Arun asked me a year ago, and I've just opened my messages now. I've done the rough math, so I figure the community should benefit from it:
Arun Vajpey said:
Mr Turner, I want to ask how much this effect would have affected any people trapped and alive in the stern section of the Titanic. There is a chance that a few of them would have been trapped in relatively small compartments at the extreme stern well within the bowels of the ship. In that case, those areas would have taken longest to flood and some air pockets probably still remained until the stern imploded at about 500 feet depth. Also, if there were no large loose objects there, the persons might have remained relatively uninjured and conscious.
If you hypothesise this scenario, what would be the air temperature gradient in those 70 seconds (I think you said elsewhere) that it took the stern to reach 500 feet?
Sorry, I have been gone for some time. - nearly a year.
If you are still interested, here is my answer:
The water pressure increase would have increased as the depth increased. This would have been nearly instantaneous (within probably 10 or 15 seconds, but hard to say). The water would have, in turn, pressurized the air. This would have been even more nearly instantaneous (a second or less). So the pressure of the air in the Titanic would have been almost the same as the water around it.
So the speed of pressure increase would have been the speed the Titanic sank. Seawater pressure is 1/2 pound of pressure per square inch every foot, as a general thumb rule (14.5 psi every 33 feet to be more exact). At 33 feet, the pressure is 14.5psi, at 100 feet it is 43.5 psi. (14.7 psi is the pressure humans are accustomed to from the weight of Earth's atmosphere, and we call this pressure "1 atmosphere" of pressure.)
We don't know for certain, but as general knowledge, the Titanic sank at about 30 mph and took about 5 or 6 minutes to reach the bottom. The Titanic is 12500 feet down, which means that it descended at about 2500 feet per minute, or 41.6 feet per second. That means the air pressure would have increased proportionally (41.6 divided by half) which gives us about 20.8 pounds per second.
We also need to remember that the Titanic didn’t fall at an even speed. It would have started slowly, picking up speed, until it hit Terminal Velocity of about 30mph in sea water (if you remember from High School physics, this is the speed at which the force of friction will equal the force of weight of the object, so downward acceleration stops. Friction is greater in water than air, so terminal velocity is much lower for a sinking ship than a falling airplane). https://en.wikipedia.org/wiki/Gas_compressor#Temperature
Using the formula:
T2 = T1 (P2 / P1) ^((k-1)/k)
We can calculate the temperature. We must also use absolute temperature and pressure, so what we consider normal air pressure is actually 14.7 psi, and we must add the weight of this air to all our water figures, too (because the atmosphere is pressing down on the ocean water). We also will need absolute temperature. I’ll use Rankine degrees, because the math is simpler conversion to Fahrenheit (just add about 460).
K is a constant, which is about 1.4 for air.
If we assume the air temperature below decks on Titanic was 50 Fahrenheit degrees (+460 = 510 Rankine degrees), and 14.7 psi at the surface. At 500 feet (reached in about 12 seconds) pressure would have been about 220 psi (plus 14.7 atmosphere). We can calculate:
T2 = T1 * (P2 / P1) ^((k-1)/k)
T2 = 510 * (234.7 / 14.7) ^((1.4-1)/1.4)
T2 = 510 * (15.96) ^(.2857)
The 15.96 is our compression ratio. A typical diesel engine has a compression ratio of about 14-22 (source wikipedia).
T2 = 510 * 2.2
T2 = 1125 Rankine degrees
-460 to convert back to Farenheit
Gives us 665 Farenheit degrees final temperature (351 C). That’s a 615 F increase over 50 F (10 C). This would have happened in about 12 seconds so, 51 F per second (28 C per second). Or using the original 70 seconds, 9 F per second (5 C per second)
Granted , this is rough math. We don’t know the interior temperature of the Titanic before the sinking, but we can assume it was mostly around 50-60 F degrees. 70 F is normal room temperature. Exterior temperature was recorded as 48 F degrees, and we know they were turning heaters on because they felt cold, in 1st class and the bridge so we can assume that 3rd class probably wasn’t too much warmer than outside air (at least, somewhere between 48 F and 70 F).
We must also remember that the seawater was near or below freezing. This would have sucked heat out of the Titanic, like putting a boiling hot pot in the freezer. It won’t stay hot for long.
Finally, the Titanic started sinking slowly, then picked up speed. It probably took a bit longer than 12 seconds to reach 500 feet. Probably closer to 30 seconds or a minute. It’s maximum speed was about 30 mph. If it was 70 seconds, that would be a bit less than 9 F per second (5 C per second).
Which means it was a very hot ride down, and if you hadn't already passed out from the pressure and heat, you were probably grateful when the ship imploded in the crushing embrace of freezing waters.
Yes, morbid, I know, but something a submarine sailor thinks about.