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Aeronautical Engineering/Wind turbine blade twist angle

Question

Consider a rotor radius of 5 meters.
The wind speed is 20 m/s.
The tip speed ratio is 8; meaning the tip of the radius is traveling at 160 m/s.
Diameter of the rotor multiplied by pi = a rotor circumference of 31.415926 meters
Tip speed divided by circumference = 5.0929582 rotations per second; multiplied by 60 for an rpm of 305.57749
Rpm multiplied by 0.1047 = 31.993963 radians per second.
Radius multiplied by radians per second = 159.96981 (we will call this "r*w")
Wind speed squared = 400 m/s (we will call this "m/s2")
We now square r*w to get a value of 25590.340 (we will call this "r*w2)
We now add r*w2 and m/s2 together to get a value of 25990.340 (we will cal this "Val1")
The square root of Val1 = 161.21519 (we will call this "Val2")
We divide r*w by Val2 to get 0.9922750 (we will call this "Val3")
Arccosine of Val3 in degrees = 7.12634508 (we will call this "Val4")
The instructor of this video states in the next video that Val4 should equal 8!!!!

Where am I going wrong?

Brad - I think you are confusing a ratio with an angle. What you call Val4 is the angle between the tip speed vector and the resultant vector comprised of the tip speed and wind speed. It is the angle whose cosine s 160/161.245. You made a small error in squaring the tip speed: it should be 160^2 = 25,600; you got 25,590.34. But that doesn't matter much because your value for Val4 = 7.126 is very close to the value I got of 7.124. In any case, that is the angle described above. The ratio of tip speed to wind speed is 8, which I think you are confusing with Val4. I confess that I didn't have time to watch the whole video, but the only place I saw the author use the value 8 was in describing the ratio of tip speed to wind speed.

Paul

Aeronautical Engineering

Volunteer

Paul Soderman

Expertise

Aeronautics, Aerodynamics, Fluid Mechanics, Aeroacoustics, Noise Control, Muffler Design, Wind Tunnel Research.... I know nothing about India - do not ask about schools, jobs, application requirements, career choices, etc. for India. Please, no text message verbiage; I prefer full words in full sentences. Thanks.

Experience

38 years as research engineer at NASA

Publications
AIAA, NASA

Education/Credentials
B.S. and M.S. Aeronautical Engineering - U. of Washington, Graduate work Standford U.

Awards and Honors
AIAA Associate Fellow (American Institute of Aeronautics and Astronautics)