A flat plate is in a solar orbit 150,000,000 km from the sun. It is always oriented normal to the rays of the sun and both sides of the plate have a finish which has a spectral absorptance of 0.95 at wavelengths shorter than 3 ?m and 0.06 at wavelengths longer than 3 ?m. Assuming that the sun is a 5550 K blackbody source with a diameter of 1,400,000 km, determine the equilibrium temperature of the plate.

Question

GIVEN

• A flat plate in solar orbit oriented normal to the rays of the sun

• Distance from the sun (R) = 150,000,000 km = 1.5 × 1011 m

• Spectral absorplatace (??) of both sides ??1 = 0.95 for ?1 < 3 ?m ??2 = 0.06 for ?1 > 3 ?m

FIND

• The equilibrium temperature of the plate (Tp)

ASSUMPTIONS

• The sun is a black body at Ts = 5550 K

• Sun diameter (Ds) = 1,400,000 km = 1.4 × 109 m

SKETCH

Answer 1

For the sun ?1 T = (3 × 10–6 m) (5550 K) = 16.7 × 10–3 m K



The absorptance of the plate is



Let qp = the flux incident on the plate

Energy leaving the sun surface = energy crossing sphere of radius R



Performing the energy balance on the plate

Energy absorbed from sun = energy emitted form both sides of the plate



This can be solved iteratively. For a first guess, let TP = 500 K.



Repeating this procedure



COMMENTS

Note that the total absorptance (?) = 0.929 but the total emittance (?) = 0.088.