🎓 Lesson 5
D3
Geometric, Kinematic, and Dynamic Similarity in Scale-Up
Geometric, kinematic, and dynamic similarity are rules that let engineers test small-scale models (like lab blasts or fluid flow rigs) and confidently predict how full-size mining systems—such as blast patterns or slurry transport lines—will behave in the real world.
🎯 Learning Objectives
- ✓ Calculate geometric scale ratios between laboratory blast models and field-scale operations
- ✓ Analyze kinematic similarity by verifying consistent Froude and Reynolds number scaling across model–prototype pairs
- ✓ Design dynamically similar blasting experiments by selecting appropriate charge mass, confinement, and rock simulant properties
- ✓ Explain why matching only geometric and kinematic similarity is insufficient for predicting fragmentation or ground vibration without dynamic force equivalence
- ✓ Apply Buckingham Pi theorem to derive similarity criteria for multiphase flow in ore pass transport systems
📖 Why This Matters
In mining, you cannot safely or economically test full-scale blast designs or slurry transport configurations on-site before implementation. Engineers rely on scaled-down physical models—from 1:50 explosive fracture tests in rock simulants to 1:10 hydraulic flumes for chute flow—to validate performance. But if the model doesn’t behave *like* the prototype—not just look like it—you’ll mispredict fragmentation, flyrock, energy efficiency, or pipe erosion. Understanding and enforcing geometric, kinematic, and dynamic similarity is what separates guesswork from predictive engineering.
📘 Core Principles
Geometric similarity is foundational: every length in the model (Lₘ) must equal the prototype length (Lₚ) multiplied by a constant scale factor λ = Lₘ/Lₚ. Kinematic similarity builds on this by requiring velocity (V), acceleration (a), and time (t) to scale as Vₘ/Vₚ = λ^(1/2) (for Froude similarity) or Vₘ/Vₚ = λ (for kinematic viscosity dominance). Dynamic similarity is the culmination: it mandates that all force ratios (e.g., inertial/gravity, inertial/viscous, inertial/explosive impulse) match between model and prototype—enforced by equating dimensionless numbers. For blasting, the key Pi groups include the Froude number (Fr = V/√(gL)), Reynolds number (Re = ρVL/μ), and the blast-specific ‘explosive similarity number’ Πₑ = W / (ρ₀ R³), where W is charge energy, ρ₀ is ambient density, and R is characteristic radius—derived from energy balance arguments in explosive hydrodynamics.
📐 Dynamic Similarity Criterion for Blast Scaling
For explosive-driven rock fracture, dynamic similarity requires matching the non-dimensional explosive energy density. This ensures equivalent stress wave propagation, fracture mechanics, and comminution response between model and prototype.
💡 Worked Example
Problem: A lab-scale blast uses 0.2 kg of ANFO (energy density = 3.0 MJ/kg) in a 0.5 m diameter granite simulant block (density = 2.4 g/cm³). The field-scale blast targets a 10 m diameter ore zone. What charge mass ensures dynamic similarity?
1.
Step 1: Compute model energy density ratio: Πₑₘ = Wₘ / (ρₘ Rₘ³) = (0.2 × 3.0×10⁶ J) / (2400 kg/m³ × (0.25 m)³) = 600,000 / (2400 × 0.015625) = 600,000 / 37.5 = 16,000
2.
Step 2: Set Πₑₚ = Πₑₘ → Wₚ = Πₑₘ × ρₚ × Rₚ³ = 16,000 × 2650 kg/m³ × (5.0 m)³
3.
Step 3: Calculate: (5.0)³ = 125; 16,000 × 2650 × 125 = 16,000 × 331,250 = 5,300,000,000 J ≈ 1767 kg ANFO (since 3.0 MJ/kg → 5.3 GJ / 3.0 MJ/kg)
Answer:
The dynamically similar charge mass is ~1767 kg ANFO, confirming that simple linear scaling (×20 in diameter → ×8000 in volume) overestimates required energy unless density and confinement are preserved.
🏗️ Real-World Application
At BHP’s Olympic Dam copper–uranium mine, researchers used 1:40 geometrically scaled concrete–gravel simulants in shock tube experiments to optimize ring-drill blast sequencing for sublevel caving drawpoints. By matching both Froude (gravity-driven rockfall) and explosive energy density (Πₑ) criteria—and validating against full-scale microseismic event clustering—they reduced oversize production by 22% and improved draw column stability. Crucially, they discovered that ignoring dynamic similarity (i.e., using only geometric + kinematic scaling) led to underprediction of tensile fracture radius by 35%, resulting in excessive muck pile segregation in early trials.
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