How Efficient Can a Radiative Heat Engine Be for a Moon Base?

[Physgeek64]

Homework Statement

The emission of radiation from the Sun’s disc is observed to peak at 0.5 μm wave- length and that from the Moon’s disc at 10.0μm. A heat engine to power a Moon base is to be constructed using radiation collected from the Sun. What is the maximum theoretical efficiency of such an engine? Comment on whether it would be practical to achieve this.

Homework Equations

The Attempt at a Solution

$T\lambda_{max}=constant$
$T_{moon}\lambda_{moon max}=T_{sun}\lambda_{sun max}$
$\frac{T_{moon}}{T_{sun}}=0.05$

$\eta=1-\frac{Q_2}{Q_2}$
$\frac{Q_2}{Q_1}=\frac{T_2}{T_1}$
$\eta=1-0.05=0.95$

does this seem right? I feel like assuming the radiation is the same temperature as the sun is wrong

but i don't know how to relate this all together. Any help would be greatly appreciated. Many thanks

 

[Charles Link]

I believe you computed it correctly. One item of interest is the constant in the Wien's law formula is $2898 \, \mu m \, K$ making the temperature of the (surface of the} sun $T_{sun} \approx 6000 \, K$ , and that of the moon $T_{moon} \approx 300 \, K$. $\\$ Note: The question also asks to comment on whether this is practical. To answer this question really requires some additional background material, so let me provide some of that info: $\\$ In order for light from the sun to be able to heat an object up to $T$ anywhere near $6000$ K , you would need to use lenses to focus the light onto the surface, (with the surface in the focal plane of the lens), and the lens system would need to fill a good portion of the hemisphere surrounding the surface being heated. Is this practical? $\\$ Without the use of optics, the sun only subtends an angle of $\Delta \theta=.01$ radians, (as seen from the Earth or moon), and would basically heat up surfaces that are placed on the moon to the same temperature as the surface of the moon. (That's why the moon is at the temperature that it is. The moon's surface, (assuming emissivity 1.0, but this same dynamic equilibrium also holds for a greybody), is radiating away energy as a blackbody at temperature $T_{moon}$, at the same rate that it is receiving energy from the sun). $\\$ Additional question: Would it perhaps be somewhat more practical to use lenses to make a heat engine system where (smaller) lenses heated the material to a temperature in the neighborhood of $T= 1000 \, K$ ?