Tip Speed Ratio: How to Calculate and Apply TSR to Blade Selection
The Tip Speed Ratio (TSR) is used by wind turbine designers to properly match and optimize a blade set to a particular generator (i.e. the permanent magnet alternator). This is important to answer one of the most common questions we get: What size blades should I choose to match with my generator?
We attempt to help you answer this question by focusing on explaining the simple physics behind calculating the Tip Speed Ratio!
Understanding Tip Speed Ratio
By definition, TSR is the speed of the blade at its tip divided by the speed of the wind. For example, if the tip of a blade is traveling at 100 mph (161 kph) and the wind speed is 20 mph (32 kph or 9 m/s), then the TSR is 5 (100 mph/20 mph). Simply put, the tip of the blade is traveling five times faster than the speed of the wind.
Now, you must be wondering why this is important. For a particular generator, if the blade set spins too slowly then most of the wind will pass by the rotor without being captured by the blades. If the blades spin too fast, then the blades will always be traveling through used/turbulent wind. This is because the blades will always be traveling through a location that the blade in front of it just traveled through (and used up all the wind in that location). It is important that enough time lapses between two blades traveling through the same location so that new/unused wind can enter this location. Thus, the next blade that passes through this location will be able to harness fresh/unused wind. In short, if the blades are too slow they are not capturing all the wind they could and if they are too fast, then the blades are spinning through used/turbulent wind. For this reason, TSR’s are employed when designing wind turbines so that the maximum amount of energy can be extracted from the wind using a particular generator.
Without going into details, physics and research have shown that the approximate optimal TSR’s for a given blade rotor are:
|TSR||Number of Blades|
There are many important conclusions one can draw from analyzing TSR’s. For the do-it-yourselfer that is putting together their own wind generator, let’s go over a few of the most basic and important points:
- Rotors with many blades (i.e. 11 blades) are generally not a good idea. An 11 bladed rotor would have an optimal TSR which is very low. This means an 11 bladed rotor would operate most efficiently at extremely low rpm’s. Because nearly all generators (permanent magnet alternators) are not optimized for extremely low rpm’s, there is no advantage or reason to use a rotor with many blades. Remember, rotors with lots of blades are capturing used/turbulent wind at high TSR’s and are thus extremely inefficient if used as a high-rpm blade set. This is a very important point because many people intuitively think that more blades equal a faster and more efficient blade set. But, the laws of physics say that this is not true.
- If you already have a generator or a motor and it requires high rpm’s to reach charging voltage, then your best bet is a two or three blade rotor. These rotors operate more efficiently at high rpm’s. Also, keep the blades as short as pragmatically possible because shorter blades obviously spin faster than longer blades.
- Last but not least, keep in mind the Tip to Speed Ratio! If your wind generator rotor is operating at a low TSR compared to the optimum value, then your wind turbine’s blades will tend stall before hitting maximum power/efficiency. If the wind turbine’s blades are spinning above the recommend TSR, then the blades will be traveling through turbulent wind. Not only is this inefficient, the turbulent wind puts your blades and entire wind turbine under unnecessary stress and fatigue.
How to Measure TSR
Measuring the TSR of a blade set is fairly easy. To accomplish this measurement you will need two things:
- A digital tachometer. These are available online for about 25 USD and can be used to measure the rpm’s of a blade set.
- A anemometer. A digital anemometer can be purchased online for fairly cheap (~20 USD) and is used to measure the wind speed.
With these two items, you can obtain the necessary measurements to calculate TSR’s. But, one question does remain. How do we calculate the speed at the tip of a wind turbine blade, if we only know the rpm at the tip of the blade from our tachometer measurement? Well, we have to do a little math. Let’s break down this calculation step by step:
Distance the tip of the blade travels to complete one revolution = circumference of a circle with radius r = (2)(Π)(r)
where r = the length of the blade.
What distance does a one meter blade travel to complete one revolution?
Answer: Distance = (2)(Π)(r) = (2)(Π)(1 meter) = 6.28 meters
Now, let’s assume we measure an rpm of 450 at the tip of the blade using our digital tachometer. How far does the tip of the blade travel in one hour?
Answer: 450 rpm = 450 (rotations)/(minute) = 450 r/min
(450 r/min) x (60 min/hour) = 27000 rotations per hour = 27000 r/hour
(27000 r/hour) x (1 hour) = 27000 rotations
(27000 rotations) x (6.28 meters/rotation) = 169,560 meters
Note: we know that the blade tip travels 6.28 meters in one rotation because this is the first calculation we did!
So, now we know that the tip of the blade travels 169,560 meters in one hour. Now, let’s convert the meters to miles:
169,560 meters x (1 mile)/(1609 meters) = 105 miles
Alright, we are almost finished. Now we have to calculate the speed at the tip of the blade. This is easy because we know the tip of the blade traveled 105 miles in one hour. See calculation below:
Distance = (rate) x (time) and rate = (distance)/(time)
Rate = (105 miles)/(1 hour) = 105 miles/hour = 105 mph
That’s it! The tip speed of this particular blade is 105 mph at 450 rpm. So what if the wind was blowing at 20 mph when we measured 450 rpm. What is the TSR? That’s easy:
TSR = (Blade tip speed)/(wind speed) = (105 mph)/(20 mph) = 5.3