Isentropic Flow Calculator
Calculate isentropic flow ratios — temperature, pressure, density, and area — for compressible nozzle flow at any Mach number and specific heat ratio.
Isentropic Flow
Isentropic flow is an idealized model of compressible gas flow through a nozzle where no heat transfer or friction occurs — entropy remains constant. It applies to supersonic nozzle design, wind tunnels, and rocket engine analysis.
Governing Parameter
The Mach number M = flow velocity / local speed of sound fully determines all flow ratios.
Key Ratios (static to stagnation)
| Ratio | Formula |
|---|---|
| T/T₀ | 1 / (1 + (γ−1)/2 × M²) |
| 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))) |
Where T₀, P₀, ρ₀ are stagnation (total) conditions and A* is the throat area at M=1.
Specific Heat Ratio γ
| Gas | γ |
|---|---|
| Air (standard, diatomic) | 1.400 |
| Monatomic gas (He, Ar) | 1.667 |
| Hot combustion gas | ~1.300 |
| CO₂ | 1.289 |
Physical Interpretation
At M=0: all ratios equal 1 (static = stagnation). At M=1 (sonic throat): P/P₀ = 0.528 for air — the critical pressure ratio. Beyond M=1 (supersonic): temperature, pressure, and density all drop rapidly.
Applications
Rocket nozzle sizing uses A/A* to find the throat and exit areas. Wind tunnel test sections are designed for target Mach numbers using these ratios. Pitot tubes in aircraft use stagnation-to-static pressure to infer airspeed.
Choking, and why the throat stops listening
The critical pressure ratio is the single most useful number here. Drop the pressure downstream of a converging nozzle and flow increases, but only until the throat reaches Mach 1. That happens when the back pressure falls to 0.528 of the supply pressure for air. Below that point nothing further happens: the throat is choked, mass flow is fixed by the supply conditions and the throat area alone, and lowering the downstream pressure another 50% changes it not at all.
This is not a curiosity. It is why a compressed-air line has a predictable leak rate through a given hole regardless of what is on the other side, why safety relief valves are sized on upstream conditions, and why a rocket engine’s thrust does not depend on the ambient pressure at the throat. Information cannot travel upstream faster than sound, so once the throat is sonic the downstream world simply cannot tell the throat anything.
A/A has two answers, and geometry will not tell you which*
Every area ratio above 1 corresponds to two Mach numbers: one subsonic, one supersonic. A converging-diverging nozzle with an exit-to-throat ratio of 1.6875 can run at Mach 0.37 or at Mach 2.00, and the same piece of hardware does both depending on the pressure it is fed. Get the throat choked and hold the back pressure low enough and you land on the supersonic branch. Otherwise the diverging section behaves as a diffuser and slows the flow back down.
Picking the wrong root is the classic first mistake in nozzle design, and the arithmetic gives no warning at all: both answers satisfy the equation exactly.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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