Affinity laws provide key insights about how centrifugal pumps work. However, as with all shortcuts, you must understand their limitations to prevent unexpected problems.
Let's first consider the head affinity law. It states that pump head, h, varies with impeller diameter, d, squared:
h1/h2 = d12/d22 (1)
This affinity law gives a quick estimate of head change when an impeller gets cut down to a reduced diameter. Experienced engineers know the affinity law is only an approximation. It generally understates the head reduction when using a smaller impeller. You must consult the pump manufacturer to get a more accurate idea of the actual changes resulting from the cut-down impeller.
Why does a real pump's performance differ from the affinity law prediction? Many factors go into this but we'll look at only one of them. Pump head depends upon impeller size and shape. Cutting down the impeller changes both. The affinity law takes into account the consequences of a diameter change — but not of a shape change.
Figure 1 shows that trimming an impeller alters both shape and diameter. It depicts, from left to right, the original impeller, the impeller trimmed 10% of its diameter by cutting its end off, the impeller proportionally scaled down by 10%, and the trimmed impeller superimposed on the scaled impeller — with blue and green highlighting the differences.
Normally, only the original pump manufacturer will have the data that allow for accurately predicting how impeller performance will change when trimmed. Applying the affinity law for diameter change provides an educated guess on the new pump head — but it's still only an estimate.
Nonetheless, we can predict the general trend via pump specific speed, NS, a parameter centrifugal pump manufacturers use in detailed pump design:
NS = N(Q)0.5/h0.75 (2)
where N is rotational speed in rpm, Q is flow rate in gpm, and h is head in ft. Specific speed is evaluated at the flow rate and head delivered at the pump's best efficiency point for that impeller. As long as dimensions are fully scaled, pumps with different size, but the same shape, impellers will have the same specific speed. This means that pump curve shifts due to impeller diameter changes can be predicted accurately.
The most efficient pumps have specific speeds in the range of 2,000–3,000 rpm. Higher heads and lower flow rates shift pump selection toward lower specific speeds, and vice versa.
Figure 2 compares pump impeller shape versus specific speed. Low-specific-speed pumps have relatively little change in impeller shape when trimmed. In contrast, high-specific-speed pumps have dramatic changes in impeller shape when trimmed. Thus, the lower the pump specific speed, the less deviation there is from the affinity law when cutting the impeller diameter down.
So far, we've focused on single-stage pumps. The same logic applies to multiple-stage pumps as well. However, performance of a multiple-stage pump depends upon the exact shape of every stage. Some multiple-stage pumps use different impeller shapes in various stages — making behavior of these pumps very complex when trimming impellers.
As a summary, the affinity law gives a good approximation of pump performance with diameter changes. For relatively small diameter changes (<5% of diameter) and for low-specific-speed pumps (<1,500 rpm), it provides a reasonably close answer. However, even those pumps often may undergo surprising head changes. Pumps getting larger trims or operating at higher specific speeds may exhibit dramatically different performance than expected. Always consult the pump manufacturer when considering significant pump-impeller cuts.
ANDREW SLOLEY is a Chemical Processing contributing editor. You can e-mail him at ASloley@putman.net