A large Darrieus vertical axis wind turbine was built by the...

A large Darrieus vertical axis wind turbine was built by the U.S. Department of Energy near Sandia, New Mexico $[48] .$ This machine is $18 \mathrm{m}$ tall and has a $5-\mathrm{m}$ radius; the area swept by the rotor is over $110 \mathrm{m}^{2}$. If the rotor is constrained to rotate at $70 \mathrm{rpm},$ plot the power this wind turbine can produce in kilowatts for wind speeds between 5 and 50 knots.

Key Concepts

Unit Conversion

Proper unit conversion is essential when working with wind energy calculations, especially when wind speed is given in units like knots rather than the standard meters per second. Converting these units accurately is crucial to apply the wind power formula correctly and ensure that all related calculations are dimensionally consistent.

Rotor Dynamics and Tip Speed Ratio

When a turbine rotor operates at a constrained rotational speed, such as 70 rpm, the ratio of the blade tip speed to the wind speed (tip speed ratio) becomes a fundamental parameter. This ratio influences the aerodynamic efficiency and overall performance of the turbine, affecting how effectively the rotor converts wind energy into mechanical energy.

Swept Area

The swept area is the effective area through which wind passes and from which energy is extracted by the turbine. It is a critical factor in wind power calculations because the amount of kinetic energy available in the wind is directly proportional to this area. A larger swept area generally means more wind energy can be harnessed.

Vertical Axis Wind Turbine

This type of wind turbine has its main rotor shaft arranged vertically, which allows it to capture wind from any direction without adjustments. Its aerodynamic design is different from horizontal axis turbines, and it often faces different challenges and advantages regarding efficiency, maintenance, and installation. The Darrieus design is a well-known example within this category, utilizing curved blades to extract energy from wind.

Wind Power Formula

Wind power is calculated using the formula P = ½ × ? × A × v³, where ? is the air density, A is the swept area, and v is the wind speed. This formula illustrates that wind power increases with the cube of the wind speed, making wind speed a very sensitive factor in energy production. Efficiency losses are typically factored in through a performance or power coefficient.

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