Formula & Calculator
Isentropic Temperature Ratio
Relates static temperature to stagnation temperature for isentropic compressible flow as a function of Mach number.
Interpretation
Isentropic temperature ratio: T₀/T = 1 + ((γ−1)/2)·M², where T₀ is stagnation temperature, T is static temperature, M is Mach number. It relates temperature increase due to flow deceleration. Example: M=2, γ=1.4 → T₀/T = 1+0.2×4=1.8.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| T0/T | Stagnation-to-static temperature ratio | |
| γ | Ratio of specific heats | |
| M | Mach number |
What it means
The isentropic temperature ratio gives the change in temperature when a flow is brought to rest isentropically (stagnation). It is derived from the energy equation for an ideal gas. This ratio is used in compressible flow calculations, such as in gas turbines, nozzles, and shock waves. It shows that at high Mach numbers, stagnation temperatures become significantly higher than static temperatures, affecting material selection and cooling requirements. The ratio appears in the definition of total temperature and is used to compute stagnation properties. Understanding this relation is essential for gas dynamics and propulsion.
Worked example
Isentropic Temperature Ratio – Two Examples
Real‑World| Parameter | Value |
|---|---|
| γ | 1.4 |
| M | 0.5 |
| Parameter | Value |
|---|---|
| M | 2.0 |
Common mistakes
- Isentropic temperature ratio: T₀/T = 1 + ((γ−1)/2)·M².
- γ: Specific heat ratio (cp/cv).
- M: Mach number.
- Stagnation temperature T₀: Total temperature – higher than static T.
- Applicable to isentropic, steady, compressible flow.
Applications
The isentropic temperature ratio, T₀/T = 1 + ((γ−1)/2)·M², relates the stagnation temperature (total temperature) to the static temperature in compressible flow. This is used in gas turbine and nozzle analysis to compute the temperature rise across a compressor or the temperature drop across a turbine. Engineers use it to design engine components, to calculate air properties, and to assess the effects of high‑speed flight. The ratio also determines the heat transfer to surfaces. By understanding isentropic relationships, aerospace engineers can accurately model and design propulsion and aerodynamic systems.
- Gas turbine engine component design (compressor, turbine)
- Nozzle and diffuser flow analysis
- High‑speed flight aerodynamics (boundary layer heating)
- Total temperature measurements and corrections
- Engine performance modelling (thermodynamic cycles)
Frequently Asked Questions
It relates the static temperature to the stagnation temperature for isentropic compressible flow as a function of Mach number. It is used in gas dynamics, nozzle design, and high‑speed aerodynamics.
T0 = stagnation (total) temperature (K)
T = static temperature (K)
γ = specific heat ratio (dimensionless)
M = Mach number
It determines the heating of high‑speed vehicles and the performance of engines and nozzles. At high Mach numbers, stagnation temperatures can be very high.
As M increases, T0 increases relative to T. For M = 2, T0/T ≈ 1.8; for M = 5, T0/T ≈ 6.
- Using γ=1.4 for hot combustion gases where γ is significantly different (closer to 1.2–1.3).
- Applying the isentropic relation across a shock wave, where the flow is no longer isentropic.
- Confusing static and stagnation temperatures.
Air at M = 2 has static temperature T = 220 K, γ = 1.4. T0 = 220 × (1 + 0.2×4) = 220 × 1.8 = 396 K. The stagnation temperature is 176 K higher.
At high Mach, the stagnation temperature can exceed the melting point of materials. This drives the need for thermal protection systems (TPS).
Similar relations exist: p0/p = (1 + ((γ−1)/2)M²)^(γ/(γ−1)) and ρ0/ρ = (1 + ((γ−1)/2)M²)^(1/(γ−1)).
With a stagnation temperature probe (thermocouple) that brings the flow to rest. The measured temperature is T0.
The temperature rise from compression dictates the combustion efficiency and thrust. The ratio determines the maximum achievable temperature and pressure.