Library / Thermodynamics / THE-17
Thermodynamics
Temperature after a polytropic process
Temperature of an ideal gas after a polytropic process, which follows from the process law and the ideal-gas law.
- Ref
- THE-17
- Rev
- v1
- Field
- Thermodynamics
- Result
- T2 [K]
For an ideal gas undergoing a polytropic process pVn = constant, the temperature follows T₂ = T₁(V₁/V₂)n−1. For n = 1 the temperature is constant, and for larger n the gas heats more on compression, up to the reversible adiabatic value at n = γ. Temperature must be absolute (kelvin).
Temperature of an ideal gas after a polytropic process, which follows from the process law and the ideal-gas law. Compression with n above 1 raises the temperature.
Calculator
- Temperature rise, T₂ − T₁259.8 K
- Final temperature in °C286.7 °C
- Final temperature if adiabatic (n = 1.4)689.2 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.
- Compressing from 1 L to 0.125 L with n = 1.3 heats the gas from 300 K to 559.8 K (287 °C), a rise of 260 K.
- With no heat loss at all (n = 1.4 for air) the same compression would give 689.2 K: this is why intercoolers and cooled cylinder walls are used to limit the discharge temperature.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T1 Initial absolute temperature | 200 to 1,000 K | 373.2 | 1,866 | 1,493 | +10% | +10 % |
| n Polytropic exponent | 0.5 to 1.67 | 106.1 | 1,208 | 1,102 | +10% | +31 % |
| V1 Initial volume | 0.1 to 10 L | 280.6 | 1,117 | 836.4 | +10% | +2.9 % |
| V2 Final volume | 0.05 to 10 L | 150.4 | 736.9 | 586.6 | +10% | −2.82 % |
Sweep of Final volume
| Final volume (L) | Final temperature (K) |
|---|---|
| 0.05 | 736.9 |
| 1.294 | 277.7 |
| 2.537 | 226.9 |
| 3.781 | 201.3 |
| 5.025 | 184.8 |
| 6.269 | 173 |
| 7.512 | 163.8 |
| 8.756 | 156.5 |
| 10 | 150.4 |
Sweep one variable
Variables and how each one contributes
Final temperature
What the equation gives you. Everything else feeds it.
Initial absolute temperature
The initial absolute temperature, in kelvin. Raises the final temperature in direct proportion. Raising it by 10 K (to 310 K) moves the result by +3.33 %, from 559.8 to 578.5 K.
Initial volume
The initial volume, in litres. Raises the final temperature, because the volume ratio is larger. Raising it by 0.1 L (to 1.1 L) moves the result by +2.9 %, from 559.8 to 576.1 K.
Final volume
The final volume, in litres. Lowers the final temperature, because the gas is compressed less. Raising it by 0.025 L (to 0.15 L) moves the result by −5.32 %, from 559.8 to 530 K.
Polytropic exponent
The polytropic exponent. For n = 1 the temperature does not change. Raises the final temperature on compression: a larger exponent means less heat escapes during compression. Raising it by 0.05 (to 1.35) moves the result by +11 %, from 559.8 to 621.2 K.
Worked example with the values above
- Volume ratio: V₁/V₂ = 8, exponent n − 1 = 0.3.
- Raise to the power: 80.3 = 1.866.
- T₂ = 300 × 1.866 = 559.8 K.
Assumptions and limits
- An ideal gas following pVn = constant, with absolute pressure and temperature.
- A closed system with a fixed amount of gas, and a reversible, quasi-equilibrium process.
Common mistakes
- Using gauge pressure or Celsius temperature: both must be absolute.
- Using γ for n when the process loses heat: n is lower than γ for a cooled compressor.
Statistics and uncertainty
The temperature rise is sensitive to n − 1, which is small and so has a large relative uncertainty: an error of 0.05 in n changes the temperature rise at a volume ratio of 8 by about 24 %. Measuring the discharge temperature is a good way to find n.
Where it is used
- Discharge temperature of a compressor or the end-of-compression temperature in an engine.
- Checking whether a compression process is closer to isothermal or adiabatic.
Maths you need
- Ideal gas law: pV = mRT
- Polytropic process: pVn = constant, with n set by the heat exchange