Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T18:36:15.243Z Has data issue: false hasContentIssue false
  • English
  • Français

The Pe ~ 1 regime of convection across a horizontal permeable membrane

Published online by Cambridge University Press:  17 May 2011

G. V. RAMA REDDY  and
BABURAJ A. PUTHENVEETTIL
G. V. RAMA REDDY*
Affiliation:
Heat Transfer and Fluid Flow, Corporate R&D, Bharat Heavy Electricals Ltd., Hyderabad 500093, India
BABURAJ A. PUTHENVEETTIL
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, India
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In natural convection, driven by an unstable density difference due to a heavier fluid (brine) above a lighter fluid (water) across a horizontal permeable membrane, we discover a new regime of convection, where the Sherwood number (Sh) scales approximately as the Rayleigh number (Ra). Inferring from the planforms of plume structure on the membrane and the estimates of velocity through the membrane, we show that such a regime occurs when advection balances diffusion in the membrane, i.e. the Péclet number based on the membrane thickness (Pe) is of order one. The advection is inferred to be caused by the impingement of the large-scale flow on the membrane. Utilizing mass balance and symmetry assumptions in the top and the bottom fluids, we derive an expression for the concentration profile in the membrane pore in the new regime by solving the convection–diffusion equation in the membrane pore; this helps us to obtain the concentration drops above and below the membrane that drive the convection. We find that the net flux, normalised by the diffusive flux corresponding to the concentration drop on the side opposite to the impingement of the large-scale flow remains constant throughout the new regime. On the basis of this finding, we then obtain an expression for the flux scaling in the new regime which matches with the experiments; the expression has the correct asymptotes of flux scaling in the advection and the diffusion regimes. The plume spacings in the new regime are distributed lognormally, and their mean follows the trend in the advection regime.

JFM classification

Convection: Buoyancy-driven instability Convection: Plumes/thermals Turbulent Flows: Turbulent convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 679 , 25 July 2011 , pp. 476 - 504
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adrian, R. J., Ferreira, R. T. D. S. & Boberg, T. 1986 Turbulent thermal convection in wide horizontal fluid layers. Exp. Fluids 4, 121–141.CrossRefGoogle Scholar
Deardorff, J. W. 1970 Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci. 27, 1211–1213.2.0.CO;2>CrossRefGoogle Scholar
Dworecki, K., Slezak, A., Ornal-Wasik, B. & Wasik, S. 2005 Effect of hydrodynamic instabilities on solute transport in a membrane system. J. Membr. Sci. 263, 94–100.CrossRefGoogle Scholar
Goldstein, R. J., Chiang, H. D. & See, D. L. 1990 High Rayleigh number convection in a horizontal enclosure. J. Fluid. Mech. 213, 111–126.CrossRefGoogle Scholar
Graham, W. J. & David, F. J. 1986 The permeability of fibrous porous media. Can. J. Chem. Engng 64 (3), 364–374.Google Scholar
Kerr, R. M. 1996 Rayleigh number scaling in numerical convection. J. Fluid. Mech. 310, 139–179.CrossRefGoogle Scholar
Krishnamurti, R. 1970 On the transition to turbulent convection. J. Fluid. Mech. 42, 295–307.CrossRefGoogle Scholar
Lide, D. R. 2003 CRC Handbook of Chemistry and Physics, 84th edn., chap. 5. CRC.Google Scholar
Puthenveettil, B. A. 2004 Investigations on high Rayleigh number turbulent free convection. PhD thesis, Indian Institute of Science, Bangalore, http://etd.ncsi.iisc.ernet.in/handle/2005/140.Google Scholar
Puthenveettil, B. A., Ananthakrishna, G. & Arakeri, J. H. 2005 The multifractal nature of plume structure in high-Rayleigh-number convection. J. Fluid. Mech. 526, 245–256.CrossRefGoogle Scholar
Puthenveettil, B. A. & Arakeri, J. H. 2005 Plume structure in high-Rayleigh-number convection. J. Fluid. Mech. 542, 217–249.CrossRefGoogle Scholar
Puthenveettil, B. A. & Arakeri, J. H. 2008 Convection due to an unstable density gradient across a permeable membrane. J. Fluid. Mech. 609, 139–170.CrossRefGoogle Scholar
Radiometer Analytical SAS 2006 Conductivity Theory and Practice Manual. Radiometer Analytical.Google Scholar
Rama Reddy, G. V. 2009 Investigations on convection across a horizontal permeable membrane. Master's thesis, Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai.Google Scholar
Rotem, Z. & Classen, L. 1969 Natural convection above unconfined horizontal surfaces. J. Fluid. Mech. 39 (1), 173–192.CrossRefGoogle Scholar
Slezak, A., Dworecki, K. & Anderson, J. E. 1985 Gravitational effects on transmembrane flux: The Rayleigh–Taylor convective instability. J. Membr. Sci. 23, 71–81.CrossRefGoogle Scholar
Slezak, A., Grzegorczyn, S., Jasik-Slezak, J. & Michalska-Malecka, K. 2010 Natural convection as an asymmetrical factor of the transport through porous membrane. Transp. Porous Med., 84 (3), 685–698.CrossRefGoogle Scholar
Sparrow, E. & Husar, R. 1969 Longitudinal vortices in natural convection flow on inclined plates. J. Fluid. Mech. 37, 251–255.CrossRefGoogle Scholar
Theerthan, S. A. & Arakeri, J. H. 1994 Planform structure of turbulent Rayleigh–Bénard convection. Intl Commun. Heat Mass Transfer 21, 561–572.CrossRefGoogle Scholar
Theerthan, S. A. & Arakeri, J. H. 1998 A model for near wall dynamics in turbulent Rayleigh–Bénard convection. J. Fluid. Mech. 373, 221–254.CrossRefGoogle Scholar
Theerthan, S. A. & Arakeri, J. H. 2000 Planform structure and heat transfer in turbulent free convection over horizontal surfaces. Phys. Fluids 12, 884–894.CrossRefGoogle Scholar
Townsend, A. 1959 Temperature fluctuations over a heated horizontal surface. J. Fluid. Mech. 5, 209–211.CrossRefGoogle Scholar
Xia, K. Q., Lam, S. & Zhou, S. Q. 2002 Heat flux measurement in high-Prandtl-number turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 88 (6), 064501.CrossRefGoogle ScholarPubMed

Rama Reddy et al. supplementary movie

Planform view of the dynamics of the sheet plumes just above the membrane in the Pe~1 regime at Ra=4.6x10^11 and Pr=601. The area of view is 15cm x 15cm

Download Rama Reddy et al. supplementary movie(Video)
Video 8.6 MB

Rama Reddy et al. supplementary movie

Planform view of the dynamics of the sheet plumes just above the membrane in the Pe~1 regime at Ra=4.6x10^11 and Pr=601. The area of view is 15cm x 15cm

Download Rama Reddy et al. supplementary movie(Video)
Video 9.5 MB