In nozzles, diffusers and wind tunnels, gas flows smoothly without losses. These isentropic relations, tabulated in every gas-dynamics book, show how temperature and pressure fall as the flow speeds up and how a nozzle must widen to go supersonic.
How to use the Isentropic Flow Calculator
- Mach number from
- Mach number M
- Velocity
- Speed of sound
- Heat capacity ratio γ
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the Mach number (or velocity and speed of sound) and γ.
Tips
- Use absolute temperature (K) in gas laws and efficiency formulas.
- Temperature differences are the same in K and °C.
- Heat = mass × specific heat × temperature change when no phase change occurs.
- Phase changes need latent heat at a constant temperature.
Common mistakes to avoid
- Using °C in pV = nRT.
- Forgetting latent heat during melting or boiling.
- Confusing heat capacity (J/K) with specific heat (J/(kg·K)).
Inputs explained
| Field | What to enter |
|---|---|
| Mach number from | Choose one: Direct value, Velocity and speed of sound |
| Mach number M | A number |
| Velocity | Units: m/s, km/h, mph, ft/s, knots, km/s |
| Speed of sound | Units: m/s, km/h, mph, ft/s, knots, km/s |
| Heat capacity ratio γ | A number |
What each result means
- Temperature ratio T/T₀ = 1÷[X]
- Pressure ratio p/p₀ = [X]^(-[Heat capacity ratio γ]÷([Heat capacity ratio γ] – 1))
- Density ratio ρ/ρ₀ = [X]^(-1÷([Heat capacity ratio γ] – 1))
- Area ratio A/A* = 1÷[Mach number]×(2÷([Heat capacity ratio γ] + 1)×[X])^(([Heat capacity ratio γ] + 1)÷(2×([Heat capacity ratio γ] – 1)))
Example values
How the main result changes when Mach number M changes and everything else stays at the example values above:
| Mach number M | Temperature ratio T/T₀ |
|---|---|
| 1 | 0.8333 |
| 1.5 | 0.6897 |
| 2 | 0.5556 |
| 2.5 | 0.4444 |
| 3 | 0.3571 |
| 4 | 0.2381 |
Formula
T/T₀ = (1 + (γ−1)/2·M²)⁻¹, p/p₀ = (T/T₀)^{γ/(γ−1)}, ρ/ρ₀ = (T/T₀)^{1/(γ−1)}A/A* = (1/M)·[(2/(γ+1))·(1 + (γ−1)/2·M²)]^{(γ+1)/(2(γ−1))}Worked example
Inputs — Mach number from: Direct value; Mach number M: 2; Heat capacity ratio γ: 1.4.
Results — Mach number = 2; Temperature ratio T/T₀ = 0.555556; Pressure ratio p/p₀ = 0.127805; Density ratio ρ/ρ₀ = 0.230048; Area ratio A/A* = 1.6875; Mach angle = 30 °.
Key terms
- Isentropic
- Adiabatic and reversible (constant entropy).
Frequently asked questions
What is stagnation temperature?
The temperature if the flow were brought to rest without losses.
Why does a supersonic nozzle widen?
Beyond Mach 1, area must increase for the flow to accelerate.
What is A*?
The throat area where M = 1.
Is real flow isentropic?
Approximately in nozzles; shocks and friction add losses.
What formula does the Isentropic Flow Calculator use?
T/T₀ = (1 + (γ−1)/2·M²)⁻¹, p/p₀ = (T/T₀)^{γ/(γ−1)}, ρ/ρ₀ = (T/T₀)^{1/(γ−1)}.