A windmill, such as that in the opening photograph of this...

A windmill, such as that in the opening photograph of this chapter, turns in response to a force of high-speed air resistance, $R=\frac{1}{2} D \rho A v^{2}$ . The power available is $9^{p}=R v=$ $\frac{1}{2} D \rho \pi r^{2} v^{3},$ where $v$ is the wind speed and we have assumed a circular face for the windmill, of radius $r$ . Take the drag coefficient as $D=1.00$ and the density of air from the front endpaper. For a home windmill with $r=1.50 \mathrm{m},$ calculate the power available if $(a) v=8.00 \mathrm{m} / \mathrm{s}$ and (b) $v=24.0 \mathrm{m} / \mathrm{s}$ . The power delivered to the generator is limited by the efficiency of the system, which is about 25$\%$ . For comparison, a typical home needs about 3 $\mathrm{kW}$ of electric power.

Key Concepts

Wind Energy Conversion Efficiency

Efficiency in wind energy conversion refers to the fraction of the available wind energy that is actually converted into electrical power. In practical systems, losses due to mechanical, aerodynamic, and electrical factors reduce the raw energy available, and this efficiency is critical for estimating the actual power output for uses like residential power supply.

Circular Area of a Wind Turbine

For a wind turbine, the cross-sectional area exposed to the wind is crucial for power calculation. When the turbine has a circular face, the area is calculated as ? times the square of the radius. This area directly affects the amount of wind power that can be harnessed.

Drag Force

The drag force is the force exerted by the air on an object moving through it, dependent on factors like the drag coefficient, air density, cross-sectional area, and the square of the velocity. It quantifies the resistance encountered due to the medium, which in this context, is the wind acting on the windmill’s face.

Wind Power Calculation

Wind power calculation involves determining the energy available from the moving air. The power available is a product of the drag force and wind velocity, which leads to a dependence on the cube of the wind speed. This relationship implies that even a small increase in wind speed results in a significant increase in power output.

Please give Ace some feedback