Determining mass flow rate and exit velocity of compressible flow

Question

This one's a bit of a doozy, so please feel free to comment on any one single part or parts to this 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: https://physics.stackexchange.com/questions/131032/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. 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.

Answer 1

You are correct that the Bernoulli equation will not work here because you're in the compressible flow regime. As you already learned, the flow through the orifice will become sonic when the pressure in the bottle exceeds about 30 psi. What you have here is probably mostly adiabatic expansion of the air in the bottle (no heat transfer), however the problem will be easier to solve if you assume isothermal (constant temperature) expansion instead. If isothermal doesn't produce accurate results, I can elaborate on how to do adiabatic instead.

As for calculating the velocity of the gas: I usually use the choked flow equation found on Wikipedia for this type of problem:

$$ \dot m=C_dA\sqrt{k \rho_0 P_0 \left( \frac{2}{k+1} \right)^\left( \frac{k+1}{k-1} \right)} $$

where $\dot m$ is the mass flow rate, $C_d$ is the discharge coefficient, $A$ is the cross-sectional area of the hole, $k$ is the heat capacity ratio (1.4 for air), $\rho_0$ is the upstream density and $P_0$ is the upstream pressure. Pressure is known and density can be calculated as:

$$ \rho_0 = \frac{P_0}{RT} $$

where $R$ is the specific gas constant (not to be confused with the universal gas constant) and is equal to $287 \frac{J}{kg \cdot K}$ for air. You can probably estimate the hole diameter by measuring the diameter of the thumb tack that fits in it. That just leaves $C_d$ - from experience, a value of 0.7 should work for this problem. You can adjust as necessary as you collect more empirical data.

For sub-sonic flow, I use the following equation, which I got from here:

$$ \dot m = A \cdot \sqrt{\frac{2g \gamma}{R(\gamma -1)}} \frac{P_0}{\sqrt{T_0}} \frac{\left[ \left( \frac{P_0}{P} \right)^\frac{\gamma-1}{\gamma}-1 \right]^\frac{1}{2}}{\left( \frac{P_0}{P} \right)^\frac{\gamma +1}{2\gamma}} $$

$P_0$ and $T_0$ are the upstream pressure and temperature, $P$ is downstream pressure, and $g$ is the gravitational constant. I used $\gamma$ for heat capacity ratio instead of $k$ in this equation.

As you mentioned in your question, the pressure in the bottle is decreasing with time, which changes the rate of mass flow. What you need to do is solve the choked flow equation over time. It is a non-linear ordinary differential equation, but is fairly easy to solve numerically using the forward-Euler method and a spreadsheet program. The general solution method is to calculate the pressure, then the rate of mass transfer, then total mass remaining for several discrete time steps.

First calculate the initial mass of air in the bottle using the ideal gas equation of state:

$$ m=\frac{PV}{RT} $$

$R$ is again the specific gas constant here. You know the initial pressure; plug it in to the choked flow equation to get the rate of mass loss. Now multiply this rate by a small time step, 0.1 second might be a good initial guess. This gives you the approximate mass of air that escaped over the first 0.1 second after the stopper was removed. Now subtract this mass off the initial mass that you calculated already. Using the new mass, recalculate the pressure, recalculate the rate of mass transfer, calculate the new mass, and repeat until the pressure drops below 30 psi. Below 30 psi, you're not at choked flow any more, so you should switch to the equation for subsonic flow. Repeat the above until the pressure drops to atmospheric, or the length of time covers the time interval of interest. This may sound complicated, but it's actually fairly easy to implement with a spreadsheet.