Stagnation Temperature Calculator

Formula used

For adiabatic, isentropic compressible flow, stagnation temperature relates to static temperature by:

In energy form (useful when velocity and c_p are known):

Notes: γ is the specific heat ratio, M is Mach number, V is flow speed, and c_p is specific heat at constant pressure.

How to use this calculator

1. Select a calculation mode that matches your known variables.
2. Pick units for temperature and velocity to match your data.
3. Enter γ, then provide the required inputs for the selected mode.
4. For energy modes, enter c_p or provide R to compute it.
5. Click Calculate to view results above the form.
6. Use Download CSV or Download PDF after computing.

Example data table

Sample values are illustrative for typical gas dynamics problems.

Stagnation temperature in compressible flow

1) What stagnation temperature represents

Stagnation temperature (T₀) is the total temperature a moving gas would reach if it were brought to rest adiabatically with no external work. It is a convenient “energy level” for a flow because it stays constant across isentropic ducts and nozzles even when static temperature changes.

2) Key isentropic relation and typical ratios

For an ideal gas in isentropic flow, the calculator uses T₀/T = 1 + (γ−1)M²/2. With air (γ ≈ 1.40), common ratios are: M = 0.3 → T₀/T ≈ 1.018, M = 0.8 → ≈ 1.128, and M = 2.0 → 1.800. These values help sanity‑check inputs.

3) Velocity-based energy form

When you know speed instead of Mach number, the energy form is practical: T₀ = T + V²/(2c_p). For air near room conditions, c_p is often 1004–1007 J/(kg·K). At T = 293.15 K and V = 250 m/s with c_p = 1005 J/(kg·K), T₀ rises by about 31.1 K.

4) Selecting realistic γ and c_p

γ depends on gas composition and temperature. Air is commonly modeled as 1.40 at moderate temperatures, while CO₂ is closer to 1.30. If you provide the specific gas constant R, the calculator can estimate c_p from c_p = γR/(γ−1), which is useful when a consistent property set is needed.

5) Inverse calculations for design work

Engineering problems often require inverses: solving for T from T₀ and M, estimating M from T₀/T, or finding V from a measured temperature rise. This tool supports those workflows with consistent validation (for example, it enforces T₀ ≥ T for real solutions).

6) Measurement context and recovery

In wind tunnels and intakes, a probe may not recover the full total temperature because of heat transfer and viscous effects. The computed T₀ is the ideal isentropic value. If you use measured probe temperatures, treat them as “recovered” totals and interpret differences as losses or recovery factor effects.

7) Units and conversion consistency

Temperature inputs can be entered in K, °C, °F, or °R and are converted internally to Kelvin. Velocities can be entered in m/s, km/h, mph, or ft/s and are converted to m/s. Keeping units consistent prevents hidden errors, especially in the V² term.

8) Practical ranges and interpretation

For subsonic ventilation or HVAC ducts, M is typically below 0.3, so T₀ is only slightly above T. In nozzles and high‑speed jets, M can exceed 1 and T can drop significantly while T₀ remains nearly constant for isentropic sections. Use the ratio T₀/T as a quick indicator of compressibility intensity.

FAQs

1) Is stagnation temperature the same as static temperature?

No. Static temperature is the local thermodynamic temperature of the moving gas. Stagnation temperature includes the kinetic energy effect and is higher than static temperature whenever the flow speed is nonzero.

2) When should I use the Mach-number mode versus velocity mode?

Use Mach-number mode when compressible flow conditions are known or derived from gas dynamics. Use velocity mode when you have measured speed and a reliable c_p (or R and γ) for the operating temperature range.

3) What values of γ are reasonable for air?

A common engineering value is γ = 1.40 for air near room temperature. At higher temperatures, γ can decrease. For accuracy-critical work, use property tables or a model that accounts for temperature-dependent specific heats.

4) Why does the calculator require T₀ ≥ T for some modes?

In the ideal relations used here, T₀/T = 1 + (γ−1)M²/2, which is always at least 1. If T₀ is less than T, there is no real Mach number or velocity solution.

5) Can I compute c_p automatically?

Yes. If you provide R and γ, the tool estimates c_p using c_p = γR/(γ−1). This is a useful approximation for ideal-gas calculations with constant properties.

6) Does stagnation temperature change across a shock wave?

Across a normal shock, the flow is not isentropic. Stagnation temperature is approximately conserved if there is no external heat transfer, but stagnation pressure drops due to irreversibility. Interpret results accordingly.

7) How accurate is the velocity-based temperature rise?

Accuracy depends on c_p being representative and on negligible heat transfer and shaft work. For moderate speeds and small temperature changes, it works well. For very high temperatures, variable properties should be considered.