Here's my analysis from maybe wrong principles. If you have slightly more dense air beneath slightly less dense air, the air will experience a net force upwards. If this net force is stronger than gravity, then the air will experience upwards acceleration. This will continue as long as there is a difference strong enough. At the top of the tube, there is no more force since the density will be the same since the air will spread out after exiting. (If it's not already at the same density after going through the tube).
Looking at it this way this seems sound to me, am I wrong?
Air is compressible. And because of that the mass of air that is being pulled toward Earth by the gravitational attraction of the rest of the planet means it compresses as much as it can as close to the surface as it can get, and then becomes less and less dense as you increase the distance from the center of the Earth.
This is nominally a stable state. Air doesn't move.
When you inject energy into air, that increases the kinetic energy of the air molecules. They bounce off each other and push themselves apart. As a result they have "more space" between them that air molecules than ones that aren't currently heated. What will happen then is the 'less energetic' molecules will 'fill in the spaces' left by the more energetic ones bouncing apart. You might visualize this like sand filling in a hole you are digging by throwing other sand up into the air.
If you do nothing, more energetic molecules end up higher up, and less energetic ones end up lower. Colloquially, "hot air rises". And if you constrain it in an envelope of some form (a balloon for example) then you can create a mass of air that has a lower average density than the air around it and that results in a lifting force.
Here is the rub, as long as you put heat into the air it will stay less dense and your balloon will stay in the air. However, stop adding heat (energy) and the balloon cools and begins to sink. Finally, there is a point where you cannot add any more energy to the air to get it less dense than the air around it. In balloonist terms that is your balloon's ceiling height. 2004 record was 4.1 miles[1]. This is a balloon where you have a super hot flame shooting up into it, and it won't go up any more because you have reached the point where you cannot put enough energy into the air to make it less dense than the surrounding air.
As a result, if you surround a column of air, it might initially rise because energy inside the column is unable to diffuse into the air around the column, but it will only do that until it reaches a new equilibrium point. Early on in the process not being able to spread out allows the air to keep its heat, but at it moves up in the tower/chimney the chimney prevents it from becoming less dense, so relative to the air above it, it gets heavier and heavier per cubic foot. These two effects balance out and the air stops moving.
I feel like a bunch of people are skirting right around this but this was unclear to me.
The design calls for an absolutely enormous taper at the bottom called the collector. That land area is covered surface, so there is a huge volume of hot air that isn't mixing around the column near the base. I assume this is how they get around the symmetry issues of a uniform cylinder.
I had to go to one of the publications to find a close up photo with much clarity, maybe this will help others to notice.
There are sites promoting that idea, but TFA (or the video on the front page, anyway) claims the idea will work without any kind of energy collector at the base.
But isn't the idea that you would end the column well before the air in it reaches equilibrium, and that way you would have a major temperature pressure gradient from the exit into the surrounding atmosphere, which would create a continuous flow?
Sure you would want to, but there aren't any solutions to the equations where those conditions are met.
Think if it in terms of energy in versus energy out. Where does the energy come from? Well the author is attempting to use the latent heat in the air. So you have to ask how much energy is there? You can start with Carnot and his principles of heat engines, but you also have to consider the ideal gas law, the temperature of the air goes down with a decrease in pressure. How much? Well if you look at the typical math the loss in temperature from the decrease in pressure between the top and the bottom of the chimney is exactly equal to the temperature difference measured at the top and bottom of the chimney. That situation is true because there is no net input of energy to raise the temperature of the air. So this relationship holds from altitude 0 to the bottom of the stratosphere. At the start of the stratosphere the atmosphere gets hotter as you go up[1].
Solar energy due to ground warming is contributed evenly for a first approximation across solid ground (note that cities are hotter than vegetation etc). There is no way to 'preferentially capture' just the warm air at the base of the chimney (see Maxwell's Demon[2])
At the end of the day, there is no "excess energy" to harvest there.
[1] "Within the stratosphere temperatures increase with altitude (see temperature inversion); the top of the stratosphere has a temperature of about 270 K (−3°C or 26.6°F).[5] This vertical stratification, with warmer layers above and cooler layers below, makes the stratosphere dynamically stable: there is no regular convection and associated turbulence in this part of the atmosphere." -- https://en.wikipedia.org/wiki/Stratosphere
That doesn't quite add up to me. After all, we extract energy from heating differentials all the time in the form of windmills. This really just seems like a scheme for channeling what is basically the same air circulation. The argument being that ground-warmed air doesn't typically rise in a straight line, and so by preventing lateral convection, you'll get a much faster flow up the chimney.
The use of it for wind energy is less interesting to me, though, than the idea of getting hot air radiating further from the surface of the Earth. If that is indeed significant in how the greenhouse effect works, that alone seems worth considering.
My layman's understanding is that wind is caused by pressure differences that result from different regions on the ground heating up differently in the sun. But the pressure difference between the ground and air higher up is just right to balance out gravity, so on average you should expect no movement at all.
Additionally, windmills work without any kind of tunnel, so you should expect this tower to work just as well if you just pointed a few wind turbines downward on a pole. But I think the horizontal winds would be much stronger, so you should readjust the direction a bit, which leaves you with a really tall windmill.
My understanding is that this balancing out with gravity would be true, but for horizontal mixing that occurs, as well as downdrafts of cold air and radiation (which is significant enough to motivate the chimney idea, according to the author). So you have this very complex voyage of ground-heated air upward, which is responsible for most of what we think of as local weather.
I understand the idea is to streamline the updraft, essentially controlling weather in the immediate vicinity.
Looking at it this way this seems sound to me, am I wrong?