# Numerical Analysis of Natural Convection with Thermal Radiation in a Partially Heated 3D Cavity

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## Abstract

In this study, the finite volume and the discrete ordinate methods are used to study the phenomenon of radiation and natural convection in a cubical cavity for both transparent and participating media. The cavity is heated by six identical isothermal elements placed on one of its vertical walls. Three possible arrangements named cases I, II, and III of the six elements are studied. In case I, the number of these elements increases with the height of the cavity: one, two then three elements (1 × 2 × 3). Case III is the opposite with 3, 2, and then 1 elements passing from the bottom to the top of the hot wall. Case II corresponds to the uniform arrangement 2 × 2 × 2. The simulation results are obtained for different values of the Rayleigh number in the range of \( 10^{3} \le {\text{Ra}} \le 10^{7} \), the common emissivity of the walls (0 ≤ ε ≤ 1) and the optical thickness of the fluid (0 ≤ τ ≤ 10^{3}). The thickness of the cavity varies from 2 cm to 12 cm, and the horizontal and vertical distances separating the six isothermal elements are in the range of \( 0.25 \le {\text{D}}_{\text{H}} \le 0.75 \) and \( 0.208 \lesssim {\text{D}}_{\text{V}} \le 0.375 \), respectively. The number of Prandtl and the cold wall temperature are fixed at \( { \Pr } = 0.71 \) and \( {\text{T}}_{\text{C}} = 293\,{\text{K}} \). The obtained results show that the case II leads to the largest global heat transfer and the radiation contribution becomes negligible if τ is too high.

## Keywords

Arrangement Natural convection Partial heating Participating media Radiation Tri-dimensional## List of Symbols

- \( {\text{B}} \)
Dimensionless size of the heating element

- \( {\text{D}} \)
Dimensionless distance between elements

- \( {\text{g}} \)
Gravitational acceleration, \( {\text{m}} {\cdot} {\text{s}}^{ - 2} \)

- \( {\text{H}} \)
Height of the cavity, \( {\text{m}} \)

- \( {\text{I}}_{\text{R}} \)
Dimensionless radiative intensity

- \( {\text{I}}_{\text{Rb}} \)
Dimensionless blackbody emission

- \( {\text{L}} \)
Width of the cavity, \( {\text{m}} \)

- \( {\text{k}} \)
Thermal conductivity, \( {\text{Wm}}^{ - 1} {\cdot} {\text{K}}^{ - 1} \)

- \( {\text{Nu}} \)
Nusselt number

- \( {\text{P}} \)
Pressure, \( {\text{Pa}} \)

- \( { \Pr } \)
Prandtl number

- \( {\text{Pl}} \)
Planck number

- \( {\text{Q}} \)
Dimensionless heat flux

- \( {\text{Ra}} \)
Rayleigh number

- \( {\text{T}} \)
Temperature, \( {\text{K}} \)

- \( {\text{T}}_{\text{R}} \)
Temperature ratio

- \( {\text{U}} \)
Dimensionless velocity

- \( {\text{W}} \)
Depth of the cavity, m, dimensionless vertical component of the velocity

- \( {\text{x}},{\text{y}} \)
Cartesian coordinates, \( {\text{m}} \)

- \( {\text{X}}, {\text{Y}} \)
Dimensionless cartesian coordinates

- α
Thermal diffisuvity, \( {\text{m}}^{2} {\cdot} {\text{s}}^{ - 1} \)

- β
Thermal expansion coefficient, \( {\text{K}}^{ - 1} \)

- \( \Delta {\text{T}} \)
Temperature difference, \( {\text{K}} \)

- δ
Kronecker symbol

- ε
Emissivity of the radiative surface

- ν
Kinematic viscosity, \( {\text{m}}^{2} {\cdot} {\text{s}}^{ - 1} \)

- ξ
Direction cosines

- θ
Dimensionless temperature

- σ
Stefan–Boltzmann constant, \( {\text{W}} {\cdot} {\text{K}}^{ - 4} {\cdot} {\text{m}}^{ - 2} \)

- τ
Dimensionless time, optical thickness

- ω
Scattering albedo

## Subscripts

- C
Cold, convection

- H
Hot, horizontal

- i, j
Directions

- L
Local

- m
Mean

- R
Radiative, ratio

- inc
Incident

- V
Vertical

## Notes

## References

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