Quenching Simulation
Theory: the surface cools, the core lags
Overview
What should a quenching simulation calculate first?
Temperature history. Whether steel hardens depends on how fast each point passes through the transformation range. The surface cools at once in contact with the quenchant, but core heat can only escape by conducting through the steel. The Biot number measures that lag: the ratio of how readily heat leaves the surface to how readily it moves through the interior. The larger it is, the wider the surface–core gap. Temperature gives microstructure through cooling rate and thermal stress and distortion through temperature differences.
Transient conduction in a cylinder
Biot number
Conditions: conductivity 25 W/mK, density 7,800 kg/m³, specific heat 600 J/kgK, initial 850 °C, quenchant 40 °C (air 25 °C). Heat transfer coefficients are held constant: water 5,000, oil 1,000, air 50 W/m²K.
The Jominy test and cooling curves
In 1938 Jominy and Boegehold devised the end-quench test to compare how deeply steels harden (hardenability). Water is sprayed on one end of a bar and hardness is measured along it. Distance from the end corresponds to cooling rate, so linking it to the cooling rate at each point of a part predicts hardness. Quenching simulation extends this idea to the whole part geometry.
Worked examples
Example 1: 20 mm bar
| Quenchant | Bi | Centre to 500 °C | Centre to 300 °C | Centre rate at 700 °C | Max surface–centre ΔT |
|---|---|---|---|---|---|
| Water | 2.0 | 6.3 s | 10.4 s | 79 °C/s | 375 ℃ |
| Oil | 0.40 | 16.9 s | 31.6 s | 25 °C/s | 126 ℃ |
| Air | 0.020 | 262 s | 519 s | 1.4 °C/s | 8 ℃ |
Example 2: 50 mm bar
| Quenchant | Bi | Centre to 500 °C | Centre to 300 °C | Centre rate at 700 °C | Max surface–centre ΔT |
|---|---|---|---|---|---|
| Water | 5.0 | 28.6 s | 45.6 s | 18 °C/s | 548 ℃ |
| Oil | 1.0 | 55.9 s | 98.3 s | 8.4 °C/s | 249 ℃ |
| Air | 0.050 | 669 s | 1,316 s | 0.6 °C/s | 20 ℃ |
Air cooling has Bi below 0.1, so the bar cools almost uniformly. The lumped formula gives τ = ρcR/(2h) = 468 s (20 mm), so the time to 500 °C is τ·ln(825/475) = 258 s, close to the computed 262 s.
At 2.5 times the diameter, even water cools the centre less than a quarter as fast.
At large Biot numbers cooling is governed by conduction inside the steel, not the surface. Heat takes time proportional to radius squared to reach the centre, so 2.5 times thicker takes about 6 times longer, and a stronger quenchant barely speeds the core. At 50 mm, switching from water to oil only halves the centre rate (18 to 8.4 °C/s) but cuts the surface–centre difference from 548 to 249 °C. Water on a thick part may fail to harden the core and merely raise cracking risk, which is why thick parts use alloy steels with high hardenability. Real calculations add the vapour-film stage, transformation heat and volume change, and give the heat transfer coefficient as a function of temperature.
Simulation workflow
- Estimate with the Biot number whether surface–core differences matter.
- Give the heat transfer coefficient as a function of temperature (vapour film, nucleate boiling, convection).
- Vary properties with temperature and phase; include transformation heat.
- Derive microstructure and hardness from cooling rate, and stress and distortion from temperature and transformation.
- Calibrate the heat transfer coefficient against measured cooling curves (thermocouples).
“It wasn't supposed to crack”
A stepped shaft water-quenched cracked at the junction of its thin and thick sections. The simulation had used a constant heat transfer coefficient, and temperature differences stayed within limits. In real water quenching, the thick section stays hot longer under an insulating vapour film, while the thin section breaks through the film early and cools rapidly. Recomputing with a temperature-dependent coefficient produced a large difference at the junction, matching the crack location.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Constant heat transfer coefficient | Local rapid cooling missed | Use temperature-dependent h |
| Ignoring transformation heat | Cooling too fast | Include latent heat |
| Relying on water for thick parts | Soft core, crack-prone | High-hardenability steel |
| Coarse mesh at surface | Wrong surface temperature | Refine at surface |
| No comparison with measurement | Uncertain coefficients | Calibrate on cooling curves |
I'd like to learn related topics.
Related pages include quench heat treatment, tempering, induction hardening, casting thermal analysis and furnace simulation.
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