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Pumps & Systems, November 2007

I was pleased to receive a letter from one of our readers concerning my two-part article "Affinity Laws for Pumping Systems (ALPS)" (Beyond the Flanges, July-August 2007). To satisfactorily address all of this reader's queries, this month's column addresses them in detail for the benefit of our entire audience.

ALPS is a simplified method of predicting pump and system interactions for the most general system curve relationship: H = H_static + KQⁿ, where n can be any exponent. (Normally, n = 2, but any value can be used.)

The one key requirement of this analysis is that the pump performance curve be linearized so that it is in the form: H = H_o + mQ, where H_o 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. The slope "m" represents the instantaneous slope at the point of interest. The friction only and general system curves are also linerarized at the point of interest.

With these simplifications, ALPS can arrive at solutions for Q_system and H_system, the expected operating point at the new conditions.

To use ALPS, go to Pump-zone.com and click on the PumpCalcs button. Once you enter PumpCalcs, click on the APLS link under the expert calculators.

Reader:

Robert, I enjoyed perusing your ALPS articles. Once again you present unique insights and a useful analysis for predicting pump operation after a diameter or speed change. I never considered the possibility of confusion concerning where the new operating point should be located after one or both of the above changes to a pump.

Figure 2 (in July) and Figure 3 (August), including the effect of system static head, clearly illustrate why the affinity-law projection is not necessarily the new operating point. Since the pump performance curve is used to estimate H_o for a linear prediction model, I wonder why the pump curve wasn't used to construct a polynomial curve model?

The intersection of a model pump curve that passes through (Q_aff = r x Q₁, H_aff = r² x H₁) with the actual system curve may more closely approximate a new operating point (Q_sys, H_sys). Of course, a few more points from the existing performance curve are required for a new performance model. So, I used the given data from your article to confirm the ALPS output and to generate a curve-intersection model.

Since the initial performance curve was not shown, additional points were estimated to generate a model curve that is tangent to the linearized performance curve at (Q₁, H₁). Surprisingly, for both the diameter and the speed change cases, the estimated new operating points (Q_sys, H_sys) predicted by both the all-curve and ALPS models are very close!

What about other possible models? Besides the above polynomial curve and system curve model, a performance line and system curve model and a performance line and system line model were also developed. The later performance line and system line model would be the same as ALPS except that a linearized affinity curve was not used and it was developed to actually go through, or include, the affinity-predicted point (Q_aff = r x Q₁, H_aff = r² x H₁).

The system predictions calculated from the performance line and system curve model are close to the ALPS output, while the system values from the performance line and system line model are the farthest from expected (Q_sys, H_sys) values.

RXP:

You are correct in thinking there are many model combinations that will work satisfactorily. All the methods you mention here are valid as long as you understand how pumps interact with system curves. Let's go through all the possibilities:

1. There's always the old fashion graphical method that requires the generation of a new pump curve using the affinity laws. You must then plot the system curve along with the "new" performance curve to see where they intersect. This method can be conducted with pencil and paper or using a spreadsheet program such as Excel.
2. Polynomial method is the most complex of these three. First, you must use the affinity laws to determine where your pump performance will be at the new conditions. Next, you need curve fitting software to determine the best fit for your "new" performance data. Finally, you must analytically arrive at the intersection of the performance and system curve.This was the approach I took early on, but I discovered several difficulties with this method. I found that most pump curves required 3^rd (or higher) order polynomial fits for acceptable curve fits. (Flat pump curves may require higher order polynomials for a proper fit.) I soon decided against this since cubic equations have three solutions (or roots). I realized not all pump users want to deal with 3^rd and 4^th order polynomial equations.This is not to say this is not a valid approach. I welcome you or our other readers to provide a user-friendly general solution using higher order polynomials for performance curves.
3. A model based on a linearized pump curve and a system curve also works. In fact, I was torn between this model and the linear pump and linear system model. For simplicity, I decided to go with the final ALPS version - I'm a big fan of simplicity. 
4. Finally, ALPS "classic" was developed as a practical field tool for pump users that requires only a pump curve, straight edge, and some basic knowledge about the system conditions. You can consider ALPS to be a screening tool. If you are surprised by the initial results, you can confirm or refine those results with any of the methods mentioned above. For comparison, I plotted all the various analysis methods together (see Figure 1) and found they all intersect at approximately the same point. This suggests to me that the linear ALPS model will probably provide a solution within engineering accuracy (± 5 percent) for most applications. If a more accurate answer is required, there's no reason why one of the other ALPS models cannot be used.