Library / Thermodynamics / THE-12
Thermodynamics
Adiabatic compression temperature
Temperature after rapid compression with no heat exchange.
- Ref
- THE-12
- Rev
- v1
- Field
- Thermodynamics
- Result
- T2 [K]
T₁ is the starting temperature in kelvin, r the compression ratio (starting volume over final volume) and γ the ratio of specific heats (1.4 for air). The compression is assumed to happen so fast, or be so well insulated, that no heat leaves, and with no friction (isentropic). The same relation gives T₂/T₁ = (p₂/p₁)(γ−1)/γ.
Temperature after rapid compression with no heat exchange. A higher compression ratio r gives a hotter charge.
Calculator
- Temperature rise T₂ − T₁389.2 K
- T₂ in degrees Celsius416.1 °C
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.
- Air at 300 K compressed 8 to 1 reaches 689 K, which is 416 °C. At the 16 to 1 ratio typical of a diesel it reaches about 909 K, well above the temperature at which diesel fuel ignites, so no spark is needed.
- The same heating limits the temperature rise in the compressor stage of a gas turbine and shows why intercooling can save work.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T1 Initial temperature | 100 to 1,000 K | 229.7 | 2,297 | 2,068 | +10% | +10 % |
| r Compression ratio | 1 to 30 | 300 | 1,169 | 869.4 | +10% | +3.89 % |
| γ Ratio of specific heats | 1.1 to 1.67 | 369.3 | 1,208 | 839 | +10% | +33.8 % |
Sweep of Compression ratio
| Compression ratio | Final temperature (K) |
|---|---|
| 1 | 300 |
| 4.625 | 553.6 |
| 8.25 | 697.8 |
| 11.88 | 807.2 |
| 15.5 | 898 |
| 19.13 | 976.7 |
| 22.75 | 1,047 |
| 26.38 | 1,111 |
| 30 | 1,169 |
Sweep one variable
Variables and how each one contributes
Final temperature
What the equation gives you. Everything else feeds it.
Initial temperature
The temperature before compression, in kelvin. Directly proportional: the temperature after is a fixed multiple of the one before.
Compression ratio
The compression ratio V₁/V₂. Power law with exponent γ − 1 = 0.4: a 10 % rise in r raises T₂ by about 3.9 %.
Ratio of specific heats
The ratio of specific heats Cp/Cv (about 1.4 for air, 1.67 for a monatomic gas). Strong: a gas with a higher γ heats up much more. Raising γ by 10 % (1.4 to 1.54) raises T₂ by about 34 %.
Worked example with the values above
- Exponent: γ − 1 = 1.4 − 1 = 0.4.
- Temperature ratio: rγ−1 = 80.4 = 2.2974.
- T₂ = 300 × 2.2974 = 689.2 K.
Assumptions and limits
- An ideal gas with constant γ.
- A reversible adiabatic (isentropic) process with no heat transfer and no friction.
Common mistakes
- Entering T₁ in °C.
- Using the pressure ratio in place of the volume ratio.
Statistics and uncertainty
T₂ ∝ T₁ rγ−1, so the compression ratio’s error is weighted by 0.4 for air, while the error in γ is weighted more heavily (it is in the exponent): a 1 % error in γ is about 3.4 % in T₂ at r = 8. Use a γ appropriate to the mean temperature, since γ falls from about 1.40 at 300 K to 1.33 near 1000 K.
Where it is used
- Why diesel engines ignite fuel by compression alone.
- Estimating the delivery temperature of a compressor stage.
Maths you need
- Ideal gas law: pV = nRT
- Ratio of specific heats: γ = Cp/Cv