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I'm trying to model the resultant temperature drop in a bottle of high-pressure CO2 as gas is allowed to escape (the application is for a cold gas propulsion system and for supply to air bearings). Right off the bat, I understand that cooling of a

*single-phase ideal gas*as it is allowed to expand is due to adiabatic expansion (aka Joule expansion), but I don't think this model will work for my case since CO2 in a bottle is a two-phase liquid-vapor mixture. Furthermore, some liquid is evaporating due to the removal of mass from the system, so I presume cooling is due to the absorption of energy from the system in order to vaporize some of the liquid to maintain equilibrium.

My end goal is to graph the time-rate of change of pressure and temperature in the CO2 as mass is removed at a steady (or unsteady?) rate. For my first iteration I will ignore heat transfer from the surrounding air (I can add it in later--I think it's pretty trivial compared to the rest of the model). I originally went into this thinking I could numerically model it after finding some differential equations that model the process, but it's not nearly that simple.

The way I'm thinking about it is like this:

- The control volume consists of 9 oz (0.255 kg) of compressed CO2 at room temp (25C, or 77F), and since it's a two-phase mixture, that automatically gives us the pressure (6.4342 MPa, or 933.20 PSI)
- There is
*some**total enthalpy (H)*associated with the system at this initial point, depending on the*quality*of the mixture. Since I'm starting with full bottles, then for the sake of simplicity I'll say that it's entirely liquid. Therefore, by definition, quality is 0%. The mixture is entirely a saturated liquid at 25C, the specific enthalpy is 274.78 kJ/kg, meaning the total enthalpy of my system is (274.78 kJ/kg)(0.255 kg), or**H = 70.1 kJ**. - When I remove some small amount of mass from the system, then the total enthalpy of the system must drop (in effect I am removing energy in the form of matter). Some liquid will evaporate in order to re-establish equilibrium (and oddly enough, the quality
*increases*). The temperature will drop due to this evaporation, and the overall pressure will be slightly lower. I now have a two-phase liquid/vapor mixture of CO2 that is at some slightly lower temperature/pressure and of some reduced total mass.

*constant*(average) mass flow rate out of the system which I have measured empirically at 0.0396 g/s. I presume it's a valid assumption because 1) the change in pressure differential is presumably small compared the total pressure, and 2) the flow is regulated by a pressure regulator downstream anyway

At this point, I would think that the problem is entirely independent of time. If I remove a little mass, then the temperature/pressure will drop a little bit. If I remove a lot of gas, then the temperature/pressure will drop a lot. Again, this is entirely neglecting heat transfer from outside the bottle.

Is there some sort of relationship that I can derive between how much the temperature changes vs. how much mass is removed from the system? Are the assumptions I've made even valid? Would I have to approach this using something like Gibb's Free Energy function or something to determine the new equilibrium temperature after some mass is removed?

Any help would be appreciated! Even just pointing me in the right direction would be a great start. Thank you in advance!