Energy produced by an infinite waterfall

[jack]

A 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?

Comments

[simont]

I remember a conversation many years ago – possibly when Portal first came out, or then-abouts, and probably at a post-pizza or similar gathering – in which we tried to work out how you could add portals to the rules of physics while violating as few as possible of the existing constraints.

To begin with, conservation of momentum: when something enters a portal it transfers its momentum to the surface the portal is attached to, and when something exits, equal and opposite momentum is gained by the surface attached to the portal it came out of. So if you put portals on two surfaces facing each other, and walk into one and out of the other, then the surfaces are pushed apart by a force. Of course if they're buildings then they can probably withstand the force of pedestrian traffic – but if you planned to use the same portals for high-speed road or rail, you might have to reinforce their supporting structures considerably more.

Angular momentum I can't remember exactly how we dealt with. I have a vague recollection that we started by dividing the process of portal-transit into three conceptual parts rather than two: (a) object enters portal and becomes temporarily one with it (hence portal gains their combined momentum as above); (b) object is teleported from one portal to the other; (c) object exits other portal. And I think (b) is the point where angular momentum becomes interesting, because that teleportation of mass applies some sort of moment, but I forget what exactly we decided to do about it.

But for these purposes, the interesting one is conservation of potential energy. And the answer, we decided, is that gravitational potential does not transit portals, or if it does, it only diffuses out by a short distance. So if you opened a portal from Cambridge to geostationary Earth orbit, for example, you'd find that six inches from this side of the portal there would be the normal Cambridge gravitational potential, and six inches from the other side, the normal GEO potential. Hence, the portal would be traversable in principle, but it wouldn't be a short cut to orbit, or at least not in any sense but the literal: to move an object that single foot of distance through the portal would require you to put in all the same energy you'd more usually apply by rocketry. And therefore, you also wouldn't get free energy out of it.

Of course, that's not how the Portal portals themselves work; it's clear just from ordinary play that a portal between different levels of gravitational potential (such as floor and ceiling) can be traversed just as if there were no potential difference between its endpoints, and indeed, this does lead to free energy. (To say nothing of the very end of Portal 2.)

[jack]

Oh yes, I remember that, I think we didn't have completely definite conclusion, but I thought your assumptions about transferring momentum and energy gradients were sensible ones.

Isn't angular momentum just a special case of momentum?

In this case, I decided to wonder what would happen if you _do_ allow breaking the conservation of energy, even though I agree not doing that makes something more likely to be a consistent physical model. And leave relativity out of it else I didn't think I'd get an answer at all :) And yet, indeed, I think the conclusion that if you allow non-conservation of energy at all, you can get an unbounded amount out, suggests that allowing that really doesn't lead to any sensible set of physics.

But yes, if you assume the conservation-of-energy model, you could imagine a portal to orbit but not infinite free energy. In fact, in many ways, that might solve orbital travel for you even if you still need the energy input -- after all travelling 100km along isn't that difficult, it's doing that while supporting yourself hovering which is prohibitive, so even if you have to build a train that spends several hours traversing your the last foot through your portal, it might be worth it.

The implications for conservation of energy are interesting. I guess if something is just through the portal, it would be in orbit -- but it would tend to fall back up through the portal?

Come to think of it, did we work out what happens if you try to touch the portal edge-on? I guess it might be similar to the conservation of energy -- it's still like an infinitely sharp knife, but the push you have to give still has to do the work of prising the electrons apart?

[simont]

The possibility of accidentally dropping something back down the portal-to-orbit is quite scary, now you mention it – it would arrive on Earth with the same kinetic energy it would have gained during a conventional deorbit, but without most of it having been taken away by aerobraking. All at once, delivered right where someone was probably trying to do something useful. Bang!

[jack]

Ouch! Apparently trying to cheat the laws of physics can be painful :)

Although now I'm not sure I understand this set-up as much as I thought I did. If you're in orbit and "drop" something I think it just gently drifts away, not plunges to earth. So if you're in orbit, can you drop something into the portal? Or do you need to push it? Or does it depend how you set the portal up..?

[simont]

Coming back to this a long time later: I wanted to reply to it at the time, but couldn't quite think how best to explain my intuitive visualisation. But now I realise that the right visualisation is the one Greg Egan used in Diaspora (in the bit with the exhibition of 5-dimensional orbital mechanics). So I'm coming back to post it, and I hope it's still interesting :-)

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.

[damerell]

My rationalisation was that Aperture's vast infrastructure provides the necessary energy so as not to violate conservation laws. We know, after all, that the facility contains at least one nuclear reactor.

My answer to the question was similar, but to drop a solid steel bar through the portals. If the portals are 2m^2 and we drop the bar at a very modest 10 m/s we get out 160MW.

It's trivial to extract this energy; electric railway locomotives have all the engineering we need. The Class 91 is an obsolete design now, but each bogie's 4 driving wheels can move nearly 250kW from one steel surface to another (an electric motor pretty well runs in reverse as a generator, the frictional considerations are the same, etc). We can also obviously drop the bar at the Class 91's c. 60 m/s top speed, generating just under a gigawatt from 16,000 steel wheels pressed against the bar. (Sanity check; each wheel then has just over 300 cm^2 of bar surface to itself). The bar is safely held by the infrastructure of wheels, electric generators, and the supporting framework for same; it can be sped up or slowed down by adjusting the load.

Obviously one could do much better with custom designed equipment, but this is clearly possible because it uses existing - 1980s - technology.

It's counterintuitive that the speed of the bar makes one get more free energy out, but it does make sense. Cyclists (who are also nerds) are familiar with this effect because, although power to overcome air resistance varies with the cube of speed, power input from gravity varies with speed. Another way of looking at it is that our figure for work done by gravity on the bar (which is where that 160MW comes from) should match our figure for potential energy created by teleporting stuff. 2m^2 (cross-section) * 10 m/s (speed of bar) * 8000 kg/m^3 (density of steel) * 100m (height teleported) * 10 J/kg/m (energy imparted to mass by being teleported up against gravity) is indeed 160MW. Now it's clear why a faster moving bar works better; more mass is being teleported every second.

I don't know what the air resistance losses are - although I hope they're low for a polished steel cylinder - but in principle we could run the mechanism in a low-pressure environment.

[gerald_duck]

On thinking about the physics a little, I believe you can produce unbounded energy from such a system. While normally this would indicate something had gone very wrong with the calculations, you've already assumed something impossible, so are getting what you deserve. :-p

Assume you have your system up and running in a steady state. The amount of energy you're getting out must be the same as the amount entering the system via the use of the portals.

Energy arrives by mass being magically moved from the bottom of the column to the top, thereby gaining gravitational potential energy. How much energy? So far as I can see that depends solely on how many kg you stuff down the hole.

So you don't just let the water fall, you fire it through the lower portal using a water cannon with as much clout as you can muster. I'm not sure there's much physical limit on how much mass you can move per second, if you're prepared to spend energy like it's free. Which, of course, it is. Especially as you can recover most of it at the exit from the other portal.

Or, simpler, you set up your water column in a frictionless tube and only withdraw some of the potential energy using your turbine. The water will speed up indefinitely.

[pseudomonas]

Or use a frictionful tube and run a heat engine off it.