There is, of course, a standard theory of wind turbines, similar to propeller and helicopter theories. This is known as actuator disk theory, and goes roughly like this:
At any point along any of the blades, it is possible to work out the local apparent wind, which comes mainly from the rotational speed and partly from the oncoming wind, since the turbine will be rotating faster than the wind is blowing. From this you can calculate the local lift and drag forces on the blade at that point, which vary in size and direction all over the rotor. If you then add all these up, you can get the overall forces and the power output on the turbine.
Unfortunately, the wind speed at the turbine is not the same as the wind speed upstream, since extracting power from the wind slows the wind down. Therefore, you do not actually know what the wind speed is in order to do these calculations. However, using basic principles of physics such as conservation of mass, you can also work it out in terms of the change of the momentum of the wind as it passes through the rotor. Then you use the fact that the axial thrust from one set of equations is equal to the change of axial momentum in the other set, and so on. When you balance them, the unknown losses can be found and a bit of maths then gives you the thrust and the power.
Of course, it is more complicated than that really, with the downwind blades passing through the wake of the upwind blades, and various losses at the tips and the hub, but I will not go into all that here and now.
The simplest form of the theory involves assuming that the losses are equal everywhere, which is not very accurate but gives you a very simple expression for the thrust and power in one equation. However, using such a rough method to design the blades is not such a good idea, and calculating back to find the blade stresses with this method will not be very accurate either.
Different forms of the theory involve either making assumptions about how the losses are distributed, or performing the whole force balance calculation separately at a whole lot of simultaneous points - which is no problem if you are doing it by computer.
I have been developing the theory for turbines at an angle to the wind, and my results so far agree fairly well with the small amount of measured data that I have been able to get hold of, so maybe I am on the right lines. But when it comes to coned turbines, I have no measured data to compare my theory with. As for cones at an angle to the wind - it really needs a series of model tests with accurate force measurements.
For those members who have joined more recently than 1989 and have not seen any of those, do not despair because I think that Joe Norwood, who has written the soon-to-be-released AYRS 120, (due out in Jan 1996) has put in a chapter on wind turbine boats in which the same theory appears yet again, in yet another different notation.
These theories should all give the same answers, because they are all the same standard theory just restated in different notation, and are all based on a wind turbine going straight into the wind. None of them make any mention of non-axial flow.
I suppose I shall be publishing my non-axial, coned turbine theory somewhere at some time, but I want to test it out against actual measurements first, in case I have got something wrong.
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