# What happens if the mass flow rate is increased continuously in a converging nozzle?

I recently studied about nozzles where I came to know that the maximum velocity one could attain using a converging nozzle is the sonic velocity at the exit of the converging nozzle. An further decrease in the exit area would only decrease the mass flow rate (as area is decreased) and the velocity at exit would remain sonic.

So, what happens if we keep on increasing the mass flow rate through a converging nozzle after attaining sonic velocity at exit?

My intuition says the mass gets accumulated inside the nozzle and the nozzle would explode eventually.

• Gas is compressible。 Sep 16, 2022 at 10:39
• How is the mass accumulating inside the nozzle, which explodes eventually not a necessary outcome, the only doubt being 'how much' in any parameter? Sep 16, 2022 at 20:34

This subject can indeed be quite confusing, as I got confused a bit myself as well reviewing the theory. So lets review the basics first:

## The Basics

The complete formula for the mass flow through a convergent-divergent nozzle, assuming ideal compressible flow, is as follows (see this NASA link for the derivation):

$$\dot{m}=\frac{A\ p_0}{\sqrt{R\ T_0}} \sqrt{\gamma}\ M\bigg(1 + \frac{\gamma-1}{2} M^2 \bigg)^{-\frac{\gamma+1}{2(\gamma-1)}}$$

With $$A$$ the cross-sectional area, $$p_0$$ the upstream total pressure, $$R$$ the mass specific gas constant, $$T_0$$ the upstream total temperature, $$\gamma$$ the ratio of specific heats and $$M$$ the Mach number.

It is important to realize that our previous formula doesn't specify at which point in the nozzle we evaluate the mass flow. We don't really care as we know the mass flow should be the same throughout the nozzle through the conservation of mass: mass cannot just disappear. Nonetheless for $$A$$ and $$M$$ it does matter and they should be given for the same point. Which we'll see is important later.

Now if you take the derivative of this equation with respect to $$M$$ and set the result to zero, one finds the maximum value for $$\dot{m}$$ is found when $$M$$ is 1 (again, see NASA link). So, one says: the maximum mass flow rate is found when the velocity is sonic and we call this maximum flow: choked flow.

If we fill in $$M=1$$, we get the following formula for choked flow:

$$\dot{m}_{max}=\frac{A\ p_0}{\sqrt{R\ T_0}} \sqrt{\gamma}\bigg(\frac{\gamma+1}{2}\bigg)^{-\frac{\gamma+1}{2(\gamma-1)}}$$

Now we know that the flow velocity in a choked nozzle is sonic at the throat*, so $$A$$ and $$\dot{m}$$ are both evaluated at the throat**. Also, for clarity, let's replace all $$\gamma$$ terms by the Vandenkerckhove-function, $$\Gamma$$, to end up at the final formula for the mass flow through a convergent-divergent nozzle:

$$\dot{m}_{max,\ throat}\ \ \ =\frac{A_{throat}\ \ p_0\ \Gamma}{\sqrt{R \ T_0}}$$

Having arrived at this final formula it is quite clear that decreasing the throat area indeed reduces the mass flow rate, as you already stated, while the flow velocity at the throat (the end of the converging part of the nozzle) would remain sonic.

However, nothing is preventing us from increasing the upstream pressure and thus increasing the mass flow rate. Inversely, "increasing the mass flow rate" (not sure how one would do this directly) would increase the upstream pressure, all else being equal. So mass wouldn't "accumulate", but your intuition isn't wrong: increasing the pressure would eventually explode your nozzle!

Additionally, our flow velocity at the throat would remain sonic, so the density of the flow must increase to increase the mass flow rate. This is possible, because we've assumed compressible flow at the start!

Hope this answer helps, just comment if it still isn't clear and welcome to the space stack exchange!

* Choked flow, and consequently sonic conditions at the throat, only happens if a certain pressure-ratio is achieved (see below), otherwise the flow through the whole nozzle is subsonic

** We know the mass flow is equal throughout the nozzle, so in general one can say $$\dot{m} = \dot{m}_{max,\ throat}$$, but I've kept it intentionally specific to prevent misunderstanding

$$\frac{p_e}{p_0}=\bigg(\frac{\gamma+1}{2}\bigg)^{-\frac{\gamma+1}{2(\gamma-1)}}$$

• So, the pressure ratio (Pe / P0) is the only factor which influences the sonic condition at the nozzle exit ryt? Sep 17, 2022 at 12:50
• If that's true, then this explanation clears everything. Sep 17, 2022 at 17:05
• Sorry for the late reply, but basically yeah. The required ratio itself is clearly dependent on the heat capacity ratio, $\gamma$, but apart from that only the pressure-ratio determines whether the flow at the throat is sonic or subsonic Sep 21, 2022 at 8:49