🎓 Lesson 11
D5
Runaway Reaction Prediction Using Semenov and Frank-Kamenetskii Criteria
It’s a way to predict when a chemical reaction in a hot material (like explosives or stored chemicals) will suddenly speed up and overheat—potentially causing an explosion—because heat is generated faster than it can escape.
🎯 Learning Objectives
- ✓ Calculate the Frank-Kamenetskii parameter (δ) for a given explosive emulsion pile using thermal conductivity, activation energy, and ambient temperature
- ✓ Analyze whether a specified charge geometry (e.g., 1.2 m diameter cylindrical borehole charge) is thermally stable under adiabatic storage conditions using the critical δ_c value
- ✓ Explain the physical significance of the Semenov number (Nu) and its relationship to Biot number and Damköhler number in thermal runaway onset
- ✓ Apply both criteria to compare thermal safety margins for ANFO vs. emulsion explosives at elevated mine ambient temperatures (e.g., 45°C)
- ✓ Design safe maximum charge diameters for temporary surface storage of sensitized emulsions based on Frank-Kamenetskii critical dimensions
📖 Why This Matters
In mining, runaway reactions in bulk explosive storage, misfired holes, or heated emulsion charges have caused catastrophic incidents — including the 2013 West Fertilizer explosion and multiple underground mine detonations triggered by spontaneous heating. Predicting thermal runaway isn’t just theoretical: it directly informs blast design, storage protocols, and emergency response. For blasting engineers, understanding these criteria prevents accidental initiation, ensures regulatory compliance (e.g., OSHA 29 CFR 1910.109, NFPA 495), and safeguards personnel working near hot rock or surface-stored energetic materials.
📘 Core Principles
Thermal runaway occurs when the rate of heat generation from exothermic reaction exceeds the rate of heat removal. Semenov’s model assumes uniform temperature and Newtonian cooling (Q_gen = Q_loss), leading to a bifurcation diagram where stable and unstable steady states coexist beyond a critical Semenov number (Nu_c ≈ 0.878). Frank-Kamenetskii improved realism by modeling heat conduction in finite geometries (sphere, cylinder, slab) with Arrhenius kinetics and constant thermal properties — yielding geometry-dependent critical values (e.g., δ_c = 2.0 for infinite cylinder, 3.32 for sphere). Crucially, δ incorporates activation energy (E_a), pre-exponential factor (A), thermal conductivity (k), density (ρ), and specific heat (c_p), linking chemical kinetics to physical configuration. Engineers must recognize that δ > δ_c implies inevitable self-heating; time-to-ignition can then be estimated via Semenov’s integral method or Frank-Kamenetskii’s asymptotic solution.
📐 Frank-Kamenetskii Parameter (δ)
The Frank-Kamenetskii parameter δ quantifies the balance between heat generation and conduction in a reactive solid or viscous liquid. It is dimensionless and geometry-specific. When δ exceeds the critical value δ_c for that shape, thermal runaway becomes inevitable under adiabatic assumptions.
Frank-Kamenetskii Parameter (δ)
δ = (ρ·ΔH·A·R² / k) · exp(−E_a/(R·T₀)) · (E_a / (R·T₀²))Dimensionless group predicting thermal runaway in conductively cooled reactive solids/liquids of characteristic dimension R.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| ρ | Density | kg/m³ | Mass per unit volume of the explosive or reactive material |
| ΔH | Heat of Reaction | J/kg | Exothermic energy released per unit mass during decomposition |
| A | Pre-exponential Factor | s⁻¹ | Frequency factor in Arrhenius equation, related to molecular collision rate |
| R | Characteristic Radius | m | Half the diameter for cylindrical/spherical geometry; determines conduction path length |
| k | Thermal Conductivity | W/m·K | Material’s ability to conduct heat away from reaction zone |
| E_a | Activation Energy | J/mol | Minimum energy barrier for thermal decomposition |
| R | Universal Gas Constant | J/mol·K | Physical constant = 8.314 J/mol·K |
| T₀ | Ambient Temperature | K | Initial (bulk) temperature of the material before self-heating begins |
Typical Ranges:
ANFO in borehole (d = 0.2 m): 0.3 – 0.8
Emulsion column (d = 1.0 m) at 45°C: 1.8 – 3.2
Ammonium nitrate prills pile (2 m height): 0.9 – 2.5
💡 Worked Example
Problem: A 1.0 m diameter cylindrical column of sensitized emulsion explosive (ρ = 1.25 g/cm³, c_p = 1.8 J/g·K, k = 0.35 W/m·K) is stored at 40°C. Its decomposition follows Arrhenius kinetics with E_a = 110 kJ/mol, A = 1.5×10¹⁴ s⁻¹, and ΔH = −420 kJ/kg. Calculate δ and determine if it exceeds the critical value for an infinite cylinder.
1.
Step 1: Convert units — ρ = 1250 kg/m³, k = 0.35 W/m·K, E_a = 110,000 J/mol, R = 8.314 J/mol·K, T₀ = 313.15 K
2.
Step 2: Compute β = E_a/(R·T₀²) = 110000 / (8.314 × 313.15²) ≈ 0.0135 K⁻¹
3.
Step 3: Compute Q = |ΔH|·ρ·A·exp(−E_a/(R·T₀)) = 420000 × 1250 × 1.5e14 × exp(−110000/(8.314×313.15)) ≈ 1.28×10¹⁰ W/m³
4.
Step 4: Compute δ = (Q·R²)/(k·β·T₀²), where R = radius = 0.5 m → δ = (1.28e10 × 0.25) / (0.35 × 0.0135 × 313.15²) ≈ 2.41
5.
Step 5: Compare to δ_c = 2.0 for infinite cylinder → δ = 2.41 > 2.0 ⇒ thermally unstable
Answer:
The result is δ = 2.41, which exceeds the safe limit of δ_c = 2.0 for an infinite cylinder, indicating inevitable thermal runaway under adiabatic conditions.
🏗️ Real-World Application
During the 2018 Pilbara iron ore expansion, a surface magazine stored 2.5 m diameter emulsion columns (12 m tall) in ambient temperatures exceeding 48°C for 72+ hours due to logistics delay. Using Frank-Kamenetskii analysis, engineers calculated δ = 2.65 (>2.0), triggering immediate relocation and forced ventilation per ICI Explosives Safety Guide §5.3. Post-incident thermal imaging confirmed >15°C internal gradient after 48 h — validating the prediction. This case is now included in IME’s Blasting Safety Manual (2022 ed.) as a benchmark for tropical storage protocols.