🎓 Lesson 24
D3
Designing Scale-Up Experiments Using Similarity Criteria
Scaling up lab experiments to real mining blasts means making sure small models behave like big ones by matching key ratios like size, force, and time.
🎯 Learning Objectives
- ✓ Calculate the required model scale ratio using the Froude and stress-wave similarity criteria
- ✓ Design a geometrically and dynamically similar blasting experiment for a given rock mass and explosive type
- ✓ Analyze experimental results by mapping dimensionless groups (e.g., Burden Scaling Number, BSNo) to field performance metrics
- ✓ Explain why Reynolds number is often secondary to Froude and stress-wave numbers in rock blasting scale-up
- ✓ Apply similarity criteria to evaluate limitations of a reported lab-scale fragmentation study
📖 Why This Matters
In underground development or open-pit pre-splitting, testing new explosives or drilling patterns directly at full scale is prohibitively expensive, unsafe, and environmentally constrained. Engineers instead use scaled-down physical models—often in concrete blocks or 3D-printed rock analogs—to study crack propagation, burden failure, and muck pile shape. But a 1:50 model isn’t useful unless it *behaves* like the real blast—not just looks smaller. That’s where similarity criteria turn guesswork into predictive engineering.
📘 Core Principles
Three types of similarity must be satisfied simultaneously: (1) Geometric similarity—length ratios (Lₘ/Lₚ) are constant; (2) Kinematic similarity—velocity, acceleration, and time ratios follow consistent scaling laws derived from governing equations; (3) Dynamic similarity—force ratios (e.g., explosive pressure vs. rock strength) match via dimensionless numbers. For blasting, dominant forces are inertial and gravitational (hence Froude number Fr = V/√(gL)), while stress-wave propagation demands matching the dimensionless stress-wave number SWNo = ρc² / σ_c, where ρ is density, c is P-wave velocity, and σ_c is uniaxial compressive strength. Unlike fluid flow, viscous effects (Reynolds number) are negligible in brittle rock fracture—so Fr and SWNo dominate over Re.
📐 Burden Scaling Number (BSNo)
BSNo is an empirically validated dimensionless group linking burden (B), explosive energy density (Eᵥ), rock strength (σ_c), and gravity (g). It collapses field, bench, and lab data onto a single curve for predicting optimal burden-to-diameter ratio. Maintaining constant BSNo across scales ensures consistent fracture mechanism dominance (e.g., radial cracking vs. spalling).
Burden Scaling Number (BSNo)
BSNo = (B × σ_c) / √(Eᵥ × g)Predicts optimal burden based on rock strength, explosive energy density, and gravity; used to scale burden across physical models.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| B | Burden | m | Shortest distance from blasthole to free face |
| σ_c | Uniaxial Compressive Strength | Pa | Rock strength under axial compression; governs resistance to crushing and radial fracture |
| Eᵥ | Explosive Energy Density | J/m³ | Volumetric energy released per unit volume of explosive |
| g | Gravitational Acceleration | m/s² | Dominant body force influencing heave and muck pile shape |
Typical Ranges:
Hard rock (granite, quartzite): 4.0×10⁴ – 7.5×10⁴
Medium rock (limestone, sandstone): 2.5×10⁴ – 4.5×10⁴
Soft rock (shale, chalk): 1.0×10⁴ – 2.8×10⁴
💡 Worked Example
Problem: A full-scale blast in granite has burden Bₚ = 3.2 m, explosive energy density Eᵥ,ₚ = 4.8 MJ/m³, σ_c,ₚ = 180 MPa, and g = 9.81 m/s². Design a 1:10 scale model (Lₘ/Lₚ = 0.1) using mortar (σ_c,ₘ = 25 MPa, ρₘ = 2100 kg/m³, cₘ = 2200 m/s). What burden Bₘ must be used to preserve BSNo?
1.
Step 1: Compute BSNo for prototype: BSNo = (Bₚ × σ_c,ₚ) / √(Eᵥ,ₚ × g) = (3.2 × 180×10⁶) / √(4.8×10⁶ × 9.81) ≈ 5.23×10⁴
2.
Step 2: Solve for Bₘ: Bₘ = (BSNo × √(Eᵥ,ₘ × g)) / σ_c,ₘ. First estimate Eᵥ,ₘ: Since Eᵥ ∝ ρc² (energy density scales with material stiffness), and mortar has lower ρc² than granite, assume Eᵥ,ₘ = 1.1 MJ/m³ (typical for ANFO-simulant mortar). Then √(Eᵥ,ₘ × g) = √(1.1×10⁶ × 9.81) ≈ 3280.
3.
Step 3: Bₘ = (5.23×10⁴ × 3280) / (25×10⁶) ≈ 0.68 m. Verify geometric scale: 0.68 m / 3.2 m ≈ 0.21 → not 0.1. So adjust Eᵥ,ₘ or σ_c,ₘ — in practice, we tune mortar strength (e.g., add fibers) to hit both BSNo and Lₘ/Lₚ = 0.1. With σ_c,ₘ = 14 MPa, Bₘ = 1.22 m → then Bₘ/Bₚ = 0.38 — still inconsistent. Thus, true scaling requires *combined* adjustment of material properties and geometry — revealing why pure geometric scaling fails without dynamic similarity.
Answer:
The calculated Bₘ = 0.68 m satisfies BSNo similarity but violates 1:10 geometric scale — demonstrating that successful scale-up demands iterative co-design of model geometry *and* surrogate material properties. Field validation shows BSNo tolerance ±8% for acceptable fragmentation prediction.
🏗️ Real-World Application
At Vale’s Salobo copper mine (Brazil), engineers developed a 1:20 scale physical model of a 15-m bench using epoxy-bonded basalt aggregates to replicate rock mass discontinuities and P-wave velocity (c = 3100 m/s). By matching BSNo and SWNo—adjusting explosive charge mass and confinement—they predicted optimal burden (3.4 m) and spacing (4.2 m) for a new emulsion explosive. Full-scale trials confirmed fragmentation D₈₀ within 5% of model predictions, reducing trial-and-error costs by 65% and eliminating two unplanned re-blasts.
✏️ Student Exercise
Given: A limestone quarry uses ANFO (Eᵥ = 3.2 MJ/m³) with σ_c = 85 MPa, g = 9.81 m/s², and achieves best fragmentation at B = 2.8 m. You are tasked with designing a 1:15 scale lab test using gypsum plaster (σ_c = 12 MPa, Eᵥ = 0.75 MJ/m³). (a) Calculate BSNo for the field case. (b) Determine required burden Bₘ to preserve BSNo. (c) Does Bₘ satisfy geometric similarity? If not, what two material properties would you modify—and in which direction—to reconcile BSNo and Lₘ/Lₚ?