Turbine go boom
Wind 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.
Tags: boom, energy, explodo, kinetic energy, power, rotation, turbine, wind
February 29th, 2008 at 3:15 pm
That was wicked.
March 1st, 2008 at 8:37 am
That’s damned amazing. Surprised someone just happened to be there to film it.
March 8th, 2008 at 11:01 am
@Cecily: I know! I thought the same thing! Talk about being in the right place at the right time.
I’ll think about this next time I book a vacation to Aarhus.
“But there are three blades, each with an enormous 308,200 kg · m2 of inertia. The total is 924,500 kg · m2!”-But wouldn’t it be 924,600?
Which reminds me, how do I do HTML in here? It didn’t work last time, and I really wanted to put some italics on the “6″ this time.
March 20th, 2008 at 7:09 pm
Wow, I was going to write a post about this video, it makes you think about near a turbine. The regulated size for a wind turbine that size is around 2000 feet, but that still seems to be kind of close.
March 21st, 2008 at 5:15 am
I just realized that when I type on my laptop, some of the words get cut out, because of my hand rubbing on the pointer pad, so let me correct my previous post:
Wow, I was going to write a post about this video, it makes you think about LIVING near a turbine. The regulated SEPARATION DISTANCE for a wind turbine that size is around 2000 feet, but that still seems to be kind of close.
Okay, I feel better now!
March 21st, 2008 at 10:57 am
@Rob: It is kind of close, but it doesn’t look like any really large pieces of debris went that far… for scale, the diameter of the plane in which the blades rotate is 140 feet. It doesn’t look like the debris was thrown more than about 2-3 times that distance.
Wind is still by far one of the safest and most environmentally-responsible power sources out there, regardless of this little accident anyway.
Regardless, we have to admit, it was crazy-cool to watch. :-)
March 21st, 2008 at 5:22 pm
Too bad wind is only about 40% efficent as a power source..
March 22nd, 2008 at 4:49 pm
@Rob
What’s your baseline for the 40% number? Total lifetime energy recouped versus energy going into production of the turbine?
If it’s energy out versus the wind’s kinetic energy in, I’d say 40% is pretty good. Though when you’re not paying for the energy on the way in, even 2% would be good: no one’s affected by the wind slowing down a little bit, so every bit of energy we get out might as well be free.