Tempering-calculator software based on the Larsen-Miller equation facilitates analysis of the heat-treating process by integrating tempering factors, which helps in the design and operation of tempering equipment and improves process control.

Tempering information currently is available in reference handbooks in the form of hardness-temperature curves, a different curve being required for a change in time or material chemical composition. For example, Fig. 1 shows the hardness-temperature relationship for a one-hour tempering curve for AISI 1050 steel. By comparison, tempering-related formulas and equations can be combined into a software program that can facilitate the selection and the analysis of a tempering cycle.

Thermal effect

The relationship between time and temperature can be described as a logarithmic function in the form of the Larsen-Miller equation, which shows that the thermal effect (TE) is dependent on the temperature and the logarithm of time:

whereTis temperature,tis time andRis a constant specific to the material. The thermal-effect also is known as the tempering parameter.

A material tempered at a temperature of 740°F for one hour has the same TE as a material treated at 800°F for 6 minutes, and the results of tempering will be the same (Fig. 2). Thus, increasing the tempering temperature by, for example, 20°F significantly reduces the necessary tempering time as shown in Fig. 3. In practice, the operating windows are reduced as the temperature is increased; therefore, it is essential to tighten the time and temperature controls.

To illustrate the logarithmic nature of the time-temperature relationship, software was initially developed to correlate time-temperature factors by means of the thermal effect concept. A display of the first tempering calculator is shown in Fig. 4. The time and temperature is set by means of sliders, then TE is calculated and possibilities are tabulated. Using the proper control and equipment, it is now possible to accelerate a tempering process for a known material to achieve specific properties.

Hardness characteristic

The contribution by Hollomon and Jaffe to the thermal-effect concept was to add the composition-properties of steel for a tempering cycle to the time-temperature relationship. Hardness (expressed as a HRC value) is an optimum value (a theoretical characteristic hardness), which varies with the carbon content (Fig. 5). The optimum hardness is 92 HRC for 0.50% carbon 98 HRC for 0.80% carbon.

The thermal effect will reduce the hardness to the required value, called the hardness-reduction parameter. The following formula is used to determine the desired hardness in a tempering process:

Measured Hardness = Characteristic Hardness minus Hardness Reduction Parameter

For example, AISI 1050 steel has a characteristic hardness of 92 HRC. The hardness reduction parameter for a one hour temper at 600°F is 42 HRC for a TE of 20.1 (Fig. 6). The shaded area represents the hardness reduction due to the thermal effect with increasing temperature. Figure 7 illustrates the hardness reduction with the temperature factor (from Hollomon and Jaffe). For the tempered AISI 1050 steel example, the measured hardness after tempering is 50 HRC.

The thermal effect is effective for all types of processing, from batch to induction tempering, including the PYRO high-speed technology shown in Fig. 8. A comparison of the time ratio varies from 1 to 10 to 100. It is possible to select cycle times in seconds, minutes and hours.

A well-known adage in the heat treating industry states that for the same tempering time, a temperature variation of 25°F will change the hardness by 1 HRC (Fig. 9). Thus, the time must be corrected by a factor 10 to change the hardness by 2 HRC in a tempering cycle for the same temperature (Fig. 10).

The tempering calculator

To simplify the selection of the correct tempering parameters, the formulations described above were combined and integrated into a software program called the Tempering Calculator. Constant and variable thermal effects are considered.

Constant thermal effect.This approach allows setting the carbon content of the steel, the hardness required and the tempering time, which then provides the temperature with a constant thermal effect value (Fig. 11). The main feature is allowing the choice of any combination that you need to analyze, select or improve the tempering cycle.

Variable thermal effect.A different approach is to fix the temperature and select the time to influence the thermal effect and hardness as shown in Fig. 12. Alternative parameters can be obtained through the use of simple keyboard or mouse operations.

The constant and variable thermal effect situations interrelate time-temperature-hardness-composition for carbon and low-alloy steels. TheLarsen-Millerequation also applies to other steels. Introducing the values of a known tempering cycle that results in the desired properties of the tempered steel permits the generation of an infinite number of equivalent cycles (Fig. 13).

PYROTEMP tempering simulation

The Tempering Calculator provides time at temperature. To have the total cycle time requires adding the respective oven ramp up time. Pyromaitre recently integrated the Tempering Calculator in its PYROGRAPH heat transfer software used for over 15 years to operate high speed ovens. The new version, called PYROTEMP, enables the heat treater to generate and optimize process recipes in PYRO high speed ovens.

In the gear simulation shown in Fig. 14, a four-minute cycle time would not provide enough time for the thick section to reach temperature; the PYROTEMP software detects nonuniform tempering (Fig. 15). The software also provides operating cost figures, a valuable tool for the heat treater quoting new business.

The Tempering Calculator should only be used as a guide, because results may be affected by variations in chemical composition and the thermal history of the material. Also, it is necessary to use equipment that is capable of providing the required heat input to reach the selected temperature for the equivalent time.

However, the calculator should be used as a guide, because results may be affected by variations in chemical composition and the thermal history of the material. Also, it is necessary to use equipment that is capable of providing the required heat input to reach the selected temperature for the equivalent time.


  • Hollomon, H. and Jaffe, L.D., Time-Temperature Relations in Tempering of Steel, 1945
  • Understanding the Larson-Miller Parameter, Scripta Met., Vol. 11, p 193-196, 1977
  • Tempering of Steel, Metals Handbook, 8th Ed., Vol. 2, p 47, 1964
  • Induction Tempering, ASM Handbook, Vol. 4, p186, 1991
  • Rapid Stress Relief and Tempering, Gear Solutions, May, p 26-33, 2005
  • Grenier, M., et.al., Rapid Tempering and Stress Relief Via High-Speed Convection Heating, May 2003

For more information:Pyromaitre Inc., 1081 Chemin Industrial, St. Nicolas, PQG7A 1B3 Canada; Tel: 418-831-2576; fax: 418-831-3206; e-mail:pyro@pyromaitre.com; Internet:www.pyromaitre.com.

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