Revisiting the Affinity Laws for Pumping Systems

Written by Robert X. Perez

Page 2 of 2

Linearized ALPS Model with Polynomial ALPS Model

Reader:

The ALPS "new performance line" falls below (Q_aff, H_aff); it does not pass through. It appears this may be the result of utilizing a linearized affinity curve. The Q_aff = r x Q₁ derivation substitution intersects the linearized affinity curve below H_aff. However, this lower positioning of the ALPS "new performance line" actually positions the predicted (Q_sys, H_sys) intersection closer to the polynomial curve model expectation! Brilliant! Was this intentional?

RXP:

Brilliant? No, just lucky. The brilliance was supplied by Sir Isaac Newton. By using his calculus, I was able to derive a generalized tool that is both simple and useful.

Reader:

The specific speed (Ns) for both published examples is around 1000. These are low-Ns radial flow impellers. The (Q_aff/Q₁) = (D₂/D₁), (H_aff/H₁) = (D₂/D₁)² , and (HP_aff/HP₁) = (D₂/D₁)³affinity relationships should give a good representation for the published examples.

However, pump impellers with Ns values > 1000 may require larger exponents for affinity relations involving impeller diameter size changes. For the users of higher Ns pumps, Figure 2.20 in Vertical Turbine, Mixed Flow, & Propeller Pumps by John L. Dicmas shows relationship curves for diameter ratio exponents as a function of Ns. There are also techniques for determining these exponents from lab tests of the same pump with two different impeller trims. The Hydraulic Institute may recommend consulting a pump manufacturer before making a diameter change > 5 percent.

The standard affinity law exponents for speed change relationships are good since efficiency should remain about constant.

Based on the above brief analysis for the published examples, ALPS appears to be a good, non-complex estimator for system performance resulting from low-Ns impeller diameter or speed changes that are less than 10 percent. It was interesting to see an exponent other than 2 used for a system curve.

RXP:

I asked Pumps & Systems Contributing Editor Joe Evans to respond to your comments. He says, ". . . my experience has been with pumps with a specific speed that ranges from about 900 to around 3000. My experience has been that the affinity laws do a pretty good job at these speeds. I also find nothing in the Pump Handbook that mentions the affinity laws change with respect to specific speed."

I agree with Joe's comments. We both agree that the classic affinity laws are good engineering approximations for pumps with specific speeds between 900 to 3000. This should cover most pumps used in our industry.

Reader:

Within the Appendix derivation, the Q system equation (before simplification) shows a " - r (A - C)" term in the numerator. I assume this should actually have shown " + r (A - C)."

RXP:

You are correct! I am amazed you found this typo. It should read:

Q system equation

Reader:

An additional thought concerning the August ALPS article: On page 24, the first caveat contains the restriction "ALPS can only be applied to a single pump ‘riding' on a system curve . . ." I assume the term "single pump" is meant to exclude cases where several pumps may be running in parallel OR series.

Of course, a "pump in series" application might also be interpreted to include a "single pump" with multiple stages. Only if all the impellers of a multistage single-pump system are identical and trimmed the same will a diameter ratio change be okay to use in the current ALPS analysis.

However, it may be common practice not to trim the upstream impellers, if possible, so that NPSH requirements aren't affected. In that non-identical trim situation, you probably would not use ALPS. A speed ratio change should certainly be okay to use for this multistage single pump example.

RXP:

You are correct in stating that ALPS may be used to predict the effects of small speed changes in multistage pumps. I guess when I wrote this caveat I was thinking more about multistage impeller trimming. As you know, it is more desirable to destage (as opposed to trimming) to reduce head requirements in multistage pumps. Destaging is a more energy-efficient modification. As you mentioned, for ALPS to be valid you must trim all impellers equally, which is rarely done.

I hope I have satisfied all your questions. Thanks again for your interest in the ALPS methodology and your insightful feedback.

Robert X. Perez, the website editor for PumpCalcs.com, has over 25 years of rotating equipment experience in the petrochemical industry, holds a BSME from Texas A&M University in College Station, a MSME degree from the University of Texas at Austin, a Texas PE license, and is an adjunct professor at Texas A&M University-Corpus Christi, teaching the Engineering Technology Rotating Equipment course.

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