# Wind Turbine Design

## Tom Parise December 10, 2011

### Submitted as coursework for PH240, Stanford University, Fall 2011

 Fig. 1: This image displays wind turbine blades before installation. To estimate the size of the blades, compare to the semi-truck slightly to the fore of blades. Source: Wikimedia Commons.

## Introduction

As issues have been raised over the sustainability of fossil fuels and the impact of their use, attention has increased for alternatives such as solar and wind. Wind power is an attractive option due to its wide range of possible sites but faces engineering challenges such as intermittency.

## Power

Wind passing over the blades of a wind turbine cause the blades to spin. This spinning rotates a generator, generating electricity. The source of this electrical energy is the kinetic energy of the wind passing over the blades. The rate of kinetic energy, or power, of the wind passing over the blades is

P=½πρl2uw3

where ρ is the density of the air, l is the length of the turbine blades, and uw is the speed of the wind incident on the blades. [1] Thus, the power passing through the blades of the turbine, and thus in turn the power of the electricity the turbine generates, is proportional to the square of the length of the blades: a doubling of blade length leads to a quadrupling of wind power passing over the blades. Further, the power passing over the blades is proportional to the cube of the speed of the wind passing over the blades: a doubling of the wind speed results in an eight-fold increase in the power. Wind speed varies with height above the ground, as given in the equation

uw=uw0(h/h0)p

where uw0 is the wind speed at height h0 above the ground, h is the height above the ground, and p is some parameter, typically varying between 0.2 and 0.8. This increase of wind speed with increasing height is due to both low level obstructions and no slip boundary conditions, a common assumption in fluid problems, at the ground. Substituting the second equation above into the first yields

P=½πρl2uw03(h/h0)3p

which shows a 3p exponential relationship between power and height above the ground. Taking a value of p=0.5, the power passing over the turbines will be higher by a factor of 1.5 with respect to height. Thus, doubling the height of the turbine will result in the power passing over the turbine increasing by a factor of 2.8. Combining the effects of height and blade length, doubling the height and doubling the blade length will result in an increase of power over 22 fold.

## Challenges

The benefits of increased power from increased height and blade length have costs. Longer turbine blades experience greater forces and thus need to be made of stronger materials. The longer blades have more mass and thus greater kinetic energy, which can have catastrophic effects. If the forces on the blades exceed the material strengths of the blades, the blades can fly apart, launching high energy projectiles in all directions. This tremendous kinetic energy must be dissipated on impact and can have explosion-like results. In one case in Northern Ireland, blades from a broken blade "shattered" the roof of a nearby home. [3] Thus, designers of wind turbines must reduce the chances for catastrophic failure through such measures as shut down in high winds and stronger blades. Other tactics for mitigation of the effects of broken blades include frequent safety checks and siting in remote areas away from residences.

## Conclusion

Wind power is a promising source of renewable energy whose efficacy can be improved through the use of longer blades mounted on taller turbines. However, designers of wind turbines must take steps to reduce the possibilities of damage caused by broken blades on the larger turbines.

© Thomas Parise. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

## References

[1] J. Twidell and T. Weir, Renewable Energy Sources, 2nd ed. (Taylor & Francis, 2004).

[2] G. A. Demarrais, "Wind-speed Profiles at Brookhaven National Laboratory," J. Meteorology 16, 181 (1959).

[3] M Connellan, "Spinning to Destruction," The Guardian, 3 Sep 08.