The maximum mechanical power available on the shaft of a windmill is given by the formula:
P = 0,49 CP D²V3
D being the diameter of the windmill and V the wind-speed.
Optimize the power of a windmill requires then to design it to get an optimum power-coefficient Cp. The theory states that Cp cannot exceed 0.6 (Betz limit 60%)
It comes from the windmills theory that rotational-speed exchanges with torque and vice-versa. Since in this project we need to implement a powerful water-friction-brake to heat water, we will naturally prefer a high-torque to stirr larger quantities of water. Consequently the windmill will be of the low-speed type.
On this sketch we see that the american-multiblade windmills "B" have the highest torque. Their tip-speed ratio lambda is about "1", meaning that they get their best efficiency when the linear tangential speed at the end of the blades equals the actual wind-speed.
The other types of windmills with 2 or 3 blades must be driven to very higher rotational speeds to deliver their maximum power, and their torque is less important.
The more the cumulated area of the blades, the less the tip-speed ratio lambda, as confirmed by the Hütter study dealing with the "solidity" of the windmills.
Nevertheless a trade-off has to be done, since increasing the number of the blades increases its cost in terms of material. The weight of the windmill and the drag on the supporting tower are also increased.
A better approach of the Cp coefficient requires to look into the theory developped by G. Schmitz in 1953, that takes into account the effects of the downstream turbulences generated by the windmill.
The losses in terms of efficiency are all-the-more important that the solidity is high and the tip-speed ratio is low.
So, by using a low-speed windmill, we will have to accept to operate with a reduced Cp coefficient, on the left part of the curve.
We will see now, what is the effect of the profile of the blades on the Cp coefficient, since implementing a large number of sophisticated blades could result in a prohibitive cost.
These next curves are similar to the previous one, but show in details the influence of the parameter sigma on the Cp coefficient.
Sigma = Cl/Cd is the ratio between the lift and the drag coefficients characteristics of a given profile. The highest the sophistication of the profile, the highest the sigma figure, and the highest the cost of implementation.
The curves show that with a low tip-speed ratio, and provided that the number of blades is high, the Cp does not vary very much with the sigma (between 0.38 and 0.47 in blue).
Therefore we will accept a rather low sigma in our design. Like most of the american windmills, the blades will be made ot of mere bended steel-sheets. The expected sigma is around 14, which will result in Cp=0.4
These curves correspond to various bendings of the sheets. They are calibrated with the angle of attack alpha.
We have selected the fourth one with a ratio 0.05 between the deflection "f"and the chord "L".
The optimum sigma corresponds to the tangent to the curve from the origin O. It meets the curve for the optimum angle of attack alpha=5° and the corresponding sigma= Cz/Cx=0.7/0,05=14
The next step is to work-out the length of the chord "c" for each radius "r" along the blade, together with the corresponding static pitch angle "beta". We will use the formulas from G. Schmitz:
The rows in pink correspond to the positions where the blade is maintained in position by a mechanical device.
An internal portion with a diameter of 0.8m is kept without blades and almost hollow to reduce the drag on the windmill, where anyhow the contribution to the overall efficiency is low.
Between these constrained points the pitch angle settles itself as shown in the last blue column, very near from the theory.
The implemented blade chord is chosen linearly increasing from the base to the 1.5m radius, then constant from 1.5m to the end of the blade (6th blue column).
An example of device to maintain the blades in their middle is shown on this drawing. We propose to add an external ring to better constrain the bending and pitch at the end of the blade.
Note: The formulas and most of the technical illustrations come from the book "Wind Power Plants" by R. Gasch & J. Twele edited by Solarpraxis in 2002
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