Solar radiation at 600 W/m2 is absorbed by a black roof inclined at 30°C as shown. If the underside of the roof is well insulated, estimate the maximum roof temperature in Tinf= 20°C air.

GIVEN

• Inclined roof, well insulated on the underside

• Incline angle (?) = 30 degrees

• Air temperature = 20°C

• Solar radiation absorbed (qs) = 600 W/m2

FIND

• The maximum roof temperature

ASSUMPTIONS

• The roof behaves as a black body (? = 1.0)

• The sky behaves as a black body at 0 K

SKETCH



PROPERTIES AND CONSTANTS

The Stephan-Boltzmann constant (?) = 5.67 × 10–8 W/(m2 K4).


The maximum roof temperature will occur when the air is quiescent. Since the air properties must be evaluated at the mean of the ambient and surface temperatures, as iterative procedure must be used. Iteration #1 Let Ts = 60°C = 333 K

for dry air at the mean temperature of 40°C

Thermal expansion coefficient (?) = 0.00319 1/K

Thermal conductivity (k) = 0.0265 W/(m K)

Kinematic viscosity (?) = 17.6 × 10–6 m2/s

Prandtl number (Pr) = 0.71

The Grashof number is



The average convective heat transfer coefficient for this geometry. Although GrL Pr cos ? is slightly larger than 1011, will be extrapolated by this problem



For steady state, the solar gain must equal the convective and radiative losses



Checking the units then eliminating them for clarity



By trial and error: Ts = 314 K.

Repeating this procedure for another iteration



The maximum roof temperature:



COMMENTS

The procedure converges quickly because of the 1/4 power in the Nusselt number correlation.

Physics & Space Science

You might also like to view...

Siddhartha saved up money from his summer job to buy a powerful home theater system. When shopping for the best buy in a home theater system the sales representative informs Siddhartha that the more expensive models are worth the money because they ____.

A) amplify sound up to 160 dB B) boost frequencies below 80 Hz C) buffer sounds that reach 60 dB D) filter out frequencies below 10000 Hz

Physics & Space Science

Nanostructured materials can have unique properties. For materials with dimensions measured in nanometers the amount of surface energy is comparable to the bonding energy in the material.

(a) Compare the magnitude of the total surface energy to the total bond energy for a sphere of Ni atoms with one atom at the center surrounded by twelve nearest neighbors. The surface energy of Ni is 1.90 J/m2, the cohesive energy of Ni is –4.435 eV/atom, and the lattice parameter of FCC Ni is 0.352 nm. Assume that the total radius of this sphere is equal to 1 atomic diameter. (b) Is this 13-atom cluster of atoms stable?

Physics & Space Science

Intelligence may be subject to convergent evolution because it

A) is inevitable B) is a fundamental characteristic of life C) has survival value D) is required for survival

Physics & Space Science

An electron and a positron, which are two massive particles, can annihilate when they come together to produce two photons, which have no mass. This pair annihilation is an example of

a. time dilation. b. length contraction. c. the equivalence of mass and energy. d. the principle of relativity. e. none of the above.

Physics & Space Science