Is the Betz Limit Calculation Based on Inaccurate Assumptions?

[QwertyXP]

Betz Limit

The amount of power in wind can be expressed as: 0.5*M*(V^2), where M=mass flow rate.

As per my knowledge, we assume M to be constant regardless of where we are measuring it, downstream, upstream or close to the wind turbine.

If measuring close to the wind turbine, the expression for M would be:
M=ρ*S*Vb

where S=the area sweeped by the wind turbine blades and Vb=velocity of wind at the blades


Now, When finding the power coefficient, why do we take the reference power for the Betz efficiency calculation to be equal to 0.5*ρ*S*(Vd^3), where Vd=velocity of wind downstream

The above expression for reference power (or total power available in the wind) could also be written as:
0.5*(ρ*S*Vd)*Vd^2

So here we are assuming that the cross-sectional area of wind downstream is the same as the cross-sectional area of wind close to the turbine, which appears to be wrong because then the mass flow rate would no longer be constant!

It seems to me that the reference power (or total power available in the wind) should actually be:
Pref=0.5*M*Vd^2
=0.5*ρ*S*Vb*(Vd^2)

But then, this expression is useless because we don't know the value of Vb..except that Vb=0.5*(Vd+Vu)

 

[AlephZero]

The total power available from the wind is the power "flowing" throigh the area of the turbine, when the turbine is not there. That is what your first fornula 0.5*M*(V^2) says.

Of course there is no way to extract some of the power without changing the wind flow in some way. That's why you can't make a turbine that is 100% efficient.

 

[QwertyXP]

Thanks a lot! This problem was troubling me quite a bit.