🎓 Lesson 13
D5
Nusselt Number Correlations for Forced Convection
The Nusselt number tells us how well heat moves from a surface to a flowing fluid—like cooling a hot pipe with water or air.
🎯 Learning Objectives
- ✓ Calculate the Nusselt number for internal flow in circular pipes using appropriate empirical correlations
- ✓ Analyze flow regime (laminar vs. turbulent) and select the correct correlation based on Reynolds and Prandtl numbers
- ✓ Apply Nusselt number correlations to estimate convective heat transfer coefficients for real mining ventilation ducts or heat exchangers
- ✓ Explain the physical significance of Nu, Re, and Pr in the context of thermal management in underground mine cooling systems
📖 Why This Matters
In deep mining operations, equipment like diesel-powered loaders and rock drills generate intense heat—and inadequate cooling can lead to thermal failure, reduced efficiency, and unsafe working conditions. Forced convection (e.g., fan-driven airflow in ventilation ducts or coolant flow in heat exchangers) is the primary method for managing this heat. The Nusselt number is the key link between measurable flow conditions (velocity, duct size, fluid type) and actual heat removal rates—making it indispensable for designing reliable thermal control systems in mines.
📘 Core Principles
Heat transfer in forced convection is governed by three dimensionless groups: Reynolds number (Re), quantifying inertial vs. viscous forces; Prandtl number (Pr), expressing the relative thickness of velocity and thermal boundary layers; and Nusselt number (Nu), directly relating surface heat flux to temperature gradient. For a given geometry (e.g., pipe, flat plate, tube bundle), empirical correlations—derived from experimental data and validated across decades—express Nu as a function of Re and Pr. These correlations differ significantly between laminar (Re < 2300) and turbulent (Re > 10⁴) regimes, and account for entrance effects, wall heating conditions (constant temperature vs. constant heat flux), and fluid property variations. In mining applications, air (Pr ≈ 0.71) and water (Pr ≈ 4–6) are most common, and ducts often operate in turbulent flow—so correlations like Dittus–Boelter dominate practical design.
📐 Key Calculation
For turbulent flow inside smooth circular pipes with uniform wall temperature and moderate temperature differences (ΔT < 5°C), the Dittus–Boelter equation is the standard correlation. It applies for 10⁴ ≤ Re ≤ 1.2×10⁵ and 0.7 ≤ Pr ≤ 120. Heating uses exponent n = 0.4; cooling uses n = 0.3.
Dittus–Boelter Correlation
Nu = 0.023 \, Re^{0.8} \, Pr^{n}Empirical correlation for turbulent forced convection in smooth circular pipes under uniform wall temperature conditions.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| Nu | Nusselt number | dimensionless | Ratio of convective to conductive heat transfer |
| Re | Reynolds number | dimensionless | Ratio of inertial to viscous forces in fluid flow |
| Pr | Prandtl number | dimensionless | Ratio of momentum diffusivity to thermal diffusivity |
| n | Exponent | dimensionless | 0.4 for heating, 0.3 for cooling |
Typical Ranges:
Mine ventilation ducts (air): 200 – 500
Chilled water heat exchangers (water): 2,000 – 10,000
💡 Worked Example
Problem: A ventilation duct in an underground copper mine carries air at 25°C (ρ = 1.184 kg/m³, μ = 1.86×10⁻⁵ Pa·s, k = 0.0261 W/m·K, Pr = 0.71) at 8 m/s. The duct inner diameter is 0.8 m. Calculate the convective heat transfer coefficient h.
1.
Step 1: Compute Reynolds number: Re = ρVD/μ = (1.184)(8)(0.8)/(1.86×10⁻⁵) ≈ 408,000 → turbulent flow (Re > 10⁴)
2.
Step 2: Apply Dittus–Boelter: Nu = 0.023 × Re⁰·⁸ × Pr⁰·⁴ = 0.023 × (4.08×10⁵)⁰·⁸ × (0.71)⁰·⁴
3.
Step 3: Calculate: (4.08×10⁵)⁰·⁸ ≈ 12,450; (0.71)⁰·⁴ ≈ 0.91; so Nu ≈ 0.023 × 12,450 × 0.91 ≈ 261
4.
Step 4: Solve for h: h = Nu·k/D = 261 × 0.0261 / 0.8 ≈ 8.5 W/m²·K
Answer:
The convective heat transfer coefficient is 8.5 W/m²·K, which falls within the typical range of 5–25 W/m²·K for turbulent air flow in large mine ducts.
🏗️ Real-World Application
At the Kidd Creek Mine (Ontario, Canada), engineers redesigned the chilled water heat exchanger system for the main ventilation air stream to counteract geothermal heating at 3 km depth. Using the Gnielinski correlation (a more accurate extension of Dittus–Boelter for transitional and high-Pr fluids), they recalculated Nu for water-cooled copper tubes (Re = 3.2×10⁴, Pr = 5.2, roughness ε/D = 0.001). This yielded h = 3,200 W/m²·K—enabling precise sizing of tube length and flow rate to achieve a required 12°C air cooling delta without oversizing pumps or chillers. The correlation’s accuracy directly impacted capital cost savings of CAD $1.4M and improved thermal reliability during summer peak loads.
📋 Case Connection
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