Jun 13, 2019 9:00 AM

Estimate the Energy of an Utterly Massive Wind Turbine

The Hornsea Wind Farm is expected to be the largest offshore wind farm in the world. Here's how much energy is generated from one rotation of a turbine.

[#video: https://www.youtube.com/embed/TgRPjCQn7Tw

It's amazing that we can get electrical power just from the wind, but that's exactly what happens with a wind farm. It's a collection of wind turbines in a particular location with abundant wind. And as with many things, bigger is better. The Hornsea Wind Farm, for example, is being built 75 miles off the coast of Yorkshire, England, and upon completion it is expected to be the biggest offshore wind farm in the world. These are some big wind turbines: The video lists the turbine diameter as 154 meters (more than 500 feet).

But here's the part I'm interested in. They claim that just one turn of these giant wind turbines can generate enough energy to power a house for a whole day. You know what comes next right? An estimation.

Of course, I'm not an expert on renewable energy. I'm just a plain physicist. This means that I can take some very basic fundamental ideas along with some rough estimates to calculate the energy from one turn of the turbine. Sure, the numbers might be off—but that's what makes this so much fun.

Oh, and don't worry. I'm going to include my calculations in a python script below. If you don't like my estimations, you can even put in your own values.

I suppose we should start with the fundamental physics principles that make this wind turbine create electricity. Like most other methods of generating electricity, it involves turning a coil of wire in a magnetic field. When a changing magnetic field passes through a coil it produces an electric current. But then how do you get that coil to turn? For a wind turbine, the air moving into the large blades pushes them in a rotating motion.

Of course you don't have to use wind to turn a turbine. You could use running water by placing some type of propeller-based blades in moving water like a stream or from a dam (hydroelectric power plants do this). Another popular option is to heat up water and let the steam turn a turbine. This is how coal and natural gas power plants work. It's also how a nuclear power plant operates. The only difference between nuclear and coal power production is that one burns a fossil fuel to heat up water and the other uses a nuclear reaction. The turbine parts are essentially the same. It's crazy if you think about it. Really, the only power generation method that's different is a solar panel. That works with a different method.

That's great and all, but how do you estimate the power (and then the energy) from a wind turbine? All of the energy comes from the kinetic energy of moving air. Kinetic energy can be calculated as:

Rhett Allain

In this expression, m is the mass of air and v is the air speed. Let's assume that the air enters the turbine with a speed of v₁ and then leaves at a slower speed of v₂. This decrease in speed means a decrease in kinetic energy—and it is that energy that eventually turns into electricity. The speeds might be easy to estimate, but what about the mass of the air?

Suppose that we have a large cylinder of air that enters the turbine. The cross-sectional area of this cylinder is a circle the same size as the rotating props and the length of the cylinder is just a value—for now I will call it L. This diagram should help.

Rhett Allain

What is the mass of this giant cylinder of air? Well, if I know the density of air (I do) and the volume of a cylinder (yup), I can calculate the mass of air. Remember, a good estimate for the density of air (ρ) is around 1.2 kilograms per cubic meter.

Rhett Allain

There is still one thing missing—the length of this cylinder of air (L). If the air is moving with a speed of v₁ for some interval of time (Δt), then the length of this cylinder would be:

Rhett Allain

This just comes from the definition of velocity in one dimension. Now instead of estimating the length, I can estimate the time interval. Remember we are trying to find the energy from one rotation of that wind turbine. I can just look at the video and see an actual rotating turbine. My estimate is that it takes about 4 seconds to make one rotation.

There are only a few more things to estimate:

Initial air velocity = 5 meters per second (11.2 mph)
Final air velocity = 2.5 meters per second (5.6 mph)
Efficiency (fraction of kinetic energy that goes to electrical energy) = 0.25
Average power consumption for a house = 2,000 watts.

Now for the calculation. Here it is (you can change the values if you disagree with my estimates). If you want to see and change the code, click the pencil icon. Click the Play button to run it.

So, that didn't work. Using my values, one turn of the wind turbine creates 291 watt-hours (a unit of energy), but a house uses about 48,000 watt-hours. Well, I should add that energy is an approximation for a US house. In the UK, it is significantly lower at around 12,000 watt-hours. But I'm still off by a factor of about 40—that's quite a bit. That means you would need 40 turbines (or 40 rotations) to get enough energy to run a house for a day. OK, so what went wrong? There are a couple of options. Obviously, my estimations could be off. But why would they be off by so much? Even if I double the wind speed and put the final air speed at 0 m/s, it's still not enough energy.

But there is another way to approach this physics problem. The rated power production for the turbine in the video is 7 MW (that’s 7 million watts). On closer inspection the rotation time for a single blade might be closer to 6 seconds (earlier I'd said 4), so let's use that number. If we take that rated value and multiply by it by a rotation time of 6 seconds, you'd get an energy of 11.7 kilowatt-hours. This is just about the right value for the energy needed to run a UK house for one day.

Why are the two values different? The most obvious answer is that my rough estimation was just that, an estimation. I had ballpark guesses to some of the parameters, such as windspeed and efficiency. But the physics is still valid—and still fun (at least for me).

Updated 06-19-2019, 3 pm EDT: This story has been updated to clarify the author’s calculations and to include the typical energy usage of a UK-based home.