Wind Turbines

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Can wind turbines actually be used to harness a substantial amount of power?

Big Ideas: 
  • There is kinetic energy in moving air (wind) which can be harnessed to produce useful power.

The total energy available in wind and the power that can feasibly be extracted from it will be determined using the fundamentals of kinematics. The limitations of location and machinery of wind turbines that restrict the amount of power that can be harnessed will also be examined, before comparing the impact of wind energy with various other common sources of energy.

How is wind energy created?

Wind energy has been proposed as an alternative energy source, although it is currently in an early stage of large scale development1. Windmills were seen in Persia as early as 500 AD and were used to grind grain and pump water2. It is only in modern times that humanity has attempted to extract raw power from wind in the form of electricity. The question to ask in this early developmental stage is whether or not it is possible to extract a useful amount of raw energy from the wind. To do so, we will consider the broad energetics of wind turbines to determine if harnessing wind energy could be a viable option given constraints of time, location and machinery. The first thing to consider is whether or not there is enough energy in wind to make it possible to extract a useful amount of power. It is important to note that although wind may possess a lot of kinetic energy, that is the energy resulting from the movement of masses, the rate at which this energy can be extracted limits the amount of useful power available, as power in its most rudimentary form is defined as the rate of doing useful work. To discover whether or not a useful amount of power is available, we must first discuss where the wind comes from and how much power is available in the wind. Wind energy ultimately comes from a series of energy transformations from solar energy (radiation) to wind energy (kinetic), where about 2% of the solar energy absorbed by the Earth goes into wind3. Solar radiation is absorbed by the surface of the Earth and heats it unevenly4. Different areas of the globe receive varying amounts of the incident solar intensity (W/m2) due to the angle of the Sun; the equator receives a greater percentage of solar intensity (and hence becomes hotter) than the poles4. Also, during the day the land heats up faster than the sea does, while at night the water retains heat longer than the land does4. Wind is a direct result of solar heating and the earth's rotation as they generate changes in temperature. As the air gets warmer, it rises and cooler air must rush in to take its place, producing wind! In all, location effects how much wind energy is available (Fig. 1).

Figure 1. A wind energy map of Canada showing the average power (in W/m2 of turbine cross-section area) that can theoretically be extracted from the wind5.

How much energy can be harnessed by wind?

The mean intercepted solar intensity at the top of the Earth's atmosphere is 350 W/m2. Given that 2% is converted to wind, this results in 7 W/m2 going into wind energy6. This wind energy is spread out over the Earth's atmosphere, with 35% of the energy (2.45 W/m2 of land area) dissipating in the first kilometre above Earth's surface6. Over a period of one year, the wind energy is approximately

<br />
\begin{eqnarray}<br />
   \textnormal{wind energy} &=& \textnormal{(intensity)(Earth's surface area)(seconds per year)}\nonumber \\<br />
   &=& (2.45 \textnormal{ W/m}^2)(5.1 \times 10^{14} \textnormal{ m}^2)(3.2 \times 10^7 \textnormal{ s})\nonumber \\<br />
   &=& 4.0 \times 10^{22} \textnormal{ J}\nonumber<br />
   \end{eqnarray}<br />

which is 200 times larger than our energy consumption on Earth, estimated to be 2 x 1020 J6. Now we can calculate the energy and power harnessed from the wind. To use the basic equation for kinetic energy, KE = ½mv2, we will need the rate at which air passes through the rotor of the wind turbine. The rotor is made up of the blades that spin on a wind turbine, and the total area of the rotor can be approximated by a circle as this is the area swept by the blades. To do this, we can imagine we are holding a hoop up in the air, and measure the mass of air travelling through the hoop in time Δt (Fig. 2).

Figure 2. At time t = 0, the mass of air is just about to pass through the hoop, but Δt later, the mass of air has passed through the hoop. The mass of this piece of air is the product of its density ρ, area A, and length vΔt.

From this, we can see that the mass is

<br />
\begin{eqnarray}<br />
   \textnormal{mass} &=& \textnormal{density} \times \textnormal{volume} \nonumber \\<br />
   &=& \rho A v \Delta t \nonumber<br />
   \end{eqnarray}<br />

where ρ is the density of the air (1.2 kg/m3 for standard atmospheric pressure (1 atm) and temperature (0°C) at sea level), v is the velocity of the air and Δt is the length of time for a unit of air to pass through the loop7. The area A is the area swept by the blades, not the blade area. This is because the blade moves much faster than the air and so each particle of air is affected by the blade. Therefore, the kinetic energy is found to be

<br />
\begin{eqnarray}<br />
   K &=& \dfrac{1}{2} m v^2 \nonumber \\<br />
   &=& \dfrac{1}{2} \rho A t v^3 \nonumber<br />
   \end{eqnarray}<br />

while the power of the wind passing through our hoop is

<br />
\begin{eqnarray}<br />
   P &=& \dfrac {\dfrac{1}{2} \rho A t v^3}{t} \nonumber \\<br />
   &=& \dfrac{1}{2} \rho A v^3 \nonumber<br />
   \end{eqnarray}<br />

But this is not the actual power produced by the turbine as turbines can't extract all of the kinetic energy of the wind. Why not? If this was the case the air would stop as soon as it passed through the blades and no other wind would be able to pass through. An analysis by Betz (1919) shows that you cannot capture any more than c.60% of the wind's energy8. In addition, there are also small losses due to friction and turbulence. So ideally you want the turbine to slow the wind down by 2/3 of its original speed (as the maximum of P/P0 = 0.593 is found at v2/v1 ≈1/3). For more information, click here. The power produced by one turbine is found by

<br />
\begin{eqnarray}<br />
   \textnormal{Power} &=& \textnormal{(efficiency)(power)} \nonumber \\<br />
   P &=& \dfrac{1}{2} \eta \rho A v^3 \nonumber \\<br />
   &=& \dfrac{1}{2} \eta \rho v^3 \pi r^2 \nonumber<br />
   \end{eqnarray}<br />

where d is the diameter of the circle covered by the rotor. The v3 term found here emphasizes the need to have a high wind speed in order to capture an useful amount of power. What we have just derived is based on a single wind turbine in constant wind conditions. In real life, however, wind conditions change. So what local conditions must be satisfied in order to make the use of wind turbines feasible? The location of the wind turbine must be carefully selected. Wind turbines only work efficiently when wind moves uniformly in the same direction. Turbulence, the unsteady flow caused by buildings, trees, and land formations, causes an inconsistent air flow which makes harnessing power inefficient and places increased stress on the rotor. The edge of a continental shelf, high ground and tundra are the best locations to build a turbine9. This is because their geography lacks any large obstructions that may create turbulence. Local wind is also an important factor, and should be, on average, at least 7 m/s at 25 m above the Earth's surface in order to make harnessing wind from it worthwhile3. One must also keep demand and dependability in mind when considering the extraction of energy from the wind. First of all, since wind is not locally predictable in the short term, the use of wind energy should be limited to only fulfil 5 - 15% of the total energy demand of the area3. To overcome this problem and make wind energy more reliable, turbines need to be set up in many different locations so that the power available averages out3. In other words, on one day one of three of the locations may not have enough wind to operate, but the next day, that location may be in operation while the other two locations are not. Despite the local wind patterns, however, globally there is always a relatively constant amount of wind energy being harnessed at any one moment.

How do wind turbines work?

The machinery of a wind turbine also has its limitation on how much power can be extracted from wind. To begin, let us cement some of the terminology used in describing the structure of wind turbines, before looking at the actual mechanics of converting wind energy into electricity.

Figure 4. A turbine is composed of a foundation, a tower, a nacelle and a rotor consisting of 3 blades.

Principle components of a wind turbine unit are a foundation, a transformer, a tower, a rotor and a nacelle (Fig. 4). The wind turns the rotor, which turns the generator to produce electricity. The electricity is then transmitted to a transformer at the base of the tower before going to a substation. To maximize the power extracted, the nacelle, which connects the rotor to the tower and houses the generator, can be rotated into the direction of the wind. Rotors' diameters range from 27 m for a 225 kW generator to 80 m for a 2500 kW generator, and depend on the desired power output, location limitations et cetera (Fig. 5)8. A 1 MW turbine has a rotor diameter of 54 m, a tower standing 80m tall, and works in wind speeds ranging from 3 - 25 m/s (10 - 90 km/h)7.

Figure 5. The dimensions and characteristics of a typical smaller sized turbine.

The power produced by a wind turbine depends on rotor area, air density, wind speed, and wind shear. Air density increases with colder temperatures, decreased altitude, and decreased humidity. The molar mass of air (29 g/mol) is greater than the molar mass of water (18 g/mol) and so the less moisture in the air, the denser the air is. Wind shear is a difference in wind speed and direction over a short distance and is caused by mountains, coastlines and weather patterns8. Wind speed increases the farther you get away from the ground (Fig. 6)7. To maximize the power output of wind turbines, rotors are tilted slightly upwards. Why do you think this is?

Figure 6. As you get higher off the ground, the air speed increases, corresponding to a longer arrow. The rotors are tilted slightly upwards so that each part of the rotor is exposed to the same speed.

While it may be possible to use a single wind turbine for personal energy demands, entire cities and countries need huge wind farms to satisfy their energy needs. To optimize energy production in a wind farm, turbines are spread 5 - 9 rotor diameters apart in the prevailing wind direction and 3 - 5 rotor diameters apart in the perpendicular direction (Fig. 7)7.

Figure 7. On a wind farm, turbines must be spaced out enough so that they do not interfere with each other. As the wind passes through the turbine it slows down, and so there is no point in putting a turbine in the region where the air is guaranteed to be slow. One common way of spacing them out is ensuring there is at least 5 rotor diameters between each turbine.

When the turbines are placed on a square grid, the power per unit land area is

<br />
\begin{eqnarray}<br />
   \dfrac{\textnormal{power}}{\textnormal{land area}} &=& \dfrac{ \dfrac{1}{8} \eta \rho v^3 \pi d^2} {(nd)^2} \nonumber \\<br />
   &=& \dfrac{ \dfrac{1}{8} \eta \rho v^3 \pi}{n^2} \nonumber<br />
   \end{eqnarray}<br />

where n is the number of turbine diameters between turbines. The average power of a wind turbine farm is the product of the capacity of the farm and the fraction of the time when the wind conditions are near optimal. The capacity factor is usually around 15 - 30%7.

How does wind energy compare to other energy production alternatives?

Now that it is established that wind is a possible source of power, the benefits and drawbacks need to be considered. Why use wind power in lieu of other energy sources? The harnessing of wind power does not produce hazardous wastes, use non-renewable resources or cause significant amounts of damage to the environment1. Some CO2 is produced in the manufacturing of the turbines, but as demonstrated in the problem set, it is much less than the emissions from burning an energy-equivalent amount of coal or natural gas. In addition, the use of wind power can reduce hidden costs such as those related to pollution and the consequential healthcare impacts, and in the longer term, climate change1. Also, wind turbines use less space than traditional power stations, because you can farm around them, so they can be built without extensively reducing agricultural land and locally-owned wind farms create income for communities1. So why, in light of these positive elements, is there so much resistance against wind turbines? Arguments against include the following fears of damages from collapsing turbines, noise, a less attractive skyline, an unreliable power source, unnecessarily high bird fatality, and significantly modifying the Earth's wind patterns. First of all, the noise of a typical turbine is 45 dB at 250 m away10. This level is lower than the background noise at an office or a home11. Secondly, the reliability of wind energy increases depending on location and how many farms are operating in a variety of sites within the area. In regards to bird deaths, in the US less than 40,000 are said to die from turbine blades while hundreds of millions are said to die from domestic cats12!  Finally, in regards to modifying the Earth's climate, it is plausible that one would see local climate change surrounding areas with a high concentration of wind farms, but the large-scale climatic effects will likely be negligible13. In addition, since wind turbines will be replacing coal-fired power plants, if anything, we anticipate a considerable reduction in CO2 emissions. While currently wind turbines are the accepted machinery used to extract wind energy, technology can change quite easily as people discover new and more efficient ways to harness power. Currently, wind cells are being investigated by Accio Energy, which takes a completely different approach to the method of extracting wind energy than the tradition wind turbine14.

Above:Making full use of the wind: Wolfe Island Wind Farm, Lake Ontario

Resources
Lecture Notes: 
Wind Turbines Lecture Notes
Take Home Experiment: 
Wind Turbines Take-Home Experiment
Multiple Choice Problems: 
Wind Turbines Multiple Choice Questions

Comments

The area swept by the blades

The area swept by the blades is πr^2=πd^2/4.

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