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Imagine there is a micro-city built on the ocean floor in the pacific ocean covered with a durable dome with a 1 mile radius to keep water out. At this moment let us assume that the dome holds up beautifully thanks to the super-tough reinforced concrete developed to stand up to TSAR bombardment.
The problem is: how do submersibles get in and out of this dome city? Air pressure inside dome is 1 atm and the water pressure is approximately 1000 times stronger.
You can use the concept of a decompression chamber with two doors, one to the outer, one to the inner environment.
When it is open to the outer environment it is closed to the inner. Submersible can enter the environment from the outer (or leave to it).
When the outer door is closed the pressure can be adjusted (and some water removed) to match the inner pressure and the door can be opened on the inner side, allowing the crew to reach it.
I suggest flooding your dome.
No, wait, hear me out.
Your subs, essentially, want to be able to get in and out of the dome. Moving them in and out of water is messy and inefficient.
Your dome, too, will have to pump out water occasionally from a ‘sump’ (I’d call it a bilge, but this isn’t a ship), so you must have pumps capable of moving water out of the dome. Trouble with these pumps is theyll have to do an awful lot of work to pump out into 1000 atm, which will make them slow.
If the base of your dome is flooded with water at sea level pressure then you give the pumps a bit of a buffer that can happily absorb a sudden influx of water.
Your subs can then simply sail from the outside into a water lock with an inner and outer door (preferably ones that open sideways, or if you’re not just opening the doors to move water around, outwards). Once the sub is in you can close the outer doors and use the water pressure inside the lock to force water into the flooded base of your dome. If you allow the water out through carefully chosen vents you can minimise currents against the sub. If you want an example of this try looking up canal locks.
Once the pressures (very quickly) equalise you can open the inner doors and let the sub in. Your sump pumps can then take their time with the extra water, and thanks to how incompressible water is the level of your sub marine submarine pen won’t even rise that much (say your water lock is 20m by 10m by 10m you’ll be adding just under 100l of water to a lake a mile wide; or less than 0.1 mm rise).
Operating in reverse is similar, but this time you let water in from the outside, taking care that either your sub is strong enough to deal with the watery jackhammer you’re applying (hydraulic shock is not the same as water pressure) or that you let the water in slowly. Keep the lock open for when you want to return.
Plus you get a lagoon under the sea, and that’s pretty cool.
There are a few options:
Basically a dry dock air/water lock. Your dome would need a gate carved into the side. The gate would open allowing subs into a chamber. The gate would close and water would be pumped out of the chamber and filled with atmosphere. The chamber could then be accessed by personnel. This method has the added advantage that external maintenance could then be easily performed on the sub.
A Docking system just like space shuttles and the ISS, you could have a standardized docking system that connects the dome to a port on the sub, thus docking subs outside your dome.
The other method that is not possible with your system:
Bottom entry sub pen if you pressurize your dome to the point water cannot flow in from the bottom (much like a bucket of air pulled into a pool) subs could come in from the bottom, surface and be docked.
All of these methods are currently used one or the other today in marine technologies.
I suggest two types of docking systems:
Personnel docking: the submarines may dock via "door to door" system which allows personnel and cargo to pass through a connection while the submarine stays outside. Think of astronauts docking to a space station. This is the most efficient docking system as maintenance is seldom needed. That's why this docking method will be used most of the time.
decompression chamber large enough for a submarine: a second dome contains an air pocket at depth pressure and may remain open at the "seaside". When personnel leaves the submarine through this large chamber, two steps are necessary: the seaside entry and the habitat-side are sealed. Decompression is achieved by pumping air into a storage tank and then the habitat door is open. The water is barely compressible and is pumped out (into another tank) only when drydock maintenance operation is needed.
Points of interest:
1- i have found that using storage tanks for the air in the airlock is easier than having to pump water in and out into the open.
2- I based my assumption on water compressibility as explained in this Wikipedia article. https://en.m.wikipedia.org/wiki/Properties_of_water#Compressibility
Have you tried this thing as a kid, when you are swimming in a pool or at a lake, where you take a small bucket, turn it upside down, and competed for who could push it down the farthest? It is hard, since the bucket will hold air inside itself, which wants to float. However, it is possible to force it under and hold it down. The air does not come out of the bucket unless you flip the bucket around.
Your underwater bio-dome is basically a giant semi-spherical bucket. Its structure gives it weight, plus it requires a handful of ultra-strong cables anchoring it to the sea floor, probably all the way to bed rock. The air may need to scrubbed (as on the ISS) and replenished from time to time but it will not escape. Here is a crude doodle that I made in paint:
My doodle omits the "city" part, but I would expect buildings could either be on stilts from the ocean floor, or suspended from the top of the dome, or both.
Now that we see this, the answer to your question is almost trivial. The submersibles enter and exit from underneath.
You have two transitions to make here, if you want the vehicle to come into the dome. One is high pressure to low pressure, the other is air to water. Thus, you would need a chamber big enough to hold the vehicle, and then pump water out and air in. You then have the decision to make - do you pump the water back into the sea (high pressure) or into the dome (low pressure)? When the vehicle leaves, the problem is just one of engineering - designing the chamber so that it would handle the inrush of high-pressure sea water of its own accord.
So your decision is an external docking method, like the ISS, or an internal hanger/bay system, like Star Trek. The external system would be faster, and energy-efficient, but the second system allows for maintenance of the vehicle and loading external weapons.
But a consideration - is the dome truly a dome? Is it a sphere, or a hemisphere? Is it half a mile high, or a mile high? A hemisphere (actually, a distorted hemisphere, somewhat like a Hershey's kiss) would handle the external stresses better - take them to ground. It makes a difference, as the water pressure at the top of the dome would be considerably less than at the base, and would be the better place to put your docking mechanism. Less stress.
