It's not really an electricity source, but recently I was curious about pumped hydro storage, and how to develop any kind of intuition about the relationship be

It's not really an electricity source, but recently I was curious about pumped hydro storage, and how to develop any kind of intuition about the relationship between volume of water, height difference, and how much energy you can actually store that way. I used lake Mead and Hoover dam as an example. I wanted to know how much stored energy is there. Apparently the U.S. uses 3.9 trillion kwh per year. Prelude> (3.9 * 1000 * 1000 * 1000 * 1000) / 365 1.0684931506849316e10 That comes out to about 10 billion kwh per day. We'll come back to that number later. The amount of stored gravitational energy of a thing in joules is its mass in kg times height in meters times 9.8 m/s^2 (on Earth). There are 3,600,000 joules in a kilowatt hour. Lake Mead is about 29,000,000 acre feet. An acre foot weighs about 2,718,000 pounds. There's 2.2 pounds per kilogram. The "hydraulic" height of Hoover dam is 567 feet, according to wikipedia. I'm assuming that's when it's full (it isn't, but let's pretend it is). Not all the water would be at the top though. We'll say that the average water molecule is at 3/4 of the height. (It's more than half because the lake has a cross section like a martini glass). I don't know what the real number is, but 3/4 seems good enough for an estimate. We can convert feet to meters by dividing by 3.28. Prelude> (9.8 * ((5670.75)/3.28) ((29000000 * 2718000)/2.2)) / 3600000 1.2644999688192902e10 Huh, that number is pretty close to the one from earlier. If I did my math right, there's about twelve and a half billion kilowatt hours stored in a full lake Mead. So, we could, in theory, maintain a buffer capable of powering the United States for an average day with the amount of stored energy in a full lake Mead. We could also do the same thing with 1/10th the water and an altitude difference that's 10 times bigger. If one wanted to actually build something like this to buffer out day/night variations in solar power, the amount of pumps and turbines you'd need to move a lake Mead worth of water every day would be staggering. This is relevant to the question of land use, because a dedicated pumped hydro system would probably have two reservoirs, or a reservoir and a large natural body of water like the ocean, and the amount of energy you can store scales up with the water volume, which corellates with horizontal surface area. It's kind of an interesting problem to think about... where are the best pumped hydro sites?