Library / Aerospace / AER-45
Aerospace
Exit Mach of a convergent nozzle versus back pressure
Exit Mach number of an ideal convergent nozzle with 0 < pb ≤ pt.
- Ref
- AER-45
- Rev
- v1
- Field
- Aerospace
- Result
- Me
For an ideal convergent nozzle the exit static pressure equals the back pressure while the flow is subsonic, and isentropic flow gives Me = √{2[(pt/pb)^((γ−1)/γ) − 1]/(γ−1)}. When the pressure ratio exceeds the critical value, the exit is sonic and cannot go faster, so Me = 1. With pb = pt there is no flow and Me = 0.
Exit Mach number of an ideal convergent nozzle with 0 < pb ≤ pt. Below the critical pressure ratio the exit is sonic and its static pressure stays at the critical value, so the result is capped at 1. At pb = pt the Mach number is zero. Reverse flow is excluded.
Calculator
- Critical back pressure ratio, pb,crit/pt0.5283
- Actual back pressure ratio pb/pt0.8
- Exit static temperature ratio, Te/Tt0.9382
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.
- With a pressure ratio of 1.25 the exit Mach number is 0.5737, subsonic, so the nozzle is not choked.
- The critical pressure ratio is 1.893, which corresponds to a back pressure of 79.242 kPa for this total pressure.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| pt Total pressure | 120,000 to 400,000 Pa | 0 | 1 | 1 | +10% | +20.3 % |
| pb Back pressure | 40,000 to 150,000 Pa | 0 | 1 | 1 | +10% | −24.8 % |
| γ Heat-capacity ratio | 1.1 to 1.67 | 0.5287 | 0.6402 | 0.1115 | +10% | −4.3 % |
Sweep of Back pressure
| Back pressure (Pa) | Exit Mach number |
|---|---|
| 40,000 | 1 |
| 53,750 | 1 |
| 67,500 | 1 |
| 81,250 | 0.9784 |
| 95,000 | 0.8349 |
| 108,800 | 0.6937 |
| 122,500 | 0.5458 |
| 136,300 | 0.3732 |
| 150,000 | 0 |
Sweep one variable
Variables and how each one contributes
Exit Mach number
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 −4.3 %, from 0.5737 to 0.549.
Total pressure
The total pressure upstream of the nozzle, in Pa. Raises the exit Mach number up to the choked value of 1. Raising it by 10 % (to 165,000) moves the result by +20.3 %, from 0.5737 to 0.6901.
Back pressure
The back pressure outside the exit, in Pa. Lowering it raises the exit Mach number until the nozzle chokes. Raising it by 10 % (to 132,000) moves the result by −24.8 %, from 0.5737 to 0.4313.
Worked example with the values above
- Pressure ratio: pt/pb = 150,000 / 120,000 = 1.25; critical value 1.89293.
- Isentropic Mach: √(2[(pt/pb)^((γ−1)/γ) − 1]/(γ − 1)) = 0.573723.
- Me = min(1, 0.573723) = 0.573723.
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