Turbine go boom
Friday, February 29th, 2008Wind turbines sit quietly on mountaintops, silently spinning away. Stealing a little bit of the wind’s kinetic energy, these turbines convert the energy the wind imparts to electrical energy. As power generation goes, wind is one of the most environmentally-safe sources we have. There is essentially zero carbon emission, other than that required to make the turbines in the first place.
Last week, a windstorm in Aarhus, Denmark caused the braking mechanism for one particular turbine to fail.
This was the result:
In this particular accident, it’s theorized that one of the rotor blades struck the tower as the turbine whirled out of control. In turn, the entire system was knocked out of balance and the remaining blades disintegrated immediately thereafter. Danish site Jyllands-Posten has an article on the collapse (in English).
Wind turbines that spin too fast are a very bad thing. In order to reduce the rotational kinetic energy, most turbine systems use a braking mechanism to slow the rotors down to a manageable speed. Often, the heat generated as a result of braking is used to heat the tower itself.
To find the rotational kinetic energy of a wind turbine, assume that it’s made up of three slender rods coming out of a central hub. According to windturbines.ca, the mass of a Nordtank 600 kW turbine’s rotor blade is 2,000 kg and the rotor diameter is 43.0 m. We’ll assume that the length of one of the blades is half that, or 21.5 m. Its rated rotational speed is 27 revolutions per minute (though we’ll convert to 2.83 radians/second).
The variable I is known as the rotational inertia; basically, the resistance to rotational acceleration. The variable ω (Greek letter omega) is the angular velocity. We have to calculate the rotational inertia of each blade. We’ll use the formula I = (1/3)mL2, where I is the rotational inertia, m is the mass of the rotor blade and L is its length. This makes I = (1/3)(2000 kg)(21.5 m)2 = 308,200 kg · m2. But there are three blades, each with an enormous 308,200 kg · m2 of inertia. The total is 924,500 kg · m2! For reference, the rotational inertia of a bicycle’s tires is about 1 kg · m2.
Now that we have the rotational inertia, the kinetic energy is easy. The kinetic energy is one-half of the rotational inertia, times the angular velocity squared (or, speaking math, KE = ½Iω2). Plug in all the numbers above, and we find that the kinetic energy is a whopping 3.7 million joules at rated speed. For reference, this is about the energy of two mid-size cars colliding head-on at 110 miles per hour!
But what about the speed at which the turbine was actually turning at the time of failure? Watching the video, I’ve been trying to count the number of turns per second and it’s really hard. The best I can do is count to 25 as blades pass by the vertical. Since there are three blades, seeing 25 blades means 8 1/3 rotations. I timed it three times and got 3.60 s, 3.53 s, and 3.50 s. Close enough. Take the average of these three (3.54 s) we find that the speed of rotation is about 2.4 revolutions per second, or 14.8 radians per second.
Notice that the energy varies as the square of the speed at which it rotates. If you were to double its rotational speed, this particular turbine would have four times the kinetic energy. In our case, since we quintupled its rotational speed, this turbine’s energy will be twenty-five times its previous value.
The rotational kinetic energy is 101 million joules! This is the same as the energy of a collision in which two cars collide head-on, each one traveling 580 miles per hour. That’s about three-quarters the speed of sound!
I’m glad I wasn’t standing under the thing.