Energy produced by an infinite waterfall
Sep. 4th, 2014 06:06 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackA friend asked on twitter, if you could place two connected portals anywhere on earth (but not in space), what would you do with them. Personally, linking Cambridge and Liv's campus flat would be nicest! Geopolitically, probably linking two disparate regions might be most useful.
But of course, the question turned to free energy. Suppose the portal is 1mx2m, laid horizontally, one at ground level and one 100km up at the edge of space, and you diverted sufficiently much water into the lower one to get an endless waterfall. How much energy would you get out?
I'm not sure I have these equations right any more, but under those assumptions (and that the density of water and the gravitational constant for the whole height are rounded to the nearest power of ten and that the square root of 2 is 1.5), and assuming that extracting the energy from the falling water slows it to rest again, I tried to calculate the speed and the energy. I think freefall from that height (assuming you can somehow construct an airtight tube) lands at 1500m/s. And that translates to 3x10^12 Watts = 3TW. Can anyone confirm if I have that right?
It's hard to tell from wikipedia, but it looks like that world energy consumption was 15 TW the last time someone worked it out and updated that page. So this device would make a worthwhile dent in it, but not obsolete everything else.
Of course, that assumes there are sensible engineering solutions to "build an airtight tube to the edge of space" and "slow water from mach 5 safely without wasting any energy" which there probably aren't.
You could dig the tunnel _down_ instead, but you'd have to actually dig it, although you could put one portal down there and dig up through the other one, until the rock fell freely. And you'd have to be careful not to go down too far because if your experiment starts spewing pressurised magma you've invented your own personal volcano, and you have to hope that it solidifies and buries it well enough to withstand ten atmospheres of pressure.
Of course, if you did the above experiment with rock instead of water, the energy involved would be 10 times greater, although presumably the engineering challenge would still be the limiting factor.
However, leaving the practical impossibilities aside, something bothered me about the physics. It seems like, if you don't slow the falling substance completely but let it go through the portal already at high speed, you get correspondingly more energy out. I guess because the limiting factor for energy is that each atom going through a portal to 100km up creates a certain amount of potential energy, so you get the more energy the more you cause that to happen. It seems dodgy that the energy production could just keep on growing in that case, but I guess the assumptions violated physics, so there's no reason not to expect that to violate a wide number of other principles. Have I actually got that right?
But of course, the question turned to free energy. Suppose the portal is 1mx2m, laid horizontally, one at ground level and one 100km up at the edge of space, and you diverted sufficiently much water into the lower one to get an endless waterfall. How much energy would you get out?
I'm not sure I have these equations right any more, but under those assumptions (and that the density of water and the gravitational constant for the whole height are rounded to the nearest power of ten and that the square root of 2 is 1.5), and assuming that extracting the energy from the falling water slows it to rest again, I tried to calculate the speed and the energy. I think freefall from that height (assuming you can somehow construct an airtight tube) lands at 1500m/s. And that translates to 3x10^12 Watts = 3TW. Can anyone confirm if I have that right?
It's hard to tell from wikipedia, but it looks like that world energy consumption was 15 TW the last time someone worked it out and updated that page. So this device would make a worthwhile dent in it, but not obsolete everything else.
Of course, that assumes there are sensible engineering solutions to "build an airtight tube to the edge of space" and "slow water from mach 5 safely without wasting any energy" which there probably aren't.
You could dig the tunnel _down_ instead, but you'd have to actually dig it, although you could put one portal down there and dig up through the other one, until the rock fell freely. And you'd have to be careful not to go down too far because if your experiment starts spewing pressurised magma you've invented your own personal volcano, and you have to hope that it solidifies and buries it well enough to withstand ten atmospheres of pressure.
Of course, if you did the above experiment with rock instead of water, the energy involved would be 10 times greater, although presumably the engineering challenge would still be the limiting factor.
However, leaving the practical impossibilities aside, something bothered me about the physics. It seems like, if you don't slow the falling substance completely but let it go through the portal already at high speed, you get correspondingly more energy out. I guess because the limiting factor for energy is that each atom going through a portal to 100km up creates a certain amount of potential energy, so you get the more energy the more you cause that to happen. It seems dodgy that the energy production could just keep on growing in that case, but I guess the assumptions violated physics, so there's no reason not to expect that to violate a wide number of other principles. Have I actually got that right?
no subject
Date: 2014-10-01 08:50 am (UTC)Imagine graphing the gravitational potential at each point around the planet. You get the classic 'rubber sheet' diagram, with a big well in the middle, and potential proportional to 1/r everywhere (giving force proportional to 1/r^2).
Now transition into a rotating frame of reference. That means you now have to pretend there's a centrifugal force acting in addition to gravity; centrifugal force is proportional to r, so if you think of that in terms of a potential surface too, it's one where the potential is proportional to r^2.
So the combined potential surface looks like r^2+1/r, which tends to infinity at both r=0 and r=infinity, and in between it has a minimum. That minimum is the radius of the stable circular orbit with whatever angular velocity you picked for your rotating frame. And if you imagine dropping a marble on to that potential surface, it will probably end up rolling back and forth in the groove, which corresponds (back in non-rotating coordinates) to an elliptical orbit with the same period. That gives an intuitive visualisation of why nudging or gently dropping an object in orbit causes it to just drift into a slightly different orbit rather than plummeting to earth – you may have nudged the marble a little, but ultimately, it's still rolling back and forth in a groove, so even a perturbed orbit is basically stable.
Now we introduce our portal, which artificially distorts the potential surface: at some point somewhere in the orbital region, there's an absolutely massive downward spike in the gravitational potential, because within the space of a few inches, the potential falls off from its normal orbital value to the value at the planet's surface. And the point is that that downward spike is so enormous that even in the rotating frame, with the centrifugal force 'potential' added back in, it still creates not only a local minimum of potential but a local minimum with terrifyingly steep sides and a hell of a long way down.
So if your marble is rolling back and forth in its orbital groove but is somewhere near that pit, everything will be fine until it gets just a little too close and slips over the edge – and then it will plummet, and arrive on the planet's surface with all the kinetic energy it had in orbit plus what it gained from the fall, without even the decency to have shed a big chunk of that against the atmosphere on the way down.