Deep Seed Vault: Heat Leak and Warm-Up

What is modelled

A long cold tunnel sits in rock that is warmer than the storage temperature, because the ground warms with depth. Heat flows radially inward through the insulation and the surrounding rock, and the refrigerator has to remove that leak plus the internal heat. If the plant stops, the thermal mass of the contents and walls slows the warm-up, which is why a big cold store can ride through an outage.

Equations

• Rock temperature Tg = Ts + g z; radial resistance R = ln(r2/r1) / (2π k L) for each layer, in series.
• Cooling load Q = (Tg − Tc)/R + Qint, power W = Q / COP, COP = η Tc/(Th − Tc).
• Warm-up C dT/dt = U (Tg − T) + Qint, so T(t) approaches its limit with time constant C/U.

Checks run in the page

• Series resistance against a radial finite-volume solution of the conduction equation.
• The warm-up curve against the stepped equation after one time constant.
• The hold time to the alert temperature, closed form against stepping.
• Energy and entropy balance of the refrigerator.

Honest simplifications

The rock is a steady cylinder with a fixed temperature at 60 m radius, with no seasonal or transient ground response, no air exchange and no moisture, and all stored mass is at one temperature. Refrigeration quality is one number, a fraction of the Carnot COP. This illustrates the heat balance and is not a cold-store design.

• Press Thin insulation and watch the heat leak and the electric power climb.
• Press Volcanic ground, hot rock to see how the warm rock raises the cooling load.
• Press Large hall, light contents to see how little thermal mass shortens the hold time.