The physics of windmills

Wind turbines are now part of the European landscape. They have become the models of renewable energy, their only rivals being photovoltaic panels, to the point that we seem to have forgotten the massive structures connected with hydroelectricity. Nuclear reactors are a daily feature of the media, but rarely in a positive way. Technically-aware French people have some idea of the operating principles of reactors, hydroelectric dams and photovoltaic cells. Paradoxically, while windmills are among the oldest mechanical energy generation devices, the operating principles of wind turbines are largely unknown. Betz’s law , which links the power of the wind turbine to the wind speed and the area swept by the blades, which is a generalised approach to the wind turbine, is quite well known by the initiated. At the other extreme, you have reference books such as that of Cunty (physics very close to that of wind propulsion), which explain how a blade in isolation responds to the forces exerted by wind on its surface. However, it is difficult to find out how to reconcile these seemingly contradictory approaches . Much more astonishing, while the forces exerted by the  wind are handled correctly, those due to air resistance are not. This article aims to present a unified and analytical presentation of the set of phenomena involved in the transformation of wind energy into electricity. The derivation of Betz’s law provides a maximum efficiency value for wind turbines. The traditional presentation of the interaction between the wind and a wind turbine blade