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Consider two rectangular surfaces perpendicular to each other with a common edge which is $1.6 \mathrm{~m}$ long. The horizontal surface is $0.8 \mathrm{~m}$ wide and the vertical surface is $1.2 \mathrm{~m}$ high. The horizontal surface has an emissivity of $0.75$ and is maintained at $400 \mathrm{~K}$. The vertical surface is black and is maintained at $550 \mathrm{~K}$. The back sides of the surfaces are insulated. The surrounding surfaces are at $290 \mathrm{~K}$, and can be considered to have an emissivity of $0.85$. Find the net rate of radiation heat transfers between the two surfaces, and between the horizontal surface and the surroundings.
Solution
Verified$\textbf{Given:}$

Two perpendicular rectangles of length: $W = 1.6$ m;

The horizontal surface $A_1$ width: $L_1 = 0.8$ m;

The vertical surface $A_2$ height: $L_2=1.2$ m;

The horizontal surface emissivity: $\varepsilon_1=0.75$;

The horizontal surface temperature: $T_1 = 400$ K;

The vertical surface emissivity: $\varepsilon_2 = 1$;

The vertical surface temperature: $T_2=550$ K;

The emissivity of the surrounding surfaces: $\varepsilon_3 = 0.85$;

The temperature of the surrounding surfaces: $T_3=290$ K;
$\textbf{Required:}$
a) The net rate of radiation heat transfer between the two surfaces: $\dot Q_{12}$;
b) The net rate of radiation heat transfer between the horizontal surface and the surroundings: $\dot Q_{13}$;
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