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The high water has some energy E = mgh. In the basic school, we were told that hydropower plants extract the energy of "falling water". Now, that banned principle says that if your water falls upon still turbines, you will loose energy. The water will turn into a foam, vapor and heat up until it slows down to the speed of rotating turbine. At this point, half of the "falling energy" is lost. This means that speed of turbines must match the speed of the water stream.

Now, the waterfall has the highest speed at the basement and, in order to capture all the energy, you need to stop the water completely because water escaping at speed v, carries energy $E = mv^2/2$, where mass m grows linearly with time (your losses accumulate over time). These losses are proportional to water density m and speed v. Which means we loose more energy with each time unit if outbound stream speed v is higher. The only way to achieve 100% efficiency is to stop the water completely, to v=0. However, this implies no flow and no electricity produced.

All these considerations visited me when I did some switching power supplies. Switching power supply achieves 100% efficiency by operating in one of 2 modes: full power, when water input is 100% open to accelerate the turbine and itself, and zero power, when fresh water input is closed and previously accelerated water altogether with massive turbine slow down by the energy consumer (assume a spring that we charge against its force -- it will slow down the stream with turbine). In electronics, the role of moving water and rotor is played by constant inductor, whose inductance is constant, unlike the water, which would arrive all the time and accumulate at v whenever you open the switch. I therefore would stop the water completely in the lowland, before releasing it there, achieving 100% extraction. But, it seems that in real hydro plants and wind mills, turbine rotates constantly, without any switching.

So, what theoretical efficiencies can wind and hydro turbines achieve?

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closed as unclear what you're asking by Olin Lathrop, Wasabi, Chris Mueller, Mahendra Gunawardena, hazzey Dec 7 '15 at 14:44

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    $\begingroup$ @ValentinTihomirov Hypothetical musings don't fit well into our format. Perhaps you should try one of the many engineering forums? $\endgroup$ – Chris Mueller Dec 7 '15 at 12:46
  • $\begingroup$ @ChrisMueller I provided them for a question, not to announce anything, as you see. You seem to insist that the questions must come out of nothing and nobody should draw attention to the matter that really bothers. Particularly, pointing out the way energy is wasted and how can it be not wasted, is bad idea when I want to ask about the efficiency limits. Doing so is "nonsensical". I am no blind, I see what you mean. $\endgroup$ – Valentin Tihomirov Dec 7 '15 at 12:54
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The theoretical maximum efficiency of a wind turbine is the Betz Limit, which is $\frac{16}{27}$ ( ~ 59.3%).

The result dates back to 1919.

The Betz limit is, as you note, speed invariant. To get an intuition as to why, consider the system speeded up or slowed down. This changes the fluid's velocity, but does not change any of the issues around the dispersion of the fluid after the rotor. This means that the limit is driven not by absolutely velocity, but rather the ratio of the speed after the rotor, to the speed before it. That ratio is 1:3 - the fluid loses two-thirds of its speed in the perfect rotor.

It's different in hydro where you've got a vertical drop. There, the turbine can take almost all of the speed out of the water. That's because there's a reservoir both before and after the turbine, and a vertical drop after the turbine. That means that gravity can do the work of dispersing the water after a turbine, and there's enough volume of space immediately behind the rotor for the water to travel into slowly. And as water is much denser, and much less compressible, than air, it displaces it easily. So we get real-world hydro efficiencies of ~90%.

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  • $\begingroup$ Curiously, it displays no v-dependence on speed of downstream. How do you rule it out? $\endgroup$ – Valentin Tihomirov Dec 7 '15 at 12:09
  • $\begingroup$ Thanks, but I also want to hear about water. I have demonstrated that it is possible to achieve 100% with switching power at least in hydro, where water is not flowing in real time and, thus, you can stop it completely. So, what is theoretical maximum there and why switching power is not applied in hydro, where you can stop the stream completely? $\endgroup$ – Valentin Tihomirov Dec 7 '15 at 12:17
  • $\begingroup$ OK, basically, you say that mills are doing good job reducing the speed to the point where more complex (switching) solution is irrelevant. I have accepted because you seem to explain that width of out pipe is a solution. You say generator basically should work as funnel to achieve high efficiency. But, I do not understand where in the funnel do you place your generator. It seems to me that stream speed is the same at both front and back planes of the blades, which means that that stream naturally slows down in the funnel but blades don't extract anything. $\endgroup$ – Valentin Tihomirov Dec 7 '15 at 15:10
  • $\begingroup$ If stream speed were the same at the front and back planes of the blades, then the blades would have extracted zero energy. Don't think of hydro as a funnel: it's more like an hourglass egg-timer, with the turbine in the narrow waist. $\endgroup$ – EnergyNumbers Dec 7 '15 at 15:16
  • $\begingroup$ But telling that it looks like a single dot does not explain anything. I am asking how that single dot can achieve high $v_{in}/v_{out}$ ratio, how can front blades experience higher water speed than back end, if hourglass egg-timer is symmetrical (input channel is as wide or narrow as the output channel) and channel widening begins only behind the generator? $\endgroup$ – Valentin Tihomirov Dec 7 '15 at 15:27

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