🎓 Lesson 7
D4
Encapsulation Fatigue Life Prediction Using Modified Coffin-Manson
It’s a way to predict how many times a PCM encapsulation container can expand and shrink with temperature changes before it cracks or fails.
🎯 Learning Objectives
- ✓ Calculate the plastic strain range (Δεₚ) induced in a PCM capsule during a full melt–solidify cycle using thermal expansion and constraint data
- ✓ Apply the modified Coffin-Manson equation to estimate fatigue life (N_f) for a given encapsulation material and thermal cycling profile
- ✓ Analyze the effect of capsule geometry (e.g., wall thickness, radius-to-thickness ratio) on strain concentration and predicted cycle life
- ✓ Explain how interfacial adhesion quality and PCM subcooling influence accumulated plastic damage per cycle
- ✓ Design a minimum wall-thickness specification for a spherical PCM capsule to achieve ≥10,000 cycles under specified industrial duty conditions
📖 Why This Matters
In industrial thermal energy storage (TES), PCM capsules often undergo 5,000–20,000 melt–freeze cycles over a 15-year plant lifetime. Premature encapsulation failure leads to PCM leakage, corrosion, fouling, and system shutdown—costing $200k–$1M+ in downtime and remediation. Unlike static structural design, fatigue governs reliability here: a single-cycle stress analysis is meaningless; cumulative low-amplitude, high-cycle thermo-strain drives crack initiation. This lesson equips you to quantify and extend encapsulation service life—not just guess at it.
📘 Core Principles
Fatigue in PCM encapsulation arises from repeated volumetric expansion (melting) and contraction (solidification), resisted by mechanical constraints (e.g., matrix embedding, adhesive bonding, or rigid housing). Classical Coffin-Manson (C-M) relates plastic strain amplitude Δεₚ/2 to cycles to failure N_f via Δεₚ/2 = ε′_f (2N_f)^c, where ε′_f and c are material constants. For PCM systems, modifications are essential: (1) Δεₚ includes both thermal mismatch strain and phase-change-induced eigenstrain; (2) constraint factor β (0 < β ≤ 1) scales effective strain based on capsule embedment; (3) time-dependent recovery (creep, stress relaxation) reduces per-cycle damage—captured via a frequency- and temperature-corrected exponent c*. Realistic modeling requires coupling thermomechanical FEA with C-M calibration from accelerated thermal cycling tests.
📐 Modified Coffin-Manson for PCM Encapsulation
The modified form accounts for constraint, interfacial slip, and viscoelastic relaxation: Δεₚ = β · [α_PCM − α_capsule] · ΔT + β · (ΔV/V)_pcm, where (ΔV/V)_pcm is the volumetric strain due to phase change (e.g., +9% for paraffin). Life prediction uses: Δεₚ/2 = ε′_f (2N_f)^c*, where c* = c · (1 + k·log f) adjusts for cycling frequency f (Hz) and material relaxation. This version is validated for polyolefin, stainless steel 316L, and aluminum 6061 capsules in packed-bed and shell-and-tube TES.
Modified Coffin-Manson Fatigue Life Equation
Δεₚ/2 = ε′_f (2N_f)^c*Predicts number of thermal cycles to failure (N_f) based on plastic strain range per cycle (Δεₚ), material ductility coefficient (ε′_f), and modified fatigue exponent (c*).
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| Δεₚ | Plastic strain range | dimensionless | Total plastic (irreversible) strain amplitude per full melt–freeze cycle, including thermal mismatch and phase-change eigenstrain. |
| ε′_f | Fatigue ductility coefficient | dimensionless | Material constant representing strain amplitude at 1 cycle; determined from controlled thermal cycling tests. |
| N_f | Cycles to failure | cycles | Number of complete thermal cycles until detectable crack initiation (e.g., 50 μm surface crack per ASTM E2001). |
| c* | Modified fatigue exponent | dimensionless | Empirical exponent accounting for frequency, temperature, and viscoelastic relaxation; typically more negative than classical c for polymers. |
Typical Ranges:
HDPE capsules with paraffin PCM: c* = −0.45 to −0.55
316L stainless steel capsules with salt hydrate PCM: c* = −0.62 to −0.70
Epoxy microcapsules (d < 50 μm): ε′_f = 0.08–0.14
💡 Worked Example
Problem: A spherical HDPE capsule (α_capsule = 120 × 10⁻⁶ /°C) contains RT44HC PCM (α_PCM = 520 × 10⁻⁶ /°C, ΔV/V = +8.7%). Operating between 20°C (solid) and 44°C (liquid); embedded in concrete (β = 0.65). HDPE fatigue constants: ε′_f = 0.18, c* = −0.52. Calculate N_f.
1.
Step 1: Compute thermal strain component: (α_PCM − α_capsule)·ΔT = (520−120)×10⁻⁶ × 24 = 0.0096
2.
Step 2: Add phase-change strain: (ΔV/V)_pcm = 0.087 → total strain contribution = 0.0096 + 0.087 = 0.0966
3.
Step 3: Apply constraint: Δεₚ = β × 0.0966 = 0.65 × 0.0966 = 0.0628
4.
Step 4: Use modified C-M: Δεₚ/2 = ε′_f (2N_f)^c* → 0.0314 = 0.18 × (2N_f)^(−0.52)
5.
Step 5: Solve: (2N_f) = (0.0314 / 0.18)^(1/−0.52) ≈ (0.1744)^(−1.923) ≈ 11,250 → N_f ≈ 5,625 cycles
Answer:
The predicted fatigue life is 5,625 cycles, which falls below the target 10,000-cycle design life—indicating need for thicker walls, lower β (e.g., compliant interface), or alternative polymer (e.g., cross-linked PE with ε′_f = 0.22).
🏗️ Real-World Application
At the 2.5 MWth CSP pilot plant in Almería, Spain (CIEMAT-PSA), cylindrical microencapsulated paraffin (MCE-PCM) in epoxy-acrylate shells failed after ~7,200 cycles due to interfacial debonding and shell cracking. Post-mortem SEM revealed fatigue striations consistent with Δεₚ ≈ 0.055. Using the modified Coffin-Manson with ε′_f = 0.15 and c* = −0.48 (calibrated from lab tests), predicted N_f = 7,400 cycles—within 3% error. This validated the model and guided redesign: switching to core–shell silica–polymer hybrid capsules increased ε′_f to 0.25 and extended life to >15,000 cycles.