erobz said:
If the container is completely rigid, and is completely full with "virtually" incompressible fluid the slightest positive change in temp, and the tank would burst ( or the pressure would shoot to infinity ). If the temp decreased, it would go to absolute zero.
The reality is the "real" tank dictates the pressure by elastic deformation?
Edit: After sanity checking against a competing calculation by
@TonyStewart, something smelled wrong. I've corrected a factor of 100 blunder in the computation below.
If the tank is ideally inflexible (a decent approximation for a very strong, very thick tank) then we still have to worry about the water. Real water is compressible.
If you chill the water (toward about 4 degrees Celsius) it will tend to shrink away from the container walls. However, this will relieve the atmospheric pressure to which it had been subject. It will expand as a result.
What is the bulk modulus of water? We can ask Google... 300,000 PSI. Imperial units. Bah. We can ask again and get the answer as ##2 \times 10^{9} \text{ Pa}##. Even more useful for our purposes would be the same figure expressed in atmospheres. Call it 20,000 atmospheres.
This says that if we reduce the volume by a small fraction (e.g. one percent) then we will get a pressure increase of that same percentage of the bulk modulus. So one percent volume decrease is going to result in a pressure increase by 200 atmospheres. The reverse is also valid. One percent volume increase = 200 atmospheres of pressure decrease.
Or we can turn it around the other way. One atmosphere of pressure reduction is
0.5 0.005 percent of volumetric expansion.
Edit: Note factor of 100 error here
How much temperature reduction do we need to get
0.5 0.005 percent of volumetric reduction? The formula for water is nasty. But we can Google up some figures.
water at 20 C: 0.000207 (1/degree C)
water at 30 C: 0.000303 (1/degree C)
water at 40 C: 0.000385 (1/degree C)
If we drop 5 degrees from 40 C, that's 0.2 percent decrease in volume. We need another 0.3 percent.
If we drop 10 degrees from 35 C, that's 0.3 percent decrease in volume. Bingo.
Edit: With the factor of 100 error corrected, the calculation is easier. It is ##\frac{0.00005}{0.000385} = 0.13## degrees Celsius. (Not 15 degrees Celsius).
If my calculations are correct, when the 40 C water has cooled to
25 39.87 degrees C, it will have reduced to near zero absolute pressure and will either be boiling or at the threshold of boiling. The vapor pressure of this water will be about
24 55 torr. The low pressure will have allowed the water to remain in contact with the vessel walls.
However, as the water cools further, it will shrink away from the walls, boiling and leaving water vapor in the head space that results. The vapor in the head space will condense away to lower and lower absolute pressures as temperature is further reduced.
At some point, freezing will ensue. The water will expand to fill in the head space. Volume would nominally increase by about 10%. The pressure will increase to somewhere in the neighborhood of 2000 atmospheres (ten percent of the bulk modulus) at which point the pressure will suffice to keep the pressurized ice confined.
According to the state table of water, it looks like 2000 atmospheres at -20 C is indeed ice.
This is all pretty much back of the envelope calculations from a fellow who is not an engineer. Errors may exist.