Page 38 - Fluid Power Engineering

16 Chapter Tw o

FIGURE 2-7 Illustration
of rotor acting as an
impenetrable wall.
y

ν 2 = 0 x

ν 0

speed is the same and pressure is the same on both sides of the rotor,
then no energy is extracted by the rotor.
The second question is: Can a rotor of a wind turbine extract all
the kinetic energy from wind and make v 2 = 0?
To answer this question, see Fig. 2-7. If v 2 is zero, then there is no
wind passing through the rotor. The rotor acts like an impenetrable
wall and the wind flows around the wall. Since no wind is passing
through the rotor, there is no energy extraction.

Derivation of Betz Limit
In 1919, Albert Betz postulated a theory about the efficiency of rotor-
based turbines. Using simple concepts of conservation of mass, mo-
mentum,andenergy,hepostulatedthatawindturbinewithadisc-like
rotor cannot capture more than 59.3% of energy contained in a mass
of air that will pass through the rotor. The Betz limit is derived next.
Applying conservation of mass, Eq. (2-9), in control volume A 0 ,
A r , and A 2 with constant density (see Fig. 2-5):

A 0 v 0 = A r v r = A 2 v 2
where v 2 is the average wind speed at A 2 . Applying Newton’s second
law from Eq. (2-12), force exerted on rotor by wind:

F = ˙m r (v 0 − v 2 ) = ρA r v r (v 0 − v 2 ) (2-13)

The force exerted on the rotor is also because of the pressure difference
across the rotor:

2
0
F = A r (p − p ) (2-14)
r
r