Pumps: The Affinity Laws and Beyond

The affinity laws below are well known and used daily by pump people.

(Q1 /Q2) = (N1/N2) x (D1/D2)

(H1 /H2) = (N1/N2)2 x (D1/D2)2

(P1 /P2) = (N1/N2)3 x (D1/D2)3

(NPSH31 /NPSH32) = (N1/N2)2

Per standard nomenclature, Q is flow rate, H is head, P is power, and NPSH3 is Net Positive Suction Head for a 3 percent head drop. 

Now for some modifications.

For single stage pumps, these equations work very well. Some corrections need to be made when modifying multistage pumps, particularly when changing the diameter. When trimming impellers, the increased vane tip clearance will lower the efficiency and performance a couple of percentage points. Usually about 1.5 percent per stage. If you are trimming impellers on a 10-stage pump, you will need to reduce the amount of trim by roughly 15 percent to avoid reducing the capacity and head below what you would originally expect from the affinity laws. The power saved would also be lowered because of the reduced efficiency.

NPSH3 is not affected by changing the diameter because the impeller eye does not change. However, real world fluids do not behave exactly according to the affinity laws. For hydrocarbons, an exponent of 1.8 will give better predictions than the theoretical 2. Some chemicals, such as caustic, can use and exponent of 2.1 to 2.2 for better predictions. User experience with each liquid will refine your technique. Unfortunately, industry does not run tests for these affects, and the few that do, do not share or publish the results.

How friction power modifies the equation

From the equations above, you would not understand why the various methods of achieving high head vary so greatly in efficiency and power requirements. Why is a large diameter impeller turning at motor speed so much less efficient than the smaller impeller turning much faster and a multistage pump more efficient still? The answer is the friction developed by the impeller disc as it moves through the liquid. The friction power modifies the power equation by the following:

(P1 /P2) = (D1/D2)5 

The 5 for an exponent means the efficiency decreases very rapidly as the diameter increases so a large diameter impeller requires much more power than a smaller impeller. From the original power equation we learn that speed carries a cubic exponent. So a multistage pump running at a lower RPM will be more efficient than a pump with a small impeller turning at high RPM which will be more efficient than a low speed pump with a larger diameter.

This information does not have much affect on low power applications. However, when a multistage pump may be greater than 75 percent efficient and a high speed pump is 25 percent efficient and a large diameter pump would run below 10 percent efficiency the cost of power over the life of the application can make a significant difference in the life cycle cost of the pump system.