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).

  • $\begingroup$ 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 [...]" $\endgroup$ – Bianfable Jan 25 at 12:52
  • $\begingroup$ @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. $\endgroup$ – DrZ214 Jan 25 at 14:05
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    $\begingroup$ For airflow < 100 m/s it can be treated as incompressible, which seems fine for wind turbines, but not for gas turbines. $\endgroup$ – Organic Marble Jan 25 at 18:18
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    $\begingroup$ @DrZ214, see 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. $\endgroup$ – Jan Hudec Jan 26 at 13:10
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    $\begingroup$ 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. $\endgroup$ – mins Feb 11 at 18:14

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