Water & Wastewater
Pump Affinity Laws
How flow, head, and power change with pump speed or impeller diameter, and the energy saving a variable frequency drive can deliver.
Calculated reference results
- Speed ratio
- 0.8000
- New flow
- 400.0gpm
- New head
- 38.4ft
- New power
- 6.14hp
- Power change
- -48.8%
- Static head as a share of total
- 0%
1,400 / 1,750 rpm
80.0% of original
64.0% of original
51.2% of original
Mostly friction, so the affinity laws apply reasonably.
The arithmetic
- Speed ratio = 1,400 / 1,750 = 0.8000
- Flow = 500.0 x 0.8000 = 400.0 gpm
- Head = 60.0 x 0.8000 squared = 38.4 ft
- Power = 12.00 x 0.8000 cubed = 6.14 hp
What this does not account for
- The affinity laws assume a system curve that is entirely friction and passes through the origin. Where static lift dominates, as at most lift stations, reducing speed does not reduce power anywhere near the cube law and can stop the pump delivering entirely.
- Below a minimum speed the pump cannot overcome static head and delivers nothing while still drawing power. Establish that minimum speed and enforce it in the control logic.
- Efficiency is not constant with speed. The laws predict the operating point, not the efficiency at it.
How this is calculated
The affinity laws state that flow varies directly with speed, head varies with the square of speed, and power varies with the cube of speed. Reducing a pump to 80% speed gives 80% flow, 64% head, and 51% power, which is why variable speed pumping saves energy where the system allows it.
Formulas
Q2 / Q1 = N2 / N1
H2 / H1 = (N2 / N1) squared
P2 / P1 = (N2 / N1) cubed
Assumptions and limitations built into this calculator
- Friction-dominated system curve passing through the origin.
- Constant efficiency between the two operating points, which is an approximation.
- Applies to speed change. Impeller trimming follows similar but not identical relationships.
Frequently asked questions
- Why is my VFD not saving the energy the cube law predicts?
- Because your system has static head. The cube law assumes all head is friction. At a lift station lifting sewage thirty feet, most of the head does not go away when you slow down, so the savings are far smaller.
Direct contact
Have a controls question?
Reach Eric Sullivan directly about anything on this site, a controls or automation topic, or one of his personal projects.