How much power would a moderate-sized 50 %- efficient wind...

How much power would a moderate-sized $50 \%-$ efficient wind turbine produce whose radius is $10 \mathrm{~m}$ at wind speeds of $5 \mathrm{~m} / \mathrm{s}$, $10 \mathrm{~m} / \mathrm{s}, 15 \mathrm{~m} / \mathrm{s}$, and $20 \mathrm{~m} / \mathrm{s}$ ? Express the answers in $\mathrm{kW}$ or $\mathrm{MW}$, depending on what is most natural.

Key Concepts

Turbine Efficiency

Turbine efficiency accounts for the fact that not all kinetic energy in the wind can be converted into electrical energy. This efficiency factor, expressed as a percentage, represents losses due to aerodynamic, mechanical, and electrical factors, and is used to calculate the actual power output relative to the ideal power available.

Cubic Relationship of Wind Speed

The power available in wind scales with the cube of the wind speed. This means that even modest increases in wind speed can lead to significant increases in the energy that can be harnessed by a wind turbine.

Wind Kinetic Energy

This concept involves recognizing that moving air carries kinetic energy which can be converted into electrical energy. The power available from the wind is calculated using the kinetic energy of the air, based on its density and the cube of its velocity.

Swept Area of the Turbine

The swept area is the circular area that the wind turbine's blades cover as they rotate. The power intercepted by the turbine is directly proportional to this area, which is calculated using the formula A = ?r², where r is the radius of the blades.

Transcript

00:01 Okay, in this problem we have a wind turbine with an efficiency of 50%.

00:07 So this is going to give us something called a power coefficient for a wind turbine equal to 0 .50.

00:18 So that's just the efficiency really.

00:21 And then we're told that the turbine has a radius of 10 meters and it operates at four different wind velocities.

00:32 So the first one is 5 meters per second.

00:35 The second is 10 meters per second.

00:39 The third is 15 meters per second.

00:44 And the last velocity is 20 meters per second.

00:48 And we're also going to need the density of air for this problem.

00:53 And that's about 1 .225 kilograms per cubic meter.

00:59 Now we want to find the power at each of these different wind speeds.

01:06 So the power is given by 1 half times the density of air times pi r squared, which is just the area that the turbine covers.

01:23 And that's multiplied by the velocity of the wind cubed.

01:29 Now the efficiency is 50%.

01:32 So we're actually going to get about half of the power that this equation gives.

01:38 So the equation we have to use is going to be 0 .5 times 1 half times the density of air times pi r squared times the velocity cubed.

01:54 We'll go ahead and find the power at each of these different velocities.

02:00 So we'll say p1 is equal to 1 half times 0 .5 times the density of air, 1 .225 kilograms per meter cubed times pi times r, the radius, squared times the velocity of 5 meters per second cubed.

02:30 Now just to make it simpler to find the other powers, this part of the equation is going to stay the same for the others and just the velocity is changing.

02:40 So when we multiply all of this together, we get about 96 .21.

02:46 And then we're multiplying this by 5 meters per second cubed.

02:50 And that's going to give us a power of about 12 ,026 watts...

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