🎓 Lesson 13
D5
Reynolds Analogy and Its Limitations
Reynolds Analogy is a simple rule that says how easily heat moves through a fluid is closely related to how easily momentum (like flow resistance) moves through it — so if you know one, you can estimate the other.
🎯 Learning Objectives
- ✓ Explain the physical assumptions and limitations underlying the Reynolds Analogy
- ✓ Calculate the Stanton number and friction factor using experimental pipe flow data and compare them to assess analogy validity
- ✓ Apply the modified Reynolds Analogy (Chilton–Colburn) to estimate heat transfer coefficients in ventilation ducts of underground mines
- ✓ Analyze why the analogy fails in high-Pr fluids (e.g., water in cooling circuits) or low-Pr gases (e.g., molten sulfur vapor in processing vents)
📖 Why This Matters
In mining operations, thermal management of ventilation air, explosive detonation gases, and ore processing streams relies on accurate prediction of heat transfer — yet direct measurement is often impractical underground. The Reynolds Analogy provides a rapid, equipment-free bridge between widely measured pressure drop data (e.g., from anemometer traverses and manometers) and required heat transfer rates. Misapplying it — especially in wet, dusty, or high-velocity mine airways — leads to undersized cooling systems, unsafe gas temperature excursions near explosives, or inaccurate blast fume dispersion models.
📘 Core Principles
The analogy originates from Reynolds’ 1874 observation that turbulent eddies transport momentum and heat similarly. For fully turbulent, incompressible flow over a flat plate or inside smooth pipes, the nondimensional shear stress (friction factor) and nondimensional heat flux (Stanton number) become numerically equivalent when Pr = 1. Later, Chilton and Colburn extended it to Pr ≠ 1 via the j-factor: j_H = f/2 × Pr^{−2/3}, restoring accuracy for common mining fluids (air: Pr ≈ 0.71; water: Pr ≈ 6–7). Crucially, the analogy breaks down where molecular diffusion dominates (laminar flow), property gradients are steep (e.g., hot exhaust mixing with cold intake air), or surfaces are roughened by dust deposition — all routine in mine ventilation networks.
📐 Key Calculation
The Chilton–Colburn analogy modifies Reynolds’ original form to account for Prandtl number effects, enabling reliable estimation of convective heat transfer coefficient (h) from friction factor (f) measured in field airflow tests.
Chilton–Colburn j-factor relation
j_H = f/2 × Pr^{−2/3}Extended analogy valid for 0.6 < Pr < 100; relates j_H (heat transfer factor) to friction factor and Prandtl number.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| j_H | Colburn j-factor for heat transfer | dimensionless | j_H = St × Pr^{2/3} = h/(G c_p) |
| f | Darcy friction factor | dimensionless | From Moody chart or Colebrook equation |
| Pr | Prandtl number | dimensionless | Pr = μ c_p / k |
Typical Ranges:
Mine ventilation air (20–40°C): 0.70–0.73
Water in cooling loops (10–30°C): 5.5–7.2
💡 Worked Example
Problem: A 1.2-m-diameter ventilation duct in a deep copper mine carries air at 35°C (Pr = 0.707) and Re = 2.4×10⁵. A pitot-static traverse yields a Darcy friction factor f = 0.018. Estimate the convective heat transfer coefficient h (W/m²·K) assuming duct length L = 50 m and k_air = 0.0269 W/m·K.
1.
Step 1: Compute j_H = f/2 × Pr^{−2/3} = 0.018/2 × (0.707)^{−2/3}
2.
Step 2: Calculate exponent: (0.707)^{−2/3} ≈ (0.707)^{−0.6667} ≈ 1.225 → j_H = 0.009 × 1.225 = 0.01103
3.
Step 3: Relate j_H to h: j_H = h / (G × c_p), where G = mass velocity (kg/m²·s); assume average air velocity V = 8 m/s, ρ = 1.14 kg/m³ → G = ρV = 9.12 kg/m²·s; c_p = 1007 J/kg·K → h = j_H × G × c_p = 0.01103 × 9.12 × 1007 ≈ 101.5 W/m²·K
Answer:
The estimated h = 102 W/m²·K, which aligns with typical forced-convection values for turbulent mine air (70–130 W/m²·K).
🏗️ Real-World Application
At the Bingham Canyon Mine (Utah), engineers used the Chilton–Colburn analogy during design of the ‘Cooling Airway Retrofit’ project. Pressure drop measurements across 200 m of existing 3.6-m-diameter concrete-lined haulage drift (f ≈ 0.014, Re ≈ 1.8×10⁶) were paired with ambient and exhaust air temperatures to back-calculate h without installing thermocouple arrays. This confirmed that existing rock-cooling capacity was insufficient during summer operation — prompting installation of chilled-water heat exchangers. Post-installation validation showed <8% error between predicted and measured outlet air temperature, validating analogy use under controlled, clean-air conditions.
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