I live in British Columbia, where we get a lot of our electricity from hydroelectric dams. One of the simplest forms of power generation, hydro leverages the gravitational potential energy of water. It's one of the oldest energy sources; waterwheels work on the same principle. As for where the energy comes from originally? The sun evaporates water, moving it up into the air. After the water is lifted up thousands of meters from sea level, it falls almost all of those thousands of meters as rain; the associated potential energy turning into heat. It loses hundreds more meters of altitude flowing over terrain and into a reservoir. Then the hydroelectric dam turns a grand total of maybe 100 meters worth of potential energy into electricity. Though the turbines of a big hydroelectric dam are indeed awesomely powerful, they can capture only a fraction of the power at nature's disposal.

Imagine boiling some water in the kettle. The water in the kettle might get you thinking about the water that was dropped to generate the electricity to run the kettle. How much water do we have to run through the dam in order to boil a cup (250mL) of water?

To answer this, we need to know the heat capacity of water. (You might think we also need to know the latent heat of vaporization, but when we talk about "boiling" some water, the goal is not actually (usually) to turn all of that water into steam. We just want to heat the water to 100°C, while only a small fraction is boiled off.) The specific heat capacity of water is 4184 J/kg°C. Note the kg in the denomiator of the unit. That means that if we double the amount of water, it takes twice the energy to raise its temerature the same amount. That means there exists some fixed ratio between the amount of water boiled and the amount of water lost from the reservoir in order to boil it.

The exact value of this ratio depends on the height of the dam and how full its reservoir is at the time. So, here's an even simpler thing we can calculate: Imagine I'm visiting a planet like Earth but with no atmosphere. I'm at the top of a cliff carrying a bucket of water. I dump the water over the edge of the cliff and it falls all the way down before striking the ground at the bottom of the cliff. How high up do I have to be before the water reaches 100°C from the sheer violence of its collision with the ground?

If we assume that the water can't lose any energy to external sources then:

* A mass m falling a height h yields an energy of mh (9.8m/s

2

).

* A mass m takes an energy of roughly m (75°C) (4184 J/kg°C) to heat from 25°C to boiling.

Setting these equal and solving for h gives h = 32000 m, an altitude that would be well into the stratosphere on Earth.

Given that most dams aren't 32 kilometers high, it's clear that we'll have to use much more water to generate the energy than the amount we want to boil. Specifically, 320 times more water for a 100 meter dam with a full reservoir, and the ratio is even more extreme for shorter dams or less-full reservoirs. This is kind of a shocking ratio if you haven't thought about it before, or at least that was my reaction.

One other thing that I thought was interesting about these calculations is that the water floating around near the top of the reservoir carries much more extractable energy than the water floating around near the bottom. Simply because it's higher up. So if the reservoir is low, that might be when you most desire incoming water from a scarcity perspective, but incoming water actually brings with it the most extractable energy when the reservoir is nearly full. This is related to how it takes more energy to add charge to a capacitor the more charge it already has on it so that stored energy goes as the square of the stored charge.

One thing this means is that if you're digging out a reservoir to make it larger, you should mostly focus on increasing the volume just underneath the maximum water level of the reservoir. I.e. you should shallowly dig a large area rather than deeply digging a small area.