Combined Convection, Radiation, and Heat Flux
Consider the south wall of a house that is L = 0.2 m thick. The outer surface of the wall is exposed to solar radiation and has an absorptivity of \alpha = 0.5 for solar energy. The interior of the house is maintained at T_{\infty 1} = 20^{\circ} C , while the ambient air temperature outside remains at T_{\infty 2} = 5^{\circ} C , The sky, the ground, and the surfaces of the surrounding structures at this location can be modeled as a surface at an effective temperature of T_{\text {sky }} = 255 K for radiation exchange on the outer surface. The radiation exchange between the inner surface of the wall and the surfaces of the walls, floor, and ceiling it faces is negligible. The convection heat transfer coefficients on the inner and the outer surfaces of the wall are h_{1} = 6 W / m ^{2} \cdot{ }^{\circ} C and h_{2} = 25 W / m ^{2} \cdot{ }^{\circ} C , respectively. The thermal conductivity of the wall material is k = 0.7 W / m \cdot{ }^{\circ} C , and the emissivity of the outer surface is \varepsilon_{2} = 0.9 Assuming the heat transfer through the wall to be steady and one-dimensional, express the boundary conditions on the inner and the outer surfaces of the wall.