🎓 Lesson 11
D5
Scaling Laws for Pump and Compressor Performance
Scaling laws let engineers predict how a pump or compressor will perform when its size, speed, or fluid changes—without building every version.
🎯 Learning Objectives
- ✓ Calculate dimensionless coefficients (flow, head, power) for a given pump or compressor operating condition
- ✓ Apply affinity laws to predict new flow, pressure, and power when impeller diameter or rotational speed changes
- ✓ Analyze whether geometric and dynamic similarity is satisfied between a lab-scale model and full-size mine dewatering pump
- ✓ Design a scaled-down test rig for a high-pressure compressor used in compressed air drilling, justifying similarity criteria
📖 Why This Matters
In mining, selecting or retrofitting pumps for dewatering deep shafts—or compressors for pneumatic rock drills—requires reliable performance predictions before installation. Building full-scale prototypes for every scenario is prohibitively expensive and time-consuming. Scaling laws bridge this gap: they allow engineers to test small, low-cost models in labs and confidently project performance at mine-scale conditions—ensuring safety, energy efficiency, and regulatory compliance. A miscalculated scaling error once caused catastrophic cavitation in a 500 L/s dewatering pump at a South African gold mine—delaying production by 11 weeks.
📘 Core Principles
Scaling laws rest on three types of similarity: geometric (identical shape, all dimensions scaled by ratio λ), kinematic (streamlines identical; velocity ratios constant), and dynamic (forces proportional—requiring matching dimensionless numbers). For turbomachinery, the most critical are the flow coefficient (φ = Q/(N D³)), head coefficient (ψ = gH/(N² D²)), and power coefficient (π = P/(ρ N³ D⁵)). When Reynolds number (Re = ρND/μ) is high (>10⁶), viscous effects become secondary, and the Euler number (Eu = Δp/(ρN²D²)) governs similarity. However, for compressors at high Mach numbers (>0.3), compressibility requires Mach number (Ma = U/a) matching—often necessitating air instead of water in model tests.
📐 Affinity Laws & Dimensionless Coefficients
The affinity laws are practical scaling rules derived from dimensional analysis assuming geometric similarity and constant Reynolds/Mach numbers. They relate changes in speed (N) or diameter (D) to flow (Q), head (H), and power (P). Dimensionless coefficients normalize performance across machines and enable comparison in pump/compressor catalogs.
💡 Worked Example
Problem: A mine ventilation compressor delivers 8.2 m³/s at 120 kPa pressure rise and 420 kW input power at 1450 rpm. If speed is increased to 1680 rpm (same impeller), what are the new flow, pressure rise, and power?
1.
Step 1: Compute speed ratio r = N₂/N₁ = 1680/1450 = 1.1586
2.
Step 2: Apply affinity laws — Q ∝ N → Q₂ = Q₁ × r = 8.2 × 1.1586 = 9.50 m³/s
3.
Step 3: H ∝ N² → H₂ = 120 × (1.1586)² = 120 × 1.342 = 161.0 kPa
4.
Step 4: P ∝ N³ → P₂ = 420 × (1.1586)³ = 420 × 1.555 = 653.1 kW
Answer:
New flow = 9.50 m³/s, pressure rise = 161.0 kPa, power = 653 kW — all within safe motor and casing limits per ASME PTC-10.
🏗️ Real-World Application
At the Bingham Canyon copper mine (Utah), engineers needed to upgrade the primary dewatering system for a new 300-m-deep sump. Instead of commissioning a full-scale 1200 mm centrifugal pump, they tested a 1:5 geometrically similar model (240 mm) at 2900 rpm using water. By matching Reynolds number via viscosity adjustment (using glycerol-water mixture) and verifying constant φ and ψ coefficients, they validated the full-scale prediction: 3200 m³/h at 85 m head and 920 kW. Field measurements deviated <2.3%—within ASME PTC-11 tolerance—and enabled 7-week schedule acceleration.