Choked Flow

What choking is, physically

Drive a gas from a high-pressure reservoir through a converging nozzle into a region of lower pressure. As long as the flow is subsonic, lowering the downstream (back) pressure pulls more mass through. But there is a wall you cannot push past. At a definite back-pressure the velocity at the throat — the smallest cross-section — reaches the local speed of sound, Mach 1. Beyond that, dropping the back-pressure even to a hard vacuum does nothing to the mass flow rate. The nozzle is choked.

The cause is the finite speed of sound. Pressure disturbances propagate through a gas as sound waves, and the speed of sound relative to the moving gas is fixed. When the throat flow reaches Mach 1, a pressure wave trying to travel upstream against the flow makes exactly zero progress — it is pinned at the throat. The gas upstream of the sonic point therefore never "hears" that the downstream pressure has dropped, and it cannot respond. The throat is acoustically isolated from everything downstream of it.

The area-Mach relation

Why the throat and not somewhere else? The governing relation for steady, isentropic, quasi-one-dimensional compressible flow connects area change to velocity change through the Mach number:

dA/A = (M² − 1) · dV/V

  A = local cross-sectional area
  V = local velocity
  M = local Mach number = V / a
  a = local speed of sound = √(γRT)

Read off the sign of the bracket:

• Subsonic (M < 1): (M²−1) is negative, so to accelerate the gas (dV > 0) the area must shrink (dA < 0). A converging duct speeds up subsonic flow — the familiar incompressible intuition.
• Supersonic (M > 1): (M²−1) is positive, so to accelerate the gas the area must grow. A diverging duct speeds up supersonic flow — completely counter-intuitive.
• Sonic (M = 1): the only way to satisfy the relation with finite dV is dA = 0, i.e. an area extremum. The flow can reach exactly Mach 1 only where the area is at a minimum — the throat.

This is the whole story of the de Laval (converging-diverging) nozzle: converge to accelerate subsonic flow up to Mach 1 at the throat, then diverge to keep accelerating it supersonically. A purely converging nozzle has no diverging section, so its maximum exit Mach number is 1, achieved at the throat once choked.

The critical pressure ratio

From the isentropic relations, the static-to-stagnation pressure where M = 1 is the critical pressure ratio:

p*/p₀ = (2 / (γ + 1))^(γ / (γ − 1))
T*/T₀ = 2 / (γ + 1)

For air, γ = 1.4:
  p*/p₀ = (2/2.4)^(1.4/0.4) = (0.8333)^3.5 = 0.528
  T*/T₀ = 2/2.4                              = 0.833
  ρ*/ρ₀ = (2/2.4)^(1/0.4)   = (0.8333)^2.5  = 0.634

So the moment the back-pressure falls below 52.8 % of the upstream stagnation pressure, an air nozzle chokes. If your reservoir sits at 1.0 bar absolute, the nozzle is already choked exhausting to anything below 0.528 bar — including ordinary atmosphere if the reservoir is above about 1.9 bar. This is why compressed-air lines, regulators and pneumatic tools are nearly always operating choked at their orifices.

Choked mass flow rate

Once choked, the mass flow rate through the throat is fixed by upstream conditions alone:

ṁ = C_d · A* · p₀ · √(γ / (R T₀)) · (2/(γ+1))^((γ+1) / (2(γ−1)))

  C_d = discharge coefficient (0.95–0.99 for a smooth nozzle)
  A*  = throat area (m²)
  p₀  = upstream stagnation pressure (Pa)
  T₀  = upstream stagnation temperature (K)
  R   = specific gas constant (287 J/kg·K for air)
  γ   = ratio of specific heats (1.4 for air)

Three consequences are worth burning into memory:

• ṁ is independent of back-pressure. The downstream pressure does not appear in the equation. This is the defining property of choked flow.
• ṁ scales linearly with upstream pressure and throat area. Double p₀ and you double the flow; double A* and you double the flow.
• ṁ falls as upstream temperature rises — proportional to 1/√T₀. A hotter gas has higher sonic speed but lower density, and density wins.

Worked example: a sonic air nozzle

A reservoir holds air at p₀ = 5.0 bar absolute and T₀ = 300 K, discharging through a converging nozzle with a 5 mm throat diameter to the atmosphere (1.0 bar). Is it choked, and what is the flow?

Back-pressure ratio: 1.0 / 5.0 = 0.20  <  0.528  →  CHOKED

Throat area:  A* = π(0.0025)² = 1.963 × 10⁻⁵ m²
Constant:     (2/2.4)^((2.4)/(0.8)) = (0.8333)^3 = 0.5787
√(γ/RT₀):     √(1.4 / (287 × 300)) = √(1.626×10⁻⁵) = 4.032×10⁻³

ṁ = 0.98 × 1.963×10⁻⁵ × 5.0×10⁵ × 4.032×10⁻³ × 0.5787
   = 0.98 × 1.963×10⁻⁵ × 5.0×10⁵ × 2.334×10⁻³
   ≈ 0.0225 kg/s   (about 22.5 g/s, ~1100 standard litres/min)

Now drop the downstream pressure to a vacuum: the mass flow does not change. Raise the reservoir to 10 bar and it exactly doubles to ~45 g/s. That linearity in p₀, and the deafness to back-pressure, is precisely why sonic nozzles make excellent flow standards and metering devices.

Subsonic, choked, and over-expanded compared

	Subsonic (unchoked)	Just choked	Choked + diverging (under-expanded)
Back-pressure ratio p_b/p₀ (air)	> 0.528	= 0.528	< 0.528
Throat Mach number	< 1	= 1	= 1 (locked)
Effect of lowering p_b	ṁ increases	ṁ at its max	ṁ unchanged
Where extra expansion goes	inside nozzle	—	outside, as expansion fans / shock cells
Exit jet	pressure-matched	pressure-matched	under-expanded supersonic plume
Typical example	HVAC duct, low-Δp valve	onset in a relief valve	rocket exhaust, air-tool orifice

Converging vs converging-diverging nozzles

	Converging only	Converging-diverging (de Laval)
Max exit Mach number	1.0 (sonic)	> 1 (supersonic, set by area ratio)
Where M = 1	at the exit (= throat)	at the throat, between converge and diverge
Behaviour below critical p_b	under-expands outside the nozzle	expands supersonically inside, then matches
Area ratio A_exit/A*	1 (no diverging section)	e.g. 77 on the Space Shuttle Main Engine
Use	flow meters, valves, blow-down jets	rocket engines, supersonic wind tunnels

Where engineers meet choked flow

• Rocket nozzles. A liquid rocket runs its throat choked from ignition to shutdown; the choked throat sets the chamber pressure-to-flow relationship, and thrust follows from the supersonic expansion in the diverging bell. The throat is the single most thermally loaded point in the engine.
• Critical-flow (sonic) venturis. Used as primary flow standards in calibration labs precisely because, once choked, the flow depends only on upstream stagnation conditions and the throat area — quantities you can measure to a fraction of a percent.
• Safety relief valves and burst discs. Sizing standards (API 520) assume choked flow because it caps the maximum discharge rate; if the valve weren't choked, a falling vessel pressure could keep pulling ever more flow.
• Turbocharger and gas-turbine nozzles. Turbine nozzle guide vanes commonly choke at design power, fixing the engine's corrected mass flow — a designed-in feature, not a fault.
• Gas pipeline orifices and regulators. Pressure-reducing stations often choke across the regulator orifice; the loud hiss and the audible "snap" to a steady tone is the sonic condition setting in.
• Leaks. A pinhole in a 10-bar line vents at a fixed, choked rate set by the hole area — which is why leak rates are predictable enough to design alarms around.

Failure modes and trade-offs

• Throat erosion and thermal damage. The sonic throat sees the highest heat flux and shear in the whole device. Rocket throats are regeneratively cooled or use ablative/graphite inserts; as the throat erodes, A* grows, ṁ rises, and chamber pressure drifts — a self-feeding problem.
• Over-expansion and flow separation. A C-D nozzle run with too low a chamber pressure (e.g. a sea-level start of a vacuum-optimized engine) over-expands; a shock forms inside the diverging section and the flow separates from the wall, causing side loads that can crack the bell.
• Discharge-coefficient uncertainty. Real nozzles aren't ideal; sharp-edged orifices have C_d near 0.6–0.85 while rounded sonic nozzles reach 0.99. Using the wrong C_d is the most common error in choked-flow sizing.
• Condensation and real-gas effects. Rapid expansion of steam or moist air can drop the static temperature below the dew or saturation point, forming a fog that releases latent heat and shifts the effective γ — the textbook ideal-gas numbers no longer hold.
• Assuming the flow is choked when it isn't (or vice versa). Below the critical pressure ratio the back-pressure is irrelevant; above it, it is everything. Mis-classifying the regime gives wildly wrong flow predictions — the single most important check is the pressure-ratio test against 0.528.