# How to determine the choke point in a compressed gas system?

I have a pressure regulator bringing an inlet pressure of 150 bar down to 10 bar which is then lead into a plenum followed by a nozzle of area 340 mm^2, exposed to atmospheric conditions.

Weirdly enough at both points the necessary condition of pressure ratio for choking is satisfied! The critical exit pressure below which choking occurs (p*), is 5.8 bar at the nozzle exit and 79.7 bar at the regulator exit. How do I explain the fact that the flow chokes only at the nozzle?

• There is an independent maximum flow through both of the locations where choking can occur. The actual flow through the system will be determined by the smaller maximum. The choked location that could have a greater flow will "adjust" to whatever that lower flow is. Remember that the pressure regulator presents a variable sized orifice until it's fully open which provides one means for the system to adjust. Since you're using a regulator to ensure the pressure drop is critical, I suspect that the area of the regulator adjusts so that the flow through it and out of the system match. – Byron Wall Jul 6 '16 at 17:44
• For fixed orifice sizes, you will see the adjustment occur with pressure. Since these are critical pressure drops, this means you will see more or less pressure lost to turbulence. In this case, it's also possible that at the lower flow rate, the pressure drop through one of the restrictions is low enough to switch from critical to non-critical flow. You have to consider how the flow and pressure profile in the system develop once the maximum flow through the system has been determined. – Byron Wall Jul 6 '16 at 17:50
• Thanks Byron, that is a brilliant way of thinking about it. The regulator orifice could indeed have a more gradual profile of pressure drop! Please post this as an answer so that I can accept it and others can refer to it in future. – DBTKNL Jul 7 '16 at 8:03
• It's not that there is a difference in profile at the regulator (there is still a sudden pressure drop with energy wasted as turbulence), but instead the orifice area of the regulator adjusts so that at its choked (i.e. sonic) velocity the flow through the regulator orifice matches the flow out of the system to atmosphere. I'll post a proper answer later instead of comments with some equations and a diagram. – Byron Wall Jul 7 '16 at 14:14

## 1 Answer

How do I explain the fact that the flow chokes only at the nozzle?

To explain why the flow only chokes at a single point, it is important to remember that for a system at steady state, there can only be a single flow. All of the fluid entering the system must leave. Since there can only be a single flow rate through the system, you must then consider where that maximum flow can occur.

The following equations describe the flow through a frictionless nozzle where the expansion occurs adiabatically and isenthropically. They are from Perry's page 6-23. The actual flow through an orifice is usually handled by a flow coefficient since the flow through an orifice will be less than a frictionless nozzle.

\begin{align} \frac{p^*}{p_0} &= \left(\frac{2}{k+1}\right)^{k/(k-1)} \\ \frac{T^*}{T_0} &= \frac{2}{k+1} \\ \frac{\rho^*}{\rho_0} &= \left(\frac{2}{k+1}\right)^{1/(k-1)} \\ G^* &= p_0 \sqrt{\left(\frac{2}{k+1}\right)^{(k+1)/(k-1)}\left( \frac{kM_w}{RT_0} \right)} \\ w^* &= G^*A \\ V = V^* = c^* &= \sqrt{\frac{kRT^*}{M_w}} \end{align}

For the formulas above, the $^*$ represents the condition at choked flow and the $_0$ condition is inlet. The other variables are defined as:

• $p$ = pressure
• $T$ = temperature
• $\rho$ = density
• $G$ = mass velocity (mass flow per unit area)
• $w$ = mass flow
• $A$ = nozzle exit area
• $V, c$ = exit velocity
• $R$ = gas constant
• $M_w$ = molecular weight

Choked flow occurs when the downstream pressure is less than the critical pressure or the pressure ratio is less than the critical ratio. This is shown in equation 1 and repeats your initial question. Once you know the flow will be choked, you can then use the remaining equations. Looking at the equation for the mass velocity, $G^*$, you can see that choked flow is a function of gas composition $(k,M_w)$ and inlet conditions $(T_0,p_0)$ and that changing downstream conditions has no effect on the mass velocity. To get to the mass flow rate $w^*$ you must also consider the orifice area $A$. With those variables known, you can determine which orifice will create the limiting flow rate. This can become an iterative process as changing upstream conditions may then limit downstream components.

I have a pressure regulator bringing an inlet pressure of 150 bar down to 10 bar

Since you have a pressure regulator, this tells you something about the dynamics of the system. The pressure regulator presents a variable sized orifice to the process until it is fully opened. Once it is fully open, it behaves like a fixed orifice size. Since the pressure regulator is able to adjust to maintain a downstream condition, it will not be the limiting component until it is wide open.

When the regulator is partially open, the system has established a steady state condition wherein:

• the flow out of the system (to atmosphere) is the maximum flow through the system
• the position of the regulator (e.g. what % open) is such that its exposed flow area provides exactly the same flow as the outlet to atmosphere; this flow is a function of the temperature, upstream pressure, and composition of the gas at that choke point
• Choked flow occurs when the downstream pressure is less than the critical pressure or the pressure drop ratio is greater than the critical ratio Don't you mean when the pressure drop ration is smaller? – idkfa Jul 13 '16 at 8:00
• @idkfa, thanks for noticing that. I was being sloppy with terms since I used pressure drop ratio and then critical ratio. I will clean it up. – Byron Wall Jul 13 '16 at 14:08