According to the centenary Betz‐Joukowsky law, the power extracted from a wind turbine in open flow cannot exceed 16/27 of the wind transported kinetic energy rate. This limit is usually interpreted as an absolute theoretical upper bound for the power coefficient of all wind turbines, but it was derived in the special case of incompressible fluids. Following the same steps of Betz classical derivation, we model the turbine as an actuator disk in a one dimensional fluid flow but consider the general case of a compressible reversible fluid, such as air. In doing so, we are obliged to use not only the laws of mechanics but also and explicitly the laws of thermodynamics. We show that the power coefficient depends on the inlet wind Mach number , and that its maximum value exceeds the Betz‐Joukowsky limit. We have developed a series expansion for the maximum power coefficient in powers of the Mach number that unifies all the cases (compressible and incompressible) in the same simple expression: .

中文翻译：

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风力发电机的最大热力学功率系数

根据百年纪念Betz-Joukowsky法则，从风力涡轮机中以开放流提取的功率不能超过风能传输动能率的16/27。该限制通常被解释为所有风力涡轮机功率系数的绝对理论上限，但它是在不可压缩流体的特殊情况下得出的。遵循Betz经典推导的相同步骤，我们将涡轮建模为一维流体流中的致动器盘，但考虑了可压缩可逆流体（例如空气）的一般情况。为此，我们不仅必须使用力学定律，而且还必须明确使用热力学定律。我们证明了功率系数取决于进气风的马赫数，并且其最大值超过Betz-Joukowsky限制。我们针对最大功率系数（以马赫数为单位）开发了一系列展开式，以相同的简单表达式统一了所有情况（可压缩和不可压缩）。