A house has a black tar, flat, horizontal roof. The lower surface of the roof is well insulated, while the upper surface is exposed to ambient air at 300K through a convective coefficient of 10 W/m2-K. Calculate the roof equilibrium temperature for a) a clear sunny day with an incident solar radiaton flux of 500 W/m2 and the ambient sky at an effective temperature of 50K and b) a clear night with an ambient sky temperature of 50K.

Answer a) roof equilibrium temperature is 400K

Explanation:

The roof surface is sorrounded by ambient air whose temperature is 300K and which has a convective coefficient h of 10W/m2-K.

This means that heat will be conducted to the roof from the air around at an heat Flux of 300K x 10W/m2-K = 3000w/m2

For a clear solar day with solar heat Flux of 500W/m2, total heat Flux on roof will be

Q = 500 + 3500 = 3500W/m2

Q = h (Tr-Ta)

Ts is temperature of roof,

Ta is temperature of air

3500 = 10 (Tr - 50)

350 = Tr - 50

Tr = 400k

Answer b) : roof equilibrium temperature will be 350K

Explanation:

At night total heat Flux is only due to hot ambient air sorrounding the surface of the roof.

Q = 3000W/m2

Q = h (Tr-Ta)

3000 = 10 (Tr-50)

300 = Tr - 50

Tr = 350K