A plane layer of coal of thickness \(L=1 \mathrm{~m}\) experiences uniform volumetric generation at a rate of \(\dot{q}=10 \mathrm{~W} / \mathrm{m}^{3}\) due to slow oxidation of the coal particles. Averaged over a daily period, the top surface of the layer transfers heat by convection to ambient air for which \(h=\) \(8 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(T_{\infty}=30^{\circ} \mathrm{C}\), while receiving solar irradiation in the amount \(G_{S}=500 \mathrm{~W} / \mathrm{m}^{2}\). Irradiation from the atmosphere may be neglected. The solar absorptivity and emissivity of the surface are each \(\alpha_{S}=\varepsilon=0.95\).

(a) Write the steady-state form of the heat diffusion equation for the layer of coal. Verify, by direct substitution, that this equation is satisfied by a temperature distribution of the form \[T(x)=T_{s}+\frac{\dot{q} L^{2}}{2 k}\left(1-\frac{x^{2}}{L^{2}}\right)\]
From this distribution, what can you say about conditions at the bottom surface \((x=0)\) ? Sketch the temperature distribution and label key features.

(b) Obtain an expression for the rate of heat transfer by conduction per unit area at \(x=L\). Applying an energy balance to a control surface about the top surface of the layer, obtain an expression for \(T_{s}\). Evaluate \(T_{s}\) and \(T(0)\) for the prescribed conditions.

(c) Daily average values of \(G_{S}\) and \(h\) depend on a number of factors, such as time of year, cloud cover, and wind conditions. For \(h=8 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), compute and plot \(T_{S}\) and \(T(0)\) as a function of \(G_{S}\) for \(50 \leq G_{S} \leq\) \(500 \mathrm{~W} / \mathrm{m}^{2}\). For \(G_{S}=500 \mathrm{~W} / \mathrm{m}^{2}\), compute and plot \(T_{S}\) and \(T(0)\) as a function of \(h\) for \(5 \leq h \leq 50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\).

Transcribed Image Text:

L- Ambient air Th Coal, k.q