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Research Papers: Natural and Mixed Convection

Heat Transfer Due to Natural Convection in a Periodically Heated Slot

[+] Author and Article Information
J. M. Floryan

e-mail: mfloryan@eng.uwo.ca
Department of Mechanical and
Materials Engineering,
The University of Western Ontario,
London, ON, N6A 5B9, Canada

1Current address: Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET), Dhaka-1000, Bangladesh.

2Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received November 28, 2011; final manuscript received August 14, 2012; published online December 28, 2012. Assoc. Editor: Sujoy Kumar Saha.

J. Heat Transfer 135(2), 022503 (Dec 28, 2012) (11 pages) Paper No: HT-11-1539; doi: 10.1115/1.4007420 History: Received November 28, 2011; Revised August 14, 2012

Heat transfer resulting from the natural convection in a fluid layer contained in an infinite horizontal slot bounded by solid walls and subject to a spatially periodic heating at the lower wall has been investigated. The heating produces sinusoidal temperature variations along one horizontal direction characterized by the wave number α with the amplitude expressed in terms of a suitably defined Rayleigh number Rap. The maximum heat transfer takes place for the heating with the wave numbers α = 0(1) as this leads to the most intense convection. The intensity of convection decreases proportionally to α when α→0, resulting in the temperature field being dominated by periodic conduction with the average Nusselt number decreasing proportionally to α2. When α→∞, the convection is confined to a thin layer adjacent to the lower wall with its intensity decreasing proportionally to α−3. The temperature field above the convection layer looses dependence on the horizontal direction. The bulk of the fluid sees the thin convective layer as a “hot wall.” The heat transfer between the walls becomes dominated by conduction driven by a uniform vertical temperature gradient which decreases proportionally to the intensity of convection resulting in the average Nusselt number decreasing as α−3. It is shown that processes described above occur for Prandtl numbers 0.001 < Pr < 10 considered in this study.

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References

Bénard, H., 1900, “Les tourbillons cellulaires dans une nappe liquide,” Rev. Gen. Sci. Pures Appl., 11, pp. 1261–1271.
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Kelly, R. E., and Pal, D., 1976, “Thermal Convection Induced Between Non-Uniformly Heated Horizontal Surfaces,” Proceedings of Heat Transfer and Fluid Mechanics Institute, Stanford University Press, pp. 1–17.
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Kelly, R. E., and Pal, D., 1978, “Thermal Convection With Spatially Periodic Boundary Conditions: Resonant Wavelength Excitation,” J. Fluid Mech., 86, pp. 433–456. [CrossRef]
Pal, D., and Kelly, R. E., 1978, “Thermal Convection With Spatially Periodic Non-Uniform Heating: Non-Resonant Wavelength Excitation,” Proceedings of 6th International Heat Transfer Conference, Toronto, pp. 235–238.
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Figures

Grahic Jump Location
Fig. 1

Sketch of the flow configuration

Grahic Jump Location
Fig. 2

Streamlines (solid lines) and isotherms (dash lines) for the long wavelength heating (α = 0.01, Rap = 400) for fluids with Prandtl numbers Pr = 0.01, 10. Values of the stream function and the temperature are normalized with the respective maxima; ψmax for Pr = 0.01, 10 are 4.186, 4.186 × 10−3, respectively, and θmax = (2Pr)−1.

Grahic Jump Location
Fig. 3

Variations of the maximum of the stream function ψmax (Fig. 3(a)) and the average Nusselt number Nuav (Fig. 3(b)) as a function of the heating wave number α for Rap = 100, 400. Solid and dash-dot lines are for the Prandtl numbers Pr = 10 and 0.01, respectively, dotted lines denote the corresponding asymptotes. ψmax ∼ α1, Nuav ∼ α2 for α→0, ψmax ∼ α−3, Nuav ∼ α−3 for α→∞.

Grahic Jump Location
Fig. 4

Streamlines (Fig. 4(a)) and isotherms (Fig. 4(b)) for convection generated by the short wavelength heating (Rap = 2000, α = 15). Lines for Pr = 10 and 0.01 overlap. hv denotes thickness of the convection layer. ψmax for Pr = 0.01 and 10 are 2.003 and 2.004 × 10−3, respectively.

Grahic Jump Location
Fig. 5

Variations of the roll strength ψmax as a function of the heating wave number α and the Rayleigh number based on the heating wavelength Rap3. Results for Pr = 10, 0.71, 0.1, 0.01 are displayed in Figs. 5(a)–5(d), respectively. The dash lines correspond to the largest Rap considered in this study. The dotted lines describe asymptotes (24). Thin solid lines identify conditions where the difference between the actual value of ψmax and the value computed from Eq. (24) is equal to 1%.

Grahic Jump Location
Fig. 6

Variations of the thickness of the convection layer hv as a function of the heating wave number α and the Rayleigh number Rap. Solid and dotted lines correspond to Pr = 10 and 0.01, respectively.

Grahic Jump Location
Fig. 7

Variations of the thickness hv of the convection layer as a function of the Prandtl number Pr for Rap = 2000

Grahic Jump Location
Fig. 8

Variations of the average Nusselt number Nuav as a function of the heating wave number α and the Rayleigh number based on the heating wavelength Rap3. Results for Pr = 10, 0.71, 0.1, 0.01 are displayed in Figs. 8(a)–8(d), respectively. The dash lines correspond to the largest Rap considered in this study. The dotted lines describe asymptotes (26). Thin solid lines identify conditions where the difference between the actual value of Nuav and the value computed from Eq. (26) is equal to 1%.

Grahic Jump Location
Fig. 9

Changes of the vertical position yc of the roll center as a function of the heating wave number α for the Prandtl numbers Pr = 10 (solid lines) and Pr = 0.01 (dash lines)

Grahic Jump Location
Fig. 10

Variations of the roll strength ψmax as a function of the heating wave number α and the Rayleigh number Rap. Results for Pr = 10, 0.71, 0.1, 0.01 are displayed in Figs. 10(a)–10(d) respectively. Dotted lines describe asymptotes (22) and (24).

Grahic Jump Location
Fig. 11

Variations of the roll strength as measured by the maximum of the stream function ψmax for Rap = 2000

Grahic Jump Location
Fig. 12

Topology of the temperature field for α = 1, Rap = 2000 for Pr = 0.01 (Fig. 12(a)) and Pr = 10 (Fig. 12(b))

Grahic Jump Location
Fig. 13

Variations of the average Nusselt number Nuav as a function of the heating wave number α and the Rayleigh number Rap. Results for Pr = 10, 0.71, 0.1, 0.01 are displayed in Figs. 13(a)–13(d), respectively. Dotted lines describe asymptotes (23) and (26).

Grahic Jump Location
Fig. 14

Variations of the average Nusselt number Nuav as a function of the Prandtl number Pr for Rap = 2000

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