If you have a cylinder of gas under pressure, and it is released into a closed system, how do we calculate the immediate temperature and pressure without assuming adiabatic expansion?
We can use the ideal gas law, PV=nRT, which simplifies to PV=T, but we’ve only got one equation (PV/T = constant), and two unknowns (pressure and temperature.
I seem to recall needing some type of Z number, and it’s got to involve the rate of release somehow.
Bonus question: How would I find the maximum rate of release of said cylinders via damage in the head (as opposed to body) of the cylinder?
Some questions: Is your closed system already full of gas? I.e. are we talking some kind of real-world situation, e.g. a nitrogen cylinder knocks its valve off in a sealed room full of air?
You might be able to do something with heat capacities and energy. Your escaping gas may or may not be doing work, but that work will still end up as heat in your closed system.
Your “bonus question” is a flow problem, and very likely to be turbulent flow at that. So you assume laminar flow, assume the properties of the hole, get an answer and be aware that’ll have almost no basis in reality…
That’s exactly the accident scenario I’m trying to model.
Would you be able to point me towards some sources with some relevant equations, or that give me the information I need to derive said equations?
I think I can tackle that with those assumptions. As far as having a basis in reality, I know that assuming isothermal expansion comes really close to the safety threshold, so I’m trying to come up with a hard number that shows a realistic case would have an overpressure of far less than the simplistic, isothermal case to develop an argument for the safety basis.
It would be back to the library for me at this point, I’m afraid! But it’s not really relevant anyway - only noble gases at low pressures approximate to ideal gases, and high-pressure gas leaks are governed by something called the “choked” condition. (This was news to me, but I lucked out with Google.)
The second link has some models and worked examples. It seems to me that you could generate a spreadsheet with one of the models, showing mass flow rates at various time increments. Treat the incremental gas additions to the room as isothermal ideal gas, and you can plot an approximate pressure change with time. I hope that’s some use to you!
matt, I can’t tell you thanks enough! I’ve been looking for that equation for a month (sporadically). It didn’t occur to me that it’d be supersonic flow (although it makes sense), and even though I don’t have much experience compressible fluid flow, the equations in the second cite are very straightforward. I’ll just use the log mean temperature of the state on either side of the interval when I use the IGL. I’ll code it in MatLab, so it’ll be pretty easy to use tenth or hundredth second intervals. I’ll probably use a discharge coefficient of 1, just to make things standard. Thanks again!
The EPA and other agencies have several models that are free and fairly easy to use. Based on your posts, I’m not sure any will apply to your scenario, but you may find some of the models linked from this page useful. The level of complexity and ease of use for these ranges all over the place.