Imagine a planet like ours with an atmosphere whose clouds reflect 30% of the Sun's light (albedo: 0.30) but that does not have a green-house effect (it does not absorb infrared radiation from the surface of the planet) and does not transport entropy convectively of conductively. The planet is mostly covered by water. For the subsequent models, take a body of water having a surface of 1 m^2 that absorbs all the light of the sun down to a depth of 10 m (take it to be well mixed). This body of water losed heat only upward and only by radiation (no downward or lateral transports of entropy). Assume the water to stay liquid with a constant specific entropy capacitance even if temperatures change beyond the normal range for liquid water. Also take the absorber of a solar collector (a sheet of blackened copper, 1 m^2 having a thickness of 5 mm) covered by glass (density 3000 kg/m^s, thickness 8 mm) with no air between the absorber and the glass cover. The glass absorbs all the radiation from the absorber. The bodies considered are at the equator; the Sun shines vertically down upon the equator.
a. If the sun were shining constantly upon the body of water, what would be the steady-state temperature of the water? What would it be if you assumed the radiation to be the average radiation for 24 hours? b. Assume the absorber not to have a glass cover. Repeat the calculations according to problem a for the absorber. | | c. Assume there to be a glass cover for the absorber. Again repeat the calculations of problem a (or b) for the absorber and the glass cover. d. Prepare a dynamical model for the calculations described in a, b, and c. Create a function for the energy current of solar radiation falling upon the bodies (having a surface area of 1 m^2) during 24 hours (use a simple sine or cosine function with a maximum derived from 1370 W/m^2 and the albedo of the planet). Introduce a continuous function for an arbitrary legth of time. e. Create a dynamical model for the body of water. Use a sensible initial temperature. Interpret the result for the changes of temperature. f. Repeat problem e for the absorber of the collector without the glass cover. Do the following calculation by hand to estimate the change of temperature of the absorber: At the beginning of the evening, the absorber has the temperature(s) calculated in b. How far will it cool down during the night? g. Repeat problem f for the collector with absorber and glass cover. h. Why are the average temperatures of the bodies involved so low? Why is the absorber hotter with the glass cover? |
