Modeling the Twisted

Savonius Wind Turbine

with Geometry Expressions

By Nick Halsey

Background:

Windmills have been in practical use since the 7^th century. They have been used for generation of power in both mechanical and electrical ways as they those needs have evolved. Modern day windmills are almost all used to generate electrical power, bringing about the modernized name of wind turbines. These were generating approximately 160,000 Mega-Watts (MW) as of 2009, according to the World Wind Energy Association ^[1] and the numbers are ever growing with an expected 200,000+ MW in 2010 and approaching 2,000,000MW in 2020.

There are many different types of wind turbine. The overwhelming majority are Horizontal Axis Wind turbines, or HAWTs, which follow the same basic design as the original windmills from the 12^th century. They consist of a vertical tower, with the axle and the gears on top, oriented horizontally. Usually, there are three blades attached to this horizontal axle in modern turbines. The other major type of wind turbine is Vertical Axis Wind Turbines (VAWTs). These always have a tower that also acts as the axle. The moving axle can be atop a longer tower or supported by frame that goes around and above the blades. The generator, gearbox and other electronics are near the ground in this design which is an advantage over HAWTs. There are two major subtypes of VAWTs: lift-based Darrieus designs which utilize airfoils and drag-based Savonius turbines which use large cups for blades. Both designs were first developed in the early 20^th century. The animation below depicts the basic operation of the three primary types of modern wind turbines. The figures were created with Geometry Expressions.

Savonius VAWTs get their name from the first known man to successfully create a VAWT of this type, Sigurd Savonius. Because they are so much newer than HAWTs and are therefore less developed, Savonius turbines are rare and only used for specific applications. In particular, they are known for their reliability but also generally low power output. The basic operational principle of the turbine is that the differential of the wind 's force being deflected off the back of one blade and the force being deflected into and captured by the front of the other blade causes the turbine to spin. Because of the way they operate, Savonius wind turbines are potentially ideal for operation in turbulent environments where HAWTs are extremely ineffective. One of these turbulent places is around buildings. If not for the turbulence issue this would be the ideal place for wind turbines because the power would be generated near the location of its use; reducing losses in transporting the energy. Higher wind speeds are also found at higher altitudes, so placing wind turbines on the roofs of buildings would have many benefits if those turbines could capitalize on the turbulent wind environment. It would also be vital that the turbines didn't create vibrations or pulsations, which could easily be channeled into the building, not only damaging the building but also potentially the people inside.

Many studies have shown that a helical twist (about the axle) of a simple Savonius turbine greatly increases performance and efficiency. This is essentially because the wind is pushing on the blade for more of each rotation, so it can extract more of the power from the wind. This twist also continues to improve the performance in turbulent winds. Arguably the most important benefit, though, is that twisting the turbine enough for constant pushing of the wind can remove the potential for pulsations. This would allow these twisted Savonius wind turbines to be placed on buildings, therefore capturing more energy and virtually eliminating transmission costs and losses. The only negative is that the shape of a twisted Savonius wind turbine is extremely complex and therefore very expensive to manufacture. In this project I will model the shape of the twisted Savonius wind turbine from a two-dimensional top view and a pseudo-three-dimensional angular side view, both in Geometry Expressions' 2D plain. I will also use the Computer Algebra System (CAS) Maple and Geometry Expressions to approximate the “unrolled” shape of the blade and its surface area.

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