Nozzle Design and Thrust — theory

Overview

A rocket nozzle is a de Laval, or converging–diverging, duct. Hot gas leaves the chamber at stagnation pressure p_c and temperature T_c, reaches sound speed at the throat and keeps accelerating in the diverging section. The thrust is written as

where A_t is the throat area and C_F the thrust coefficient, which depends only on the gas, the nozzle shape and p_a/p_c. The calculator answers three design questions:

• What shape? It draws the wall from the chamber to the exit — a cone, or the parabolic approximation of Rao’s thrust-optimised bell (Contour).
• What happens inside? It solves the one-dimensional flow along that wall: area ratio, Mach number, pressure, temperature and density (Isentropic flow), with a normal shock if the ambient pressure forces one (Shocks).
• Is it safe at this ambient pressure? It checks whether the wall boundary layer separates (Separation) and gives the verdict, the thrust, the mass flow and the specific impulse (Performance).

Assumptions: a perfect gas with constant (frozen) \gamma and molar mass, steady quasi-one-dimensional flow (each cross-section uniform), no friction or heat loss, and the chamber pressure taken as the stagnation pressure.

One-dimensional isentropic flow

Between the chamber and any section without a shock, the flow is isentropic and every ratio to the chamber state is a function of the local Mach number alone (NACA 1135):

and the Mach number follows from the local area ratio

Each area ratio has a subsonic and a supersonic root. Upstream of the throat the calculator takes the subsonic one, downstream the supersonic one (both by bisection to machine precision). The Nozzle and the flow along it plot draws the nozzle and, below it on the same axial scale (zero at the throat), the area, Mach number, pressure, temperature and density; the throat, shock and separation lines run through every panel. Hovering gives each value with its unit and its ratio to the sonic (critical) value at the throat, A/A^*, p/p^*, T/T^*, \rho/\rho^*.

At the throat M=1 and p^*=p_c\,(2/(\gamma+1))^{\gamma/(\gamma-1)}, about 0.56 p_c for \gamma=1.2.

The gas entering the convergent moves at the subsonic Mach number of the inlet-to-throat area ratio. If the inlet is narrow, this is no longer small, p_c is no longer the stagnation pressure, and the calculator warns (above about Mach 0.3, a rule of thumb).

Nozzle contour: cone or bell

You enter the nozzle as a machinist would draw it: inlet, throat and exit diameters d_{in}, d_t, d_e, and the convergent and divergent lengths L_c, L_d. The expansion ratio is \varepsilon=(d_e/d_t)^2. The angles below are solved from the lengths by bisection. A length that no angle in the accepted range can produce is refused, and the message gives the achievable range: convergent 10–80°, cone 3–45°, bell 50–120 % of a 15° cone.

The convergent carries slow gas, so its shape costs little (Huzel & Huang): a cone of half angle \theta_c (usually 20–45°) runs into a circular arc of radius 1.5\,r_t that meets the throat, so L_c=1.5\,r_t\sin\theta_c+\frac{r_{in}-r_t-1.5\,r_t(1-\cos\theta_c)}{\tan\theta_c}. Downstream, a second arc of radius R_2=0.382\,r_t starts the expansion.

Cone. After the arc, a straight wall at half angle \alpha (usually 12–18°) runs to the exit radius r_e=r_t\sqrt\varepsilon. Its length is Huzel & Huang’s eq. (4-7):

The gas leaves along rays from the cone’s apex, so only part of its momentum is axial. The thrust efficiency is Malina’s factor, as given by Huzel & Huang eq. (4-8): \lambda=(1+\cos\alpha)/2, 0.983 for 15°.

Bell (Rao, parabolic approximation). A bell expands the gas fast just after the throat and then turns it back towards the axis, so the exit flow is nearly parallel in a shorter length. Huzel & Huang (fig. 4-13) approximate Rao’s optimum contour by the 0.382\,r_t arc up to an angle \theta_n, then a parabola tangent to it that reaches the exit at angle \theta_e. Its length is a fraction L_f of a 15° cone with the same throat, arc and expansion ratio, and the angles come from their fig. 4-14:

These values were digitised from the scanned chart in this project (curves read to about ±0.2°). Between grid points the calculator interpolates linearly in L_f and in \ln\varepsilon. Outside the chart (ε 5–50, L_f 60–100 %) it holds the edge values and warns. The parabola is drawn as the quadratic Bézier curve through the two end points and the intersection of their tangents.

The bell’s thrust efficiency comes from Huzel & Huang fig. 4-12: about 97 % at 60 % length, 98.6–98.9 % at 80 % and 99.15 % at full length, against 98.3 % for a full 15° cone. Beyond about 80 % a longer bell gains little, which is why 80 % is common. The Divergence efficiency against nozzle length plot redraws that figure for your expansion ratio.

Thrust coefficient and ideal expansion

The first term is the jet momentum, reduced by the divergence efficiency \lambda; the second is the pressure difference on the exit area. With p_e>p_a the nozzle is under-expanded (a longer nozzle would add thrust); with p_e<p_a it is over-expanded. The one-dimensional value peaks at p_e=p_a, and the calculator reports that expansion ratio. A nozzle sized for sea level wastes performance at altitude, and one sized for altitude is over-expanded at launch.

Shocks inside the nozzle

With the throat sonic, the exit pressure of the supersonic solution, p_e, does not depend on p_a. As the ambient pressure rises past it, the ideal (inviscid) flow passes through these regimes:

1. p_a<p_e: under-expanded; expansion waves outside.
2. p_e<p_a<p_{a,2}: over-expanded; oblique shocks outside, the flow inside unchanged. Here p_{a,2}=p_e\,\frac{2\gamma M_e^2-(\gamma-1)}{\gamma+1} is the pressure behind a normal shock standing at the exit (NACA 1135 eq. 93).
3. p_{a,2}<p_a<p_{a,1}: a normal shock inside the divergent. Across it the Mach number drops to M_2^2=\frac{(\gamma-1)M_1^2+2}{2\gamma M_1^2-(\gamma-1)} (eq. 96) and the stagnation pressure to p_{c2}/p_c from eq. (99). Behind it the flow is subsonic and decelerates to the exit. Because the stagnation pressure fell, the sonic area grows to A_2^*=A_t\,p_c/p_{c2}. The calculator moves the shock (bisection on its area ratio) until the subsonic exit pressure equals p_a.
4. p_a>p_{a,1}: p_{a,1} is the exit pressure of the fully subsonic solution. Above it the throat is no longer sonic and the nozzle is a venturi; the calculator stops and asks for a higher chamber pressure.

The flow plot marks the inviscid shock with a vertical line through every panel. The Mach number, static pressure, static temperature and density jump there; the area does not, and neither does the stagnation temperature (the flow is adiabatic), only the stagnation pressure. A test checks that the mass flux \rho VA is the same at every station, through the shock.

A real nozzle rarely reaches this regime. The boundary layer separates first, well upstream (next section).

Flow separation

In an over-expanded nozzle the pressure rise at the exit travels upstream through the slow wall boundary layer. When the wall pressure has fallen to a fraction of p_a, the boundary layer separates and the jet leaves the wall. Separation is uneven around the circumference and moves about: it causes side loads on the nozzle and the motor mount, and the thrust differs from the attached value. Schmucker (NASA TM-77396) reviewed rocket-engine separation data and the criteria that fit it:

• Summerfield, eq. (7): p_i/p_a=0.4, a guide for low pressure ratios (p_c/p_a about 15–20) that decreases at higher ones;
• Kalt and Bendall, eq. (10): p_i/p_a=0.667\,(p_c/p_a)^{-0.2};
• Schmucker, eq. (11): p_i/p_a=(1.88\,M_i-1)^{-0.64}, where M_i is the wall Mach number where separation begins. Fitted to averaged chemical-engine data, which lie near M_i 3–4.6; not to be used above M_i=5.

Schmucker found the criterion lowered by a higher Mach number at separation, and little affected by divergence angle, contour (bell or plug), wall temperature or propellant. The calculator uses his eq. (11) by default. It walks down the attached wall-pressure distribution and reports the first station where p_{wall}\le p_i. The Wall pressure against the separation pressure plot shows the crossing.

Verdict. The calculator compares the attached exit pressure with the separation pressure at the exit Mach number:

The 1.15 band is this project’s choice (DECISIONS D21), sized to the scatter between Schmucker’s criteria and data. It is not a published limit. When the design is not safe, the calculator gives the largest expansion ratio and the lowest chamber pressure that restore the margin.

Thrust after separation. Downstream of separation the wall pressure is close to p_a, so the nozzle acts roughly like one cut at the separation point. The calculator reports that estimate and marks it as such.

Mass flow, exhaust velocity and specific impulse

With the gas constant R=R_u/\mathcal{M}, the choked mass flow and the characteristic velocity follow from the isentropic relations at the sonic throat:

These are ideal values. Two-phase flow (particles in aluminised or sugar propellants), friction, heat loss and incomplete combustion lower them by several percent in real motors. Take \gamma, T_c and \mathcal{M} for your propellant from a thermochemistry code (NASA CEA, ProPEP, OpenMotor). The calculator’s defaults are round example values, not a particular propellant.

Verification

tests/nozzle.test.js checks the following:

• every value of the RS&T THRUSTCF example (\gamma=1.4, \varepsilon=6, 15°, 600 psia into 14.7 psia): M_e=3.3679, C_F=1.4316, also through the calculator as a cone;
• Huzel & Huang’s worked example (80 % bell, \varepsilon=14, r_t = 12.45 in): 15° cone length 128 in, L_n = 102.4 in, \theta_n = 27.4°, \theta_e = 9.8° (the chart gives 27.5° and 9.9°), point N at (2.19, 12.99) in;
• normal shock at M_1=2: M_2=0.5774, p_2/p_1=4.5, p_{c2}/p_c=0.7209 (NACA 1135 table II);
• Schmucker eq. (11) at M_i=3 (0.374, as in his fig. 25);
• with a shock inside, the exit pressure equals p_a and the mass flux is conserved at every station; separation lies upstream of the inviscid shock;
• the suggested expansion ratio and chamber pressure restore the 1.15 margin exactly, and \dot m\,c^*=p_cA_t;
• area and pressure ratios against the NACA 1135 tables, and the sonic and infinite-expansion limits of C_F.

Notes on the method

• Separation criterion (changed). The previous version used p_e<0.4\,p_a (Summerfield) at the exit only. The default is now Schmucker’s Mach-number criterion along the wall, which agrees better with rocket data at high pressure ratios (Schmucker). Summerfield and Kalt–Bendall remain selectable.
• Choking limit (corrected). The previous version refused any ambient pressure above the throat pressure p^*. A converging–diverging nozzle stays choked up to the subsonic-exit pressure p_{a,1}, which is higher. The range in between (normal shock inside) is now solved.
• Bell efficiency is read from Huzel & Huang’s chart, not computed by the method of characteristics.
• Exit Mach number. It is solved to machine precision. The original example’s 50-step iteration is 6 × 10⁻¹⁰ high, and the thrust coefficients agree to 10⁻¹¹.

References

1. Ames Research Staff, “Equations, Tables, and Charts for Compressible Flow,” NACA Report 1135, 1953. NTRS
2. D. K. Huzel and D. H. Huang, Design of Liquid Propellant Rocket Engines, NASA SP-125, 2nd ed., 1971, §4.3. NTRS
3. R. H. Schmucker, “Flow processes in overexpanded chemical rocket nozzles, Part 1: Flow separation,” NASA TM-77396, 1984. NTRS
4. F. J. Malina, “Characteristics of the Rocket Motor Unit Based on the Theory of Perfect Gases,” Journal of the Franklin Institute 230, 1940 — cited through Huzel & Huang eq. (4-8); not read directly.
5. C. P. Hoult, nozzle thrust coefficient calculation (THRUSTCF), Rocket Science and Technology. Source file