You sized a centrifugal pump for 1750 RPM and need 60% flow. Do you throttle a valve (wasting energy as heat) or slow the pump down? The affinity laws tell you exactly what happens to flow, pressure, and power when you change pump speed or impeller diameter — and they're the reason VFDs on pumps deliver such dramatic energy savings.

For a centrifugal pump at constant impeller diameter, three relationships govern everything:

• Flow (Q) scales linearly with speed: Q₂/Q₁ = N₂/N₁
• Head (H) scales with the square of speed: H₂/H₁ = (N₂/N₁)²
• Power (P) scales with the cube of speed: P₂/P₁ = (N₂/N₁)³

That cubic power relationship is the headline. Drop pump speed to 80% and you get 80% flow, 64% head, and just 51% power. That's why VFD retrofits on oversized pumps routinely pay back in under two years.

Real-world example — building chilled water pump: A 50 HP pump runs flat-out at 1750 RPM delivering 800 GPM at 100 ft head. Building load drops to 60% in shoulder season. Throttling a valve to 480 GPM keeps the motor near 50 HP (you're just burning the excess as turbulence). Slowing the pump to 1050 RPM (60% speed) gives you 480 GPM at 36 ft head, drawing only 50 × 0.6³ ≈ 10.8 HP. Annual savings: tens of thousands of kWh.

The catch — affinity laws shift the operating point along a parabola, not the system curve. Your pump curve moves down and left as you slow it, but the system curve (pipes, fittings, static head) doesn't change. If your system has significant static head (lifting water up a building), slowing the pump too far means head falls below static head and flow drops to zero. Affinity laws work cleanly only for friction-dominated systems where the static component is small.

Rule of thumb: The cube law on power assumes pump efficiency stays roughly constant. Below about 50% of rated speed, efficiency drops noticeably and bearings/seals may suffer from reduced lubrication flow. Don't run VFD-driven pumps below 30% speed for extended periods.

For impeller trimming (machining the impeller smaller instead of changing speed), the same exponents apply — Q ∝ D, H ∝ D², P ∝ D³ — but only over modest trim ranges (within ~10–15% of original diameter) before the geometry changes too much.