Answer:
a) 87 C
b) 47 C
Explanation:
We assume that the roof has finished heating and is at equilibrium, so the energy it receives is the same it dissipates.
Psolar = Pconv
The energy it dissipates through convection is per unit of area is:
Pconv = h * (troof - tair)
troof - tair = Pconv/h
troof = Pconv/h + tair
troof = Psolar/h + tair
troof = 800/12 + 20 = 87 C
If it has an emissivity of 0.8
Psolar = Pconv + Prad
The equation for thermal radiation per unit of surface is:
Prad = ε * σ * Troof^4
Where
ε: emissivity
σ: Stefan-Boltzmann constant (5.67*10^-8 W*m^-2*K^-4)
Troof: absolute temperature of the roof
Then:
Psolar = h * (Troof - Tair) + ε * σ * Troof^4
Tair = 293 K
-h * Tair - Psolar + h * Troof + ε * σ * Troof^4 = 0
-12 * 293 - 800 + 12 * Troof + 0.8 * 5.67*10^(-8) * Troof^4 = 0
-4316 + 12*Troof + 4.54*10^(-8)*Troof^4 = 0
Solving this equation we get that it has two complex roots, one negative root and one positive root. SInce this is an absolute temperature it cannot be complex or negative, so we take the only positive solution
Troof = 320 K = 47 C
