🎓 Lesson 35
D5
Ergun Equation Derivation and Application Limits
The Ergun equation is a formula that tells us how hard it is for fluid (like air or water) to flow through packed rocks, gravel, or blast debris — like predicting pressure drop across a pile of broken rock.
🎯 Learning Objectives
- ✓ Calculate pressure drop across a fragmented blast muck pile using the Ergun equation
- ✓ Analyze when the Ergun equation is valid versus when Darcy’s law or turbulent approximations should be used
- ✓ Explain the physical significance of void fraction and particle diameter in blast-induced permeability
- ✓ Apply the Ergun equation to estimate airflow resistance in post-blast ventilation design
📖 Why This Matters
After blasting, rock fragments form a highly porous 'muck pile' — a dynamic porous medium critical for ventilation, drainage, and subsequent loading. If airflow resistance is underestimated, toxic fumes linger; if overestimated, fans are oversized and energy-wasted. The Ergun equation is the industry’s go-to tool to quantify this resistance — not just for lab columns, but for real-scale blast heaps where safety, efficiency, and environmental compliance hinge on accurate momentum transfer prediction.
📘 Core Principles
Fluid flow through porous media transitions from viscous-dominated (low Re, linear Darcy regime) to inertial-dominated (high Re, quadratic Forchheimer regime). The Ergun equation unifies both by summing two terms: one proportional to fluid viscosity and velocity (laminar), and another proportional to fluid density and velocity squared (turbulent). Its derivation starts from force balances on a representative elementary volume (REV), incorporating drag coefficients calibrated from experimental data on packed spheres. Crucially, it assumes isotropic, monodisperse, rigid spheres — so real blast muck requires careful characterization of effective particle diameter and void fraction to maintain fidelity.
📐 Key Calculation
The Ergun equation expresses pressure gradient (ΔP/L) as a function of superficial velocity, particle properties, and fluid properties. It is used to size ventilation ducts, evaluate leach pad permeability, and model gas migration through fractured waste rock. Validity requires 1 < Reₚ < ~2000 — outside this, Darcy (Reₚ < 1) or Forchheimer (Reₚ > 2000) models are preferred.
Ergun Equation
ΔP/L = 150·μ·vₛ·(1−ε)² / [dₚ²·ε³] + 1.75·ρ·vₛ²·(1−ε) / [dₚ·ε³]Predicts pressure gradient across a fixed porous bed accounting for both viscous and inertial losses.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| ΔP/L | Pressure gradient | Pa/m | Total pressure drop per unit bed height |
| μ | Dynamic viscosity | Pa·s | Fluid resistance to shear (e.g., air ≈ 1.86×10⁻⁵ Pa·s at 25°C) |
| vₛ | Superficial velocity | m/s | Volumetric flow rate divided by total cross-sectional area |
| ε | Void fraction | dimensionless | Fraction of bed volume occupied by fluid (typically 0.35–0.45 for blast muck) |
| dₚ | Sauter mean particle diameter | m | Volume-surface mean diameter; dₚ = Σ(nᵢ·dᵢ³) / Σ(nᵢ·dᵢ²) |
| ρ | Fluid density | kg/m³ | Mass per unit volume of flowing fluid (e.g., air ≈ 1.18 kg/m³ at 25°C) |
Typical Ranges:
Blast muck piles (hard rock): 0.10 – 0.30 m
Leach pad crushed ore: 0.005 – 0.02 m
Void fraction in well-fragmented muck: 0.38 – 0.44
💡 Worked Example
Problem: A post-blast muck pile has average fragment size dₚ = 0.15 m, void fraction ε = 0.42, height L = 8 m. Air at 25°C (ρ = 1.184 kg/m³, μ = 1.86 × 10⁻⁵ Pa·s) flows upward at superficial velocity vₛ = 0.35 m/s. Calculate ΔP across the pile.
1.
Step 1: Compute particle Reynolds number: Reₚ = ρ·vₛ·dₚ / [μ·(1−ε)] = (1.184)(0.35)(0.15) / [(1.86e−5)(1−0.42)] ≈ 572 → within Ergun validity range.
2.
Step 2: Apply Ergun equation: ΔP/L = 150·μ·vₛ·(1−ε)² / [dₚ²·ε³] + 1.75·ρ·vₛ²·(1−ε) / [dₚ·ε³]
3.
Step 3: Plug values: First term = 150·(1.86e−5)·0.35·(0.58)² / [(0.15)²·(0.42)³] ≈ 14.2 Pa/m; second term = 1.75·1.184·(0.35)²·0.58 / [0.15·(0.42)³] ≈ 28.9 Pa/m → total ΔP/L ≈ 43.1 Pa/m. So ΔP = 43.1 × 8 = 345 Pa.
Answer:
The result is 345 Pa, which falls within the safe operational range for auxiliary ventilation fans (typically designed for 200–1000 Pa static pressure across muck piles).
🏗️ Real-World Application
At Newmont’s Boddington Gold Mine (Western Australia), engineers used the Ergun equation to redesign post-blast ventilation for 18-m-high muck piles. By measuring in-situ void fraction (ε = 0.38–0.45 via gamma densitometry) and sieve-analysis-derived dₚ,eff = 0.12 m, they recalibrated fan curves and reduced fan runtime by 22% — cutting diesel consumption and improving CO clearance time from 42 to 28 minutes, directly supporting WA Mines Safety Standard 2022 §8.3.2 on hazardous gas removal.
🔧 Interactive Calculator
🔧 Open Fluid Flow and Transport Phenomena Calculator📋 Case Connection
📋 Ethylene Oxide Absorption Column Design Optimization
Low mass transfer efficiency causing solvent over-circulation and high energy use
📋 Pneumatic Conveying of Catalyst Powder in Fluidized Bed Reactor Feed System
Catalyst attrition and line plugging due to intermittent slug flow and particle segregation
📋 Ventilation System Redesign for Lithium-Ion Battery Dry Room
Moisture ingress hotspots near doorways and equipment penetrations due to buoyancy-driven convection currents