Heat Treatment (Tempering) Thermal Analysis
Theory: combining temperature and time into one value
Overview
The spec says temper the quenched gears at 550 °C for 1 hour, but the furnace is busy — could we run 500 °C longer instead? How many hours would match?
Tempering lets carbon gradually leave the over-hardened martensite from quenching, lowering hardness and raising toughness. It depends strongly on temperature and only weakly on time, so the tempering parameter, which combines the two, lets you convert. At 500 °C you would need about 20 hours — dropping 50 K multiplies the time by 20.
Tempering parameter (Hollomon–Jaffe)
$T$ is absolute temperature (K), $t$ holding time (h) and $C$ a material constant, commonly about 20 for carbon and low-alloy steels. The time $t_2$ at temperature $T_2$ giving the same $P$ follows from the second formula. Because time enters only logarithmically, a tenfold longer hold has only the effect of a small temperature rise.
What happens during tempering
| Approximate range | Microstructural change | Properties |
|---|---|---|
| 150–250 °C | Fine carbides begin to precipitate | Keeps most hardness, reduces brittleness |
| 250–400 °C | Retained austenite decomposes | Some steels embrittle in this range |
| 400–650 °C | Carbides coarsen, matrix recovers | Hardness falls, toughness rises |
1945: a formula for trading temperature against time
The tempering parameter was proposed in 1945 by Hollomon and Jaffe in the United States, who organised the hardness of steels tempered at many temperatures and times. Plotting against this parameter instead of temperature collapsed scattered hardness data onto a single curve. The same idea applied to creep life of materials used long at high temperature is known as the Larson–Miller parameter. The notion that a thermally activated change is governed by the combination of temperature and time remains basic to heat-treatment process design.
Worked examples
Example 1: equivalent temperatures and times
Times giving the same tempering parameter as 550 °C for 1 hour (P = 16,463, C = 20):
| Temperature | Equivalent time |
|---|---|
| 500 ℃ | about 19.7 h |
| 525 ℃ | about 4.2 h |
| 550 ℃ | 1 h |
| 575 ℃ | about 15 min |
| 600 ℃ | about 4 min |
Just 25 K changes the time by a factor of four.
Yes. Put the other way, a furnace just 10 K hot gives the same effect in about 35 minutes (58%), and 20 K hot in about 20 minutes (34%). Furnace non-uniformity of 10–20 K is not unusual, which is why hardness scatters with position. A thermal analysis of the furnace shows which positions risk over- or under-tempering. The parameter is empirical, though: C varies by steel, and it can break down for double tempering or very short times.
Example 2: time for parts to reach furnace temperature
A long round steel bar at 25 °C placed in a 550 °C furnace, heated by convection (20 W/(m²·K)) and radiation (emissivity 0.8), treating the part as a single temperature:
| Diameter | Biot number | Time to 540 °C |
|---|---|---|
| 50 mm | 0.04 | about 0.55 h |
| 100 mm | 0.08 | about 1.1 h |
| 200 mm | 0.15 | about 2.2 h (core lags further) |
Heat-up time is roughly proportional to diameter. At 200 mm the Biot number exceeds 0.1, the surface–core difference is no longer negligible and the core lags further. Counting a 1-hour hold from loading means the core of a 100 mm part is barely held at all. In real furnaces, densely packed parts shield each other from radiation and take longer still.
How to approach the analysis
Procedure
- Obtain the furnace temperature distribution; burner or heater placement and loading pattern cause differences of 10 K or more.
- Compute surface and core temperature histories for each part.
- Convert temperature histories to a tempering parameter (or equivalent) and predict hardness.
- Compare with measured part hardness to check the constant and model.
Cautions
- Temperature-dependent properties: steel specific heat changes sharply near the transformation around 700 °C, and even in the tempering range differs noticeably from room temperature.
- Radiation: it dominates heat transfer in hot furnaces; account for parts shielding each other.
- Residual stress relaxation: tempering also relieves quench stresses; model thermo-mechanical coupling to predict distortion.
“Same settings, different hardness”
A common case: parts tempered in the same furnace under the same settings, yet those in the middle of the basket came out too hard. Surrounded by other parts, they received less radiation, reached temperature later and effectively got a shorter hold. With the thermocouple near the furnace wall, the furnace can read correctly while the parts are much cooler. Attach thermocouples to parts, or use analysis to find the slowest position and set the hold time from it.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Lowering temperature and extending time slightly | Under-tempered, too hard | Convert with the parameter |
| Counting hold from loading | Core of large parts under-tempered | Compute core heat-up |
| Assuming uniform furnace temperature | Hardness varies with position | Analyse and measure the distribution |
| Not checking constant C | Conversion off | Check with hardness data per steel |
| Ignoring radiation | Heat-up overestimated | Include radiation and shielding |
I'd like to try tempering and hardness calculations myself.
Try the heat treatment simulator for heat-treatment temperature histories, the Jominy hardenability tool for hardenability, the hardness conversion tool for hardness conversion and the Larson–Miller parameter tool for the same idea applied to creep life. Neighbouring processes are covered in quenching thermal analysis, induction hardening and furnace simulation.