, 5811–5822 (2008).

  20. M. D. Mikhailov and M. N. Ozisik, On general solution of heat conduction in an anisotropic medium, Lett. Heat MassTransf., 8, 329–335 (1981).

    Article  Google Scholar 

  21. D. Ramkrishna and N. R. Amundson, Linear Operator Methods in Chemical Engineering, Prentice-Hall, New Jersey (1985).

    Google Scholar 

  22. R. M. Cotta, B. P. Cotta, C. P. Naveira-Cotta, and G. Cotta-Pereira, Hybrid integral transforms analysis of the bioheat equation with variable properties, Int. J. Therm. Sci., 49, No. 9, 1510–1516 (2010).

    Article  Google Scholar 

  23. C. P. Naveira-Cotta, R. M. Cotta, H. R. B. Orlande, and O. Fudym, Eigenfunction expansions for transient diffusion in heterogeneous media, Int. J. Heat Mass Transf., 52, Nos. 21–22, 5029–5039 (2009).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to D. C. Knupp.

Additional information

Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 93, No. 1, pp. 65–77, January–February, 2020.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Knupp, D.C., Cotta, R.M. & Naveira-Cotta, C.P. Conjugate Heat Transfer: Analysis Via Integral Transforms and Eigenvalue Problems. J Eng Phys Thermophy 93, 60–73 (2020). https://doi.org/10.1007/s10891-020-02091-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10891-020-02091-x

Keywords

Navigation