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The fundamental theory of design and operation of wind turbines is derived based on a first principles approach using conservation of mass and conservation of energy in a wind stream. A detailed derivation of the “Betz Equation” and the “Betz Criterion” or “Betz Limit” is presented, and its subtleties, insights as well as the pitfalls in its derivation and application are discussed. This fundamental equation was first introduced by the German engineer Albert Betz in 1919 and published in his book “Wind Energie und ihre Ausnutzung durch Windmuhlen,” or “Wind Energy and its Extraction through Wind Mills” in 1926. The theory that is developed applies to both horizontal and vertical axis wind turbines. The power coefficient of a wind turbine is defined and is related to the Betz Limit. A description of the optimal rotor tip speed ratio of a wind turbine is also presented. This is compared with a description based on Schmitz whirlpool ratios accounting for the different losses and efficiencies encountered in the operation of wind energy conversion systems. The theoretical and a corrected graph of the different wind turbine operational regimes and configurations, relating the power coefficient to the rotor tip speed ratio are shown. The general common principles underlying wind, hydroelectric and thermal energy conversion are discussed.

Wind Turbines Theory - The Betz Equation and Optimal Rotor Tip Speed Ratio