# How to calculate a gas leakage rate from a small hole of cylinder which contains a compressed liquid?

for example LPG cylinder having leakage

Question

for example LPG cylinder having leakage

- The gas leakeage could be calculated as a restrictor disk. That is q=Cd*A*SQRT(2*g*(Pmax-Pmin). Where Cd es un discharge coefficient, A the hole surface, g in the gravity ant Pmax is the pressure in the cylinder and Pmin is the atmosferic pressure. All in SI units.
- I think we should know more about the hole and the fluid. If the hole is very narrow and the wall is thick, you may have laminar flow and can use the Hagen–Poiseuille law. If the Reynolds number is too large, you have to use laws for turbulent flow. If the pressure drops below the vapour pressure of the fluid, you will get evaporation and possibly two-phase flow.

If the leak is big enough, you will see a massive temperature drop, the steel of the container will become brittle, the container will rupture, and your leakage rate will be something like 100%/millisecond. :-) - thanks both of you Ulrich Deiters and Carlos for your reply

Let's consider Cl2(Chlorine) is leaking from a tonner (inside the tonner pressure is 32kg/cm2g, temp.30deg.C) diameter of the hole is around 0.1mm. There is a possiblity of forming two phase flow, if this is the case how do we calculate leakge rate .....? sir - A first approach: The phase change will be done in the atmosferic part. That is the leak will be a liquid leak. So the aproach as a restriction plate is still valid.
- … certainly a reasonable approach, if the wall is thin But we should consider the thickness of the wall and its heat conductivity, which have not been specified, yet. For a thick wall and a straight cylindrical bore, one might assume capillary flow. Then the flow rate would be proportional to the pressure difference (assuming constant temperature and no phase change).
- My suggestion is that you check what kind of a thermodynamic process would your leakage gas under go.....refer any gas dynamics text for a basic idea to calculate the mass flow rate using the pressure ratio....this doesnt consider any flow physics...you will be indirectly using energy conservation law....later you may stop at a proper point to check the issues with phase change if any....but i think ,assuming no phase change within the small hole...you will definitely get the gas mass flow rate outlet...i suggest not to incorporate uncessary heat transfer calculations..
- Dear Surya,

if the leak is from pressurized liquid Cl2 to the atmosphere at ambient temperature, a phase change must occur somewhere. If the flow is rapid, the evaporation will take place outside the vessel, and then Carlos' “restriction disk” approach might work. If it is a slow flow through a narrow bore, evaporation will set in somewhere in the middle. Finding out where is a nice optimization problem. - For low speed flows, the flow rate is limited by the liquid part of the flow. When this reaches the saturation pressure of the fluid, it will start to vapourise (and therefore cool). So do the calculation based on fluid flow through a hole. Then bolt on the thermodynamics.
- Mr.Ulrich....your example is correct....i donot say that phase change does not occur....it surely occurs ...as u rightly mentioned i was referring to the case where the phase change occurs outside the bore.i.e. for a rapid flow..i.e..higher pressure ratio....but please correct me if my assumptions are a bit harsh...
- Dear Suriya,

you assumptions are not “a bit harsh”. The problem is that the author of the question did not tell us anything about the wall thickness and wall material (-> heat conductivity). Without these informations we can continue this discussion forever. - I came across this problem very recently and by a simple omission have plunged myself deeply into a mess. Using the discharge coefficient equation - same Carlos has mentioned - results I was getting were completely wrong. The point is that in the contraption I was developing pressurized gas was bleeding via several diam 0.8mm orifices into a packing chamber of a low pressure-drop turbine. The gas flow rate I was calculating initially was obviously excessive, say, three time over, at least . Upon digging out some of very old textbooks I have used as a lecturer many, many years ago I found (re-found) that upon using truncated nozzle equation (otherwise the 'choked Laval nozzle'), applicable because the pressure drop accros the steering wall was in excess of 2.5 bar, flow rates diminish. Upon using this approach the results quickly went back in line with the experients. Wall was thick enough for the gas to form a nozzle-type flow and the pressure drop forced choking, so the flow rate was restrickted. Perhaps, I should mention that my gas stream contained some airborne water droplets, and adjusting for the cross section taken up by water and upon ajdusting for critical pressure and temperature for the water vapor I got the gas flow rate right on the money. Wall thickness was merely 1.8 mm and yet the flow was choked.

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