(Fluid mechanics) How to find the flow rate of a gas from a vessel

[Geith]

TL;DR

How can I find out how long the air in a pressurised container (known pressure, volume, temperature and size of hole) will take to leave, assuming it is opened under atmospheric pressure?

I am working on a project where I have to calculate various results relating to the motion of a water bottle rocket being launched. I am currently stuck on trying to find how long the thrust period of the rocket is. The model for the rocket is as follows: It is a 2L (0.002m³ bottle filled with air at a pressure of 7 Bar, one third of the bottle is to be filled with water, however for simplicity's sake I am willing to assume that the volume of the air in the bottle is a constant 2L, and that the temperature remains constant (room temp. 293K). I also know that the area of the hole in the bottle is about 3×10^-4 m².

Using this information, is it possible to determine the time that it will take for the pressure in the bottle to reach equilibrium? I am assuming it will follow a somewhat inverse exponential model, where the pressure will never truly reach equilibrium, but will get exceptionally close at a certain time (similar to time constants in a capacitor).

Feel free to ask for additional information.

Thanks.

 

[anorlunda]

Fun problem. I moved it to Mechanical Engineering because it might get more useful answers here.

The volume of air in the bottle and the pressure aren't close to constant. As all the water is ejected, the volume will expand to be 100% of the bottle, and the pressure will decrease to atmospheric.

I am not a M.E., but I wager that some of our M.E. can help you.

 

[Geith]

Fun problem. I moved it to Mechanical Engineering because it might get more useful answers here.

The volume of air in the bottle and the pressure aren't close to constant. As all the water is ejected, the volume will expand to be 100% of the bottle, and the pressure will decrease to atmospheric.

I am not a M.E., but I wager that some of our M.E. can help you.

I understand that the volume is not constant, but I am willing to ignore the presence of the water, and assume that the entire volume of the bottle is occupied by the air, as an approximation. The pressure however will not be constant, it will start at 7 bar and decrease to 1 bar (Atmospheric).

 

[jrmichler]

Using this information, is it possible to determine the time that it will take for the pressure in the bottle to reach equilibrium?

Yes. This is easiest to solve using numerical simulation, which can be done in a spreadsheet. The steps are as follows:

1) Choose a time step.
2) Calculate the water exhaust velocity from Bernoulli's Equation.
3) Calculate the water flow rate from the velocity and hole area.
4) Calculate the amount of water that flows out for one time step.
5) Calculate the amount of water remaining (and the new air volume) after one time step.
6) Calculate the air pressure.
7) Loop back to Step 2 until the water is empty.
After the water is empty, decide if you want to calculate the time for the air to exhaust. If so, study up on choked flow because the calculations get more complex.

You find the optimal time step by iteration. Too long causes inaccurate results, too short takes longer. To do it in a spreadsheet, put the equations for Steps 2 thru 6 in Columns A thru E, then just copy down until the simulation finishes.

 

[Geith]

Thanks, that was very helpful. I was able to put the info into Matlab and get some promising data. I was able to use 0.01 second iterations since I am using Matlab. I have posted my code, as well as the data. The almost linear relationship of the volume of water surprised me.

I am also wondering if it is possible to write out this iteration as a differential equation? I assume using calculus would give more exact results.

 

[jrmichler]

The next step is to verify your simulation. Anchor your water bottle rocket, video it in action, then measure the time of discharge by counting frames in the video. If you used the correct numbers, and your code is correct, then the simulated discharge time should match the measured time within 5% or so.

If the difference is larger than 5 to 10%, then recognize that your simulation is only as accurate as your assumptions. Did you include entrance loss in the nozzle? Are you assuming that air expansion as the pressure drops is isothermal, adiabatic, or in between? Friction loss in the nozzle? Does the water make a vortex as it discharges? In this type of simulation, errors due to faulty assumptions and simplifications are normally much larger than numerical errors.

I suggest adding code to calculate the acceleration, velocity, and position of the rocket. And verify that code. The last part, of course, is to use your simulation to find the optimal amount of water to get the greatest height. Then, knowing exactly how to get the maximum height, challenge your friends to a water rocket height contest.

 

[Lnewqban]

Summary:: How can I find out how long the air in a pressurised container (known pressure, volume, temperature and size of hole) will take to leave, assuming it is opened under atmospheric pressure?

In this link, look for "Excess pressure in container":
https://www.engineeringtoolbox.com/flow-liquid-water-tank-d_1753.html

That equation includes height of water column and a velocity coefficient of 0.97 for water.

You can also find some practical tips and comparisons here:
http://www.aircommandrockets.com/water.htm