Formula & Calculator
Stagnation Temperature
Temperature a moving fluid would reach if brought to rest isentropically, important for high-speed vehicle heating.
Interpretation
Stagnation temperature: T₀ = T·(1 + ((γ−1)/2)·M²). It is the temperature when the flow is brought to rest isentropically. Example: T=300 K, M=2 → T₀ = 300×(1+0.2×4)=540 K.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| T0 | Stagnation temperature | K |
| T | Static temperature | K |
| γ | Ratio of specific heats | |
| M | Mach number |
What it means
Stagnation temperature (also called total temperature) is the temperature that a fluid would have if it were decelerated to zero velocity isentropically. It is a conserved quantity in adiabatic flows and is used in gas turbine engines, nozzle design, and high‑speed aerodynamics. The formula shows that at high Mach numbers, stagnation temperature can be significantly higher than static temperature, which affects material thermal limits. This temperature is measured by a total‑temperature probe. Understanding T₀ is essential for energy balance and for designing cooling systems in high‑speed flight.
Worked example
Stagnation Temperature – Two Examples
Real‑World| Parameter | Value |
|---|---|
| T | 216.5 K |
| γ | 1.4 |
| M | 2.0 |
| Parameter | Value |
|---|---|
| T | 288 K |
| M | 0.5 |
Common mistakes
- Stagnation temperature: T₀ = T · (1 + ((γ−1)/2)·M²).
- T: Static temperature.
- M: Mach number.
- T₀ > T for M>0.
- Constant during isentropic stagnation.
Applications
Stagnation temperature, T₀ = T·(1 + ((γ−1)/2)·M²), is the temperature the gas would reach if brought to rest isentropically. It is a crucial parameter in engine design, representing the total thermal energy available. Engineers use it to compute heat transfer to engine components, to design cooling systems, and to assess the thermal limits of materials. In high‑speed flight, stagnation temperatures become very high, requiring thermal protection. By understanding stagnation temperature, aerospace engineers can size cooling passages, select materials, and ensure that engines can withstand the thermal environment.
- Gas turbine engine component thermal design (turbine blades, combustor)
- Thermal protection system design for hypersonic vehicles
- Inlet and intake thermal assessment
- Engine performance modelling (energy balance)
- Heat exchanger and cooling system design
Frequently Asked Questions
It is the temperature a moving fluid would reach if brought to rest isentropically. It is important for high‑speed vehicle heating, engine performance, and thermodynamic calculations.
T0 = stagnation (total) temperature (K)
T = static temperature (K)
γ = specific heat ratio
M = Mach number
At hypersonic speeds (M > 5), the stagnation temperature can exceed 2000 K, leading to thermal stresses and the need for thermal protection systems.
- Neglecting stagnation temperature rise (aerodynamic heating) when estimating skin temperatures at high Mach.
- Using static temperature instead of stagnation in energy balances.
- Assuming the stagnation temperature is constant across a shock (it is not; it remains constant across a normal shock for adiabatic flow).
Air at M = 3, T = 220 K, γ = 1.4. T0 = 220 × (1 + 0.2×9) = 220 × 2.8 = 616 K. The temperature rise is 396 K.
In a turbojet, the stagnation temperature at the compressor inlet determines the compression work and the overall temperature rise, affecting thrust and efficiency.
Total temperature includes the kinetic energy contribution; static temperature is the thermodynamic temperature of the fluid in its local frame.
With a stagnation temperature probe (thermocouple or thermistor) that brings the flow to rest, measuring T0.
For an adiabatic shock, the stagnation temperature remains constant (since no heat is added). However, the static temperature increases.
For a perfect gas, h0 = h + V²/2, and h = cpT, so T0 = T + V²/(2cp).