### Pumped energy storage analysis

2011-Nov-15, Tuesday 17:16**mindstalk**

*Do The Math*looks at pumped hydro power. http://physics.ucsd.edu/do-the-math/

He has his math; I'll try mine.

E=mgh

E=energy in Joules, m= mass in kilogram, g=10 m/s^2 (gravity), h = height of the mass.

For a column of water of height h, a mass of water m on top has E=mgh, the bottom has E=0, and I think the total is an average, E=Mgh/2, M=the total mass of the column.

Say we want to store US power usage of 1e12 Watts (lower than real) for 1e6 seconds (week and a half, higher than Murphy's 7 days), that's 1e18 J. Divide by g, and Mh=2e17.

h=100 m, M=2e15 kg, also we have 1e5 kg/m^2 (100 meter column of water), so we need 2e10 m^2, or 20,000 square kilometers. Volume of water is 2000 km^3, compared to 480 for Lake Erie, or 3500 for Lake Huron, both with a smaller average depth.

h=1000 m, M=2e14 kg, 1e6 kg/m^2, so 2e8 m^2, or 200 square kilometers, and 200 km^3 volume.

As simple areas, those don't sound so big. If 2% of the US's 1e7 km^2 are urbanized, that's 200,000 km2, which at 1000 people/km2 is 200 million people, so that passes OOM (order of magnitude) sanity check.

OTOH, most of the US doesn't naturally support a 100 meter -- a kilometer! -- high table of water. Murphy tries to think about semi-appropriate natural sites that can be dammed off. An alternative would be building giant concrete tanks or artificial lakes where we need them, but that would be epic. Murphy's already positing unprecedented amounts of concrete. Also, even "let's double the developed area of the US quickly" is daunting.

Also we need to double the area numbers again, to provide a place to store the water in the low-energy state.

But wait, that provides an idea. Could we excavate the pits needed for the low-energy storage, and use the earth from that to create an artificial elevated lake to store the high-energy water, without having to mess with concrete? Rammed earth instead of concrete? Maybe. Still pretty epic; for 100 m or less we're talking about excavating a new Great Lake, while for 1000 meters we're talking about an artificial structure a kilometer tall, as well as a pit a kilometer deep; unprecedented on both ends.

For comparison, assuming our 3 TW of power comes from fossil fuels, at about 3e7 s/year and 3e7 J/kg that's 3e12 kg of coal and oil that we handle every year (so, a couple mountains worth, every year, being shipped around and then going up in smoke); building the storage means moving 100-1000 times as much of that in dirt. Or double that, since dirt is denser than water.

Real US power usage is 3e12 W, so we might triple those numbers. OTOH, a lot of that energy is waste heat in converting to electricity or motive work, or simply burned for heat, so a a country that's converted to renewable might be using less. Today: 1 TW burned to make 300 GW electricity, 1 TW burned for transportation, 1 TW used for heat. Tomorrow: 300 GW of direct electricity, 300 GW for additional transportation electricity, 300 GW for yet more electricity powering heat pumps that move 1 TW of heat. 0.9 TW, on the other hand you need a lot of that heat in the winter, so ideally we'd have annual leveling, not weekly...

A commenter suggested http://82.80.210.34/gravitypower/

"10m storage shaft

3m return pipe

1000-2000m deep

Up to 150 MW for 4 hrs per shaft

Up to 210,000 tonnes/shaft

Up to 2400 MW in 2.5 acres"

What's a kilometer shaft among friends? Storage shaft area is 5^2*pi so 75 m^2. Call it 100, say a pure iron piston of 10 tons/m^3, the piston is 21,000 m^3, or 200 meters tall, 10-20% of the height of the shaft. Storage for 2km is 2e8 kg*10*2e3m = 4e12 J. 150 MW for 4 hours = 1.5e8*4*3.6e3 = 2e12 J. Right OOM, anyway. A narrow shaft might be a lot easier to build than an artificial mountain of water, and use of a dense piston instead of water is a neat trick.

Concrete is 40 kg/dollar. Bulk iron is maybe 2 kg/dollar. 210,000 tons is 2e8 kg or between $100 million and $5 million just for the raw material of the piston. 1 TW / 150 MW = roughly 10,000 of these, or $50 billion to $1 trillion. That's for four hours or roughly 1e4 seconds; 1e6 seconds would mean $5 trillion to $100 trillion.

Oops.

Conclusion: I'm not sure power storage is impossible or impractical, but it's at the least epic. Pure iron-based storage for a week is, it's just too expensive. Concrete based is sort of doable, so the dirt and water scheme might be as well. There's also a question of whether we need 1e6 seconds of storage -- though as Murphy points you, you don't need to think of that as a week and a half of no renewable input at all, it could be 3 weeks of 50% sunlight and wind for some reason. And sunlight varies with the year, going down right when heating needs go up -- so a high-latitude solar powered economy could probably be thinking about months of storage, not weeks.