Wärme - und Stoffübertragung

, Volume 22, Issue 1–2, pp 69–77 | Cite as

Natural convection in an inclined rectangular porous cavity with uniform heat flux from the side

  • P. Vasseur
  • L. Robillard
  • I. Anochiravani


The problem of natural convection in an inclined rectangular porous layer enclosure is studied numerically. The enclosure is heated from one side and cooled from the other by a constant heat flux while the two other walls are insulated. The effect of aspect ratio, inclination angle and Rayleigh number on heat transfer is studied. It is found that the enclosure orientation has a considerable effect on the heat transfer. The negative orientation sharply inhibits the convection and consequently the heat transfer and a positive orientation maximizes the energy transfer. The maximum temperature within the porous medium can be considerably higher than that induced by pure conduction when the cavity is negatively oriented. The peak of the average Nusselt number depends on the Rayleigh number and the aspect ratio. The heat transfer between the two thermally active boundaries is sensitive to the effect of aspect ratio. For an enclosure at high or low aspect ratio, the convection is considerably decreased and the heat transfer depends mainly on conduction.


Heat Transfer Heat Flux Aspect Ratio Nusselt Number Natural Convection 
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.



aspect ratio of the cavity,H′/ L′


gravitational acceleration, m/s2


thickness of the cavity, m


effective thermal conductivity, W/m-°K


permeability of porous material, m2


length of the cavity, m


Nusselt number, (q′/ΔT′)L′/k


heat flux, W/m2


Rayleigh number,g β KH′2 q′/α v k


time, s


temperature, °K


reference temperature, °K


characteristic temperature difference,q′ L′/k


horizontal dimensionless velocity


vertical dimensionless velocity


horizontal dimensionless coordinate


vertical dimensionless coordinate

Greek symbols


effective thermal diffusivity, m2/s


coefficient of thermal expansion, °K−1


kinematic viscosity, m2/s


angle of inclination of enclosure, rad


density, Kg/m3


dimensionless stream function


dimensionless time



dimensional quantities

average value



maximum value

Naturkonvektion in einem geneigten, rechteckigen, porösen Hohlraum mit gleichmäßigem seitlichen Wärmefluß


Numerisch wird das Problem der Naturkonvektion in einer geneigten, rechteckigen, porösen, eingeschlossenen Schicht studiert. Der Raum ist auf der einen Seite beheizt und auf der anderen durch konstanten Wärmeabzug gekühlt, während die beiden anderen Wände isoliert sind. Es wird der Einfluß des Längen-Seiten-Verhältnisses, des Neigungswinkels und der Rayleigh-Zahl auf den Wärmeübergang studiert. Es stellte sich heraus, daß die Neigung des Hohlraumes beträchtlichen Einfluß auf den Wärmeübergang hat. Negative Orientierung behindert die Konvektion stark — und damit auch den Wärmetransport — und eine positive Neigung führt zu einem Maximum im Wärmetransport. Bei negativer Orientierung kann die maximale Temperatur in dem porösen Medium beträchtlich höher sein als wie sie eine reine Wärmeleitung hervorrufen würde. Das Maximum in der über die Fläche gemittelten Nusselt-Zahl hängt von der Rayleigh-Zahl und dem Längen-Seiten-Verhältnis ab. Der Wärmetransport zwischen den beiden thermisch aktiven Grenzen ist empfindlich auf den Einfluß des Längen-Seiten-Verhältnisses. Für große oder kleine Seitenverhältnisse nimmt die Konvektion stark ab und der Wärmetransport beruht hauptsächlich auf Leitung.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bejan, A.: On the boundary layer regime in a vertical enclosure filled with a porous medium. Lett. Heat Mass Transfer 6 (1979) 93–102Google Scholar
  2. 2.
    Weber, J. E.: The boundary layer regime for convection in a vertical porous layer. Int. J. Heat Mass Transfer 18 (1975) 569–673Google Scholar
  3. 3.
    Chan, B. K. C.; Irey, C. M.; Barry, J. M.: Natural convection in enclosed porous media with rectangular boundaries. ASME J. Heat Transfer 2 (1970) 21–27Google Scholar
  4. 4.
    Hichox, C. E.; Gartling, D. K.: A numerical study of natural convection in a horizontal porous layer subjected to and end-to-end temperature difference. ASME J. Heat Transfer 103 (1981) 797–802Google Scholar
  5. 5.
    Burns, P. J.; Chow, L. C.; Tien, C. L.: Convection in a vertical slot filled with porous insulation. Int. J. Heat Mass Transfer 20 (1976) 919–926Google Scholar
  6. 6.
    Shiralhar, G. S.; Haajizadeh, M.; Tien, C. L.: Numerical study of high Rayleigh number enclosure in a vertical porous enclosure. Numer. Heat Transfer 6 (1983) 223–234Google Scholar
  7. 7.
    Poulikakos, D.; Bejan, A.: Natural convection in vertically and horizontally layered porous media heated from the side. Int. J. Heat Mass Transfer 26 (1983) 1805–1814Google Scholar
  8. 8.
    Blythe, P. A.; Simpkins, P. G.; Daniels, P. G.: Thermal convection in a cavity filled with a porous medium: a classification of limiting behaviours. Int. J. Heat Mass Transfer 26 (1983) 701–708Google Scholar
  9. 9.
    Blythe, P. A.; Daniels, P. G.; Simpkins, P. G.: Thermally driven cavity flows in porous Media. I. The vertical boundary layer structure near the corners. Proc. Royal Society London, A 380 (1982) 119–136Google Scholar
  10. 10.
    Seki, N.; Fukusaho, S.; Inaba, H.: Heat transfer in a confined rectangular cavity packed with porous media. Int. J. Heat Mass Transfer 21 (1978) 985–989Google Scholar
  11. 11.
    Hoist, P. H.; Aziz, K.: A theoretical and experimental study of natural convection in a confined porous medium. Canad. J. Chem. Eng. 50 (1972) 232–241Google Scholar
  12. 12.
    Klarsfeld, S.: Champs de température associés au mouvement de convection naturelle dans un milieu poreux limité. Rev. Gen. Therm. 9 (1970) 1403–1424Google Scholar
  13. 13.
    Bories, S. A.; Combarnous, M. A.: Natural convection in a sloping porous layer. J. Fluid Mech. 57 (1973) 63–79Google Scholar
  14. 14.
    Burns, P. J.; Tien, C. L.: Natural convection in porous media bounded by concentric spheres and horizontal cylinders. Int. J. Heat Mass Transfer 22 (1979) 929–939Google Scholar
  15. 15.
    Bejan, A.: Natural convection in a vertical cylinder filled with porous medium. Int. J. Heat Mass Transfer 23 (1980) 726–729Google Scholar
  16. 16.
    Bejan A.: The boundary layer regime in a porous layer with uniform heat flux from the side. Int. J. Heat Mass Transfer 26 (1983) 1339–1346Google Scholar
  17. 17.
    Prasad, V.; Kulacki, F. A.: Natural convection in a rectangular porous cavity with constant heat flux on one vertical wall. ASME J. Heat Transfer 106 (1984) 152–157Google Scholar
  18. 18.
    Caltagirone, J. P.; Bories, S.: Solutions and stability criteria of natural convective flow in an inclined porous layer. J. Fluid Mech. 155 (1985) 267–287Google Scholar
  19. 19.
    Vlasuk, M. P.; Convection heat transfer in a porous layer. All-Union Heat Mass Transfer Conf., 4th, Minsk., (1972)Google Scholar
  20. 20.
    Ozoe, H.; Sayama, H.; Churchill, S. W.: Natural convection in an inclined rectangular channel at various aspect ratios and angles-experimental measurements. Int. J. Heat Mass Transfer 18 (1975) 1425–1431Google Scholar
  21. 21.
    Roache, P.: Computational fluid dynamics. Hermosa 1982Google Scholar
  22. 22.
    Peaceman, D. W.; Rachford, H. A.: The numerical solution of parabolic and elliptic difference equations. J. Society Indust. Applied Mathematics 3 (1955) 28–43Google Scholar
  23. 23.
    Robillard, L.; Nguyen, T. H.; Vasseur, P.: Free convection in a two dimensional loop. ASME J. Heat Transfer 108 (1986) 277–283Google Scholar
  24. 24.
    Haajizadeh, A.; Ozgug, A. F.; Tien, C. L.: Natural convection in a vertical porous enclosure with internal heat generation. Int. J. Heat Mass Transfer 27 (1984) 1893–1902Google Scholar
  25. 25.
    Vasseur, P.; Nguyen, T. H.; Robillard, L.; Tong, T. V. K.: Natural convection between horizontal concentric cylinders filled with a porous layer with internal heat generation. Int. J. Heat Mass Transfer 27 (1984) 337–349Google Scholar
  26. 26.
    Vasseur, P.; Robillard, L.: The Brinkman model for boundary layer regime in a rectangular cavity with uniform heat flux from the side. Int. J. Heat Mass Transfer 30 (1987) 717–727Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • P. Vasseur
    • 1
  • L. Robillard
    • 1
  • I. Anochiravani
    • 1
  1. 1.Department of Mechanical EngineeringEcole PolytechniqueMontrealCanada

Personalised recommendations