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A wind turbine rotor produces power from the torque generated by the rotor blades. This torque arises from forces on blade elements which in turn are the consequence of pressure differences on each side of the aerofoils. (source)

Thermodynamic theory states that when the pressure of a gas does work, the gas cools down. In this case the air pressure does work on the rotor blades. Microscopically, the air molecules impacting the blade on its high pressure side move the blade and transfer kinetic energy onto it which lowers their average kinetic energy and decreases the temperature of the air.

Mathematically there should be a temperature drop at the rotor plane although it is probably quite small. What do you say?

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    $\begingroup$ You could have an isothermal case and just decrease the density, could you not? By the way, both my hypothetical and your thought experiments are only valid for ideal gases (which isn't necessarily a horrible approximation). $\endgroup$ – honeste_vivere Apr 20 '16 at 18:32
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    $\begingroup$ "Quite small", even in the ideal case, would be the operative phrase here and one doesn't gain much by analyzing a wind turbine using thermodynamics. Air, in this case, is a transport medium for (kinetic) energy, it's not the working fluid of a thermodynamic engine. The question of "Where is the thermodynamic engine that causes the wind in the first place?" is very valid, though. Unfortunately, that's more meteorology than physics, I am afraid. $\endgroup$ – CuriousOne Apr 20 '16 at 18:42
  • $\begingroup$ @CuriousOne: I disagree here; the air is the working fluid. You have to consider that a wind turbine rotor does not convert the air's kinetic energy but the air's pressure energy. There is no kinetic energy extracted from the fluid at the rotor plane but pressure energy is extracted. This is what the actuator disk model says. And when gas pressure does work, the gas cools down. $\endgroup$ – Chris Apr 21 '16 at 14:05
  • $\begingroup$ The air pressure is the same before and after the rotor and the formula for the power output of a wind turbine clearly uses the wind velocities. When the wind speed is zero, there is no wind energy to be had. A wind turbine is not a high pressure turbine. I have a feeling that Bernoulli is raising its ugly head again with the false theory of why planes fly. You are, of course, welcome to take a temperature measurement before and after your house fan. Let me know what Carnot predicts for the thermodynamic efficiency. $\endgroup$ – CuriousOne Apr 21 '16 at 15:21
  • $\begingroup$ My instinct is "yes, but an unmeasurably small amount". $\endgroup$ – Flyto Apr 21 '16 at 21:36
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There is no significant temperature drop. For the conditions encountered in a typical wind turbine installation, pressure differences on either side of the rotor are on the order of a kPa or so.

Now look at the formula for an adiabatic process in air:

$$ {\displaystyle P^{-2/5}T^{7/5}={\text{constant}},} $$

A change in pressure of 1 kPa over atmospheric corresponds to just a 0.3% change in temperature, or about 1 C at 300 K.

This analysis is actually an overestimate because the pressure change is not just a simple adiabatic process, it's also isothermal to a degree (the blades exchange heat with the air, and the air is also continuously mixing and this homogenizes the temperature). So in practice you never get close to a 1 C temperature change.

Further, as the comments pointed out, this is just the maximum temperature difference possible along the surfaces of the airfoil, not the temperature difference between incoming and outgoing air.

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There's a PNAS article on measuring the difference in temperature downwind of one wind farm called the San Gorgonio Pass Wind Farm. Apparently it's one of the three major wind farms in California. Here's a picture taken by Kit Conn (Wikipedia handle FarWestern):

enter image description here

Here are the relevant figures from the cited paper:

enter image description here

enter image description here

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I have to agree that a temperature drop must occur if energy is extracted. The energy equation does not allow the pressure to change without a temperature change. Bernoulli's equation should not be applied to wind turbines. Because of this temperature drop Betz's limit is invalid and it only requires a few hundredths of a degree in temperature drop to significantly increase the performance of a wind turbine. Furthermore if the airflow is forced by the turbine to rotate, this rotational kinetic energy must come from the temperature drop.

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