Heat and Mass Transfer

, 43:907 | Cite as

Conjugate heat transfer in square enclosures

  • Fu-Yun Zhao
  • Di Liu
  • Guang-Fa TangEmail author


Building elements represented by square vertical enclosures encircled with finite walls or with centered solid body, could maintain the equivalent fluid volumes through the volume ratio scale. Present work aims to investigate the fluid flow and heat transfer in these two building elements. Complete two-dimensional numerical simulation of the conjugate heat conduction and natural convection occurring in both enclosures is carried out. An analytical expression for the minimum size of the inserted body at which the body begins to suppress the natural convection flow is proposed and validated by the numerical results. The fluid flow and heat transfer characteristics are analyzed through the streamlines, heatlines, and total heat transfer rates across both enclosures. Results reveal that heat transfer rates across both enclosures are complex functions of the volume ratio scale, Rayleigh number, and the relative thermal conductivity.


Heat Transfer Nusselt Number Natural Convection Rayleigh Number Heat Transfer Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Symbols


isobaric specific heat (J/K kg)


gravitational acceleration (m/s2)


heat function


thermal conductivity (W/m K)


thermal conductivity ratio


thermal conductivity ratio of solid to fluid


length of square enclosure (m)


normal direction


local Nusselt number


pressure (N/m2)


dimensionless pressure


Prandtl number, ν/α


total heat transfer rate (W/m2)


Rayleigh number, gβ(t ht c)L 3/να


distance (m)


temperature (K)


dimensionless temperature


velocity component in x directions (m/s)


velocity component in y directions (m/s)


dimensionless velocity component in X


dimensionless velocity component in Y


length of inner cavity or body (m)

x, y

Cartesian coordinates (m)

X, Y

dimensionless Cartesian coordinates

Greek Symbols


thermal diffusivity (m2/s)


volume expansion coefficient (1/K)


volume ratio scale (°)


void fraction of enclosure A


solid-to-fluid volume ratio of enclosure B


kinematics viscosity (m2/s)


density (kg/m3)


stream function



average value


enclosure encircled with finite walls


enclosure inserted with solid body


cold wall


value of the fluid domain


hot wall






value of the solid domain






The authors gratefully acknowledge the financial support of National Natural Science Foundation of China (No. 50578059). The authors are also grateful to the anonymous referees who provided detailed and constructive comments.


  1. 1.
    Kim DM, Viskanta R (1984) Study of the effects of wall conductance on natural convection in differently oriented square cavities. J Fluid Mech 144:153–176CrossRefGoogle Scholar
  2. 2.
    Kim DM, Viskanta R (1985) Effect of wall heat conduction on natural convection heat transfer in a square enclosure. J Heat Transfer 107:107–146CrossRefGoogle Scholar
  3. 3.
    Costa VAF (1997) Double diffusive natural convection in a square enclosure with heat and mass diffusive walls. Int J Heat Mass Transfer 40:4061–4071zbMATHCrossRefGoogle Scholar
  4. 4.
    Costa VAF (1999) Unification of the streamline, heatline and massline methods for the visualization of two-dimensional transport phenomena. Int J Heat Mass Transfer 42:27–33zbMATHCrossRefGoogle Scholar
  5. 5.
    Baytas AC, Liaqat A, Grosan T, Pop I (2001) Conjugate natural convection in a square porous cavity. Heat Mass Transfer 37:467–473CrossRefGoogle Scholar
  6. 6.
    Ben Yedder R, Bilgen E (1995) Turbulent natural convection and conduction in enclosures bounded by a massive wall. Int J Heat Mass Transfer 38:1879–1891zbMATHCrossRefGoogle Scholar
  7. 7.
    Misra D, Sarkar A (1997) Finite element analysis of conjugate natural convection in a square enclosure with a conducting vertical wall. Comput Methods Appl Mech Eng 141:205–219zbMATHCrossRefGoogle Scholar
  8. 8.
    Sun YS, Emery AF (1997) Effects of wall conduction, internal heat sources and an internal baffle on natural convection heat transfer in a rectangular enclosure. Int J Heat Mass Transfer 40:915–929zbMATHCrossRefGoogle Scholar
  9. 9.
    Costa VAF (2002) Laminar natural convection in differentially heated rectangular enclosures with vertical diffusive walls. Int J Heat Mass Transfer 45:4217–4225zbMATHCrossRefGoogle Scholar
  10. 10.
    House JM, Beckermann C, Smith TF (1990) Effect of a centered conducting body on natural convection heat transfer in an enclosure. Numer Heat Transfer, Part A 18:213–225Google Scholar
  11. 11.
    Oh JY, Ha MY, Kim KC (1997) Numerical study of heat transfer and flow of natural convection in an enclosure with a heat-generating conducting body. Numer Heat Transfer, Part A 31:289–304Google Scholar
  12. 12.
    Ha MY, Jung MJ, Kim YS (1999) Numerical study on transient heat transfer and fluid flow of natural convection in an enclosure with a heat-generating conducting body. Numer Heat Transfer, Part A 35:415–433CrossRefGoogle Scholar
  13. 13.
    Ha MY, Jung MJ (2000) A numerical study on three-dimensional conjugate heat transfer of natural convection and conduction in a differentially heated cubic enclosure with a heat-generating cubic conducting body. Int J Heat Mass Transfer 43:4229–4248zbMATHCrossRefGoogle Scholar
  14. 14.
    Deng QH, Tang GF (2002) Numerical visualization of mass and heat transport for conjugate natural convection/heat conduction by streamline and heatline. Int J Heat Mass Transfer 45:2373–2385zbMATHCrossRefGoogle Scholar
  15. 15.
    Merrikh AA, Lage JL (2005) Natural convection in an enclosure with disconnected and conducting solid blocks. Int J Heat Mass Transfer 48:1361–1372Google Scholar
  16. 16.
    Zhao FY, Zhang L, Tang GF, Lu JL (2003) Numerical simulation of airflow in partitioned enclosures at high Rayleigh numbers. In: Proceedings of energy and environment 2003 conference, Science Press, Beijing, pp 139–143Google Scholar
  17. 17.
    Zhao FY, Tang GF, Liu D (2006) Conjugate natural convection in enclosures with external and internal heat sources. Int J Eng Sci 44:148–165CrossRefGoogle Scholar
  18. 18.
    Zhao FY (2003) Numerical simulation of thermal environment in urban residential district. M.Sc. Thesis, Hunan University, Changsha, ChinaGoogle Scholar
  19. 19.
    Bejan A (1995) Convection heat transfer, 2nd edn. Wiley, New YorkGoogle Scholar
  20. 20.
    Ben Yedder R, Bilgen E (1997) Laminar natural convection in inclined enclosures bounded by a solid wall. Heat Mass Transfer 32:455–462CrossRefGoogle Scholar
  21. 21.
    Zhao FY, Tang GF, Liu D (2005) Mixed convection and conjugate heat transfer in multi air ducts of thermoelectric refrigerator. HV&AC 35:12–17Google Scholar
  22. 22.
    Yucel N, Ozdem AH (2003) Natural convection in partially divided square enclosures. Heat Mass Transfer 40:167–175CrossRefGoogle Scholar
  23. 23.
    Turkoglu H, Yucel N (1996) Natural convection heat transfer in enclosures with conducting multiple partitions and side walls. Heat Mass Transfer 32:1–8CrossRefGoogle Scholar
  24. 24.
    Patankar SV (1980) Numerical heat transfer and fluid flow. McGraw-Hill, New YorkzbMATHGoogle Scholar
  25. 25.
    Thakur S, Shyy W (1993) Some implementational issues of convection schemes for finite-volume formulations. Numer Heat Transfer, Part B 24:31–55Google Scholar
  26. 26.
    Kalita JC, Dalal DC, Dass AK (2001) Fully compact higher-order computation of steady-state natural convection in a square cavity. Phys Rev E 64(066703):1–13Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.College of Civil EngineeringHunan UniversityChangshaPeople’s Republic of China

Personalised recommendations