Good.
Once the pipeline has been filled to its standard volume, (the pipe dimension tables figures are based on 60ºF) at 0 psig, the pipe expands elastically with wall stress/Young's modulus = [σ]/E = P*D/2/t/E. If the pipe is not restrained, there is axial shortening = Poissons ratio * [σ], however usually, during hydrotesting, pipelines are considered to be mostly axially restrained, due to continuous ground contact. (You can multiply by a factor Fr, less than 1, to include an estimate of axial restraint).
where 1 = full restraint.
New Area (from radial pressure expansion),
((?/4)^0.5*D +(?/4)^0.5* P*D^2/2/t/E-(?/4)^0.5*2*t )^2
New Length = L (1- (v*[σ])*(1-Fr)/E)
[&sigma] = P*D/2/t
v = Poisson's ratio [default = 0.3]
New Volume due to pressure = Vp = new Length * new Area
Usually of more significant concern is the volume change of the pipe due to temperature.
For steel this is New Volume, Vt = Vo (1 + 3*? * [δ]T)
where ? = 0.000006 in/in/Fº
[δ]T = change in temperature Fº
The volume of the Pipe must be equal to the volume of water, which also changes with pressure and temperature. The water volume factors for T and P can be found in tables, or there are some equations which are also around.
Now just find P such that volume of water @ P & T = volume of pipe @ P & T.
Usually the Pressure to Water volume response observed during an actual hydrotest depends more on how much air was left in the pipe, the temperature of the water entering vs the temperature of the water in the pipeline already and the temperature in the morning and during course of air temperature change into the afternoon than the actual pressure response of pipe expansion. For that reason, one can see a lot of different effects, depending mostly on the weather, and how long your pipeline has had a chance to stabilize before beginning the test.
Note, the equations are based on an initial volume at 60ºF, so if your test temperature is different, you must first translate your initial pipe and water volumes to your base test temperature.
If you get large volumes of water added and small changes in pressure, you usually have air that needs to be vented. If you make a running plot of volume water added vs pressure, its pretty obvious how long you need to settle and vent air.
Without all those variable conditions considered, and for a 400 mm pipe diameter, you would probably only need around 1 to 1.5 cubic meters of water added to raise the pressure 7000 kPa. Throw in a temperature expansion allowance of 20ºF change volume for day-night temperature difference.
BigInch
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-born in the trenches.
