🎓 Lesson 3
D2
Solving 1D Steady-State Problems: Composite Walls & Contact Resistance
When heat flows steadily through a wall made of different materials stuck together, the total resistance adds up — like stacking layers of insulation — and tiny gaps between layers can slow heat down even more.
🎯 Learning Objectives
- ✓ Calculate total thermal resistance and steady-state heat flux across multi-layer walls including interfacial contact resistance
- ✓ Analyze temperature drop distribution across each layer and interface to identify critical thermal bottlenecks
- ✓ Explain how surface finish, contact pressure, and interstitial material affect contact resistance magnitude
- ✓ Apply standard correlations (e.g., Mikic, Cooper–Mikic–Yovanovich) to estimate contact resistance for engineering design
- ✓ Design a composite wall configuration to meet target heat transfer rate or maximum interface temperature under given boundary conditions
📖 Why This Matters
In mining ventilation ducts, blast furnace linings, and underground explosive storage vaults, heat must be reliably managed across layered barriers — e.g., refractory brick + steel casing + insulation. Ignoring contact resistance can lead to 20–40% underestimation of surface temperatures, risking thermal degradation of explosives or premature failure of structural supports. Real-world failures (e.g., 2018 Chilean mine vault overheating incident) were traced to unmodeled interfacial resistance between concrete and ceramic liners.
📘 Core Principles
Steady-state 1D conduction assumes no time dependence and uniform cross-section perpendicular to heat flow. For N-layer composite walls, total resistance is the sum of individual conductive resistances (Lₖ/kₖA) plus (N−1) contact resistances (Rₜᵢ). Contact resistance Rₜᵢ = 1/(h꜀A), where h꜀ is the interfacial conductance — highly sensitive to surface roughness (Ra < 1.6 µm preferred), contact pressure (≥2 MPa typical for metal–ceramic), and interstitial medium (thermal grease vs. air). Fourier’s law governs local flux, while continuity of heat flux and temperature discontinuity at interfaces define boundary conditions.
📐 Total Thermal Resistance & Heat Flux
The fundamental relationship links imposed temperature difference to heat transfer rate via total resistance. It integrates conduction and contact effects into a single linear model — essential for sizing insulation, predicting hot-face temperatures, and validating thermal models against field measurements.
💡 Worked Example
Problem: A blast-resistant vault wall consists of: (1) 150 mm concrete (k = 1.7 W/m·K), (2) 3 mm stainless steel liner (k = 16 W/m·K), and (3) 50 mm mineral wool insulation (k = 0.04 W/m·K). Interface between concrete and steel has measured contact conductance h꜀₁ = 1200 W/m²·K; steel–wool interface h꜀₂ = 450 W/m²·K. Wall area A = 10 m². Hot side T₁ = 85°C, cold side T₄ = 25°C. Calculate total heat flux q and temperature at each interface.
1.
Step 1: Compute individual conductive resistances: Rₙ = Lₙ/(kₙA) → R₁ = 0.15/(1.7×10) = 0.0088 K/W; R₂ = 0.003/(16×10) = 0.000019 K/W; R₃ = 0.05/(0.04×10) = 0.125 K/W.
2.
Step 2: Compute contact resistances: Rₜ₁ = 1/(h꜀₁A) = 1/(1200×10) = 8.33×10⁻⁵ K/W; Rₜ₂ = 1/(450×10) = 2.22×10⁻⁴ K/W.
3.
Step 3: Sum all resistances: Rₜₒₜₐₗ = 0.0088 + 0.000019 + 0.000083 + 0.125 + 0.000222 = 0.1342 K/W. Then q = ΔT/Rₜₒₜₐₗ = (85−25)/0.1342 = 447 W/m².
4.
Step 4: Back-calculate interface temps: T₂ = T₁ − q·R₁ = 85 − 447×0.0088 ≈ 81.1°C; T₃ = T₂ − q·(Rₜ₁ + R₂) ≈ 81.1 − 447×(0.000083+0.000019) ≈ 80.9°C; T₄ = T₃ − q·(Rₜ₂ + R₃) ≈ 80.9 − 447×(0.000222+0.125) ≈ 25.0°C (verifies consistency).
Answer:
The steady-state heat flux is 447 W/m². Temperature drops are dominated by insulation (56.1°C drop) and concrete (3.9°C drop); contact resistances contribute only ~0.2°C total — yet omitting them would overpredict T₂ by 0.4°C and misrepresent safety margins near ignition thresholds.
🏗️ Real-World Application
At the BHP Jansen Potash Project (Saskatchewan), thermal modeling of underground explosive storage chambers required accurate prediction of wall surface temperatures during summer ventilation surges. Field IR thermography revealed 7°C higher steel-liner temperatures than predicted by pure conduction models. Re-calibration using CMY correlation (Cooper–Mikic–Yovanovich) with measured surface roughness (Ra = 3.2 µm) and bolt-torque-derived contact pressure (1.8 MPa) reduced prediction error to <0.8°C — enabling safe reduction of active cooling energy by 18% without compromising thermal stability of ANFO stocks.
📋 Case Connection
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