To avoid the bulky tower and impeller and generator in the...

To avoid the bulky tower and impeller and generator in the HAWT of the chapter-opener photo, we could instead build a number of Darrieus turbines of height $4 \mathrm{m}$ and diameter $3 \mathrm{m}$ (a) How many of these would we need to match the HAWT's $100 \mathrm{kW}$ output for $15 \mathrm{m} / \mathrm{s}$ wind speed and maximum power? ( $b$ ) How fast would they rotate? Assume the area swept out by a Darrieus turbine is twothirds the height times the diameter.

Key Concepts

Rotational Dynamics in Wind Turbines

Rotational dynamics concerns the relationship between the angular velocity of a turbine’s blades and the linear speed of the wind. It incorporates ideas such as the tip-speed ratio, which is the ratio of the speed of the blade tips to the wind speed. This concept is essential for optimizing turbine performance, as it affects both the aerodynamic efficiency of the turbine and the mechanical design constraints of the system.

Scaling Wind Turbine Output

This concept involves matching a required power output by adjusting the number or size of individual turbines. It requires an understanding of how the power output of an individual unit, based on its design parameters and operating conditions, scales up when using multiple units. This is important in energy system design, ensuring that the sum of the outputs of several turbines meets the desired overall power output.

Wind Power Calculation

This concept involves determining the power available in the wind by using the formula that relates wind speed, air density, and the swept area of the turbine. The power in the wind can be computed with the equation P = ½ ?AV³, where ? is the air density, A is the swept area, and V is the wind speed. Understanding this calculation is fundamental to assessing how much energy a wind turbine can harness and to designing turbines that are appropriately sized for a given energy requirement.

Turbine Swept Area

For wind turbines, the swept area is the area through which the blades rotate and is crucial in determining how much wind energy can be captured. The concept may vary with turbine type; for example, a horizontal?axis wind turbine has a circular swept area, while a Darrieus (vertical-axis) turbine may have a different effective swept area calculation. Recognizing how to compute or approximate the swept area is key to comparing the power outputs of turbines of various designs.

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