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How to use
Pick a preset or choose a steel grade and tweak the composition sliders. The bar starts at the austenitising temperature; the jet cools the bottom end. Watch the glow fade from the jet end upward, then see hardness, microstructure and cooling rate appear along the bar as each row finishes transforming.
Try: compare 1045 with 4140 (press Reset between runs: the previous curve stays as a grey dashed ghost in graph 1). Slide chromium from 0 to 1.5 % and watch the C-curve in graph 2 move right. Weaken the jet or heat the water and see the whole curve shift. Set a coarse grain (ASTM 3) to see why grain size matters. Move the probe to see its cooling curve against the C-curves: if it clears the nose, you get martensite.
Press Finish quench to compute the result instantly. Drag to orbit, wheel to zoom. Space pauses, R resets.
3D rig: Blender 5.2 built the quench stand, the stainless water nozzle, the 25.4 mm bar with 100 rings, the box furnace and tongs, the Rockwell hardness tester and the wall chart; the bar glows with the live temperature field and the hardness ribbon on the ground flat is painted from the live microstructure. Use the camera buttons and the guided tour.
The science
The Jominy end-quench test (ASTM A255 / ISO 642) measures hardenability: how deep a steel hardens, not how hard it can get. A 25.4 mm diameter, 100 mm long bar is heated to austenite, then one end is sprayed with a standard water jet. Cooling speed falls steadily with distance from that end, so one bar samples every cooling rate at once, and the hardness measured along a ground flat is the steel's fingerprint.
1. Heat flow. The bar is axisymmetric; the model solves the heat equation on an r-z grid of 8 x 100 finite volumes with temperature-dependent conductivity and heat capacity:
ρ cp(T) ∂T/∂t = (1/r) ∂/∂r ( r k(T) ∂T/∂r ) + ∂/∂z ( k(T) ∂T/∂z )
end face: −k ∂T/∂z = hjet (T − Tw); sides and free end: h = 12 ffan + εσ(T2+Ta2)(T+Ta), ε = 0.8
with ρ = 7800 kg/m3, k = 51.5 − 0.0275 (T − 20) W/m/K and cp = 450 + 0.2 (T − 20) plus a Gaussian bump near 745 C that stands in for the Curie point and the A1 latent heat. The scheme is explicit with a time step set by the stability limit (about 10 ms); the surface flux includes the half-cell conduction resistance so a cell-centred value gives a surface boundary condition.
2. What forms. The transformation of austenite is tracked at the outer ring of every row with Scheil additivity: on a continuous cooling path the sum Σ Δt / τ(T) of the isothermal "1 % start" times reaches 1 when a transformation begins. Two C-curves are used:
τF(T) = τn exp[((T − TnF)/120)2] / (1 − exp(−(Ae1 − T)/15)), τB(T) = 2.5 τn exp[((T − TnB)/100)2]
X = 1 − exp(−0.01005 s2) (Avrami, n = 2, s = Σ Δt/τ), XM = XA(Ms) [1 − exp(−0.011 (Ms − T))]
The last relation is the Koistinen-Marburger equation for athermal martensite. Ferrite+pearlite can only form between Bs and Ae1, bainite between Ms and Bs, and whatever austenite is left when the metal reaches Ms turns into martensite as it cools below it. The transformation temperatures use the classic linear fits of Andrews (1965) and Steven and Haynes (1956):
Ms = 539 − 423 C − 30.4 Mn − 17.7 Ni − 12.1 Cr − 7.5 Mo; Bs = 830 − 270 C − 90 Mn − 37 Ni − 70 Cr − 83 Mo
Ae1 = 723 − 10.7 Mn − 16.9 Ni + 29.1 Si + 16.9 Cr; Ae3 = 912 − 203 √C + 15.2 Ni + 44.7 Si + 31.5 Mo
3. Why alloys help. Alloying elements delay the formation of ferrite and pearlite, which pushes the C-curve to longer times, so a slower cooling rate still misses the nose. The size of the shift is computed from the Grossmann ideal critical diameter DI (the largest round bar that has 50 % martensite at its centre in an ideal quench), built from a carbon-and-grain-size base and the ASTM A255 style multiplying factors:
DI = 0.35 √C · 1.1(7−G) · (1 + 3.35 Mn)(1 + 0.7 Si)(1 + 0.363 Ni)(1 + 2.16 Cr)(1 + 3.0 Mo) [inches]
τn = 0.2 s × (DI / 0.97 in)1.7
The prefactor 0.2 s and exponent 1.7 are not from a handbook: they were tuned once so that the plain-carbon and alloy steels in the presets give Jominy curves with the right shape and ceilings. Treat the absolute distances as illustrative (about +-5 HRC and a few millimetres), and the trends as the lesson.
4. Hardness. Each phase has a hardness, mixed linearly: martensite HRCM = min(67, 20 + 60 √C), ferrite+pearlite HRCP = 6 + 34 C + 4 (Mn + Cr + Mo + Ni/2), bainite 55 % of the way up from P to M, retained austenite HRCM − 10. This captures the main fact that hardness of martensite depends on carbon and not on alloy, while the depth depends on alloy and grain size.
Numerical checks run in the page. (1) The solver reproduces the exact solution of a semi-infinite solid with a convective surface, T = T0 − (T0 − Tw) [erfc(x/2√αt) − exp(hx/k + h2αt/k2) erfc(x/2√αt + h√αt/k)], to 0.004 in normalised temperature; (2) enthalpy lost by the bar equals the heat the surface fluxes removed; (3) Avrami additivity at the nose; (4) the Koistinen-Marburger half-point; (5) the Andrews Ms for 4140 against its published range; (6) cooling rate is monotone with distance.
Simplifications. Austenite is assumed to be homogeneous and fully dissolved; no grain growth, no segregation, no distortion, no self-tempering of early martensite and no transformation latent heat other than the bump in cp. The jet is described by a single heat-transfer coefficient (real jet cooling has a stagnation zone, nucleate boiling and wetting front); radial sampling uses the outer ring cell rather than a ground flat 0.38 mm deep. Compositions above about 0.8 % C (undissolved carbides) are outside the model.