# Optimal size of a windmill

1. Sep 18, 2014

### McLaren Rulez

Hello folks,

I tried googling this but found surprisingly little on it.

Here is the problem: Assume that you have some constant wind speed. I want to run a windmill but I need to decide how big a windmill I want. The size is characterized by the length of the blades, r. There are no engineering constraints (I can build perfect, balanced windmills of any size). However, the mass of the windmill goes as r^2 and there is some constant friction coefficient on the axis. Given these conditions, is there an optimal r? The goal is to generate as much electricity as possible with the set windspeed.

Okay, that was the physics part. If your answer was that the windmill should be built as large as possible, then what are the common engineering problems that occur as we scale up? In other words, why are real windmills not bigger than they currently are? And why are some smaller while others are bigger?

Thanks!

Last edited: Sep 18, 2014
2. Sep 18, 2014

### rcgldr

3. Sep 18, 2014

### McLaren Rulez

The wiki article does not answer the question of optimal size, or even if there is an upper limit to this optimal size.

https://en.wikipedia.org/wiki/Wind_turbine_design#Turbine_size tells you that it is limited by the strength of the material or the stiffness of the blades but assuming that those are not constraints, is it correct that bigger is always better?

4. Sep 18, 2014

### A.T.

Did you mean r^3? Or is it actually r^2 because the blades are hollow with constant wall thickness?

Is the radius of the axis (the radius at which the friction acts) proportional to r?

5. Sep 18, 2014

### 256bits

I suppose one does not get into a wing so large that the difference in the wind speed ( for example with elevation ) is such that increases in swept area produce minimal returns in power.

6. Sep 18, 2014

### A.T.

The wind gradient is already a problem for existing sizes of windmills. But I guess the OP asks about an idealized uniform wind field of arbitrary size.

7. Sep 18, 2014

### CWatters

Last edited: Sep 18, 2014
8. Sep 18, 2014

### McLaren Rulez

Thanks, the NatGeo article is very nice. So in some sense, bigger is better, no?

AT, yes I assumed constant thickness blades so it would be r^2? And yes, I wanted to assume a uniform wind field too.

9. Sep 18, 2014

### cjl

Bigger will always generate more power, all else equal. However, you are limited by structural constraints, manufacturing constraints, transportation (blades on modern multi-megawatt wind turbines are enormous), and that sort of thing.

10. Sep 18, 2014

### CWatters

In england most new onshore wind turbines are 126m tall. There are a few at 130 and possibly 140m.

One issue is that wind shear can cause the noise they make to become amplitude modulated and people find this more annoying.

11. Sep 18, 2014

### A.T.

So assuming a constant coefficient, friction will also scale with r^2, just like the aerodynamic force on the blade. Then I don't see how friction could start to catch up, and therefore should be no finite size that maximized the net gain under these assumptions.

12. Sep 18, 2014

### AlephZero

If you ignore the constraints that you don't like (and make wild assumptions like "I can build perfect, balanced windmills of any size") you can get any answer you want.

In the real world, if you start from the basic assumption that the maximum possible power available is proportional to the cross section area swept out by the turbine blades, eventually you reach a size where building two turbines each of area A is cheaper than building one of area 2A.

There are other constraints apart from the mechanical design. Where I live, a local water company built two large turbines (130m high) on its site to sell the electricity. After installing them, they discovered they were messing up the radar system at an airport about 10 miles away. They have been sitting there doing nothing for almost 12 months. They are just starting some low-speed testing, after spending about £500,000 upgrading the radar.

Last edited: Sep 18, 2014
13. Sep 18, 2014

### Staff: Mentor

Square-cube law - your windmill mass cannot scale with r^2 for arbitrary sizes, in the same way you cannot scale up an ant to the size of a human to carry hundreds of tons. Lighter materials allow larger turbines and taller towers, but this is always an engineering/physics issue to consider, and it limits the size of turbines.

14. Sep 19, 2014

### A.T.

Yeah, there is no way around it. When you use hollow elements with constant wall thickness to get m ~ r^2, the crossectional area supporting the load is just ~ r, the the stresses still increase ~r.

15. Sep 20, 2014

### enorbet

In any practical application, I have problems with the very first assumption "Assume that you have some constant wind speed". AFAIK this just doesn't exist. All sizes and designs have optimal operating conditions, both high and low limitations as referenced to wind velocity. Some designs, such as the Savonius Rotor Turbine will begin to turn and produce some power at even nominal wind velocities but become inefficient as velocity increases. Catenary design has the highest power coefficient but takes a very substantial wind velocity for startup. TANSTAAFL