Betz Law & Submerged Turbines: Is it Applicable?

[aladinlamp]

Hi, is Betz law fully applicable for hydro, hydrokinetic submerged turbines, since air is compressible but water is not ?

 

[Lnewqban]

Copied from
https://en.wikipedia.org/wiki/Betz's_law

"Betz's law applies to all Newtonian fluids.
Water and air can be assumed to be Newtonian for practical calculations under ordinary conditions."

 

[cjl]

Betz's law actually is based on incompressible flow. Air can be treated as effectively incompressible below mach 0.3 or so (depending on the level of accuracy you need), so for most everyday aerodynamics, you can assume that the compressibility of air is negligible. As a result, Betz's law definitely applies to water turbines as well (in the case that you're trying to capture the energy from a moving current - this is not the case if you can confine the entirety of the flow to travel through the turbine such as with a dam).

 

[aladinlamp]

in case of wind turbines, exiting air has larger area and lower velocity than incoming, how is this applied to water in fixed crossection open channel, water cannot expand at exit, Q in has to be same as Q out , so velocity in should be same as velocity out

 

[cjl]

That's why I said it only applies if the flow isn't confined to travel through the turbine. If the flow is confined, you still need to have an energy loss somehow - either the cross section will change, or the flow will have to descend and lose gravitational potential energy (or there will be a substantial pressure gradient across the turbine).

Is there a specific scenario you have in mind here?

 

[aladinlamp]

how can velocity out be smaller than velocity in

 

[cjl]

So first of all, just to be clear, that would count as a constrained or confined flow, so Betz's law wouldn't apply.

As for the details of the flow, either the depth after the turbines is higher, thus providing a larger cross section and slower flow, or there's a drop in water level across the turbines and thus the energy extraction comes from gravitational potential energy. One or the other must be true though - you can't extract energy from a flow and also have the outlet flow with the same velocity, cross section, and water level as the inlet.

EDIT: If it were fully confined to a pipe (which is not the case above, but I just want to cover all the scenarios here), you could have both the inlet and outlet with the same velocity and cross sectional area and nevertheless extract energy. In that case though, you'd see a pressure drop across the energy extracting turbine. For the generalized case, you'll always see a drop in gravitational potential, pressure, or velocity if you extract energy from a fluid.

 

[aladinlamp]

ok so you are saying , in this scenario,, if Q in = Q out, and V in = V out, there has to higher water level at inlet than at outlet, there is no other explanation

 

[cjl]

Correct.

 

[cjl]

Also, did you edit your post to remove the example we're talking about here? Please re-edit and add it back, since that removes a lot of the context to our discussion.