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Aerospace
Critical back pressure of a convergent nozzle
Back pressure at which the exit of a purely convergent nozzle just reaches Mach 1, fed from total pressure pt.
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- AER-44
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- v1
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- Aerospace
- Result
- pb,crit [Pa]
A convergent nozzle accelerates the flow to sonic speed at its exit when the back pressure falls to pb,crit = pt[2/(γ+1)]^(γ/(γ−1)), which is 0.528 pt for air. At lower back pressure the flow stays choked and the exit pressure stays at this value, with the expansion happening outside the nozzle. Above this value the flow is subsonic and the exit pressure equals the back pressure.
Back pressure at which the exit of a purely convergent nozzle just reaches Mach 1, fed from total pressure pt. For air it is 0.528 pt. Lowering the back pressure further leaves the ideal mass flow unchanged. Convergent-divergent nozzles have different back-pressure regimes.
Calculator
- Critical pressure ratio, pb,crit/pt0.5283
- Total pressure needed to choke against 101 325 Pa191,800 Pa
- Static temperature ratio at the sonic exit, T*/Tt0.8333
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 nozzle chokes when the back pressure is below 105.66 kPa, 52.83 % of the total pressure.
- Discharging to the atmosphere at 101.3 kPa, the total pressure must exceed 191.8 kPa to choke the exit.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| pt Total pressure | 50,000 to 2,000,000 Pa | 26,410 | 1,057,000 | 1,030,000 | +10% | +10 % |
| γ Heat-capacity ratio | 1.1 to 1.67 | 97,330 | 116,900 | 19,600 | +10% | −4.26 % |
Sweep of Total pressure
| Total pressure (Pa) | Critical back pressure (Pa) |
|---|---|
| 50,000 | 26,410 |
| 293,800 | 155,200 |
| 537,500 | 284,000 |
| 781,300 | 412,700 |
| 1,025,000 | 541,500 |
| 1,269,000 | 670,300 |
| 1,513,000 | 799,000 |
| 1,756,000 | 927,800 |
| 2,000,000 | 1,057,000 |
Sweep one variable
Variables and how each one contributes
Critical back pressure
What the equation gives you. Everything else feeds it.
Total pressure
The upstream total pressure, in Pa. Scales the critical back pressure in direct proportion. Raising it by 10 % (to 220,000) moves the result by +10 %, from 105,700 to 116,200 Pa.
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 −4.26 %, from 105,700 to 101,200 Pa.
Worked example with the values above
- Ratio: 2/(γ + 1) = 0.833333.
- Exponent: γ/(γ − 1) = 3.5; ratio^exponent = 0.528282.
- pb,crit = 200,000 × 0.528282 = 105,656.4 Pa.
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