Stagnation Temperature Calculator
Solve for total temperature, static temperature, Mach number or γ. Includes unit conversions, isentropic tables, gas comparison and live graphs.
Calculator
Results
Temperature conversion of results
What this tool offers
How to use
- Pick a modeBasic finds T₀ from T and Mach. Advanced unlocks every variable, gases and extra outputs.
- Choose your temperature unitValues convert automatically; the formula always uses absolute temperature.
- Enter known valuesType the static temperature T and Mach number M (or use the slider). In Advanced, choose the unknown, then set γ or a gas.
- Read the resultsResults, tables and graphs update instantly. The amber dot on each graph is your operating point.
- ExploreChange Mach or gas and watch the curves. Check the Mach table for standard reference values.
Graph 1 · T₀ vs Mach number for different gases
Graph 2 · T₀ vs static T at fixed Mach
Graph 3 · Isentropic ratios (log scale)
Mach table (current T and γ)
Temperature unit conversion
Gas properties
Theory & related concepts
Derivation
From the steady adiabatic energy equation, h₀ = h + V²/2. For a calorically perfect gas h = cₚT, and V = M·a, a² = γRT, cₚ = γR/(γ−1). Substituting gives T₀/T = 1 + (γ−1)/2 · M².
Inverse forms
T = T₀ / (1 + (γ−1)M²/2)M = √[2(T₀/T − 1)/(γ−1)]γ = 1 + 2(T₀/T − 1)/M²
Related relations (isentropic)
p₀/p = (T₀/T)^(γ/(γ−1)),ρ₀/ρ = (T₀/T)^(1/(γ−1))- Speed of sound
a = √(γRT); velocityV = M·a; temperature riseΔT = V²/(2cₚ) - Adiabatic wall (recovery) temperature:
T_aw = T(1 + r(γ−1)M²/2), with r ≈ √Pr (laminar ≈ 0.85) or ∛Pr (turbulent ≈ 0.89)
Assumptions & limits
- Adiabatic flow, calorically perfect gas (constant γ). T₀ is constant along an adiabatic flow even across shocks; p₀ is not.
- Above roughly M 5 in air, γ varies with temperature (vibrational excitation, dissociation), so real T₀ is lower than predicted.
- Always use absolute temperature (K or °R) inside the formula; the calculator converts for you.
Where it is used
Aircraft and re-entry thermal loads, pitot/total-temperature probes, gas turbine and nozzle design, wind tunnel stagnation chambers, and supersonic inlet analysis.
Why is total temperature higher than static?
Decelerating the flow to rest converts kinetic energy into enthalpy; the gas heats up. Sensors measure something between T and T₀ depending on recovery factor.
Can T₀ be less than T?
No. With M ≥ 0 and γ > 1, T₀ ≥ T. At M = 0, T₀ = T.
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
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