The cut in and cut out wind speeds of a 3 mw wind turbine...

The cut-in and cut-out wind speeds of a 3 MW wind turbine are 3 m/s and 25 m/s, respectively. The rated wind speed is 15 m/s. Sketch the representation of the wind turbine power curve (power delivered against speed). wind In the representation, indicate the power and wind speed values for the main characteristic points of the curve.

The power output of a wind turbine depends on many factors. It can be shown using physical principles that the power $P$ generated by a wind turbine is modeled by $$ P=k A v^{3} $$ where $v$ is the wind speed, $A$ is the area swept out by the blades, and $k$ is a constant that depends on air density, efficiency of the turbine, and the design of the wind turbine blades. (a) If only wind speed is doubled, by what factor is the power output increased? (b) If only the length of the blades is doubled, by what factor is the power output increased? (c) For a particular wind turbine, the length of the blades is $30 \mathrm{~m}$ and $k=0.214 \mathrm{~kg} / \mathrm{m}^{3} .$ Find the power output (in watts, $\mathrm{W}=\mathrm{m}^{2} \cdot \mathrm{kg} / \mathrm{s}^{3}$ ) when the wind speed is $10 \mathrm{~m} / \mathrm{s}$, $15 \mathrm{~m} / \mathrm{s}$, and $25 \mathrm{~m} / \mathrm{s}$

Calculus: Early Transcendentals, Metric Edition

Functions and Models

Mathematical Models: A Catalog of Essential Functions

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