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Chapter 15: Problem 65

At a hearing on a proposed wind farm, a wind-energy advocate says an installation of 800 turbines, with blade diameter \(95 \mathrm{m}\) could displace a \(1-\) GW nuclear power plant. You're asked if that's really possible. How do you answer, given an average wind speed of \(12 \mathrm{m} / \mathrm{s}\) and a turbine power output that averages \(30 \%\) of the theoretical maximum?

Short Answer

Expert verified
The answer depends on the calculated average power output. If it is greater than or equal to 1GW, then the advocate's claim is true, otherwise it is false.

Step by step solution

01

Calculate the Swept Area of a blade

The area swept out by the blades can be calculated assuming they scribe out a circle as they turn. For a circle, \( A = \pi r^{2} \), where \( r \) is the radius of the circle. The diameter of a blade is given as 95m, we take half to get the radius. Hence, \( r = 95/2 = 47.5 m \). Thus, \( A = \pi (47.5)^{2} m^{2} \)
02

Calculate the total potential power of all turbines

The total potential power that all the turbines can get from the wind is \( P_{total} = \frac{1}{2} \rho A v^{3} \) times the number of turbines. Here air density \(\rho = 1.225 kg/m^{3}\) and wind speed \(v = 12 m/s\), calculated area \(A = \pi (47.5)^{2} m^{2}\) and number of turbines = 800. Plug these values in the formula and calculate for \( P_{total} \)
03

Calculate the average power output

The turbine power output averages 30% of the theoretical maximum. Therefore, the average power output of the farm is \( \frac{30}{100} \) times \( P_{total} \). Calculate this to find efficient wind energy generated.
04

Compare with nuclear plant output

Compare the calculated average power output with the power output of a 1-GW nuclear power plant. If our calculated power output is greater than or equal to 1GW, then the advocate's statement is correct.

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