Page 40 - Fluid Power Engineering

18 Chapter Tw o

The pressure difference across the rotor creates the force that

performs the work and generates power. This is counterin-
tuitive. One would expect the wind speed to drop abruptly
across the rotor. In fact, the wind speed does not drop, instead,
there is an abrupt drop in pressure and the pressure energy is
transferred to the rotor. Note, an abrupt drop in wind speed
would cause large undesirable acceleration and force.
In the downstream volume, the static pressure rises from

2
p to p 0 . Again assuming there is no mass transfer across
r
the flow boundary, this pressure rise is because of transfer of
kinetic energy from wind to static pressure. The wind speed,
therefore, reduces from v r to v 2 .
The net effect is that the flow starts with p 0 as the upstream pres-
sure and ends with p 0 as the downstream pressure. In the middle, the
1
2
pressure rises to p in front of the rotor, drops abruptly to p behind
r r
the rotor, and then climbs back to p 0 . During this same interval, kinetic
energy of wind is extracted, converted to pressure energy, and deliv-
ered to the rotor. In fact, theoretically, 62.5% of the energy delivered to
an ideal rotor is extracted from kinetic energy in front of the rotor. The
rest of the 37.5% is extracted from kinetic energy in the wake of the
rotor. In other words, although 100% of the pressure energy is deliv-
ered to energy extraction device at the rotor, only 62.5% of this energy
came from kinetic energy to pressure energy conversion in front of
the rotor. This means that at the rotor, the pressure energy goes into a
“deficit” by delivering more energy to rotor than was imparted to it
by kinetic energy of wind. This deficit in pressure energy is recovered
in the wake of the rotor.
In all the discussions above, v 2 is the wake wind speed at a distance
from the rotor where the pressure is restored to p 0 . It is assumed that
until this imaginary point in the wake is reached where average wake
speed is v 2 and pressure is p 0 , there is no mass transfer from the sur-
rounding air. Beyond this imaginary point and further downstream,
p 0 will remain the same, but the wind speed will start increasing and
eventually reach the free-flow speed of v 0 . This will happen because
the air around slower wake will cause the wake to accelerate through
either shear force or mass transfer.
Note this is an idealized rotor and no reference is made to the
blades and the aerodynamics of the blades of the rotor. In the next
chapter, the behavior of wind at the rotor, that is, interactions of wind
with the blades (aerodynamics) is discussed.
The power delivered to the idealized rotor by the wind is:

0 2
P = Fv r = (p − p )A r v r (2-21)
r r