The roof of a house consists of a 22 cm thick concrete slab (k = 0.72 W/m·?K) that is 15 m wide and 20 m long. The emissivity of the outer surface of the roof is 0.9 and the convection heat transfer coefficient on that surface is 15 W/m2·?K. The inner surface of the roof is maintained at 15°C. On a clear winter night, the ambient air is 10°C while the night sky temperature for radiation heat transfer is 255 K.
a) Considering both radiation and convection heat transfer, determine the outer surface temperature and the rate of heat transfer though the roof.
b) If the house is heated by a furnace with an efficiency of 85% and the unity cost of fuel is $1.20/therm (1 therm = 105,500 kJ of energy content), determine the money lost through the roof that night during a 14 hour period.
c) The roof is now insulated with a spray on urethane foam that reduces the effective thermal conductivity of the roof to k = 0.15 W/m·?K. What is the money lost through the roof during the 14-hour period now?