Helium Reverse-Brayton Refrigerator

What is modelled

High-pressure helium is cooled in a counterflow heat exchanger, expanded through a turbine that makes it colder still, and then flows past the cold load before returning through the heat exchanger. The heat exchanger is never perfect, so a share of the cold is lost; with an ideal gas the cold-end temperature follows in closed form from the balance of the loop. The 3D page shows the plant hardware; the thermodynamic loop is computed here with helium as an ideal gas.

Equations

• Turbine outlet Te = g Ta with g = 1 - ηe (1 - r-0.4); exchanger Ta = Th - ε (Th - TL).
• Load balance Q = m cp (TL - Te), so TL = (Q/(m cp) + g Th (1 - ε)) / (1 - g ε).
• Compressor work Wc = m R Th ln r / ηc, turbine work m cp (Ta - Te), half of it recovered (assumed).
• COP Q / Wnet against Carnot TL/(Th - TL).
• Cooldown M c(T) dT/dt = Q - Qc(T) with Qc = m cp (T (1 - g ε) - g Th (1 - ε)).

Checks run in the page

• Fixed-point iteration of the loop against the closed-form cold-end temperature.
• Runge-Kutta integration of the adiabatic relation against T r^-0.4, and non-negative expander entropy.
• Cooldown integration against the exponential for constant heat capacity.
• Reversible minimum work against the machine work during cooldown, and COP below Carnot.

Honest simplifications

Ideal-gas helium (cp = 5193 J/kg/K, gamma = 5/3), one stage, no pressure drops, balanced counterflow, 300 K warm end, an isothermal compressor efficiency of 0.6, half of the turbine work recovered, and a copper-like power-law heat capacity. Below roughly 20 K real helium departs from the ideal gas and real plants use several expanders; the numbers here teach the trends.

• Press Excellent exchanger and see the cold-end temperature fall sharply: effectiveness is the most sensitive setting.
• Press Poor exchanger: most of the cooling power is lost to the warm end.
• Raise the heat load and watch the cold end warm; the cooldown graph shows how long the load takes to come down.