🎓 Lesson 20
D5
Geometric, Kinematic, and Dynamic Similarity Principles
Geometric, kinematic, and dynamic similarity are rules that let engineers test small-scale models (like lab blasts or simulations) and confidently predict how full-size mining blasts will behave.
🎯 Learning Objectives
- ✓ Calculate geometric scale ratios for blast design models
- ✓ Analyze kinematic similarity conditions using dimensionless numbers (e.g., Froude, Reynolds, and Pi-blast numbers)
- ✓ Apply dynamic similarity criteria to validate scaled laboratory blasting experiments
- ✓ Design a dynamically similar small-scale blast test using explosive energy, rock strength, and confinement parameters
- ✓ Explain why violating dynamic similarity leads to erroneous fragmentation or flyrock predictions
📖 Why This Matters
In mining, testing full-scale blast designs is expensive, unsafe, and environmentally restricted. Engineers rely on scaled-down physical models (e.g., 1:50 concrete analogs) and numerical simulations—but these only work if similarity principles are rigorously applied. A single mismatch in dynamic scaling—like ignoring gravity’s role in crater formation—can cause overprediction of throw distance by 300%, leading to hazardous flyrock or poor muck pile geometry. This lesson bridges lab-scale insight to safe, efficient production blasting.
📘 Core Principles
Geometric similarity is foundational: every length (burden, spacing, hole diameter) must scale linearly by factor λ (e.g., λ = L_model / L_prototype). Kinematic similarity extends this to motion: velocities scale as √λ (due to gravity dominance), times as √λ, and accelerations remain invariant under Froude scaling. Dynamic similarity is the most demanding—it requires all dimensionless groups to match. For blasting, the key groups are the Froude number (Fr = v/√(gL)), the Pi-blast number (Π_B = ρ_r v_e² / σ_c, linking explosive energy density to rock compressive strength), and the confinement number (Cn = σ_confinement / σ_c). When all are matched, stress wave propagation, fracture initiation, and fragment size distribution become scalable.
📐 Pi-Blast Number for Dynamic Similarity
The Pi-blast number (Π_B) quantifies the ratio of explosive energy density to rock strength—critical for predicting fracture intensity and fragment size. Matching Π_B between model and prototype ensures dynamically similar breakage mechanics.
Pi-Blast Number
Π_B = (ρ_exp × v_e²) / σ_cDimensionless metric for explosive-rock energy coupling; matching Π_B ensures scalable fracture mechanics.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| ρ_exp | Explosive density | kg/m³ | Density of the energetic material |
| v_e | Detonation velocity | m/s | Speed of detonation front propagation |
| σ_c | Unconfined compressive strength | Pa | Rock strength measured in uniaxial compression |
Typical Ranges:
Hard granite (full-scale): 12 – 25
Weak shale (full-scale): 3 – 8
💡 Worked Example
Problem: A full-scale limestone quarry blast uses ANFO (v_e = 4200 m/s, ρ_explosive = 0.85 g/cm³) against rock with σ_c = 80 MPa. A 1:25 geometric scale model uses PETN (v_e = 7900 m/s, ρ_explosive = 1.76 g/cm³) in gypsum (σ_c = 12 MPa). Does the model satisfy dynamic similarity via Π_B?
1.
Step 1: Compute prototype Π_B = ρ_exp × v_e² / σ_c = (850 kg/m³) × (4200 m/s)² / (80 × 10⁶ Pa) = (850 × 17,640,000) / 80,000,000 ≈ 18.7
2.
Step 2: Compute model Π_B = (1760 kg/m³) × (7900 m/s)² / (12 × 10⁶ Pa) = (1760 × 62,410,000) / 12,000,000 ≈ 91.9
3.
Step 3: Compare: 18.7 ≠ 91.9 → Not dynamically similar. To match, reduce model explosive energy density or increase confining stress.
Answer:
The model Π_B (91.9) is over 4.9× higher than prototype (18.7); without correction, it will over-fragment and under-predict confinement effects. Adjusting charge mass or adding lateral confinement can restore similarity.
🏗️ Real-World Application
At BHP’s Olympic Dam copper mine (South Australia), scaled physical modeling of drawpoint caving blasts used 1:40 geometric models in cemented sandstone analogs. Engineers matched Π_B and Froude number by tuning explosive mass (reduced by λ³ = 1/64,000) and applying hydraulic confining pressure (scaled by λ = 1/40) to simulate in-situ stress. This validated the transition from bench blasting to sublevel caving fragmentation—reducing field trial iterations by 70% and avoiding $2.3M in unplanned ground support costs.
📋 Case Connection
📋 Bioethanol Fermentation Bioreactor Scale-Up with Inhibition Kinetics
Ethanol inhibition caused premature cessation at large scale despite matching nominal conditions