Possibility of Using a Refrigerator to Cool the Earth

I understand that the Earth as a whole cannot be cooled directly by radiating energy away for theoretical reasons. Waste heat has low entropy; effectively raising its entropy to radiate it must cost more than it gains.

I thought about building an impossibly-high tower and pumping refrigerant, ultimately making a mechanical connection between the Earth and space where black-body radiation will be effective to do the cooling. What if we wanted to cool the Moon instead, where there is little or no air to interfere? Could we build a moderately-high platform, using mirrors underneath and insulation, and ultimately lower the average temperature of the Moon significantly? This seems similar in principle to what is actually done to cool spacecraft.

Let us suppose we have a large platform, adequately insulated from the Earth or Moon and with a movable shield for the Sun. Is there a theoretical reason we could not use a refrigerator, powered (mostly) from the platform, and use black-body radiation in space to get net cooling on the attached moon or planet?

** Possible Configurations**

Platform perhaps a kilometer above the Moon. The hot reservoir could be connected to a buried structure to increase amount of heat transferred.

Platform above a (most likely impossible) hundred-kilometer tower. Too much pumping needed near bottom of tower is an obvious problem.

Platform above Mt Everest, hot reservoir exchanges heat via a deep drilled hole. Refrigerant may or may not be pumped through the hole. Less efficient due to residual atmosphere.

If there is a possibility this could be done, even though at huge cost, I would like to consider theoretical limits on how much heat could be transferred as a function of the size of the platform and refrigeration system.

  • $\begingroup$ I know heat is random energy, and an active radiator must convert it to directed energy, but I couldn't explain it well. My idea seems to be cheating somehow, but hey, doesn't Hubble do what I'm speculating about? I hope there's a basic principle of Physics that can be clearly explained. $\endgroup$
    – Bit Chaser
    May 23 '20 at 22:50

This will probably work, at least a little bit.

It looks like the maximum temperature of satellites exposed to the Sun in low-Earth orbit is around 123 degrees Celsius*. If you can get your radiator hotter than around that temperature, then it'll radiate energy away even when it's in full sunlight. It looks like digging about 20-25 km into the crust will give you a natural reservoir that's about that temperature**.

The temperature of a satellite in low-Earth orbit varies from roughly -170 degrees Celsius to 123 degrees Celsius. We can model this roughly as a sinusoid with a period of 90 minutes:

$$T_{ambient}(t)=325\text{ K}+(222\text{ K})\sin\left(\frac{t}{2\pi(90\text{ min})}\right)$$

We'll take this as the temperature that the radiator would be at if it wasn't heated by your heat exchanger. The net radiated power as a function of time would then be:

$$P(t)=\sigma A(T_{radiator}^4-T_{ambient}(t)^4)$$

where $A$ is the surface area of the radiator and $\sigma$ is the Stefan-Boltzmann constant. We can average this radiated power over one 90-minute period, which gives us:

$$\langle P\rangle=\sigma A(T_{radiator}^4-\langle T_{ambient}^4\rangle)=\sigma A(T_{radiator}^4-(1.38\times 10^{10}\text{ K}^4))$$

If we assume that you can get your radiator to 150 degrees Celsius, using a passive heat exchanger buried deep in the crust and without any heat loss on the way up, then we get:

$$\langle P\rangle = (1035\text{ W/m}^2)A$$

So you could radiate about a kW of heat energy per square meter of radiator. This isn't really that much at all (a space heater outputs more power in far less space), and your radiator would have to be absolutely gigantic in order to actually have a significant effect on the temperature of the Earth. For reference, the Earth is currently absorbing somewhere around 250 TW more energy than it's emitting***, so you would need around $250,000\text{ km}^2$ of radiator, all heated to the same temperature, before you could hope to have much of an effect on that imbalance. For reference, that's a radiator about as large as the state of Wyoming, heated to above the boiling point of water, sitting in space.

*(source: https://www.oreilly.com/library/view/diy-satellite-platforms/9781449312756/ch01s05.html)

**(source: https://www.researchgate.net/figure/Temperature-distribution-with-depth-in-the-crust-for-different-values-of-advective_fig3_241463986)

***(source: https://www.giss.nasa.gov/research/briefs/hansen_16/)

  • $\begingroup$ Thanks! Maybe we can get a prototype ready for the Moon in 2024 <grin>. If we can get power and at least partial shade up there, and do all the pumping from a kilometer above the surface, then maybe there's hope for the Mt. Everest idea. Unfortunately we will need twice the area of Wyoming because of the partial atmosphere! $\endgroup$
    – Bit Chaser
    May 24 '20 at 1:59

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