# Isentropic Flow Calculator

URL: https://speedycalculator.com/calculators/physics/isentropic-flow/
Last updated: 2026-10-30

> The isentropic flow calculator finds the static-to-stagnation temperature, pressure and density ratios, the area ratio A/A* and the Mach angle for an ideal compressible flow at a given Mach number.

## Inputs

- Mach number from
- Mach number M
- Velocity (m/s, km/h, mph, ft/s, knots, km/s)
- Speed of sound (m/s, km/h, mph, ft/s, knots, km/s)
- Heat capacity ratio γ

## Results

- Mach number
- Temperature ratio T/T₀
- Pressure ratio p/p₀
- Density ratio ρ/ρ₀
- Area ratio A/A*
- Mach angle

## 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))}
```

## Explanation

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

FieldWhat to enterMach number fromChoose one: Direct value, Velocity and speed of soundMach number MA numberVelocityUnits: m/s, km/h, mph, ft/s, knots, km/sSpeed of soundUnits: m/s, km/h, mph, ft/s, knots, km/sHeat capacity ratio γA 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 MTemperature ratio T/T₀10.83331.50.689720.55562.50.444430.357140.2381

## 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).

## FAQ

**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)}.


## Sources

- [NASA Glenn – Isentropic flow](https://www.grc.nasa.gov/www/k-12/airplane/isentrop.html)
- [NACA Report 1135](https://ntrs.nasa.gov/citations/19930091059)