Choked Nozzle (Mach 1 Throat)

Choked mass flow, throat conditions, and critical pressure ratio for a converging nozzle running sonic at the throat.

Inputṁ = A*·P₀·√(γ/(R·T₀))·(2/(γ+1))^((γ+1)/(2(γ−1))), P*/P₀ = (2/(γ+1))^(γ/(γ−1)), T*/T₀ = 2/(γ+1)

Stagnation pressure P₀ (absolute) (kPa)or enter psia — converts to kPaStagnation temperature T₀ (K)or enter °F — converts to KSpecific heat ratio γMolar mass (g/mol)Throat area (mm²)or enter in² — converts to mm²Back pressure (absolute, choking check) (kPa) — optionalor enter psia — converts to kPa

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The engineering

When the back pressure on a converging nozzle drops below the critical ratio — about 52.8% of the stagnation pressure for air — the throat locks at Mach 1 and the mass flow hits a ceiling set entirely by upstream conditions and throat area. Pulling a harder vacuum downstream changes nothing upstream of the throat. This card gives the choked flow rate, the sonic-throat state, and the critical pressure ratio; if you enter a back pressure above critical, it tells you the nozzle isn't choked. Reach for it when sizing relief paths, cold-gas thruster orifices, purge restrictors, or sonic-nozzle flow meters.

Two classic traps: everything here is absolute pressure and stagnation (total) temperature, not gauge and not static — a 60 psig air line is P₀ ≈ 515 kPa absolute. And the flow scales linearly with P₀ but only with 1/√T₀, so a supply that warms 30 K costs you ~5% flow while a 5% pressure sag costs the full 5%. Sanity check for air at room temperature: ṁ ≈ 0.0404·A·P₀/√T₀ in SI units.

Where this math comes from

In 1839 Barré de Saint-Venant and Pierre Wantzel worked the compressible-flow equations for gas escaping an orifice and hit a result nobody expected: below a certain pressure ratio, dropping the receiver pressure further did not increase the discharge. The flow had a ceiling. The result sat as a curiosity until Gustaf de Laval, chasing higher steam-turbine speeds in 1888, added a diverging section past the throat and turned the choke point into a supersonic accelerator.

Aurel Stodola's painstaking nozzle traverses at ETH Zürich, published in 1903, mapped the pressure along the axis and confirmed the sonic throat experimentally — the plots still appear in gas dynamics texts, including Anderson's, where the relations on this card are standard fare. The same physics now runs in reverse as a metrology tool: because choked flow depends only on upstream conditions and throat area, ISO 9300 critical-flow venturi nozzles serve as primary transfer standards for gas flow measurement.

1839Saint-Venant & WantzelDerive compressible orifice flow and discover the mass-flow ceiling below the critical pressure ratio.
1888Gustaf de LavalAdds the diverging section — the convergent-divergent nozzle — to drive steam turbines supersonic.
1903Aurel StodolaPublishes nozzle pressure traverses experimentally confirming the sonic throat.
1926Robert GoddardFlies a de Laval nozzle on the first liquid-fueled rocket.
1990ISOISO 9300 standardizes critical-flow venturi nozzles as precision gas-flow references.

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