I'm looking for a formula to arrive at a pump curve for a centrifugal pump: $H(Q, n)$ - total head delivered as a function of flow and speed (rpm), and of course some pump specific parameters. I find no formula I can transform into this. For actual pump selection I would of course use the pump curves from the supplier, but a formula like this would be interesting for simulation purposes of pumping systems where VFD is used.

As both head and flow are mostly a function of speed, impeller size and shape I think a non purely empirical fomrlua for a pump curve should exist.

p.s. Alternativly, I considered using a simple polynomial or similar curve and fitting it numerically to an existing pump curve. Pump simulators, is this an actually useful approach lacking an analytically correct curve?

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    $\begingroup$ The boundary would be input work = volumetric flow rate * differential pressure (assuming incompressible fluid). The difference between that boundary and the actual performance curve is based on the efficiency. I am also currently trying to figure out how to estimate the efficiency... $\endgroup$ – ericksonla Aug 28 '18 at 19:45
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    $\begingroup$ There are a million different ways to do this.. Good Question. It's pretty much empirical for a few data points and expanded out from there. I'll add this to my list of things to come back and answer. $\endgroup$ – Mark Aug 28 '18 at 22:42
  • $\begingroup$ It sounds like you are looking for the pump affinity laws. en.wikipedia.org/wiki/Affinity_laws $\endgroup$ – mkeith Aug 29 '18 at 7:35
  • $\begingroup$ I've looked at those. They give me an idea how to adjust a pump curve to different speeds, but you need a pump curve to start. $\endgroup$ – mart Aug 29 '18 at 7:56
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    $\begingroup$ Using curve fitting, yes that always works and to some degree you can parameterize the fit to provide dP at some speed and flow or some flow at dP and speed. But I've also been looking for an analytical basis for at least predicting the shape of such curves. I believe @ericksola has the right idea. For a fixed speed you have a fixed total energy that exists as a combination of potential (pressure) and kinetic (flow) energies. But nowhere to be found. The curves look like circles or ellipses, so $ap^2 + bq^2 = K \omega^2$ or the like might work. $\endgroup$ – docscience Dec 5 '18 at 0:41

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