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Research Papers: Heat and Mass Transfer

Effects of Nonuniform Heating and Wall Conduction on Natural Convection in a Square Porous Cavity Using LTNE Model

[+] Author and Article Information
A. I. Alsabery

School of Mathematical Sciences,
Faculty of Science and Technology,
Universiti Kebangsaan Malaysia,
Bangi 43600, Selangor, Malaysia
e-mail: ammar_e_2011@yahoo.com

A. J. Chamkha

Mechanical Engineering Department,
Prince Mohammad Bin Fahd University,
Al-Khobar 31952, Saudi Arabia;
Prince Sultan Endowment for Energy
and Environment,
Prince Mohammad Bin Fahd University,
Al-Khobar 31952, Saudi Arabia

I. Hashim

School of Mathematical Sciences,
Faculty of Science and Technology,
Universiti Kebangsaan Malaysia,
Bangi 43600, Selangor, Malaysia

P. G. Siddheshwar

Department of Mathematics,
Bangalore University,
Jnana Bharathi Campus,
Bangalore 560 056, India

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received March 6, 2017; final manuscript received May 26, 2017; published online August 1, 2017. Assoc. Editor: Amy Fleischer.

J. Heat Transfer 139(12), 122008 (Aug 01, 2017) (13 pages) Paper No: HT-17-1124; doi: 10.1115/1.4037087 History: Received March 06, 2017; Revised May 26, 2017

The effects of nonuniform heating and a finite wall thickness on natural convection in a square porous cavity based on the local thermal nonequilibrium (LTNE) model are studied numerically using the finite difference method (FDM). The finite-thickness horizontal wall of the cavity is heated either uniformly or nonuniformly, and the vertical walls are maintained at constant cold temperatures. The top horizontal insulated wall allows no heat transfer to the surrounding. The Darcy law is used along with the Boussinesq approximation for the flow. The results of this study are obtained for various parametric values of the Rayleigh number, thermal conductivity ratio, ratio of the wall thickness to its height, and the modified conductivity ratio. Comparisons with previously published work verify good agreement with the proposed method. The effects of the various parameters on the streamlines, isotherms, and the weighted-average heat transfer are shown graphically. It is shown that a thicker bottom solid wall clearly inhibits the temperature gradient which then leads to the thermal equilibrium case. Further, the overall heat transfer is highly affected by the presence of the solid wall. The results have possible applications in the heat-storage fluid-saturated porous systems and the applications of the high power heat transfer.

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References

Figures

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Fig. 1

(a) Physical model of convection in a porous square cavity together with the coordinate system and (b) grid-point distributions in the conducting wall (j ≤ ND + 1) and porous cavity (j > ND + 1)

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Fig. 2

Streamlines (a)—Saeid [7] (left) and present study (right) and isotherms (b)—Saeid [7] (left) and present study (right) for Ra = 103, Kr = 0.1, and S = 0.2

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Fig. 3

Streamlines, isotherms for fluid phase, and isotherms for solid phase: Baytaş and Pop [15] (a) and present study (b) for Ra = 103, γ = 10, H = 1, and S = 0

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Fig. 4

Streamlines (a)—Varol et al. [26] (left) and present study (right) and isotherms (b)—Varol et al. [26] (left) and present study (right) for Ra = 102 and S = 0

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Fig. 5

Variation of the streamlines (left), isotherms of the global fluid phase (middle), and isotherms of the global porous phase (right) with modified conductivity ratio (γ) and Rayleigh number (Ra) when the bottom wall heated uniformly (θ = 1) for H = 1, Kr = 1, and S = 0.2: (a) Ra = 102 and (b) Ra = 103

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Fig. 6

Variation of the streamlines (left), isotherms of the global fluid phase (middle), and isotherms of the global porous phase (right) with modified conductivity ratio (γ) and Rayleigh number (Ra) when the bottom wall heated nonuniformly (θ = sin(2πX)) for H = 1, Kr = 1, and S = 0.2: (a) Ra = 102 and (b) Ra = 103

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Fig. 7

Variation of the streamlines (left), isotherms of the global fluid phase (middle), and isotherms of the global porous phase (right) with thermal conductivity ratio (Kr) when the bottom wall heated nonuniformly (θ = sin(2πX)) for Ra = 103, γ = 1 H = 1, and S = 0.2: (a) Kr = 0.44, (b) Kr = 2.40, (c) Kr = 9.90, and (d) Kr = 23.8

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Fig. 8

Variation of the streamlines (left), isotherms of the global fluid phase (middle), and isotherms of the global porous phase (right) with the solid wall thickness (S) when the bottom wall heated nonuniformly (θ = sin(2πX)) for Ra = 103, γ = 10, H = 1, and Kr = 1: (a) S = 0, (b) S = 0.1, (c) S = 0.3, and (d) S = 0.5

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Fig. 9

Variation of the weighted-average Nusselt number with Ra for different (a) γ, (b) S, and (c) Kr when the bottom wall heated nonuniformly (θ = sin(2πX)) for γ = 1, H = 1, Kr = 1, S = 0.2, and φ=0.4

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Fig. 10

Variation of the weighted-average Nusselt number with S for different (a) Ra, (b) γ, and (c) Kr when the bottom wall heated nonuniformly (θ = sin(2πX)) for Ra = 103, γ = 1, H = 1, Kr = 1, and φ=0.4

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Fig. 11

Variation of the weighted-average Nusselt number with γ for different (a) H, (b) S, and (c) Kr when the bottom wall heated nonuniformly (θ = sin(2πX)) for Ra = 103, H = 1, S = 0.2, Kr = 1, and φ=0.4

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