### Underwater Compressed Air Maths

The energy stored in a normal, constant volume compressed air reservoir is

pvln(p/a)

where p is the pressure in the reservoir, v is the volume of the reservoir, ln means natural logarithm and a is atmospheric pressure.

It’s more in our underwater constant pressure reservoir:

pv(1 + ln(p/a))

– not only are you compressing the air into the vessel, you’re also displacing that volume of water at that pressure; or looked at another way, the pressure delivered remains constant, rather than decreasing as the vessel empties.

These formulae are correct if the heat generated during compression is dumped to the surroundings or a heat store, and heat is supplied from the surroundings or a heat store during decompression (isothermal process). Less energy is stored if the heat is retained in the compressed air (adiabatic process). In the isothermal case, the thermal resistance of the heat sink/source is a major determinant of the system efficiency; in the adiabatic case, storage time and the effectiveness of the insulation around the reservoir are the major determinants. Thus an isothermal system would be the preferred type for long-term storage, whereas an adiabatic system might be preferred for short-term storage (for reasons too detailed for this essay, you probably wouldn’t use submarine compressed air for short term storage in the UK – although you might use the same system in lakes).

It’s worth noticing that the mgh of the pumped hydro system is the same as the pv of the isothermal underwater compressed air system:

pv = hρg * v = ρv * hg = mgh

where ρ is the density of the water. Thus, if ln(p/a) is large (that is, the reservoir is deep underwater), the volume required for an underwater compressed air reservoir will be very much less than the volume of a reservoir for a pumped storage system, for the same value of h. (p/a is close to (depth in metres)/10.)