# How effective and efficient are wind turbines

## Wind power plants

### From the windmill to the wind turbine

Over a thousand years ago, windmills were in operation in the Persian region and in China, see Book of Synergy. In the 12th century, the post mill was introduced in Europe, which could be turned in the direction of the wind. In the 16th century the cap windmill was added, which was also called the Dutch windmill. For areas with less wind in America, the wind turbine known as the Western Mill was introduced around 1854 with a large number of blades to drive pumps. After that, however, windmills were increasingly replaced by steam engines and combustion engines. Of the approximately 200,000 mills that existed in Europe in the middle of the 19th century, only one in ten was preserved after a hundred years. The specimens that have survived to this day hardly ever grind or pump. But the number of wind turbines has been growing steadily since the end of the last century and has taken on an important role in electricity generation in many countries.

### Physics of the wind turbine

For all wind turbines, the wind power is proportional to the third power of the wind speed, as we will now show. Wind energy is the kinetic energy of moving air. The kinetic energy of a mass m with speed v is

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

determine. With that we get

We consider a small period of time Δtin which the air particles make their way s = v Δt flow through. Let's multiply the distance with the rotor area A. of the wind turbine results in a volume of

that drives the wind turbine during this small period of time. The result is the wind power

The wind power increases with the third power of the wind speed. In other words: a doubling of the wind speed results in eight times the wind power. Choosing a "windy" location is therefore very important for a wind turbine.

The effectively usable wind power is less than indicated by the above equation. The wind speed behind the wind turbine cannot go to zero, since then no air could flow in. So only part of the kinetic energy can be used. We consider the following picture:

The wind speed in front of the wind turbine is greater than that after it. Because the flow must be continuous, the area is A.2 after the wind turbine larger than the area A.1 before. The effective power is the difference between the wind power:

If the difference between the two speeds is zero, we have no useful power. If the difference is too great, the air flow through the rotor will be hindered too much. The performance coefficient cp characterizes the relative power consumption:

When deriving the above equation from the equations above, it was assumed that A.1v1 = A.2v2 = A (v1+v2) / 2 is. The relationship v2/v1 we have on the right side of the above equation with x designated.

An extreme value consideration of this equation (zeroing the first derivative according to x ) results for x = 1/3 a maximum.

Maximum power is drawn from v2 = v1 / 3. The ideal coefficient of performance at this ratio is

Another wind turbine located too close behind a wind turbine would only be driven by the slower air. For this reason, a minimum distance of eight times the rotor diameter or at least four times the distance perpendicular to it must be observed in the main wind direction in wind farms. The usual diameters of wind turbines are 50 m with an installed capacity of 1 MW and 126 m with a 5 MW wind turbine. The latter is mainly used in the open sea (off shore) used.

The installed power or nominal power of a wind power plant corresponds to the electrical power output at the nominal speed between 12 and 16 m / s, i.e. under optimal wind conditions. At higher wind speeds, for which the system is still designed, no greater power is generated for safety reasons. The systems are switched off in the event of a storm. Inland, an annual average occupancy rate of 23% can be achieved. This value increases to 28% on the coast and to 43% for off-shore systems.

At the end it will be explained why the wind turbines have lost a wing compared to the windmills. The mechanical rotor performance Pmech = 2π Mn is proportional to the torque acting on the shaft M. and to the speed n. The latter is determined by the high speed number λ affects that according to λ = vu / v from the ratio of peripheral speed (blade tip speed)vu of the rotor and the wind speed v1 calculated.

Now the torque is growing M. with the number of wings. It is therefore the largest for the multi-wing western mill and larger for the four-winged than for the three-winged. With increasing speed, each wing reduces the amount of wind available for the following wing in the direction of rotation. This "slipstream" has a stronger effect, the more blades a rotor has. The optimal high speed number is therefore only about one for the Western Mill, hardly exceeds two for the four-blade windmill and is 7 to 8 for the three-blade rotors. These achieve a value of at their optimal high speed number cp = 48% and thus come to the above-mentioned ideal power coefficient of the wind energy yield of cp = 59% closer than four-winged aircraft. For two-wing aircraft or single-wing aircraft balanced with weights, the yield is also lower because of the lower torque, despite the higher high-speed speeds. That is why wind turbines have three blades.

### Bush wind

Wilhelm Busch described the economic problems involved in operating a wind turbine as follows:

The miller looks out of the mill,
who wants to grind so much,
the wind becomes quieter and quieter
and the mill stands still.
It’s always the way I think
exclaims the miller in anger,
if you have corn so it is missing on the wind,
if there is wind, there is no grain.

For the wind turbines, the peak times of the electricity demand take over the role of the grain for the miller. Since peak times and optimal wind conditions cannot be correlated, other electrical energy producers with short start-up times have to supplement wind power generation in a distribution network equipped for this purpose.

Incidentally, Wilhelm Busch did not paint the right picture for the above verse. It is the illustration of a verse (Busch: Der Bauer und der Windmüller) about conflicts between operators and residents of wind farms at the time.

### Wind statistics

The Annual Wind Report of the Global Wind Energy Council, GWEC, shows that around 30,000 wind energy plants (WEA, or WT = wind turbines) with a total output of around 61 GW were installed in Germany in 2019. Offshore plants accounted for a good 7 GW. In the case of offshore systems, the difficult connection to the consumer network via submarine cables is often the reason for a longer period of time between installation of the system and connection to the network. New offshore turbines have a mean rotor diameter of 155 m in 2019. The mean value of the turbines newly built on land in 2019 has a diameter of 122 m, a hub height of 133 m and an output per turbine of 3.3 MW. The average power at sea is around 7 MW. Until 2008, Germany was the world leader in terms of installed capacity, but was then overtaken by the USA (around 135 GW in 2019) and China (around 236 GW in 2019), see GWEC. Germany only leads the way when it comes to wind power per capita. The contribution of wind energy to gross electricity consumption in Germany was around 22% in 2019, while the world average is still below 3%.

The Bundesverband WindEnergie and DEWI (UL International GmbH, former German Wind Energy Institute) offer up-to-date information on wind energy. Technical details can be found in books called Wind Power Plants or Wind Turbines.

Last change: 04/03/2020