Why the yolk stays hot after the shell goes cold.

Two heat-transfer mechanisms are at work, not three: from water to shell, heat moves by convection — boiling water and running tap water are both turbulent enough that the shell's surface sits within a degree of the water temperature within moments, in this model treated as instant. From the shell inward to the yolk, there is no fluid motion to carry heat, so it moves by conduction alone through egg white and yolk — far slower, and the process this whole page actually models. Radiation is negligible here: at these temperatures and this size, radiative heat exchange is orders of magnitude smaller than either of the other two and is left out entirely. That two-step picture (fast convection outside, slow conduction inside) is exactly why it's reasonable to treat the shell as clamped to the bath temperature and solve only for how long conduction takes to catch up at the yolk.

The equation above is physicist Charles D. H. Williams' published egg-boiling formula (University of Exeter), reproduced as written rather than rearranged. It comes from the standard one-term solution for transient conduction in a sphere whose surface is held at a fixed bath temperature, evaluated at about 69% of the way from the centre to the shell — not the exact geometric centre, but where a real egg's yolk–white boundary sits — and simplifies to the form above once the sphere's radius is rewritten in terms of mass, r = (3M⁄4πρ)^1⁄3. Internally, this page draws its temperature-vs-time chart from the equivalent temperature form T(t) = T_w + 0.76(T₀−T_w)·exp(−π²αt⁄r²), with α = K⁄(ρc) — the same equation, just solved for temperature at any moment instead of for the moment a target temperature is reached. Exeter's own page doesn't publish numeric values for c, ρ and K, but a table of Williams' constants — c = 3.7 J⁄g°C, ρ = 1.038 g⁄cm³, K = 0.0054 W⁄cm°C, and separately c = 2.7, ρ = 1.032, K = 0.0034 for the yolk itself — is published, with a citation back to him, at calculatinghistory.com's egg slide rule. This page uses the egg-white (albumen) values, since it's the white the heat actually has to cross to reach the yolk boundary being tracked, not the yolk's own properties. That same source gives Williams' own target yolk-boundary temperatures — 62°C soft, 65°C intermediate, 69°C hard-boiled — which this page's doneness picker uses directly; with those numbers, this formula reproduces three of Exeter's own four worked fridge-egg examples (57 g: 4.4 min vs. his stated 4.5; 47 g: 3.8 vs. 4; 67 g: 4.9 vs. 5) to within a few percent, though his one room-temperature example runs about 13% off — consistent with his figures being rounded to the nearest half-minute in a popular-science write-up rather than stated to full precision. The exact same equation, with the water and starting temperatures swapped, runs the cooling phase — the "ouch" and "safe to peel" readouts are the same t solved for T_y→45°C and T_y→35°C instead of T₀→T_y. The plotted yolk curve sums the first three terms of the full Fourier series rather than just one, so it starts smoothly at the true initial temperature instead of jumping straight to 76% of the way there; the closed-form readouts still use the one-term version, which is accurate everywhere except the first few seconds. Since M ∝ r³ at fixed density, M^2⁄3 ∝ r² — doubling the egg's radius itself, not just its mass, would quadruple every time on this page.

The altitude picker changes T_w because that is the physical mechanism: lower atmospheric pressure at altitude (about 80 kPa at 2,000 m versus 101 kPa at sea level) lowers the temperature at which water boils, via the Clausius–Clapeyron relation — there is no separate pressure variable here because pressure's only effect on this model is that one temperature. The shell line on the chart is deliberately drawn as a simple fast exponential (12-second time constant) toward the bath temperature, illustrating the convective assumption rather than a separately measured or derived surface response — it is qualitative, unlike the yolk curve. The model also treats the egg as a uniform sphere dropped straight into water already at a full boil (or a tap already at its set temperature): it ignores the egg's true ellipsoidal shape, the insulating shell and air pocket, and the slower cold-start method of bringing water up to temperature with the egg already in the pot, all of which push real cooking times a little higher, especially for very large eggs like an ostrich's. A still bowl of cooling water (rather than running water) would also cool the yolk more slowly than shown, since the bath itself would warm as it absorbs the egg's heat instead of staying clamped at the set temperature.