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At the critical back pressure, an ideal converging nozzle reaches Mach 1 at its throat; lowering back pressure further does not increase the modeled mass flow.
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Formula
p in Pa; T in K; R in J/(kg·K); area in m2; mass flow in kg/s; γ dimensionless
| Symbol | Meaning | Unit |
|---|---|---|
| p* | Critical static pressure at the throat | Pa |
| Upstream stagnation pressure | Pa | |
| Stagnation temperature | K | |
| Throat area | m2 | |
| Specific-heat ratio cp/cv | dimensionless | |
| Specific gas constant | J/(kg·K) | |
| Choked mass flow | kg/s |
\frac{p^*}{p_0}=\left(\frac{2}{\gamma+1}\right)^{\!\gamma/(\gamma-1)},\quad \dot m=A_t\frac{p_0}{\sqrt{T_0}}\sqrt{\frac{\gamma}{R}}\left(\frac{2}{\gamma+1}\right)^{\!\frac{\gamma+1}{2(\gamma-1)}}Applies to steady, one-dimensional, isentropic flow of a calorically perfect gas with constant γ through a converging nozzle; M=1 at the critical section. Losses, boundary layers, real-gas effects and nonuniform profiles change the discharge. Below critical back pressure, the ideal model predicts constant mass flow; a divergent nozzle is required for supersonic expansion.
Applies to steady, one-dimensional, isentropic flow of a calorically perfect gas with constant γ through a converging nozzle; M=1 at the critical section. Losses, boundary layers, real-gas effects and nonuniform profiles change the discharge. Below critical back pressure, the ideal model predicts constant mass flow; a divergent nozzle is required for supersonic expansion.
At the critical back pressure, an ideal converging nozzle reaches Mach 1 at its throat; lowering back pressure further does not increase the modeled mass flow.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Critical nozzle pressure ratio and choked mass flow” and check its unit and magnitude.
For air with γ=1.4, p*/p0=(2/2.4)^3.5≈0.528.
At p0=500 kPa the critical throat pressure is about 264 kPa.
With T0=300 K, area 1.0×10−4 m2 and R=287 J/(kg·K), the mass-flow relation gives about 0.1166 kg/s.
Dimensional check yields kg/s.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: p in Pa; T in K; R in J/(kg·K); area in m2; mass flow in kg/s; γ dimensionless.
Formula cluster
At the critical back pressure, an ideal converging nozzle reaches Mach 1 at its throat; lowering back pressure further does not increase the modeled mass flow.
For air with γ=1.4, p*/p0=(2/2.4)^3.5≈0.528. At p0=500 kPa the critical throat pressure is about 264 kPa. With T0=300 K, area 1.0×10−4 m2 and R=287 J/(kg·K), the mass-flow relation gives about 0.1166 kg/s. Dimensional check yields kg/s.
Check critical flow and maximum throughput in nozzles, valves and gas releases.
Here is how to work through a typical Critical nozzle pressure ratio and choked mass flow (p*/p₀; ṁ bei M=1 am Düsenhals) task step by step:
Task
Work through the numerical example for “Critical nozzle pressure ratio and choked mass flow” and check its unit and magnitude.
Solution path
For air with γ=1.4, p*/p0=(2/2.4)^3.5≈0.528.
At p0=500 kPa the critical throat pressure is about 264 kPa.
With T0=300 K, area 1.0×10−4 m2 and R=287 J/(kg·K), the mass-flow relation gives about 0.1166 kg/s.
Dimensional check yields kg/s.
First compute p*/p0 from γ. If back pressure exceeds p*, an ideal converging nozzle remains subsonic; at p* the throat becomes sonic. Lower back pressure does not increase the idealized mass flow further. The ratio compares critical static pressure with stagnation pressure and is not a generic pressure-drop relation for arbitrary valves. Assume a calorically perfect gas, constant γ and one-dimensional isentropic throat flow. The critical state begins at M=1. Application: Check critical flow and maximum throughput in nozzles, valves and gas releases. The numerical example For air with γ=1.4, p*/p0=(2/2.4)^3.5≈0.528. At p0=500 kPa the critical throat pressure is about 264 kPa. With T0=300 K, area 1.0×10−4 m2 and R=287 J/(kg·K), the mass-flow relation gives about 0.1166 kg/s. Dimensional check yields kg/s anchors the idea in a concrete measurement case.
At choking, local density is set by p* and T*, and throat velocity equals sound speed. Mass flow is therefore ρ*a*At. Increasing throat area raises ideal flow proportionally when gas and stagnation state stay fixed. Real discharge coefficients reduce the value because of losses and should be included for component design. Assume a calorically perfect gas, constant γ and one-dimensional isentropic throat flow. The critical state begins at M=1. The corresponding units are p in Pa; T in K; R in J/(kg·K); area in m2; mass flow in kg/s; γ dimensionless. For air with γ=1.4, p*/p0=(2/2.4)^3.5≈0.528. At p0=500 kPa the critical throat pressure is about 264 kPa. With T0=300 K, area 1.0×10−4 m2 and R=287 J/(kg·K), the mass-flow relation gives about 0.1166 kg/s. Dimensional check yields kg/s. This checks that the reference quantities are compatible.
R here is the specific gas constant for the gas in J/(kg·K), not the universal molar constant in J/(mol·K). The square-root factor then gives the mass units needed for mass flow. Using the molar constant requires a consistent molar-flow formulation with molar mass. Confusing the two creates a major unit and numerical error. Assume a calorically perfect gas, constant γ and one-dimensional isentropic throat flow. The critical state begins at M=1. Scope: Applies to steady, one-dimensional, isentropic flow of a calorically perfect gas with constant γ through a converging nozzle; M=1 at the critical section. Losses, boundary layers, real-gas effects and nonuniform profiles change the discharge. Below critical back pressure, the ideal model predicts constant mass flow; a divergent nozzle is required for supersonic expansion. A practical application is Check critical flow and maximum throughput in nozzles, valves and gas releases.
An ideal converging nozzle accelerates only up to Mach 1. Controlled further expansion to supersonic speed requires a divergent section after the throat. Pressure matching then depends on area ratio and back pressure; shocks can occur away from the design condition. The critical mass-flow equation describes throat throughput, not the entire supersonic field. Assume a calorically perfect gas, constant γ and one-dimensional isentropic throat flow. The critical state begins at M=1. Check the calculation with this case: For air with γ=1.4, p*/p0=(2/2.4)^3.5≈0.528. At p0=500 kPa the critical throat pressure is about 264 kPa. With T0=300 K, area 1.0×10−4 m2 and R=287 J/(kg·K), the mass-flow relation gives about 0.1166 kg/s. Dimensional check yields kg/s. Keep the model limits and units in view.
For γ=1.4, the power (2/2.4)^(1.4/0.4)≈0.528, so p*≈264 kPa from 500 kPa. Substitute R=287 J/(kg·K), T0=300 K and area 10−4 m2; the result is about 0.117 kg/s. Check that pressure is in Pa, area in m2 and temperature in kelvin. Assume a calorically perfect gas, constant γ and one-dimensional isentropic throat flow. The critical state begins at M=1. The relation describes At the critical back pressure, an ideal converging nozzle reaches Mach 1 at its throat; lowering back pressure further does not increase the modeled mass flow. This helps distinguish its target quantity from neighbouring quantities.
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