# Affinity laws for centrifugal pumps

There are occasions when you might want to vary the amount of fluid you are pumping or change the discharge head of a centrifugal pump. There are four ways you could do this:

- Change the speed of the pump.
- Change the diameter of the impeller. Replace it with a larger impeller, or cut down the one you have.
- Purchase a different pump with the operating curve you need.
- Valve or orifice the discharge of the pump to get the capacity or head you need. Unfortunately this can cause the pump to operate off of its best efficiency point (BEP).

Of the four methods the first two are the only sensible ones unless you are prepared to buy a new pump. In the following paragraphs we will learn what happens when we change either the pump speed or impeller diameter, and as you would guess, other characteristics of the pump are going to change along with the speed or diameter.

To determine what is going to happen, you begin by taking the new speed or impeller diameter and divide it by the old speed or impeller diameter. Since changing either one will have approximately the same affect,

I will be referring only to changing the speed in this part of the discussion.

As an example:

The capacity or amount of fluid you are pumping will vary directly with this number.

100 Gallons per minute x 2.0 = 200 Gallons per minute50 Cubic meters per hour x 0,5 = 25 Cubic meters per hour

The head produced by the pump varies by the square of the number.

A 50 foot head x 4 (2

^{2}) = 200 foot headA 20 meter head x 0,25 (0,5^{2}) = 5 meter head

The horsepower required changes by the cube of the number

A 9 Horsepower motor was required to drive the pump at 1750 rpm. How many horsepower is required now that you are going to 3500 rpm?

9 x 8 (2

^{3}) = 72 Horsepower is now required.Likewise if a 12-kilowatt motor was required at 3000 rpm. and you decreased the speed to 1500 the new kilowatts required would be:

12 x 0,125 (0,5

^{3}) = 1,5 kilowatts required for the lower rpm.

The following relationships are not exact, but they give you an idea of how speed and impeller diameter affects other pump functions.

The net positive suction head required by the pump manufacturer (npshr) varies by the 1.5 power of the number at the BEP.

- Example : A 3 meter NPSHR x 2.8 (2
^{1.5}) = 8.5 meter N.P.S.H.R. - Or : 10 foot NPSHR x 0.35 ( 0.5
^{1.5}) = 3.5 foot N.P.S.H.R.

The amount of shaft run out (deflection) varies by the square of the number

As an example: If you put a dial indicator on the shaft and noticed that the total run out at 1750 rpm. was 0.005 inches, then at 3500 rpm the run out would be 0.005″ x 4 or 0.020 inches.Likewise if you had 0,07 mm. run out at 2900 rpm. and you slowed that shaft down to 1450 rpm the run out would decrease to 0,07 mm x 0,25 or 0,018 mm.

The amount of friction loss in the piping varies by 90% of the square of the number. Fittings and accessories vary by the square of the number.

As an example: If the system head loss was calculated or measured at 65 meters, at 1450 rpm. the loss at 2900 rpm. would be:65 meters x 4 = 260 x 0.9 = 234 Meters

If you had a 195 foot loss at 3500 rpm. the loss at 1750 rpm. would be: 195 x 0.25 = 48.75 / 90% = 54.17 feet of head loss.

The wear rate of the components varies by the cube of the number.

At 1750 rpm. the impeller material is wearing at the rate of 0.020 inches per month. At 3500 rpm the rate would increase to:0.020 ” x 8 or 0.160 inches per month. Likewise a decrease in speed would decrease the wear rate eight times as much.

I started this discussion by stating that a change in impeller speed or a change in impeller diameter has approximately the same affect. This is true only if you decrease the impeller diameter to a maximum of 10%. As you cut down the impeller diameter, the housing is not coming down in size so the affinity laws do not remain accurate below this 10% maximum number.

The affinity laws remain accurate for speed changes and this is important to remember when we convert from stuffing box packing to a balanced mechanical seal. After the conversion to a mechanical seal we sometimes experience an increase in motor speed rather than a drop in amperage. The affinity laws will help you to predict the final outcome of the change.

The affinity laws also explain the effect on capacity and head when you change motor speed with a variable frequency motor (variable speed driver).

You can use the following formulas to supplement the Affinity Laws. Please keep in mind that these numbers are based on the fluid flowing through the correct size clean pipe.

Product build-up and pipe roughness are variables that will affect the final figures so consider the following “ball park” rather than exact numbers.

Please use these keys when you read the following ratios:

- hf
_{1}The friction loss in the piping, valves and fittings before the change in flow. - hf
_{2}The friction loss in the piping, valves and fittings after the change in flow. - Q
_{1}The pump capacity before the change in flow. - Q
_{2}The pump capacity after the change in flow. - H
_{1 }The pump head before the change. - H
_{2}The pump head after the change. - D
_{1}The impeller diameter before the change. - D
_{2}The impeller diameter after the change.

If you are not familiar with raising a number to some power, please look at the following examples:

3

^{2}means 3 x 3 = 93

^{5}means 3 x 3 x 3 x 3 x 3 = 24333

^{2.5}means to multiply the square of 3 (9) by the square root of 3 (1.732) = 15.6.

The piping friction loss will vary as the square of the capacity ratio

Example: assume you looked at the friction loss charts and learned that 300 gpm. flowing through a pipeline will suffer 20 feet of friction head loss. Then 500 gpm through the same line will lose:

= 56 feet of head loss.

The pump’s capacity varies as the square root of the head on the liquid

Example: if a 160 foot head would deliver 300 gpm. through a specified pipeline, a 100 foot head would deliver:

= 237 gpm

The friction loss in the piping is inversely proportional to the fifth power of the pipe diameter ratio

Example: Assume a 3 inch diameter pipe can handle 300 gpm with a 20 foot friction loss. The same flow rate through a 2 inch diameter pipe would create:

= 152 foot loss

The same flow through a 4 inch line would create:

= 5 foot loss

The capacity of a pipe would vary as the 2.5 power of the diameter ratio

Example: assume that a 3 inch diameter discharge pipe delivers 300 gpm. under a specified head. Under the same head, a 2 inch pipe will deliver:

= 109 gpm.

When using these affinty laws, keep in mind

- The affinity laws predict pump performance changes, not changes in pump/ system interactions. You’ll have to plot your “before and after” pump curves with your system curve to estimate where your pump will operate.
- The problem is “Static head. ” It’s not changing with impeller speed or diameter. The higher the percentage of static head, the bigger the miscalculation.