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

Question

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?

Answer 1

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