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Betz's Law says the extractable power from wind is only 59% (16/27) of its original power.

$$P_{max} = \frac{16}{27} \frac{1}{2} \rho A v^3$$

Rho is the gas density, A is the cross section area of gas flow, and v is the original flow velocity (before it goes through the turbine stages).

Note that power is force times velocity, so the force component is $\frac{1}{2} \rho A v^2$ and then multiplied by another v makes it $v^3$. This is a little confusing because it looks very similar to the formula for kinetic energy ($\frac{1}{2} m v^2$). However, Betz's Law deals with power.

Note: I'm asking about the gas flow after it has been combusted. So it starts out hot and fast with some speed and temperature. And we assume the total heat energy remains constant (it's an adiabatic process) because again, this is considering the gas after combustion.

So does Betz's Law apply to it? If not, what are the exceptions invalidating it? I'm interested in any turbine-based engine (turboprop, turbojet, turbofan).

DrZ214
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  • Did you read the Assumptions section of the Wikipedia article? Particularly point 3 does not seem to hold for jet engines: "The flow is non-compressible. Density remains constant [...]" – Bianfable Jan 25 '21 at 12:52
  • @Bianfable Yes i read those 4 assumptions, and point 3 is very confusing to me. Air is very, very compressible. Yet Betz's Law applies to wind turbines, and what is wind? Moving air. So i find that very strange. – DrZ214 Jan 25 '21 at 14:05
  • Betz Law does not have anything to do with gas turbines, it is about extracting kinetic energy purely from the wind itself, without any additional sources of energy (i.e. fuel). – Juan Jimenez Jan 25 '21 at 16:55
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    For airflow < 100 m/s it can be treated as incompressible, which seems fine for wind turbines, but not for gas turbines. – Organic Marble Jan 25 '21 at 18:18
  • @JuanJimenez The gas turbine extracts energy from air. This air is at higher temps, density, and pressure, but as far as i can tell the working principle is the same. Those turbine blades are airfoils and you deal with lift and drag just like the blade of a traditional wind turbine. – DrZ214 Jan 25 '21 at 23:59
  • @OrganicMarble I should prolly ask another q about how fast is the air in a core of a turbine engine. – DrZ214 Jan 26 '21 at 00:01
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    @DrZ214, see https://aviation.stackexchange.com/q/29279/524: about 300–400 m/s throughout the turbine, taking energy more from the pressure and temperature than the velocity. – Jan Hudec Jan 26 '21 at 13:10
  • @DrZ214 No, your understanding of how a gas turbine works is totally incorrect. You are confusing the combustion process with the generation of thrust. A gas turbine does not extract "energy from air," it gets its energy from the combustion of fuel with air. A wind turbine, which is what Betz Law addresses, gets its energy purely from the movement of air against the blades. – Juan Jimenez Jan 27 '21 at 00:26
  • @JuanJimenez Someone deleted my comment. (EDIT: oops, wrong post, it's in Sanchises answer). I wouldn't say that. The point of a turbine is to extract power from a fluid (which either decreases its velocity or makes it expand). The point of jet exhaust is to increase the velocity of a fluid to make thrust. So they are kinda at odds with each other. I think you are confusing the different stages of an entire engine and trying to think of it as one big thing. My question is only dealing with the turbine. A turbine extracts energy from a fluid. A compressor applies energy to a fluid. – DrZ214 Jan 27 '21 at 01:16
  • @drz214 your statement still makes little sense. I suggest you look up "gas turbine" and "wind turbine" on Wikipedia. The gas turbine gets its energy from the combustion, not "wind." – Juan Jimenez Jan 27 '21 at 01:42
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    Betz limit applies to open systems, not to ducts. It is relevant for water too, but tidal turbines using channels are able to extract a lot more energy than the Betz allows, see Exceeding the Betz Limit with Tidal Turbines. Turbine engines have ducted paths too. – mins Feb 11 '21 at 18:14

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