Aviation Calcs

C-D nozzle

Convergent-Divergent Nozzle Calculator

A convergent-divergent (de Laval) nozzle is designed so the throat is sonic and the exit is supersonic. Enter the exit-to-throat area ratio — or a design exit Mach — and a ratio of specific heats (default γ = 1.4). The page reports the isentropic exit Mach, pe/p0 and Te/T0, the ideal one-dimensional thrust coefficient, and, if you enter a back pressure, whether the nozzle is over-, perfectly, or under-expanded. Optional chamber pressure and throat area give dimensional thrust F = Cf p0 At.

I want to enter

Must be 1 or greater. Ae/At = 1 is a sonic exit (a converging nozzle). For γ = 1.4, Ae/At = 1.6875 is a design Mach of 2.

default 1.4 (air, diatomic)
optional, any unit
optional, same unit as p0
optional
optional, for dimensional thrust

p0, pa, and At may use any consistent unit system. Thrust is then Cf × p0 × At in those units (for example pascals and square metres give newtons). If you enter pa/p0 only, Cf and the expansion regime are still reported.

Exit Mach

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Exit Mach from the area ratio

For isentropic flow of a perfect gas, the area ratio and Mach number are tied by the same NACA Report 1135 identity used on the isentropic-flow calculator:

Ae/At = (1/Me) [(2/(γ+1)) (1 + ((γ−1)/2) Me²)]^{(γ+1)/(2(γ−1))}

A C-D nozzle is designed on the supersonic root. For γ = 1.4, a design Mach of 2 has Ae/At = 1.6875 exactly and pe/p0 ≈ 0.12780. Ae/At = 1 is a sonic exit. The subsonic root at the same area ratio is shown for contrast; it is the exit Mach of a purely converging duct with that area, not of a started C-D nozzle.

Ideal thrust coefficient and expansion

The ideal one-dimensional thrust coefficient (Sutton / NASA) is

Cf = √{(2γ²/(γ−1)) (2/(γ+1))^{(γ+1)/(γ−1)} [1 − (pe/p0)^{(γ−1)/γ}]} + ((pe − pa)/p0) (Ae/At)

The first term is the momentum contribution at the isentropic exit (and equals Cf when the nozzle is perfectly expanded). The second term is the pressure thrust. Then F = Cf p0 At. Vacuum Cf uses pa = 0. Compared with the back pressure: pe > pa is underexpanded, pe = pa is perfectly expanded, and pe < pa is overexpanded. A strongly overexpanded nozzle can have shocks or separation in the diverging section that this ideal model does not include.

Sources

NACA Report 1135 (isentropic-flow tables); NASA Glenn Beginner's Guide (nozzles, area–Mach relation); Sutton and Biblarz, Rocket Propulsion Elements (thrust coefficient); Anderson, Modern Compressible Flow. For Isp and exhaust velocity, see specific impulse; for Δv from Isp, see the rocket equation.

Frequently asked questions

What exit Mach does Ae/At = 1.6875 give for air?

Mach 2, for γ = 1.4, on the supersonic branch. The isentropic exit pressure ratio is pe/p0 ≈ 0.12780.

What is the ideal thrust coefficient at Mach 2, γ = 1.4, perfectly expanded?

About 1.208. The pressure term vanishes when pe = pa, so Cf equals the momentum term of the Sutton formula.

Does this include shocks in an overexpanded nozzle?

No. The page will label the nozzle overexpanded when pa > pe, but the Mach number and Cf are still the isentropic, shock-free values. Separated or shock-containing flow needs a different model.