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Question : Combined Convection, Radiation, and Heat Flux Consider the s...

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.

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SOLUTION  We take the direction normal to the wall surfaces as the x-axis with the origin at the inner surface of the wall, as shown in Figure. The heat transfer through the wall is given to be steady and one-dimensional, and thus the temperature depends on x only and not on time. That is, T = T (x). The boundary condition on the inner surface of the wall at x = 0 is a typical convection condition since it does not involve any radiation or specified heat flux. Taking the direction of heat transfer to be the positive x-direction, the boundary condition on the inner surface can be expressed as

 

                                         –  k  \frac{d T(0)}{d x} = h_{1}\left[T_{\infty 1} –  T(0)\right]

 

The boundary condition on the outer surface at x = 0 is quite general as it involves conduction, convection, radiation, and specified heat flux. Again taking the direction of heat transfer to be the positive x-direction, the boundary condition on the outer surface can be expressed as

 

  – k  \frac{d T(L)}{d x} = h_{2}\left[T(L)-  T_{\infty 2}\right] + \varepsilon_{2} \sigma\left[T(L)^{4} – T_{ sky }^{4}\right] –  \alpha \dot{q}_{ solar }

 

where \dot{q}_{\text {solar }} is the incident solar heat flux. Assuming the opposite direction for heat transfer would give the same result multiplied by 1, which is equivalent to the relation here. All the quantities in these relations are known except the temperatures and their derivatives at the two boundaries.

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