Page 36 - Fluid Power Engineering

14 Chapter Tw o

There is no heat exchange.

There is no mass transfer.

Relative position of fluid with respect to the earth’s surface

doesnotchange,thatis,thepotentialenergyremainsconstant.
The first two assumptions define an ideal fluid. The above assump-
tions lead to Bernoulli’s equation:
v 2
Total energy per unit volume = ρ + p = constant (2-11)
2
v 2
ρ is the kinetic energy term, which is also called the dynamic pres-
2
sure, and p is the static pressure.
Bernoulli’sequation,therefore,statesthatalongastreamlinewhen
speed increases, then pressure decreases and when speed decreases,
then pressure increases. The magnitude of change in pressure is gov-
erned by the quadratic relationship.
Note that Bernoulli’s law can be applied from A 0 to the left of the
rotor; and then from right of the rotor to A 2 (see Fig. 2-4). Bernoulli’s
law cannot be applied across the device that extracts energy; the con-
stant in Eq. (2-11) will be different for the two regions.

Conservation of Momentum
Since the wind rotor is a machine that works by extracting kinetic
energy from wind, the wind speed is reduced. Since momentum is
mass multiplied by speed, there is a change in momentum. According
to Newton’s second law, the rate of change of momentum in a control
volume is equal to the sum of all the forces acting. In order to simplify
the equations, the following assumptions are required:
There are no shear forces in the x-direction.
The pressure forces on edges A 0 and A 2 are equal.
There is no momentum loss or gain other than from A 0 and A 2 .
The equation for Newton’s second law along the x-axis becomes:
˙ m 0 v 0 − ˙m 2 v 2 = F (2-12)
Because of change in momentum in the control volume, there must
be external force acting. In this case, rotor provides the external force.
According to Newton’s third law, there must be an equal, but opposite,
force that acts on the rotor. This force is exerted by wind.
Because wind is exerting a force on the rotor, there must be a
pressure difference across the rotor equal to the force divided by the
area of rotor. Since the rotor hinders the flow of air, the pressure at
0
the front of the rotor (p ) is higher than the free-stream pressure (p 0 );
r
2
the pressure at the back surface of rotor (p ) is below the free-stream
r
pressure (see Fig. 2-5).