# Affinity Laws for Pumping Systems (ALPS) - Part Two

Written by:
Robert X. Perez

Pumps & Systems, August 2007

Conclusion of our look at a new tool that predicts how pumps and systems interact.

Last month, I opened this discussion by explaining how the affinity laws are used to estimate the effect of speed and impeller diameter changes on pump performance.

I emphasized that these laws convey simple relationships that are only meant to be used on pump curve predictions, or when system curves fit the general form of H = KQ2, 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.

## Affinity Laws for Pumping Systems

Once I realized this potential error, I decided to determine if the affinity laws could be modified so that they could be used to predict pumping system interactions. This quest became an obsession.

After numerous failed attempts, I was finally able to arrive at a simplified method to predict real-world pump and system interaction for the most general system curve relationship: H = Hstatic + KQn, where n can be any exponent. (Normally, n = 2, but any value can be used.)

The one key requirement was that I had to linearize the pump performance curve so that it has the form: H = Ho + mQ, where Ho is the point where a line tangent to pump curve at the location of interest (Q1, H1) intersects the vertical axis, i.e. the ordinate (see Figure 3). The slope "m" represents the instantaneous slope at the point of interest.

I also had to linerarize the friction only and general system curves at the point of interest. With these simplifications, I was able to arrive at solutions for Qsystem and Hsystem, the expected operating point at the new conditions. More details of my analysis can be found in the appendix.

## ALPS Example with Diameter Change

Let's first work through an example with a change in impeller diameter. Initially, we have to input all the required data in the lime-green boxes as shown in Table 1.

In this analysis, we have chosen to change the impeller diameter from 10-in to 10.75-in while maintaining constant pump speed. (Note: The input variable called "Ho" is that value where the pump curve tangent line intersects the Y-axis. In this case, its value is 375-ft.)

You can see that the head, flow, and horsepower values are different for affinity law predictions vs. the system curve predictions. If you look at Figure 4, you can see the ALPS prediction line (green) superimposed on for the affinity law curve and system curve.

Notice that the prediction made by ALPS is not perfect. It slightly passes the system curve. The ALPS prediction for flow is 576.61-gpm vs. 537.61-gpm for the affinity law prediction. Here, ALPS predicts a substantially greater flow that the traditional affinity laws!

Continuing this same example, let's see what effect this diameter change has on power consumption. You can see that power consumption increased from 57.24-hp to 75.38-hp, instead of the 71.11-hp the horsepower affinity law predicts. If you had a 75-hp motor driver, this disparity would make the difference between having a few horsepower of margin and running your motor at full load.

## VFD Analysis Example

Now let's look at an example with changing speed. Assume we are changing speed from 3560-rpm to 3360-rpm, while maintaining a constant impeller diameter. Again, we first have to input our data into the ALPS calculation table (see Table 2).

With an assumed static head Hs equal to 200-ft, the values for head, flow, and horsepower are markedly different for the affinity law predicted values versus the ALPS predicted values. The system curve flow is estimated at 443.82-gpm and the flow for the affinity law prediction is 471.91-gpm. The graphical version of these results are shown in Figure 5.

Once again the ALPS prediction line does not exactly match the system curve. However, for small (<10 percent) changes in speed, the prediction is close enough for engineering purposes.

Another benefit of the ALPS methodology is its ability to determine if flow control by speed modulation is viable. Let's continue with the example above, where speed is varied from 3560-rpm down to 3360-rpm. ALPS determines that system flow will drop from the original 500-gpm to 443.82-gpm, or a total of 56.18-gpm. This means we can expect 0.2809-gpm per rpm (56.18-gpm/200-rpm) change in pump speed.