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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?

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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.

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  • $\begingroup$ 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, 2014 at 3:43
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    $\begingroup$ @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.) $\endgroup$
    – N. Virgo
    May 16, 2014 at 13:03
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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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Arguably, the Betz limit is not an instance or offshoot of the Carnot limit. Indeed, when we ask the question "Given a uniform airflow with speed $v$ just above a rigid firm static infinitely-massive surface, how much electrical energy could we in principle extract from a given volume of air $V$ with contained mass $m=\rho V$?" the answer is "100%" and does not involve the Betz limit. This should not be surprising since macroscopic kinetic energy is organized energy (similar tto the potential energy of the water in the hydro-electric dam) and organized energy can be converted in principle to another form of organized energy with 100% efficiency, exactly like in the case of the hydro-electric dam.

Let me sketch a thought experiment where we approach the 100% energy extraction from wind that was promised before, which at the same time will clarify why the Betz limit may be interesting and significant after all. Imagine having a kind of lightweight (infinitely thin) airtight foil that can unfold itself at your command and enclose a given volume of open space until a balloon-like closed conformation has been reached (complete without any leaky seams). If you were to let this foil deploy itself in the aforementioned uniform air flow in a comoving sense, it is clear that this deployment doesn't require any work from the foil. So after this foil has transformed into a balloon enclosing the volume $V$, we arrange for an additional lightweight anchor on this contraption. Next we arrange a hook connected to a chain connected to a gear connected to a generator on the static ground surface. When the balloon drifts by at relative speed $v$, the anchor attaches to the hook and the generator brakes the balloon until it and the air inside have come to a halt (in actuality it may perhaps brake to a small speed $v_f<<v$ since the wind keeps on pushing the arrested balloon. But this $v_f$ as well as the time-span of the arrest-process can be made smaller by arranging for a heavier gear + generator setup). This arrest-motion is in principle 100% efficient so the initial kinetic energy $\rho V v^2/2$ of the balloon will have been converted into electricity inside the generator. After the balloon has been arrested, the foil can refold or deflate (an action which doesn't require any work), we transport the folded foil upwind (no work required) and repeat the whole process.

I think I don't need to explain why this whole thought experiment has no practical implications and why a $<100\%$ limit may appear for more practical devices.

I remember a paper by a Portuguese author first pointing out this matter, I'll link it as soon as I remember where I found it.

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