Plugging q=1/3 into the above equations, we find
that the maximum efficiency Pextr/P0 is
This is known as Betz's limit. It means the maximum
possible efficiency of a windmill is simply and concretely 59%. In
thermodynamics, the Carnot engine sets the theoretical maximum for
efficiency. In this case, we have reached a similar universal statement.
In reality though, just as a Carnot cycle does not make a good engine for
your car, it is impractical to create a windmill with this efficiency.
Instead, most windmills operate at efficiencies of roughly 40%, as noted
by Ragheb . This falls well under the universal limit we found using
just simple equations.
Maximizing Power with Betz's Formula
Now that we know the maximum efficiency for a
windmill, we can abandon the "black box" idea and begin looking at
specific designs. We narrow our analysis to the horizontal-axis turbines,
the most common variety. What is the best design for such a wind turbine?
For example, we can imagine varying the radius of the turbine, the height
of the tower, and the number of blades. What optimization rules have led
to the canonical design? Using the analysis of the previous section, we
can write the formula governing power-output of an ideal windmill.
This is known as Betz's formula , and is the
maximum theoretically allowed power output, using Betz's limit. We see
that the power output depends on three factors: the air density, the wind
speed, and the area swept out by the turbine. The first of these is not
something we can vary as we design out windmill, so we focus on optimizing
the final two variables.
That the power depends upon the wind velocity is no
surprise. In stronger wind, windmills spin faster. However, the strength
of this relation is at first surprising. The power depends upon the cube
of the wind speed. Thus, we see the most important factor in power output
is finding a location with strong winds . This is in fact done, and is
why windmills are generally found along ridges instead of deep in
However, we can do more to optimize wind speed than
just the placement of the windmill. The height of our windmill is an
important component. As discussed in Ragheb , the wind speed is a
function of height. Starting from a vanishing wind speed at the ground,
the wind speed increases as one goes higher. This is familiar to those
who have stood atop a tall tower. This dependence can be approximated
In this equation, we find the wind speed based on
comparison with the known wind speed at a height h0. The
constant a depends on the environment, but is on the order of .1 .
Thus, the velocity increases slowly with increasing height. Doubling the
height here would result only in a roughly 10% increase in speed.
However, with the cubic dependence on wind speed in Betz's formula, the
modest gain in wind speed results in a significant increase in power
Thus, placing the wind turbines high above the
ground leads to an increase in power. This is one reason that windmills
are placed on tall towers. A windmill stationed low to the ground would
be ineffective. Just looking at Betz's formula, we see that, in fact, the
higher the windmill the better. This is, of course, not true due to
practical constraints. At a certain height, it becomes difficult or
uneconomical to build taller windmills.
We now turn to the final variable in Betz's formula,
which is the area of the turbine. This is merely given by
πR2, where R is the length of the blades on the windmill.
An increase in the length of the blades will lead to a quadratic increase
in the power output. As it turns out, the power output actually depends
more sensitively on the wind speed than the turbine radius, which may be
surprising. Again, though naive optimization using Betz's formula would
lead us to thinking the bigger the turbine the better, practical and
monetary considerations interfere with this conclusion. For instance, at
the very least, we do not want the blades reaching down to the ground.
Number of Blades
The blades of a windmill are perhaps the most studied
part of the turbine design, and are well summarized by Adkins and Liebek
 and Jischke . For instance, the blade design is closely related to
the design of propellors on aircraft. However, we will focus on a simpler
component of blade-design: the number of blades. In the current wind
turbines used for generating electricity, there are three, and always
three blades. But, why? The old Dutch mills had four blades, and the
small metal farmer mills have roughly a dozen blades.
As outlined in , there are several considerations
used when deciding how many blades a turbine should have. The first is in
the efficiency of the turbine. A very large number of blades would
decrease the efficiency of a turbine. Intuitively this is clear. The
drag caused by 100 spinning blades would be much greater than that caused
by 3. In trying to find the optimum number of blades, we are led to
choose a low number.
However, there is a second consideration, which is
unrelated to efficiency. Should a turbine have an odd or even number of
blades? At first, it may not seem to matter much whether there are two
blades or three. However, upon further inspection , a simple
difference is revealed. The balance and stability of a windmill greatly
depends on the number of blades.
First, let us consider a windmill with two blades.
Imagine the instant that the blades are vertical. At that moment, the
bottom blade is passing in front of the tower, drastically cutting down
the wind force on the blade. On the other hand, the top blade is fully
exposed to the maximum wind speed - recall that wind speed increases with
height. There is a greater force on the top blade than the bottom blade.
The net effect is that there is a torque on the turbine. It has a
tendency to be tilted backwards slightly. This wobble leads to an
undesired instability in the windmill, and leads to engineering headaches
when it comes to designing the hub of the windmill.
|Fig. 3: Windmill with Three Blades. As the lower blade
passes the tower, the other blades will be getting more force
from the wind. This causes a backwards torque, but with an
odd number of blades this torque is considerably less than
with an even number.
Now consider a windmill with three blades. In this
case, when one of the blades passes the tower, the other two are each at
30° above the horizontal. They do indeed get more force from the wind
than the lower blade. However, this situation is better than the
two-blade case for two reasons. First, since the blades are not at their
zenith, the force on each is not at a maximum. Secondly, most of the
forces cancel between these two blades, since they are nearly opposite one
another. Only a fraction of the force is directed at tilting the turbine
"backwards." This case is far more stable than the two-blade case.
Thus we see that, in general, an odd number of
blades is easier to deal with when designing a windmill. There will be a
less pronounced wobble than in cases where there are an even number of
By looking at the physics behind windmills, we now
have a better understanding of why windmills look the way they do.
Current windmills are first and foremost power plants. From Betz's
formula, we find the important variables affecting power output. The wind
speed is the most important factor, and this is optimized both by choosing
good locations and by placing the turbines on tall towers. We see that
the length of the blades is related to the power output; specifically the
power is proportional to the square of the radius. This, too, can be
optimized in designing windmills. Finally, the number of blades is an
important design consideration, if not for the efficiency, but for the
stability of the windmill.
The increasing concerns over climate change and
environmental awareness will increasingly make wind power an appealing
source of energy. More and more windmills will pop up on hillsides, and
it is important to know the physics behind their design. Windmills do not
have an arbitrary design, but rather a design dictated by the physics of
© 2007 Andy Hall. The author grants permission to
copy, distribute and display this work in unaltered form, with attribution
to the author, for noncommercial purposes only. All other rights,
including commercial rights, are reserved to the author.
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