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RESEARCH PAPERS: Natural and Mixed Convection

# Analytical Study of Natural Convection in a Cavity With Volumetric Heat Generation

[+] Author and Article Information
Mandar V. Joshi, U. N. Gaitonde

Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India

Sushanta K. Mitra

Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, Indiaskmitra@me.iitb.ac.in

J. Heat Transfer 128(2), 176-182 (Apr 13, 2005) (7 pages) doi:10.1115/1.2137761 History: Received September 27, 2004; Revised April 13, 2005

## Abstract

An analytical study has been conducted for natural convection in a cavity of different aspect ratios with uniform volumetric heat generation. Two different boundary conditions are investigated for the cavity, viz., all walls are isothermal; two horizontal walls are adiabatic and two vertical walls are isothermal. A stream function vorticity formulation is used where the variables are expanded in terms of Rayleigh number, defined as $Ra=gβh5q‴∕ανk$. The governing equations are reduced, to biharmonic equations, and these biharmonic equations are solved using one of the methods, available in the literature. It is observed that the horizontal component of velocity is smaller than the vertical component near the center and the vertical walls of the cavity. The results for velocity profiles are compared with the simulations obtained from Fluent and they are found to be in good agreement.

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## Figures

Figure 1

Schematic of a cavity with volumetric heat generation

Figure 2

Streamlines for square cavity with isothermal walls

Figure 3

Streamlines for square cavity with horizontal walls adiabatic

Figure 4

Streamlines for cavity of aspect ratio 3 with isothermal walls

Figure 5

Streamlines for cavity of aspect ratio 3 with horizontal walls adiabatic

Figure 6

Vertical velocity profile along the horizontal centerline for cavity with isothermal walls

Figure 7

Vertical velocity profiles along the horizontal centerline for cavity with horizontal walls adiabatic

Figure 8

Horizontal velocity profile along y=0.25 for cavity with isothermal walls

Figure 9

Horizontal velocity profile along y=0.25 for a cavity with horizontal walls adiabatic

Figure 10

Isotherms for a square cavity with horizontal walls adiabatic

Figure 11

Variation of dimensionless temperature θ along horizontal centerline

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