# Efficiency model for centrifugal pumps, using relations from Europump's Ecopump project

Context:

I'm trying to create a mathematical model for centrifugal pumps. This model should describes how the operational variables of centrifugal pumps $(Q, \Delta P, ...)$ related to it's design parameters $(Stages, Dimensions, ...)$ and it's use-case $(\rho, \mu, ....)$ .The intention behind this job is to develop an educational simulation environment. Anyway, after spending reasonable time reading+hacking various references I didn't find an idiomatic, integrated, already provided model to use. And I decided to assemble it using different parts from different sources.

Currently I think that:

• One important part of the model is to write $f(\eta,Q,...)=0$ or finding relation between Efficiency, Flow + design parameters on allowable, preferred or at list rated flow range.

Proposal:

To overcome the above bulleted item of my modeling problem, I picked mandatory conditions of an acceptable Ecopump project's centrifugal pump. Reference: Assessing the Energy Efficiency of Pumps and Pump Units By: Bernd Stoffel

Ecopump is the name of the successful flagship project that Europump launched in 2004 to bring the pump industry’s commitment to energy savings to the fore.

Following equations from the referred handbook, describes minimum $Efficiency$ required at these three different flow rates: 1-$BEP$, 2-part-load$(PL)$ and 3-over-load$(OL)$ , in terms of design parameters:

$\eta_{BEP,min\, requ} = -11.48*(\log(n_s))^2-0.85*(\log(Q_{BEP}))^2- 0.38*\log(n_s)*\log(Q_{BEP})+88.59*\log(n_s)+13.46*\log(Q_{BEP})\qquad (7.4)$

$\eta_{PL,min\, requ} = 0.947*\eta_{BEP,min\, requ}\qquad (7.5)$

$\eta_{OL,min\, requ} = 0.985*\eta_{BEP,min\, requ}\qquad (7.6)$

$#Where:$

$Q_{PL} = 0.75*Q_{BEP}$

$Q_{OL} = 1.1*Q_{BEP}$

Although the above conditions are three inequality rule that define a floor for a real pump $\eta$ value, I make an important suggestion here about a conceptual minimum Ecopump which exactly overlays those minimum requirements so $\eta_{real}==\eta_{min}$ ,and using those three points for $\eta$ it is possible to interpolate a second order polynomial and find $\eta(Q)$ for that conceptual minimum Ecopump with specific design parameters.

I expect that this model along with its numerical results could help instructors to cover following topics in a classic pump course:

1. How efficiency of a pump is related to other parameters.
2. How much a pump efficiency should be, to fulfill mandatory items of standards.
3. How does $\eta$ change related to $Q$
4. How to deal with modern standards about pump efficiency

Questions:

• Part One:
Am I on the right way? Or is(are) there any better way(s) to achieve the mentioned four goals.

Also:
The original $(7.4)$ equation have an additional $-C$ constant with the following description:

The constant C(in [%]) depends on MEI and additionally on the type and nominal speed of the size.

• Part Two:
From the handbook it is not clear for me how to calculate this constant, but for my approximate educational model I simply put it aside, Is this approximation acceptable? How much it affects my model? How to substitute more accurate value for $C$?

Any idiomatic points to create a better model are appreciated.