There was an analysis done on the theoretical (Carnot/ 2nd Law) limits of using Earth's infrared emissions in this way: https://www.pnas.org/content/111/11/3927.abstract (Roughly 4 W/m2 for a system that purely exploited the radiative mismatch between outgoing and incoming long-wavelength radiation from the sky.

The bigger limit in our case is that we're using a thermoelectric generator - and achieving a relatively small temperature difference. We argued in the paper it might be possible with improved engineering and more favorable weather conditions to push performance to 0.5 W/m2.

In general, solar gets you far more power than this method ever will. The only advantage to combining the two might be to provide incremental power at night that improves the overall energy economics of the footprint associated with the solar panel.

And yes, a heat source would improve the power output. This has been the approach of an entire field of research that one might term 'waste heat recovery'. This encompasses everything from industrial sources to the human body or a campfire. The advantage, such as it is, of what we've done is that you don't need a source of heat besides the air itself.

> Is there a theoretical limit to the device's performance?

Yes.

Let the night-time equilibrium temperature be T_C (temperature_cold). Let the heat reservoir temperature be T_H (temperature_hot). The maximum theoretical efficiency is equal to 1 - T_C / T_H. This is from Carnot’s theorem and the 2nd law of thermodynamics.

The wasted energy is radiated off into space. You can calculate this with the Stefan–Boltzmann law. At 10°C we get 4.6 mW/m^2. (Edit: Whoops, bad arithmetic. Ignore these numbers. Do the math yourself.)

If your heat reservoir is 25°C and your cold temperature is 10°C then you have an efficiency of 5.0%. So you would generate 0.24 mW/m^2 at maximum theoretical efficiency.

You can even solve here for the optimum night-time temperature. Too cold and not enough heat is radiated. Too hot and the efficiency suffers. There is a maximum in the middle (but I am not going to do the math).

There are other interesting calculations I’m sure you can do to figure out maximum and minimum reservoir temperatures, but the challenge here is that you don’t want to harness sunlight to heat up your reservoir—you want to use existing heat that you have lying around.

Apparently, with our atmosphere we can achieve something like 40°C cooling in ideal conditions, and it is claimed that 60°C is possible. Back-of-the-envelope math suggests that you would achieve maximum theoretical power at around ~60°C difference.

With a reservoir temperature of 25°C my estimate is around 40W maximum power (with the correct arithmetic). You can get more power with a hotter reservoir.