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Scaling

Wind Power

Page 3:  Scaling
Lift

Wind turbine rotor blades use lift to extract kinetic energy from the wind.

According to this principle, the rotor blade shape causes air to form a pocket of low pressure next to a rotor blade to draw the rotor blade into it (the blade moves to fill a partial vacuum, like being drawn toward a vacuum cleaner). This is also how airplane wings work.

This pressure difference, causing lower pressure on one side of the blade (or wing) is created by the air flow having to travel further on one side of the blade than the other. The side that requires air to travel further effectively stretches out the air (making the air less dense on that side, lowering air pressure, causing the vacuum cleaner pulling effect on that side).

It is better to make wind generators larger, because more air crosses the larger rotor area, providing more air from which more kinetic energy can be extracted, and also because the efficiency of a wind generator partially relies on the wing tip speed (the speed of the blade tip in a circular direction on the rotor plane). Think of it as like a lever. The longer the rotor blade, the less rotations are required for a particular wing tip speed.

Wing tip speeds may be less noticable for larger wind turbines at a distance, which may appear to be turning slowly but actually have a fast wing tip speed (if you could zoom in to a wing tip).


Betz Limit

It is not possible to extract all of the kinetic energy of the wind, because that would stop the wind (which would stop the wind generator from working).

Air that passes through the wind turbine rotor has to slow down becuase some of its kinetic energy is harvested by the turbine. This slowing down of the air causes it to bunch up and therefore have to spread out:

Figure 3.1  Stream (tube) of wind, flowing (left to right) with velocity v through effective wind turbine rotor area A. The wind velocity v1 upstream is greater than the velocity v2 downstream, and the cross sectional area A1 of the utilized wind upstream is less than the cross sectional area A2 downstream.

Taking everything into account, researchers like German physicist Albert Betz have calculated that, at best possible efficiency, a wind turbine can only extract up to 16/27ths (59.3%) of the kinetic energy of the wind.

Thus, increasing the cross sectional area of the wind stream tube is important to increase extraction of kinetic energy from wind.


Wind Speed

Wind speed in the atmosphere is faster at heights further away from the ground. Ground obstacles interrupt air flow, causing wind to slow down near the ground.

Figure 3.2  Wind speed increases rapidly at heights above a given height near the ground. At much higher heights, wind speed still increases for additional heights, although with less rapid increase.

Ground obstructions are referred to as “roughness” and may include tall grass, fences, trees, buildings, etc.

The difference of wind speed at different heights (greater wind speed at taller heights) is referred to as “wind shear”. Following is a common wind shear formula for calculating wind speed of a height above ground relative to a lower reference height (Danish Wind Industry Association):

The following more simplified formula may also be used (NOAA):

See References below for examples of how to use these formulas.


Scaling Muliplier

The amount of electric power P generated by a wind turbine for an air density ρ depends on the cross sectional area A of the utilized wind stream tube and on the cube of the wind speed v, giving wind power a dramatic scaling multiplier:

Scaling refers to the ability of an energy generating technology to be scaled to produce larger amounts of energy. For example, an energy generating technology that produces twice as much energy when it is doubled in size is considered to have scaling.

A scaling multiplier is an increasing rate of change (acceleration) of the scaling. For example, a technology that produces three times the energy when it is doubled has a scaling multiplier.

The scaling multiplier for wind energy is very dramatic. Making a wind turbine system twice as large produces much more than twice as much electricity.

At the very least, the scaling is squared, because increasing the size of a wind power system increases the cross section area of the wind stream tube by the square of the stream tube radius (from elementary geometry: Area = π × radius^2).

Further substantial scaling acceleration may be accomplished by increasing wind speed as the wind turbine tower increases in height, since electricity generated is proportional to the cube of the wind speed.


References for this page:

 1.  “Roughness and Wind Shear”, Danish Wind Industry Association.

 2.  “Wind Shear Formula (Power law)”, National Oceanic and Atmospheric Administration (NOAA). html

 3.  Tom Parise, “Wind Turbine Design”, Stanford 2011. html

 4.  Bethany Kuhn, Julie Marquis, Hilary Rotatori, “Wind Turbine Design and Implementation”, Worcester Polytechnic Institue (WPI) 2010. pdf


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