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The Trieste submarine reached the bottom of the Mariana trench using some clever techniques like gasoline ballast, but let's assume the ocean just kept going to the core of the earth (or maybe the entire planet is water!). Can they just keep making the pressure vessel thicker?

How deep could we go before weird problems started happening? For instance, the atmosphere on Jupiter transitions from a gas to a "metallic soup". I would imagine even titanium would begin to break down (or perhaps dissolve?) in some way at high enough H2O pressure.

Assume temperature is constant, or doesn't increase much. This is a question about water pressure (unless water changes at high enough pressure)

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    $\begingroup$ The Trieste was a bathyscaphe not a submarine. Submarines have much more restricted maximum operating depths, with 1000 meters being a record; that was the ill-fated titanium-hulled Soviet submarine K-278 Komsomolets. (The difference is that a submarine is capable of independent operation, whereas a bathyscaphe or submersible is dependendent on a support ship.) $\endgroup$
    – AlexP
    Commented Jul 21, 2023 at 0:54
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    $\begingroup$ I'm not sure there's a worldbuilding quesiton here. It sounds like a physics question. How can we assume temperature can be ignored when pressure causes temperature? Deep hard rock mines are hot because of pressure, not nearness to the core (they're nowhere near that deep). Further, what's the point of the question? How deep we can go with current tech? The answer is not very far. $\endgroup$
    – JBH
    Commented Jul 21, 2023 at 1:04
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    $\begingroup$ @JBH Pressure does not by itself cause high temperature (except in stars, where it causes fusion). (Or you may be thinking of compressing an ideal gas, but in that case it's the work done during the change in volume that causes the heat, and the gas will soon cool down if you leave it under compression). The Earth's interior is hot due to radioactive decay, left-over heat from the Earth's formation, and a few other reasons. Pressure alone is not one of them. See en.wikipedia.org/wiki/Geothermal_gradient $\endgroup$
    – causative
    Commented Jul 21, 2023 at 1:59
  • $\begingroup$ Look up the concept of “supercritical water”… fascinating things happen at high-temp-but-higher-pressure depths. $\endgroup$
    – SRM
    Commented Jul 21, 2023 at 4:55
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    $\begingroup$ ## Titanium Probably Works ## Some common alloys of Titanium have yield strengths of 1,000 MPa, meaning that LDutch's phase transition diagram definitely comes into play. It is likely possible to build a pressure vessel that can be submerged in a deep enough water column that "ceasing to be in liquid water" becomes the main design constraint. $\endgroup$
    – codeMonkey
    Commented Jul 21, 2023 at 15:58

2 Answers 2

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For starters, your assumption of constant temperature is not too far off: considering that water has a maximum density at around 4 C, the bottom of any ocean, far from any appreciable heat source, would be around that temperature.

Now, coming to your question, the maximum depth at which a manufactured item would be able to dive would be the maximum pressure at which water is still liquid.

If you look at the water phase diagram

enter image description here

you see that at 4 C the max pressure at which water is still liquid is about 630 MPa or 6.3 kbar, which, considering that on Earth 1 bar is about the pressure below 10 meters of water, would mean 63 km of depth.

A more precise calculation points to 65 km of depth under freshwater, or 63 km under saltwater.

To go deeper than that you would need more than just a sturdy enough contraption.

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    $\begingroup$ While this sets the theoretical limit at which something could dive in water, which is tbh quite interesting, it doesn't answer how deep some pressured hull could go and survive. At 600 km deep, titanium might just start to deform and flow, regardless of how thick the hull is. $\endgroup$
    – Cyrus
    Commented Jul 21, 2023 at 9:18
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    $\begingroup$ @Cyrus if you want "design a system to withstand arbitrarily high pressures from diving", you need to start specifying your requirements very carefully. $\endgroup$
    – fectin
    Commented Jul 21, 2023 at 12:00
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    $\begingroup$ I would love to see a device intended to mine through pressure-formed-ice. Perhaps you could 'suck' your way through it by locally reducing in the direction you want to go. $\endgroup$
    – sdfgeoff
    Commented Jul 21, 2023 at 13:09
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    $\begingroup$ "Sturdy enough contraption" would be quite the challenge in its own right. 630 MPa is roughly the yield strength of high-strength steel. You're probably looking at making the pressure vessel from exotic alloys or things like single-crystal diamond. $\endgroup$
    – Mark
    Commented Jul 21, 2023 at 22:25
  • $\begingroup$ @Mark can a high-strength steel hull withstand 630MPa on one side, and 629MPa on the other? Then, you "just" need 630 nested hulls, which makes this an engineering challenge rather than a fundamental limit. $\endgroup$ Commented Jul 22, 2023 at 7:49
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Alright time to add my 2 cents.

Its not that simple. By which i mean you cant make any definite statements here because so many factors come into play. Everything from Gravity, Water salinity, Buckling forces, Vessel purpose and more. Its not a trivial thing to answer.

For example, if you classify as Submarine as a sphere that can sink to the bottom of a 30km deep ocean, you can build that. Make a sphere 10 meters in radius with a small cavity in the middle and you are good to go. But that's not a very useful DSV now is it ?

Water Density

Water is not incompressible. It is just very hard to compress. But that little bit of compression is responsible for 99% of your problems. For instance, why do Submarines implode supersonically ? After all, if your pressure vessel fails the Water should just "fall" in right ? Well no, because water compresses it becomes a spring. When you have a pressure vessel, what it is actually resisting is the spring force of all the water around it. The instance the pressure vessel fails, this spring force is released, which happens at the speed of sound in Water. ~1500 m/s. Which is why implosions are instant.

But ok, how do you model fluid density at depth ? This equation;

$$p_d = \frac{1}{1-\left(\frac{pgh}{B}\right)}p$$

gives us a good starting point. It computes the density of Water or really any medium at depth. The B is the bulk modulus, or how hard it is to compress something. Now there is an immediate problem with this, to figure out the pressure at depth, we have to run this equation for many different depths and add up the total. Which still does not take into account Temperature or salt contents. So while this equation is a good start, its not perfect.

enter image description here

So here is a graph i made. You can see that the Pressure up to a depth of 100km varies widely between the classical $P = phg$ and more accurate approach. To the point where at the maximum model depth the pressure suggested by the classical model is ~25% smaller than what it would more realistically be. Of course, this has an exponential nature to it. If the depth is 200 kilometers the projected pressure by the classical model is ~$2GPa$, where as in actuality it would be closer to $5.5 GPa$

This does of course leave out phase changes, which will occur. My main point here is that pressure is not a simple subject. And Water would never under natural circumstances turn into any other phase. Only in very weird situations could this even begin to happen. The diagram L.Dutch showed is correct but imo a bit misleading in presentation. Because they leave out the density differences. For Instance Ice II is less dense than liquid water and would just shoot up the moment it was made, to quickly dissolve again. Afaik, we don't know the exact densities for other forms of Ice or the bulk modulus. So it is a bit hard to be super accurate here. But from the Diagram, where Ice VII turns into Ice X after a 100 fold pressure increase, we can guess the bulk modulus is pretty high.

Oceans cannot achieve these pressures

Ill skim over this a bit, but these pressures are entirely theoretical for one main reason. Gravity. Oceans are just valleys. And while they can get quiet deep, they cant get massively deeper as the largest mountain on a planet is. On Earth, you cant get a 20km deep ocean because the ground couldn't support that depth. Same reason why you cant get a 20km tall Skyscraper.

If you want deeper oceans, you need lower gravity. But since gravity is a term in both models, if you make it lower the pressure decreases. I talked about this in a different comment but the TLDR is that you cant get exotic forms of Water in really any natural configuration. Either the oceans cant physically get this deep, or the temperature is to high or the gravity is to low.

Generally speaking, take the Gravity of your world, divide it by Earths gravity and that fraction is how high your tallest mountain and deep your deepest ocean should be.

For example, lets look at Ganymede. Ganymede's surface gravity is 1.4 m/s² While Earth is obviously 9.8 m/s². The fraction of this (Earth/Ganymede) is 7. The deepest point on Earth is ~12 kilometers deep, so this would suggest an Ocean depth for Ganymede of like 80-90 kilometers. Which is around the right ballpark. We estimate the depth to be ~100 kilometers. Note, the Pressure at the bottom of Ganymede deepest ocean is going to be about the same as in the Challenger Deep. Obviously with some variations because of the different temperature and salinity etc.

So, to answer the question, we can build DSVs for basically any Earth like ocean that will work. Duo the likely certainty that it really does not get more extreme than on Earth. Sure the oceans might be deeper which can lead to issues where it takes a week to get down there. But those are no physical limitations. If you took the Trieste and dropped it into Ganymede's oceans, it would probably still fail because the descend rate is to low but conceptually it could survive on the bottom.

Other Planets

So this is all for Oceans on planets. On Gas giants / dropping a sub into one you will get ripped apart by the winds long before reaching any solids. The Temperature will also become an issue. Titanium is pretty tough, as long as it is cold. Titanium is also surprisingly moldable once it gets hot.

Sadly we just dont have any data on this. We dont know what the temperature deep inside a gas giant looks like.

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    $\begingroup$ "Ripped apart by the winds" in gas giants. Not so sure. It doesn't matter how fast the wind is if there is no ground to stand on. Velocity gradients may be an issue to the (very low-density) floating habitats. But on Earth balloons generally don't get torn apart when they enter the jet-stream. Of course, a dense heavy submarine would plummet to it's crushing and then melting doom. $\endgroup$ Commented Jul 21, 2023 at 21:31
  • $\begingroup$ @KevinKostlan I was more refering to debris in the winds such as ice coming from updrafts. $\endgroup$
    – ErikHall
    Commented Jul 21, 2023 at 21:54
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    $\begingroup$ If you are also moving along with the updrafts why would the debris hit you any faster than your sub's terminal velocity? (which I think exceeds the speed of the updrafts since they are slower than the prevailing winds) $\endgroup$ Commented Jul 21, 2023 at 22:03
  • $\begingroup$ @KevinKostlan because i would think a free falling submarine that weighs a lot more than ice balls is probably going to fall in updrafts that can lift ice but not a sub. $\endgroup$
    – ErikHall
    Commented Jul 21, 2023 at 22:08

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