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Transport in Porous Media

, Volume 76, Issue 3, pp 363–375 | Cite as

Experimental Study on Oscillatory Natural Convection in a Hele-Shaw Cell due to Unstably Heated Side

  • A. H. AbdelkareemEmail author
  • S. Kimura
  • T. Kiwata
  • N. Komatsu
Article

Abstract

The oscillatory motion of natural convection in a porous medium has been investigated experimentally using a Hele-Shaw cell technique. The cell has been heated on the lower half and cooled on the upper half along the same vertical sidewall. Flows have been visualized using the pH indicator method. Photographs of natural convection patterns as well as average Nusselt number data have been presented for different Rayleigh numbers. Oscillatory motion of natural convection has been observed for large enough Rayleigh numbers and the critical Rayleigh number has been estimated to be between 120 and 450. Scaling analysis has been conducted to understand the heat transfer and the oscillating mechanism. According to the scaling analysis, it has been found that the average Nusselt number is proportional to the square root of the Rayleigh number, and that the oscillation frequency is proportional to the Rayleigh number. Obtained experimental data support the scaling analysis.

Keywords

Natural convection Porous media Geothermal energy Hele-Shaw cell Oscillatory convection 

Nomenclature

A

Cross section area of the cell

C

Specific heat

d

Cell gap space

f

Oscillation frequency

g

Gravitational acceleration

H

Cell height

K

Permeability

k

Thermal conductivity

Nu

Nusselt number

n

Positive Integer in Eq. 6

q

Heat transfer

r

Ratio of the clamp thickness to the contact length with the copper plate; a/b

Ra

Rayleigh number

w

Width of the acrylic clamps

T

Temperature

t

Time

Y

Cell width

Greek symbols

α

Thermal diffusivity of porous medium

β

Thermal expansion coefficient

δ

Vertical boundary layer thickness

ε

Thermal boundary layer thickness

ν

Kinematic viscosity of the fluid

ρ

Density of the fluid

τ

Dimensionless time

Subscripts

f

Fluid

w

Wall

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References

  1. Aidun C.K., Steen P.H.: Transition to oscillatory heat transfer in a fluid-saturated porous medium. J. Thermophys. Heat Transf. 1, 268–273 (1987)CrossRefGoogle Scholar
  2. Babushkin I.A., Demin V.A.: Experimental and theoretical investigation of transient convective regimes in a Hele-Shaw cell. Fluid Dyn. 41, 323–329 (2006)CrossRefGoogle Scholar
  3. Babushkin I.A., Demin V.A.: Capture of convective vortices by an unsteady-state flow in a Hele-Shaw cell. J. Eng. Phys. Thermophys. 80, 107–113 (2007)CrossRefGoogle Scholar
  4. Baker D.J.: A technique for the precise measurement of small fluid velocities. J. Fluid Mech. 26, 573–575 (1966)CrossRefGoogle Scholar
  5. Bejan, A.: Chap. 15 Porous media. In: Bejan, A., Kraus, A. D. (eds.) Heat Transfer Hand Book, pp. 1131–1180. Wiley (2003)Google Scholar
  6. Caltagirone J.P.: Thermoconvective instabilities in horizontal porous layers. J. Fluid Mech. 72, 269–287 (1975)CrossRefGoogle Scholar
  7. Cheng P.: Heat transfer in geothermal systems. Adv. Heat Transf. 14, 1–105 (1978)Google Scholar
  8. Hartline B.K., Lister C.R.B.: Thermal convection in a Hele-Shaw cell. J. Fluid Mech. 79, 379–389 (1977)CrossRefGoogle Scholar
  9. Horne R.N., O’Sullivan M.J.: Oscillatory convection in a porous medium heated from below. J. Fluid Mech. 66, 339–352 (1974)CrossRefGoogle Scholar
  10. Hossain M.A., Rees D.A.S.: Natural convection flow of a viscous incompressible fluid in a rectangular porous cavity heated from below with cold sidewalls. Heat Mass Transf. 39, 657–663 (2003)CrossRefGoogle Scholar
  11. Ingham, D.B., Pop, I. (eds.): Transport Phenomena in Porous Media III. Elsevier (2005)Google Scholar
  12. Kimura S., Schubert S., Straus J.M.: Route to chaos in porous medium thermal convection. J. Fluid Mech. 166, 305–325 (1986)CrossRefGoogle Scholar
  13. Kimura S., Schubert S., Straus J.M.: Instabilities of steady, periodic and quasi-periodicand modes of convection in porous media. J. Heat Transf. 109, 350–355 (1987)CrossRefGoogle Scholar
  14. Kimura, S., Okajima, A., Kiwata, T.: Natural convection in a saturated porous section due to an unstably heated side wall. In: Tyvand, P.A. (ed.) Time-dependent Nonlinear Convection, pp. 115–138. Computational Mechanics Publications (1998)Google Scholar
  15. Kimura S., Okajima A., Kiwata T.: Natural convection heat transfer in an anisotropic porous cavity heated from side (2nd Report, Experimental using a Hele-Shaw cell. Heat Transf. Asian Res. 31(6), 463–474 (2002)CrossRefGoogle Scholar
  16. Küblbeck K., Merker G.P., Straub J.: Advanced numerical computation of two-dimensional time-dependent free convection in cavities. Int. J. Heat Mass Transf. 23, 203–217 (1980)CrossRefGoogle Scholar
  17. Moffat R.J.: Describing uncertainties in experimental results. Exp. Thermal Fluid Sci. 1, 3–17 (1988)CrossRefGoogle Scholar
  18. Nield, D.A., Bejan, A.: Convection in Porous Media. Springer (2006)Google Scholar
  19. Poulikakos D., Bejan A.: Natural convection in a porous layer heated and cooled along one vertical side. Int. J. Heat Mass Transf. 27(10), 1879–1891 (1984)CrossRefGoogle Scholar
  20. Schubert G., Straus J.M.: Transitions in time-dependent thermal convection in fluid-saturated porous media. J. Fluid Mech. 121, 301–313 (1982)CrossRefGoogle Scholar
  21. Turcotte, D.L., Schubert, G.: Geodynamics. Wiley (1982)Google Scholar
  22. Yoon D., Kim M.C., Choi C.K.: The onset of oscillatory convection in a horizontal porous layer saturated with viscoelastic liquid. Transp. Porous Media 55, 275–284 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • A. H. Abdelkareem
    • 1
    Email author
  • S. Kimura
    • 2
  • T. Kiwata
    • 2
  • N. Komatsu
    • 2
  1. 1.Department of Mechanical Power EngineeringEl-Minia UniversityEl-MiniaEgypt
  2. 2.Department of Mechanical Systems EngineeringKanazawa UniversityKanazawaJapan

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