The potential energy in the water at the top of the mountain gets eaten away as it flows downstream whether there is a hydro dam or not, due to the massive friction of the water flowing downstream.
This is what I mean by terminal velocity of the river. The water does not just keep speeding up and speeding up as it flows downstream.
At first the water starts to flow faster, and indeed it is converting potential energy into kinetic energy. But as the water starts to flow, the kinetic energy starts getting sucked out through massive amounts of friction in the flow. Ever hear a river roaring? That’s the audio track of massive amounts of kinetic energy draining out into the surrounding earth. (A frictionless river would flow silently)
As the water keeps speeding up as it flows downstream it reaches a point where the friction is costing as much energy as the elevation drop is adding. Terminal velocity. At that point, every foot of elevation drop keeps the river flowing but does not increase net kinetic energy in the river.
Take one kilogram of water and pour it down a chute at the top of a 100 foot high drop.
Take another kilogram of water and pout it down a 200ft high chute. A 1,000ft high chute.
At the bottom of the chutes we measure the speed of the water. You will find the water is flowing at exactly the same speed in all three cases. Why?
For a typical hydroelectric dam, kinetic energy is not particularly important (although smaller kinetic hydroelectric plants do exist). The relevant factors are pressure (determined by the hydraulic head) and volumetric flow.
In ideal conditions, if you have two sections of identical pipe, and a turbine in the middle, water flowing through the pipe will not change velocity (incompressible fluid, and none is gained or lost), yet the turbine can harvest one Joule per cubic metre per Pascal of pressure.
This is the predominant principle by which a dam works; it creates a large pressure differential, and then drives a turbine at a relatively much slower velocity than the maximum possible by converting all the GPE to KE.
To adapt this to your chutes idea. Water will not pour down the longer chute significantly quicker than the shorter one, yet if you fill them, the longer chutes will have a proportionately larger pressure at the bottom. Thus a flow of 1m^3/s will deliver proportionately more energy across a turbine which reduces the pressure to ambient.
The potential energy in the water at the top of the mountain gets eaten away as it flows downstream whether there is a hydro dam or not, due to the massive friction of the water flowing downstream.
This is what I mean by terminal velocity of the river. The water does not just keep speeding up and speeding up as it flows downstream.
At first the water starts to flow faster, and indeed it is converting potential energy into kinetic energy. But as the water starts to flow, the kinetic energy starts getting sucked out through massive amounts of friction in the flow. Ever hear a river roaring? That’s the audio track of massive amounts of kinetic energy draining out into the surrounding earth. (A frictionless river would flow silently)
As the water keeps speeding up as it flows downstream it reaches a point where the friction is costing as much energy as the elevation drop is adding. Terminal velocity. At that point, every foot of elevation drop keeps the river flowing but does not increase net kinetic energy in the river.
Take one kilogram of water and pour it down a chute at the top of a 100 foot high drop.
Take another kilogram of water and pout it down a 200ft high chute. A 1,000ft high chute.
At the bottom of the chutes we measure the speed of the water. You will find the water is flowing at exactly the same speed in all three cases. Why?