The idea is to build a stirling engine powered by temperature difference between high altitude air and ground level air.

It is very unlikely I'm ever going to even try building one, but I want to do some feasibility estimates and think about design pathways.

I'm pretty sure it isn't a new idea, but I can't seem to find any info (as usual, I'm probably missing the right keywords), and I have never heard anyone discussing it, so here we go!

Details

Why is the air at high altitude colder?

I mean, in a typical small-scale case, hot air rises to the top. If it's hot below and cold above, convection should turn all the air upside down real quick, so it's extremely hard to keep such an upside-down state. So why does it happen in nature?

Sunlight is absorbed mostly by ground / sea, that is where the air is heated; the air high above radiates its heat into space, as infrared radiation. So, temperature gradient buildup is to be expected.

But why is it not equilibrated by convection? That is because of pressure. If you take a bunch of air and pull it up into the stratosphere, it will have to expand to match with lowering pressure around it, and thus get cooler. So, convection won't start before that adiabatic temperature change is overcome. Or if that adiabatic temperature change is reduced by condensing moisture - then, a thunderstorm is likely to form.

So, we could use that temperature difference to generate power. As a byproduct, we heat the stratosphere, which will then radiate more heat into space, eventually cooling our Earth =)

Let's
assume we put one heatsink on top of Everest, and another one down at sea
level. Yes, this is better than a best-case scenario, but it should at least
give some insights, how stupid this idea really is =)

Let's aim
for output power of 1 GW, which is about the power of a good nuclear reactor
(electrical output).

We have a
temperature drop of about 60 K. I think it's best to let half of that drop to
be left to heatsink-vs-air drops, and half to the engine (this comes from the
idea, that load resistance should match the internal resistance a battery to
get as much power on load as possible).

So the
engine will operate with heater temperature of about 285 K, and cooler
temperature of 255 K. So, a best possible engine efficiency (carnot cycle) is
11.7%. (this is low, so essentially all the heat absorbed by the ground
heatsink will flow into the strato-heatsink)

For 1 GW
output power, we thus need 8.5 GW of heat flux.

For our
temperature drop of 60 K, to achieve that heat flux, we need total air-to-air
thermal resistance be 7.06e-9 K/W. So the thermal resistance of heatsink has to
be 1/4th that, 1.76E-09 W/K.

It’s hard
to calculate the actual size of the heatsink needed, but I can calculate, how
much air does it have to pass through itself, assuming the air comes out with
the temperature of heat carrying liquid.

Heat
capacity of air, Cp, can be calculated from ideal gas laws. It is k*(gamma)/(gamma-1)
per molecule, where gamma is 1.4 for diatomic gases (air is made mostly of diatomic
molecules, so it applies). See Wikipedia. After I did all conversions, I got:

On top: it
takes 4.38E+02 J to heat 1 m3 of air by 1 K.

On ground: 1.17E+03
J/(m3 K)

So, a heatsink
will need minimum airflow of 1.30E+06 m3/s on the mountain-top side, and 4.86E+05
m3/s on the ground.

The average
wind speed on top of Everest seems to be circa 28m/s (100 kph), so the heatsink
must be sweeping an area of 4.66E+04 m2. For ground one, I took a more-or-less
random value I found on the internet for wind speed, 10 m/s (that’s quite high,
very good for wind turbines). The sweep area for ground one is almost the same,
4.86E+04 m2.

Just to put it into perspective, if we are to make the heatsink of square
cross-section, the side of the square is circa 220 m (720 ft). It does sound
like pretty big, but not impossibly big. Constructing such a thing on top of Everest
is going to be super challenging, though.

This looks like about the right size:

So, this was the best case. And it seems pretty impractical already. Now if
I start listing things that I missed, it feels hopeless.

* there is no sea-level place next to Everest, we won’t have this
temperature drop

* there is only one Everest on earth. And the world is not packed with mountains
approaching it, not even remotely!

* I assumed, heat can be transferred to top of Everest with negligible
temperature loss. I have not tried to estimate such a system yet, but I have a
feeling it is going to be epic impractical.

* I assumed constant wind direction, and no wind resistance of the heatsink,
and perfect temperature exchange

* I assumed perfect carnot-cycle heat engine,

* and 100%-efficient electric generator,

* and no power loss for pumping liquid to top of Everest.

* and I bet I overlooked some other losses I didn’t even think of.