All Kinds of Turbines

I got an interesting question, related to the heat-from-the-tunnel project:

Has anyone considered the fact that the water can be used to first drive turbines and then distributed to supply the input source for the heat pumps?

I am a water turbine fan, and every time I spot a small hydro power plant on a hiking map, I have to find it.

Small Pelton turbine
Pelton turbine. The small regional utility has several of them, the flow rate is typically a few 100 liters per second. The NSA should find an image of myself in the reflections.

This does not mean I have developed intuition for the numbers, so I have to do some cross-checks.

You can harvest either kinetic or potential energy from a flowing river in a hydro power plant. Harvesting kinetic energy could be done by something like the ‘under-water version’ of a wind turbine.

The tunnel produces a flow of 300 liters per second but this information is not yet sufficient for estimating mechanical power.

The kinetic energy of a mass m moving at velocity v is:  \frac{mv^{2}}{2}. From the mean velocity in a flow of water we could calculate the energy carried by flow by replacing m in this expression by mass flow.

If 300 liters per second flow through a pipe with an area of 1 m2, the flow velocity is equal to  0,3 m3/s divided by this area, thus 0,3 m/s. This translates to a kinetic energy of:

\frac{300 ^{.} 0,3^{2}}{2} W = 13,5 W

… only, just enough for a small light bulb.

If the cross-section of the pipe would be ten times smaller, the power would be 100 times larger – 1,35 kW.

(Edit: This is just speculating about typical sizes of the natural pipe determined by rocks or whatever. You cannot create energy out of nothing as increasing velocity by a sort of funnel would decrease pressure. I was rather thinking of a river bed open to ambient air – and ambient pressure – than a closed pipe.)

On the other hand, if that water would be allowed to ‘fall’, we could harvest potential energy:

Mill Wheel
Also this mill wheel is utilizing potential energy from the height difference of a few meters. (Critically inspected by The Chief Engineer, photo by elkement)

This is how commercial hydro power plants work, including those located at rivers in seemingly flat lowlands.

The potential energy of a point mass at height h is mgh, g being the acceleration due to gravity (~ 10m/s2). Assuming a usable height of 10m, 300kg/s would result in about

300 . 10 . 10 W = 30kW – quite a difference!

Of course there are huge error bars here but the modest output of kinetic energy is typical for the topography of planet earth.

Mass flow has to be conserved, and it enters both expressions as a factor. If I am interested in comparing potential and kinetic energies relative to each other, it is sufficient to compare \frac{v^{2}}{2} to gh.

Cross-checking this for a flow of water we know more about:

The Danube flows at about 3-10 m/s, so

\frac{v_{Danube}^{2}}{2} = 4,5 – 50m2/s2

But we cannot extract all that energy: The flow of water would come to a halt at the turbine – where should the water go then? For the same reasons there is a theoretical maximum percentage of wind power that turbines can harvest, even if perfectly frictionless.

In addition, such a turbine would need to be much smaller than the cross-section of the river. Mass flow needs to be conserved: when part of the water slows down, it gets spread over a larger cross-section.

So the realistic \frac{v_{Danube}^{2}}{2} will be smaller.

I have stumbled upon an Austrian startup offering floating turbines, designed for operations in larger rivers and delivering about 70kW at 3,3m/s flow velocity (Images on the German site). This is small compared to the overall kinetic energy of the Danube of about several MW, calculated from 2.000m3/s (mass flow near Vienna) and about 3m/s. The first hydro power plant at the Danube in Austria was built in 1959 – an icon of post World War II reconstruction. The plant is currently modernised, the rated power will be increased by 5% to 250MW. Utilized difference in height: 10m.

So the whole kinetic energy – that cannot be extracted anyway – is still small compared to the rated power of typical power plants which are several 100MW!

If the water of the Danube ‘falls’ about 10m then

gh_{Danube} ~ 100

… which is much larger than realistic values of \frac{v_{Danube}^{2}}{2}! Typical usable kinetic energies are lower than typical potential energies.

So if tunnel drain water should drive a turbine, the usable height is crucial. But expected powers are rather low compared to the heat power to be gained (several MW) so this is probably not economically feasible.

I was curious about the largest power plants on earth: Currently the Chinese Three Gorges Dam delivers 22GW. I have heard about plans in Sweden to build a plant that could deliver 50GW – a pumped hydro storage plant utilizing a 50km tunnel between two large lakes, with a difference in altitude of 44m (See the mentions here or here.)

12 Comments Add yours

  1. In Canada–and yes, here in NL–hydro is quite feasible. The present Churchill falls installation in NL does 5GW and there are plans to add 4 GW in two stages. I am always amazed, though, at the level of political interference that goes into the planning of these things. In our case, for example, getting the energy out of Labrador meant building a power corridor through Quebec. We never did come to proper terms, for a whole host of complicated reasons, and in the end would up selling it to Hydro Quebec for a few tenths of a cent/kWh. Amazing!
    That said, once in place a hydro facility should produce cheap energy for millenia. You’d.You’d really only need to keep the dame up in shape as the turbines and generators are constantly being maintained and upgraded anyway. From time to time, one assumes, major work would need to be done on the dame, though.
    It’s often not mentioned, though, the two kinds of pollution that can happen with hydro. First, when the land is flooded, besides the loss of habitat space there’s also a large release of CO2 as the living material decomposes. Second, there’s the issue of thermal pollution. Some of the potential/kinetic energy released by the falling water goes to raise the temp a bit and this can be a problem for fish both in terms of direct effects of the temperature as well as unwanted algae growth.
    When I saw the picture of the dam in China I was reminded of a new problem faced by developing countries not necessarily related to hydro. It seems that many of the rivers in Asia now are completely tapped dry before they reach the sea owing to vastly increased demands for fresh water.

    1. elkement says:

      Thanks for your comment, Maurice! Again it motivates me to do more research of energy generation in different countries. It is so interesting to learn what works and what does not work – often for non-technical reasons.
      In Austria we have plenty of hydro power, but the potential is also very well exploited. It is an interesting challenge for environmentalists: The Green Party in Austria was basically born out of a movement that prevented building the last “missing” large power plant at the Danube.

  2. When 300 liters of water is in free fall, it reaches a terminal speed of 14 m/s, i.e. 50 km/hr. We rarely see rivers flow that fast so from that angle, it makes sense to go for height drops to collect energy from water. But to be honest, this never occurred to me until you did the maths :)

    1. elkement says:

      That’s also a great way to compare gh und v^2/2 directly, thanks! But you mean 10m, not 300 liters, right? … as velocity only depends on height.

      1. Yes I meant the 300 liters from 10 meters. Or any weight – indeed :)

  3. The Snowy Mountains river scheme in Australia has a number of hydro-electric power plants. I used to live 20km from the Hume Dam power station which has a peak output capability of 58MW and was built in 1957 (before I was born). Putting the potential energy capacity into perspective, the water stored is 5 times that of Sydney Harbour!

    The huge Snowy mountain hydro scheme, built between 1949 – 1979, has a total power generation capability of 3.772GW! That’s 3772MW of peak power….

    Compared to the single three gorges project in China, the Australian Snowy scheme (built ~50 years ago), dwarves the rest in both size and peak power output.

    Find out about the Hume dam here:

    Find out about the Snowy Mountain river hydro scheme here:

    1. elkement says:

      Thanks a lot for your comment – absolutely impressive!

      I have tried to find a more authoritative list than Wikipedia now:

      But it seems in those lists such schemes as Snowy Mountain “get lost” and only individual plants are counted.

  4. Wow. Impressive. The calculations, which are casual fun for you, would take me a finite among of time to unravel. I hadn’t thought about the fact that water from the tunnel would have no altitudinal drop. And your calculations for in-flow turbines yield little power. When you say, “If the cross-section of the pipe would be ten times smaller, the power would be 100 times larger – 1,35 kW.” Perhaps, but now you’re talking about a pipe only just 15cm in diameter? Could you support flow rates through that which would yield your 1.35kW. Even if you could … what’s 1.35kW when compared to 50MW? I did migrate to the German site to look at the floating turbines … very fun. Thanks for taking the time to investigate this. For now, I’ll not be digging into the hillside next to me to release water to drive an in-line turbine! I would, however, like a nice big 100kW wind turbine in my backyard! Maybe someday! D

    1. elkement says:

      Ha – I hoped somebody would ask about the cross-section :-) It seems you are an engineer at heart! I had a sentence in a previous version “Unfortunately this effect cannot be used to ‘generate’ additional energy”. I deleted it though as I found it difficult to explain briefly, and I would have opened another can of worms. What I have not mentioned is that there is actually a third “sort of energy” which is the pressure in the water. When the pipe gets narrower in some sort of funnel, the pressure gets smaller so that the total energy remains the same.

      I should probably not have called it a pipe, but I was thinking of an open riverbed (same pressure everywhere dictated by ambient air) and just speculating about how wide this natural pipe might probably be. I’ll a small remark.

      Another idea for a future post perhaps :-)

  5. M. Hatzel says:

    This subject is so far a field of my understanding or experience that I’m unable to say anything… except that you make a good teacher of complex things. :) I think I’ve discovered some appreciation for hydro power.

    1. elkement says:

      Thanks, Michelle! What I find fascinating is that we (incl. myself here) often have no well-developed intuition for numbers related to such tangible things – I am not speaking of quantum physics and the Higgs boson.

      I am always trying to keep those cross-checks in mind – but if I don’t work with some technology hands-on for a while or solve problems and calculate something, I forget it, too. But I try to optimize and re-learn all the time! So I thanks for commenting here – as I am never sure if such crude estimates and number games might be somewhat interesting.

      1. M. Hatzel says:

        They are interesting. It’s like Dave’s posts that begin “do people know where food really comes from?” The answer isn’t the grocery store, just as the answer to the same question for energy isn’t “the furnace” or “the light switch.” It’s a healthy exercise to be reminded of even that simple fact!

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