You would like to put a solar hot water system on your roof, but you\'re not sur

Question

You would like to put a solar hot water system on your roof, but you're not sure it's feasible. A reference book on solar energy shows that the ground-level solar intensity in your city is 800 W/m2 for at least 5 hours a day throughout most of the year.

Assuming that a completely black collector plate loses energy only by radiation, and that the air temperature is 20 C, what is the equilibrium temperature of a collector plate directly facing the sun? Note that while a plate has two sides, only the side facing the sun will radiate because the opposite side will be well insulated.

Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

It doesn't matter how big the plate is, but for the sake of making computations easy, let's assume it is 1 m^2 in area. Then the power of the sun hitting it is 800 watts.

As the plate is neither heating nor cooling, the power that the plate is radiating must be equal to 800 watts.

If you don't remember the Stefan-Boltzmann law, you can look it up here

https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law

From that, P = s * T^4, where P is power, s is the S-B constant, and T is absolute temperature. Set the left side to 800, and you can solve for temperature. Then convert it to Celsius for a more useful answer.

T = (800 / 5.67 x 10-8) 0.25  = 344.65 K = 71.15 Co

I have ignored the temperature of the air on the assumption that when a solar radiation figure is given, it includes all sources, including the air and blank sky. And also, the problem states (unrealistically) that there is no heat loss by conduction or convection.

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