Cryostat Heat Budget

What is modelled

Heat reaches the cold stage in three ways: by radiation from the warm wall, by conduction along the supports, and by the thin gas that is still in the vacuum. N floating shields between wall and stage each settle at their own temperature and split the radiation gap into N + 1 smaller ones. The warm and cold temperatures, emissivity, support area, conduction path length and radiating area are those of the 3D page; the shield count, gas pressure and boil-off are added here.

Equations

• Radiation with shields Qrad = σ A (Tw4 - Tc4) / [(N + 1)(2/ε - 1)]; shield temperatures Ti4 = Tc4 + (Tw4 - Tc4)(N + 1 - i)/(N + 1).
• Fourier conduction Qcond = k As ΔT / L with k = 0.22 W/m/K, L = 0.33 m.
• Free-molecular gas conduction Qgas ≈ Λ p A ΔT, Λ = 1.2 W/m2/K/Pa, valid when the mean free path exceeds the gap.
• Boil-off of liquid nitrogen Q / (199 kJ/kg × 0.808 kg/L) per hour.

Checks run in the page

• Shield chain relaxed numerically so every gap carries the same flux, against the formula.
• The Planck spectrum integrated over wavelength against sigma T⁴.
• The Maxwell speed distribution integrated for the wall-hitting flux against p / sqrt(2 pi m k T).
• Steady heat equation on 40 nodes against Fourier's law.

Honest simplifications

The 3D formula uses one constant conductivity and one radiating area for the whole chamber, which this page keeps. Emissivity is the same on every surface and does not change with temperature, shields are ideal thin plates with no extra support or gap leaks, the gas term is only valid at low pressure and is shown beyond its range at the top of the pressure axis, and cool-down time and the cold-stage heat capacity are not modelled.

• Choose No shields and then Six shields: the radiation falls roughly as 1 over N + 1.
• Choose Poor vacuum: the gas term overtakes radiation and conduction.
• Raise the cold stage temperature and watch the total load fall as the temperature difference shrinks.