*Pumps & Systems*, July 2007

When I first discovered the affinity laws, I was impressed by their simplicity, versatility, and power. For me, they held a certain allure. I believed I could use these three simple relationships to solve any pump problem:

Here Q represents *flow*, N is *rotational speed*, D is *impeller diameter*, H is *pump differential head*, and BHP is *required pump brake horsepower*.

My problem was that, unknowingly, I misapplied these laws early in my pump career because of a common misunderstanding: *these simple relationships are only meant to be used on pump curve predictions or when system curves fit the general form of H = KQ ^{2}, i.e. friction only system curves!* Using them to directly determine the effects impeller or speed changes for system curves with static head can result in serious calculation errors.

The affinity laws are primarily used to estimate the effect of speed and impeller diameter changes on pump performance. For example, let's look at Figure 1.

**
Figure 1**.

To use the affinity laws to determine the effect of increasing speed from 3400-rpm to 3600-rpm, we must first calculate the effect of speed on flow and head for each performance point and replot the points as seen in Figure 1.

Using the classical affinity laws, the points for the new pump curve (blue) were determined by knowing the head and flow values of the original pump curve (violet) and the change in pump speed. Once you have the two pump curve plots, you can plot your system curve on top of the pumps curves to predict how the pump and system will interact.

For the friction-controlled curve shown in Figure 1, you can see that the intersection of the pump performance and the system curve points at "A" and "B" match perfectly at both speeds. This visually confirms that the affinity laws can be used to forecast where your pump and system will operate, assuming your systems follows the H = KQ^{2} relationship.

This convenient relationship *does not* hold true for pumping systems with static head, as seen in Figure 2.

**
Figure 2.**

While the revised pump curve is still valid, the affinity laws cannot predict how the pump and system will interact under the new conditions when static head is added to the mix. Notice that at point "A" both system curves converge on the 3400-rpm equation, while at point "B" the curve for "friction + 200 feet of static head" no longer matches the static-only system curve points at the higher pump speed.

This mismatch worsens with higher values of static head (H_{s}). (Please note that the green and black curves in Figure 2 are quite different. The black "affinity law" curve is defined by the relationship H = K_{1}Q^{2} that exactly intersects points "A" and "B" in Figure 2. An "affinity law" curve was used to define all the new points on the blue 3600-rpm curve. The green curve is the pumping system curve defined by the relationship H = H_{
s }+ K_{2}Q^{2}, where H_{
s} is the static head present in the system and K_{
2} is the constant required to allow the system curve to intersect the 3400-rpm pump curve at point "A." At speeds higher and lower than 3400-rpm, the "affinity law" curve and pumping system curve diverge from one another.)

I want to stress that there is nothing wrong with using the affinity laws. They are perfectly capable of helping you create modified pump curves at different speeds and impeller diameters. What I am saying here is that to draw valid conclusions about how changes in speed or impeller diameters will change system flows or pressures, you must pursue the problem graphically.

You must plot your "before" and "after" pump curves along with your system curve to estimate where your system will operate. Simply stated: *the affinity laws predict pump performance changes - not changes in pump/system interactions*.

Next month we will conclude this discussion by looking at affinity laws for pumping systems, an ALPS example involving diameter change and another with changing speed.

**References**

- Karassik, Igor, et al, Pump Handbook, New York, McGraw-Hill Book Company, 1976, pp 2.135-2.142.
- Lobanoff, Val, Ross, Robert, Centrifugal Pump - Design and Application, Houston, Gulf Publishing, 1992, pp 12-14.