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Isentropic nozzle area–Mach ratio
Ratio of the local flow area to the sonic area A* needed to pass the same mass flow from the same stagnation state, at Mach number M.
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- AER-34
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- v1
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- Aerospace
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- A/A*
Mass conservation in isentropic flow links the local flow area to the Mach number: A/A* = (1/M){[2/(γ+1)][1 + (γ−1)M²/2]}^((γ+1)/(2(γ−1))). A* is the sonic area for the same mass flow and stagnation state. The ratio is at least 1, and it reaches 1 at M = 1, the minimum area, so a given ratio above 1 has one subsonic and one supersonic solution. A supersonic nozzle needs a throat.
Ratio of the local flow area to the sonic area A* needed to pass the same mass flow from the same stagnation state, at Mach number M. It is at least 1, and reaches 1 at M = 1. Inverting it needs a choice of the subsonic or supersonic branch.
Calculator
- Stagnation-to-static pressure ratio7.824
- Stagnation-to-static temperature ratio1.8
- Stagnation-to-static density ratio4.347
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.
- At Mach 2 the flow area is 1.6875 times the sonic area, so the exit of a nozzle that expands to this Mach number must be 1.688 times the throat area.
- Mach 2 is supersonic, and the same area ratio would also occur at the matching subsonic Mach number.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| M Mach number | 0.1 to 5 | 1.005 | 25 | 23.99 | +10% | +18.8 % |
| γ Heat-capacity ratio | 1.1 to 1.67 | 1.53 | 2.032 | 0.5021 | +10% | −5.46 % |
Sweep of Mach number
| Mach number | Area ratio |
|---|---|
| 0.1 | 5.822 |
| 0.7125 | 1.086 |
| 1.325 | 1.077 |
| 1.938 | 1.603 |
| 2.55 | 2.763 |
| 3.163 | 4.942 |
| 3.775 | 8.749 |
| 4.388 | 15.05 |
| 5 | 25 |
Sweep one variable
Variables and how each one contributes
Area ratio
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 −5.46 %, from 1.688 to 1.595.
Mach number
The local Mach number (above 0). The area ratio falls to 1 at M = 1 and rises on both sides, steeply at high M and as M → 0. Raising it by 10 % (to 2.2) moves the result by +18.8 %, from 1.688 to 2.005.
Worked example with the values above
- Bracket: [2/(γ + 1)] × [1 + (γ − 1)M²/2] = 0.833333 × 1.8 = 1.5.
- Exponent: (γ + 1)/(2(γ − 1)) = 3.
- A/A* = 3.375 / 2 = 1.6875.
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