A plane layer of coal of thickness L = 1 m experiences uniform volumetric generation at a rate of q = 20 W/m3 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 = 5 W/m2 ∙ K and T∞ = 25°C, while receiving solar irradiation in the amount Gs = 400 W/m2. Irradiation from the atmosphere may be neglected. The solar absorptivity and emissivity of the surface are each aS = ε = 0.95.

(a) Write the steady-state form of the heat diffusion equation for the layer of coal. Verify that this equation is satisfied by a temperature distribution of the form T(x) = Ts + qL2/2k (1-x2/L2) from this distribution, what can you say about conditions at the bottom surface (x = O)? 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 Ts. Evaluate Ts 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 = 5 W/m2 ∙ K, compute and plot Ts and T (0) as a function of Gs for 50 ≤ Gs ≤ 500 W/m2. For Gs = 400 W/m2, compute and plot Ts and T (0) as a function of h for 5 ≤ h ≤ 50 W/m2 ∙ K.

Transcribed Image Text:

Ambient air T h L- Coal, k, 4