What the Affinity Laws Let You Predict and Where They Stop

Why this matters

The cube law is the most repeated number in mechanical services: slow the pump down a fifth and the power falls by about half. It is true, and it is true about the pump. It is not automatically true about the job the pump is doing, because the laws map one point on a pump curve to another point on a specific parabola, and only some systems live on that parabola. On a closed loop they do. On anything that lifts liquid to a height or pushes into a held pressure, they do not, and the same 20 percent speed cut buys a fraction of the saving while giving up far more flow than anyone expected. This article runs one stated gate against two systems that answer it oppositely, so you can tell in about a minute which one you are standing in front of.

Before you write a speed into a drive

Changing speed to see what happens is an instruction that reaches into the process, so it gets the same treatment as an instruction that touches the machine. Before a reduced speed is written into a drive on a live system: confirm the reduced flow will not take the pump below the minimum flow its manufacturer publishes, because a centrifugal pump run below that turns its own shaft power into heat inside the casing; confirm no check valve in the loop will slam closed and dead-head the pump; and confirm nothing downstream loses a safety function when flow drops, which means a loop that proves flow for a combustion appliance, a compressor, a heat exchanger with a freeze exposure, or any relief or protective function is not a loop you speed-test on. On those, the test is done with the machine off and the schedule reviewed on paper. On a motor cooled by a fan on its own shaft, sustained low speed also reduces its own cooling, so a speed floor is a motor question as well as a hydraulic one and the motor manufacturer owns that number. Work on the drive itself, including opening it to land a control wire, is electrical: lock the supply disconnect, prove dead against a known live source, and respect the drive's stated bus discharge time, because a drive's capacitors hold a lethal charge after the supply is opened (29 CFR 1910.333(b)(2); NFPA 70E-2021, 120.5; 29 CFR 1910.147 for the stored energy).

The three relations and the conditions they were derived under

For the same pump, the same impeller diameter, the same liquid, and two operating points that are hydraulically similar (meaning the flow has changed in the same proportion as the speed, so the liquid meets the vanes at the same angles), and assuming efficiency is unchanged between them:

• Flow varies with speed.
• Head varies with the square of speed.
• Shaft power varies with the cube of speed.

Every one of those conditions is load-bearing. Drop the similarity condition and the relations still describe the pump, but they stop describing where the pump ends up working.

There is a fourth relation with the same conditions: required net positive suction head varies roughly with the square of speed between corresponding points. It is useful and it is fragile in exactly the same way, for exactly the same reason.

A diameter change is not a speed change. Trimming an impeller is often written with the same square and cube exponents and it does not obey them as well, because a trim alters the vane exit while a speed change does not. The impeller trim article owns that relationship and this one defers to it.

The gate: what fraction of the head at duty is static

Put the pump's total head at duty into two buckets.

• Static head is the part that does not change with flow: the elevation the liquid has to climb from the suction level to the discharge point, plus any pressure held at the far end.
• Friction head is the part that rises with flow, roughly with the square of it.

If static head is zero, the system curve is a parabola through the origin, which is exactly the parabola the affinity laws move along, and the operating point tracks them perfectly. If static head is most of the total, the system curve starts high and rises gently, the two curves cross somewhere the laws never predicted, and the operating point walks off the parabola immediately.

To run the arithmetic you need one more number off the curve sheet: the pump's shutoff head, the head it makes at zero flow. A workable field model of a centrifugal curve is head equals shutoff head minus a constant times flow squared, with the constant set by making the model pass through the measured duty point.

Worked example: one 20 percent flow reduction, two systems

Both systems run a pump that at full speed delivers 100 percent flow at 90 ft of head, with shutoff head 120 ft off the curve sheet and 78 percent efficiency at duty. So the curve model at full speed is head = 120 - 30 x (flow fraction) squared, because 30 x 1.0 squared is the 30 ft between shutoff and duty. Scaled to a speed fraction s, that model becomes head = 120 s squared - 30 x (flow fraction) squared.

Both are asked the same question: hold 80 percent of design flow, on speed rather than on a valve.

System A, a closed chilled water loop. Static head is zero, because the liquid returns to where it started and gains no elevation. Friction is the whole 90 ft.

• 80 percent flow needs 80 percent speed, because the operating point stays hydraulically similar.
• Head becomes 0.8 squared x 90 = 57.6 ft.
• Power becomes 0.8 cubed = 51.2 percent of full-speed power.
• The correction the general section demands: efficiency is assumed unchanged, and in a real machine it slips a little at reduced speed. So 51.2 percent is a floor on the power and the saving is an upper bound: the saving is < 48.8 percent, written with one inequality and not as an interval, because nothing here gives a lower bound.

System B, a transfer pump filling an elevated tank. Measured static head is 60 ft of lift from the suction tank level to the discharge point, with nothing held at the far end. Friction is therefore 90 - 60 = 30 ft at design flow, so the system curve is head = 60 + 30 x (flow fraction) squared.

• Set the two models equal at 80 percent flow: 120 s squared - 30 x 0.64 = 60 + 30 x 0.64. That is 120 s squared = 60 + 19.2 + 19.2 = 98.4, so s squared = 0.82 and s = 0.906. Holding 80 percent flow takes 90.6 percent speed, not 80 percent.
• Head at that point is 60 + 19.2 = 79.2 ft.
• Power scales with flow times head divided by efficiency, so relative to full speed: (0.8 x 79.2) / (1.0 x 90) = 0.704 before any efficiency correction.
• The efficiency correction, printed: the new point sits left of where it was on the curve, and the curve sheet gives about 76 percent there against 78 percent at duty, so the power ratio multiplies by 78/76 = 1.026, landing at 0.722. Power is 72 percent of full speed, a saving of about 28 percent, not 49.
• The static-term correction, printed: 60 ft was measured with the suction tank near full. At minimum tank level the lift is 66 ft. Re-running the same equality: 120 s squared = 66 + 19.2 + 19.2 = 104.4, so s squared = 0.87 and s = 0.93. The speed schedule has to be built on the worst static condition the pump actually sees, not on the day you measured, or the pump will fall short of the duty exactly when the tank is low.

The direct comparison, which is the payload. Same pump, same 20 percent flow reduction, same cube law quoted in both rooms. System A saves under 49 percent of the power. System B saves about 28 percent. The gap is not an error in the laws; it is that the laws describe the pump and the static head describes the job, and the job wins.

What the naive version would have produced on System B. A tech who applies 0.8 speed on System B gets a very different answer, and it is worth working because it is the mistake people actually make. At s = 0.8 the pump model gives head = 76.8 - 30 q squared, and the system needs 60 + 30 q squared, so 16.8 = 60 q squared, q squared = 0.28, and flow collapses to 53 percent. Not 80. The operator sees the tank filling far too slowly, decides the pump is worn, and starts a diagnosis of a machine that is doing exactly what it was told.

The suction check, and why it is not scaled here. Required suction head varies with the square of speed only between corresponding points, and System B's new point at 80 percent flow and 90.6 percent speed is not a corresponding point. So the required figure is read off the curve sheet at the new speed and flow, not calculated by multiplying by 0.906 squared. On System A, where the point stayed similar, the square-of-speed scaling is legitimate and the suction margin genuinely improves.

The floor check. System B's 53 percent flow in the naive case, and any deep turndown on either system, has to be compared against the pump's published minimum flow before it is allowed to run there. That comparison belongs to the minimum flow article and it is not optional.

What would change the reading

A system holding a pressure at the far end behaves exactly like a static lift even with no elevation change at all: a loop that must maintain a pressure at a remote sensor, or a pump feeding into a vessel held at pressure, has a system curve that starts high, and it sits on the System B side of the gate.

A drive holding a differential setpoint rather than a speed inverts the whole question, because the drive is choosing the speed and the thing you are looking at is the speed command. On those, a speed command that has crept up over a season is a fouling signal.

A very deep turndown, past roughly half speed, is where the constant-efficiency assumption stops being a small error and starts being the answer, and where the motor's own cooling and the pump's minimum flow are usually the real limits rather than the hydraulics.

How to verify the prediction held

Measure the same three things at the new speed that you measured at the old one, on the same ports, and compare each against what you predicted: speed off the drive display, differential head off the two gauges, and delivered flow. If flow lands well under the prediction on a system you called friction-dominated, you have static head you did not account for, and the place to look is the elevation between the suction liquid level and the discharge point, or a pressure being held somewhere downstream that nobody mentioned. If power lands well over the prediction, check where the new point sits relative to best efficiency before you suspect the motor.

References

• Manufacturer's published pump curve sheet at multiple speeds, which owns shutoff head, efficiency and required suction head at any given point
• ANSI/HI 9.6.1, Hydraulic Institute guidance on net positive suction head margin, in the edition your engineering specification adopts
• 29 CFR 1910.333(b)(2) with NFPA 70E-2021, 120.5 for proving a drive dead, and 29 CFR 1910.147 for the drive's stored bus energy
• See related: How an Operating Point Moves; What a Pump Curve and a System Curve Do Together; What Impeller Trim Changes and What It Does Not; Why Minimum Flow Exists and What Happens Below It