Here’s the premise.

When you compress a gas, it experiences adiabatic heating. For a given reduction in volume, the most work you can do would be adiabatic compression, where none of the heat can dissipate. The least work you can do for the same compression is isothermal compression, where all of the heat due to compression is dissipated as the gas is compressed. If your compressed air tank holds air at scuba-tank pressure (200-300 bar) at room temperature, the compressor would have to dissipate a huge quantity of heat from the compression of the gas; that molar quantity of gas, if compressed adiabatically, would be over thousand degrees. (In fact, at compression above 250 bar, the ideal gas law no longer works as an approximation of the behavior of air.)

All of the heat dissipated during compression is energy lost. The storage tank will be hot after being filled with compressed air, and it will dissipate energy as it cools down until it matches ambient temperatures. However, there are even more losses during decompression: gases experience adiabatic cooling when decompressed, especially when decompressed too quickly to absorb significant quantities of heat from the surrounding, such as when driving pistons in a compressed air engine. Your compressed air storage tank will get extremely cold as you let air out. As you know from thermodynamics, adiabatic expansion is the *least* work a volume of gas can do, yet the entire operation of a compressed air engine is in the realm of adiabatic expansion (if you’re not going to use fuel to heat the compressed air).

If you put heat absorption fins on your compressed air tank to absorb as much ambient heat as possible as it decompresses and somehow manage to achieve near isothermal expansion, at best you lose all the energy dissipated as heat during compression. If your tank is insulated, or doesn’t have a high surface area to volume ratio, at worse you lose all the energy between the adiabatic expansion curve (on a PV diagram, starting at ambient temperature and the compressed volume of a full tank) and the adiabatic compression curve (starting at the intersection of ambient temperature, the original volume of all the air that went into the tank). This is a huge quantity of energy lost. A casual inspection of a graph of these curves shows that much more energy is lost than is stored and recoverable even in the best case scenario.

(If you have a Mac, I can send you the Grapher file with my calculations.)

Have you taken the losses due to dissipation of heat into account?

Let me drop one intriguing tip to consider. If it entices you, you know where to contact me. ^_^

Use a graphite piston perfectly fitted into a smooth fused quartz cylinder. (They have virtually the same thermal expansion coefficients in operational temperature ranges.) No seals. No static friction.

(Chill the piston in dry ice or liquid nitrogen to get it to shrink so you can get it into the cylinder.)

This is a great age to be working on real work, when many people our age have no idea what they want to do, and older people have already given up on making anything useful. Onward ho!