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Sonic speed at a choked throat
Speed of sound, and also flow speed, at an isentropic sonic section fed from a reservoir at stagnation temperature Tt, for gas constant R and heat-capacity ratio γ.
- Ref
- AER-35
- Rev
- v1
- Field
- Aerospace
- Result
- a^* [m/s]
At the throat of a choked nozzle the flow is sonic, M = 1, and the flow speed equals the local sound speed a*. In terms of the reservoir stagnation temperature Tt, the static temperature at the throat is T* = 2Tt/(γ+1), so a* = √(γRT*) = √(2γRTt/(γ+1)). For air this is about 91 % of the sound speed in the reservoir.
Speed of sound, and also flow speed, at an isentropic sonic section fed from a reservoir at stagnation temperature Tt, for gas constant R and heat-capacity ratio γ. It is about 91 % of the sound speed in the reservoir for air. Use absolute temperature and constant properties.
Calculator
- Sound speed in the reservoir, √(γRTt)491 m/s
- Ratio a*/at, √(2/(γ + 1))0.9129
- Static temperature at the throat, 2Tt/(γ + 1)500 K
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.
- The throat speed is 448.22 m/s, 91.29 % of the reservoir sound speed of 491 m/s.
- The static temperature at the throat is 500 K, so the gas has cooled by 100 K from its stagnation value in accelerating to sonic speed.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Tt Stagnation temperature | 250 to 3,000 K | 289.3 | 1,002 | 712.9 | +10% | +4.88 % |
| R Gas constant | 200 to 500 J/(kg·K) | 374.2 | 591.6 | 217.4 | +10% | +4.88 % |
| γ Heat-capacity ratio | 1.1 to 1.67 | 424.7 | 464.1 | 39.39 | +10% | +1.95 % |
Sweep of Stagnation temperature
| Stagnation temperature (K) | Sonic speed (m/s) |
|---|---|
| 250 | 289.3 |
| 593.8 | 445.9 |
| 937.5 | 560.3 |
| 1,281 | 655 |
| 1,625 | 737.6 |
| 1,969 | 811.9 |
| 2,313 | 879.9 |
| 2,656 | 943.1 |
| 3,000 | 1,002 |
Sweep one variable
Variables and how each one contributes
Sonic speed
What the equation gives you. Everything else feeds it.
Heat-capacity ratio
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 +1.95 %, from 448.2 to 457 m/s.
Gas constant
The specific gas constant of the gas, in J/(kg·K). Raises the speed with the square root. Raising it by 10 % (to 315.7) moves the result by +4.88 %, from 448.2 to 470.1 m/s.
Stagnation temperature
The reservoir stagnation temperature, in K. Raises the speed with the square root. Raising it by 10 % (to 660) moves the result by +4.88 %, from 448.2 to 470.1 m/s.
Worked example with the values above
- Numerator: 2γRTt = 2 × 1.4 × 287 × 600 = 482,160.
- Divide by γ + 1 = 2.4: 200,900.
- a* = 448.219 m/s.
Assumptions and limits
- Steady, adiabatic, reversible flow of a calorically perfect gas, with no shocks or friction.
- A constant heat-capacity ratio, and quasi-one-dimensional flow where an area enters.
Common mistakes
- Using the subsonic branch of the area–Mach relation for a supersonic nozzle, or the reverse.
- Entering static temperature where the formula needs stagnation temperature.
- Applying the area-velocity relation at M = 1, where it is singular.
Statistics and uncertainty
Isentropic relations are exact for a perfect gas. Real nozzle flow departs from them: boundary layers and non-equilibrium chemistry make mass flow a few per cent lower, and for hot combustion gas γ is a function of temperature, not a constant.
Where it is used
- Sizing nozzles, inlets and diffusers.
- Referring engine data to standard conditions for comparison.
Maths you need
- Stagnation conditions: the state reached if the flow is slowed to rest isentropically
- Choking: the flow is sonic at the minimum area and the mass flow is maximal
- Mach number: M = V/a