Advertisement

Journal of Mathematical Chemistry

, Volume 53, Issue 3, pp 911–924 | Cite as

DRBEM solution of natural convective heat transfer with a non-Darcy model in a porous medium

  • B. Pekmen
  • M. Tezer-Sezgin
Original Paper

Abstract

This study presents the dual reciprocity boundary element (DRBEM) solution of Brinkman–Forchheimer-extended Darcy model in a porous medium containing an incompressible, viscous fluid. The governing dimensionless equations are solved in terms of stream function, vorticity and temperature. The problem geometry is a unit square cavity with either partially heated top and bottom walls or hot steps at the middle of these walls. DRBEM provides one to obtain the expected behavior of the flow in considerably small computational cost due to the discretization of only the boundary, and to compute the space derivatives in convective terms as well as unknown vorticity boundary conditions using coordinate matrix constructed by radial basis functions. The Backward-Euler time integration scheme is utilized for the time derivatives. The decrease in Darcy number suppresses heat transfer while heat transfer increases for larger values of porosity, and the natural convection is pronounced with the increase in Rayleigh number.

Keywords

DRBEM Natural convection Porous medium  Brinkman–Forchheimer-extended Darcy model 

Mathematics Subject Classification

76S99 76R10 76D05 65M38 

References

  1. 1.
    K. Al-Farhany, A. Turan, Non-Darcy effects on conjugate double-diffusive natural convection in a variable porous layer sandwiched by finite thickness walls. Int. J. Heat Mass Transf. 54, 2868–2879 (2011)CrossRefGoogle Scholar
  2. 2.
    C. Beckermann, S. Ramadhyani, R. Viskanta, Natural convection flow and heat transfer between a fluid layer and a porous layer inside a rectangular enclosure. Trans. ASME J. Heat Transf. 109, 363–370 (1987)CrossRefGoogle Scholar
  3. 3.
    M. Bhuvaneswari, S. Sivasankaran, Y.J. Kim, Effect of aspect ratio on convection in porous enclosure with partially active thermal walls. Comput. Math. Appl. 62, 3844–3856 (2011)CrossRefGoogle Scholar
  4. 4.
    X.B. Chen, P. Yu, S.H. Winoto, H.T. Low, Free convection in a porous wavy cavity based on the Darcy–Brinkman–Forchheimer extended model. Numer. Heat Transf. Part A 52, 377–397 (2007)CrossRefGoogle Scholar
  5. 5.
    S. Das, R.K. Sahoo, Effect of Darcy, fluid Rayleigh and heat generation parameters on natural convection in a porous square enclosure: a Brinkman-extended Darcy model. Int. Commun. Heat Mass Transf. 26, 569–578 (1999)CrossRefGoogle Scholar
  6. 6.
    R. Jecl, L. Skerget, Boundary element method for natural convection in non-Newtonian fluid saturated square porous cavity. Eng. Anal. Bound. Elem. 27, 963–975 (2003)CrossRefGoogle Scholar
  7. 7.
    M. Karimi-Fard, M.C. Charrier-Mojtabi, K. Vafai, Non-Darcian effects on double-diffusive convection within a porous medium. Numer. Heat Transf. Part A 31, 837–852 (1997)CrossRefGoogle Scholar
  8. 8.
    K.M. Khanafer, A.J. Chamkha, Mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium. Int. J. Heat Mass Transf. 42, 2465–2481 (1999)CrossRefGoogle Scholar
  9. 9.
    K. Khanafer, B. Al-Azmi, A. Marafie, I. Pop, Non-Darcian effects on natural convection heat transfer in a wavy porous enclosure. Int. J. Heat Mass Transf. 52, 1887–1896 (2009)CrossRefGoogle Scholar
  10. 10.
    S.A. Khashan, A.M. Al-Amiri, I. Pop, Numerical simulation of natural convection heat transfer in a porous cavity heated from below using a non-Darcian and thermal non-equilibrium model. Int. J. Heat Mass Transf. 49, 1039–1049 (2006)CrossRefGoogle Scholar
  11. 11.
    D.S. Kumar, A.K. Dass, A. Dewan, Analysis of Non-Darcy models for mixed convection in a porous cavity using a multigrid approach. Numer. Heat Transf. Part A 56, 685–708 (2009)CrossRefGoogle Scholar
  12. 12.
    P.A.K. Lam, K.A. Prakash, A numerical study on natural convection and entropy generation in a porous enclosure with heat sources. Int. J. Heat Mass Transf. 69, 390–407 (2014)CrossRefGoogle Scholar
  13. 13.
    H.Y. Li, K.C. Leong, L.W. Jin, J.C. Chai, Analysis of fluid flow and heat transfer in a channel with staggered porous blocks. Int. J. Therm. Sci 49, 950–962 (2010)CrossRefGoogle Scholar
  14. 14.
    A. Narasimhan, Essentials of Heat and Fluid Flow in Porous Media (CRC Press; Taylor & Francis Group, 2013)Google Scholar
  15. 15.
    D.A. Nield, A. Bejan, Convection in Porous Media (Springer, Berlin, 2006)Google Scholar
  16. 16.
    P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Natural convective heat transfer in a fluid saturated variable porosity medium. Int. J. Heat Mass Transf. 40, 3955–3967 (1997)CrossRefGoogle Scholar
  17. 17.
    P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Effect of porosity on natural convective heat transfer in a fluid saturated porous medium. Int. J. Heat Fluid Flow 19, 56–58 (1998)CrossRefGoogle Scholar
  18. 18.
    P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Finite element modelling of flow, heat and mass transfer in fluid saturated porous media. Arch. Comput. Methods Eng. 9, 3–42 (2002)CrossRefGoogle Scholar
  19. 19.
    W. Pakdee, P. Rattanadecho, Unsteady effects on natural convective heat transfer through porous media in cavity due to top surface partial convection. Int. J. Heat Mass Transf. 26, 2316–2326 (2006)Google Scholar
  20. 20.
    P.W. Partridge, C.A. Brebbia, L.C. Wrobel, The Dual Reciprocity Boundary Element Method (Computational Mechanics Publications, Elsevier, Southampton, London, 1992)Google Scholar
  21. 21.
    B.V. Rathish Kumar, Shalini, Free convection in a non-Darcian wavy porous enclosure. Int. J. Eng. Sci 41, 1827–1848 (2003)CrossRefGoogle Scholar
  22. 22.
    H. Saleh, A.Y.N. Alhashash, I. Hashim, Rotation effects in an enclosure filled with porous medium. Int. Commun. Heat Mass Transf. 43, 105–111 (2013)CrossRefGoogle Scholar
  23. 23.
    B. Sarler, J. Perko, D. Gobin, B. Goyeau, H. Power, Dual reciprocity boundary element method solution of natural convection in Darcy–Brinkman porous media. Eng. Anal. Bound. Elem. 28, 23–41 (2004)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsAtilim UniversityAnkaraTurkey
  2. 2.Department of Mathematics and Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey

Personalised recommendations