Gas flow from one chamber to another

[geologic]

Hi,

I don't know a ton of fluid mechanics, and haven't been able to mathematically define this problem, so I was hoping someone might have an idea.

The problem I want to solve is the time it takes for gas to flow from one chamber (with finite pressure) to another (vacuum). The problem is non-equilibrium and fluid velocity is time-dependent (when the pressures are almost equal, the fluid flow should be slow (I would think)). So the known quantities would be the initial pressures, volumes and temperature and I want to calculate the time to equilibrium (or, effectively, the time-constant).

Thanks,
geo

 

[SteamKing]

You will also need to know the details of the connection between the two chambers (i.e., the internal diameter, the length, any fittings like valves, etc.) You will also need to determine if the flow will be isothermal or adiabatic.

 

[geologic]

Ok, so let's say I know the details of the valves, tubes, etc.

I'm not sure if the flow is adiabatic (I'm not putting in any heat, the experiment is done at room temperature). If there is cooling/heating upon expansion, then heat could transfer in through the metal chamber. Any ideas how I can ensure that the flow is one or the other? It isn't possible to thermally isolate the system I'm using. How can I estimate how important these effects are?

 

[SteamKing]

To make the process adiabatic (or reasonably so), wrap your chambers in insulating material, if this is practical. Another line of reasoning would be to assume that if the entire process occurs in a short amount of time, no significant quantity of heat could have entered or exited the chambers in that brief period.

 

[boneh3ad]

Well, you could do this in a much less complicated manner than SteamKing is suggesting depending on how accurate your answer needs to be. If you just need a reasonable estimate, consider the following:

If your filled tank is reasonably high-pressure, then the most dominant effect on the time to reach equilibrium will be the amount of time the flow through the connection is choked, which will be most of the time for any reasonably high starting pressure. While the flow is choked, the effect of the length of the connecting pipe and roughness and the like is going to be very minor, even negligible. The important factors there are the pressure in the reservoir, smallest cross-section of your connection line and total temperature in your reservoir. This will likely represent the largest portion of time in reaching equilibrium.

The above is very easy to calculate assuming your process is adiabatic. Your process, for all intents and purposes, will be adiabatic. There will be some slight heat transfer going on as the tank discharges and cools, but it won't likely be a lot since it will be simply by conduction. You can make it closer to adiabatic by wrapping the tanks in insulation, but it will be pretty darn close to adiabatic even without in most cases for the purposes of a reasonably accurate time estimate.

After the flow is no longer choked, then the geometry of the connection line becomes more important, but you can probably neglect a lot of it without any major effects and just use an unsteady Bernoulli-type equation with a correction factor such as the Darcy-Weisbach equation to set up a differential equation for the flow through the connection. That leaves you with a choked-flow differential equation to start out with known initial conditions and whose final conditions provide the initial conditions for your Bernoulli-type equation for the rest of the time. That will get you pretty darned close to the right answer analytically.