Library / Aerospace / AER-16
Aerospace
Temperature indicated by a recovery probe
Temperature read by a probe that brings the flow to rest imperfectly, from static temperature T, Mach number M and recovery factor r.
- Ref
- AER-16
- Rev
- v1
- Field
- Aerospace
- Result
- Tr [K]
A probe that brings the air to rest recovers the stagnation temperature Tt = T[1 + (γ−1)M²/2], but in practice it recovers only a fraction r of the rise: Tr = T[1 + r(γ−1)M²/2]. A recovery factor r = 1 gives the full stagnation temperature. Typical total-air-temperature probes have r of 0.95 to 0.99, and thermometers in the skin of the aircraft about 0.85 to 0.9. Conduction, radiation and installation errors are separate.
Temperature read by a probe that brings the flow to rest imperfectly, from static temperature T, Mach number M and recovery factor r. With r = 1 it reads the full stagnation temperature. Constant specific heats; conduction, radiation and installation errors are excluded.
Calculator
- Stagnation temperature, T[1 + (γ − 1)M²/2]282 K
- Temperature rise indicated, Tr − T28.8 K
- Error against stagnation temperature, Tt − Tr3.2 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 probe reads 278.8 K, 28.8 K above the static temperature of 250 K; the full stagnation rise would be 32 K.
- To find the static temperature from the probe reading, invert the formula: T = Tr/[1 + r(γ − 1)M²/2], which needs the Mach number from the pitot-static system.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| M Mach number | 0 to 3 | 250 | 655 | 405 | +10% | +2.17 % |
| T Static temperature | 180 to 320 K | 200.7 | 356.9 | 156.1 | +10% | +10 % |
| γ Heat-capacity ratio | 1.1 to 1.67 | 257.2 | 298.2 | 41.04 | +10% | +3.62 % |
| r Recovery factor | 0.7 to 1 | 272.4 | 282 | 9.6 | +10% | +1.03 % |
Sweep of Mach number
| Mach number | Indicated temperature (K) |
|---|---|
| 0 | 250 |
| 0.375 | 256.3 |
| 0.75 | 275.3 |
| 1.125 | 307 |
| 1.5 | 351.2 |
| 1.875 | 408.2 |
| 2.25 | 477.8 |
| 2.625 | 560.1 |
| 3 | 655 |
Sweep one variable
Variables and how each one contributes
Indicated temperature
What the equation gives you. Everything else feeds it.
Static temperature
The static air temperature, in K. Raises the indicated temperature in direct proportion. Raising it by 10 % (to 275) moves the result by +10 %, from 278.8 to 306.7 K.
Recovery factor
The recovery factor of the probe (1 gives the full stagnation temperature). A higher factor recovers more of the temperature rise. Raising it by 10 % (to 0.99) moves the result by +1.03 %, from 278.8 to 281.7 K.
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 +3.62 %, from 278.8 to 288.9 K.
Mach number
The local Mach number. Raises the temperature rise with M², from zero at rest. Raising it by 10 % (to 0.88) moves the result by +2.17 %, from 278.8 to 284.8 K.
Worked example with the values above
- Full rise factor: (γ − 1)M²/2 = 0.2 × 0.64 = 0.128.
- Recovered part: r × 0.128 = 0.1152.
- Tr = 250 × (1 + 0.1152) = 278.8 K.
Assumptions and limits
- Calorically perfect gas with a constant heat-capacity ratio, and isentropic deceleration to the pitot tube.
- Subsonic flow for the pitot and calibrated-airspeed formulas; a shock forms ahead of the probe above Mach 1.
Common mistakes
- Using the incompressible ½ρV² formula at high Mach number, where it underestimates impact pressure.
- Confusing true, equivalent and calibrated airspeed.
- Treating a recovery temperature as a static temperature.
Statistics and uncertainty
Air-data conversions are exact for the stated models, so errors come from the instrument: pitot and static ports carry position errors of 1 % to 2 % in speed, and a poorly designed temperature probe can have a recovery factor that differs from 0.9 by several per cent.
Where it is used
- Converting instrument readings to speed, Mach number and static temperature.
- Understanding what the airspeed indicator and temperature probes really measure.
Maths you need
- Stagnation point: where the flow is brought to rest
- Dynamic pressure: q = ½ρV²
- Speed of sound: a = √(γRT)