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Aerospace
Turbine total pressure ratio from actual temperature drop
Outlet-to-inlet total pressure ratio of a turbine from its measured total temperature drop and total-to-total isentropic efficiency ηt.
- Ref
- AER-51
- Rev
- v1
- Field
- Aerospace
- Result
- pt5/pt4
A turbine’s actual temperature drop is related to the ideal one by its isentropic efficiency: Tt4 − Tt5 = ηt Tt4[1 − (pt5/pt4)^((γ−1)/γ)]. Solving gives the pressure ratio pt5/pt4 = [1 − (Tt4 − Tt5)/(ηt Tt4)]^(γ/(γ−1)). It is below 1, since the gas expands. The bracket is the ideal exit-to-inlet temperature ratio and it must be positive.
Outlet-to-inlet total pressure ratio of a turbine from its measured total temperature drop and total-to-total isentropic efficiency ηt. The bracket is the ideal outlet-to-inlet temperature ratio and must be positive. The page needs T4 ≥ T5.
Calculator
- Ideal temperature ratio, bracket0.75
- Actual temperature ratio, Tt5/Tt40.8
- Expansion ratio, pt4/pt52.737
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 turbine expands the gas by a factor of 2.737 in total pressure, so the outlet pressure is 36.54 % of the inlet value.
- If the turbine were ideal (ηt = 1) the same temperature drop would need only an expansion ratio of 2.184, so the inefficiency costs 0.553 in expansion ratio.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Tt5 Outlet total temperature | 600 to 1,200 K | 0.03229 | 1 | 0.9677 | +10% | +55 % |
| Tt4 Inlet total temperature | 1,000 to 2,000 K | 0.02537 | 0.8357 | 0.8103 | +10% | −36.4 % |
| γ Heat-capacity ratio | 1.1 to 1.67 | 0.04224 | 0.4882 | 0.446 | +10% | +20.5 % |
| ηt Isentropic efficiency | 0.7 to 0.95 | 0.308 | 0.4372 | 0.1292 | +10% | +11 % |
Sweep of Isentropic efficiency
| Isentropic efficiency | Turbine pressure ratio |
|---|---|
| 0.7 | 0.308 |
| 0.7312 | 0.3268 |
| 0.7625 | 0.3448 |
| 0.7937 | 0.362 |
| 0.825 | 0.3784 |
| 0.8562 | 0.3941 |
| 0.8875 | 0.4091 |
| 0.9187 | 0.4235 |
| 0.95 | 0.4372 |
Sweep one variable
Variables and how each one contributes
Turbine pressure ratio
What the equation gives you. Everything else feeds it.
Inlet total temperature
The inlet total temperature, in K. For a fixed temperature drop, hotter gas has a smaller relative drop and so a milder expansion ratio. Raising it by 10 % (to 1,320) moves the result by −36.4 %, from 0.3654 to 0.2324.
Outlet total temperature
The outlet total temperature, in K. A cooler outlet means more work taken and a lower pressure ratio. Raising it by 10 % (to 1,056) moves the result by +55 %, from 0.3654 to 0.5662.
Isentropic efficiency
The total-to-total isentropic efficiency. A lower efficiency needs a larger expansion for the same temperature drop, lowering the pressure ratio. Raising it by 10 % (to 0.88) moves the result by +11 %, from 0.3654 to 0.4056.
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 +20.5 %, from 0.3654 to 0.4402.
Worked example with the values above
- Actual drop: Tt4 − Tt5 = 1,200 − 960 = 240 K.
- Bracket: 1 − 240 / (0.8 × 1,200) = 0.75.
- pt5/pt4 = 0.75^3.5 = 0.365354.
Assumptions and limits
- Constant specific heats in each stream, adiabatic components and total-to-total efficiencies.
- Steady operation at one operating point, with no bleed and no leakage unless the page says so.
Common mistakes
- Mixing total and static states.
- Using a total-to-static efficiency where a total-to-total one is needed.
- Forgetting the fuel added to the turbine mass flow.
Statistics and uncertainty
Constant-cp cycle calculations are rough: cp rises from 1005 J/(kg·K) in cold air to about 1150 J/(kg·K) in hot gas, which shifts results by 5 % to 10 %. Component efficiencies from maps are good to 1 % to 2 % at design and worse off design.
Where it is used
- Matching compressors, combustors and turbines and finding fuel flow and power.
- Sizing engines and checking cycle designs.
Maths you need
- Isentropic efficiency: ideal work divided by actual work (or the reverse for a turbine)
- Total temperature: the temperature the gas would reach at rest
- Brayton cycle: compression, heat addition, expansion