A plane layer of coal of thickness L = 1 m experiences uniform volumetric generation at a rate of q̇= 40 ?/?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 = 10 W/m2·K and T∞= 20 °C, while receiving solar irradiation in the amount GS = 1360 W/m2. Irradiation from the atmosphere may be neglected. The solar absorptivity and emissivity of the surface are the same, each αS = ε = 0.90. a) Write the steady-state form of the 1D Cartesian heat diffusion equation (HDE) for the layer of coal. Using the HDE, determine the temperature distribution as a function of x and constants given above. b) Sketch the temperature distribution and label key features. c) 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.
A plane layer of coal of thickness L = 1 m experiences uniform volumetric generation at a rate of q̇= 40 ?/?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 = 10 W/m2·K and T∞= 20 °C, while receiving solar irradiation in the amount GS = 1360 W/m2. Irradiation from the atmosphere may be neglected. The solar absorptivity and emissivity of the surface are the same, each αS = ε = 0.90.
a) Write the steady-state form of the 1D Cartesian heat diffusion equation (HDE) for the layer of coal. Using the HDE, determine the temperature distribution as a function of x and constants given above.
b) Sketch the temperature distribution and label key features.
c) 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.
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