A rocket engine nozzle is a propelling nozzle (usually of the de Laval type) used in a rocket engine to expand and accelerate combustion products to high supersonic velocities.

Simply: propellants pressurized by either pumps or high pressure ullage gas to anywhere between two and several hundred atmospheres are injected into a combustion chamber to burn, and the combustion chamber leads into a nozzle which converts the energy contained in high pressure, high temperature combustion products into kinetic energy by accelerating the gas to high velocity and near-ambient pressure.

History

Simple bell-shaped nozzles were developed in the 1500s. The de Laval nozzle was originally developed in the 19th century by Gustaf de Laval for use in steam turbines. It was first used in an early rocket engine developed by Robert Goddard, one of the fathers of modern rocketry. It has since been used in almost all rocket engines, including Walter Thiel's implementation, which made possible Germany's V-2 rocket.

Atmospheric use

The optimal size of a rocket engine nozzle is achieved when the exit pressure equals ambient (atmospheric) pressure, which decreases with increasing altitude. The reason for this is as follows: using a quasi-one-dimensional approximation of the flow, if ambient pressure is higher than the exit pressure, it decreases the net thrust produced by the rocket, which can be seen through a force-balance analysis. If ambient pressure is lower, while the force balance indicates that the thrust will increase, the isentropic Mach relations show that the area ratio of the nozzle could have been greater, which would result in a higher exit velocity of the propellant, increasing thrust. For rockets traveling from the Earth to orbit, a simple nozzle design is only optimal at one altitude, losing efficiency and wasting fuel at other altitudes.

Just past the throat, the pressure of the gas is higher than ambient pressure and needs to be lowered between the throat and the nozzle exit by expansion. If the pressure of the exhaust leaving the nozzle exit is still above ambient pressure, then a nozzle is said to be underexpanded; if the exhaust is below ambient pressure, then it is overexpanded.^[1]

Slight overexpansion causes a slight reduction in efficiency, but otherwise does little harm. However, if the exit pressure is less than approximately 40% that of ambient, then "flow separation" occurs. This can cause exhaust instabilities that can cause damage to the nozzle, control difficulties of the vehicle or the engine, and in more extreme cases, destruction of the engine.

In some cases, it is desirable for reliability and safety reasons to ignite a rocket engine on the ground that will be used all the way to orbit. For optimal liftoff performance, the pressure of the gases exiting nozzle should be at sea-level pressure when the rocket is near sea level (at takeoff). However, a nozzle designed for sea-level operation will quickly lose efficiency at higher altitudes. In a multi-stage design, the second stage rocket engine is primarily designed for use at high altitudes, only providing additional thrust after the first-stage engine performs the initial liftoff. In this case, designers will usually opt for an overexpanded nozzle (at sea level) design for the second stage, making it more efficient at higher altitudes, where the ambient pressure is lower. This was the technique employed on the Space Shuttle's overexpanded (at sea level) main engines (SSMEs), which spent most of their powered trajectory in near-vacuum, while the shuttle's two sea-level efficient solid rocket boosters provided the majority of the initial liftoff thrust. In the vacuum of space virtually all nozzles are underexpanded because to fully expand the gas's the nozzle would have to be infinitely long, as a result engineers have to choose a design which will take advantage of the extra expansion (thrust and efficiency) whilst also not adding excessive weight and compromising the vehicle's performance.

Vacuum use

For nozzles that are used in vacuum or at very high altitude, it is impossible to match ambient pressure; rather, nozzles with larger area ratio are usually more efficient. However, a very long nozzle has significant mass, a drawback in and of itself. A length that optimises overall vehicle performance typically has to be found. Additionally, as the temperature of the gas in the nozzle decreases, some components of the exhaust gases (such as water vapour from the combustion process) may condense or even freeze. This is highly undesirable and needs to be avoided.

Magnetic nozzles have been proposed for some types of propulsion (for example, Variable Specific Impulse Magnetoplasma Rocket, VASIMR), in which the flow of plasma or ions are directed by magnetic fields instead of walls made of solid materials. These can be advantageous, since a magnetic field itself cannot melt, and the plasma temperatures can reach millions of kelvins. However, there are often thermal design challenges presented by the coils themselves, particularly if superconducting coils are used to form the throat and expansion fields.

de Laval nozzle in 1 dimension

The analysis of gas flow through de Laval nozzles involves a number of concepts and simplifying assumptions:

• The combustion gas is assumed to be an ideal gas.
• The gas flow is isentropic; i.e., at constant entropy, as the result of the assumption of non-viscous fluid, and adiabatic process.
• The gas flow rate is constant (i.e., steady) during the period of the propellant burn.
• The gas flow is non-turbulent and axisymmetric from gas inlet to exhaust gas exit (i.e., along the nozzle's axis of symmetry).
• The flow is compressible as the fluid is a gas.

As the combustion gas enters the rocket nozzle, it is traveling at subsonic velocities. As the throat constricts, the gas is forced to accelerate until at the nozzle throat, where the cross-sectional area is the least, the linear velocity becomes sonic. From the throat the cross-sectional area then increases, the gas expands and the linear velocity becomes progressively more supersonic.

The linear velocity of the exiting exhaust gases can be calculated using the following equation^[2][3][4]

${\displaystyle v_{\text{e}}={\sqrt {{\frac {TR}{M}}\,{\frac {2\gamma }{\gamma -1}}\left}}}$

where:

${\displaystyle T}$,	absolute temperature of gas at inlet (K)
${\displaystyle R}$	≈ 8314.5 J/kmol·K, universal gas law constant
${\displaystyle M}$,	molecular mass or weight of gas (kg/kmol)
${\displaystyle \gamma }$	${\displaystyle =c_{\text{p}}/c_{\text{v}}}$, isentropic expansion factor
${\displaystyle c_{\text{p}}}$,	specific heat capacity, under constant pressure, of gas
${\displaystyle c_{\text{v}}}$,	specific heat capacity, under constant volume, of gas
${\displaystyle v_{\text{e}}}$,	velocity of gas at the nozzle exit plane (m/s)
${\displaystyle p_{\text{e}}}$,	absolute pressure of gas at the nozzle exit plane (Pa)
${\displaystyle p}$,	absolute pressure of gas at inlet (Pa)

Some typical values of the exhaust gas velocity v_e for rocket engines burning various propellants are:

• 1.7 to 2.9 km/s (3800 to 6500 mi/h) for liquid monopropellants
• 2.9 to 4.5 km/s (6500 to 10100 mi/h) for liquid bipropellants
• 2.1 to 3.2 km/s (4700 to 7200 mi/h) for solid propellants

As a note of interest, v_e is sometimes referred to as the ideal exhaust gas velocity because it based on the assumption that the exhaust gas behaves as an ideal gas.

As an example calculation using the above equation, assume that the propellant combustion gases are: at an absolute pressure entering the nozzle of p = 7.0 MPa and exit the rocket exhaust at an absolute pressure of p_e = 0.1 MPa; at an absolute temperature of T = 3500 K; with an isentropic expansion factor of γ = 1.22 and a molar mass of M = 22 kg/kmol. Using those values in the above equation yields an exhaust velocity v_e = 2802 m/s or 2.80 km/s which is consistent with above typical values.

The technical literature can be very confusing because many authors fail to explain whether they are using the universal gas law constant R which applies to any ideal gas or whether they are using the gas law constant R_s which only applies to a specific individual gas. The relationship between the two constants is R_s = R/M, where R is the universal gas constant, and M is the molar mass of the gas.

Specific impulse

Thrust is the force that moves a rocket through the air or space. Thrust is generated by the propulsion system of the rocket through the application of Newton's third law of motion: "For every action there is an equal and opposite reaction". A gas or working fluid is accelerated out the rear of the rocket engine nozzle, and the rocket is accelerated in the opposite direction. The thrust of a rocket engine nozzle can be defined as:^[2][3][5][6]

${\displaystyle {\begin{aligned}F&={\dot {m}}v_{\text{e}}+\left(p_{\text{e}}-p_{\text{o}}\right)A_{\text{e}}\\&={\dot {m}}\left,\end{aligned}}}$

the term in brackets is known as equivalent velocity,

${\displaystyle F={\dot {m}}v_{\text{eq}}.}$

The specific impulse ${\displaystyle I_{\text{sp}}}$ is the ratio of the thrust produced to the weight flow of the propellants. It is a measure of the fuel efficiency of a rocket engine. In English Engineering units it can be obtained as^[7]

${\displaystyle I_{\text{sp}}={\frac {F}{{\dot {m}}g_{\text{o}}}}={\frac {{\dot {m}}v_{\text{eq}}}{{\dot {m}}g_{\text{o}}}}={\frac {v_{\text{eq}}}{g_{\text{o}}}},}$

where:

${\displaystyle F}$,	gross thrust of rocket engine (N)
${\displaystyle {\dot {m}}}$,	mass flow rate of gas (kg/s)
${\displaystyle v_{\text{e}}}$,	velocity of gas at nozzle exhaust (m/s)
${\displaystyle p_{\text{e}}}$,	pressure of gas at nozzle exhaust (Pa)
${\displaystyle p_{\text{o}}}$,	external ambient, or free stream, pressure (Pa)
${\displaystyle A_{\text{e}}}$,	cross-sectional area of nozzle exhaust (m2)
${\displaystyle v_{\text{eq}}}$,	equivalent (or effective) velocity of gas at nozzle exhaust (m/s)
${\displaystyle I_{\text{sp}}}$,	specific impulse (s)
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