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The science
What is stored. In adiabatic compressed-air energy storage (A-CAES) surplus electricity runs a multi-stage compressor. The air leaves each stage hot; instead of throwing that heat away, an intercooler moves it into a thermal store. The cooled air goes into an underground salt cavern. When power is needed the air is released, reheated from the store and expanded through turbines.
Compression stage. For an ideal gas with k = (γ-1)/γ = 0.2857 and a total pressure ratio π = pcavern/p0 split equally over N stages, r = π1/N:
Tout = Tin [ 1 + (rk - 1) / ηc ], w = cp (Tout - Tin)
The intercooler has effectiveness ε: Tcooled = Ta + (1 - ε)(Tout - Ta). Heat cp(Tout - Tcooled) per kg is stored. The electric power is P = ṁ Σw / ηm.
Cavern. A rigid cavern of volume V holds mass m at temperature T. The first law for an open system with inflow enthalpy cpTin and heat flow hA(T - Twall) to the rock is
m cv dT/dt = ṁ (cp Tin - cv T) - hA (T - Tw) (charge), m cv dT/dt = - ṁ R T - hA (T - Tw) (discharge), p = m R T / V
The pressure therefore slides between pmin and pmax and the machines work at the instantaneous ratio. Charging heats the gas, discharging cools it, which is why the wall heat transfer matters.
Discharge. Air from the cavern is reheated to Tin = max(T, Ta + ε(Tstore - Ta)) before each turbine stage and expands to Tout = Tin[1 - ηt(1 - r-k)], giving w = cp(Tin - Tout). The store is stratified by cavern pressure: the heat taken from stage j at a given pressure is returned at the same pressure, as in a real packed-bed or oil store. It also decays toward ambient with time constant τ.
Round-trip efficiency is ηRT = Eout/Ein over a charge-hold-discharge cycle. In the reversible limit (η = 1, ε = 1, no store loss, isothermal cavern) every stage is run backwards, so ηRT = 1. Everything below that is lost to machine irreversibility (ηc, ηt, ηm), to heat that the exchangers fail to capture (the (1 - ε) share dissipates in the cavern and the rock) and to store decay.
Checks (run in the page). (1) the isentropic compression work from the integral of v dp against cpTa(πk - 1); (2) the cavern first law over a charge: the change of internal energy against the integrated enthalpy inflow minus wall heat; (3) the reversible limit giving ηRT = 1; (4) p = mRT/V after a full cycle.
Simplifications. Air is a calorically perfect ideal gas (cp = 1004.7 J/kg/K); pipe and valve pressure drops, humidity, compressor surge limits, a part-load efficiency curve, thermal stress on the salt and brine creep are ignored; the cavern is one lumped node at uniform T; the thermal store is a stratified tag table, not a full heat-transfer model; the equal pressure ratio per stage is an assumption (optimal for equal efficiencies). Real projects target 60 to 70 percent. The model is educational and is not affiliated with any plant or vendor.