Pump and fan affinity laws
Flow, head and power at a new speed or impeller size — and the yearly saving from slowing a pump or fan with a VFD.
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The Pump System Curve and Operating Point Excel tool does everything here and more: system curve from one known point or from the pipe, the operating point for 1–4 pumps in parallel or series, and throttle vs VFD vs impeller trim to a target flow, with the yearly energy cost. Print-ready report, any units, visible formulas, no macros.
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How this calculator works
For the same pump or fan at a new speed N: flow ∝ N, head (or pressure) ∝ N², power ∝ N³. Slowing a pump to 80 % speed cuts the power to about half (0.8³ = 0.51). The input power includes the losses of the variable-speed drive. For small impeller trims the same laws are used with the diameter ratio.
The example is a pump giving 500 m³/h at 50 m with 90 kW shaft power at 1,480 rpm. At 1,184 rpm (80 %) it gives 400 m³/h at 32 m and 46 kW; with a 97 % efficient drive the saving is about 42 kW, or USD 25,000 a year over 6,000 hours.
References: Perry's §10; Hydraulic Institute; Karassik, Pump Handbook; US DOE pump and fan system sourcebooks.