Yup, there's a big penalty to pay for high altitude.
You can take some comfort that forces are reduced on the tower.
That comfort goes stale when you also notice that the furling tail needs those forces to work, too. So the force that keeps it facing the wind is also reduced, making the two effects cancel out.
What I wanted to add to the stuff I posted before is where the curves come from. Making the graphs is really just number-crunching for the computer, thankfully, so here is the general method.
It's called Blade Element Analysis. There's a simplified form that you can cram onto one page of paper if you just want one design point. The key to simplified BEA is to do the calculations at the 3/4 radius point on the rotor, and extrapolate the rest. It works surprisingly accurately, and you can use it for aircraft propellors just as well if you want.
Start with the basic geometry
Radius = 4.5 feet
3/4 Radius = 3.4 feet
chord varies between 5 inches at the root and 2 at the tip
chord at the 3/4 radius = 3.0 inches
The airfoil could be anything so just as a guess:
slope of the lift curve = 0.090 1/degree
angle of zero lift = -2 degrees
incidence of blade twist = 2 degrees
Now let's suppose the wind is 30 kph, and it's turning at 500 RPM.
This is a point on the graph above so you can check my work.
The tip speed ratio is 8.6
500 RPM = 52.4 radian/sec
30 kph = 27.3 ft/sec
To get the angle of attack of the blade, you use this:
AoA = arctan (1/TSR) = arctan (1/8.6) = arctan (0.12) = 6.6 degrees
But we want it at the 3/4 radius point, which is different:
AoA = arctan [1/(TSR*3/4)] = arctan [1/8.6*3/4)] = arctan (0.155) = 8.8 degrees
Here comes the tricky part. This doesn't often come up in discussions of building and testing, but there's an "inflow factor" that also has to be considered to do this right. Good flow through a wind turbine reduces the wind exiting down to 33% of the incoming wind. The wind speed passing the disk is then 67% of the free wind speed. This factor changes the angle of attack of the blades. To include this angle in the angle of attack:
AoA = arctan [1/TSR*3/4*(1+0.33)/2] = arctan (1/8.6*3/4*0.67) = arctan (0.060) = 5.9 degrees
This gets the coefficents of lift and drag at the point 3/4 out the radius:
CL = 0.090 1/degree * (5.9 + 2.0 - 2.0) = 0.54
CD = 0.020 + (5.9 + 2.0 - 2.0)^2 / 2.1 = 0.025
With these coefficients we can find the forces acting at this point on the radius:
We also need the density of air now. This is where the altitude is a penalty:
rho, sea level = 0.002378 slug / cubic foot
rho, 10,000 ft = 0.001800 slug / cubic foot
The local airspeed at this point on the radius is
V,local,3/4 = 3.4 feet * 52.4 radian/sec = 178 ft/sec
dL = (0.001800 slug / cubic foot) / 2 * (178 ft/sec)^2 * (3.4 ft) * (0.54)
dL = 17 pounds
dD = (0.001800 slug / cubic foot) / 2 * (178 ft/sec)^2 * (3.4 ft) * (0.025)
dD = 0.8 pounds
Each of these forces acts ALMOST at right angle to the direction the blade is moving, but not quite. The tiny remaining angles give the lift the ability to create useful torque, while the drag mostly resists it.
Torque, lift = 3 blades * dL * R * 0.75 * sin(AoA) = 3 * 17 Lb * 3.4 ft * sin(5.9deg) = 208 Lb*in
Torque, drag = 3 blades * dD * R * 0.75 * cos(AoA) = 3 * 0.8 Lb * 3.4 ft * cos(5.9deg) = 96 Lb*in
Net Torque = 208 - 96 = 112 Lb*in
Power = Torque * speed = 112 Lb*in * 52.4 radian/sec = 664 Watt
Which is basically how any of the points on the curves above are calculated.
I grant that I skipped a lot of info about the inflow factor. It's related to the coefficient of power Cp and the factor used for calculation is Cp=0.59.
The reason I picked the point that I did for this calculation is that it's the point of max Cp on the curves above, so that it would make this part of the calculations straightforward. If I was to work on a different point on the graph, less or more RPM, then I'd need to figure out the effect on Cp and then change the inflow factor. That causes you to dig in to the Lift/Drag ratio of the airfoil which, in this case, peaks at 6 degrees.
