# Physics of Wind Turbines

Over a thousand years ago, windmills were in operations in Persia and China, see TelosNet and Wikipedia. Post mills appeared in Europe in the twelfth century, and by the end of the thirteenth century the tower mill, on which only the timber cap rotated rather than the whole body of the mill, had been introduced. In the United States, the development of the water-pumping windmill was a major factor in allowing the farming and ranching of vast areas in the middle of the nineteenth century. The wind pumps (sometimes called Western mills) are still common in America and Australia. They have a rotor with about 30 vanes (or blades) and the ability to turn itself slowly. Of the 200,000 windmills existing in Europe in the middle of the nineteenth century, only one in ten remained after 100 years. The old windmills have been replaced by steam and internal combustion engines. However, since the end of the last century the number of wind turbines (WT) is growing steadily, and is beginning to take an important role in power generation in many countries.

We first show that for all wind turbines, wind power is
proportional to wind speed cubed. Wind energy is the kinetic energy
of the moving air. The kinetic energy of a mass *m* with the
velocity *v* is

The air mass m can be determined from the air density ρ and the air volume V according to

Then,

Power is energy divided by time. We consider a small time, Δ*t*,
in which the air particles travel a distance *s* = *v*
Δ*t* to flow through. We multiply the distance with the rotor
area of the wind turbine, *A*, resulting in a volume of

which drives the wind turbine for the small period of time. Then the wind power is given as

The wind power increases with the cube of the wind speed. In other words: doubling the wind speed gives eight times the wind power. Therefore, the selection of a "windy" location is very important for a wind turbine.

The effective usable wind power is less than indicated by the above equation. The wind speed behind the wind turbine can not be zero, since no air could follow. Therefore, only a part of the kinetic energy can be extracted. Consider the following picture:

The wind speed before the wind turbine is larger than after.
Because the mass flow must be continuous, the area *A*_{2}
after the wind turbine is bigger than the area *A*_{1}
before. The effective power is the difference between the two wind
powers:

If the difference of both speeds is zero, we have no net
efficiency. If the difference is too big, the air flow through the
rotor is hindered too much. The power coefficient c_{p}
characterizes the relative drawing power:

To derive the above equation, the following was assumed: *A*_{1}*v*_{1}
= *A*_{2}*v*_{2} = A (*v*1+*v*2)
/ 2. We designate the ratio *v*2/*v*1 on the right
side of the equation with *x*. To find the value of *x*
that gives the maximum value of C_{P}, we take the
derivative with respect to *x* and set it equal to zero. This
gives a maximum when *x* = 1/3. Maximum drawing power is then
obtained for *v*_{2} = *v*_{1} / 3,
and the ideal power coefficient is given by

Another wind turbine located too close behind would be driven only by slower air. Therefore, wind farms in the prevailing wind direction need a minimum distance of eight times the rotor diameter. The usual diameter of wind turbines is 50 m with an installed capacity of 1 MW and 126 m with a 5-MW wind turbine. The latter is mainly used off shore.

The installed capacity or rated power of a wind turbine corresponds to an electrical power output of a speed between 12 and 16 m/s, with optimal wind conditions. For safety reasons, the plant does not produce greater power at the high wind conditions than those for which it is designed. During storms, the plant is switched off. Throughout the year, a workload of 23% can be reached inland. This increases to 28% on the coast and 43% off-shore.

More details can be found in the Internet pages wind-works.org and in the pages of the American Wind Energy Association.

The installed capacity of wind power in the United States was in 2014 about 70 GW. This capacity is exceeded only by China (over 90 GW). The Alta Wind Energy Center in California is the largest wind farm in the United States with a capacity of 1.32 GW. The electricity produced from wind power in the United States amounted to about 180 TWh (terawatt-hours) in 2014, or 4.3% of all generated electrical energy. The U.S. Department of Energy’s report 20% Wind Energy by 2030 envisioned that wind power could supply 20% of all U.S. electricity, which included a contribution of 4% from offshore wind power. Detailed information about the present state in the US can be found in Wikipedia.

A crucial point about wind power is that the times of peak electricity demand and the times of optimal wind conditions rarely coincide. Thus, other electric power producers with short lead times and a well developed electricity distribution system are necessary to supplement wind power generation.

Why have the wind turbines of today lost one blade in comparison
to the old four-blade windmills? The rotor power P_{mech}=
2π *M n* is proportional to the torque *M* acting on
the shaft and the rotation frequency *n*. The latter is
influenced by the tip speed ratio λ, which is calculated according
to λ = *v*_{u} / *v*_{1} from the
ratio of peripheral speed (tip speed) *v*_{u} of the
rotor and the wind speed *v*_{1}. The torque *M*
increases with the number of blades. It is therefore largest for the
many-vaned Western mills, smaller for wind mills with four blades,
and smallest for today’s wind turbines with 3 blades. However, every
blade, as it rotates, reduces the wind speed for the following
blades. This "wind shadow" effect increases with the number of
blades. The optimal tip speed ratio is about one for the Western
mill, barely over 2 for the four-bladed type, and 7−8 for the
three-bladed rotors. At their optimal tip speed ratio, three-bladed
rotors achieve a c_{p} value of 48% and come closer to the
ideal value of 59% than wind turbines with 4 blades. For wind
turbines with two blades or weight-balanced one-bladed rotor
configurations, the yield is smaller in spite of a higher tip speed
ratio, because of the smaller torque *M*. Therefore, wind
turbines today have three blades.