The limit of compressors in series What is the theoretical limit of putting compressors in series if each subsequent stage of compressor lives inside the compression tank the last is pumping into?
The gauge pressure experienced by each compressor is surely the same. The outer air tank would need to be very large because you are nesting the system like Russian dolls.
Would each subsequent inner tank need extra strengthening because of the high absolute pressure or would the fact that each tank's gauge pressure is small mean the walls of each tank could be the same?
With the greater than being a compressor into the next tank, the following diagram is kind of what I'm asking:
|--------------|
| |----------| |
| | |------| | |
> > >      | | |
| | |------| | |
| |----------| |
|--------------|

 A: In this idealized picture, there are two basic failure modes to consider. Since you're asking for the theoretical limit, let's suppose that the compressors are running very slowly, so that the heat they generate has plenty of time to dissipate, and that the mechanism doing the compressing works at arbitrarily high pressures. Also assume the air never solidifies, which would happen eventually.
As long as each compressor ultimately generates the same pressure difference across inside and outside, the inner compressor isn't more likely to explode than the outside one, as the net outward force per unit area on the compressor walls is the same. However the walls of the inner compressor are under a much greater total pressure, meaning that they are being crushed very hard from both sides (think putting a solid plate of metal at the bottom of the ocean). Things would go wrong when that total pressure becomes so high that the wall material would fail (maybe it would become brittle, or get squished so thin that it becomes weaker, I'm not sure).
However the most likely scenario is actually explosion of the outer-most compressor, because it would have to be so big! Even though the net outward pressure on it's walls would be the same as the inner compressor, the total force would be huge, and you would need thicker walls to support it.
A: Adding more layers will fail when the nature of the matter of some component changes to such an extent that the compressor (or some other part) no longer functions as required by the compression process or the pressure-holding nature of the wall. And when you cannot compensate for that change within your budget.
Quite interestingly, at reasonably attainable pressures at least, air stays gaseous at room temperature. Some other mechanism will cause failure before air becomes liquid. If your compressors require the working material to stay gas, that will probably not be the limiting process.
There is a hard limit at the point pressures exceed the electron pressure in matter. That is, you could not exceed the pressure that turns ordinary matter into degenerate matter such as in a neutron star. That gives you an upper bound. Regardless of what kind of ordinary matter you made the compressor or container, you could not get above that. The molecules won't stay molecules. The atoms won't stay atoms. Everything gets mushy at those pressures.
However, there are very likely going to be failures long before that. The compression strength of steel is 250 MPa. (Roughly 2470 atmospheres.) That means, at those pressures, steel will begin to compress and get denser. Depending on the nature and construction of your compressors, somewhere round about this point, the compressor won't be able to maintain a seal on the compression mechanism. Whatever shape or design it has will distort under the pressure and move out of the way of the gas being compressed.
Suppose it was based on a piston in a cylinder, with various valves. The piston will get smaller because it is compressed from the outside. The cylinder wall will get thinner because it is compressed inside and out. The inside radius of the cylinder will increase while the outside radius decreases. Any seals on the piston will also compress. So the piston will lose the ability to keep a seal. It will lose the ability to hold back pressure.
Consider a design that only functions when it is compressed. I can't work out what that might be. Maybe you must use remote handling of some kind to assemble the compressors in the high-pressure zones. You build all the parts at ordinary pressure and pre-size them so that when they compress they will fit correctly. You will have to do the same thing for the containers. If they are welded, for example, the welds will compress. You would need to either make sure the welds compress the same way as the surrounding material or weld it at pressure.
You would also need your handling equipment at each stage to be pre-built and assemble it at each stage. The articulations in the moving parts won’t articulate correctly if you assemble them at normal pressure. Plus, the electronics (or whatever) that controls the handling equipment is a challenge. I don't know what maximum pressure electronics can be made to survive.
This process becomes steadily much more difficult. Each layer must be large enough to accommodate all the handling equipment and components of the next layer in. I suppose you only need handling equipment etc. on one side and only big enoough to account for the equipment and the compressor. But it all has to be inside the boundary of the next layer out. It gets bigger, and so more expensive, very fast with layer number.
So, this becomes a question of how much time, energy, and money you are prepared to put into it. Eventually you will exceed some material property, or you will exhaust your budget.
