"Twist" of centrifugal pump impeller

Question

I suppose this is a logical extension of my earlier question here - in fact, this is probably what I was meaning to ask in the first place. I posted as "centrifugal pump", although the same concept can be observed in other (radial or axial) pumps.

Getting to it, pretty much any photo of a high-efficiency impeller you can find shows that the blade "twists", or has three-dimensional curvatures that are much more complex than taking a flat blade profile and simply extruding it, i.e. this Francis turbine runner:

First part of this question being, is this for the same reason that an airplane propeller twists - (forgive me, rather generalizing here) due to flow field conditions and the radial velocity effects? Are the same principles at work here?

Second part being, how does one go about constructing structures like this? There must be some math behind it that isn't completely CFD-based, as you can see the same ideas in the Rocketdyne F-1 turbopumps constructed in the '60s. However, no matter how many pump design sources I try to find, I can't find any that treat the impeller vanes as anything other than a two-dimensional chord on the impeller back shroud (see below image).

So, in essence, how would I start to go from a basic pump design (inlet/outlet velocity triangles, simple impeller) to a more "advanced" or "efficient" design like an impeller of the above nature?

Answer 1

The reason behind the twist is that the absolute inlet velocity (in your notation $V$) is constant from hub to shroud. However, the linear velocity (in your notation $U$) varies from hub to shroud. This means the relative velocity (in your notation $V_r$) should change to accommodate a constant absolute velocity. The modification in $V_r$ is enforced through blade angle $\beta$.

The same velocity triangle principle shown in the original post is applied to multiple planes likes the hub plane and shroud planes to get a distribution of $\beta$; hence, a twist appears whenever a considerable change in radius is present. This is usually the case for a pump or compressor inlet, a Francis turbine outlet, a propeller, a wind turbine, etc. It should be mentioned that a 3D design considers many other factors and often uses a mixture of analytical, empirical and numerical relations to define the final blade twist.