Thermal expansion of water

[klugesmith]

It's been a daily habit to boil water using this glass jar in a microwave oven.
Decided to see how much the level rises, on heating from 10 or 15 °C to 100 °C.
More than I expected, in fact dramatically visible without mechanical amplification.


There is a sort of 3x amplification when we observe bulk expansion of a liquid, compared to linear expansion of a solid in 3 axes.  Familiar in thermometers for hundreds of years.
In this example the water volume is close to 500 ml.  The expansion is roughly 15 or 20 ml.

If anyone wants to get analytical here, with literature values for water density and thermal expansion of glass, you can then carry on with puzzle number 2:

Here is a refillable propane tank, as used for barbecue grills and outdoor heaters.  The size is popularly called 20 pound, or 5 gallon.  It's been retired from service, and I want to do a hydrostatic pressure test.
How much extra water must be forced in, to bring pressure from ambient to 300 psig (20 bar)?
Any guesses as to what fraction of that is from elastic expansion of the steel tank, and what fraction is from bulk compression of water at room temperature?

[rikkitikkitavi]

assumption no 1- the tank is completely filled with water, no airpockets what so ever. Probably easiest done by first adding a little bit of water, then apply vaccum for an extended period to evacuate air and then water vapour eventually.

In practice this is impossible and the smallest air pockets inside makes the volume a bit elastic , which in practice makes hydrostatic pressure testing feasible without blowing the tank immediately...

Caveat- I might missremember the exact valuse for compressibility and metal yield strength  aa bit, but I think I am correct to the right magnitude atleast

The added water volume for 20 bar pressure is aproximately 0,1% of the total volume assuming the tank is completely rigid , ie no reversible expansion of the propane tank.

Accounting for reversible mechanical expansion of the propane tank, I have no clue about US design codes for these and cant say how much the stress in the shell affects this. But to make a guestimate:

Assuming a safety factor of 3 to the burst pressure , we are operating about a third of the yield strength where the typical expansion would be less than 1,5% so I would say that roughly the vessel will be 0,5% larger, give or take.

Hence the expansion of the vessel is in reality the larger portion of the amount of liquid used for increasing the pressure during hydrostatic pressure testing of vessel, of which I have done a few. Quite a bit larger actually.

Amazing when you find a weld pore at 200 bar Eye protection is quite mandatory. We did hydrostatic pressure with air once, due to the vessel was not designed to be water filled when installed in the steel structure, and we used low pressure (about 0,5 barg or 7,5 psi). Safety distance 100m...
Remote operated compressor with dump valves...

I could say we where quite nervous, because the vessel had been subjected to a local fire internally and hence we where not sure about its structural integrity.

But no boom.

[klugesmith]

Rikki, glad to see someone else address the puzzle.   
I tackled it on paper a few days ago; details below.

You have real experience with hydro testing of metal vessels.   My experiment was prepared but not completed 10 years ago: Unscrewed propane valve from cylinder and substituted a threaded steel pipe nipple.  Made a sight glass with some very thick-walled Lexan (polycarbonate) tubing.   Got a hand-pumped grease gun for developing pressure. Plan A was to begin with water high in the sight glass, and apply compressed air above it.

Analytically, and trying not to be overly precise:

We both got the same answer for bulk compression of water.
It reaches 0.1% (22 ml in my tank) at pressure of about 21 bar or 310 psi, depending slightly on temperature and on which handbook we open.

We agree that cylinder volume expansion will be significantly more than that, for any given pressure.   Figured using Strength of Materials, without knowing the wall thickness or estimating it from weight.  Here's my attack:

Suppose we chose a wall thickness to make elastic volume expansion match the water compression, 0.1% at that pressure.   From linear strain of about 0.02% in height and 0.04% in diameter, because axial stress is half of hoop stress.  Using the literature value for E, we find a hoop stress of 80 MPa (11,600 psi).

Design maximum pressure for DOT standard propane cylinders in USA is actually about 20% lower:  17 bar or 250 psi.  So in that hypothetical tank whose volume stiffness matches that of water, hoop stress would reach 64 MPa (9300 psi) at maximum working pressure.  Factor of safety about 4 even with very conservative value for yield strength of mild steel.

It wouldn't surprise me if the actual walls are half that thick, doubling the amount of tank expansion and doubling the stress in steel.  Portable cylinders appear to be made from deep-drawn sheet steel, with machine-welded joint between top and bottom.   
I found wall thickness given in mechanical drawing for a large pad-mounted propane tank (ASME code instead of DOT), whose 250 psi working pressure would develop 18,000 psi  (124 MPa) of hoop stress in the wall.
Hey, look what Interne search found:

[klugesmith]

On the nominal Subject, I found that water expands about 4 % in volume.
Water level change will be diminished by about 0.25 % because the glass jar expanded.
Around 9 ppm/degree linear, 27 ppm/degree volume for soda-lime glass, as charted with wrong label.


On the propane cylinder hydro test, I got an opportunity to see the material thickness after someone else's destructive test.

Wall is a little thicker than a nickel, or about 1.5 dimes.  Say, 1.9 mm (0.075 inches).
Then at max working pressure of 250 psi (17.3 bar), hoop stress would be about 20 ksi (140 MPa).   
Water bulk compression would be about 0.08 %.
Tank volume expansion would be about 0.17%, twice the magnitude of water compression.
Altogether about  0.25%, requiring injection of about 55 ml of water for the hydro test.