How do 3 blade turbines come so close to the Betz Limit?

I have seen many diagrams (see attached) that indicate that modern 3 blade turbines are nearly 50% efficient, which is about 80% of the Betz limit.

Betz calculated his limit based on a theoretical infinite number of blades, or an idealized "actuator disk".

How do 3 blades that only account for a fraction of the area of the disk reach so close to the Betz limit?

Clarification (after first 2 answers): The air hitting the blades at any one instant in time, should only be dependent on the blades' size, not on their speed too. Or am I incorrect in thinking this?

• Are you just changing your question so you don't need to accept an answer? Commented Jul 26, 2021 at 15:18
• Solar Mike, I clarified the question as I see it. If the blade speed is the reason then I would accept NMech's answer. Commented Jul 26, 2021 at 17:13

The Betz limit is based on an extremely crude model for an actuator disk. If you want to compare the performance of the product of a billion dollar industrial development program, let's at least use a respectable benchmark.

Image lifted from: Ideal Optimum Performance of Propellers, Lifting Rotors and Wind Turbines - https://openscholarship.wustl.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=2174&context=etd

See eqs. 46,47,48 for a comparison of Betz's ideal spanwise circulation vs the Galerkin ideal spanwise circulation.

Accounting for profile drag (and possibly a tip factor) yields the following graph for an infinite-bladed rotor.

So the reality is that for an ideal rotor in real air that creates a uniform wake around the entire cylindrical shell at all radii from hub to tip, accounting for profile drag, induced drag, swirl, and real tip vortex core phenomena, The optimum power coefficient is about 0.55 at a tip speed ratio in the five to six range.

A three bladed rotor can't produce a uniform wake because some of the air in the swept disc never gets very close to a blade. For example, take a 95.5 foot diameter turbine with three blades. Along the perimeter of the swept disc, the tips are 100 feet apart, so in the middle, the air is fifty feet from the nearest tip, yet it is scored in the swept area.

To understand what is going on, change you point of view to a stream tube flowing downstream behind the turbine. If the tip speed ratio is 5, the blade tip travels 5 feet for every 1 foot of free stream wind movement. So the blades chop the stream tube every 20 feet. That's a lot better than 100 feet. And when you account for the wake induction and deflection, it is less than that. So a stream tube has an infinite string of blade wake vortexes running downstream about 20 feet apart. When you integrate this half-infinite string of vortices to get the wake's real velocity profile, it ends up being pretty smooth over the entire wake. The vortices from far down stream have a significant smoothing effect on the overall wake.

Having said this, it is still better to have smaller vortices closer together than bigger ones further apart. And we can see that the tip speed ratios of real machines are designed to be a bit higher than the ideal tip speed ratio. This helps with efficiency and it helps with the mechanical gearbox and it reduces bending loads on the blades.

Here are two plots of the axial wake factors computed using a pretty good and a very good model of wakes. The second uses a full free vortex model for a long distance downstream.

Images lifted from: A Simplified Free Vortex Wake Model of Wind Turbines for Axial Steady Conditions - https://www.mdpi.com/2076-3417/8/6/866/pdf

The wind turbines basically extract energy from the moving air.

The Betz limit is essentially the maximum percentage of energy that is possible to extract from the air (the limit has to do with the fact that the air need to keep moving).

When you use 1 blade, when energy is extracted from one part of the disk the other side is relatively unaffected. In order to sweep and extract the maximum from the whole disk, one bladed wind turbines need to sweep faster. (That is why the optimal tip-speed-ratio is greater compared to three wind turbines). (Of course there is a limit where the speed is so great that other phenomena kick in and the extraction efficiency drops).

As you increase the number of blades, in the same time it is possible to extract more energy from the disk of moving air that intercepts the blades. Essentially you can do that with lower rotational speed, thus the tip-speed ratio is lower.

The three blade turbine is essentially the best compromise (that is one of the reason for its dominance in the wind generator market).

• I added this to the question, but wanted to get your input via comments here. The air hitting the blades at any one instant in time, should only be dependent on the blades' size, not on their speed too. Or am I incorrect in thinking this? Commented Jul 26, 2021 at 17:15
• In my understanding, the speed has two effects. 1) it changes (and optimises) the angle of attack so that it maximizes the $C_L$ value, 2) if the blades are static then there is air passing through the blades which is not interacting with the turbine (i.e. its relatively unaffected), as the blade rotates then the blade interacts with a larger volume of air with velocity $v_\infty$. At least that's how I understand it.
– NMech
Commented Jul 26, 2021 at 17:23