Target-Wall power-

Cryogenic Cooling Cascade — 2D Ladder

A cryostat is a ladder of plates at 300 K, 77 K, 4.2 K, 1.2 K, 100 mK and 2 mK. Heat leaks down the wires and by radiation, and every stage must pump its own load plus all the heat dumped on it from below, paying Carnot work divided by a second-law efficiency.

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Presets

Load

Target stage (1 = 77 K ... 5 = 2 mK)

Load at the target (log10 W)

Heat leaks

Wires into the cold space

Shield effective emissivity (log10)

Refrigerators

Second-law efficiency scale

ideal (reversible, no leaks) flow animation

Live readouts

Checks

The science

Carnot limit. To lift heat Q_c from T_c to T_h an ideal machine needs W = Q_c (T_h/T_c - 1). Real machines reach a fraction eta of that (second-law efficiency); here W_i = Q_c,i (T_(i-1)/T_i - 1) / eta_i.

Cascade. Stage i dumps Q_c,i + W_i onto stage i-1, so the heat to be lifted is accumulated from the cold end upwards: Q_c,i = P_i + (Q_c,i+1 + W_i+1), where P_i is the parasitic load on plate i (plus the external load on the target plate).

Parasitic loads. Wires with k = k1 T conduct (N A / L) k1/2 (T_h^2 - T_c^2). Radiation between shields is sigma eps A (T_h^4 - T_c^4); eps is an effective value for multilayer insulation. Plate areas shrink with temperature. Below 1 K the real wiring uses superconductors and the thermal-link physics differs, so the numbers are indicative.

Efficiencies. Defaults (0.20 at 77 K down to 0.005 at 100 mK) are typical orders of magnitude for pulse-tube, Joule-Thomson, pumped-He and dilution stages, not measured data; the last stage (adiabatic demagnetisation) is treated as a single-shot refrigerator with the same formula for comparison.

Reading the result. The wall power grows roughly with T_h/T_c, so a colder target costs orders of magnitude more per watt of cooling. Reversible limit: an ideal cascade needs the same work whatever the number of stages.