What the calculation gives you
A centrifugal pump's catalogue curve is measured at one speed with one impeller. The affinity laws tell you what the pump does at another speed, or with a trimmed or larger impeller, without a new test. You use them to size a variable-speed drive, to decide how much to trim an impeller to hit a duty point, to estimate the saving from slowing a pump instead of throttling it, and, as the worksheet below does, to predict a full-size pump from a scale model.
The laws
For the same pump at two speeds N1 and N2, or for geometrically similar pumps of impeller diameters D1 and D2 at the same speed:
Q2 / Q1 = (N2 / N1) (D2 / D1)3
flow goes with the speed and with the cube of the size
H2 / H1 = (N2 / N1)2 (D2 / D1)2
head goes with the square of the tip speed, N D
P2 / P1 = (N2 / N1)3 (D2 / D1)5 (ρ2 / ρ1)
P shaft power · ρ fluid density
They follow from dimensional analysis: at the same flow coefficient Q / (N D3) and head coefficient g H / (N2 D2), two similar pumps work at the same point of the same dimensionless curve, with the same efficiency. The power law is the product of the other two with the density. The specific speed Ns = N √Q / H3/4, computed at the best efficiency point, is the number that identifies the family: pumps with the same Ns are the same design at a different scale.
Speed change versus impeller trim
A speed change keeps the geometry, so the laws hold closely over a wide range; a pump at 80% speed gives 80% of the flow, 64% of the head and takes 51% of the power, which is why a variable-speed drive saves so much against a throttling valve. An impeller trim changes the geometry (the outlet width and blade angle no longer match the volute), so the laws hold only for small trims, about 10 to 15% of the diameter at most; beyond that the measured head falls faster than the square and the efficiency drops. Manufacturers give trim curves for that reason, and the practical rule is to trim a little less than the laws suggest and measure.
Where the laws stop holding
The laws assume dynamic as well as geometric similarity. They break down when the Reynolds number of the two cases differs so much that friction and leakage no longer scale, which is the concern in a model test: a half-scale model at the same speed has a quarter of the Reynolds number ρ ω D2 / μ. As long as both are well into the turbulent range the error is a few percent, and the usual correction is to expect the full-size pump to be slightly more efficient than the model (Moody's formula is the common estimate). Cavitation does not scale with the laws either: the required NPSH goes roughly with the square of the speed, so a pump sped up may cavitate at a flow that was fine before.
Worked example
The worksheet below starts from a half-scale prototype of a centrifugal pump with a 2.125 in impeller, with the water density and viscosity, the speed and the measured flow and pressure of the model. It writes the Reynolds number as a function of the diameter, compares the two Reynolds numbers to judge whether the scaling is trustworthy, and then solves the affinity relations for the full-scale pump of twice the diameter at the same speed: eight times the flow and four times the pressure rise. Change the full-scale diameter or the speed and the solved values follow.
Common mistakes
- Moving a duty point along the laws. The laws move a point on the pump curve to a new pump curve; they do not say where the pump will run. The new operating point is where the new curve meets the system curve, and a system with much static head does not follow the Q2 shape, so the flow will not drop in proportion to the speed.
- Power scaled with the square. It is the cube of the speed and the fifth power of the diameter. The motor must be checked at the highest speed intended, not the nominal one.
- Trimming by the laws alone. Use the manufacturer's trim correction, and never trim below the minimum diameter on the curve sheet.
- Density forgotten in the power. Head in metres of fluid is independent of density; the power and the pressure are not.
- NPSH left at the old value. Check the required NPSH at the new speed before raising it.
Questions
- Do the laws apply to fans?
- Yes, identically: fan laws are the same relations with pressure in place of head. They apply to any rotodynamic machine, pumps, fans and turbines alike, and not to positive-displacement pumps, whose flow is proportional to speed at any head.
- How accurate is a scale-model prediction?
- Within a few percent on flow and head when both Reynolds numbers are above about 106 and the model is accurate to scale, including surface finish and clearances. Efficiency is usually under-predicted for the larger machine.
- Can I use them to pick a pump for a different fluid?
- For density, yes, through the power law. For viscosity well above water's, no: a viscous liquid lowers the head and flow and raises the power at the same speed, and the Hydraulic Institute viscosity corrections are needed instead.
