Cooling Power of a Dilution Refrigerator
The cooling power at the mixing chamber depends on helium-3 circulation rate and temperature, following an approximately quadratic law that sets what loads a fridge can absorb.
A predictable law
The useful cooling a dilution refrigerator can provide at its coldest stage is not arbitrary; it follows from the enthalpy difference helium-3 gains crossing into the dilute phase. To a good approximation the cooling power at the mixing chamber equals a constant times the helium-3 molar circulation rate times the difference between the square of the mixing-chamber temperature and the square of the temperature of the helium-3 entering the chamber.
The working formula
A widely used form is that the cooling power in watts is about 84 times the circulation rate in moles per second times the mixing-chamber temperature squared in kelvin, once the incoming helium-3 is well precooled. The T-squared dependence is the crucial feature: cooling power falls off rapidly as the temperature drops, which is why base temperature is limited by residual heat leaks. Doubling the circulation rate doubles cooling power, but the returning stream must still be cooled adequately by the heat exchangers.
A worked estimate
At 100 millikelvin, T squared is 0.01. With a circulation rate of, say, 500 micromoles per second, cooling power is about 84 times 0.0005 times 0.01, roughly 420 microwatts. This is the order of magnitude of what a large fridge provides at 100 millikelvin, and it must accommodate all wiring and device dissipation at that stage.
def cooling_power_uW(ndot_umol_s, T_mK):
ndot = ndot_umol_s * 1e-6 # mol/s
T = T_mK * 1e-3 # K
return 84 * ndot * T**2 * 1e6 # microwatts
print(round(cooling_power_uW(500, 100), 1)) # ~420 uWDesign implications
Because the returning helium-3 must be precooled to near mixing-chamber temperature for the formula to hold, heat-exchanger quality and circulation rate together bound performance. A fridge with excellent exchangers but a small pump, or a strong pump but poor exchangers, will underperform. Balancing the two is the essence of designing for a given cooling power.
- Cooling power grows with circulation rate and temperature squared
- Roughly 84 times circulation rate times T squared in watts
- Falls off rapidly as temperature drops toward base
- Set jointly by pump capacity and heat-exchanger quality