🎓 Lesson 2
D2
Fundamentals of Steady-State Flow in Pipes
Steady-state flow in pipes means water moves at a constant speed and pressure over time — like a smoothly running conveyor belt for water.
🎯 Learning Objectives
- ✓ Calculate volumetric flow rate using continuity and Darcy-Weisbach equations
- ✓ Analyze head loss across pipe segments and identify dominant contributors (friction vs. minor losses)
- ✓ Design pipe diameter for a given flow rate and allowable pressure drop within regulatory limits
- ✓ Explain how Reynolds number determines flow regime and influences friction factor selection
- ✓ Apply energy grade line (EGL) and hydraulic grade line (HGL) concepts to diagnose system performance
📖 Why This Matters
In mining water distribution networks — from dewatering systems to dust suppression and processing plant supply — reliable, predictable water delivery is non-negotiable. Steady-state flow analysis forms the foundation for sizing pipelines, selecting pumps, ensuring adequate pressure at remote blast-hole rigs or slurry lines, and avoiding costly failures like pipe rupture or insufficient flow during critical operations. Misjudging steady-state behavior leads directly to under-designed infrastructure, energy waste, or operational downtime.
📘 Core Principles
Steady-state flow rests on three pillars: (1) Conservation of mass (continuity), requiring inflow = outflow at every junction; (2) Conservation of energy (Bernoulli + losses), where total head decreases due to friction and fittings; and (3) Flow regime classification via Reynolds number, distinguishing laminar, transitional, and turbulent flow — critical because friction factor (f) depends strongly on regime. Turbulent flow dominates mining hydraulics (> Re = 4000), where roughness-dominated Colebrook-White or Hazen-Williams approximations are used. Real pipe networks also require understanding of HGL slope as the driver of flow — not just pressure.
📐 Darcy-Weisbach Head Loss
The Darcy-Weisbach equation quantifies major (frictional) head loss in circular pipes under steady flow. It is universally applicable across flow regimes and pipe materials, making it essential for rigorous design — especially where accuracy matters (e.g., high-pressure slurry mains or long-haul dewatering lines).
💡 Worked Example
Problem: A 300-mm HDPE pipeline (ε ≈ 0.007 mm) carries 180 L/s of water (ν = 1.004 × 10⁻⁶ m²/s) over 1.2 km. Calculate head loss.
1.
Step 1: Compute velocity → Q = 0.180 m³/s, A = π(0.15)² ≈ 0.0707 m² ⇒ V = 0.180 / 0.0707 ≈ 2.55 m/s
2.
Step 2: Compute Reynolds number → Re = V·D/ν = 2.55 × 0.3 / 1.004×10⁻⁶ ≈ 762,000 (turbulent)
3.
Step 3: Compute relative roughness ε/D = 0.000007 / 0.3 ≈ 2.33×10⁻⁵; use Colebrook equation or Moody chart → f ≈ 0.0135
4.
Step 4: Apply Darcy-Weisbach → h_f = f·(L/D)·(V²/2g) = 0.0135 × (1200/0.3) × (2.55²/(2×9.81)) ≈ 0.0135 × 4000 × 0.332 ≈ 17.9 m
Answer:
The head loss is 17.9 m, which falls within acceptable limits for a gravity-assisted or booster-pumped segment in a mine dewatering network (typical max h_f = 20–25 m/km).
🏗️ Real-World Application
At the Cadia East underground copper-gold mine (NSW, Australia), steady-state analysis guided the redesign of the primary dewatering trunk main — a 650-mm ductile iron pipe spanning 4.8 km from deep sumps to surface treatment. Engineers used Darcy-Weisbach with site-specific roughness and variable flow scenarios (peak dewatering: 850 L/s) to verify that existing 300-kW pumps could maintain >25 m residual pressure at the discharge header. Model calibration against field pressure transducer data confirmed ±3% accuracy — enabling deferral of a $2.1M pump station upgrade.
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