Bernoulli Equation and its application to turbines and pumps

[Shivam Sinha]

Hi, I have never found a satisfactory explanation for why the Bernoulli equation is not valid when the streamline passes through a turbine, pump or another work transferring device. I have read many books that simply state this limitation without providing a convincing reason.

Bernoulli equation can be derived by integrating the Euler's equation for a steady, inviscid, incompressible flow along a streamline. So, it should be valid for steady, inviscid, incompressible flow regardless of whether a work-transferring device is present or not, shouldn't it? (because the Euler equation is valid at all points for such flows, and the Bernoulli equation is just the intergrated version of it)

Can anyone provide me a detailed explanation for this limitation of the Bernoulli's theorem?

 

[Shivam Sinha]

I gave some more thought to this question and arrived at the conclusion that the flow through rotating blades cannot be steady. Hence, the standard Bernoulli equation is not valid in such cases.

However, if we choose a frame of reference attached to the rotor, the flow can be steady and the streamlines in this frame of reference will be along the surface of the rotating blades. The modified Euler's equation in the rotating frame can then be integrated along the streamlines to get a new Bernoulli's equation. This new equation will hold true for pumps and turbines.

I'll be very glad if someone can verify the above explanation. Thanks!

 

[anorlunda]

There are two sets of blades, one rotor static, one stator static. How is that reflected in your analysis?

 

[boneh3ad]

Flow around a turbine is also not inviscid, nor can it be readily approximated as such. Have you considered the derivation of Bernoulli's equation from the integral conservation of energy equation? It may be more clear there.

 

[Shivam Sinha]

Thanks, I agree that in the real world, flow around a turbine is not inviscid. Hence, the Euler's equation and the Bernoulli equation will provide incorrect results. And so, we use the integral conservation of energy in such cases.

Many books say that even if the flow in the turbine were inviscid, the Bernoulli equation would not hold true. Some of them explain it by saying that such devices destroy the streamlines (I don't know what this means) while others say that it is because Euler's equation does not take account of shaft work (Again, how can a momentum balance equation take account of work of any kind?). I have never seen sense in these reasons.

The purpose of my last post was to explain that even if flow through a pump or turbine were inviscid, Bernoulli equation would not be valid only because of the unsteady nature of the flow through these devices and not because of the reasons that many of the books provide.

 

[boneh3ad]

I'll ask again. Have you looked at the derivation of Bernoulli's equation via the integral energy conservation equation.

 

[Shivam Sinha]

Yes, I have looked at the derivation using the integral energy conservation.