Fill a compressed-hydrogen vessel and watch pressure, temperature and density. A real gas is denser than an ideal one, and gas rushing in heats the tank.
The science
Real gas (Abel-Noble). p (v − b) = Rs T with specific volume v = V/m and co-volume b ≈ 7.69·10⁻³ m³/kg [verify against a source]. Compressibility Z = 1 + b p/(RsT); at 700 bar and 15 °C Z ≈ 1.46, so the real density (≈ 40.5 kg/m³) is far below the ideal-gas 58.9 kg/m³. The reference value (≈ 40.2, NIST) is checked on the page.
Energy balance. For Abel-Noble gas the internal energy depends only on temperature, u = cv T, and enthalpy is h = cp T + b p. A filling tank obeys m cv dT/dt = ṁ (hin − cv T) − UA (T − Twall). With no heat loss and a constant inflow the exact solution is T(m) = hin/cv + (T₀ − hin/cv) m₀/m, which the ODE solver is checked against.
Why a tank heats. Flow work and the extra enthalpy b pin of high-pressure inflow turn into thermal energy; lower supply pressure (cascade) and pre-cooling lower the temperature. Real standards also limit the gas temperature (about 85 °C) [verify against a source].
State of charge. SOC is stored mass divided by the mass of a full tank at nominal pressure and 15 °C; a hot fill reaches the target pressure with less mass, so SOC drops as the gas cools.
Energy density. H2 lower heating value ≈ 120 MJ/kg (33.3 kWh/kg) but only ≈ 3 kWh/m³ at 700 bar; the fuel cell efficiency is taken as 50 %.
Simplifications. Constant cv, one lumped gas node and one wall node, no Joule-Thomson cooling or ortho-para conversion, assumed heat-transfer coefficients and wall mass, banks of infinite size, no real fill protocol.