Integrity Code Series — Week 8

High-Temperature Creep-Fatigue Interaction in Fired Heater Tubes

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Material: 9Cr-1Mo-V-Nb (ASME Grade T91 / P91) Anchor: Marathon Martinez CSB Report (2025-03-13), API STD 560:2026 Standards: API 530, API 579-1/ASME FFS-1 Part 10, ASME Section III Division 5, API 571

Integrity Code Series

Part of an ongoing series of physics-first integrity simulators by Felipe Rocha:

#	Repo	Domain
Week 3	integrity-code-series-week3-f1-lap-simulation	F1 lap simulation (six coupled ODEs)
Week 6	integrity-code-series-week6-smartphone-galvanic	Smartphone galvanic corrosion (Laplace + Butler-Volmer)
Week 7	integrity-code-series-week7-h2-lferw	LF-ERW H2 conversion (B31.12 + NACE TM0316)
Week 8	integrity-code-series-week8-creep-fatigue-heater	Creep-fatigue 9Cr-1Mo (Norton/Omega + Coffin-Manson) — this repo
Week 9	integrity-code-series-week9-cui	CUI thermohygro-electrochemical (3 PDEs, Strang)
Week 10	integrity-code-series-week10-nnph-scc	NNpHSCC full-physics (Chen-Sutherby-Xing + BS 7910)
Week 11	integrity-code-series-week11-erosion-corrosion-multiphase	Erosion-corrosion multiphase (NORSOK M-506 + DNV-RP-O501 + G119 + API 579 Part 5)
Bonus	Vibration-Accelerated-Corrosion-Coupled-Mechano-Electrochemical-Simulation	Vibration-accelerated corrosion (SDOF + Butler-Volmer + Archard)
Bonus	synthetic-integrity-digital-twin-piml	Physics-informed neural-network surrogate
Bonus	integrity-data-foundation	Engineering data validation baseline

Problem Statement

A fired heater tube operating in the creep regime degrades through three coupled mechanisms: time-dependent creep deformation, thermal fatigue from startup/shutdown cycles, and oxide scale growth that reduces structural wall thickness. The coupling is physical — oxide growth thins the wall, thinning increases hoop stress, higher stress accelerates creep, and the combined creep and fatigue damage is assessed against the ASME III-5 bilinear interaction envelope for Gr91 steel.

The question: at what operating conditions does the tube cross from safe to unsafe, and how sensitive is that boundary to temperature excursions?

Key Results

Condition	D_creep	D_fatigue	Margin	Status
Design (580 C, 5 MPa)	0.043	0.003	2.3x to envelope	Safe
Hot spot (+40 C)	0.647	0.003	Exceeded	Fail

A 40 C hot spot increases creep damage by 15x — exponential temperature sensitivity dominates all other parameters.

Governing Equations

view the full rendered reference

Every constant is tagged to its source standard or paper. The equations below render natively on GitHub; the full MathJax reference lives in docs/equations.html.

Norton Power-Law Creep (Secondary)

$$ \frac{d\varepsilon_{cr}}{dt} ;=; A,\sigma^{,n},\exp!\left(-\frac{Q_{cr}}{R,T}\right) $$

with $A = 7.86\times10^{-57}\ \mathrm{[1/(s\cdot Pa^{,n})]}$, $n = 10.5$, $Q_{cr} = 600\ \mathrm{kJ/mol}$. Calibrated to NIMS Creep Data Sheet No. 43A (2014) — single point $1\times10^{-8}\ \mathrm{/s}$ at 100 MPa, 873 K.

MPC Omega Method (API 579-1 Part 10, Tertiary)

$$ \dot{\varepsilon} ;=; \dot{\varepsilon}_0,\exp!\left(\Omega,\varepsilon_{cr}\right), \qquad \varepsilon_{\text{rupture}} ;=; \frac{1}{\Omega} $$

with $\Omega = 7.5$, $\dot{\varepsilon}_0 = 3.2\times10^{-10}\ \mathrm{/s}$ at 580 °C / 42 MPa, giving rupture strain $1/\Omega = 0.133$.

Larson-Miller Parameter (API 530)

$$ \mathrm{LMP} ;=; T,\bigl(C + \log_{10} t_r\bigr), \qquad C = 20 \ \text{for 9Cr-1Mo} $$

$$ \log_{10}\sigma_{\mathrm{MPa}} ;=; 0.672 + 3.80\times10^{-4},\mathrm{LMP} - 1.50\times10^{-8},\mathrm{LMP}^{2} $$

Anchor points: $\mathrm{LMP}=19000 \to 300\ \mathrm{MPa}$; $\mathrm{LMP}=21325 \to 90\ \mathrm{MPa}$; $\mathrm{LMP}=23000 \to 30\ \mathrm{MPa}$ (3-point fit of the API 530 minimum-rupture-stress curve).

Parabolic Oxide Growth (Wagner 1933)

$$ x^{2} ;=; k_p,t, \qquad k_p ;=; k_{p0},\exp!\left(-\frac{Q_{ox}}{R,T}\right) $$

with $k_{p0} = 4.08\times10^{-4}\ \mathrm{m^2/s}$, $Q_{ox} = 250\ \mathrm{kJ/mol}$ (Quadakkers et al. 2005). Effective metal loss $= 0.60,x$ [ASSUMED] (spallation estimate).

Coffin-Manson Thermal Fatigue

$$ \Delta\varepsilon ;=; C_1,N_f^{-\beta_1} ;+; C_2,N_f^{-\beta_2} $$

with $C_1 = 0.0045$, $\beta_1 = 0.085$ (Basquin elastic term) and $C_2 = 0.48$, $\beta_2 = 0.58$ (Coffin-Manson plastic term). Reference: Fournier et al. (2008), Int. J. Fatigue 30, 1797–1812.

Creep-Fatigue Interaction (ASME III-5)

$$ D_f ;=; \sum_j \frac{n_j}{N_{f,j}} \qquad\text{(fatigue damage fraction)} $$

$$ D_c ;=; \sum_k \frac{\Delta t_k}{t_{r,k}} \qquad\text{(creep damage fraction, time-fraction rule)} $$

Gr91 bilinear interaction envelope: $(1,0) \to (0.1,,0.01) \to (0,1)$. Per Code Case N-812 / ORNL/TM-2018/868.

Hoop Stress (Barlow / API 530)

$$ \sigma ;=; \frac{P,D_{\text{mean}}}{2,t_{\text{eff}}}, \qquad t_{\text{eff}}(t) ;=; t_{\text{nominal}} - 0.60,\sqrt{k_p,t} $$

The wall-thinning term couples oxidation to stress: as the oxide grows, $t_{\text{eff}}$ falls, hoop stress rises, and Norton creep accelerates. The $0.60$ metal-loss fraction is [ASSUMED].

Architecture

src/
├── config.py              Material properties, operating conditions
├── creep_engine.py        Norton, LMP (API 530), Omega (API 579-1)
├── oxidation.py           Parabolic oxide growth, wall thinning
├── fatigue.py             Coffin-Manson, thermal cycling
├── creep_fatigue.py       ASME III-5 interaction envelope
├── tube_model.py          Coupled creep-fatigue-oxidation simulation
├── surrogate.py           GBR surrogate model for fast prediction
├── monte_carlo.py         LHS parametric sweep, Spearman sensitivity
├── cybersecurity.py       STRIDE threat model, SHA-256 audit chain
└── utils.py               Unit conversions
tests/
├── conftest.py            Fixtures and path setup
├── test_creep.py          46 tests — Norton, LMP, Omega, hoop stress
├── test_oxidation.py      38 tests — parabolic kinetics, wall degradation
├── test_fatigue.py        43 tests — Coffin-Manson, thermal strain
├── test_creep_fatigue.py  36 tests — ASME III-5 envelope, damage rules
├── test_tube_model.py     37 tests — coupled simulation, failure modes
├── test_surrogate.py      21 tests — GBR training, cross-validation
├── test_monte_carlo.py    33 tests — LHS sampling, sensitivity
├── test_cybersecurity.py  37 tests — audit chain, STRIDE coverage
└── test_visualization.py  33 tests — project structure validation
docs/
└── equations.html         Rendered (MathJax) governing-equations reference

Quick Start

git clone https://github.com/felipearocha/integrity-code-series-week8-creep-fatigue-heater.git
cd integrity-code-series-week8-creep-fatigue-heater
pip install -r requirements.txt
python -m pytest tests/ -v

Sensitivity Analysis (Monte Carlo)

2,000 Latin Hypercube samples across 5 parameters:

Parameter	Spearman rho	Effect
Temperature	-0.59	Dominant — exponential via Arrhenius
Pressure	-0.50	Strong — linear via Barlow hoop stress
Wall thickness	+0.40	Protective — inverse stress relationship
Cycles/year	-0.15	Weak — fatigue secondary to creep
Delta T	-0.08	Weak — thermal strain below plastic threshold

Key Assumptions

Parameter	Value	Source	Flag
Internal pressure	5.0 MPa	Catalytic reformer	[ASSUMED]
Design temperature	580 C (853 K)	High-severity reformer	[ASSUMED]
Thermal cycle	300 K, 8 cycles/yr	Startup/shutdown	[ASSUMED]
Oxide metal loss	60% of scale	Spallation estimate	[ASSUMED]
Max wall loss	40% of nominal	Retirement criterion	[ASSUMED]
Norton A	7.86e-57	NIMS 43A single point	Calibrated
LMP polynomial	3 anchor points	Simplified API 530	Fitted
Coffin-Manson	Fournier 2008	T91 at 500-600 C	Published
Oxidation k_p	Quadakkers 2005	9Cr in air	Published

Cybersecurity (STRIDE)

STRIDE threat model with 7 assessed threats:

• Spoofing: Triple-redundant TMT with 2-of-3 voting
• Tampering: SHA-256 hash chain on all model parameters and outputs
• Repudiation: Immutable audit log for alarm overrides
• Information Disclosure: Role-based access to proprietary correlations
• Denial of Service: Local TMT data buffering for DCS communication loss
• Elevation of Privilege: Restricted alarm setpoint modification

Escalation Table

Week	Domain	Escalation
1-6	Corrosion / RBI	RBI, CUI, PINNs, minimum thickness, CCD automation
7	Hydrogen pipeline	Diffusion PDE, pit-to-crack, HA-FCG, LHS+GBR
8	Creep-fatigue	New physics domain, triple coupling, Omega nonlinearity, ASME III-5

References

1. API 530, 7th Edition — Calculation of Heater-Tube Thickness in Petroleum Refineries
2. API 579-1/ASME FFS-1 — Fitness-For-Service, Part 10 (Creep Assessment)
3. ASME Section III Division 5 — High Temperature Reactors (Code Case N-812)
4. NIMS Creep Data Sheet No. 43A (2014) — 9Cr-1Mo-V-Nb steel
5. Fournier et al. 2008, Int. J. Fatigue 30, 1797-1812
6. Quadakkers et al. 2005, Oxidation of Metals 64, 175-209
7. Maruyama et al. 2001 — Creep activation energy for ferritic steels
8. ORNL/TM-2018/868 — Messner & Sham, Creep-fatigue procedures for Gr91
9. Marathon Martinez CSB Report (2025-03-13)

Anti-Hallucination Note

Every equation and constant in this package is tagged to its origin, and the tiers are applied honestly:

• T1 (published standard / paper): the Norton, MPC Omega and Larson-Miller forms and the ASME III-5 bilinear envelope come from API 530, API 579-1 Part 10 and ASME Section III Division 5 (Code Case N-812 / ORNL-TM-2018/868); the Coffin-Manson coefficients from Fournier et al. 2008; the parabolic oxidation kinetics from Quadakkers et al. 2005; the creep single-point anchor from NIMS Creep Data Sheet 43A.
• T2 (derived / fitted): the Norton pre-exponential A is calibrated to the single NIMS 43A point, and the Larson-Miller minimum-stress polynomial is a 3-anchor fit of the API 530 curve. These are labelled Calibrated / Fitted in the Key Assumptions table.
• T3 (assumed operating / modelling): internal pressure, design temperature, the thermal-cycle count, the 60% oxide metal-loss fraction and the 40% wall-loss retirement criterion are engineering assumptions, each carried with an [ASSUMED] flag in the Key Assumptions table.

No constant, equation or citation is used that is not traceable to one of the references above; where a value is an assumption rather than a standard number it is flagged, not presented as measured.

Disclaimer

Research tool only. Not for design, fitness-for-service, or safety-critical decisions without site-specific calibration and independent PE review.

License

MIT — Felipe Rocha. See LICENSE.

How to Cite

If this software contributes to your work, please cite both the software (this repository) and the underlying methods it implements.

Software (archived release):

Rocha, F. (2026). Integrity Code Series - Week 8 - High-Temperature Creep-Fatigue Interaction in Fired Heater Tubes (Version 1.0.1) [Computer software]. Zenodo. https://doi.org/10.5281/zenodo.20172467

BibTeX:

@software{rocha_2026_week8,
  author       = {Rocha, Felipe},
  title        = {{Integrity Code Series - Week 8 - High-Temperature Creep-Fatigue Interaction in Fired Heater Tubes}},
  year         = 2026,
  publisher    = {Zenodo},
  version      = {v1.0.1},
  doi          = {10.5281/zenodo.20172467},
  url          = {https://doi.org/10.5281/zenodo.20172467}
}

The two DOIs Zenodo provides are:

DOI	What it points to
10.5281/zenodo.20172467 (concept)	Always resolves to the latest version - use this for citation.
10.5281/zenodo.20172468 (version)	Pinned to v1.0.1 specifically - use when reproducibility matters.

A machine-readable citation file is also available in CITATION.cff - GitHub will display a "Cite this repository" widget at the top right of the repo page that exports BibTeX / APA / RIS automatically.