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# Cheat Sheets: The Affinity Laws

September 26, 2010
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 Larry Bachus (a.k.a. "Pump Guy")

In an earlier article in this series on the “Cheat Sheets,” I stated, “At 1,800 RPM, the impeller diameter in inches, multiplied by itself (or squared), is approximately the shutoff head of the pump in feet.” (“Unwritten Pump Rules,” Mar. ’08, page 40). That article, and those words, sparked some comments. Among them:
• My pumps spin at 1,750 RPM, not 1,800 RPM.
• Why is 1,800 RPM important?
• What about 3,600 RPM motors?
• How does the pump behave with pulleys or a gearbox?
• How does variable speed affect the pump?

The Affinity Laws answer these and other similar questions. The Affinity Laws govern the performance of your pumps. The laws will help you extrapolate and predict pump performance at different velocities. They are also useful to modify your pumps for new and different required duties.

Because most pumps are directly coupled to a standard industrial electric motor, the pump speed is most often the motor speed. And because centrifugal pumps work with centrifugal force, the velocity and diameter of the impeller determine the head, or pressure, the pump can develop.

The formula for the speed of the electric motor is:

Speed =
2 x 60-Hz x 60-sec
Number of Poles

where:
• “2” is for alternating current (AC electricity)
• “60” is for 60-Hertz electricity. This means the electricity moves (oscillates) at 60-waves per second. (Some countries have electricity at 50-Hz.)
• “60” seconds per minute. This gives us revolutions per minute.
• The “Poles” are the positive and negative electromagnetic fields in the motor.

Standard electric motor speeds at 60-Hz electricity.
• 2-pole motor = 3,600 RPM
• 4-pole motor = 1,800 RPM
• 6-pole motor = 1,200 RPM
• 8-pole motor = 900 RPM

Standard electric motor speeds at 50-Hz electricity.
• 2-pole motor = 3,000 RPM
• 4-pole motor = 1,500 RPM
• 6-pole motor = 1,000 RPM
• 8-pole motor = 750 RPM

The speed can be adjusted by multiples just by changing to a motor with a different number of poles.

You may have noticed that a standard four-pole electric motor doesn’t exactly spin at 1,800 RPM. The motor speed is rated at 1,785, 1,750, or 1,730 RPM. This is called the “slip” factor. It is part of the motor’s design and determines if the motor has good startup torque or good running torque.

A low slip four-pole motor (1,785 RPM) is better for pumping a liquid that flows like water (low viscosity). A motor with a low slip has good running torque. A high slip four-pole motor (1,725 RPM) is better for pumping a liquid that flows like honey (high viscosity). A high-slip electric motor has good start-up torque.

The variable-speed electric motor or VFD (variable frequency drive) is gaining in popularity. The first models appeared about 30 years ago. By varying the motor’s frequency, the pump speed is also variable.

Before the VFD, industry traditionally employed gearboxes or belts and pulleys to adjust the pump speed. DC motors, steam turbines, internal combustion engines, and hydraulic motors were employed to vary the pump velocity.

The change in speed brings about a corresponding change in the head, flow, and power requirements of the pump according to the Affinity Laws. Let’s look at them.

Stated simply, the Affinity Laws say:
1. The flow (GPM) varies proportionally with the change in speed. This means that twice the speed is twice the flow. One-third speed is one-third the flow.
2. The pump head (pressure) varies with the square of the change in the speed. Two times the speed is four times (22) the head generated. Eighty percent speed is 64 percent (.802) the head generated.
3. The power requirement (horsepower or kilowatts) varies by the cube of the change in speed. Two times the speed would burn eight times (23) the power. One half the speed would require one-eighth (.503) the power to drive the pump.

Algebraically, we would write the equations as:
• New Flow = Old Flow x (new speed/old speed)
• New Power = Old Power x (new speed/old speed)3

Engineers study these laws in school, but sometimes they fail to apply their education when they take a position in industry. Just last week, a staff engineer at a chemical plant wrote to me to inform me that his company had purchased the wrong pump for a boiler. The boiler needed a feedwater pump rated at 950 ft. @ 30 GPM. Through some error, they took delivery on a pump rated at 950 ft. @ 60 GPM.

The pump was delivering twice the required flow to the boiler. The engineer sought my opinion regarding two options:
1. Install a bypass valve and divert the excess flow back into the DA tank.
2. Mate the new pump to a VFD and correct the flow by reducing the speed.

I responded that neither option seemed correct. Energy is getting expensive; why waste energy to recirculate half the excess flow back into the DA tank? We should look to reduce energy consumption, not waste it.

And, yes, the VFD at half speed would reduce the flow to 30 GPM. But it would also reduce the head by the square of the velocity reduction. Half speed would be 25 percent head (.502 = .25). The pump would only generate about 237 ft. at half speed and will deadhead against the boiler.

I told the engineer he should exchange the incorrect pump for the correct pump. I mean, if you bought the wrong socks, or battery, or picture frame, you’d exchange it for the correct one … Right?

If he can’t return the whole pump, another possible option is to exchange or modify the impeller. Regarding impellers, the diameter and the velocity of the impeller determines the head a pump can generate. In this case, the engineer in question would need to keep the same impeller diameter and speed to generate the 950 ft. of head required by the boiler.

The height of the impeller blades and the velocity will determine the flow a pump can deliver. The pump company might make a “skinnier” impeller (less height to the blades) with adapter wear bands that mate to the volute. This would deliver less flow and conserve some power. I told him to send me his pump curve and system curve with the next e-mail to explore this option.

Another set of the Affinity Laws deal with changes to the diameter of the impeller. If the speed should remain a constant, the Affinity Laws state:
1. The flow (GPM) varies proportionally with the change in impeller diameter. This means that 10 percent impeller reduction is 10 percent flow reduction.
2. The pump head (pressure) varies with the square of the change in the impeller diameter. If you reduce the impeller diameter by 20 percent (80 percent original diameter), the pump head is reduced to about 65 percent (.802 = .64) the original head generated.
3. The power requirement (BHp or Kw) varies by the cube of the diameter change. A 20 percent reduction in the impeller diameter (80 percent original diameter) would reduce the power consumption to about half (.803 = .512). This is significant.

Algebraically, we would write the equations as:
New Flow = Old Flow x (new imp. diam. / old imp. diam.)
New Head = Old Head x (new imp. diam. / old imp. diam.)2
New Power = Old Power x (new imp. diam. / old imp. diam.)3

Why are these laws important to your work with pumps? Production managers and equipment operators like variable-speed motors for the first Affinity Law. If they need 20 percent more production, they increase the motor speed by 20 percent, and they get the production increase.

However, the head generated varies by the square of the velocity or impeller diameter change. For example, increasing the pump speed by 20 percent would increase the head (pressure) by close to 50 percent. This could burst the screens in a downstream filter or overload a mechanical seal. Have you noticed an increase in mysterious seal failure since installing a VFD?

A pump’s performance is composed of both a flow element and a head (pressure) element. This is the reason VFDs seem to work well on fans and blowers, but not necessarily on all pumps. You see, a fan or blower is only a flow application; if you needed air pressure, you’d use a compressor, not a fan.

Sometimes people ask me, “How does my pump performance curve change at different motor speeds?” The profile of the curve and its elements (head, flow, power, NPSHr, efficiency) don’t change. What changes are the units of flow and head on the horizontal and vertical arms of the graph. They change by the Affinity Laws. (There may be a slight improvement in efficiency with more velocity. Also, the NPSHr may not vary precisely with the Affinity Laws because other factors participate in deriving the NPSHr.)

When I’m lecturing on pumps, I always spend plenty of time on the Affinity Laws. Either you dominate them, or they’ll dominate you. Learn to use them or they will mess with your pumps and reliability. Put them in your CHEAT SHEETS.

Larry Bachus, founder of pump services firm Bachus Company Inc., is a regular contributor to Flow Control magazine. He is a pump consultant, lecturer, and inventor based in Nashville, Tenn. Mr. Bachus is a member of ASME and lectures in both English and Spanish. He can be reached at larry@bachusinc.com or 615 361-7295.

www.bachusinc.com