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RESEARCH PAPERS: Natural and Mixed Convection

# Effect of Surface Radiation on Multiple Natural Convection Solutions in a Square Cavity Partially Heated from Below

[+] Author and Article Information
El Hassan Ridouane1

Department of Mechanical Engineering, The University of Vermont, Colchester Ave, 201 Votey, Burlington, VT 05405eridouan@cems.uvm.edu

Mohammed Hasnaoui

Department of Physics, LMFE, Faculty of Sciences Semlalia, Marrakech, Morocco

1

Corresponding author.

J. Heat Transfer 128(10), 1012-1021 (May 16, 2006) (10 pages) doi:10.1115/1.2345429 History: Received July 29, 2005; Revised May 16, 2006

## Abstract

A numerical study of natural convection with surface radiation in an air filled square enclosure with a centrally heated bottom wall and cooled upper wall is presented. The vertical walls and the rest of the bottom wall are assumed to be insulated. The problem is studied for Rayleigh numbers Ra, ranging from $103$ to $4×106$ and surfaces emissivity $ε$, varying from 0 to 1. The governing equations, written in terms of stream function-vorticity formulation, are solved using a finite difference approach. It is found that, under these heating/cooling conditions, three different steady-state solutions are possible in the ranges of the parameters considered. Results are presented detailing the occurrence of each steady-state solution and the effect of Ra and $ε$ on its range of existence. It is found that the surface radiation alters significantly the existence ranges of the solutions. For each solution, convective and radiative contributions to the global heat transfer are also quantified for various Ra and $ε$. The influence of the heated surface dimension on the fluid flow and thermal patterns is also presented by comparing the present results against those obtained by the authors in an earlier study within a square cavity totally heated from below.

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## Figures

Figure 1

Schematic of the physical system

Figure 2

Streamlines and isotherms obtained in the case of S1 for ε=0: (a)Ra=5×103 and (b)Ra=106

Figure 3

Emissivity effect on the S1 patterns at Ra=105: (a)ε=0, (b)ε=0.5, and (c)ε=1.0

Figure 4

Variations of Rac with the emissivity for both partially and totally heated cavities

Figure 5

Comparison of the global Nusselt numbers obtained for both heating conditions in the case of S1

Figure 6

Streamlines and isotherms obtained in the case of S2 for ε=0: (a)Ra=2×105 and (b)Ra=9.3×104

Figure 7

Extremum values of the stream function for the S2 type solution as a function of walls emissivity at Ra=105

Figure 8

Variations of Rai (i=1, 2, 3) with ε for both partially and totally heated bottom wall

Figure 9

Comparison of the global Nusselt numbers obtained for both heating conditions in the case of S2

Figure 10

Streamlines and isotherms obtained in the case of S3 for ε=0: (a)Ra=4×106 and (b)Ra=4×105

Figure 11

Comparison of the global Nusselt numbers obtained for both heating conditions in the case of S3

Figure 12

Phase diagram of existence of the different solutions Si for different values of Ra and ε: (a) PH conditions and (b) TH conditions

## Errata

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