Hi Hugh & Jim,
Here I seem to have sent this thread off in an unexpected direction. I think I was splitting hairs, there, but you took me seriously anyway.
This is perhaps a better explanation of what I was talking about:
Although capturing energy from the wind up to the Betz limit is a lofty goal, it is very difficult to do, and not a realistic one for the amateur builder.
My windmill has a high TSR at cut-in. The corresponding Cp is pretty high. As the wind speed increases, the blades turn faster, but the generator load also increases. The TSR goes down, and so does the Cp. At wind speeds around 10 m/sec, the Cp is about 30% and the TSR is 6.5. My design TSR was 6. Obviously not a perfect windmill, but it's actually better than I expected. I can make these assertions because I have logged data from my wind mill and anemometer. To be honest, the data is not very good, but the trends are pretty obvious.
My point is that at cut-in, I see high Cp, but that's hardly a useful design criterion. There is little power to be captured there, anyway. At higher wind speeds, the TSR will drop, bringing the Cp down with it. There's not much you can do about it. Fundamentally, increasing the CL of an airfoil brings with it an increased CD. The best one can do is improve surface smoothness, carve twist for even span-wise loading, and pay attention to details like nose cones, reflex angles of incidence at the tips, and fragile sharp trailing edges, and so on.
So when I replied to the original post, I just picked Cp=33% because it's a number that's realistic for me. If I was trying to make a model of the wind on my machine, that's what I'd use. Since Cp implies that 1-0.33 = 67% energy is captured, I just quickly estimated that the output wind speed would be sqrt(1-0.33) = 82% of the free wind speed. Then I split the difference between the start and finish, to find the wind speed at the blade itself.
You were suggesting that the wind speed out the back can be reduced by sqrt(1-0.597)=63% of the free wind speed, and hence 80% wind speed is present at the blade airfoil. That isn't likely to happen because operating at both the design TSR and the max Cp possible during the design wind speed is very difficult to achieve.
There is nothing wrong nor dangerous in designing for a higher Cp than you are likely to achieve, because it's a lot like designing for too-high TSR. The ultimate result is a higher lift coefficient, higher drag, and a slower blade, that may actually stall-regulate (a bit). This makes the tail's job easier, so the windmill is safer.
Thank you Hugh and Jim for the engaging discussion of this topic, because I was ignorant of these facts up until not long ago, and built blades last year that, luckly, aren't too far "out of whack".
