🎓 Lesson 15
D5
Diffusion vs. Convection Dominance: Péclet Number Analysis
The Péclet number tells us whether mixing in a fluid (like air or water carrying particles or gases) happens mostly by spreading out slowly (diffusion) or being swept along quickly by flow (convection).
🎯 Learning Objectives
- ✓ Calculate the Péclet number for subsurface ventilation flows using measured velocity, characteristic length, and molecular diffusivity
- ✓ Analyze whether gas dispersion in mine drifts is diffusion- or convection-dominated based on Pe value and select appropriate modeling approach
- ✓ Explain the physical implications of Pe thresholds (0.1, 1, 10) on contaminant plume behavior in underground mining environments
- ✓ Apply Pe-based criteria to design ventilation duct geometry and airflow rates that ensure adequate dilution of blasting fumes
📖 Why This Matters
After blasting, toxic gases like NO₂ and CO must be rapidly removed from underground workings. If convection dominates (high Pe), ventilation airflow efficiently sweeps contaminants away—but if diffusion dominates (low Pe), gases linger dangerously near blast zones. Understanding Pe helps engineers avoid under-ventilating high-flow zones or over-designing low-flow dead ends—saving energy while ensuring worker safety.
📘 Core Principles
Mass transfer in mining environments occurs via two parallel mechanisms: diffusion (random molecular motion, dominant at micro-scales or stagnant zones) and convection (bulk fluid motion, dominant in ventilated drifts). The Péclet number emerges naturally from scaling the advection–diffusion equation: ∂C/∂t + u·∇C = D∇²C. Dividing by D/L² yields Pe = (uL)/D, where u is characteristic velocity, L is characteristic length (e.g., drift diameter or blast cloud radius), and D is the molecular diffusivity of the species. As Pe increases, the advective term overwhelms diffusion—making concentration gradients sharp and upstream/downstream asymmetry pronounced. In practical terms, Pe governs whether a contaminant plume remains localized (low Pe) or elongates into a thin, fast-moving tongue (high Pe).
📐 Key Calculation
The Péclet number for mass transfer is calculated as Pe = (u × L) / D, where u is average fluid velocity (m/s), L is the relevant characteristic length (m), and D is the molecular diffusivity (m²/s) of the target species in air or water. Use L = hydraulic diameter for ducts or blast cloud radius for post-blast fume analysis.
Péclet Number (Mass Transfer)
Pe = \frac{u L}{D}Dimensionless number comparing convective to diffusive mass transport rates.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| u | Characteristic fluid velocity | m/s | Average or maximum velocity in the domain (e.g., mean duct velocity) |
| L | Characteristic length | m | Representative geometric scale (e.g., hydraulic diameter for ducts, cloud radius for fumes) |
| D | Molecular diffusivity | m²/s | Diffusion coefficient of solute/species in the fluid medium (e.g., NO₂ in air) |
Typical Ranges:
Ventilation drift (standard airflow): 10⁴ – 10⁶
Blast fume cloud near face (first 30 s): 10² – 10⁴
Water infiltration in fractured rock: 10⁻² – 10¹
💡 Worked Example
Problem: A 3.2-m-wide, 2.8-m-high rectangular mine drift carries ventilation air at 1.8 m/s. After a blast, nitrogen dioxide (NO₂) diffuses in air with D = 1.5 × 10⁻⁵ m²/s. Calculate Pe using hydraulic diameter as L.
1.
Step 1: Compute hydraulic diameter L = 4 × (cross-sectional area / wetted perimeter) = 4 × (3.2 × 2.8) / (2 × (3.2 + 2.8)) = 4 × 8.96 / 12 = 2.987 m ≈ 3.0 m.
2.
Step 2: Apply Pe = (u × L) / D = (1.8 m/s × 3.0 m) / (1.5 × 10⁻⁵ m²/s) = 5.4 / 1.5 × 10⁻⁵ = 360,000.
3.
Step 3: Compare to threshold: Pe = 360,000 ≫ 10 → convection dominates completely; diffusion is negligible for bulk transport.
Answer:
The result is Pe = 360,000, which falls within the convection-dominated range (Pe > 10). Diffusive mixing contributes <0.003% to total mass transfer.
🏗️ Real-World Application
At Vale’s Sudbury Basin operations, post-blast CO monitoring revealed persistent hotspots 15 m downwind of a stope face despite nominal airflow of 2.1 m/s. Engineers calculated Pe = 120,000 using drift height (2.5 m) and D_CO = 1.9 × 10⁻⁵ m²/s—confirming convection dominance. However, CFD modeling exposed recirculation eddies behind support pillars where local u dropped to ~0.05 m/s, reducing local Pe to ~80—still convective, but insufficient to clear trapped pockets. Revised duct placement eliminated stagnation, reducing CO exposure time by 73% (Vale Internal Report V-VENT-2022-08).
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