Mass air flow through a nozzle at different upstream pressures

[InquisitiveOne]

I'm interested in identifying mass airflow through a choked convergent or conical nozzle.

I've found some web info claiming that mass airflow through a nozzle becomes primarily a linear function of the inlet pressure, and doubling the inlet pressure doubles the flowrate.

Further research has led me to question the validity of the above statement, that velocity through the nozzle can not exceed the speed of sound in the pressurized medium, compressed air. That the increased density with increased upstream pressure will increase mass air flow, however given a sonic condition occurs at a lower velocity in the compressed air, mass flow doesn't increase in a linear fashion.

So, my question is: Which is it?

 

[boneh3ad]

You are incorrect. Once a converging-diverging nozzle becomes choked, the mass flow varies linearly with the upstream total pressure and inversely with the square root of the upstream total temperature.

 

[InquisitiveOne]

Thank you for the reply boneh3ad. Would I be off base to assume that mass flow through a convergent or conical nozzle would follow the same principal as a convergent-divergent nozzle? Also, how would I go about calculating mass flow at differing pressures and temperatures? For instance, a nozzle with a 1" minimum diameter choked at atmospheric 14.7 upstream pressure at 530°K and also at 132.3 absolute @ 993°K, and what if the compressed air were cooled to the original 530°K? All assuming a choked condition.

Any input/help is much appreciated.

 

[boneh3ad]

A converging nozzle, if choked, follows the same rule. It's easy to calculate, but first, are you familiar with calculus?

 

[InquisitiveOne]

I can fumble my way through equations, but can't claim any familiarity. I've tried using Bulk Modulus Elasticity and density to the power of 0.5 (Hooks Law), but the results don't seem linear with pressure.

The more layman you can convey it, the better. If I get stumped I'll take the time to figure it out before I take more of your time.

A ton of thanks for your time thus far.

 

[boneh3ad]

So basically, you can get the mass flux (rate of mass flow per unit area passing through some imaginary plane in the nozzle) at any give point in a nozzle pretty simply. It is
$$\dfrac{\dot{m}}{A} = \rho u$$
where $\dot{m}$ is the mass flow rate, $A$ is the cross-sectional area of the aforementioned plane, $\rho$ is the density and $u$. It is common to denote the the sonic flow conditions at the throat with a star, so from conservation of mass, you can also say
$$\rho u A = \rho^* u^* A^*.$$
So,
$$\dfrac{\dot{m}}{A} = \rho^* u^* \dfrac{A^*}{A}.$$
From there, we know that from the ideal gas law, $p^* = \rho^*RT^*$ and, since the flow is choked and therefore sonic at the throat, $u^* = a^* = \sqrt{\gamma R T^*}$. Here, $R$ is the specific gas constant, $\gamma$ is the ratio of specific heats (1.4 for air), and $T$ is temperature. These can be combined with the previous equation to get
$$\dfrac{\dot{m}}{A} = \sqrt{\dfrac{\gamma}{R}} \dfrac{p^*}{\sqrt{T^*}} \dfrac{A^*}{A}.$$
Using the various isentropic relations for compressible flow, this can be expanded to
$$\dfrac{\dot{m}}{A} = \dfrac{p_{01}}{\sqrt{T_{01}}}\sqrt{\dfrac{\gamma}{R}}\left( \dfrac{2}{\gamma + 1} \right)^{\frac{\gamma+1}{2(\gamma-1)}}\dfrac{A^*}{A}.$$
You'll note there that the term $A^*/A$ is always less than or equal to 1, and it is equal to 1 at the throat, so for constant parameters, the maximum mass flux is at the throat. Anyway, that obviously gets a little simpler,
$$\boxed{\dfrac{\dot{m}}{A^*} = \dfrac{p_{01}}{\sqrt{T_{01}}}\sqrt{\dfrac{\gamma}{R}}\left( \dfrac{2}{\gamma + 1} \right)^{\frac{\gamma+1}{2(\gamma-1)}}.}$$

So, that equation gives the mass flux at the throat, $\dot{m}/A^*$, as a function of the total pressure, $p_{01}$, and total temperature, $T_{01}$, which, for an isentropic flow are the same as your reservoir temperature and pressure. The rest of the terms are known constants. In nearly all situations, you can treat that portion of the flow between the reservoir and the throat as isentropic. If you want the actual mass flow rate, just move the area term over to the right side of the equation.

This only works if the flow is choked, though. If not, then the throat is not sonic and you can't use this, and whether or not your flow is choked depends on the upstream and downstream pressures and the geometry.