# Air Tank Discharging Through A Nozzle - Unsteady Flow

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.

## Answers and Replies

boneh3ad
Science Advisor
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.

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.

Thanks for the reply.

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.

boneh3ad
Science Advisor
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.

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.

boneh3ad
Science Advisor
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.

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?

boneh3ad
Science Advisor
Gold Member
I suppose if your container was flexible then yes, the volume would change with time. That becomes more complicated and you will need to model the deformation as a function of pressure. If the tank is rigid, though, then the volume is constant. The other variable there is temperature, so you will have to decide how you want to model that as well. You could assume it is isothermal, but I would be willing to bet that this isn't the case since your tank is small and so the pressure will probably change relatively quickly inside. It may be more appropriate to model it as adiabatic, at which point you could use the isentropic relations between temperature and pressure (or density, if you are working in the mass variable instead of pressure).

Ohhhh I was being stupid about the volume, yes the tank is rigid. In this case I have assumed it is adiabatic. Thanks for your help I will give this a go.