Bimetal Snap-Disc Thermostat

What is modelled

A shallow bimetal disc is bistable: it can sit dished one way or the other. Heating makes the two metal layers expand differently, which tilts the energy landscape until the disc snaps through; it only snaps back at a lower temperature, which is the hysteresis of a thermostat. The disc opens or closes a contact that switches the heater (or a load).

Equations

• Lumped heat balance C dT/dt = Pheater - h (T - Ta) - ε σ A (T4 - Ta4) with h = hnat + 0.35 · ventilation.
• Disc coordinate: q'' = ω2 (q - q3 + F) - 2 ζ ω q', double-well energy U = -q2/2 + q4/4 - F q.
• The wells merge at the fold |F| = 2/(3 √3), where the disc snaps. F grows linearly with temperature through the centre of the hysteresis band.

Checks run in the page

• Slow temperature ramp up and down: the measured snap temperatures against the fold-point prediction.
• Radiation-free heating against the exponential approach with tau = C/h and the final value T_a + P/h.
• Small-oscillation period in one well against linear theory.

Honest simplifications

The disc is a single coordinate in a cubic potential, not a finite-element shell; the force-temperature link and the thresholds are set by hand rather than computed from layer thicknesses and moduli. The bilayer curvature formula is only an estimate. Heat transfer is lumped with one temperature, and contact bounce and arcing are ignored.

• Wait for several cycles: temperature climbs to the red line, the disc snaps, and it falls to the green line before closing again.
• Raise the hysteresis width: the loop in the right-hand graph widens and the cycle slows.
• Switch off feedback: the heater stays on and the temperature settles above the opening threshold.