# Air Tank Discharging Through A Nozzle - Unsteady Flow

Andr5w5
Hey,

So I have a system where there is an air tank (0.2Litres @ 3000psi, 293K) and I have a convergent-divergent nozzle attached directly to it. What I want to know is an expression for how the mass flow rate varies with time as the mass in the tank decreases causing the pressure to decrease.
How do I go about setting up the relationship for the mass flow rate? And would I need a similar relationship for the volume in the air tank? I think I would find the pressure simply from P=MRT/V once I have expression for M and V. My guess is the mass flow rate vs time graph would look like an inverse logarithm graph.

I have already related the mass flow rate to the pressure in the air tank and so this would allow me to calculate how the thrust from the nozzle varies over time.

Thanks for any help.

Gold Member
Do you have any background in gas dynamics? Are you familiar with the concept of choked flow and the typical operation of converging-diverging nozzles? What about differential equations? I am just trying to determine your level of prior knowledge here. I wouldn't want to start throwing things out that are well above your currently background.

Andr5w5
Do you have any background in gas dynamics? Are you familiar with the concept of choked flow and the typical operation of converging-diverging nozzles? What about differential equations? I am just trying to determine your level of prior knowledge here. I wouldn't want to start throwing things out that are well above your currently background.

Im a 5th Year Mechanical Engineering Masters student and have knowledge of C-D nozzles and differential equations. I've set up a similar problem to work out the thrust from a convergent-divergent nozzle for steady flow but confused by how to establish the un-steady flow problem.

Gold Member
Well then you know the equation for choked mass flow through the throat, which depends on the total pressure upstream, and you know how that pressure relates to the mass of gas in the bottle given a few assumptions about temperature (isothermal or adiabatic, whatever you choose to justify). That ought to let you set up a pretty simple differential equation of the form
$$\dfrac{dm}{dt} = -Cm,$$
which should be easily solvable.

Andr5w5
I'm not sure I understand the differential equation you've set up there, is C some constant and 'm' mass or mass flow rate?
Previously the only differential equation I could think of setting up was -dM/dt = m where 'M' is mass flow and 'm' mass - however I don't see how this helps as it only establishes t = m/M when solved, I think? I would have imagined there needs to be a logarithmic term in the solution to represent the decay of the mass flow rate over time.

Gold Member
Call ##m## the amount of mass in the tank. You can calculate that initially, obviously, and you know the mass flow rate through a nozzle when the flow is choked, so you know that is the rate at which mass is leaving the tank. That equation explicitly depends on ##p_t## in the tank, so you simply need to substitute a relationship between ##m## and ##p_t## into the equation to make sure you are working with the one independent variable. You could set that up as an equation solving for mass in the bottle or the pressure in the bottle, but the two are essentially equivalent. When I used ##C## that is a constant. The equation for choked mass flow rate is obviously a lot more complicated than ##Cm## or ##Cp_t##, but ultimately, all of those parameters are constants so you can wrap it up into the term I called ##C##.

Regarding the logarithmic term, I'd suggest solving the simple equation I posted above and see what you get.

Andr5w5
So I have the equation M=C*Pt that I have derived from the choked flow condition. I assume I want to substitute Pt=MRTt/V into this equation to remove the pressure function. However isn't the volume, V, a function of time as well leading back to the same problem with more than one independent variable?