The roof of a house consists of a 15-cm-thick concrete slab (k = 2?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 estimated to be 15?W/m2•K. The inner surface of the roof is maintained at 15°C. On a clear winter night, the ambient air is reported to be at 10°C, while the night sky temperature for radiation heat transfer is 255 K. Considering both radiation and convection heat transfer, determine the outer surface temperature and the rate of heat transfer through the roof. If the house is heated by a furnace burning natural gas with an efficiency of 85 percent, and the unit cost of natural gas is $0.60/therm (1?therm = 105,500 kJ of energy content), determine the

Question

money lost through the roof that night during a 14-h period.

What will be an ideal response?

Answer 1

The roof of a house with a gas furnace consists of a 15-cm thick concrete that is losing heat to the outdoors by radiation and convection. The rate of heat transfer through the roof and the money lost through the roof that night during a 14 hour period are to be determined.

Assumptions 1 Steady operating conditions exist. 2 The emissivity and thermal conductivity of the roof are constant.

Properties The thermal conductivity of the concrete is given to be k = 2 W/m?*C. The emissivity of the outer surface of the roof is given to be 0.9.

Analysis In steady operation, heat transfer from the outer surface of the roof to the surroundings by convection and radiation must be equal to the heat transfer through the roof by conduction. That is,



The inner surface temperature of the roof is given to be Ts,in = 15?C. Letting Ts,out denote the outer surface temperatures of the roof, the energy balance above can be expressed as



Solving the equations above using an equation solver (or by trial and error) gives



Then the amount of natural gas consumption during a 16-hour period is



Finally, the money lost through the roof during that period is

Moneylost= (14.3 themes)($0.60 / therm ) = $8.58