🎓 Lesson 3
D2
Ohm’s Law & Skin Effect in Industrial Heating Systems
Ohm’s Law says how voltage, current, and resistance relate in a circuit—like water pressure, flow, and pipe narrowness—and the skin effect means high-frequency electricity flows mostly near the surface of a conductor, not through its center.
🎯 Learning Objectives
- ✓ Calculate AC resistance of a copper busbar at 1 kHz using skin depth and compare it to its DC resistance
- ✓ Explain how skin effect limits effective conductor utilization in medium-frequency (1–10 kHz) induction heaters used in mineral roasting
- ✓ Apply Ohm’s Law and skin depth formula to design a safe, thermally stable induction coil for a 200 kW, 3 kHz ore preheater
- ✓ Analyze voltage drop and power loss in a 50 m aluminum heating busbar operating at 400 V, 600 A, and 2.5 kHz
📖 Why This Matters
In electrified mining processes—such as induction preheating of low-grade ores before crushing or in-situ thermal recovery—electrical efficiency directly impacts energy cost, carbon footprint, and equipment lifetime. Ignoring skin effect leads to unexpected overheating and premature coil failure; misapplying Ohm’s Law results in undersized conductors, voltage drops, and fire hazards. Understanding both is non-negotiable for feasibility studies and hardware specification in modern mine electrification.
📘 Core Principles
Ohm’s Law governs steady-state DC and low-frequency AC behavior: voltage drives current against resistance. At higher frequencies (>100 Hz), electromagnetic fields penetrate conductors only to a finite depth—the skin depth δ—given by δ = √(ρ / (π·f·μ)), where ρ is resistivity, f frequency, and μ permeability. As frequency rises, δ shrinks, forcing current into a thinner annular region. This increases effective resistance (R_ac ≈ R_dc · (d/2δ) for round wires when d ≫ δ) and raises localized temperature—critical in induction coils, busbars, and electrode feeders where thermal runaway can trigger safety shutdowns or refractory damage.
📐 Key Calculations
Two interdependent formulas govern design: Ohm’s Law (V = I·R) determines voltage requirements and losses, while skin depth (δ) quantifies AC confinement. Together they inform conductor geometry, cooling needs, and frequency selection. For rectangular busbars common in industrial heating, effective AC resistance requires correction factors from IEEE Std 835 and IEC 60287.
💡 Worked Example
Problem: A solid copper induction coil conductor (ρ = 1.68×10⁻⁸ Ω·m, μᵣ ≈ 1) operates at f = 3 kHz. Its diameter is 20 mm. Calculate skin depth δ and estimate its AC-to-DC resistance ratio.
1.
Step 1: Compute skin depth: δ = √(ρ / (π·f·μ₀·μᵣ)) = √(1.68e-8 / (π × 3000 × 4π×10⁻⁷))
2.
Step 2: Simplify denominator: π × 3000 × 4π×10⁻⁷ ≈ 0.01184 → δ ≈ √(1.68e-8 / 0.01184) ≈ √1.419e-6 ≈ 1.19 mm
3.
Step 3: Compare conductor radius (10 mm) to δ: r/δ ≈ 8.4 → use approximate AC resistance ratio R_ac/R_dc ≈ 0.5 + 0.5√(r/δ) ≈ 0.5 + 0.5×2.9 ≈ 1.95 (per IEEE Std 835 Annex D)
Answer:
The skin depth is 1.19 mm; the AC resistance is ~1.95× higher than DC resistance—meaning nearly double the I²R heating for same current, requiring derating or forced cooling.
🏗️ Real-World Application
At the BHP Nickel West Kwinana smelter (Western Australia), a 250 kW, 3.5 kHz induction preheater was retrofitted to warm lateritic ore feed (−10 mm) to 120°C prior to rotary kiln reduction. Initial copper busbars (50×10 mm²) overheated at 420 A due to unaccounted skin effect: measured δ ≈ 1.1 mm led to 2.1× R_ac increase. Engineers redesigned with segmented laminated copper (4×12.5 mm strips, insulated and transposed) to restore effective cross-section—reducing surface temperature from 142°C to 86°C and extending coil life from 4 to >18 months.
🔧 Interactive Calculator
🔧 Open Industrial Process Electrification Feasibility Framework Calculator📋 Case Connection
📋 Induction-Based Ethylene Cracker Tube Electrification (US Gulf Coast)
Thermal cycling fatigue limiting tube life to <2 years; flame impingement causing hot spots
📋 Green Hydrogen-Powered Ammonia Synthesis Reactor Electrification (Saudi Arabia)
High exothermicity requiring precise temperature zoning; catalyst sintering above 520°C