Dave asked an interesting question, commenting on 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.

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 moving at velocity is: . From the mean velocity in a flow of water we could calculate the energy carried by flow by replacing in this expression by mass flow.

If 300 liters per second flow through a pipe with an area of 1 m^{2}, the flow velocity is equal to 0,3 m^{3}/s divided by this area, thus 0,3 m/s. This translates to a kinetic energy of:

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:

**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 is , being the acceleration due to gravity (~ 10m/s^{2}). 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 to .

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

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

= 4,5 – 50m^{2}/s^{2}

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 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 2.000m^{3}/s (mass flow near Vienna) and about 3m/s.

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

~ 100

… which is much larger than realistic values of ! 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.)