# Calculating the leakage of water in a pressurized vessel based on initial and final pressures?

Let's say you have a piping system containing liquid water under high pressure. One section of the pipe gets isolated, and it is initially at the same pressure as when it was isolated. we can think of this isolated portion of piping as a pressurized vessel of liquid. Assuming no change in temperature, you notice that the pressure is slowly dropping, suggesting a passing isolating valve that's at a lower pressure on the other side, or a leak in a fitting to the environment.

Would it be correct to use the difference in final and initial pressure values to calculate the initial and final densities of the water (assuming we know the temperature and it's constant), and then use the volume of isolated piping to determine the initial and final mass to calculate how much water was lost? Is this the right approach?

• I suspect that you are doing a hydrotest of a vessel to verify vessel integrity. Is there a maximum "leak rate" that is tolerable? In other words, given the fact that small temperature variations will cause various differential expansion or contraction of equipment and fluid, it's probably not reasonable to expect absolutely zero change in pressure over time. Jul 30, 2022 at 2:14

You need an equation of state $$\rho = f(P)$$ density as a function of pressure. The method you describe is valid and often used for gases, because they use ideal gas law

$$\rho = P/RT$$

But for liquids this Eqn is very difficult to determine, and practically speaking is usually:

$$\rho = constant$$

or a function of temperature only. If you leak only a tiny amount of water even at high pressure, the pressure of the rest will drop to zero almost immediately.

In practice you are better off trying to collect the leaking water to determine the amount. But in many cases it's not important to know how much is leaking, just locating any leaks and fixing them.

• To give some context on the magnitude of pressures, if it's 70 liters at 15 MPa that drops to 7 MPa, does that change the answer? Jul 29, 2022 at 22:45
• This figure researchgate.net/figure/… summarizes that at 15 MPa, the density of water increases about 0.6% from its value at 1 atm. I would say this does not change the answer. Jul 29, 2022 at 23:50