Beyond the Holding Time: The Hollomon–Jaffe Parameter in Tempering, PWHT and Cumulative Thermal Exposure – part 2

In the previous part, I wrote about the practical background of the Hollomon-Jaffe parameter and highlighted the importance of understanding the effect of any heat treatment on the mechanical properties. In real life, a pressure vessel could undergo multiple heat treatments. Each cycle adds to the cumulative metallurgical exposure. A material may fully comply with its specification in the original delivery condition, yet the combined effect of tempering, fabrication PWHT, and repair heat treatments may reduce the remaining margin in strength, hardness, toughness, or creep-related properties.

This becomes especially important when procurement specifications define a maximum permissible Hp value, when simulated heat-treated test coupons are required, or when a welding procedure qualification does not realistically reproduce the full anticipated manufacturing heat-treatment history.

The same concern continues throughout the equipment lifecycle. Repair welding, local PWHT, and repeated maintenance interventions introduce further thermal exposure, often years after the original material qualification and fabrication records were issued. At that point, the question is no longer only whether the latest PWHT complied with the applicable code. The more important integrity question is whether the cumulative heat-treatment history remains compatible with the material properties assumed in the original design and subsequent fitness-for-service assessments.

Consecutive heat-treatment cycles

Individual ⟦fx⟧ values must not be added directly:

Because time appears logarithmically, the correct approach is to convert each cycle to an equivalent time at a common reference temperature ^{[1] [2]}.

For cycle ⟦fx⟧:

where ⟦fx⟧ may represent either:

for hold-only assessment, or:

where the ramp correction is applicable

The equivalent time at the selected reference temperature is:

The equivalent times are then simply added:

and the cumulative parameter becomes:

TWI specifically notes that effective times at a common temperature must be evaluated and added, rather than adding individual parameter values ^[1]. On the other hand, Dillinger applies the same basic equivalent-condition principle in its engineering tool ^[2].

Actual material temperature versus furnace temperature

The temperature used in an ⟦fx⟧ assessment should represent the material exposure being assessed. This is not necessarily identical to:

• the furnace setpoint;
• the heating-element temperature;
• the controller output;
• the external surface temperature;
• the hottest thermocouple;
• the coldest thermocouple.

A thick-walled component may exhibit a significant delay between furnace temperature and through-wall equalization. Local PWHT can introduce axial, circumferential and through-thickness gradients.

The representative temperature may therefore be:

• the nominal procedure temperature;
• a controlling thermocouple temperature;
• the measured minimum temperature;
• the measured maximum temperature;
• a calculated representative material temperature;
• a conservative bounding temperature.

For assessing possible over-tempering, the highest credible material exposure may govern. For verifying minimum PWHT effectiveness, the lowest relevant temperature may be more important.

These are different engineering questions and should not be merged into one unqualified value.

Limitation of simple ramp corrections

The heating and cooling equations above are useful engineering approximations, but they are not a full non-isothermal kinetic model.

More general non-isothermal treatments require methods capable of accounting for the complete changing temperature history and the kinetics of the relevant transformation. Réti, Gergely and Tardy developed generalized kinetic functions and equivalent-time concepts for processes occurring under varying temperature ^[3].

A constant-rate ramp equation does not independently model:

• changing heating or cooling rates;
• furnace overshoot;
• intermediate dwell periods;
• nonlinear natural cooling;
• temperature gradients;
• different activation energies;
• overlapping tempering mechanisms;
• phase-specific transformation kinetics.

Where the measured temperature history is available, a numerical time-step method may be more appropriate. The curve can be divided into small intervals, and every interval converted to an equivalent exposure at the selected reference temperature.

However, such a model requires a validated kinetic basis. Numerical integration alone does not remove the empirical limitations of the underlying material correlation.

Short-time and rapid heat treatments

The shorter the nominal holding time, the larger the relative contribution of heating and cooling.

For a treatment lasting several hours, the ramp contribution may be comparatively small. For a treatment lasting seconds or minutes, heating and cooling can represent a major portion of the total thermal exposure.

Studies involving high-frequency induction heating have found differences in carbide size and spheroidisation compared with conventional heating ^[4]. Short heat-pulse investigations have likewise shown that rapid local heating can produce material responses that are not necessarily represented by conventional long-duration tempering assumptions ^[5].

Rapid-tempering research has demonstrated that equal hardness or approximately equal tempering parameters can coexist with different retained-austenite decomposition and toughness behavior ^[6].

The parameter is therefore most defensible where:

• the exposure lies within the validated time and temperature range;
• the material condition is comparable with the underlying data;
• the property being assessed matches the correlated property;
• extrapolation is avoided;
• heating and cooling assumptions are clearly reported.

Material dependence and the value of ⟦fx⟧

A value of ⟦fx⟧ is commonly used for C-Mn and low-alloy steels. It is not a universal constant.

The optimal value may depend on:

• carbon content;
• alloy content;
• initial microstructure;
• property being correlated;
• temperature range;
• exposure duration.

The TWI guidance notes that values around 19.5 or 20 are commonly applied to carbon and low-alloy steels, while higher values have sometimes been used for highly alloyed steels ^[1]. Nevertheless, selecting ⟦fx⟧ should be based on the applicable material-property relationship, not simply on a generic alloy category.

Investigations of hot- and warm-forging die steels have also demonstrated material-specific and bilinear softening behavior as a function of the tempering parameter ^[7] .

The same ⟦fx⟧-value should be used consistently when:

• calculating individual cycles;
• converting cycles to equivalent times;
• accumulating exposure;
• comparing the result with a critical value.

Procurement and welding procedure specification

Cumulative exposure should be considered during procurement, not only after fabrication has been completed.

The purchaser should define the anticipated manufacturing history, including where relevant:

• original material heat treatment;
• hot forming; intermediate stress relieving;
• final PWHT;
• possible repair cycles;
• heating and cooling assumptions;
• maximum cumulative ⟦fx⟧;
• simulated heat treatment of test material.

A material test coupon exposed only to the original mill heat treatment may not represent a component that later receives several PWHT cycles.

Likewise, a welding procedure qualification subjected to one PWHT cycle may not represent production welds expected to undergo two, three or more cycles.

Where cumulative exposure approaches a material-specific critical value, the purchaser and manufacturer should consider simulated heat treatment of the test material. Depending on the applicable material and code requirements, verification may include:

• tensile testing;
• yield-strength testing;
• impact testing;
• hardness testing;
• metallographic examination;
• creep-related testing.

In case of cladded equipment, the corrosion properties of the clad material should also be considered.

Lifecycle and integrity implications

Thermal exposure does not end at mechanical completion. Repair welding and local PWHT during operation add to the existing material history.

This is particularly important for creep-strength-enhanced ferritic steels. Long-term elevated-temperature exposure can reduce yield and tensile strength, meaning that thermal aging may influence the integrity margin of components throughout their intended life ^[8].

A lifecycle assessment should therefore retain:

• original heat-treatment records;
• fabrication PWHT records;
• repair histories;
• measured temperatures;
• heating and cooling rates;
• calculated cumulative equivalent time;
• cumulative ⟦fx⟧;
• test evidence supporting continued material properties.

The Hollomon–Jaffe parameter does not calculate creep damage, but it can provide a structured record of cumulative thermal aging that supports broader integrity and remaining-life assessments.

Relationship to construction-code compliance

The Hollomon–Jaffe assessment does not replace construction-code requirements.

Construction codes may independently specify:

• minimum and maximum holding temperatures;
• minimum holding time;
• control-thickness rules;
• maximum heating and cooling rates;
• local-PWHT geometry;
• thermocouple requirements;
• material-specific restrictions.

ASME B31.1, for example, establishes requirements covering materials, fabrication, examination, testing, operation, and maintenance of power-piping systems ^[9]. Similar requirements apply in Europe for unfired pressure vessels ^[10] and for piping ^[11]

A heat treatment can have an acceptable ⟦fx⟧ while failing a mandatory code provision. Conversely, every individual PWHT may comply with the code while the cumulative metallurgical exposure becomes excessive.

The checks answer different questions:

Reporting requirements – make it once, make it right

A professional Hollomon–Jaffe assessment should state:

• material grade and delivery condition;
• product form and thickness;
• selected value of ⟦fx⟧;
• formula and unit convention;
• every manufacturing and operation-stage cycle;
• holding temperature and holding time;
• heating and cooling rates;
• whether ramp corrections were included;
• the exact ramp equations used;
• reference temperature;
• equivalent time for each cycle;
• cumulative equivalent time;
• total ⟦fx⟧;
• critical value and its source;
• assumptions and exclusions.

Where heating and cooling are excluded, the report should say:

Where the corrections are included:

Conclusion – to take away

The Hollomon–Jaffe parameter provides a practical method for comparing tempering, PWHT and cumulative thermal exposure. Its greatest value is not the calculation of one isolated holding condition, but the structured assessment of the complete heat-treatment history.

The nominal holding period is only one part of a real thermal cycle.

Heating contributes because metallurgical reactions begin before the soaking temperature is reached. Cooling contributes because those reactions continue while the material remains at elevated temperature.

For the established ⟦fx⟧ approximation, these effects can be represented through equivalent additional times:

These are added to the nominal holding time before calculating the corrected parameter.

The method remains empirical. Ramp corrections are not universal:

Used within these limits, however, the Hollomon–Jaffe parameter can help identify procurement risks, improve welding procedure qualification, document cumulative fabrication exposure, and support lifecycle integrity decisions.

That is the purpose of the TT Integrity Hp Calculator: to make the complete thermal history visible and simplify the workflow – before it becomes a material, fabrication, or integrity problem.

References

1. [1]
   TWI Ltd.. What Is the Effect on C-Mn and Low-Alloy Steels of Multiple Tempering or Stress-Relieving Heat-Treatment Cycles?. TWI Technical Knowledge FAQ. Accessed July 2026..
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2. [2]
   Dillinger Hüttenwerke AG. Hollomon Parameter — Calculation of Heat-Treatment Parameters.. Dillinger E-Service Engineering Tools. Accessed July 2026.
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3. [3]
   Réti, T.; Gergely, M.; Tardy, P.. Mathematical Treatment of Non-Isothermal Transformations.. Materials Science and Technology, Vol. 3, No. 5, 1987, pp. 365–371. DOI: 10.1179/026708387790122611..
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4. [4]
   Lee, J. B.; Kang, N.; Park, J. T.; Ahn, S. T.; Park, Y. D.; Choi, I. D.; Kim, K. R.; Cho, K. M.. Kinetics of Carbide Formation for Quenching and Tempering Steels During High-Frequency Induction Heat Treatment.. Materials Chemistry and Physics, Vol. 129, 2011, pp. 365–370. DOI: 10.1016/j.matchemphys.2011.04.026..
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   Cvetkovski, K.; Ahlström, J.; Karlsson, B.. Influence of Short Heat Pulses on Properties of Martensite in Medium Carbon Steels.. Materials Science and Engineering A, Vol. 561, 2013, pp. 321–328. DOI: 10.1016/j.msea.2012.10.005..
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6. [6]
   Euser, V. K.; Williamson, D. L.; Clarke, A. J.; Speer, J. G.. Limiting Retained Austenite Decomposition in Quenched and Tempered Steels: Influences of Rapid Tempering and Silicon.. ISIJ International, Vol. 60, No. 12, 2020, pp. 2990–3000. DOI: 10.2355/isijinternational.ISIJINT-2020-263..
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7. [7]
   Virtanen, E.; Van Tyne, C. J.; Levy, B. S.; Brada, G.. The Tempering Parameter for Evaluating Softening of Hot and Warm Forging Die Steels.”. Journal of Materials Processing Technology, Vol. 213, 2013, pp. 1364–1369. DOI: 10.1016/j.jmatprotec.2013.03.003..
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8. [8]
   Foulds, J.; Ren, W.. Thermal Aging Effects on the Yield and Tensile Strength of 9Cr-1Mo-V (Grade 91).. Journal of Pressure Vessel Technology, Vol. 144, No. 6, 2022, Article 061505. DOI: 10.1115/1.4054341..
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9. [9]
   The American Society of Mechanical Engineers. ASME B31.1-2024: Power Piping.. ASME, New York, 2024..
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10. [10]
    European Norm. EN 13445-4+A1 - Unfired pressure vessles - Part 4: Fabrication. 2023.
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    European Norm. (2024). EN 13480-4 - Metallic industrial piping - Part 4: Fabrication and installation.
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