De Laval Nozzle Designer
Design a convergent-divergent rocket nozzle from a target exit Mach number or expansion ratio, with full isentropic flow output and propellant presets.
🚀 What is a De Laval Nozzle?
A De Laval nozzle (also called a convergent-divergent or CD nozzle) is a shaped duct that converts the thermal energy of high-pressure combustion gases into directed kinetic energy by first converging to a narrow throat, then diverging to a wider exit. Flow enters the converging section subsonically, reaches exactly Mach 1 at the throat (the minimum cross-section), and then accelerates to supersonic speeds in the diverging section. The pressure drops continuously from chamber to exit, converting enthalpy into velocity. This mechanism underlies virtually every liquid and solid rocket engine in use today.
De Laval nozzles are used in three main contexts. First, in launch vehicle main engines: Merlin, Raptor, RL-10, RS-25, and Vulcain all use converging-diverging nozzles to achieve exit Mach numbers of 3 to 6 and Isp values of 280 to 460 s. Second, in attitude control systems (ACS) and reaction control systems (RCS) on spacecraft and upper stages, where small cold-gas or monopropellant thruster nozzles with exit Mach numbers of 1.5 to 2.5 provide fine attitude control. Third, in wind tunnels and supersonic test facilities that generate calibrated supersonic flow for aerodynamic testing.
A common misconception is that a longer nozzle always produces more thrust. In reality, the nozzle must be matched to the ambient pressure. A nozzle optimized for vacuum (large expansion ratio, low exit pressure) becomes overexpanded at sea level, and ambient pressure forces oblique shocks back into the nozzle, reducing performance. Conversely, a sea-level nozzle (small expansion ratio) is underexpanded in vacuum, leaving kinetic energy in the exhaust plume that could have been extracted as thrust. The designer always targets Pe = Pa for the operating altitude.
This calculator implements the isentropic one-dimensional flow model used for preliminary nozzle design. Given propellant properties (gamma, Mw, Tc), chamber conditions (Pc, Tc), and either a target exit Mach or expansion ratio, it computes all exit-plane quantities, the characteristic velocity c*, the thrust coefficient Cf, and specific impulse at any altitude. The two modes let designers work either from a Mach number target (useful when sizing an altitude-compensating nozzle) or from a geometric constraint (useful when analyzing an existing nozzle or comparing designs).
📐 Formula
A/A* = (1/M) × [(2/(γ+1)) × (1 + (γ-1)/2 × M²)](γ+1)/(2(γ-1))
A = cross-section area at any station (m²)
A* = throat area at M = 1 (m²); A* = π(d*/2)²
M = local Mach number
γ = specific heat ratio of combustion gases (dimensionless)
Isentropic exit properties (given Me):
Te = Tc / (1 + (γ-1)/2 × Me²) — exit temperature
Pe = Pc × (Te/Tc)γ/(γ-1) — exit pressure
Ve = Me × sqrt(γ × R × Te) — exit velocity; R = Ru/Mw
c* = sqrt(R × Tc) / Γ — characteristic velocity
Γ = sqrt(γ) × (2/(γ+1))(γ+1)/(2(γ-1))
Cf_vac = Ve/c* + Pe × ε / Pc — vacuum thrust coefficient
Isp = Cf × c* / g0 — g0 = 9.80665 m/s²
Example: LOX/RP-1 at Me = 3, Pc = 7 MPa, d* = 100 mm gives ε = 6.174, Ve = 2710 m/s, Isp_vac = 301 s
📖 How to Use This Calculator
Steps
1
Select a propellant preset - Choose a propellant from the dropdown to auto-fill gamma, molecular weight, and chamber temperature. Select Custom to enter your own combustion gas properties.
2
Set chamber conditions and throat diameter - Enter chamber pressure in MPa and throat diameter in mm. These set the mass flow rate and absolute sizing of the nozzle. Chamber temperature is auto-filled from the preset.
3
Choose Mach Design or Expansion Ratio mode - In Mach Design mode, set your target exit Mach number using the slider and the calculator computes the required expansion ratio. In Expansion Ratio mode, enter an existing ratio and the Newton-Raphson solver finds the exit Mach number.
4
Set ambient pressure for altitude-specific Isp - Set ambient pressure to 101.325 kPa for sea level, 0 kPa for vacuum, or any value for a specific altitude. The calculator shows both vacuum Isp and Isp at the given ambient pressure.
💡 Example Calculations
Example 1 - LOX/RP-1 Sea-Level Nozzle at Me = 3
LOX/RP-1: gamma = 1.23, Mw = 22 g/mol, Tc = 3571 K, Pc = 7.0 MPa, d* = 100 mm, Pa = 101.325 kPa
1
Expansion ratio: A/A* = (1/3) x [(2/2.23) x (1 + 0.115 x 9)]^4.848 = (1/3) x 18.51 = 6.174
2
Exit temperature: Te = 3571 / (1 + 0.115 x 9) = 3571 / 2.035 = 1754 K
3
Exit pressure: Pe = 7.0 x (2.035)^(-5.348) = 7.0 x 0.02246 = 157.2 kPa
4
Exit velocity: R = 8314/22 = 377.9 J/(kg K). Ve = 3 x sqrt(1.23 x 377.9 x 1754) = 3 x 903 = 2710 m/s
5
Isp: c* = 1773 m/s, CfVac = 2710/1773 + 157200 x 6.174 / 7e6 = 1.528 + 0.139 = 1.667. Isp_vac = 1.667 x 1773 / 9.807 = 301 s; Isp_sl = 285 s (Pe > Pa, nozzle underexpanded)
Result: ε = 6.174, de = 248 mm, Ve = 2710 m/s, Isp_vac = 301 s
Try this example →Example 2 - LOX/LH2 Vacuum Upper Stage at Me = 4
LOX/LH2: gamma = 1.22, Mw = 10 g/mol, Tc = 3600 K, Pc = 6.0 MPa, d* = 80 mm, Pa = 0 (vacuum)
1
t = 1 + (1.22-1)/2 x 16 = 2.76. Expansion ratio: (1/4) x [(2/2.22) x 2.76]^5.045 = (1/4) x 99.3 = 24.82
2
Te = 3600 / 2.76 = 1304 K. Pe = 6.0 x (2.76)^(-5.545) = 6.0 x 0.003596 = 21.6 kPa
3
R = 8314/10 = 831.4. Ve = 4 x sqrt(1.22 x 831.4 x 1304) = 4 x 1150 = 4599 m/s
4
c* = 2650 m/s. CfVac = 4599/2650 + 21600 x 24.82 / 6e6 = 1.736 + 0.089 = 1.825. Isp_vac = 1.825 x 2650 / 9.807 = 493 s
Result: ε = 24.82, de = 399 mm, Ve = 4599 m/s, Isp_vac = 493 s
Try this example →Example 3 - Analyze an Existing Nozzle with Expansion Ratio 8
Expansion Ratio mode: epsilon = 8.0, LOX/RP-1, Pc = 7.0 MPa, d* = 100 mm, Pa = 101.325 kPa
1
Newton-Raphson solver finds Me = 3.197 such that A/A*(3.197, 1.23) = 8.000 (converges in fewer than 10 iterations)
2
Te = 3571 / 2.175 = 1641 K. Pe = 7.0 x (2.175)^(-5.348) = 7.0 x 0.01566 = 109.6 kPa
3
Ve = 3.197 x sqrt(1.23 x 377.9 x 1641) = 3.197 x 873 = 2792 m/s. c* = 1773 m/s
4
Pe (109.6 kPa) is very close to Pa (101.325 kPa), so this nozzle is nearly perfectly expanded at sea level. Isp_sl = 286 s, Isp_vac = 307 s
Result: M⊂e; = 3.197, de = 283 mm, Pe = 109.6 kPa, Isp_sl = 286 s
Try this example →Example 4 - Cold Gas Nitrogen Thruster
Cold N2: gamma = 1.40, Mw = 28 g/mol, Tc = 300 K, Pc = 0.5 MPa, d* = 5 mm, Pa = 0
1
At Me = 2.0: t = 1 + 0.2 x 4 = 1.8. Epsilon = (1/2) x [(2/2.4) x 1.8]^3.0 = 0.5 x [1.5]^3 = 0.5 x 3.375 = 1.688
2
R = 8314/28 = 297.0. Te = 300/1.8 = 166.7 K. Ve = 2 x sqrt(1.40 x 297 x 166.7) = 2 x 263 = 526 m/s
3
c* = sqrt(297 x 300) / 0.6847 = 298 / 0.6847 = 435 m/s (gamma = 1.40 gives Gamma_factor = 0.6847)
4
Isp_vac = (526/435 + 27800 x 1.688 / 500000) x 435 / 9.807 = (1.209 + 0.094) x 44.35 = 57.8 s
Result: ε = 1.688, Ve = 526 m/s, Isp_vac = 58 s
Try this example →❓ Frequently Asked Questions
What is a De Laval nozzle and how does it produce supersonic flow?
A De Laval nozzle is a converging-diverging duct that accelerates combustion gases from subsonic to supersonic speed. Gas enters the converging section at low Mach number, reaches Mach 1 exactly at the throat (the minimum cross-section), then continues accelerating in the diverging section as the area increases. The thermodynamic reason is that above Mach 1, increasing area causes increasing velocity rather than decreasing velocity (as in subsonic flow). The area-Mach relation A/A* = f(M, gamma) governs the geometry at every cross-section.
What is expansion ratio and how do I pick the right value?
Expansion ratio epsilon = Ae/A* is the ratio of exit area to throat area. The correct value depends on the operating altitude. For perfect expansion, set exit pressure Pe equal to ambient pressure Pa. Use the Mach Design mode and adjust Me until Pe shown equals the ambient pressure at your target altitude (101.325 kPa at sea level, 0 kPa for vacuum). Typical values are epsilon = 8 to 20 for sea-level first stages and epsilon = 40 to 200 for vacuum upper stages.
Why is vacuum Isp always higher than sea-level Isp?
Vacuum Isp = CfVac x c* / g0 uses Cf_vac = Ve/c* + Pe x epsilon / Pc. Sea-level Isp uses Cf_sl = Ve/c* + (Pe - Pa) x epsilon / Pc. The difference is Pa x epsilon / Pc. Because Pa > 0 at sea level, Cf_sl is always lower than CfVac, and so is the sea-level Isp. For Falcon 9 Merlin (epsilon = 16, Pc = 9.7 MPa), the difference is about 29 s. For an upper stage with epsilon = 80, the difference exceeds 100 s.
What happens when a nozzle is overexpanded or underexpanded?
When exit pressure Pe is less than ambient Pa (overexpanded), ambient air forces oblique shocks back into the nozzle exit, reducing thrust and potentially causing flow separation inside the nozzle. This is the typical condition for a sea-level nozzle at low altitude or for an engine with too large an expansion ratio. When Pe is greater than Pa (underexpanded), the exhaust plume expands further after the nozzle exit in expansion fans, converting pressure into velocity outside the nozzle. Both conditions produce less thrust than a perfectly expanded nozzle at that altitude.
What is characteristic velocity c* and what does it tell me?
Characteristic velocity c* = sqrt(R x Tc) / Gamma_factor, where Gamma_factor = sqrt(gamma) x (2/(gamma+1))^((gamma+1)/(2(gamma-1))). c* depends only on propellant chemistry (gamma, Mw, Tc), not on nozzle geometry. It measures the thermodynamic potential of the propellant to produce thrust. High c* requires high Tc and low Mw. LOX/LH2 achieves c* = 2650 m/s because liquid hydrogen has Mw = 10 g/mol. LOX/RP-1 reaches c* = 1773 m/s. Cold nitrogen gives only c* = 435 m/s.
How is exit velocity computed from chamber conditions?
Ve = Me x sqrt(gamma x R x Te), where Te = Tc / (1 + (gamma-1)/2 x Me^2) is the isentropic exit temperature and R = Ru/Mw = 8314/Mw is the specific gas constant. Equivalently, Ve = sqrt(2 x gamma/(gamma-1) x R x Tc x (1 - (Pe/Pc)^((gamma-1)/gamma))). Both forms give the same answer. Higher Tc and lower Mw directly increase Ve and therefore Isp, which is why LOX/LH2 with its very low Mw produces the highest chemical Isp.
What exit Mach numbers do real rocket engines use?
Most liquid-propellant rocket engines operate between Me = 2.5 and 5. The Merlin 1D sea-level engine uses epsilon = 16, corresponding to Me = 3.3 for LOX/RP-1. The RL-10B-2 upper stage uses epsilon = 250, corresponding to Me = 6.5 approximately. The Raptor LOX/methane engine uses epsilon = 40 for the sea-level variant and 80 for the vacuum version. Solid rocket boosters typically use Me = 3 to 4. Cold gas thrusters use Me = 1.5 to 2.5.
Why does the area ratio increase so rapidly with Mach number?
The area-Mach relation is nonlinear and grows rapidly because at high Mach numbers, the gas density drops sharply (from isentropic expansion) while the velocity increases only moderately. Mass continuity requires rho x A x V = constant, so A must increase rapidly as rho falls. For gamma = 1.23, going from Me = 3 to 4 increases epsilon from 6.2 to 18.5. Going from 4 to 6 jumps to 75. This rapid growth explains why very high expansion ratio nozzles have large, heavy bell structures that are practical only for upper stages in vacuum.
What is the thrust coefficient Cf and what values indicate a well-designed nozzle?
Thrust coefficient Cf = F / (Pc x A*) is a dimensionless measure of how effectively the nozzle converts chamber pressure into thrust per unit throat area. Vacuum Cf values for well-designed rocket nozzles typically range from 1.6 to 2.0. Higher expansion ratios and lower gamma produce higher Cf in vacuum. At sea level with perfect expansion, Cf is typically 1.5 to 1.7. The maximum achievable Cf in vacuum for infinitely large expansion is about 2.2 for gamma = 1.2. Values below 1.5 indicate either poor nozzle design or severe overexpansion.
How does gamma (specific heat ratio) affect nozzle performance?
Lower gamma generally produces higher Isp for the same Tc and Mw. With lower gamma, the area-Mach curve rises more slowly, meaning a given exit Mach requires a larger expansion ratio. The isentropic temperature drop is also gentler, leaving more kinetic energy in the flow at any given Mach number. LOX/CH4 combustion products at gamma = 1.19 outperform LOX/RP-1 at gamma = 1.23 partly for this reason. Air with gamma = 1.40 and cold nitrogen with gamma = 1.40 produce the lowest nozzle performance among common gases.
What is the Newton-Raphson solver used for in the Expansion Ratio mode?
In Expansion Ratio mode, the area-Mach relation A/A* = f(M, gamma) must be inverted: given epsilon = A/A*, find the supersonic Mach number M. This cannot be done in closed form, so the calculator uses Newton-Raphson iteration. Starting from an initial guess M = 1 + 1.2 x (epsilon-1), it applies M_new = M - f(M)/f'(M) repeatedly until convergence (typically 5 to 10 iterations). The derivative f'(M) is computed numerically. The algorithm converges for all expansion ratios between 1 and several thousand.
Can I use this calculator for wind tunnel nozzle design?
Yes, with some caveats. Wind tunnels use the same isentropic flow relations, so the area-Mach formula and all derived quantities are directly applicable. Set Tc to the stagnation temperature of your test gas (room temperature for a blowdown tunnel), set Pc to the settling chamber pressure, and select gamma = 1.40 for air. The calculator gives the correct throat area for a desired test section Mach number and mass flow rate. For hypersonic tunnels (Me greater than 5), real gas effects (vibrational excitation, dissociation) begin to deviate from the perfect gas model used here.