The roof of a car in a parking lot absorbs a solar radiant flux of 800 W/m2, and the underside is perfectly insulated. The convection coefficient between the roof and the ambient air is 12 W/m2·K. (a) Neglecting radiation exchange with the surroundings, calculate the temperature of the roof under steady state conditions if the ambient air temperature is 20°C. (b) For the same ambient air temperature, calculate the temperature of the roof if its surface emissivity is 0.8.

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