Nozzle Theory

The ideal rocket

Sutton begins nozzle theory with a deliberately simple model, the ideal rocket: the gas is a perfect gas of fixed composition; the flow is steady, one-dimensional and adiabatic; there is no friction and no heat loss to the walls. It sounds crude, but real engines land within 1–6% of it. That makes it the right place to build intuition — and the right yardstick for the losses real engines suffer.

In the chamber the gas is hot, dense, and almost at rest: its energy is all enthalpy. The nozzle's job is to turn that enthalpy into directed kinetic energy. Energy conservation along the flow gives the exhaust velocity directly:

$\cvel{v_e} = \sqrt{\frac{2\gamma}{\gamma - 1}\,\frac{R\,\cflame{T_c}}{\mathfrak{M}}\left[1 - \left(\frac{\cpress{p_e}}{\cpress{p_c}}\right)^{(\gamma-1)/\gamma}\right]}$

Read it slowly, because it is the most important equation in chemical rocketry. The exhaust velocity rises with the chamber temperature $\cflame{T_c}$, falls with the molecular mass $\mathfrak M$ of the gas, and rises as the nozzle expands the gas to a lower exit pressure $\cpress{p_e}$. Lesson 4 is about the first two. This lesson is about the third.

Why wider goes faster

Squeeze a subsonic flow and it speeds up: the garden-hose intuition. But a gas is compressible. As it speeds up it expands and cools, and its density drops. Which effect wins?

Below Mach 1, speed rises faster than density falls, so each square metre carries more mass: the duct must narrow. Above Mach 1 it is the other way round, and the duct must widen to keep accelerating the gas. In between, at exactly Mach 1, the mass flux per unit area reaches its maximum. That is the throatThroatThe narrowest section of a nozzle. In a working rocket the flow there is exactly sonic (Mach 1), which fixes the mass flow.. Written as a differential relation (Sutton eq. 3–20 in spirit):

$\frac{dA}{A} = \left(M^2 - 1\right)\frac{dv}{v}$

For $M < 1$, area and velocity change in opposite directions; for $M > 1$, in the same direction. A converging–diverging nozzle — de Laval's nozzle, invented for steam turbines in the 1880s — is simply the shape this equation demands.

Choked: the throat sets the flow

Once the throat reaches Mach 1, something remarkable happens. Pressure signals travel at the speed of sound, and at the throat the gas is moving downstream exactly that fast, so no information from downstream can get upstream past it. The chamber can no longer "feel" what happens outside. The mass flow is fixed by the chamber and the throat alone:

$\cfuel{\dot m} = \frac{\cpress{p_c}\,A_t}{c^*}$

This is choked flowChoked flowOnce the throat reaches Mach 1, lowering the downstream pressure no longer increases the mass flow: information cannot travel upstream faster than sound., and it is why rocket engineers talk about chamber pressure and throat area as if they were the engine. The constant of proportionality $c^*$ — the characteristic velocityCharacteristic velocity c*p_c A_t/ṁ: a figure of merit of the combustion alone (the propellant and chamber), independent of the nozzle. ≈ √(RT_c/M)/Γ. — depends only on the combustion gas, not on the nozzle.

The lab walks through every regime a converging–diverging nozzle can be in. Watch the mass-flow plot on the right: it rises as you lower the back pressure, and then, the instant the throat chokes, it stops. Everything that happens afterwards — the normal shock sliding down the nozzle, being pushed out, the plume necking in and then ballooning out — happens downstream of the throat, without the chamber knowing.

Separating the chamber from the nozzle

The thrust can be split neatly into a product of two figures of merit:

$F = C_F\;\cpress{p_c}\,A_t$

The thrust coefficientThrust coefficient C_FF/(p_c A_t): how much the nozzle amplifies the thrust that chamber pressure alone would give on the throat area. Typically 1.3–2. $C_F$ measures how much the nozzle amplifies the thrust that the chamber pressure alone would give over the throat area. It depends only on $\gamma$, the area ratio and the pressures:

$C_F = \sqrt{\frac{2\gamma^2}{\gamma-1}\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{\gamma-1}}\left[1-\left(\frac{\cpress{p_e}}{\cpress{p_c}}\right)^{\frac{\gamma-1}{\gamma}}\right]} + \frac{\cpress{p_e}-\cair{p_a}}{\cpress{p_c}}\,\frac{A_e}{A_t}$

It is typically between 1.3 and 2.0. Combine the two equations and the exhaust velocity factorises beautifully:

$\cvel{c} = C_F \times c^*$

$c^*$ is the chamber's grade; $C_F$ is the nozzle's. Test engineers use this every day: measure chamber pressure and mass flow and you get $c^*$ — if it is 97% of theory, the injector is mixing well. Measure thrust too and you get $C_F$ — if that is low, look at the nozzle.

One bell, a whole atmosphere

A nozzle is designed for one exit pressure, but a launch vehicle climbs through the whole atmosphere. At sea level a large bell is over-expandedOver-expandedExit pressure below ambient: the atmosphere squeezes the jet, shocks form at the lip, and the pressure term of thrust turns negative.: the gas leaves below ambient pressure, the atmosphere squeezes it back through oblique shocks at the lip, and the pressure-thrust term is negative. High up the same bell is under-expandedUnder-expandedExit pressure above ambient: the jet keeps expanding outside the nozzle, where it can no longer push on the walls.: the gas leaves above ambient pressure and keeps expanding outside, where it can no longer push on anything.

Push the bell too far at sea level and the pressure near the wall drops so low that the boundary layer cannot climb back up against the atmosphere: it separatesFlow separationIn a strongly over-expanded nozzle the boundary layer cannot climb the adverse pressure and peels off the wall, causing side loads.. A common rule, Summerfield's criterion, puts this at exit pressures below roughly a third of ambient. Separation is not just inefficient: it is usually asymmetric and unsteady, and the side loads it produces have broken nozzles and gimbal actuators in testing.

This is why first-stage engines have modest bells (area ratios of 15–40) and upper-stage engines enormous ones (80–280): the RL10B-2 carries a nozzle extension that unfolds only after staging.

Real nozzles

The ideal rocket is an idealization. Sutton lists the ways real nozzles fall short, each costing a percent or two:

• Divergence loss. The flow leaves a conical nozzle along the wall angle, not straight back. For a cone of half-angle $\alpha$ the thrust is multiplied by $\lambda = \tfrac12(1 + \cos\alpha)$: 0.983 for the classic 15° cone. Bell (contoured) nozzles turn the flow back towards the axis, recovering most of this in a shorter length.
• Boundary layers on the walls slow the gas and reduce the effective throat area.
• Chemistry lag. The gas may not recombine fast enough as it cools (Lesson 4).
• Particles. Aluminium oxide droplets in solid-motor exhaust lag the gas (Lesson 9).

Together these typically cost 2–6% of ideal Isp, and every engine is characterized by a measured correction factor.

Next: once the gas leaves the nozzle, what does its velocity buy? The rocket equation, gravity losses, and why every rocket to orbit is mostly propellant.