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Ideal choked mass flow through a throat

Mass flow through a sonic throat of physical area At fed from upstream total pressure pt and total temperature Tt, for ideal steady adiabatic flow with enough downstream pressure drop to choke.

Ref: AER-43
Rev: v1
Field: Aerospace
Result: ṁ^* [kg/s]

{\dot{m}}^{\cdot} = A^{\cdot} p_{t} \sqrt{\frac{γ}{R T_{t}}} [\frac{2}{γ + 1}]^{(γ + 1) / (2 (γ - 1))}

At the throat of a choked nozzle the flow is sonic, and the mass flow depends only on the upstream total conditions and the throat area: ṁ* = A*pt√(γ/(RTt))[2/(γ+1)]^((γ+1)/(2(γ−1))). It is proportional to total pressure and inversely proportional to the square root of total temperature, and independent of the back pressure once choked. Real nozzles pass 1 % to 5 % less because of boundary layers.

Mass flow through a sonic throat of physical area At fed from upstream total pressure pt and total temperature Tt, for ideal steady adiabatic flow with enough downstream pressure drop to choke. Discharge losses and boundary layers are excluded, and they reduce the real flow by a few per cent.

Calculator

Solve for

ṁ^*Choked mass flowComputed

kg/s

A_tThroat area

m²

1e-41

p_tUpstream total pressure

50,0002,000,000

γHeat-capacity ratio

1.11.67

RGas constant

J/(kg·K)

200500

T_tUpstream total temperature

2503,000

Mass flux at the throat, ṁ/A330 kg/(m²·s)
Sonic speed at the throat448.2 m/s
Stagnation density, pt/(RTt)1.161 kg/m³

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Choked mass flow (kg/s) current value 3.3 kg/s

A_tThroat areaKey driver

0.033330

p_tUpstream total pressure

0.82533

T_tUpstream total temperature

1.485.11

RGas constant

2.53.95

γHeat-capacity ratio

3.033.5

The throat passes 3.3002 kg/s, a mass flux of 330.02 kg/(m²·s), as long as the back pressure is below the critical value of 105.66 kPa.
A 10 % rise in total pressure raises the flow by 10 %, while a 10 % rise in total temperature lowers it by 4.65 %.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (kg/s)	Local step	Change in result
A_t Throat area	1e-4 to 1 m²	0.033	330	330	+10%	+10 %
p_t Upstream total pressure	50,000 to 2,000,000 Pa	0.825	33	32.18	+10%	+10 %
T_t Upstream total temperature	250 to 3,000 K	1.476	5.113	3.637	+10%	−4.65 %
R Gas constant	200 to 500 J/(kg·K)	2.5	3.953	1.453	+10%	−4.65 %
γ Heat-capacity ratio	1.1 to 1.67	3.028	3.502	0.4738	+10%	+3.3 %

Sweep of Upstream total pressure

Upstream total pressure (Pa)	Choked mass flow (kg/s)
50,000	0.825
293,800	4.847
537,500	8.869
781,300	12.89
1,025,000	16.91
1,269,000	20.94
1,513,000	24.96
1,756,000	28.98
2,000,000	33

Sweep one variable

Vary

Choked mass flow (kg/s)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

ṁ^*

Choked mass flow

Result · kg/s

What the equation gives you. Everything else feeds it.

A_t

Throat area

m² · typical 1e-4 to 1

The physical sonic throat area, in m². Scales the mass flow in direct proportion. Raising it by 10 % (to 0.011) moves the result by +10 %, from 3.3 to 3.63 kg/s.

p_t

Upstream total pressure

Pa · typical 50,000 to 2,000,000

The upstream total pressure, in Pa. Scales the mass flow in direct proportion. Raising it by 10 % (to 220,000) moves the result by +10 %, from 3.3 to 3.63 kg/s.

Heat-capacity ratio

· typical 1.1 to 1.67

The heat-capacity ratio cp/cv of the gas, about 1.4 for air. A lower γ (hotter, more complex gas) changes the compressible-flow exponents and the sizes of the shock and expansion effects. Raising it by 10 % (to 1.54) moves the result by +3.3 %, from 3.3 to 3.409 kg/s.

Gas constant

J/(kg·K) · typical 200 to 500

The specific gas constant, in J/(kg·K). Lowers the mass flow with the inverse square root. Raising it by 10 % (to 315.7) moves the result by −4.65 %, from 3.3 to 3.147 kg/s.

T_t

Upstream total temperature

K · typical 250 to 3,000

The upstream total temperature, in K. Lowers the mass flow with the inverse square root. Raising it by 10 % (to 660) moves the result by −4.65 %, from 3.3 to 3.147 kg/s.

Worked example with the values above

Flow function: √(γ/(RTt)) = √(1.4 / (287 × 600)) = 0.00285133.
Area-pressure factor: [2/(γ + 1)]^((γ+1)/(2(γ−1))) = 0.578704; At × pt = 2,000.
ṁ* = 3.30015 kg/s.

Assumptions and limits

Steady, adiabatic flow with constant γ and cp, and a nozzle that is choked where the entry says so.
Uniform properties across the throat and exit plane, unless a survey is used.

Common mistakes

Using static instead of total pressure upstream.
Using the choked formula when the pressure ratio is below the critical value.
Forgetting that a convergent-divergent nozzle has different regimes from a convergent one.

Statistics and uncertainty

Ideal nozzle formulas overestimate real performance: discharge coefficients of 0.95 to 0.99 lower mass flow by 1 % to 5 %, and nozzle efficiency of 0.9 to 0.98 lowers exit speed by 1 % to 5 %. Total conditions are the key measurement, and a 1 % error in total pressure gives 1 % in mass flow.

Where it is used

Sizing throats and predicting thrust of jet and rocket nozzles.
Estimating mass flow through orifices, valves and venturis.

Maths you need

Critical pressure ratio: (2/(γ+1))^(γ/(γ−1)), about 0.528 for air
Momentum flux: ρu²A, the momentum carried by a stream
Nozzle efficiency: actual divided by ideal kinetic energy

Related formulas

Practise this formula