Question

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

\(P = kA{v^{\bf{3}}}\)

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 m and \(k = {\bf{0}}.{\bf{214}}\;{\bf{kg}}{\rm{/}}{{\bf{m}}^{\bf{3}}}\). Find the power output (in watts, \({\bf{W}} = {{\bf{m}}^{\bf{2}}} \cdot {\bf{kg}}/{{\bf{s}}^{\bf{3}}}\)) when the wind speed is \({\bf{10}}\;{\bf{m}}{\rm{/}}{\bf{s}}\), \({\bf{15}}\;{\bf{m}}{\rm{/}}{\bf{s}}\), and \({\bf{25}}\;{\bf{m}}{\rm{/}}{\bf{s}}\).

Short answer

a. The power output is increased by a factor of 8.

b. The power output is increased by a factor of 4.

c. The power outputs of wind turbine for velocities \(10\;{\rm{m/s}}\), \(15\;{\rm{m/s}}\), and \(25\;{\rm{m/s}}\) are \(605,000\,\,{\rm{W}}\), \(2,042,000\;{\rm{W}}\), and \(9,454,000\;{\rm{W}}\), respectively.

Step-by-step solution

Find the increase in the power output when the wind speed is doubled

When the speed is doubled, the power output by the function \(P = kA{v^3}\).

\(\begin{aligned}P &= kA{\left( {2v} \right)^3}\\ &= 8kA{v^3}\end{aligned}\)

So, the power output is increased by a factor of 8.

Find the increase in the power output when the length of blades is doubled

The area of the blades is \(A = \pi {l^2}\).

So, the power output of the wind turbine can be expressed as

\(P = k\left( {\pi {l^2}} \right){v^3}\).

When the length of the blades is doubled, the power output will be

\(\begin{aligned}P &= k\pi {\left( {2l} \right)^2}{v^3}\\ &= 4k\pi {l^2}{v^3}.\end{aligned}\)

So, the power output is increased by a factor of 4.

Find the power output

The area of the blade with length \(30\;{\rm{m}}\) is

\(\begin{aligned}{c}A = \pi {\left( {30} \right)^2}\\ = 900\pi .\end{aligned}\)

The power output from the wind turbine can be expressed as

\(\begin{aligned}P &= \left( {0.214} \right)\left( {900\pi } \right){v^3}\\ &= 192.6\pi {v^3}.\end{aligned}\)

The power output for \(v = 10\) is

\(\begin{aligned}P = 192.6\pi {\left( {10} \right)^3}\\ \approx 605,000\;{\rm{W}}{\rm{.}}\end{aligned}\)

The power output for \(v = 15\) is

\(\begin{aligned}P = 192.6\pi {\left( {15} \right)^3}\\ \approx 2,042,000\;{\rm{W}}{\rm{.}}\end{aligned}\)

The power output for \(v = 25\) is

\(\begin{aligned}P = 192.6\pi {\left( {25} \right)^3}\\ \approx 9,454,000\;{\rm{W}}{\rm{.}}\end{aligned}\)

So, the power outputs of wind turbine for velocities \(10\;{\rm{m/s}}\), \(15\;{\rm{m/s}}\), and \(25\;{\rm{m/s}}\) are \(605,000\,\,{\rm{W}}\), \(2,042,000\;{\rm{W}}\), and \(9,454,000\;{\rm{W}}\), respectively.