🎓 Lesson 9
D5
Heisler Charts & Numerical Approximation for Slab/Cylinder/Sphere
Heisler Charts are pre-calculated graphs that help engineers quickly estimate how heat spreads through solid shapes like slabs, cylinders, or spheres over time—without solving complex equations.
🎯 Learning Objectives
- ✓ Calculate dimensionless temperature distribution in slabs, cylinders, and spheres using Heisler Charts
- ✓ Apply Fourier and Biot numbers to select the correct chart and interpret results quantitatively
- ✓ Compare Heisler-based predictions with numerical finite-difference approximations for validation
- ✓ Explain limitations of Heisler Charts relative to modern computational methods in transient heat transfer analysis
- ✓ Design a simplified thermal soak-out schedule for explosive cartridge storage based on slab geometry analysis
📖 Why This Matters
In mining operations, understanding transient heat flow is critical for safe handling of explosives (e.g., ANFO or emulsions), thermal stability of drill bits during deep-hole drilling, and predicting rock spalling near high-temperature blast zones. Heisler Charts allow field engineers to rapidly assess thermal response—say, how long it takes for a 0.5-m-diameter explosive column to cool after being exposed to 60°C ambient air—without needing software or coding skills. This bridges theory and practice where time, tools, and training are constrained.
📘 Core Principles
Transient conduction in solids depends on two key dimensionless groups: the Fourier number (Fo = αt/L²), representing the ratio of heat conducted to heat stored, and the Biot number (Bi = hL/k), representing the ratio of internal conduction resistance to surface convection resistance. For simple geometries with uniform properties and constant boundary conditions, the exact solution involves infinite series—but Heisler simplified these into three universal charts (slab, cylinder, sphere), each plotting θ/θ₀ (normalized temperature) vs Fo for various Bi values. The charts separate the solution into a product of centerline and off-center terms (via the 'Gurney–Heisler' method), enabling rapid interpolation. Their accuracy is ±5% for Bi > 0.1 and Fo > 0.2; below this, lumped capacitance may suffice.
📐 Key Dimensionless Groups & Chart Selection
The core calculation uses Fo and Bi to locate points on Heisler Charts. Fo determines time scaling; Bi determines geometry-specific resistance. Once Fo and Bi are computed, users read θ/θ₀ from the appropriate chart and multiply by initial temperature difference (Tᵢ − T∞) to obtain actual temperature. Critical: L is the characteristic length (half-thickness for slab, radius for cylinder/sphere).
💡 Worked Example
Problem: A limestone quarry stores 0.3-m-diameter cylindrical ANFO cartridges in shaded steel sheds. Ambient air is 35°C (h = 12 W/m²·K). Each cartridge has k = 0.35 W/m·K, α = 1.2×10⁻⁷ m²/s. If cartridge center starts at 55°C, what is its center temperature after 4 hours?
1.
Step 1: Compute characteristic length L = radius = 0.15 m
2.
Step 2: Calculate Bi = hL/k = (12)(0.15)/0.35 ≈ 5.14
3.
Step 3: Calculate Fo = αt/L² = (1.2×10⁻⁷)(4×3600)/(0.15)² ≈ 0.092
4.
Step 4: Use Heisler Cylinder Chart (Bi = 5.1, Fo ≈ 0.09) → θ₀/θᵢ ≈ 0.58
5.
Step 5: T₀ = T∞ + (Tᵢ − T∞)(θ₀/θᵢ) = 35 + (55−35)(0.58) = 35 + 11.6 = 46.6°C
Answer:
The center temperature is 46.6°C after 4 hours — well within safe storage limits (<50°C per ICI Explosives Safety Guidelines).
🏗️ Real-World Application
At the Grasberg Mine (Indonesia), thermal modeling of 1.2-m-thick concrete blast barricades was required to ensure structural integrity during repeated high-energy detonations. Engineers used Heisler Slab Charts (Bi ≈ 0.8, Fo ≈ 0.35) to estimate peak subsurface temperature rise (≈18°C) after 90 seconds—validating that no spalling would occur. This avoided costly FEA runs for preliminary screening and aligned with PT Freeport’s Thermal Management Protocol v4.2.
📋 Case Connection
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