## Fans and Wind Power

The wind, like water, is a natural source of power. With water, we use its attraction by gravity as it descends from the mountains to the sea. With wind, we use the kinetic energy of the air. The energy of either fluid per unit volume can be expressed as the sum of three terms, ρgz + p + ρV^{2}/2, corresponding to elevation, pressure and velocity. One form can be converted into another quite freely, but the sum must remain constant, or even decrease due to dissipative forces like friction. It is convenient to divide each term by ρg, the weight per unit volume or specific weight, so that they have the dimensions of length. Then, energy is called *head* of the fluid in question.

The most significant difference between air and water in regard to energy is that they are very different in density. Water is about 800 times as heavy as air, volume for volume. This means that energy is much more tightly packed in water, so that water machines can be conveniently small. Air machines, on the other hand, must handle large volumes of air, and so are large and cumbersome. Air weighs about 0.075 pcf (70°F, 29.92 inHg) or 1.293 x 10^{-3} g/cm^{3} (0°C, 760 mmHg). Air is also *compressible*, approximately obeying the ideal gas law pV = nRT, where R is the molar gas constant, 8.31441 x 10^{7} J/K-gmol or 1545.33 ft-lb/R-lbmol. K = °C + 273.15 and R = °F + 491.7. Water, on the other hand, is incompressible in normal situations. Air is about 10 times more viscous than water, its kinetic viscosity being about 0.15 cm^{2}/s. The discussion here applies directly to air, but the behavior of water is very similar.

The most popular unit of pressure for air machines in U.S. engineering has been the inch of water gauge. An inch of water is 1.867 mmHg, or 0.036 psi, or 1.489 mb (1489 dyne/cm^{2}). Gauge pressure is pressure above atmospheric, which is considered to act at all points equally, so that it vanishes from equations. To find pressure in inches of water, multiply air head in feet by 0.1442, or air head in metres by 0.473.

Flow velocity in air or water is often measured with a Pitot Tube, as shown in the figure at the left. The impact tube faces directly into the flow, while the static tube is perpendicular to the flow. Both functions can be combined in a single unit. From Bernoulli's equation, p_{t} = p_{s} + ρV^{2}/2. In U.S. engineering units, ρ = w/g, where w is pcf and g is 32.1725 ft/s^{2}. Therefore, V is proportional to the square root of the pressure difference Δp. For air, if V is in fps and p is in inches of water, then V = 66.8√Δp, assuming air weighs 0.075 pcf (in Denver, it is closer to 0.060 pcf). For water, the coefficient is 2.32. In practice, the pitot tube must be *traversed* to measure the velocity in annular areas of equal volume of a circular duct. It is often assumed that the average velocity V is 0.83 of the velocity at the centre of the duct, but this is approximate.

Henri Pitot immersed a bent glass tube in the Seine to determine the velocity of its water in 1730. The Pitot tube has been used in hydraulics since then. Perhaps its greatest common application is as an airspeed indicator for aircraft.

Let's estimate how much wind power is available to us. The average wind in Denver is 7.5 mph or 11 fps. The corresponding velocity head is (11)^{2}/(2)(32.2) = 1.879 ft of air. Multiplying by the specific weight, 0.075 pcf, the energy is 0.1409 ft-lb/ft^{3}. Assume a windmill of 50 ft diameter. The area will be 1964 ft^{2}, and so 21,600 ft^{3} of air will be gathered per second, or 3040 ft-lb/s. Since 1 hp is 550 ft-lb/s, this is 5.5 hp. The maximum efficiency of the windmill will be no greater than 65%, the efficiency of a propeller fan, so the useful output of our 50 ft windmill will be about 3.6 hp or 2.7 kWh. To capture greater power, we need either a higher wind velocity or a larger windmill. If the windmill is 150 ft in diameter, 3 times larger, and the wind speed is four times larger at 30 mph, then the power output would be multiplied by 576 to 2073 hp or 1555 kWh. This illustrates at least two important things: wind power is not very dense, and requires large machines to capture it; and it varies rapidly with the force of the wind (as V^{3}). The past month (July 2003) in Denver has been practically windless, and would generate very little power.

Water mills come to us from antiquity, but windmills do not. Wind has been used as the motive power for boats for millennia. This application uses only the pressure of the wind on a sail, and the techniques for controlling it were highly developed. It does not require machinery, except for the minor machinery for handling rigging. In this article, we consider only the use of the wind in producing mechanical power. The basic theory of turbines, of which fans and windmills are examples, is given in Turbines.

Fans are used to move air (or other gases) in large volume at low gauge pressures. A windmill is a fan in reverse. A fan consists of a wheel or *impeller*, and a *housing*. Sometimes the housing is absent, and we have just the impeller, as in an aircraft propeller. The two principal types of fans are the *axial-flow* and the *centrifugal*. We will talk mainly of propeller-type axial flow fans here, but the general principles will apply to both types. The ducting and other appurtenances associated with a fan are called the *system*, which may be absent when a fan is used in the free air just to generate a breeze. If the system is at the output of a fan, the fan is called a *blower*, while if the system is at the input, the fan is an *exhauster*. The moving part of the fan is the *impeller* or *wheel*, and the stationary part the *housing*. A propeller fan may have a housing as simple as a circular aperture, called the *shroud*, which nonetheless makes the fan more efficient. At the other limit, the fan may be enclosed in a duct and work against static pressure.

An arrangement for a fan test is shown at the right. The flow resistance of the duct can be varied at input or output. If A is the area of the duct, then Q = VA, where V is the flow velocity. If the input and output are completely unrestricted, then the pressure difference is zero and the flow is a maximum. If the duct is blocked at each end so that the flow is zero, the pressure difference Δp will be a maximum. The most important variables for a fan are its discharge at outlet in cfm or m^{3}/s, which together with the area of the fan gives the output velocity V, and the total pressure difference Δp. For propeller fans, this pressure difference is in the range 0.5 to 1.5 inches of water. The velocity pressure at the output is ρV^{2}/2, and the static pressure at the output is the total pressure less the velocity pressure. The output power is the total power in the output, W = (p_{s} + ρV^{2}/2)Q. The efficiency is this power divided by the input power W', e = W/W'.

There may be an *egg crate straightener* about 6 duct diameters from the input. This is a lattice of square passages of side 0.075 to 0.15 of the duct diameter, and three times as long as the length of a side. This and other details are mentioned in the standard specifications for fan tests of the Association of Heating and Ventilating Engineers, or the ASME, which are excellent sources of information about fans.

Fan characteristics as determined by a test in a duct are shown at the left. These are just the general shapes of the curves, not the results for any particular fan. At zero flow the fan maintains a pressure difference of p_{max}. At free discharge, the fan produces a flow of Q_{max}. The pressure curves give the results for intermediate cases. p_{s} is the static pressure, as would be measured by a manometer with an opening in the wall of the duct. p_{t} is the total pressure, as would be measured by a Pitot tube. The difference between them is p_{v}, the kinetic energy measured in pressure units. The input power and the efficiency are shown as percentages at the right. The power curve is not quite a straight line. Most problems involving fans in ducts can be solved with the use of the characteristic curves. For example, the head loss due to friction in the system is a parabola open upwards. The intersection of this curve with the p_{t} curve will give the flow Q under those conditions.

Imagine the fan blades as elements of a helix, moving the air like an Archimedean screw. In one turn, let the helix advance a distance L, called the *pitch* of the fan. If θ is the inclination of a blade, then L = πD tanθ, or L/D = π tanθ. The amount of air moved in one revolution will then be (πD^{2}/4)L, and if the impeller makes n rev/min, then the discharge Q = (πD^{3}/4)(L/D)(n) cfm. A certain propeller fan mentioned in a handbook has the characteristics n = 1150 rpm, D = 2 ft, Q = 5000 cfm. For this fan, A = 3.142 ft^{2}, and V = 26.52 fps. From these figures, L/D = 0.69, or θ = 12.4°. This is a quite reasonable figure. The actual pitch of the blades is larger, since there is back flow or *slip*. For a slip of 50%, the blade inclination would be 24°. Slip is usually less than this, around 40% to 30%. In any case, the effective L/D is probably more or less typical for the fan design.

If D is held constant, and n is varied, we see that Q is proportional to n. Since V is proportional to Q, and Δp is proportional to V^{2} (we presume that Δp is just the velocity pressure at the output in free discharge), it follows that Δp is proportional to n^{2}. Finally, the power output is proportional to Q and to V^{2}, so W is proportional to n^{3}. These relations for varying n and constant D are called "Fan Law No. 1."

The tip velocity of a fan blade is a good reference velocity for the other velocities involved. This is V' = ωD/2 = nD/19.10 fps. If we vary D, but keep the tip velocity constant, then n varies inversely with D. In these conditions, Q will be proportional to D^{2}, assuming L/D constant. Since the area is also proportional to D^{2}, the velocity will be constant, as well as Δp. The power output W will be proportional to the discharge times the square of the constant V, so it will be proportional to D^{2}. These relations are "Fan Law No. 2." Vector summation of the blade velocity at any radial position and the speed of approach of the air will give the angle of attack on the blade. Blades are usually twisted (greater pitch for smaller radius) to equalize the effect over the effective area. Small blades may be flat and thin, but large blades should have a rounded leading edqe and a feathered trailing edge. An airfoil section is often used for propeller blades. For an aircraft propeller, the blade angle is from 10° to 28° at 0.75R. The tip velocity should not exceed the local speed of sound to avoid the creation of shock waves.

We can think of other conditions as well. For example, let us vary D, but now keep the angular velocity constant, so that the tip velocity is proportional to D. In this case, Q varies as the cube of D, Δp as the square of D, and W as the fifth power. This can be combined with Fan Law No. 1 to show that when D and n are both varied independently, Q ≈ D^{3}n, Δp ≈ D^{2}n^{2} and W ≈ D^{5}n^{3}.

The frictional loss from air flow in a duct can be estimated by the pipe flow equations. If L is the length of a circular duct of diameter D, then the head loss is h' = 0.015(L/D)(V^{2}/2g), where I have chosen what seems to be a reasonable value for the constant, usually written 4f and the one found in the Moody Chart and other references. For other duct shapes, replace D by 4R, where R is the hydraulic radius (area/perimeter). For air, the "wetted perimeter" is just the perimeter, of course. A square duct of side a has R = a/4.

Torque is the rate of change of angular momentum, just as force is the rate of change of linear momentum. When a fluid exerts a torque on a turbine runner, the reaction is a change in angular momentum of the fluid. The air that leaves a fan is rotating, the reaction to the torque that turns the impeller. Fluid is given angular momentum by the guide vanes which, ideally, is destroyed by the torque exerted on the runner. With some machines, however, the water at the exit may still have considerable angular momentum, and the energy in this motion is energy that does not appear at the shaft. Where velocity in the exit fluid is part of the desired output (as with a fan), vanes to straighten out the flow help to recover some of the energy that would otherwise be lost.

Fans for use under low pressure differences generally have a small hub, with the blades occupying most of the cross-sectional area. As the pressure differential increases, it becomes more efficient to concentrate the blade area near the periphery of the impeller. The hub then becomes larger, and the blades are stubby vanes on its surface. This is seen at the forward end of a jet engine, where the fan forms the *compressor* that efficiently decelerates the air relative to the engine, raising its pressure. Energy is added by burning fuel in the compressed air. Its velocity increases as it returns to atmospheric pressure, forming a jet the reaction to whose momentum provides the thrust. The exhaust drives a turbine that extracts some energy to operate the compressor.

The centrifugal fan is a very simple device. It consists of an impeller with blades that can be simply radial, though there are certain benefits to curved blades. It is fed from the centre, and the output is taken from a scroll case on the outside. The air is simply whirled around and centrifugal force causes the pressure to rise on the outside. It is a curious machine in that, unlike most power machines, it cannot be run in reverse to produce a torque. There are, of course gas turbines that can produce work efficiently, but they are very different from a centrifugal fan run in reverse.

An aircraft propeller is a fan with free discharge, whose purpose is to add velocity to the air it encounters. A jet engine, or even a rocket engine, has an identical function, so will be included in the following discussion. The reaction to the momentum added to the air is the *thrust* of the propeller, that pulls the aircraft through the air. This propels the aircraft although it has no connection with the ground, a somewhat marvellous phenomenon. The familiar blast of air behind a rotating propeller is called the *slip-stream*. When a single-engine aircraft is moving through the air, the velocity of the air behind the propeller is greater than the speed of the aircraft, and so produces greater parasitic drag than would otherwise be expected. Propellers on the wing, or behind the fuselage, do not produce this added drag.

Let's analyze the action of the propeller, using the conservation of energy, momentum and mass. As usual, we will be able to make considerable progress. In the figure, the dotted line marks out a cylindrical region containing the air influenced by the propeller, the slip stream. At the left, velocities relative to the aircraft are shown, while at the right are the absolute velocities. V is the speed of the aircraft, and the relative velocity of approach of the air. V' is the speed of the air behind the propeller. Since V' > V for positive thrust, the area of the slip stream is smaller in the wake of the propeller than in front of it. The absolute velocity in the slip stream is ΔV = V' - V. It extends from a diameter of 0.2D on the axis to 0.8D - 0.9D.

Let us assume that the propeller produces a pressure difference Δp that is turned into a velocity difference in a short distance. The thrust can now be expressed in two ways, T = (πD^{2}/4)Δp = QρΔV. The energy equation in the relative motion gives V^{2}/2g + Δp/ρg = V'^{2}/2g. Solving for Δp, we get Δp = ρΔV(V + ΔV/2). Then, using the relation between Δp and ΔV given by the thrust formulas, we find Q = (πD^{2}/4)(V + ΔV/2). Using this in ΔV = T/Qρ, we get a quadratic equation for ΔV, with the solution ΔV = V[√(1 - K) - 1], where K = 8T/πD^{2}V^{2}. This formula relates ΔV to the thrust, propeller diameter and aircraft velocity.

It is seen that the aircraft leaves behind an energy ΔV^{2}/2g in the slip-stream. This energy soon mixes with the other air in turbulence. This loss is a necessary part of the propulsion, since if ΔV = 0 there is no thrust. The *propulsive efficiency* η of the propeller is the ratio of the total useful ouput, TV = QρVΔV, to the input energy, which will be the sum of the useful work and the energy left in the slip-stream. The result is η = 1/(1 + ΔV/2V) = 2/(1 + V'/V). V' cannot be less than V, of course, so η <>

Propeller and jet propulsion can be compared on the basis of thrust and propulsive efficiency. The thrust is T = ρQ(V' - V), so it depends jointly on the area of the slipstream and the velocity difference. If V = 200 mph, for example, V' should not greatly exceed 200 mph. If V' = 300 mph, then η will be 0.8, a reasonable figure. For an adequate thrust, this means a large flow Q, since the velocity difference will only be 100 mph. This can be achieved by a large propeller diameter D. A jet engine, however, has a much smaller area and would not be able to provide the required Q with the given V'. On the other hand, if V = 500 mph, then V' = 750 mph would give the same propulsion efficiency of 0.8 and a velocity difference of 250 mph. Now the required T can be obtained with the dimensions of a jet engine, since the higher velocity increases both factors contributing to T.

Thrust can also be expressed as T = ηP/V, where P is the power supplied to the propeller. If P is in hp and V is in mph, the thrust in pounds is T = 375ηP/V. The tip velocity of the propeller is V" = ωD/2 = πnD, where n is rps. A design ratio often used with propellers is N = V/nD, where V is in fps, D is in ft, and n is in rps. This dimensionless ratio is also N = πV/V". The tip velocity should be kept well below the speed of sound to avoid the ceation of shock waves. In practice, N is usually between 0.8 and 1.1, which implies V"/V = 3 to 4. If propellers are provided with multiple blades, the same flow Q can be obtained at a slower speed. If there are M blades, then we have V = MV"/3, taking the smaller end of the range. The speed of sound is about 340 m/s or 760 mph, so if V" is restricted to 50% of the speed of sound, V = 127M mph. For a two-bladed propeller, this means a limit of 264 mph, for a three-bladed propeller, 381 mph, and for a four-bladed propeller, 485 mph. This seems to agree with practice. The maximum velocity of flow over the propeller is greater than the tip speed.

In all of this, we have neglected the rotation of the propeller, and the vortex motion in the wake. The propeller gives angular momentum to the air in the slip stream, and the propeller tip sheds vortices in a helix. This added motion will reduce the propulsion efficiency, but does not play a large role.

If the efficiency is calculated in the relative motion, the maximum efficiency available is only 50%, since an amount of energy equal to the useful energy is contributed to the wake. In this coordinate system, the propeller is not moving, and so cannot produce useful work by means of the thrust. The energy that in the absolute system is useful work here goes into the wake. In fact, if you make an energy balance in the relative system, you will find that the energy contains two terms, one of which is exactly the thrust energy in the absolute system, and the other is the energy left in the wake. Of course, in this system both appear in the wake.

The aircraft is also supported in the air by the reaction to an air jet, in this case air forced downward by the wings. A helicopter even uses a fan for this purpose, and its analysis is the same as we have just presented. All this is possible because air is actually pretty heavy, with each cubic metre weighing about a kilogram. It seems insubstantial to us, but air has considerable inertia.

### The Windmill

Wind as a land power source does not have as long a history as water as a land power source. Its low energy density and its unreliability are sufficient reason for this. Horizontal windmills were known in Persia, perhaps by the 8th century BCE, and the idea was carried to the Far East by prisoners of Genghis Khan, where it was considerably developed in China to drive irrigation machinery. These mills have nothing to do with the European windmill. The first documentation of windmills in Europe dates from 1185 (or 1105), with only the name mentioned, though by 1300 they were becoming common in Northern Europe. They spread to other places, where they developed many local peculiarities during their adaptation. The machinery was obviously developed from that of the Roman water mill, and the availability of millwrights was necessary to its creation. Their major use was always to turn the heavy millstones for grinding grain into flour, but before electricity was available they found many other applications, such as land drainage, pumping, sawing and ore crushing.

The European windmill consisted of *sails*, usually four in number, attached to sail *stocks* that rotated a stout horizontal *windshaft*. There was a large brake wheel on the windshaft with cogs driving a *wallower*, usually a lantern gear, that rotated the *main shaft*. In early mills, the main shaft rotated the upper millstone, or runner, which rested on the bedstone. Medieval mills had symmetrical sails, with the stock in the centre supporting transverse *sail bars*, usually braced on their outer edges with *hemlaths*. The canvas sail strips were threaded above and below alternate sail bars and tied tightly. The later *common sail* was entirely on the trailing edge of the sail, and the canvas was laid on top of it and tied down. Cords allowed the sails to be reefed as necessary, depending on the wind. Setting the sails was an arduous and difficult job. Eventually, canvas was replaced by rotating slats controlled by springs. Later, the slats could even be adjusted while the sails were in motion. Previously, the mill had to be stopped for this to be done. The leading edges of the stocks were later given fairings for smoother air flow, which made a considerable improvement. Sail spans of 50 to 70 feet were common. The windmill was built entirely of timber, connected by mortises and trenails (wooden pegs), and used as little costly wrought iron as possible. Since the wood was not exposed to water, as in the case of a water mill, it did not suffer from rot.

The basic machinery of wooden wind and water mills is shown at the right. The similarity is obvious. The millstones that turn grain into flour were about 4 ft in diameter, and rotated at 120-125 rpm. The preparation of millstones by grooving was an art. They had to be very carefully adjusted to just not touch, kept apart by the grain that was being ground. Moving stones could not be allowed to run "dry." Good stone for millstones was rare. In England, the best was a hard sandstone from the Pennines known as "millstone grit." A stone from the Rhine was superior to this, as was the "French burr" made from volcanic rock. The water mill drove the runner from below, by means of a spindle whose *mace head* turned the iron *rynd* let into the runner. The windmill drove from above, as shown. A wooden *vat* surrounded the millstones (not shown). On top of it, the *horse* supported the *hopper* from which grain was discharged into the eye of the runner. To drive more than one set of millstones, a "head and tail" arrangment could be used, with two crown wheels on the wind- or watershaft. A large spur gear could be mounted on the vertical shaft, running several sets of stones around its periphery. Metal gearing allowed even more complicated ways to distribute the power, and to attach accessories such as sack lifts.

The sails had to be directed into the wind by moving the windshaft. The first mills were *post mills*, small houses called *bucks* that contained the machinery and rotated on a stout post at the centre. The floor of the buck was supported by the *crown tree*, which rotated on a *pintle* in an enclosure called the *roundhouse*, when it was not open to the air. The *tail pole* extended from the rear of the buck, and was moved to point the sails into the wind and tied to one of a ring of stakes. The simplicity of this kind of mill meant that it never entirely disappeared, especially for small mills.

Another method was to mount the windshaft in a rotating *cap*. Only the cap rotated, while the power came down the main shaft at any cap position, since it was at the centre. In the *tower mill*, which arrived in the 15th century, the timber cap was mounted on a masonry or brick tower, usually round. The mills of La Mancha that excited Don Quixote were of this type, whitewashed and with slowly rotating sails. About this time, the horizontal windshaft was replaced by a windshaft that was inclined upwards slightly. This gave a much better stress distribution in the cap, and also more clearance below. The gearing was no problem, since all that was necessary was to incline the cogs on the crown wheel. The *smock mill* is so-called because its flaring tower, which could be of shingled or weatherboarded timber and octagonal in cross-section, looked like a rural smock. As these mills became taller to reach stronger winds, it was necessary to build a gallery around the tower so the miller could attend to the sails, and also arrange the tail pole to adjust the orientation. Dutch mills had a characteristic way of bracing the tail pole from a cross-piece on the cap by outrigger-like struts. Later, some mills were turned by a worm gear or winch inside the mill. The reader is probably familiar with the appearance of these mills, many of which have been preserved (except, alas, in the United States, where I believe they are all long gone).

When a windmill was not in operation, the sails were placed in an X position to equalize forces on the stocks. When the mill was ready to begin operation, the sails would be moved vertically one by one so the miller could reach them to set the sails, as long as this was necessary. The brake wheel held the sails immovable by the action of a weight. To allow the sails to move, the brake was "pulled off." In case of emergency, the sails could be stopped quickly by releasing the rope.

It was not only necessary to face the mill into the wind just so the sails would be turned efficiently, but also for important stability reasons. The mill was built to resist force from the front, and was in danger if *tail-winded* by a strong gust or thunderstorm. If the wind rose suddenly, and the brake could not hold the windshaft, friction rapidly set the mill on fire, especially if the miller ran out of grain to put between the millstones. Unusually strong winds are dangerous to windmills today, a hazard water mills do not face.

In 1745 Edmund Lee invented the *fantail*, or fly, a small windmill with an axis perpendicular to the windshaft. If the wind had a component from the side, the rotation of this wheel drove machinery that rotated the cap accordingly. In this way the mill was automatically kept facing the wind. The fantail was widely adopted in England and on the Continent, except, curiously, for the Netherlands, where the braced tail pole was retained.

A famous and excellent kind of windmill that can still be commonly seen, though less than formerly, is the American wind pump, simply called a "windmill" in the United States. It has an *annular sail*, which is very strong and durable, composed of many radial vanes. A tail vane keeps the sail faced into the wind. this vane is hinged so that it can be latched parallel to the sail when the mill is not intended to work. A cranked windshaft moves the vertical pump rod up and down to operate the pump in the well beneath it directly. The machinery is mounted at the top of a tower made from angle iron in the better machines, of wood in the lesser. This mill pumps water for cattle in isolated locations, and will work unattended, pumping whenever there is sufficient wind from any direction. Large mills of this type even provided locomotive water for the Union Pacific (as a photograph shows) at certain locations where the installation of a steam engine was not warranted. There could be a device that folded the tail if the wind exceeded 30 mph, or even speed governors. One example of a small mill had a 6' wheel and a 19' redwood tower. Among manufacturers were the Fairbury Windmill Co. of Fairbury, Nebraska and the Chicago Aermotor Co. A Fairbury windmill with an 8' wheel and 33' tower, restored by Bill Alexander, is shown at the left. Today, electricity has taken over most similar tasks once performed by the wind. Even the provision of small amounts of electricity for battery charging is now usually done with solar cells. However, windmills are made with geared heads for driving generators. Because of the variation in speed, the control of output voltage must be carefully considered.

An 8' wheel has an area of 50.2 ft^{2}. The maximum operating wind velocity is 30 mph, or 44 fps, which gives 2.25 ft-lb/ft^{3}. The total power available in the wind intercepted is then 4976 ft-lb/s or about 9 hp. At an efficiency of 50%, this means that a maximum of 4.5 hp is available. With an average wind of 15 mph or so, about 0.56 hp should be available, which can still pump a lot of water. The rapid variation of output with wind speed is one of the difficulties in applying wind power. Windmills are most useful for winds of Beaufort Force 4 to Force 6, or 15 to 30 mph. Over this range, their power varies by a factor of 8. Weaker winds will not provide sufficient power, while stronger winds may be damaging, and require that either the vanes be feathered or the wheel turned parallel to the wind.

The horizontal windmill was mentioned above in connection with its early appearance in Persia. On a small scale, horizontal windmills are still common, in devices like the cup anemometer. The drag coefficient of a hemisphere or cone presenting its convex side to the wind is less than when it presents its concave side, so if two or more such cups are mounted on an axis, they rotate in the wind. A similar device is the ventilating stack with a rotating, S-shaped vane on top. The wind-operated prayer wheel of Central Asia seems to have been a device of this kind, possibly suggesting the Persian windmills. There was no gearing inside a Persian mill. It is a very long way between prayer wheels and wind-operated toys to a mill that can turn heavy millstones. Larger examples were in towers with walls and openings that act in the same was as jets, acting on a runner inside with fixed vanes. These devices are not efficient, but have the advantage that they do not have to be turned into the wind, working equally well with wind from any direction. Note that they are actually impulse turbines, while the European windmill is closer to being a reaction turbine.

Wind is getting its own back, however, in wind farms to generate electricity. A good example is the North Hoyle wind farm 7 km off the North Wales coast between Prestatyn and Rhyl. 30 turbines will be installed, with a total nominal capacity of 60 MW. The hub of windshaft is 67 m above the sea, and the sails are 80 m in diameter, sweeping out 5027 m^{2}. The windshaft is inclined by 6°, and the blades are coned by 2°. Each of the three glass-fibre reinforced epoxy blades is 39 m long, with a chord 3.52 m at the root and 0.48 m at the tip, and a twist of 13°. The pitch of the blades is regulated hydraulically. A blade can rotate 95° in all, and is full feathered for stopping. The windshaft is yawed by a ring gear and pinions, just like a cap mill, but electrically driven. The controls keep the windshaft rotating at a nominal 18.1 rpm, provided there is sufficient wind. The nacelle at the top of the post is also glass-fibre reinforced epoxy. It contains the gearbox and the alternator. The gearbox is planetary and helical, giving a fixed ratio of 1:92.6 for 50 Hz operation and 1:111.1 for 60 Hz operation. The alternator is a 4-pole induction generator with 690V output, and can generate either 50 Hz or 60 Hz. An induction generator is like an induction motor run in reverse. The rated speed of the rotor is 1680 or 2016 rpm. With slip, the stator field rotates at either 1500 or 1800 rpm, producing 50 Hz or 60 Hz power. The controls keep the windshaft rotating at a nominal 18.1 rpm, provided there is sufficient wind.

Unlike synchronous alternators, induction generators cannot supply reactive power, so connecting them to the grid involves control difficulties. Also, rated power can be produced only about 20% to 30% of the time, which calls for some kind of power storage. Even pumped hydroelectric has been suggested, but it is an expensive solution.

The average wind at the site is given as 10 m/s, which is a brisk 48 mph. At this speed, wind contains 61.25 J/m^{3}. The flow is 50,270 m^{2}/s, so the total power in the intercepted wind is about 3 MW. This would make the efficiency of the unit 67%, which seems a bit high. The wind speed for producing 2 MW is not given, so perhaps 2 MW is an optimistic estimate of the actual power. The turbine will stop at 25 m/s, and restart when the wind falls to 20 m/s. At 20 m/s, the power of the wind will be 24 MW, so getting 2 MW should be no problem, but the efficiency will drop to 8%. Efficiency is rather unimportant, since wind costs nothing. The efficiency could be improved somewhat by increasing the number of blades to 4 or even 6, but this would also increase the force on the structure in a high wind. Wind turbine blades are probably limited to 2 or 3 for this reason. At the North Hoyle site, the 50-year 10 minute gust is 46 m/s, and the 50-year 2-second gust is 60.3 m/s. This 60 m/s gust is 134 mph, well in the hurricane range, and something rightly to be feared.

Since the output of this wind farm is about that of a good hydroelectric turbine and alternator (80,000 hp), the economics of the enterprise are, it would seem, doubtful. Wind power burns no fuel, however, and is a hazard only to birds and ships. At sea, it is at least out of sight. Together with the higher winds over the ocean, this is an excellent reason for offshore wind farms. Modern windmills are actually very much like the older ones, but differences in materials and how the turbines are used makes them look different. The old mills were built almost entirely of wood, which determines the appearance of the structures and machinery, while new ones are metal and plastic. The sails of a modern windmill are aerofoils, and can be feathered automatically to control the torque and speed. In the older mills, this was done by reefing the sails, or later by controlling rotating slats. Fairings on the leading edges of the old sails brought them closer to aerofoils, and the angle of attack could sometimes be varied as well. The sails and windshaft of a new windmill are mounted on top of a post, just as in a post mill, with the orientation controlled by electric motors instead of a tail pole. Mechanical power does not have to be transmitted to the ground for grinding grain, but is used close to the sails by putting the alternator directly on the windshaft (with gearing). Thinner sails can be used because of the greater angular velocity; in the older mills, speed had to be kept low for several reasons. The same necessity of intercepting a large area to gather sufficient power is common to both. Modern windmills are distinctly less attractive than the old ones, verging on visual pollution of the landscape when used *en masse* on account of the low power density, but they are no worse than advertising, electric transmission wires and four-lane highways.

J. Reynolds, *Windmills and Watermills* (New York: Praeger, 1970). Very well illustrated; covers all kinds of historic mills.

S. Strandh, *A History of the Machine* (New York: A&W Publishers, 1979). pp. 108-111. Well-illustrated. The Chinese horizontal mill and the Mediterranean jib-rigged sails are shown.

R. L. Daugherty and J. B. Franzini, *Fluid Mechanics*, 6th ed. (New York: McGraw-Hill, 1965). pp. 162-165.

J. K. Salisbury, ed., *Kent's Mechanical Engineer's Handbook*, 12th ed. (New York: John Wiley & Sons, 1950). Power Volume, pp. 1-57 to 1-96; 15-18 to 15-22; 15-38 to 15-40.

The site Windmill World has many links to windpump information, most of them broken and some of the nasty American variety that leaves webturds. Be sure not to enable cookies when using this link. There are still a few honest sites and good pictures, if little technical information.

An excellent wind power website is National Wind Power.

Peter Fairley, *Steady as she Blows*, IEEE Spectrum, August 2003, pp. 35-39.

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