When a VFD slows a centrifugal pump, flow drops with speed, head with speed squared and power with speed cubed. That is why VFDs save energy. It is also why a pump slowed too far cannot lift against static head and just churns.
Pump Affinity Laws with VFD Speed
Formula
Speed ratio r = N2 / N1 (Hz or rpm) Q2 = Q1 x r H2 = H1 x r² P2 = P1 x r³ Speed for a target flow = N1 x Q target / Q1
Inputs used
- Rated (full) speed (Hz)
- New speed (Hz)
- Flow at rated speed (gpm)
- Head at rated speed (ft)
- Power at rated speed (hp)
- Shutoff head at rated speed (optional) (ft)
- Static head of the system (optional) (ft)
- Target flow (optional, finds the speed) (gpm)
Worked example (fictional plant data)
- Effluent pump rated 1,500 gpm at 80 ft and 45 hp at 60 Hz; VFD set to 48 Hz
- r = 48 / 60 = 0.80
- Q2 = 1,500 x 0.80 = 1,200 gpm
- H2 = 80 x 0.80² = 51.2 ft
- P2 = 45 x 0.80³ = 23.04 hp
- Shutoff head = 110 x 0.64 = 70.4 ft, above the 40 ft static head
- Speed for 1,000 gpm = 60 x 1,000 / 1,500 = 40.0 Hz (estimate)
Cautions
- Affinity laws move points along the pump curve. Real flow is where the scaled pump curve meets the system curve; static head makes flow drop faster than speed.
- Set a VFD minimum speed so the pump always makes flow. Running dead headed heats the water and damages seals and bearings.
- Power savings are smaller in practice than the cube law suggests, because drive and motor efficiency fall at low speed.
- Do not run above the motor or pump rated speed without the manufacturer’s OK.
References
- Hydraulic Institute, ANSI/HI 1.3 Rotodynamic Centrifugal Pumps for Design and Application (affinity rules).
- U.S. Department of Energy, Adjustable Speed Pumping Applications (Pumping Systems Tip Sheet).
Results are estimates for operator and technician use. Equations follow the references above; always check against the instrument manual, your calibration procedure and plant data.