How to determine mdot for choked airflow through a valve, given T,P

[CoryDinkle]

TL;DR Summary

Compressible, choked airflow through an orifice

Hello, I am working on a project that involves air at high pressure and temperature flowing through an orifice (valve) from one container to another. For each container, the volume, temperatures and pressures are known. The valve diameter is also known, and the fluid is air (which can be considered an Ideal Gas). I am solving this with Matlab, using a forward differencing approach with a mass balance and an energy balance. I am interpolating specific heat with a table.

I have found some equations to determine flow that accounts for both unrestricted and choked flow, but they require the Mach number, which itself requires the flow velocity. I am having difficulty determining the flow velocity, as it is essentially the only component of the mdot that is unknown (A is known, rho can be found from the T and P, leaving v as the unknown).

Any help would be greatly appreciated!

 

[boneh3ad]

If the flow is choked, you know the Mach number at the throat by definition. Mass flow at that point only depends on upstream $p_0$ and $T_0$ plus the throat area and gas properties. Have you done a search for choked mass flow?

 

[CoryDinkle]

Hello boneh3ad, thanks for the reply!

Yes, I've done a search for choked mass flow (as I mentioned in the second paragraph that I have the equations for unrestricted and choked flow), and also yes, if I only needed the mass flow rate at choked flow this would be an easy problem, as I could just set the Mach number to unity. The difficulty with this problem is that I am trying to solve for a transient process; the pressure in the upstream container will be decreasing as the pressure in the downstream container increases (clearly the temperatures will change as well).

Being as it is a transient process, and not steady-state, I need to determine the Mach number for all timesteps (as the generalized equation for both types of flow requires the Mach number). Initially, the flow will be choked due to the large pressure delta, but that will eventually minimize such that flow will be unrestricted, and I will need to determine the Mach number during this process.

Hopefully this additional information clarifies the difficulty I am describing; I need the Mach number at all timesteps to determine the transient flow. To get the Mach number, I need the fluid velocity at all timesteps. Being as I am trying to solve for the mass flow rate with T,P, and A already known, the fluid velocity is the only variable I do not have. Yes, I can determine the velocity at exactly the moment when the Mach number equals 1, but I will need more than that.

I am currently looking into using the Local Mass Flow Function from White's Fluid Mechanics, as it does not require the Mach number, I just haven't had the time to attempt it in my code yet, as this project is not a high priority. If you are familiar with this function, would you happen to think that this is a good route? I can divide the Local Mass Flow function by (sqrt(R*T_0))/(A*P_0), and that should give me mdot.

 

[CoryDinkle]

I suppose a question I have is: for the above Local Mass Flow function, would p be the downstream pressure, and p_0 and T_0 be the upstream properties?