Where is the energy loss from using flowing water for power?

A mill uses a water wheel on a river. The water flowing downhill pushes the water wheel, which then provides power to turn the milling stone. The water then continues on its merry way.

I am trying to figure out where the water loses energy in this transaction. It is slowed by the wheel so loses some kinetic energy. Then it resumes its trip along the river and intuitively seems that it would eventually regain the same speed it would have had if it had not been interrupted.

There is some sort of equilibrium here, with the flow of water limited by friction with the riverbed, since rivers do not flow ever faster without limit as the water goes downriver (or…do they? Intuition tells me no but I actually don’t know.). So is the water used to power the wheel simply able to generate a little less friction with the riverbed, ultimately taking thermal energy out of the river? Or is there some other energy transfer I’m missing?

The water is losing potential energy. Without the wheel that energy either going into accelerating the water, or into warming the water or air or riverbed.

I think that doenstream from the waterwheel the water is moving slower and there’s less water flowing, so the river doenstream has less kinectic energy than it would otherwise have. Any energy you lose due to frictiob in the gears makes the downstream river even slower and lower. That’s why it’s not effective to put a second waterwheel dowmstream from the first one.

The water that encounters the wheel loses some kinetic energy compounded by the fact only a small amount of the overall water flow (a fraction of both the width and depth of the river) encounters the wheel.

River water flows downhill, losing the potential energy of mass x height. Much of the energy winds up as heat (the water warms up slightly); some of it does work such as moving rocks, mud, etc.

When some of it is directed over a water wheel, its potential energy is converted into mechanical motion (first of the wheel itself, then of the gears / shafts / millstones, etc.). Ultimately, this is lost to friction & heat, but some useful work gets done: wheat is ground to flour / logs are sawn into boards.

The net energy of the water (kinetic energy) has been reduced because the water wheel was there.

It just happens that there is so much water we don’t really notice.

Imagine a “block” of water 10’ x 10’ moving by the water wheel at some given speed. After the water hits the water wheel the total kinetic energy of that “block” of water is less. It has been slowed down to some degree. The water behind it may speed it up again but the total energy in the system is now less. It’s just a really, really big system so you never notice the overall energy loss.

In theory, if you had water mills every 50’ along both sides of the river from the start to the end the people at the end would notice a loss in overall power.

Same thing is true for wind turbines and the wind. But that is a truly colossal system and we come nowhere near making a dent in the gross power of the overall system.

EDIT: Although I wonder if gravity is the thing constantly pumping energy back into the system? Afterall, the water is always going downhill. Or what @Northern_Piper said just below.

Plus, even if the water has slowed down a bit because of the water wheel, it normally gains kinetic energy from the drop in the river course after the mill, so it’s hard to observe any significant effect.

That can’t be since the water isn’t destroyed.

The river is on a slight downhill slope. Because the wheel acts as a resistance, water will pile up behind it slightly. Due to the larger cross-section, the water can move more slowly while transferring the same volume. The reduced friction losses go into the wheel.

Note that a water wheel just placed in a flat river will generate very little power. That’s why efficient hydroelectric systems establish a dam behind them, so they can extract the energy from a large drop in height. That energy basically comes from all of the losses along the river that the reservoir covered.

Yes, it’s a dynamic system. It’s not like the water has a limited amount of energy. The gravitational force will add to its energy.

But where does gravity get its energy from? Paging Dr E!

Maybe a better way to think about it is that filling the dam or priming the water wheel is what “charges” the system. The flow of water downstream is reduced during this charging cycle.

There’s a different system for slow-moving streams:

Ultimately, the sun. All that water that falls from the sky (comets aside) didn’t just appear there.

Some of the water is piling up behind thee waterwheel and some is “destroyed” because it soaks into the soil while being backed up, evaporates while being backed up or escapes the river bed (finding a new route down hill thst didn’t existcwithout the backup) (I’m not sure how big any of those effects are).

And after. The ongoing power still has to come from somewhere. But the reservoir behind a dam can be seen just as a high cross-section, very slow-moving “river”. The best hydroelectricity is going to be in places where there were rapids (due to a sleep slope) that was converted into a reservoir with a high water level. All the energy that was dissipated in the rapids previously can now be extracted.

That’s not what I meant. The gravity of the earth is pumping energy into the watercourse as the water flows downstream. How does gravity add energy?
Where does gravity get its energy from?

I’m not sure what you mean by ‘adding’ energy in that case.

The water possesses a certain amount of gravitational potential energy (relative to the center of the earth in this case), some of which is converted to kinetic energy as the water loses altitude. It originally gets that altitude (and potential) from the sun adding energy to the system.

It doesn’t take that many mills to cause a problem. The Blackstone river is very powerful, rapidly descending thousands of feet in a relatively short distance. Mills had to be spaced a half-mile or so apart to keep ‘robbing’ too much power from mills downstream. Mills located further upstream could create large ponds behind dams in where the terrain was along the river course was still steeply descending to keep a strong flow of water through the wheels. Further down stream was preferrable for locating the mills but the river valley is much wider there making it difficult to build dams to maintain deep reservoirs. So reservoirs allowed some buildup in potential energy but those are limited by the rate of descent of the river.

Water evaporates from all bodies of water, due to energy from the Sun, and then it rains down all over the place. Some places where it rains down are higher than where it evaporated from, so there’s a net gain in the water’s potential energy in this process.

In the natural state, as that water runs down from the high places where it fell as rain to the low oceans, some of that potential energy is converted to kinetic energy of the moving river, but most of it is lost to friction with the riverbed and converted to heat. Hydropower systems of various sorts along the river, provided they’re spaced out enough, will mostly harness that part of the energy that would otherwise have been converted to heat.

Simply put the energy is lost (& partially converted) during an elevation change. The water doesn’t have to flow anywhere horizontally; rivers just happen to be oriented that way but you can operate a mill or generator from water falling straight down. If you don’t have an elevation drop between your intake and outlet, the wheel or veins won’t turn.

A conversation related to this would be if solar energy were used to pump water up hill say to 7000 ft. The solar energy would only be used when the grid had an excess of energy. So instead of turning it down use it to pump water up hill so some of the energy can be recaptured. This would amount to a storage battery. What is the best efficiency that could be expected from this.