🎓 Lesson 2
D2
Ampacity Physics: Joule Heating, Thermal Resistance Networks & Steady-State Equilibrium
Ampacity is the maximum electric current a cable can safely carry without overheating.
🎯 Learning Objectives
- ✓ Calculate steady-state conductor temperature rise using thermal resistance networks
- ✓ Design cable sizing for underground mining power distribution by applying IEC 60287–1–1 thermal models
- ✓ Analyze the impact of burial depth, soil thermal resistivity, and grouping on ampacity derating
- ✓ Explain how Joule heating (I²R) couples with thermal conduction to establish equilibrium temperature
- ✓ Apply IEEE Std 835 and IEC 60287 correction factors to adjust ampacity for mine-specific conditions
📖 Why This Matters
In underground mines, cables often run through hot, humid, poorly ventilated tunnels or are buried in backfill with high thermal resistivity. Overheated cables can ignite methane-air mixtures, degrade flame-retardant insulation (e.g., Type MV-105), or cause unplanned shutdowns of critical ventilation or dewatering systems. Understanding ampacity isn’t just about compliance—it’s about preventing fires, ensuring blast timing integrity (for detonator circuits), and extending cable life in harsh conditions where replacement is costly and hazardous.
📘 Core Principles
Ampacity emerges from energy balance: electrical power converted to heat (Joule heating, P = I²R) must equal the rate of heat flow out of the conductor. This is modeled as a thermal resistance network—similar to Ohm’s law—where temperature difference (ΔT) drives heat flow (Q) across resistances: conductor-to-sheath (R₁), sheath-to-armor (R₂), armor-to-soil (R₃), and soil-to-ambient (R₄). Steady-state equilibrium occurs when Q = I²R_dc = ΔT / ΣR_th. Key assumptions include uniform material properties, constant ambient temperature, and negligible solar/radiative effects (valid for underground applications). Real-world complexity arises from non-uniform soil layers, cyclic loading (e.g., blast-induced ground movement affecting burial contact), and aging-induced increases in conductor resistance and insulation thermal resistance.
📐 Key Calculation
The fundamental steady-state ampacity formula per IEC 60287–1–1 relates current to total thermal resistance and permissible temperature rise. It accounts for AC resistance, dielectric losses (negligible for power cables < 3 kV), and environmental corrections.
💡 Worked Example
Problem: A 35 mm² copper XLPE-insulated cable (Type MV-105) is directly buried at 0.8 m depth in crushed rock backfill (ρ_soil = 1.2 K·m/W) in an underground development drift. Ambient rock temperature = 35°C. Max allowable conductor temp = 90°C. DC resistance at 90°C = 0.548 Ω/km. Total effective thermal resistance (R'_{tot}) = R_{cond} + R_{ins} + R_{soil} = 0.0035 + 0.52 + 0.85 = 1.3735 K·m/W. Calculate max continuous current (ampacity).
1.
Step 1: Compute permissible temperature rise: ΔT = T_max − T_amb = 90°C − 35°C = 55 K
2.
Step 2: Apply IEC 60287 simplified form for single-core: I = √[ΔT / (R'_{tot} × R_dc)] — note R'_{tot} is per unit length (K·m/W), so use R_dc in Ω/m: R_dc = 0.548 Ω/km = 5.48×10⁻⁴ Ω/m
3.
Step 3: I = √[55 / (1.3735 × 5.48×10⁻⁴)] = √[55 / 0.0007527] = √73070 ≈ 270 A
Answer:
The calculated ampacity is 270 A, which falls within the safe range of 250–290 A for this configuration per IEEE Std 835 Table 12B (600 V MV-105, 35 mm², buried in 1.2 K·m/W medium).
🏗️ Real-World Application
At Vale’s Sudbury Underground Operations, a 12 kV, 185 mm² aluminum armored cable supplying a primary crusher station repeatedly tripped during summer months. Thermal imaging revealed conductor temperatures exceeding 95°C at splices. Investigation showed that original design used generic soil ρ = 0.9 K·m/W, but actual backfill consisted of compacted silica-rich tailings (ρ = 1.8 K·m/W). Recalculation using IEC 60287 with corrected ρ reduced ampacity by 34%. The fix involved installing localized forced-air cooling ducts adjacent to the cable route and upgrading to 240 mm² conductors—achieving 315 A capacity at 90°C while meeting CSA C22.2 No. 131 requirements for Class I, Division 1 hazardous locations.
📋 Case Connection
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