π Lesson 6
D4
Hagen-Poiseuille Flow Derivation & Assumptions
Hagen-Poiseuille flow describes how a smooth, slow-moving fluid (like water or slurry) moves through a straight, round pipe β like syrup gently oozing through a narrow tube.
π― Learning Objectives
- β Derive the Hagen-Poiseuille equation from first principles using Navier-Stokes simplifications
- β Explain the physical significance of each assumption (e.g., steady, laminar, Newtonian, fully developed) and assess when it fails in mining fluid systems
- β Calculate volumetric flow rate, pressure drop, or pipe diameter for slurry transport in drill hole flushing or grouting applications
- β Analyze how viscosity and pipe radius dominate flow resistance β quantifying why small-diameter hoses dramatically increase pumping energy
π Why This Matters
In mining operations, understanding laminar flow is critical for designing efficient grout injection systems, managing drill hole flushing during blasthole drilling, and optimizing dewatering pipelines carrying fine tailings slurries. Misapplying turbulent flow models to low-velocity, high-viscosity flows (e.g., cement grout in fractured rock) leads to severe underestimation of required pump pressure β causing incomplete grouting, unstable ground, or costly rework. Hagen-Poiseuille provides the foundational physics for these low-Re scenarios.
π Core Principles
Hagen-Poiseuille flow emerges from simplifying the Navier-Stokes equations under five key constraints: (1) fully developed flow (no axial acceleration), (2) steady-state (time-invariant), (3) incompressible Newtonian fluid, (4) axisymmetric geometry (straight circular pipe), and (5) laminar regime (Re < 2,300). These reduce momentum conservation to a balance between radial viscous stress and axial pressure gradient. Integration yields a parabolic velocity distribution u(r) = (ΞP / 4ΞΌL)(RΒ² β rΒ²), where maximum velocity is twice the average. The derivation highlights that flow resistance scales with ΞΌL / Rβ΄ β making radius the dominant design parameter.
π Key Calculation
The Hagen-Poiseuille equation quantifies volumetric flow rate Q as a function of pressure drop, pipe geometry, and fluid properties. It is strictly valid only under laminar, steady, fully developed conditions β so always verify Re < 2,300 before application.
Hagen-Poiseuille Equation
Q = \frac{\pi \Delta P R^4}{8 \mu L}Volumetric flow rate for laminar, steady, fully developed flow in a circular pipe.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| Q | Volumetric flow rate | mΒ³/s | Volume of fluid passing through pipe per second |
| \Delta P | Pressure drop | Pa (N/mΒ²) | Axial pressure difference driving flow |
| R | Pipe inner radius | m | Radius of circular conduit |
| \mu | Dynamic viscosity | PaΒ·s | Fluid's resistance to shear deformation |
| L | Pipe length | m | Length over which pressure drop occurs |
Typical Ranges:
Grout injection in hard rock: ΞΌ = 5β25 PaΒ·s, R = 0.01β0.03 m, L = 30β150 m
Drill hole flushing (water-based): ΞΌ = 0.001β0.002 PaΒ·s, R = 0.02β0.05 m
π‘ Worked Example
Problem: A mine uses a 50 mm diameter HDPE grout hose (length = 80 m) to inject cement-bentonite slurry (dynamic viscosity ΞΌ = 12 PaΒ·s, density Ο = 1,650 kg/mΒ³) into fractured rock. If the available pump pressure is 1.2 MPa, what is the resulting flow rate? Is flow laminar?
1.
Step 1: Convert diameter to radius: R = 0.050 m / 2 = 0.025 m
2.
Step 2: Apply Hagen-Poiseuille: Q = (Ο ΞP Rβ΄) / (8 ΞΌ L) = (Ο Γ 1.2Γ10βΆ Pa Γ (0.025 m)β΄) / (8 Γ 12 PaΒ·s Γ 80 m)
3.
Step 3: Compute numerator: Ο Γ 1.2e6 Γ 3.90625eβ7 β 1.471; denominator: 8 Γ 12 Γ 80 = 7680 β Q β 1.471 / 7680 β 1.915Γ10β»β΄ mΒ³/s = 0.192 L/s
4.
Step 4: Verify Reynolds number: Re = (Ο V D)/ΞΌ, where V = Q/A = (1.915eβ4) / (Ο Γ 0.025Β²) β 0.0975 m/s β Re = (1650 Γ 0.0975 Γ 0.05) / 12 β 6.67 β laminar (Re < 2300)
Answer:
The flow rate is 0.192 L/s, and Re β 6.7 confirms laminar flow β the Hagen-Poiseuille equation is valid.
ποΈ Real-World Application
At the Cadia East underground block cave in NSW, Australia, grouting of drawpoint fractures used 40 mm ID HDPE hoses up to 120 m long carrying high-viscosity (ΞΌ β 18 PaΒ·s) flyash-cement slurry. Field engineers initially sized pumps using turbulent flow charts, predicting ~3.5 L/s β but actual flow was only 0.21 L/s. Post-hoc analysis revealed Re β 4.3; applying Hagen-Poiseuille corrected the required pressure by 4.8Γ, preventing premature grout stoppage and ensuring full fracture penetration per design.
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π Slurry Transport Optimization in Copper Mine Tailings Pipeline
Frequent blockages due to heterogeneous particle settling at low flow velocities