Nozzles and thrust

Draft

A combustion chamber produces hot high-pressure gas moving in every direction. The nozzle is what turns that into directed momentum, and its geometry sets a large fraction of the engine’s performance.

The thrust equation

Thrust equation

F = \dot{m}\,v_e + (p_e - p_a)A_e

The first term is momentum thrust, mass flow times exit velocity. The second is pressure thrust, arising because the exhaust leaves at pressure $p_e$ into an ambient pressure $p_a$ , and the imbalance acts over the exit area $A_e$ .

Pressure thrust is why the same engine produces different thrust at sea level and in vacuum. As $p_a$ falls to zero, the second term becomes $p_eA_e$ and is purely additive. An engine typically gains 10 to 25% thrust between sea level and vacuum for no change in propellant flow, which is also where the difference between sea-level and vacuum specific impulse comes from.

The effective exhaust velocity used in the rocket equation absorbs both terms:

v_{\text{eff}} = v_e + \frac{(p_e - p_a)A_e}{\dot{m}}, \qquad I_{sp} = \frac{v_{\text{eff}}}{g_0}

Choking and the converging-diverging shape

Gas accelerating through a converging passage reaches Mach 1 at the minimum area, the throat, and cannot exceed it there. Once sonic, further acceleration requires the passage to diverge, because in supersonic flow increasing area accelerates rather than decelerates the flow. This reversal is why every rocket nozzle has the same converging-diverging shape.

The throat is also what sets mass flow. Once the flow is choked, mass flow depends only on chamber conditions and throat area, and is completely independent of what happens downstream:

\dot{m} = \frac{p_cA_t}{c^*}, \qquad c^* = \frac{\sqrt{\gamma R_uT_c/M}}{\gamma\sqrt{\left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)}}}

The characteristic velocity $c^*$ depends only on the propellant chemistry and chamber temperature, so it measures how well the injector and chamber are doing their job. The thrust coefficient $C_F = F/(p_cA_t)$ measures how well the nozzle is doing its job. Splitting engine performance into these two independent factors is standard practice, because a test can then attribute a shortfall to the chamber or the nozzle rather than to the engine as a whole.

Expansion ratio

\varepsilon = \frac{A_e}{A_t}

A larger expansion ratio extracts more of the thermal energy as directed velocity, so $v_e$ rises and $p_e$ falls. The optimum expansion condition is

p_e = p_a

at which the pressure thrust term vanishes and thrust is maximised for the given chamber conditions. Any other exit pressure wastes performance.

The difficulty is that $p_a$ changes by five orders of magnitude during ascent, while $\varepsilon$ is fixed by the metal. A nozzle can be optimum at exactly one altitude.

Condition	Meaning	Consequence
$p_e = p_a$	Optimum expansion	Maximum thrust for the chamber conditions
$p_e > p_a$	Under-expanded	Exhaust continues expanding outside; performance left on the table. Normal for a sea-level nozzle in vacuum
$p_e < p_a$	Over-expanded	Ambient pressure compresses the plume; the visible shock diamonds
$p_e \lesssim 0.4\,p_a$	Flow separation	The boundary layer detaches from the wall, unsteadily and asymmetrically

Flow separation is the hard limit, and it is a structural problem rather than a performance one. Separation is not symmetric or steady; it moves around the nozzle wall, producing large lateral side loads that can destroy the nozzle extension or the gimbal. The Summerfield criterion, separation at roughly $p_e < 0.4\,p_a$ , is the practical bound on how large a sea-level nozzle can be.

This single constraint explains why engines come in two versions:

Engine	Role	Expansion ratio	Vacuum $I_{sp}$
Merlin 1D	Sea level	16	311 s
Merlin 1D Vacuum	Upper stage	165	348 s
RS-25	Sea level start	69	452 s
RL10B-2	Upper stage	280	465.5 s

Same propellants, same chemistry, radically different nozzles. The RL10B-2’s expansion ratio of 280 is only possible because it never operates in atmosphere, and its 465.5 s remains the highest specific impulse of any production engine flown.

Contour

A simple conical nozzle with a 15° half-angle loses about 1.7% of thrust to divergence, because exhaust leaving at an angle contributes only its axial component. It is easy to manufacture and still used where simplicity dominates.

A bell contour turns the flow sharply just after the throat and then straightens it toward axial at the exit, recovering most of the divergence loss in roughly 80% of the length of an equivalent cone. The standard method for generating the contour is Rao’s, which solves for the shape that maximises thrust for a given length. Nearly all modern engines use a Rao-optimised bell.

Altitude compensation

A nozzle that adapted its expansion to ambient pressure would recover the performance lost at every altitude except the design point. Several approaches exist, and the honest summary is that almost none of them have flown.

Extendible nozzle: A carbon-carbon extension deploys after staging, raising the expansion ratio once in vacuum. Flown successfully on the RL10B-2. The only altitude-compensating approach in routine operational use, and it compensates in one discrete step.
Aerospike: The plume expands against a shaped centrebody with ambient pressure forming its outer boundary, so it self-adjusts continuously. Tested as the XRS-2200 for X-33 but never flown; cooling the large centrebody surface is the persistent difficulty.
Dual-bell: A contour with a deliberate inflection so the flow attaches to a short bell at sea level and jumps to a longer one at altitude. Extensively studied, not flown.
Expansion-deflection: A plug in the throat region redirects flow outward, with the free boundary adjusting to ambient. Studied since the 1960s, not flown.

The reason altitude compensation keeps being proposed and keeps not flying is that the gain, typically a few per cent of mission-averaged specific impulse, is genuinely worth having but is not large enough to justify a new and thermally difficult nozzle when the alternative, simply using two different engines on two stages, already works. Staging is itself a form of altitude compensation, and it is one the industry has fifty years of experience with.

References

Sutton, G. P. and Biblarz, O. Rocket Propulsion Elements, 9th ed., Wiley, 2017, chapter 3.
Rao, G. V. R. “Exhaust Nozzle Contour for Optimum Thrust”, Jet Propulsion, 28:377, 1958.
Summerfield, M., Foster, C. R. and Swan, W. C. “Flow Separation in Overexpanded Supersonic Exhaust Nozzles”, Jet Propulsion, 24:319, 1954.
Hagemann, G., Immich, H., Nguyen, T. V. and Dumnov, G. E. “Advanced Rocket Nozzles”, Journal of Propulsion and Power, 14:620, 1998. A survey of altitude-compensating concepts and why they remain unflown.