Can we theoretically guide cold air from high altitude down to the needy cities as free AC?

Question

If you had an insulated tube (approximate size say 15k feet (5km) tall, 50ft (15m) diameter) with the top end open and bottom end closed, would the air in the tube eventually become as cold as the one at 15k ft?

If so, (or if we manage to cool the air inside through other means), when you open the bottom end of the tube, would the cold air come rushing down sucking fresh air from the top, cool everything in its path, eventually get warmer, rise up again and continue the cicle?

We know that air gets colder as it rises to higher altitudes partly because it loses pressure, but the difference in pressure does not account for the whole difference in temperature. Per weather.gov, a lot of the heat from the upper atmosphere is lost to the outer space through radiation. If that were not the case, the whole atmosphere would eventually get hot.
The way I see it, there is an active heater at the bottom of the atmosphere (the earth) and an active cooler at the top (the outer space). After losing the heat, the cold air would come back down, but the reason why it can't is because on its way down it is constantly being mixed with the warm air coming from bellow. The only thing the "insulated tube" would do is to give the air a free passage downwards (imho)

Picture source: https://www.weather.gov/images/jetstream/atmos/energy_balance.jpg.

• Additional thought experiment:
  Say at room temp you have a small tube (1m tall), bottom closed off, with air of freezing temperature inside. When you open the bottom of the tube, the air rushes down from the tube.
  If you do the same with a 10m tube, the air will come down the tube with a greater speed and force.
  Would the same not be true for a 5km tall tube?

In case the 5k vertical tube works as stated above, would changing the direction and lenght of the tube to where it would go from the valley city sideways and slightly upwards up the surrounding hills and mountains through tunnels and bridges to the top of a 5km nearby mountain also work? Would the sheer lenght of such tube (hundreds of kilometers), the angle and the winding shape be an obstacle in a significant way?

Answer 1

Believe it or not, the air at the bottom of the tube would be just as hot at the air at the same height outside the tube. In air that's well mixed, the temperature decreases as you go up at a rate called the adiabatic lapse rate. A parcel of air displaced upward expands due to the lower pressure, and cools adiabatically as it does. Similarly, displacing a parcel of air downward causes its temperature to increase. If you got air flowing in your giant tube, its temperature would increase during the trip down just because of the increasing pressure, and instead of a refreshing cool breeze coming out of the bottom, you'd have more hot air. If the atmosphere is not well mixed, but stably stratified, meaning the the temperature decreases with height more slowly, the air coming out of the bottom of the tube would be even hotter.

For an adiabatic change in pressure (adiabatic means no heat added or removed from the gas) $P^{1-\gamma}T^\gamma$ remains constant during the process, where $P$ is pressure, $T$ is temperature (in Kelvin) and $\gamma$ is the ratio of specific heats ($\frac{7}{5}$ for air). This means, if you increase the pressure of a parcel of air by a factor of 2 (say from half an atmosphere to 1 atmosphere), you change the temperature by a factor of $2^{2/7}$, or $1.219$. So, if the air at half an atmosphere has a temperature of freezing (273 K), then after pressurization to one atmosphere, the temperature will be about 333 K. That's 60 C or 140 F: quite hot.

Answer 2

I'm guessing here but I think the temperature throughout the tube would drop over time as long as the insulation was effective enough to prevent outside heat from heating the tubes contents faster than the cold air inlet at the top was able to cool it down.

If you reversed the entire system and closed off the end at the top, left the bottom open, and put a torch under it, I think it would heat the air up inside from top to bottom, assuming good insulation. A hot air balloon flies that way by using burners to heat air that rises into the balloon through an open hole in the bottom but can't excape once inside. Hot air is less dense so it rises to the top of the balloon and accumulates. It can be vented by opening the 'deflation port' at the top of the balloon but that's only used close to the ground to force the balloon to settle quickly and solidly onto the ground so it isn't blown sideways across the ground by crosswinds.

Atmospheric pressure is caused by the weight of the air due to gravity. Since the air inside the tube would be colder, it would also be heavier and thus the pressure inside would be higher than the pressure outside so the tube would try to expand if it were made of a flexible material. The whole thing would have to be supported though so, unless it was made of a very strong rigid material like a super skyscraper, it would have to be supported from the top. The only way that's currently theoretically possible, I think, would be with carbon nanotube cables attached to an orbiting satellite of sufficient mass and velocity to create the centrifugal force needed to offset the weight of the tube. That's the basis for an earth surface to space elevator but hasn't been done yet. Carbon nanotubes are supposedly one of the few or only materials with the tensional strenth/weight ratio high enough to make it possible but manufacturing rates are currently too low to be practical for something that large.

https://en.wikipedia.org/wiki/Space_elevator

Now that I think about it, one of the problems with the theoretical space elevator is developing the mechanism that interacts with the nanotube cable(s)s to crawl up the cable with a payload. They were talking about a roller type system similar to a clothes wringer that would spin the rollers and propel the mechanism with a payload upward but the rollers would have to be rotated with some form of energy until it got to the geosynchronous orbit level. That's alot of energy. If you had a nanotube cable array supporting the tube with supercooled air inside, you could craft a tall, thin balloon with helium inside the tube and the cold air would create substantially more lift thus allowing for a heavier payload though the altitude limitations would likely still be substantial unless you were able to pump cold air up and then into the tube. Instead of a ceiling of 120,000 feet or whatever the record is now, you might be able to extend that by substantial multiples but it would take energy to pump the air beyond the outer atmosphere. It would still be a free ride though for the first part of the trip. Theoretically, I guess you could pump air up and into the tube all the way to geosynchronous orbit without additional energy. At such high altitudes, there's alot of solar energy available to run solar panels to provide the electricity required to drive the air lift pumps though. Maybe another way to get a payload into space without excessive energy costs.

Expanding on that, maybe just make the tube itself out of nanotube or graphene sheet. Since it's so strong anyway, it would help contain the tremendously increased air pressure against the inside walls of the tube at lower altitudes due to the massive column of near absolute zero air temperature contained inside.

Answer 3

This would depend on details.

Is the tube so well insulating that there is no heat transfer across its walls and the gas inside does not absorb energy from EM radiation due to sun and atmosphere around? Then yes, the air inside may get colder than the air outside. This would make it denser and thus the column of air inside the tube would be heavier than what ground-level pressure could support. Air in the tube will start moving downwards, at the bottom it will be expelled on the ground, where it will heat up and rise again. The tube would create local disturbance in the troposphere, a local weather pattern.

If the tube insulation is bad, the state of the air inside will largely be influenced by state of the air outside, and it won't be much different than outside. Occasional movements in both directions could happen, just as they do in atmosphere in the course of the day.

EDIT

What Ben51 is describing is an idealized fictional case where the air that comes up only loses energy by doing work on the other air, no heat transfer to the rest of the atmosphere can happen. In that case, if the air was to somehow come back down, it would regain its original temperature on the ground. In reality, a dry hot air will lose more energy than that because of heat transfer and thermal radiation. At some point, similar amount of air will come back down, only colder.

Convection "hot air up, cold air down" will occur even without the pipe. The upper atmosphere is a kind of chiller for the air there and also for the air below it. The pipe can only make the convection more predictable and perhaps make the air a little colder at the bottom. In reality there is also water vapor and its phase change in the upper troposphere where the air is cold enough as Whit3rd noted. This change removes water from the air and releases a lot of heat. Some of this heat goes back to the cold air and this makes things for humid air more complicated and less predictable - "weather".

Answer 4

No, the density is different, the presure near the surface area is higher.

To achieve the process, one might consider compress the air into denser ball. i.e. use some evaporative bubble with high surface tension to temporary trap the air and compress or drage the air by its own weight down.

You need a higher density.