Heat a steel bar to a glowing colour, forge a horseshoe, quench it and temper it. Everything you see follows the heat-conduction and metallurgy model below.
Left = surface, right = core. Dashed = annealed hardness, top line = fully hard martensite.
The science
1. Heating: conduction inside, radiation + convection outside. The bar is a slab of thickness t heated from both faces. Inside, ρc(T) ∂T/∂x² follows Fourier’s law; I solve it with a 16-cell finite-volume mesh (backward Euler, temperature-dependent k and c including the latent-heat bump at A1).
ρ c ∂T/∂t = ∂/∂x ( k ∂T/∂x )
q = h_c (T_gas − T_s) + F ε σ (T_fire⁴ − T_s⁴)
Radiation from the coals dominates above ~600 °C. The fire temperature follows the bellows blast (about 700 to 1450 °C); the view factor F is how buried the bar is.
2. Biot number. Bi = h Lc / k with Lc = t/2. Below ~0.1 the whole bar is at one temperature (lumped capacitance, T = T∞ + (T₀−T∞) e−t/τ, τ = ρ c Lc/h). In a water quench Bi is well above 1, so the surface is far colder than the core — the reason a thick bar hardens only in a shell. The lumped formula is used only as an independent check (see Experiments).
3. Glow colour. Planck’s law B(λ,T) = 2hc²/λ⁵ · 1/(ehc/λkT−1) is integrated against the CIE colour-matching functions (analytic fits) and converted to sRGB. A hot body becomes visible near the Draper point (about 525 °C in the dark; brighter surroundings need more heat). Wien’s law λmax = 2898 µm·K / T shows that even at 1200 °C the peak (~1.97 µm) is infrared; only the red-orange tail is seen. The displayed colour includes a mild eye-adaptation lift so 1200 °C reads yellow as it does to a smith; real colour names depend on lighting and the observer.
4. Phases (simplified Fe–C). Austenite forms above A1 = 727 °C and is complete above Ac3 ≈ 910 − 203√C (above Acm for hypereutectoid steel); on heating the austenite fraction follows the lever rule with a few-second time constant. A smith checks this with a magnet: steel stops being magnetic near the Curie point (768 °C), a convenient “critical” sign.
5. Quench: martensite vs. cooling rate. Diffusional (pearlite/bainite) transformation starts when Scheil’s additivity sum ∫ dt / τ(T) = 1, with a C-shaped τ(T) whose nose sits near 550 °C. If the cooling curve passes the nose before the sum reaches 1, austenite survives until the martensite-start temperature Ms ≈ 539 − 423 C (°C) and then transforms by the Koistinen–Marburger law f = 1 − e−0.011 (Ms − T); what is left is retained austenite. Alloying (the oil-hardening grade) moves the C-curve to longer times, so a gentler oil quench is enough. Low-carbon steel has a very short nose and a low-carbon martensite, so it will not harden.
6. Quench heat transfer. h depends on the surface temperature through a boiling curve: a vapour blanket (film boiling, low h) above a Leidenfrost-like temperature, a peak in nucleate boiling, then convection. Water and brine give h of order 103 to 4×104 W/m²K, oil 102 to 2.6×103. Warmer or still baths keep the vapour blanket longer. The curves are illustrative shapes, not measured data.
7. Hardness. Martensite hardness rises with carbon (about 400 HV at 0.2 % to 830 HV at 0.8 %) — a trend table from published curves. Annealed hardness ≈ 90 + 140·C (HV). Rockwell C is converted from HV with a piecewise-linear standard-conversion trend and is meaningful only above ~20 HRC. The estimate is for a plain bar, not a tested part.
8. Tempering and colours. The Hollomon–Jaffe parameter P = T(K) (C + log₁₀ t[h]), C ≈ 20, collapses time and temperature: the same P gives the same softening. Retained hardness is a sigmoid of P (so 200 °C for an hour keeps ~90 %, 400 °C for an hour about a third of the quench excess). For changing T the model tracks an equivalent isothermal time. The oxide that grows on bright steel is a thin film; its thickness (taken as a function of the same P, calibrated to the textbook colours) gives interference colours: reflectance of a film of index 2.3 on steel, R = |(r₀₁ + r₁₂ e−2iβ)/(1 + r₀₁ r₁₂ e−2iβ)|², β = 2π n d/λ. Straw (~230 °C), brown, purple, then blue (~300 °C). Black forge scale from the fire hides these colours, so polish the bar before tempering.
9. Hammering is visual only: it changes the shape (bar to horseshoe) but not the temperature or the metallurgy. Forging below ~700 °C (blue heat) risks cracks and above ~1300 °C the steel burns — both are flagged.
Honest simplifications. One-dimensional heat flow (edges and ends ignored), fixed material fits, no phase-transformation latent heat in the quench except the A1 bump, no distortion or cracking mechanics (only a thermal-stress indicator E α ΔT / (1−ν)), no real C-curve data for any named steel, no stress-induced transformation, no grain growth model, and no decarburisation. The numbers show how the effects scale and compare; do not use them to heat-treat a real part.
3D models generated with Meshy AI and processed in Blender 5.2 (anvil, forge hearth with bellows, quench tub, horseshoe); the bar, tongs, floor and walls were built in Blender; Three.js r160 runs the live part. Independent educational model; no affiliation with any manufacturer.