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The science
Why a geyser is possible: boiling point depends on depth
Water boils when its temperature reaches the saturation temperature of the local pressure, P(z) = P0 + ρ g z. The saturation curve is computed from the Antoine equation (checked against steam tables in the checks). At Yellowstone altitude P0 is about 0.77 bar so the vent boils at about 93.5 C, while 22 m down the water must exceed about 133 C to boil. Superheated deep water is stable only while the column above holds the pressure.
Recharge and trigger
Heat enters at the bottom and spreads upward by conduction and (restricted) convection, ∂T/∂t = κ ∂2T/∂z2 + heating - surface loss. A narrow, convoluted conduit means small κ, so the deep water heats faster than it can mix away: the T(z) curve approaches the boiling curve. When any depth touches Tsat(P(z)) it flashes, pushes out the water above, pressure drops further, and the eruption propagates downward.
Eruption energy
Flashing to the surface boiling point makes a steam mass fraction x = cp(T - Tsat,0)/L (L = 2.26 MJ/kg). Jet speed is estimated as v = √(2 ε x L) with a small conversion efficiency ε (calibrated so Old Faithful-like conditions give tens of metres) and height v2/2g.
Honest simplifications: 1D water column, constant material properties, Antoine saturation curve, no dissolved gases, no side-vents or underground chambers (real Old Faithful has a complex plumbing), calibrated eruption efficiency, and an accelerated time scale during the eruption so it can be watched.
What to try
- Raise the constriction slider to 3 m2/min: free convection removes the superheat and the geyser stops.
- Lower surface pressure (altitude) and the vent boiling point falls, which changes steam quality and jet height.
- Watch the T(z) curve approach the dashed boiling curve just before each eruption.