We have Betz's law, an upper bound on power efficiency, for wind-turbines. But there is no theoretical upper limit on hydro-turbines, Why this is so?

Is it mainly due to the incompressibility of water?


Betz' limit arises from the fact that if the turbine tries to take too much energy out of the flow, the wind will divert and go around the turbine instead of through it. With a hydrothermal system the water is typically behind a dam, so the water has nowhere else to go. The turbines effectively convert gravitational potential energy into work, which can be done (in principle) without any thermodynamic losses.

  • $\begingroup$ If we replace the wind-turbine in air with water-turbine inside the sea, and let the water be the driving force (just like air do with wind-turbine) then would the Betz' law hold same in water as in air ? $\endgroup$
    – kaka
    May 16 '14 at 3:43
  • 1
    $\begingroup$ @kaka yes, indeed it would. I'm pretty sure there are some tidal power schemes that work like that, and Betz' law does apply in those cases. (I've been meaning to update my answer to say that, but I'm a bit busy at the moment.) $\endgroup$
    – Nathaniel
    May 16 '14 at 13:03

The Betz system is an open system, with equal air pressure in front of and behind the turbine. The extracted energy comes from taking momentum (or strictly, kinetic energy) from the air and using it to spin the turbine. The Betz limit comes from the fact that slowing the air too much leads to build-up of the air, and flow will go around the turbine instead.

This will work for water in an open system like the ocean in the same way, but most water systems are closed - the turbine is in a pipe, and efficiency can get up to 85%. Here, in addition to momentum of the water, you can also generate a pressure difference from the pipe before the turbine to the pipe after, and this lets you capture more of the energy.


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