Compressible Flow & Nozzle Designer

De Laval Nozzle — Contour & Flow Solution

Choose chamber conditions and expansion ratio ε=A_e/A*. The supersonic area–Mach root sets exit Mach; contour and Mach/p/T distributions are drawn along the axis.

Working fluid

R (J/kg·K)

Chamber p₀ (bar)

Chamber T₀ (K)

Expansion ratio ε

Throat radius (mm)

Ambient p_a (bar) — 0 for vacuum

━ Mach ━ p/p₀ ━ T/T₀ shaded = nozzle wall

Isentropic Relations

Stagnation ratios & static state at a given Mach.

Mach M

Speed of Sound

a = √(γRT)

R (J/kg·K)

T (K)

Flow velocity V (m/s) — for M

Choked Mass Flow

ṁ = p₀A*/√T₀ · √[γ/R·(2/(γ+1))^((γ+1)/(γ−1))]

p₀ (bar)

T₀ (K)

R (J/kg·K)

Throat r (mm)

Thrust → Throat Sizing

Size A* & throat radius for a target vacuum thrust.

Target thrust (kN)

p₀ (bar)

T₀ (K)

R (J/kg·K)

Normal Shock Jump

Property ratios across a stationary normal shock.

Upstream M₁ (>1)

Formula reference

a=√(γRT), M=V/a; T₀/T=1+(γ−1)/2·M², p₀/p=(…)^(γ/(γ−1)).

Area–Mach A/A*=(1/M)[(2/(γ+1))(1+(γ−1)/2 M²)]^((γ+1)/(2(γ−1))); choked ṁ=p₀A*/√T₀·√[γ/R(2/(γ+1))^((γ+1)/(γ−1))].

Exit velocity V_e=√(2γ/(γ−1)·RT₀[1−(p_e/p₀)^((γ−1)/γ)]); thrust F=ṁV_e+(p_e−p_a)A_e.

Normal shock M₂²=(1+(γ−1)/2 M₁²)/(γM₁²−(γ−1)/2), p₂/p₁=1+2γ/(γ+1)(M₁²−1).

Note: 1-D isentropic results are an upper bound; real nozzles lose a few % to friction, divergence & shocks.