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This website provides Euler’s equations for turbomachinery and solves a sample problem using velocity triangles. The problem prompt only provides the speed of the pump and the geometry of the pump and the final answer provides a single flow rate and head value. However, for a given pump geometry and speed, the pump can operate at a variety of head and flow rates as can be seen on any plot of a pump curve. So how is the ideal Euler line on a pump curve derived? Are the same equations and velocity triangle method from the link above used, or is there a different method. And why does the analysis in the link only arrive at a single head and flow rate value?

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You'll notice that at one point on the web page you linked, it says 'we assume that the flow enters exactly normal to the impeller, so tangential component of [lab frame] velocity [at the inlet] is zero'. That's called the "no prewhirl assumption", and, when you make that assumption, the argument on the web page leads to a single point on the graph of head against flow rate. If, on the other hand, you don't make that assumption, instead allowing the tangential component of lab frame velocity at the inlet to take any old value, then the equivalent argument from velocity triangles leads to a (straight) line on the graph of head against flow rate, which is known as the "ideal head-flow characteristic".

There's a nice run-through of the whole argument in the textbook Fluid mechanics by Douglas et al..

However, notice the word "ideal" in the name of the graph. That should clue you in that the real head-flow rate behaviour of a rotodynamic pump or turbine doesn't bear much resemblance to that predicted by the Euler turbomachinery equation: there are an awful lot of mechanical phenomena going on that the Euler turbomachinery equation doesn't take into account.

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