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While hiking through the Grand Canyon, I started wondering. Say we have a pipe for the purpose of transporting water across a canyon, with the bottom submerged in a pool of water. Like the blue line in this terribly drawn image:

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Now let's seal the two ends of the pipe, which is full of water. What would happen if the pipe broke at the bottom of the U?

Note: It shouldn't matter whether the pipe is a U or a vertical line, I simply thought about this with a U-shaped pipe.


If the ends of the pipe weren't sealed, the water inside the pipe would leak out until the height of the water outside the pipe was the same as the height of the water inside the pipe. But the ends are sealed.

A vacuum can support 10m of water, so perhaps the water will drain out of the pipe until it is 10m above the height of the water outside of the pipe, leaving a big near-vacuum behind. This would explain why the Grand Canyon's pipe has such a large number of valves.

On the other hand, we learn from XKCD's "What If: Glass Half Empty" that water boils into a vacuum. Does this leave a significant amount of water vapour behind?

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  • $\begingroup$ You are on the right track: calculate the vapor pressure of water and you'll know roughly how much water will turn to gas (assuming you know the volume of the pipe that was filled with air to begin with). But basically you've just built a rather large barometer! $\endgroup$ Commented Sep 12, 2014 at 11:39

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It would first be an exciting rush, then a poor barometer, and after centuries an almost empty pipe.

When the pipe breaks at the bottom, the water will start falling within the pipe and boiling at its top because of the almost vacuum formed above the water. The Grand Canyon is almost $2\,\mathrm{km}$ deep, so by the time the top of the water reaches the level of the pool, the water will be moving so fast that its momentum may carry it past this level before the water from the pool rushes back in. After sloshing up and down a few times, the top of the water in the pipe will settle at a level a bit less than 10 m above the water in the pool.

Assuming a mild spring day around 80°F (27 C), the water vapour pressure above the water will be about 0.5 psi, so the top of the water will be about $10\,\mathrm{m} \times 0.5\,\mathrm{psi} / 15 \,\mathrm{psi} \sim 30\,\mathrm{cm}$ lower than it would be if it was a pure vacuum above it.

At this point, you would have broken the record for the world's tallest barometer, beating the Bert Bolle Barometer - the previous champion - by a mile.

The barometer would require some calibration, however, because temperatures in the Grand Canyon can vary from below freezing to more than 40 C, corresponding to changes in water vapour pressure from 0.0060 atm to 0.0728 atm, and a 1% range in water density, so the top of the water would move up and down by tens of centimetres as the temperature varied.

As time passed, however, you would also notice that the mean water level in the pipe was slowly dropping. If you could look inside the pipe, you would discover "What is Wrong with Water Barometers? When a bunch of University of Nebraska chemists and physicists built an 11 metre tall barometer out of clear plastic tubing, they saw bubbles slowly but continuously forming inside the tubing that eventually bubbled out of the water, increasing the pressure above the water and hence lowering the water level. They carefully sealed all joints and checked for leaks, but the water kept dropping.

Eventually they realized that air was dissolving in the pool at the bottom and then "diffusing" into the pipe, forming the bubbles that were slowly filling the empty space above the water. The diffusion constant of air in water at 25 C is only $D\approx 2\times 10^{-9}\,\mathrm{m^2/s}$, so if diffusion was the actual process it might take thousands of years for the pressures to equilibrate, but convection driven by unavoidable temperature differences moves the dissolved air much, much faster. At the University of Nebraska the level fell several centimetres per day.

The empty space in Nebraska was only about a metre, compared to your 2 km pipe, so we might expect your water to fall about 1/2000 times more slowly since the "diffusing" air would have 2000 times more volume to fill. This suggests the Grand Canyon water barometer mean level might only fall about a centimetre per year, but this is likely optimistic. Compared to the indoor Nebraska barometer, the large temperature variations in the Grand Canyon would likely drive much faster water circulation in the pipe.

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  • $\begingroup$ You forgot to mention that due to evaporation the water would quickly cool, possibly to the point of freezing. Also, I wouldn't be surprised if the water would fall significantly below 10m as the evaporating water would provide some extra pressure, I'm not sure how much though $\endgroup$ Commented Aug 16, 2023 at 20:12
  • $\begingroup$ Thanks @SmallPieceOfBread. Yes, the top of the water will cool, but whether it freezes depends on how low a vacuum is dynamically generated above the falling water and whether it has time to freeze before reaching bottom. To vacuum freeze water, I believe the pressure above the water needs to be <0.083 psi (600 Pa), but in final equilibrium, the pressure is ~0.5 psi. In any event, I don't think that a small amount of ice forming at the top would greatly affect the 2km column of water falling with a terminal velocity of at least 10m/s (according to my crude estimate for 15cm diameter pipe). $\endgroup$ Commented Aug 17, 2023 at 14:14

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