Environmental Engineering Reference
In-Depth Information
Different theories (strip theory, circulation, vortex shedding) and experimental data on airfoils
are used to predict the rotor performance of wind turbines. This theoretical performance can be
checked against the measured output of models in wind tunnels, truck testing for small-diameter
units, or field testing (atmospheric) of wind turbines. At one time, a railroad flat car was used for
controlled speed testing, as a somewhat larger turbine could be mounted. Overall efficiencies include
those of the rotor, drive train, and energy converter (generator, etc.). The complete analysis on design
of wind turbines, primarily rotors and structures, can be found in more advanced texts [3-12]; how-
ever, beginning physics can be used for a qualitative understanding of rotor performance.
6.3 MATHEMATICAL TERMS
Momentum of a particle is the mass times the velocity. Boldface in an equation indicates that it is
a vector, which has both magnitude and direction. In two dimensions, it takes two components to
define a vector, and in three dimensions, three components. In an analytical representation the vec-
tor can be represented by its components along two axes (perpendicular or orthogonal axes for this
presentation).
pv
m
(6.1)
Any particle can be treated as a single particle with the mass, M , concentrated at a point (center
of mass, R ). Position vector is indicated by r .
.....
MR
mm
r
r
m
r
(6.2)
11
2
2
ii
Forces on particles make them accelerate. Newton's second law describes the dynamics or
motion; force is the change in momentum over the change in time. In other words, to change the
momentum of a particle requires a force. That could mean a change in speed or a change in direction
of the motion of the particle. There is also a force if there is a change in mass, but for this discussion,
mass is constant.
$
$ t
p
F
, newton (N)
(6.3)
Torque makes a particle turn around some point, which can be thought of as the lever arm times
the force. A larger torque can be obtained by increasing the length of the lever arm or by increasing
the force.
TrF
r , Nm
(6.4)
where the cross-product means that two vectors produce a vector whose direction is perpendicular
to the plane of the two vectors.
If a mass is attached to a rod, which is free to rotate about its end ( Figure 6.1 ) , and a force is applied,
the torque will make the mass rotate, and there will be power available. The amount of power is the
product of the torque and angular velocity (Equation 5.1). That power is available at the shaft. Most
operations of transferring shaft power try to have a large q because of structural considerations.
P T q , W
(6.5)
Also, the rotating object will have rotational kinetic energy.
KE rot 0.5 m v 2 0.5 m r 2 q 2 , J
(6.6)
where the speed of the rotating mass depends on the radius, v q * r .
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