Maximum Power of A Lift Based Wind Turbine
The derivation for a the maximum power of a lift based machine is similar, with some modifications. First we must recognize that drag is always present, thus cannot be ignored. It will be shown that neglecting drag leads to a final solution of infinite power. This result is clearly invalid, hence we will proceed with drag. As before, equations (1), (CD) and (RelativeSpeed) will be used along with (CL) to define the power below expression.
-
(LiftPower)
Similarly, this is non-dimensionalized with equations (CP) and (SpeedRatio). However in this derivation the parameter is also used:
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(LiftCP)
Solving the optimal speed ratio is complicated by the dependency on and the fact that the optimal speed ratio is a solution to a cubic polynomial. Numerical methods can then be applied to determine this solution and the corresponding solution for a range of results. Some sample solutions are given in the table below.
| Optimal | Optimal | |
|---|---|---|
| 0.5 | 1.23 | 0.75 |
| 0.2 | 3.29 | 3.87 |
| 0.1 | 6.64 | 14.98 |
| 0.05 | 13.32 | 59.43 |
| 0.04 | 16.66 | 92.76 |
| 0.03 | 22.2 | 164.78 |
| 0.02 | 33.3 | 370.54 |
| 0.01 | 66.7 | 1481.65 |
| 0.007 | 95.23 | 3023.6 |
Experiments have shown that it is not unreasonable to achieve a drag ratio of approximately 0.01 at a lift coefficient of 0.6. This would give a of about 889. This is substantially better than the best drag based machine, hence why lift based machines are superior.
In the analysis given here, there is an inconsistency compared to typical wind turbine non-dimensionalization. As stated in the preceding section the A in the non-dimensionalization is not always the same as the A in the force equations (CL) and (CD). Typically for the A is the area swept by the rotor blade in its motion. For and A is the area of the turbine wing section. For drag based machines, these two areas are almost identical so there is little difference. To make the lift based results comparable to the drag results, the area of the wing section was used to non-dimensionalize power. The results here could be interpreted as power per unit of material. Given that the material represents the cost (wind is free), this is a better variable for comparison.
If one were to apply conventional non-dimensionalization, more information on the motion of the blade would be required. However the discussion on Horizontal Axis Wind Turbines will show that the maximum there is 16/27. Thus, even by conventional non-dimensional analysis lift based machines are superior to drag based machines.
There are several idealizations to the analysis. In any lift based machine (aircraft included) with finite wings, there is a wake that affects the incoming flow and creates induced drag. This phenomenon exists in wind turbines and was neglected in this analysis. Including induced drag requires information specific to the topology, In these cases it is expected that both the optimal speed ratio and the optimal would be less. The analysis focused on the aerodynamic potential, but neglected structural aspects. In reality most optimal wind turbine design becomes a compromise between optimal aerodynamic design, and optimal structural design.
Read more about this topic: Wind Turbine Aerodynamics
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