OpenSpecEngineering

Finds the shaft power of a centrifugal pump at its duty point, then uses the affinity laws to predict its flow, head and power after a change of speed or an impeller trim. Engineers and technicians use it to size a variable-speed drive, to estimate the energy saved by slowing a pump, or to decide how much to trim an impeller. For fans, use the Fan Power & Affinity Laws calculator.

Equations

P_h = ρ·g·Q·H     P = P_h/η

P_h is the hydraulic power given to the liquid, ρ its density, g the acceleration of gravity, Q the flow and H the pump’s total head. P is the shaft (brake) power and η the pump efficiency [1], [2]. In US units this is BHP = GPM × ft × SG/(3960 × η), with SG the specific gravity. You can enter the shaft power instead, from a pump curve or a motor reading. Picking a water temperature fills in ρ [4].

For a change of speed N or impeller diameter D, the affinity laws [2], [3] give

Q₂/Q₁ = (N₂/N₁)·(D₂/D₁)
H₂/H₁ = [(N₂/N₁)·(D₂/D₁)]²
P₂/P₁ = [(N₂/N₁)·(D₂/D₁)]³

Subscript 1 is the known point and 2 the new one. The efficiency is taken as unchanged. The Live Pump Curve panel draws an assumed curve through your one point (shutoff head 1.25×H₁) and the same curve scaled by the affinity laws. Only the scaling is exact; the shape is illustrative.

Limits and common mistakes

• The laws move the duty point along a parabola through zero flow and zero head. A system with static head (a lift, or a pressure to hold) does not follow it, so a speed cut lowers the flow more than the laws predict. Find the new point where the scaled pump curve crosses the system curve.
• The diameter law is only approximate. Trim no more than about 10 to 20 % of the impeller, and expect some loss of efficiency [3].
• Check the new point’s NPSH required and the motor size before speeding a pump up.

Worked example

A chilled-water pump delivers 300 GPM at 80 ft of head, at 1,750 rpm with a 75 % efficient 10 in. impeller. A drive will run it at 1,450 rpm. What flow, head and power result?

1. Enter Original Flow 300 GPM and Original Head 80 ft.
2. Choose Original Power Calculate from flow, head, fluid and efficiency, then Fluid Water, 10°C, and enter Pump Efficiency 75.
3. Enter Original Speed 1750 rpm and New Speed 1450 rpm.
4. Enter Original Impeller Diameter and New / Trimmed Impeller Diameter both 10 in, and press Solve. The Solution shows: Original Shaft Power (P₁ = P_h/η): 6.03278 kW
   Speed ratio N2/N1: 0.828571
   New power P2: 3.43168 kW
   New flow Q2 (US): 248.571 GPM
   New head H2 (US): 54.9224 ft
   Original power P1 (US): 8.09009 hp
   New power P2 (US): 4.60196 hp

Check with the US formula: 300 × 80 × 1.0007/(3960 × 0.75) = 8.09 hp. The speed ratio cubed is 0.8286³ = 0.569, so a 17 % cut in speed saves 43 % of the shaft power, if the system has no static head.