## Economics

Mick Sagrillo

©1993 Mick Sagrillo

Wind generators live on tall towers. And for good reason. Their "fuel" is way up there. As we'll see, the quality of your wind resource improves radically with height.

Physics

The power available to the rotor (that is, the spinning blades) of a wind generator is defined by the equation:

where P is the power at the rotor, d is the density of the air, A is the swept area of the rotor, and V is the velocity of the wind.

We can increase the power available to the rotor of a wind generator three ways — by increasing any variable in the power equation: d, A or V. Each variable in the power equation has its own effect on the power available to the rotor. Let's look at why these factors influence the power equation, and what our options are.

Density (d)

Wind generator blades spin because air molecules are moving past them. The more molecules we can move past the blades, the faster the blades will spin, and the more electricity the wind system will produce.

Density refers to the number of molecules in a given volume of air. Air is more dense in winter than in the summer. Therefore, a wind generator will produce more power in winter than in summer at the same wind speed. However, density of air is one variable that we can't do anything about.

Swept area (A)

Area of the rotor is included in the power equation because the rotor is, in essence, the collecting device for the wind generator. The rotor "captures" the power in the molecules that are moving past it. It makes sense that the larger the collecting device (that is, the rotor), the more electricity we can produce.

Increasing rotor area is not as simple as putting bigger blades on a wind generator. Many a manufacturer has learned this lesson the hard way. The swept area of the rotor is defined by the equation: A = nr2

Because we square the radius (which is the length of one blade), doubling the diameter of the rotor has the effect of quadrupling the swept area. For example, let's increase the rotor diameter of a wind generator from 10 feet to 20 feet. The 10 foot rotor has a radius, or blade length of 5 feet. Squared, this becomes 25 square feet. Multiplied by n, and we get a swept area of 78.5 sq. ft. If we double the rotor diameter to 20 feet, the radius becomes 10 feet. Squared, this is 100 sq. ft. Multiplied by n, we get 314 sq. ft.!

At first glance, this appears to be a very easy way to increase the amount of energy that a wind generator can capture. And it is. But by increasing the swept area to the tune of 400%, we have also increased all of the stresses on the wind system by that same 400% at any given wind speed. In order to compensate for this change and have our wind system survive, we must make all of the mechanical components 400% stronger. While this can be done, obviously this approach is going to get very expensive very quickly.

### Velocity Velocity Velocity

Increasing wind velocity increases the number of air molecules passing the rotor, so increasing wind speed will also have an effect on the power output of the wind system. But because velocity is cubed in the power equation, wind speed is the one variable that has the greatest impact on the power equation.

As an example, let's take a look at what happens when we double the wind speed for a given wind generator. At 5 mph, the V3 part of the power equation is 125 units of something (5 x 5 x 5) that is multiplied by density and swept area. Doubling the wind speed to 10 mph gives us 1000 units to multiply by density and swept area (10 x 10 x 10). This is an 800% increase in power output from the same wind generator!

From all of this we can conclude that we can get the biggest bang for our buck with our wind generator, not by fiddling with air density or increasing swept area, but by somehow increasing the speed of the air that the rotor sees. (For a more in-depth discussion of this topic, read "Wind Generator Tower Height" in HP #21.)

Fluid Dynamics

Like water, air is a fluid. We can learn some lessons about how the air moves by sitting on a stream bank and watching the water go by.

If we throw a twig into a stream near its center, we will see that twig move rather rapidly downstream, depending on how fast the current is flowing. If we throw another twig in the water near the stream's bank, we'll observe the twig move rather lazily downstream. Why is this?

Near its edge, the stream is slowed down because of friction between the bank and the water. As we move towards the center of the stream, the effect of the bank's friction diminishes. The laminar flow of the water moving over water allows the stream to pick up speed.

The same thing occurs with air masses as they pass over the surface of the earth. The face of the earth itself, as well as its vegetation, significantly reduce the speed at which the air can flow over the earth's surface. This is called ground drag. As you move away from the earth's surface, ground drag decreases and the laminar flow of air increases. Expressed another way, increased height means greater wind speeds.

### Turbulence

Let's go back to the stream. Our twig is still cruising along down the center of the stream. Up ahead, however, is a stump sticking straight up out of the water. Watch what happens.

As the twig approaches the stump in the stream, it slows down considerably as water piles up in front of this obstacle. The twig almost comes to a stop, but then passes slowly around the stump. Now behind the stump, the twig slowly spins around and around, until it gradually moves back into the swift flowing stream.

What we have witnessed is fluid turbulence. The same thing will happen with our wind generator if it is sited too close to trees or buildings. Turbulent air robs our wind generator of the energy available in laminar flowing air. The quality of our "fuel" has depreciated!

The obvious difference between the twig and a wind generator is that the stationary wind generator cannot escape the effect of turbulence on it as the moving twig did. Which brings us to a major rule of thumb: Wind generators must be sited at least 30 feet above anything within 500 feet.

### Picture this

Graph 1 (above right) depicts the increase in wind speed as a function of height above the earth's surface (actually above a relatively frictionless surface, such as a body of water.)

Wind speed increases significantly at first, but then the rate of increase begins to diminish with height. An example will clarify what this graph is telling us.

Let's assume that we measure the wind speed with a hand-held meter at shoulder height, 5 feet above the ground. Our measurement is 10 mph. How fast is the wind blowing at 100 feet above the ground?

Graph 1

Reading Graph 1, at 5 feet the increase factor is about 1.25. If we divide 10 mph by 1.25, we get a wind speed of 8 mph at ground level. At 100 feet, the increase factor is about 1.93. If we multiply 8 mph by 1.93, we find that the wind is blowing at 15.8 mph at 100 feet. That's a significant increase!

How significant, you ask. Remember that the power available to a wind generator rotor is a function of the cube of the wind velocity. Graph 2 below illustrates the increase in power available to the rotor as a function of height above relatively open ground. (The "windspeed" curve is the same one depicted in Graph 1 only compressed.)

Remember that this graph only depicts increase in power over relatively open ground, such as that

Increase factor Graph 2

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