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Pump Affinity Laws and VFD Speed

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)

  1. Effluent pump rated 1,500 gpm at 80 ft and 45 hp at 60 Hz; VFD set to 48 Hz
  2. r = 48 / 60 = 0.80
  3. Q2 = 1,500 x 0.80 = 1,200 gpm
  4. H2 = 80 x 0.80² = 51.2 ft
  5. P2 = 45 x 0.80³ = 23.04 hp
  6. Shutoff head = 110 x 0.64 = 70.4 ft, above the 40 ft static head
  7. 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.

This calculator also appears on ActivatedSludge.com, our consulting site. Results are for training and operator use; always follow your plant’s procedures.