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The Conduction and Convection Regimes in a Cavity with Evenly Heated and Cooled Vertical Walls

DOI: 10.1615/ICHMT.2005.AustHeatMassTransfConf.370
4 pages

Geordie D. McBain
School of Aerospace, Mechanical, & Mechatronic Engineering, The University of Sydney,

Steven W. Armfield
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia

S. Jiracheewanun
School of Aerospace, Mechanical, & Mechatronic Engineering, The University of Sydney,

Abstract

We treat natural convection in a plane vertical rectangular cavity with one vertical wall evenly heated, the other cooled, and an adiabatic floor and ceiling. We predict the temperature difference across the cavity by simple analysis, and present numerical solutions verifying the results.
If the cavity is infinitely tall, there is an exact steady one-dimensional solution depending on a single parameter: the stratification. This quantity is not prescribed by the boundary conditions, but can be determined analytically from an energy balance on a suitable control volume. This allows the calculation of the cross-cavity temperature difference (the Nusselt number−Rayleigh number relation) over the entire laminar range of Rayleigh numbers, for all Prandtl numbers, and for all sufficiently large aspect ratios. The result links the trivial conduction limit to the Kimura−Bejan boundary layer approximation, which is also shown to be asymptotically correct.
Numerical solutions verify the conduction−convection transition for the Nusselt number. They also indicate the dependence on the parameters of the problem (Rayleigh and Prandtl numbers) of the bound on the aspect ratio for the applicability of the approach: that is, the minimum separation of ceiling from floor which permits a fully developed region.

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