Chapter 26 The wind-powered generator

 

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The wind-powered generator

          Overview

          Greenbird, the speed record holder for land yachts

          The thermodynamic process

          The big picture

          The wind available to drive a wind-powered turbine

          The stagnation pressure of the wind

Types of wind turbine

The horizontal axis wind turbine

          The basic arrangement of a wind turbine

          Energy transfer to the wind turbine

          Modelling the velocity diagrams

          The aerofoil rotor

          The wind turbine as a thermodynamic device

          Feathering

Conclusion

Appendix 1

Appendix 2


Overview

In order for human beings to continue to exist it is necessary for them to keep warm and to cook. Both requirements need heat. In the long ago stages of human affairs the number of people was low enough for wood to be burnt without causing deforestation. As the population increased and wood was seen to be too useful to burn, it became necessary to find an alternative fuel and we found coal first, then oil, and later gas. These finds permitted a population explosion that resulted in an ever-increasing demand for fossil fuel not only to keep warm and cook but also to permit personal mobility and transport of goods.

 

Burning fossil fuels has released carbon dioxide into the atmosphere in sufficient quantities to change the heat balance of the whole planet irreversibly. It is all too evident that we need to find sources of heat and power that are renewable and to avoid adding to the burden of carbon dioxide and methane in the atmosphere. We have searched in vain. We can see that energy in small quantities is available from water power, solar power, wave power and wind power but none of them are remotely practical to provide sufficient power reliably and continuously. They are all very expensive to exploit when compared with fossil fuel. Only nuclear power produces heat that can be used to generate electricity. This is a catastrophic situation to be in but is an outcome of evolution which is mindless.

 

Politicians, who seem not to understand the true magnitude of our problem, are expected to do something about this when nothing can be done. So they choose to make token attempts to solve the problem. Of the possible sources of renewable power, wind-power is the most visible and the most easily installed but is far from ideal. It is appropriate to examine the wind-powered turbine (wind turbine or windmill for short) to see just what is possible and to assess its limitations. It is all that we have really.

 

It goes without saying that the function of a wind turbine is to extract energy from the atmosphere. This means that we need to have some idea of what is possible and how the extraction process actually works.

 

I start with the wing-sailed land yacht because, as I was writing this chapter, it seemed to me that, in the horizontal-axis, wind-turbine, I was describing a modus operandi that I had not encountered before and that it was difficult to explain. Then I recalled the wing-sailed land yacht that had set a new speed record that works in the same way. It is much easier to analyse than the wind-turbine and I give my analysis of that as an introduction to wind-turbines.

 

Text Box:  
Fig 26-1
Greenbird, the speed record holder for land yachts

This land yacht[1] works in the same way as the wind turbine. Both use aerofoils interacting with the natural wind to drive them, the land yacht goes in a straight line, the rotor of a wind turbine rotates. It is much easier to understand the land yacht than the wind-turbine.

 

Figure 26-1 is of the holder of the world land speed record at 126 mph in a wind gusting from 30 to 50 mph[2] (13 to 22 m/s). The yacht is driven by a rigid wing that is about twice as tall as the part-wing shown here. The wing is free to rotate about a vertical axis and it is controlled by the green stabiliser attached to the wing on a boom that also extends forward of the wing to carry a balance weight. The stabiliser, that is just like the stabiliser of an aeroplane, has a moving control surface like the elevator of an aeroplane. The elevator is controlled by the pilot and sets the angle of attack of the wing to the relative velocity of the wind. The relative velocity is, of course, the vector sum of the wind speed and the yacht speed.

 

The object of the provision for the wing to rotate under control of the stabiliser is to let the wing respond automatically to the swinging of the wind that is inevitable if the wind is gusting.

 

The body of the yacht runs on three wheels, one, at least, in the body, a steerable one in the streamlined pod on a boom placed forward and the third in the streamlined pod on the long outrigger. The yacht is designed to sail on the port tack and the outboard wheel resists the heeling force on the wing.

 

Film of the yacht during the record run shows considerable random bending of the outrigger boom and of the wing. The wind was gusting and inevitably veering and backing. But despite the fact that the stabiliser is two or three chord lengths aft of the wing it appears to control the attitude of the wing quite well.

 

We can analyse the wing/stabiliser system to find out its characteristics.

 

This wing has a very high aspect-ratio and presumably will have been made to a high standard of accuracy. The high aspect-ratio ensures that span-wise flow at the tips will have little affect on the measured data for the lift of the aerofoil in two-dimensional flow and the high standard of accuracy will keep drag to a minimum. The force generated on the wing will be very nearly equal to the aerodynamic lift and act almost at right angles to the relative velocity.

 

Before we can proceed we have to consider the direction of the yacht relative to the direction of the wind. There is no reason why we should not devise a model that allows for all points of sailing but I can restrict the sheer scale of the exploration by working only for the yacht sailing at right angles to the true wind. That is reaching. This would give a model that would be applicable to the rotor of a wind turbine.

 

If we suppose that the stabiliser does indeed maintain a constant angle of attack, the force on the wing will depend on the wind speed and the speed of the yacht and, for a given yacht speed, fluctuate only with the speed of the wind.

 

If we ignore the fluctuation in the speed of the wind the force will vary with the speed of the yacht.

 

We can calculate the force using :-  force =  where  is the coefficient of lift,  is the density of air,  are the velocities of the yacht and the wind and  is the square of the relative velocity of the air flowing over the wing.  is the area of the wing.

 

Figure 26-2 shows all this diagrammatically. It also shows that :-

Text Box:  
Fig 26-2
                             the driving force = force on wing × .

 

This is all that is needed to plot graphs and explore this mechanism.

 

I will do so using a wing area of 50 square metres, a range of  from 10 m/s to 30 m/s and a range of yacht speed of 3 to 80 m/s.

 

Figure 26-3 shows the outcome. The traces are for wind speeds of 10, 15, 20, 25 and 30 m/s. The wind speed for the record was 13 gusting to 22 m/s so the relevant traces are green and dark blue. I have added a marker at 56 m/s which is the record speed.

 

These graphs are telling us that this wing can generate, by interacting with the wind, a force that drives the land yacht to increase the speed that increases the force and this increases the speed and so on indefinitely. It seems too good to be true yet the Greenbird contrived to sail at 126 mph in a wind of 30 mph so it is a viable mechanism.

Text Box:  
Fig 26-3

The limit of speed comes when the force available to drive the yacht is equal to the sum of the rolling resistance and wind resistance of the yacht. If we suppose that this total resistance varies with the square of the speed of the yacht a graph can be plotted of the resistance. I have added a plot of  in black where 0.008 happens to give a suitable plot. This graph cuts across the other graphs at a useful angle so that the top speed will be well defined.

 

Greenbird is to be tested on a frozen lake where the ice, if it is smooth, will offer less drag and there is no physical reason why it should not go faster still.

 

Text Box:  
Fig 26-4
The exploitation of the aerofoil in this way has few applications, indeed the wing-sailed land yacht and the wind turbine are the only ones that I can think of. Its success depends on being able to take advantage of the very low drag of high quality aerofoils at small angles of attack.

 

In this range a good aerofoil can have a  ratio of 100 but aberrations that seem to be small can drop this ratio to 10.

 

Figure 26-2 was drawn for a  ratio of about 20 and we can draw another similar diagram, figure 26-4, to show the effect of a change from having zero drag to having a value of  of 10. The driving force is very nearly halved. It will get worse as the speed of the yacht increases.

 

If, to make a mechanism feasible, aerofoils must be made to high standard they must be protected from damage and anyone wanting to set speed records will look after the wing with great care. Even so, insect debris alone can destroy the operation of a wing as an aerofoil. In the case of wind turbines the blades cannot be protected against erosion from rain and hail nor from ice or sea salt.

 

The aerofoil used in this way on the Greenbird must not stall or the drag will be prohibitively high. The use of a swivelling wing controlled by the stabiliser is all that makes this mechanism possible.

 

The mechanism is interesting but is not really an every day workable device. Somehow it must be adapted for use in the design of wind turbines. It is not used for record breaking on water-born yachts.

 

The wind-powered generator

The atmosphere and the natural wind

We must start with the character of the air movement on a large scale.

 

Text Box:  
Fig 26-5
Figure 26-5 is a synoptic chart for the North Atlantic and Europe. The lines on the chart are isobars, which are lines joining points in the atmosphere at the same pressure. The most obvious feature of these isobars is that there are several groups of isobars that form closed loops. We are looking at circulating systems that are really distorted vortices that have formed naturally in the atmosphere and, as in all vortices, the pressure in the centre is lower than that at the outer regions of the vortex. There is a nonstop process of generating these circulating systems from incident radiation from the Sun. This process is the result of non-uniform heating of the atmosphere by the Sun to create density differences and buoyant forces in the air to sustain air movement that becomes organised as vortices which then move in a higgledy-piggledy way from West to East through the temperate region. The much-distorted vortices become circulating systems containing kinetic energy that, in the case of the flow across the North Atlantic, is created over the sea off the coast of North America and is ultimately dissipated in Eastern Europe.

 

It should be noted that even though there are pressure variations in the atmosphere caused by the Sun the atmospheric pressure is produced by gravity acting on the mass of the gases in the atmosphere and is fundamentally constant and uniform at any given altitude. The circulating systems are large, typically 2,000 metres in extent, and this is very large when compared with a wind generator. The atmospheric pressure of the air during its interaction with a wind generator is essentially constant and only affected in the regions that are close to the blades and in places where the flow is organised into a vortex.

 

Before I leave the isobaric chart I note that the chart makes no attempt to present the small scale details of the wind. The black lines show the general direction of the wind and where the lines are close together the speed of the wind is high. The wind speed is low where the lines are far apart. In this chart the wind is very light over Spain so Spanish wind-powered generators will not be producing much power if any. Nor will much power be generated from the wind over Eastern Europe where no doubt it is cold.

 

It is clear from the outset that, if we are to use the wind as the driving force of a device to extract energy from the atmosphere, the sheer variation of the wind speed is not at all what one would choose to use. All one can say is that everywhere there will be opportunities to gather power from the atmosphere but not continuously. This is at odds with our dependence as a civilisation on a constant supply of electrical power. Wind was not ideal for the wind-powered flour mills of old and nothing has changed. We must hope that we can find a way to operate our fossil-fuel power stations to utilise the power generated by the wind and, at the same time, give a steady supply of electricity.

 

The thermodynamic process

The wind powered generator is really an engine extracting energy from the internal energy of the atmosphere and utilising the kinetic energy of the wind to do so. A thermodynamic process is one where heat is converted to work. This needs explanation in terms that make sense to an engineer. We need to know what is normally meant by heat and by work.

 

The only practical way we know of running a thermodynamic process is to store heat in the molecular structure of air or in water in it gaseous phase, that is as steam, and then contrive to extract some of that heat as work. We think up ways of making the gas or steam exert a force on a rigid but moveable surface and let the surface move so that the force does work. During this process the gas or steam gives up some of its internal energy, that is, it gives up some heat. Heat is converted to work.

 

In the wind turbine we have the gas, atmospheric air, and it already moves as a result of a pre-existing thermodynamic process, we have the rigid moveable surface, the rotor. The net result of the interaction between the air and rotor is the creation of electrical power and a drop in temperature of the air together with the creation of angular kinetic energy to add to the existing linear kinetic energy of the wind. Our interest as mechanical engineers must be in the interaction between the air and the rotor to find out how to make it work as well as it can. The analysis of the wing-sailed land yacht gave us a very useful start.

 

If the speed of the wind is the agent by which this thermodynamic process takes place we should examine the wind.

 

The wind available to drive a wind turbine.

Wind turbines are built either on land or at sea. The wind farms built at sea are not normally a long way from the coast because they cannot yet be built in deep water.

 

They all utilise wind that is influenced either directly or indirectly by their proximity to the land but the character of the wind over the sea is generally more suitable for operating wind turbines even if the environment is very harsh for an unsupervised mechanical device to operate for a protracted period.

 

There is no doubt that the natural wind that passes any given point can change its speed and change its direction in what seems like a cyclic way. The only simple explanation for this behaviour is that the wind contains eddies of significant size when compared with a turbine rotor but very small when compared with the circulating weather system containing them.

 

Sometimes, when moderate winds blow for long distances over the sea, the flow of air seems to be free from eddies which suggests that eddies do not form easily in an unobstructed flow. Eddies form when wind blows over the general contours of the land and, to a quite different scale, round isolated obstructions like buildings and trees. They also form when wind blows over waves that move obliquely relative to the wind. Thermals can easily rise through a steady wind to form thermal streets of clouds and create disturbance. Density differences caused by solar heating of the land produce onshore and later offshore winds. These effects disturb the wind and create angular momentum that persists and flowing fluid is very good at gathering angular momentum into organised, much larger, rotating systems, ie eddies.

 

Eddies may have axes in any orientation but it is likely that those rotating about horizontal axes lose their energy in friction with the ground and die away whilst those that rotate about a more or less vertical axis are very persistent because they have no asymmetrical contact with the ground. Eddies with angled axes tend to become less vigorous than eddies with a vertical axis. This rotating flow is very common in nature and once formed is persistent because the flow pattern produces very little internal loss of energy.[1] These are the eddies that produce gusting and veering,

 

So the wind that approaches a wind turbine will not be steady in speed or direction and changes take place in a relatively short time. It is a very difficult challenge to engineers to use this wind as an agent to extract energy from the atmospheric air and it is not a challenge that will be met wholly by theorising but by a combination of the application of science and a steady learning process by making and trying turbines. Unfortunately the whole problem is exacerbated by the difficulty in getting approval for the installation of wind turbines because of cost, aesthetics and noise and this provides the incentive to build ever larger turbines to make the most of an approved site. Unhappily a large turbine is more vulnerable to exceptional weather events but, it turns out that the mechanics become ever more difficult as the size increases.

 

The stagnation pressure of the wind

If we are to use the wind speed as the driving agent for the rotor of our wind turbine we need to think in terms of the stagnation pressure as the maximum pressure that can be exerted anywhere on the blades of the rotor. The stagnation pressure is, of course, the value of  Where  is the velocity of the air relative to the blade. Some fraction of this pressure exerted on the area of the rotor produces the aerodynamic force that is exerted on the blades. The blades are shaped like aerofoils and are intended to behave as aerofoils and we can use the coefficients of lift and drag that are readily available for aerofoils. The blades will work in air moving at several times the speed of the wind.

 

Text Box:  
Fig 26-6
We shall need to know the range of speed of the natural wind. Most people think of the wind in terms of descriptive terms like "gale force" and "light breeze" and these can be related to wind speed in knots or m/s using the Beaufort scale. I give it in figure 26-6.

 

The range of speed for this table is up to 60 knots which is in the violent storm range but, exceptionally, it could be much higher at over 200 knots in gusts.

 

None of us is unaware of the wind and we all know that speeds at gale force and above are troublesome to everything exposed to them because the forces produced by the wind become large and, in the term used by weather forecasters, destructive.

 

The wind turbine is a large exposed structure and somewhere around 30 knots, or 15 m/s there is a separation point between usable winds from 0 to 15 m/s and troublesome and then impossible winds at higher speeds. The most commonly occurring winds are about 7 metres/second.

 

Text Box:  
Fig 26-7
It is pertinent to gain some idea of the magnitude of the stagnation pressure produced by the wind. I have given it in figure 26-7. The magnitude of the stagnation pressure produced directly by the useable range of wind speed is trifling at about 1/200 of an atmosphere but this is not significant. What is significant is that the blades can have tip velocities of 60 m/s and the speed of the blade relative to the wind might be much more than 100 m/s.

 

 

 

 

 

 

 

 

 

 

 

Types of wind turbine

If wind-turbines are to be used as a source of power for the electricity grid then they must be large just to get large areas of blade on which the wind can create a force.

 

Vertical axis turbines do not scale up, nor do Savonious rotors or outback wind pumps. The only type of wind-turbine that can possibly be practical is the horizontal-axis turbine with three blades or possibly more.

 

I will concentrate on that device.

 

The horizontal-axis wind turbine

Figure 26-8 shows an offshore wind turbine and figure 26-9, from Wikipedia, shows blades in transit. The working assembly is carried on a tower that is most probably sunk into a hole in the sea bed. The tower is hollow with an internal ladder for access to the pod for maintenance. The cabling is inside the tower.

 

The pod must house the alternator that is connected to the rotor through an epicyclic gear box and one shaft of the gear box forms the axle of the rotor. A brake is necessary to stop the rotor in high winds and in winds that are too low to generate power and to stop the rotor for maintenance. There must be a yaw mechanism to turn the whole pod and rotor assembly to keep the rotor facing the wind. There must be a mechanism to feather the blades to suit the wind. Finally there must be a turbine management system to make all these separate systems work together to operate the turbine safely. This latter may be in the tower.

 

The blades are hollow and made of glass fibre or other composite. When the rotor is stationary the forces on them are low compared with the forces on an aeroplane wing. When the turbine is running the forces are much larger but the blades are under tension radially to provide the essential centripetal force and this force limits the amount that the blades bend under the wind load. However the blades are flexible and the uncertain balance between the radial forces and the axial forces leads to "flapping" in the axial direction in response to the random motion in the wind. Fortunately this flapping tends to damp any oscillation and not lead to vibration.

 

Of these several components the gear box is most vulnerable. Epicyclic gears boxes have two inline shafts that are linked by helical gears. For longevity the oil film between the gears must not break down. The short shaft of the gear box is subjected to random bending forces coming from the blades that operate in a wind with random velocity components. It is proving to be difficult to prevent these forces from distorting the gear box and putting the oil film at risk.

 

Text Box:  
Fig 26-10
The greatest working wind speed for these turbines is around 25 m/s that would be reached in two stages. The first stage would be up to about 15 m/s during which the output increases with wind speed and the second stage from 15 to 25 m/s during which the blade angles are adjusted to give a constant power output. The turbine is shut down for higher wind speeds.

 

There is also a lowest speed for which the turbine can generate power. That speed is about 3.5 m/s.

 

This can be gathered together in a graph as in figure 26-10

 

Energy transfer to the wind turbine

The energy transfer between the air and the rotor of a wind turbine is achieved using aerofoil sections with every effort made to preserve attached flow so that they operate aerodynamically. The blades are designed to be long and narrow and are normally tapered as is very evident in figure 26-9. They are twisted so that, at the "rated" condition, that is the design condition, of speed of rotation, wind speed and output power the blades at every radius make the same angle of attack with the wind.

Model

Rated

 power

Blade

length

Speed

 Range

Maximum

 tip speed

Rated

 wind speed

Vestas V100

2.75 MW

50 m
(164 ft)

7.2-15.3

rpm

179 mph

15 m/s
(34 mph)

 

 

 

 

 

 

 

We need to see how this blade actually works. There is growing evidence that turbines having diameters of about 100 metres are more reliable and longer lasting than those up to 150 metres. I shall work with a typical turbine of 100 metres diameter and extend it into the range up to 150 metres. Such a turbine is the Vestas W100 for which the details are given in the table above.

 

Text Box:  
Fig 26-11
The turbine will be rated at 7.2 rpm. Using this data a velocity diagram can be drawn to scale to find the speed of the wind relative to the blade at the tip.

 

It is given in figure 26-11 I have used the same notation as was used for water turbines. The blade speed  is 37.6 m/s and the speed relative to the blade  is 40.5 m/s. So for this turbine rotor at its rated condition the speed of the wind interacting with the rotor blade at the tips is 40.5 m/s or 3.37 times the wind speed and acts at an angle of 21.75°. This means that the stagnation pressure for the air flowing over the blade at the tip is about 0.01 bar compared with the stagnation pressure of about 0.0014 bar for the free stream.

 

If this relative wind is to produce a force on the blade element it must have an angle of attack  and I have shown an aerofoil at 8° to the wind. As a result of the flow over a short element of the blade at the tip, a force will be exerted on the element of the blade as shown in figure 26-11. That force has two components, one tangential, that will help to drive the blade, and one axial, that will tend to bend the blade but will do no work. The vector diagram looks exactly like the diagram for the land yacht.

 

It is now evident how the force to drive the blade element at the tip is created but suppose that the blade length were to be increased to 75 metres.

 

Figure 26-12 is a repeat of figure 26-11 for this larger rotor. The wind speed is unchanged at 15 m/s, the speed of rotation is 7.2 rpm and the blade length to 75 metres.

Text Box:  
Fig 26-12

The angle  between  is 14.9° and, in a wind of 15 m/s (30 knots, near gale force), the speed of the air relative to the blade is now 58 m/s (110 knots). This is the speed relative to the blade of the air that will interact with the blade to produce an aerodynamic force and the stagnation pressure is much higher at 0.025 bar. The tangential force that drives the blade is about 1.4 times that of the smaller turbine when the ratio of the stagnation pressures is about 2. This arises because of the change in the angle  of the blade. But that force moves much further and does more work in a given time.

 

However there is a problem. When I drew these two diagrams I had to choose a typical angle for the force on the blade element. Aerofoil data shows that, if the flow is attached, the force will be a degree or so away from the normal to the direction of the relative velocity. I also chose the angle of attack from a knowledge of aerofoil data. For the large turbine there is only an angle of 6.9° between the chord line and the plane of the rotor. As aerofoils work by diverting the flow over them and we need to know whether this small angle is adequate for the flow to be attached given the fact that the wind veers and gusts. We need to know more about the behaviour of aerofoils and the air that flows over them.

 

Text Box:  
Fig 26-13
Figure 26-13 shows a symmetrical section in a wind tunnel. The smoke traces are standard in shape but, in this case, there are blips on the traces that were inserted by jogging the smoke rake at regular intervals. They start off in line across the tunnel. We can see that the blips in the traces passing over the aerofoil move much faster than those going under the aerofoil and are way ahead when they cross the trailing edge.

 

We can see how the force is created. In the regions marked A and D the pressure of the flowing air is below the free stream pressure. In the regions marked B and C the pressure is either around or above the free stream pressure. These regions have been created by the air flowing over the aerofoil and the faster the undisturbed flow the greater are the pressure differences between the free stream pressure and the pressures in these regions. As a result of this pressure distribution a force is exerted on the aerofoil. It will be upwards and it will incline by a small angle backwards.

 

Now we must look to see what the aerofoil does to the flow of air over it. When a stream of air flows over an aerofoil set at a suitable angle to the direction of flow of the air a force is exerted on the aerofoil and Newton's concept of momentum shows us that the air must acquire momentum in the opposite direction to the force. Newton gives us the fact that the force is equal to the increase in momentum per second in the opposite direction to the force. For this to happen the aerofoil must divert the flow of air but it will not be a case of one-dimensional flow to and from the aerofoil as might be the case for a water turbine. The extent of the diversion will not be uniform throughout the flow.

 

Text Box:  
Fig 26-14
Figure 26-14 comes from Prandtl and is for an asymmetrical aerofoil. It is a paint picture created in a wind tunnel. The picture is, in fact, the negative of the illustration in Prandtl's book and I have added the tangents to some of the flow lines extending upstream and down stream. This is not an attempt to extrapolate, only to emphasise the directions of the flow. Those tangents show that the aerofoil does indeed divert the flow and that the diversion in not uniform but it is certainly not random. The diversion is greater over the top than underneath the aerofoil. The net effect is a diversion of some of the air from the direction of the free stream by an amount that is about equal to the angle of attack.

 

This has serious consequences for our application of the aerofoil to the wind turbine especially at or near to the tips.

 

We can use vector diagrams to illustrate the problem. We start by noting that, if it has been possible to draw a velocity diagram for the flow approaching the blade of our wind turbine, it should also be possible to draw a diagram for the air leaving the blade. It will not be as reliable as those that can be drawn for water turbines because the blade is operating as an isolated aerofoil and not as one of a group acting together. We need to have a basis for drawing the exit diagram. We have a good idea of the diversion of the flow and therefore of the angle of the flow relative to the blade but we cannot draw a diagram until we have a value for the axial velocity of the air leaving the blade.

 

Text Box:  
Fig 26-15
When we dealt with water turbines and fans we had no problem because both operate in closed ducts and we were able to use the concept of continuity of flow coupled with the fact that the density was constant to put the axial velocities at inlet and outlet equal. Now we have a rotor with no solid boundaries other that its own shape operating in a supposedly steady flow of wind. The mass flow is unchanged as it passes through the rotor. The interaction between the moving air and the moving rotor blades is consequent on local changes in pressure. We need an idea of the magnitude of these pressure changes. We use the expression for the aerodynamic forces in the form of force = coefficient × stagnation pressure × area. The stagnation pressure is given by  and, in figure 26-15, this is plotted in bar for velocities up to 80 m/s. Even at 80 m/s the stagnation pressure is only 0.04 bar, that is about 0.04 × atmospheric pressure. This means that in the region of the blade and during the interactions with the air pressure is never sensibly different from the free stream pressure and the density of the air is sensibly constant. Most of the air is only peripherally involved with the blades anyway and the only conclusion that can be drawn is that the volume flow is effectively unchanged whatever any other effect the blade may have on the flow. If the volume flow is unchanged the axial velocity of flow is unchanged. This opens the way to drawing exit velocity diagrams that cannot be as accurate as those for water turbines but are useful nevertheless.

Text Box:  
Fig 26-16

Figure 26-16 shows vector triangles for the blade at the tip and at the half radius. The diagrams for the tip are made up of the inlet diagram from figure 26-11 with an exit diagram added. I have taken the aerofoil to have diverted the flow by the angle of attack and made . I have used the labelling used for similar diagrams for water turbines and called the two components of the absolute velocity leaving the blade  for velocity of whirl and  the velocity of flow. The aerofoil has increased the relative velocity substantially from  which now has two components . The air leaving the blade has tangential momentum and, of course, tangential kinetic energy. It was acquired in a thermodynamic process. The second diagram is for the mid-radius of the same blade.

 

Taken together these diagrams look eminently practical just like the practical diagrams for water turbines.

 

However we need to have clearer idea of the implications of these diagrams. If we could see the air flowing through the rotor we would see air flowing at the normal speed between the blades. It would have no angular momentum. Wakes from the blades moving at anything up to ten times the speed of the wind and containing significant kinetic energy would cross this flow and, inevitably, mix with it. During the mixing the net momentum would remain unchanged but the net kinetic energy would be greatly reduced. This kinetic energy would go back into the internal energy of the air giving a new flow of air at a temperature lower than that of the free stream but higher than that of the wake. It would still have the original momentum. There would be three such streams of air and all the momentum would then be re-organised into a free vortex and the pressure and temperature at the centre of the vortex would both drop. The wake as a whole would contain this free vortex that would then increase in diameter to become disrupted by the ground or the sea as the case may be. The disrupted wake would eventually break up as an organised flow and more angular kinetic energy would be lost until ultimately the net loss of internal energy is equal to the energy used to drive the alternator. The drop in temperature would be very small.

 

We can check. If the power produced by the generator were to be used to heat the air as it approaches the rotor the net result well downstream would be to restore the temperature in the wake to the free stream temperature and the air would flow at the original velocity.[3]

 

This is all a long way from the simplistic idea due to Betz.

 

Text Box:  
Fig 26-17
I have already pointed out that the use of turbines of larger radius runs into design difficulties. We must see why. In figure 26-17 I have drawn the diagrams for the blade in figure 26-16 for the blade extended to have a radius of 75 metres. It is obvious from the change in the size of the cross-section of the blade that I have had to make a change in scale to draw this diagram in order to get it on the page. All the velocities with the exception of the wind speed get much larger. More to the point, the angle between  gets much smaller. The first thing that we should note about this is that, as this angle gets smaller, the three wakes left by the blade tips get closer and closer together and they could, by interference, start to alter the flow pattern over the blades. Also, as that angle gets smaller the relative velocity  becomes greater, as does the value of . The limit must be set by the problem of interference between the wakes and following blades.

 

The velocity diagrams in figure 26-17 have been drawn for a wind speed of 15 m/s which is much higher than the common wind speed of 7 m/s. If the diagrams were to be drawn for 7 m/s for the same speed of rotation  would make a much smaller angle with  and the whole blade would have to be turned to give an angle of attack at the tip. Then, once the blade had diverted the flow during flow over the blade the angle of  would be impossibly small. Even if the flow were to be possible the force on the blade would make a very small angle to the axial direction and the tangential component of the force would be tiny indeed. There is no escape from the need to rotate the blade to increase the angle and to also to reduce the speed of rotation. Large diameters may be desirable to interact with a lot of air but the geometry of the velocity triangles points to great mechanical difficulty. There must be some maximum diameter beyond which the returns are too small to justify further increase. It is probably much less than 150 metres.

 

It may be desirable to go on increasing rotor size but the physics says that there are limits. We need to try to find them and this will involve modelling these velocity diagrams in a maths package. But first we need to find a working value for the angle of attack to be built into the blades.

 

Text Box:  
Fig 26-18
In figure 26-18 I have taken figure 26-12 and shown the force on the aerofoil but deleted the tangential and axial components. I have then added the lift and drag components that are at right angles to and in line with .

 

Those two components can be found from test data using the usual expressions :-

Lift =  and Drag =  where all the symbols have their usual meanings.

 

For high quality aerofoils the coefficient of lift will be anything up to 120 times the coefficient of drag but even for poor aerofoils the ratio is still about 12. We need lift and drag data for typical aerofoils.

 

Text Box:  
Fig 26-20
Text Box:  
Fig 26-19
Figure 26-19 Shows the relationship between coefficient of lift and angle of attack for a typical symmetrical aerofoil. It is from reliable test data for a good quality test model. There are three regimes of flow. At angles of attack up to about 12° the flow is attached to the upper surface and the aerofoil works aerodynamically. Then there is a transition to detached flow at about 12° to 18° when the flow has totally separated from the upper surface. The coefficient of lift takes a maximum at about 1, falls by about 40% of this maximum during the stall and then recovers to a new maximum of about 1 at 45°.

 

Figure 26-20 shows how the coefficient of drag varies with the angle of attack. Note the transition from attached flow in the range 0° to about 20° to detached flow from about 13° and upwards. It is this very low drag that makes the wing-sailed land yacht and the new, that is undamaged, wind-turbine rotor feasible mechanical devices. They will be severely impaired if they are worked outside this range.

 

The blades of wind-turbines use asymmetrical aerofoil sections and these, when smooth, can have maximum values of  up to 1.5 with a consequent increase in drag. However we must be careful because wind-turbine blades operate in the open and move at speeds up to 200 km/hour. Rain and hail will degrade the surface of the blades exactly where NACA found that surface roughness has a seriously adverse effect on the maximum value of  and then the graph in figure 26-19 would be realistic for a rough asymmetrical section.

 

I have already said that the blades are twisted so that, at the "rated" condition, that is the design condition, of speed of rotation, wind speed and output power the blades at every radius make the same built-in angle of attack with the wind. This must be the basis of a first model of the velocity triangles.

 

Text Box:  
Fig 26-21
Then we can see that the forces on the blade increase with angle of attack and we would like these forces to be as large as possible so that their tangential components are as large as possible. Unfortunately, even though the blade may be at the correct angle for the mean direction of the wind the angle of attack may increase or decrease in response to the random components of the natural wind, and the greater the design value for the angle of attack the more likely it becomes that the effect of the random components will be to take the aerofoil beyond the stall and into the detached flow regime. This will lead to sudden, random changes in force on the blades that may not apply to whole blades and may not apply to all the blades simultaneously. In order to give some margin for random increase in angle of attack and so reduce the likelihood of stalling, the design value of the angle of attack must be lower than would be used if the flow were to be steady. It is possible to get an idea of the effect of gusting and veering.

 

In figure 26-21 I have redrawn the vector triangle for the rotor with a radius of 50 metres and at 7.2 rpm in a wind of 15 m/s. The worst case for gusting and veering is when the wind swings towards the oncoming blade and the wind has gusted to its maximum. This leads to a new triangle in red. The angle of attack increases by 9°.

 

I have also drawn a second diagram for the same blade extended to 75 m radius and now the angle of increases by 6°.

 

So gusting and veering affects the blade more at the smaller radii.

 

Figure 26-19 shows that the greatest angle of attack for attached flow is about 13° and if the built-in angle of attack is say 8° the actual angle of attack can only increase by 5° before the blade stalls.

 

As the outer regions of the blades generate most of the power it would be best to make a decision on the angle of attack for these regions.

 

Then it would be reasonable to use an angle of attack of 8° to give a value of  of about 0.7 and have a margin of 5° to limit the effects of random changes of speed and direction in the wind. This is not enough to eliminate the effects of gusting and veering even at the tips but it will cover smaller random changes.

 

Given this decision it is possible to model the velocity diagrams.

 

Modelling the velocity diagrams

Text Box:  
Fig 26-22
The velocity of the air that is going to flow over the blade at any radius and generate a force is the velocity relative to the blade. It is the vector sum of the wind speed and the tangential speed at any given radius. It is shown in figure 26-22 brought down from figure 26-18.

 

Somehow a shape for the blade must be determined. The variation of width with radius is primarily a matter for mechanical design for strength, flexibility etc.. The twist must be chosen on fluid flow considerations. Figure 26-23 could be redrawn for every radius using the same angle of attack. Then  would vary directly with the radius and, at any given radius, the angle of the blade to the plane of the rotor would be :-

 

                                               

The blade speed  so we can plot the variation of  with radius and the resulting angles could be used to define the twist of the blade. Of course the blade can have only one twist that is built into it at the time of construction so a decision must be taken about the design values of . I do not know what decision is normally taken but presumably one might choose the values at which the turbine is rated or the values at the most commonly encountered wind velocity. These two choices would give blades of very different shapes. I will use the rated values and work for the Vestas V100 turbine.

 

Model

Rated

 power

Blade

length

Speed

 range

Maximum

 tip speed

Rated

 wind speed

Vestas V100

2.75 MW

50 m
(164 ft)

7.2-15.3

179 mph

15 m/s
(34 mph)

 

In figure 26-23 I have plotted two pairs of graphs. The blue solid graph gives the angles for  versus radius for a speed of rotation of 7.2 rpm, the lowest speed for the V100 turbine, and the blue chain dotted graph is for .

 

The pair in red is for the top speed of 15.3 rpm giving .

 

Text Box:  
Fig 26-23
It is clear from the velocity triangle for the blade tip in figure 26-18 that there must be some lower limit for the angle between  and I have taken this to be 10° and inserted a marker in red dotted at 10° on the graph.

 

We can see that the tips have angles greater than 10° for a speed of 7.2 rpm but for 15.3 rpm the blade will have an angle of 10° at a radius of 30 metres. I have to speculate from this point because from 30 metres to 50 metres the value of  appears to increase rapidly suggesting that a lot of energy will carried away into the wake from the rotor as kinetic energy. Allowing the rotor speed to increase would shed energy into the wake.

 

We should note that these two pairs of graphs would be for rotors of different designs and it seems that it would be preferable to design for the lowest speed of 7.2 rpm and use feathering, ie turning the blade, to match it as best we can to other wind and rotor speeds.

 

I have continued with the model for a blade designed to be correct at 7.2 rpm in a wind of 15 m/s. The outcome is given in figure 26-24.

 

The first graph is for the angle of the blade versus radius as before. The second below it gives the variation of the relative velocity at exit versus radius.

Text Box:  
Fig 26-24
The third graph top right gives the velocity of whirl versus radius and the fourth the absolute velocity of the air leaving the blade versus radius.

 

If the air leaving the blade contains kinetic energy it is possible to put the kinetic energy per unit mass of air equal to the drop of internal energy per unit mass of air. Then we can write :-                where  is the specific heat of air at constant pressure = 1 kJ/kg/°K.

 

All the quantities have magnitudes that might be expected.

 

Text Box:  
Fig 26-25
Currently (2012) wind turbines may have diameters up to 150 metres. I have extended the graphs in figure 26-24 to 75 metres and given the resulting graphs in figure 26-25.

 

The graph top left shows that the design of any blade must resolve the problem of the value of  falling below 10° at a radius of 60 metres. The result of doing nothing is shown in the last graph where the drop in temperature is now over 2 °C.

 

We must remember that this model is not as accurate as those for a water turbine where the flow is guided throughout. Nevertheless it is clear that increasing the diameter of turbines brings problems with the velocity triangles but it also must increase the problems of designing for extreme winds and for freezing rain.

 

I am sure that there are lots of complex models for wind-turbine rotors. This model is easily understood and it can be explored for all sorts of combinations of wind speed, speed of rotation and angles to get a mental picture of the complex relationship that applies to a wind turbine rotor. It might lead to a better decision about the combination for which the blade shape will be designed.

 

The aerofoil rotor

Text Box:  
Fig 26-26
If we decide to design this rotor as working with attached flow we are effectively choosing the treat the natural air as moving with uniform flow and not gusting or veering. We are further choosing to find forces on the blades by using aerofoil data using some chosen angle of attack at any given radius. If we are to model the rotor in this way to estimate the power generated we must keep these decisions in mind when we use the outcome.

 

Whatever we do the calculation will be based on the velocity diagram in figure 26-26.

 

Text Box:  
Fig 26-27
I have been over this same ground with airscrews where the decision was made to give the blade a shape based on having the same helical pitch from root to tip so that the blade at any radius has a value for  that depends on the radius and then to add an angle that was the same at every radius to give a built-in uniform angle of attack. For the rotor of a wind turbine the helix angle will be  the angle between . This can be checked by imagining that a flat plate is set up in line with . It could then move at velocity  without disturbing the flowing air. We can draw a graph of  versus radius. It is the graph of  against  where N is the speed of rotation in rpm and  is the radius.

 

It makes sense to continue to model a turbine having a blade length of 50 metres rotating at 7.2 rpm in a wind of 15 m/s. Fig 26-27 shows the fundamental helix angles for the blade.

 

To these angle we have to add an angle of attack. There may be a case for having an angle of attack that changes with radius but, for this opening calculation, it is sufficient to use a uniform angle.

 

Text Box:  
Fig 26-28
I have already argued for an angle of attack of 8° but we must note that the pitch of the helix to which this is to be added will only match that produced by the speed of rotation of the blade and the speed of the wind for one ratio of these two speeds. It is implicit in this statement that a variable speed rotor is desirable even if it is not practical. This would lead to figure 26-28 where the helix angle and the blade angle are given. Note that the blade angle is = .

 

I have also shown a marker for 10° as a practical lower limit for the blade angle and this combination of radius, wind speed and rotational speed leads to a satisfactory angle. Clearly it would not for greater radii.

 

If we continue to treat the flow as uniform at approach to the rotor it is possible to go on to find values for the torque exerted on the rotor and the power extracted from the atmosphere.

 

Text Box:  
Fig 26-29
In figure 26-22 I showed how to evaluate the force on an aerofoil from lift and drag data. For a smooth clean aerofoil the lift is abut 100 times the drag. This means that the angle between the force and the normal to the undisturbed flow is small, of the order of 0.5°. It follows that the lift and the total force on the aerofoil are effectively equal. However the angle is not zero and the tangential component of the force is small compared with the force especially towards the tips where most of the power is generated. So the drag must be subtracted from the tangential component. The angle of the force to the axial direction will be about  degrees where .

I have brought down a modified version of figure 26-22 as figure 26-29 and labelled the angle  for convenience.

 

The commonly-used model is based on dividing the blades into many elements each shaped like the section of a blade at a given radius. The thickness of the element has a radial thickness . At any radius the force on an element can be evaluated and its tangential component found. This can be used to find the torque produced by the element and the total torque can be found by integration and the power from that. It is a simple model of a seemingly simple device.

 

If the force exerted on the element at any radius is numerically the same as the lift on the element  will be given by :-

                                            where  is the velocity of the air relative to the blade at radius  and  is the area of the element =  where  is the width of the blade at radius .

 

Now  and  where  is the speed of the rotor in rpm.

 

Then                                  

 

This elemental force can be deduced from the values of the lift and drag. Figure 26-23 shows that the force  will be sensibly equal to the lift. However, as I have already said, figure 26-22 also shows that, whilst the drag is small compared with the lift, it is not so small that the angle between the lift and  can be ignored. Indeed this angle will increase with any degradation of the blade due to erosion, insect debris or ice and reduce the tangential component of . If this angle is called  its value is given by :-

Then the tangential component of  is given by :- .

 

The angle

Then the tangential component of the force on the blade element is given :-

                              

                                 and the torque produced by the element will be :-

                               .

There are three blades so this torque can be multiplied by 3 to give total torque. This expression can be summed between some lower value of radius to allow for the hub and any other radius  to give torque out to that radius. If the torque is summed out to the tip radius it can be multiplied by the angular velocity in radians per second to give total power.

 

All this can be the subject of a Mathcad calculation. The result is shown in figure 26-30.

 

Text Box:  
Fig 26-31
Text Box:  
Fig 26-30
Now we must look carefully to be certain what these graphs are for. They are for a wind turbine rotor rotating at 7.2 rpm in a wind of 15 metres/second that matches a Vestas V100 turbine of 50 metres tip radius and is rated at 1.8 MW. It was chosen for no special reason other than its being fairly typical. They are for a blade that has a shape based on a helix having a pitch equal to the wind speed and a length per revolution equal to  to which is added an angle of attack of 8°. (This is the angle of attack that matches the quoted value of the coefficients of lift and drag.) If the chosen figures for wind speed and rotor speed were to be changed the model would be for a blade of a different shape.

 

It is tempting to say that these graphs can be extrapolated to larger radii but the model may break down for an increase in radius unless the speed of rotation is changed. The breakdown may arise from the angle of the blade going negative.

 

For this model the magnitude of the torque and power vary directly with the width of the blade that I have taken to be uniform. Figure 26-31 is from Wikipedia and shows how real blades vary in shape with radius. We see that blades are tapered from about 40% of the radius and most of that 40% is used the create the root of the blade to fit into the hub so that it can rotate to feather the blade. My simple model could be modified to accept the actual blade widths. I have used a figure of 2.5 metres as an average width.

 

The calculated power is just under 1.6 MW. The rated power is 2.75 MW. It is hard to know how the figure for the rated power was obtained by the manufacturer. The model is much dependent on the values used for the coefficient of lift, for the speed of rotation and for the blade width.

 

I have said that this power is not extracted from the kinetic energy of the wind but from the internal energy of the air flowing through the turbine rotor that acts as part of a thermodynamic engine. We need to look for a thermodynamic cycle.

 

The wind turbine as a thermodynamic device

If the wind-turbine is a thermodynamic device we have to find all of the parts. Its closest parallel is the de Laval impulse turbine. That turbine is steam driven. Heat from fuel is transferred to water in a boiler to convert it to steam under pressure. This steam is piped to the de Laval turbine where it passes through a nozzle to produce a jet of steam at atmospheric pressure. That high speed jet impinges on the curved blades of a turbine wheel at a relatively small angle to the plane of the wheel and, as a result, has a large velocity of whirl relative to the blades. When running properly the jet leaves the blades axially, that is, with no velocity of whirl and a torque is exerted on the wheel. Effectively the output power is extracted from the kinetic energy of the steam. From the outside the process appears to take place at constant pressure but, during the passage over the blades the steam undergoes a rapid change in direction and is subject to a large, dynamic pressure gradient between its free surface and the surface of the blade. The steam can be collected and condensed and returned to the boiler to complete the cycle. There is no doubt that, in this cycle of events, heat has been converted to work.

 

The wind-turbine operates in the atmospheric air that flows more or less in straight lines, that is, without rotation. The air first acquired internal energy more or less at constant pressure by direct heating from the sun somewhere far away from the site of the turbine. Some of this energy was converted to kinetic energy as a result of non-uniform heating of the atmosphere either by direct heating through cloud or by heating the land or the sea and then heating the air. By some complex process circulating systems of air formed and we call the flow of air in those systems wind. In effect the atmosphere has carried out the functions of the boiler and the nozzle of the de Laval turbine to give the air internal energy and kinetic energy but whereas the nozzle could direct the flow at an angle to the blades to create whirl the wind cannot be directed and must flow on to the blades without whirl and the velocities are low. Extracting power from the atmosphere in this way is a new mechanism for this book or, more correctly, a new application of a familiar mechanism. We have seen that other turbines start with the working fluid approaching a ring of blades with whirl and that whirl is removed in the blades to create a torque. Now we have a turbine where, in order to produce a torque, the air that approaches without whirl must acquire whirl as it flows through the rotor and it must do it at sensibly constant pressure.

 

The flow of air that we call the wind approaches the turbine at steady speed and pressure and some of it interacts with the blades and makes them move. We have seen that the blades divert some of the air that flows through the rotor disc and this produces a torque, that is a mechanical force resulting from pressure produced on an area multiplied by a radius, and an equal dynamic force acting on the air at that radius to give the air angular momentum and angular kinetic energy. These forces can only exist if the torque is resisted by some applied torque which in this case is an alternator.

 

It is evident from the velocity diagrams that the air arrives without whirl but leaves with whirl as it must do to satisfy the concept of torque and angular momentum. The velocity triangles, that I must take to be valid, show that the air that interacts with the blades leaves the blades with a velocity that is greater than that of the wind. The air must leave having more kinetic energy than that in the undisturbed wind. Somewhere, in the flow over the aerofoil blades, work has been done by the air as it passes over the blades of the rotor to produce this increase in kinetic energy and to produce the output to the alternator. The blades have produced equal and opposing forces, one acting on the air and one on the rotor. One force produces the torque the other gives the air kinetic energy of rotation. The torque moves and produces power and this power and the kinetic energy of the air can only have come from the internal energy in the air which interacts with the rotor. The temperature of the air that interacts with the rotor will fall by a degree or so.

 

The thermodynamic cycle is not yet complete. The air with its rotation moves into the wake where, by mixing, the kinetic energy of rotation is "lost" as mechanical work to return to the air as internal energy with a rise in temperature. This leaves the net result of the flow through the rotor being electrical power with the air having given up some internal energy that has reappeared as work.

 

Oddly, that electrical power will, almost all, end up as heat and be returned to the atmosphere to bring it back to the original temperature. Whilst there is a drop in temperature in the vicinity of the wind-turbine the overall result is that the turbine does not affect the atmosphere.

 

Feathering

Text Box:  
Fig 26-32
It is inevitable that the power output of a wind-turbine will vary with the speed of the wind for speeds lower than the rated speed. The general shape of the power/speed graph is shown in figure 26-10 and there is a non-linear relationship between power and wind speed below the rated power for a real turbine. The model above can also be used to predict the shape of that curve for the special case of having a blade of the correct shape for every wind speed.

 

The resulting graph for a turbine having the same principal dimensions as the Vestas V100 turbine is given in figure 26-32. Let me repeat that this model is for notional rotors that are the correct shape for every combination of wind speed and rotational speed with an angle of attack of 8°. A real turbine will have only one shape probably for the rated wind speed and rotational speed and feathering will be used to turn the blades about radial axes to give the best output for other combinations. This is feathering.

 

Feathering utilises what is essentially a very simple mechanism. It is wholly geometrical and, as such, can be modelled accurately with a mathematics package.

 

For the Vestas 100 wind turbine the range of rotational speed is from 7.2 to 15.3 rpm and the range of wind speed is from 3.4 to 25 m/s. If I limit the range of wind speed to 7 m/s, which is a normal speed, up to 15 m/s, which is the rated speed, suitable graphs will indicate the viability or otherwise of feathering. For convenience I have worked for rotational speeds of 7 rpm and 15 rpm.

 

I have plotted two graphs in figures 26-33 and 26-34. Starting with figure 26-33 there are two solid traces one red, one blue. They show the blade angle at every radius derived from the helix angle at any radius minus a built-in angle of attack of 8°. The red trace is for a speed rotation of 15 rpm and a wind speed of 7 m/s. The blue one is for 15 rpm and 15 m/s. If an actual blade were to be made to the red angles and it was then feathered through +10° to make it run at 15 rpm in a wind of 15 m/s the trace would be the chained dotted one in red. It is not a good approximation to the blue solid line. If this were to be repeated to adapt the blue line to the red one by subtracting 10° the result is no better.

 

Figure 26-34 is for a speed of rotation of 7 rpm. This gives two more blade designs from the basic helices plus 8°. This time 14° has been added or subtracted to try to adapt either to the other. The result is better.

 

When assessing these diagrams the "error" between the graphs must be viewed in the light of having only 5° or perhaps a little more to play with before the blade stalls. Feathering is no panacea. However figure 26-33 for the higher wind speeds it is inevitable that the outer parts of the blades will go into the detached flow regime and this will give high drag which can unload the rotor and allow it to work in the region between the wind speed for rated power and the cut-out speed

 

Conclusion

If the wind flowed at a steady speed of about 15 m/s in one direction without gusting and veering and the turbine could be protected from rain, hail, salt, ice and insect debris this wind-turbine would be a very clever device taking advantage of the special behaviour of aerofoils at low angles of attack in the same way that aeroplanes do. None of these conditions can be satisfied in practice and the wind-turbine is in fact very vulnerable to all of them. It is futile to think that refinement of design will overcome the fundamental problems. The scope is just too small in the face of such unpredictability.

 

In the past the UK has seen rogue storms that were sufficiently severe to bring down millions of trees in one night. Such storms are likely to occur more frequently

 

No engineer would choose to become involved with wind-turbines if there was any alternative. The only real advantage of the wind-turbine over the other renewables is that there is no physical limit to how many turbines can be installed where there is a limit to hydro-power, tidal power and solar power.

 

I draw attention to a report that appeared at the end of 2012 in The Sunday Telegraph newspaper. It contains much factual observation. It is given in figure 26-35. It summarises a survey of the performance of UK turbines and some European turbines The rest of my conclusions should be read in the light of this report.

 

Text Box:  
Figure 26-35
The major problem that must be solved if wind power is the be generated on a large scale lies with the electricity grid. The National Grid system in the UK is not a stand-alone system. It is linked to those of our nearest continental neighbours and with Ireland. A complex control system works continuously to match the supply to the demand and electrical energy is moved between countries. To a large extent successful maintenance of a steady supply depends on anticipating demand which is possible because the demand that fluctuates mainly with the annual and day-to-day weather. By and large the grid system reacts slowly to change. Somehow the flexibility of the grid must be altered to accommodate the rather less predictable output of wind-turbines in sites scattered over the UK and its coastal waters. If, during the passage of time, this can be done and more realistic figures are accepted for the performance of turbines exposed to the weather, wind-turbines can ultimately make a useful contribution to our power needs. (As I write the outside temperature is 0° C and the wind speed is about 3 knots and nothing is expected to change for at least five days. Over mainland Europe the isobars are as much as 500 miles apart.)

 

The electricity demand in the UK is up to about 55 GW. Wind-turbines are rated at between 1 and 3 MW. There are about 3,000 turbines in the UK that could conceivably generate 3,000 MW when they are all actually working. As yet the contribution from wind power is not significant. The most important single observation in the report is that the efficiency of the turbines falls by a half over 15 years. This is to be expected because of the weather damage to the rotor blades that changes the mode of operation from attached flow to detached flow that halves the forces acting on the blades. This means that the wind-turbine decays with time to become the same as the windmills of old that also ran with detached flow. This process gets rid of the major effects of gusting and veering and it may be that a wind-turbine designed to have detached flow would be more durable and over, say, 25 years and produce more electricity than a similar sized "modern" turbine. Such a turbine would use a section that would be streamlined but designed to be permanently stalled.

 

The report also draws attention to another serious problem which is the experience that wind-generators in wind farms produce less power per generator than isolated wind-generators at the same wind speed. Presumably designers of wind farms accepted Betz' "assumption" (a word that I would never use) that the wind-generator produces no rotation in the air. If this were to be true, and it is totally contradictory to all recognised science, then a wind farm is viable. But the air does rotate and that rotation is persistent. No matter what direction the wind may blow through a wind farm some turbines will work in the wakes from others and their output be serious impaired. This is an intractable problem and one must accept yet another reduction in the power generated. See appendix 1.

 

Appendices

 

Text Box:  
Fig 26-36
Appendix 1

Figure 26-36 shows fog forming in the wakes from wind-powered generators in a wind farm in Denmark. This picture is well on its way to becoming the iconic image of the wind-powered generator. By chance the direction of the wind was very nearly in line with rows of turbines set in a grid pattern.

 

This picture has become possible because the temperature of the mixture of air and water molecules approaching these windmills is just above the saturation temperature. What we can see are the regions in which the temperature is lower than the saturation temperature.

 

We are looking at fog and not smoke so it is not the same thing as the smoke trails made in wind tunnels. We can see that the fog forms at a distance of about a rotor diameter downstream of the first rotor in each row and, when it first forms, the diameter of the foggy wake is much smaller than the rotor. We cannot see any fog that is associated with the outer and most important parts of the rotor blades. We need an explanation.

 

The stuff that we call water can exist as a gas or as droplets in suspension in space or, when droplets combine to become drops, as rain and, of course, as liquid water. It is better to call it  and give its state. Atmospheric air shares space with  that is acting like a gas. The  molecules move freely and undergo the normal collisions that take place in gases and mixtures of gases. Energy is shared between the molecules of  and the molecules of the gases of the air so that they all have a common temperature. The atmospheric pressure is the sum of "partial" pressures attributable to the separate quantities of the gases and .

 

A quantity of  in gaseous form could be isolated in a closed container and, as a result have a fixed density. If the container were to be cooled the temperature and the pressure of the gaseous  would both decrease. The molecules of  would be moving at lower speed and, at some temperature, that speed would become too low for all the molecules to exist separately and molecules would join together to form droplets. The density, the temperature and the pressure at which this occurs are called the saturation conditions and are listed in books of thermodynamic properties (steam tables).

 

The mixture of air and  that flows towards the rotor of a wind-powered turbine effectively contains  at fixed density and, if the temperature is just above the saturation temperature for this density and the partial pressure of the air, a drop in pressure or temperature of the mixture of air and  will lead to the formation of droplets that are visible as fog.

 

Now, as a consequence of the extraction of internal energy from the flowing mixture of air and water molecules, the mixture leaves the rotor with rotation (whirl if you use turbine terminology) and therefore angular momentum. That angular momentum will not be organised as a free vortex as it leaves the rotor but it quickly changes to a free vortex with, inevitably low pressure in the middle where there will be a core that moves as a forced vortex. In this case the drop in pressure produces fog.

 

This free vortex extends into the flow from the outer parts of the blade although it does not produce fog. The vortex increases in diameter with time and distance from the wind-generator. It has angular momentum that cannot be lost only dispersed and is consequently very persistent and, because the wind-turbines have been erected in a grid pattern, it is inevitable that the wakes from the turbines through which the wind first passes will disrupt the flow for all the others. The use of a grid ensures that whatever direction the wind may blow there will be interference to a greater or lesser extent.

 

If you look at the line of turbines second from the right in fig 26-35 it is evident that the second turbine is generating loops of fog that are not present for the first. From here on we do not know what is happening because this is not smoke but fog that is really making low temperature regions visible. It is not very encouraging.

 

Appendix 2

Size specifications of common industrial wind turbines

Model

Maximum power

Blade
*length*

Hub height†

Speed

range rpm

Max blade
tip speed‡

Rated
wind
§speed§

GE 1.5s

1.5 MW

35.25 m
(116 ft)

64.7 m
(212 ft)

11.1-22.2

183 mph

12 m/s
(27 mph)

GE 1.5sle

1.5 MW

38.5 m
(126 ft)

80 m
(262 ft)

?

?

14 m/s
(31 mph)

Vestas V82

1.65 MW

41 m
(135 ft)

70 m
(230 ft)

?-14.4

138 mph

13 m/s
(29 mph)

Vestas V90

1.8 MW

45 m
(148 ft)

80 m
(262 ft)

8.8-14.9

157 mph

11 m/s
(25 mph)

Vestas V100

2.75 MW

50 m
(164 ft)

80 m
(262 ft)

7.2-15.3

179 mph

15 m/s
(34 mph)

Vestas V90

3.0 MW

45 m
(148 ft)

80 m
(262 ft)

9-19

200 mph

15 m/s
(34 mph)

Vestas V112

3.0 MW

56 m
(184 ft)

84 m
(276 ft)

6.2-17.7

232 mph

12 m/s
(27 mph)

Gamesa G87

2.0 MW

43.5 m
(143 ft)

78 m
(256 ft)

9/19

194 mph

c. 13.5 m/s
(30 mph)

Siemens

2.3 MW

46.5 m
(153 ft)

80 m
(262 ft)

6-16

169 mph

13-14 m/s
(29-31 mph)

Bonus (Siemens)

1.3 MW

31 m
(102 ft)

68 m
(223 ft)

13/19

138 mph

14 m/s
(31 mph)

Bonus (Siemens)

2.0 MW

38 m
(125 ft)

60 m
(197 ft)

11/17

151 mph

c. 15 m/s
(c. 34 mph)

Bonus (Siemens)

2.3 MW

41.2 m
(135 ft)

80 m
(262 ft)

11/17

164 mph

c. 15 m/s
(c. 34 mph)

Suzlon 950

0.95 MW

32 m
(105 ft)

65 m
(213 ft)

13.9/20.8

156 mph

11 m/s
(25 mph)

Suzlon S64

1.25 MW

32 m
(105 ft)

73 m
(240 ft)

13.9/20.8

156 mph

12 m/s
(27 mph)

Suzlon S88

2.1 MW

44 m
(144 ft)

80 m
(262 ft)

 

 

14 m/s
(31 mph)

Repower MM92

2.0 MW

46.25 m
(152 ft)

100 m
(328 ft)

7.8-15.0

163 mph

11.2 m/s
(25 mph)

Clipper Liberty

2.5 MW
(4 × 650 KW)

44.5 m
(146 ft)

80 m
(262 ft)

9.7-15.5

163 mph

c. 11.5 m/s
(c. 26 mph)

Mitsubishi MWT95

2.4 MW

47.5 m
(156 ft)

80 m
(262 ft)

9.0-16.9

188 mph

12.5 m/s
(28 mph)


*This figure is actually half the rotor diameter. The blade itself may be about a metre shorter, because it is attached to a large hub.
†Hub (tower) heights may vary; the more commonly used sizes are presented.
‡Rotor diameter (m) × π × rpm ÷ 26.82
§The rated, or nominal, wind speed is the speed at which the turbine produces power at its full capacity. For example the GE 1.5s does not generate 1.5 MW of power until the wind is blowing steadily at 27 mph or more.

 

 

 



[1] There are other designs of land yacht where the wings or sails are manually controlled by the pilot/sailor.

[2] At first sight this is astonishing.

[3] This is all very different from the usual way that wind turbines are thought to work. Betz did nothing to help when he specifically excluded rotation from his mathematical model and ignored Newton.