# Calculating mass flow rate, force of thrust produced compressed air discharge

This one's a bit of a doozy, so please feel free to comment on any one single part or parts to this physics problem.

The setup is simple--I have a container (an empty bottle) which I fill with a certain pressure of compressed air through a car tire valve assembly attached to the top of the bottle. On the bottom is an orifice made crudely by forcing a heated paperclip through the bottom of the bottle, small enough such that a thumbtack can fit snugly through it, effectively sealing the hole. The whole thing is attached to a small toy car; Removing the thumbtack releases a stream of air that acts as a thrust force. I'm running this trial with a 2 L bottle at room temperature, and for incremental pressures of 10 psi from 30 psi to 60 psi.

As an assignment, I have to determine a few things:

1. THE NATURE OF THE FLOW: My understanding is that adiabatic flow is different from isothermal flow is different from compressible/non-compressible flow, etc. Unfortunately, this is literally the first time that I've heard any of these terms, and while I researched and understand their definitions I cannot tell which of the following my setup is. Would anyone with a more professional of experience know which my setup might be, and why? This affects which of the equations I can use to find...

2. MASS FLOW RATE & THRUST FORCE: I did see this link on calculating the flow rate of air through a pressurized hole: Calculate flow rate of air through a pressurized hole

However, my understanding is that the Bernoulli equation described in the link above does not apply because if the ratio of outside pressure to bottle pressure is lower than 0.528, then the flow is choked. This, however, opened up another can of worms, in that I had trouble finding an equation that does work concisely.

I've found quite a few, but I am the world's worst researcher and am probably missing something very helpful. I need to know the BEST one given my setup. The ideal scenario would be a time-dependent one; I found this website (the equation is towards the bottom of the page) that split the mass flow rate into a time dependent equation that had a time-dependent pressure equation and time-dependent temperature equation as part of its components. I tried plugging values (pressure of 30 psi) into the pressure equation and received some pressure marginally different from 30 psi for a time of 10 seconds into the trial. This seemed to conflict with the fact that a lot of air sounded like it was coming out of the bottle in the first 10 seconds when I ran the trial. OR, if this equation is absolutely right, am I doing something wrong with plugging in my variables or failing to account for a certain variable or effect with my experimental setup? For plugging in, I used the temperature in absolute kelvin, the gas constant R = 287.058 J mol^-1 K^-1, gamma = 1.4, orifice area in m^2, volume of bottle in m^3, and pressure in psi.

Given the nature of my setup, what specific mass flow rate equation would apply best, and why? THERE ARE A LOT, and trying to look for the right one myself has gotten me nowhere the past few days. Note that my setup involves something that is time-dependent, as pressure, mass, velocity of the gas being expelled are changing over time as air is expelled.

=========

3. COEFFICIENTS OF DISCHARGE & FRICTION: The latter I know can be roughly approximated, but I was wondering if anyone had a reliable source from which I could get information on it. Rolling friction sources that I found involved very specific tires, and I am looking for a coefficient of rolling friction for small plastic toy car wheels on marble tile. Anyone know of any good sources or approximations I can cite?

For the former; I know absolutely nothing about the Coefficient of discharge, however part of the assignment required that one extrapolated beyond classroom knowledge a decent amount. Again, I was just wondering if anyone knew of the most reliable source from which I can get a coefficient, assuming that the orifice was made the way it was and is circular in nature.

4. VELOCITY OF GAS THROUGH ORIFICE: Again, I've had trouble researching this. I know that I can relate this exit velocity to mass flow rate, since:

Mass flow rate = (gas density)*(Area of orifice) *(exit velocity), so if I solve for the former three I can easily determine exit velocity. This would be made much easier, however, with an equation describing the escape velocity. This is, again, better as time-dependent and pressure-dependent, as changing pressure means changing gas density and changing exit velocity.