The main type of rocket engine nozzles used in modern rocket engines is the de Laval nozzle which is used to expand and accelerate the combustion gases, from burning propellants, so that the exhaust gases exiting the nozzles are at hypersonic velocities.

History

The de Laval nozzle was first used in an early rocket engine developed by Robert Goddard, one of the fathers of modern rocketry. Subsequently, Walter Thiel's implementation of it made possible Germany's use of the V2 rocket to bomb England during World War II.

Atmospheric use

The optimal size of a rocket engine nozzle to be used within the atmosphere is when the exit pressure equals ambient pressure, which decreases with altitude. For rockets travelling from the Earth to orbit, a simple nozzle design is only optimal at one altitude, losing efficiency and wasting fuel at other altitudes.

If the pressure of the jet leaving the nozzle is above ambient pressure then a nozzle is said to be 'underexpanded'; if the jet is below ambient pressure then it 'overexpanded'.

Slight overexpansion causes a slight reduction in efficiency, but otherwise does little harm. However, if the jet pressure is approximately 2.5 times lower than ambient then 'flow separation' occurs. This can cause jet instabilities that can cause damage to the nozzle or simply cause control difficulties of the vehicle or 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. In most cases the optimal pressure is ambient, however if most of the thrust comes from (ambient pressure) boosters at takeoff, then the trades push to using an overexpanded nozzle. This is the technique used on the Space shuttle's main engines.

Vacuum use

For nozzles that are used in vacuum or at very high altitude, it is impossible to match ambient pressure and rather larger area ratio nozzles are usually more efficient. However, a very long nozzle has significant mass and a length that optimises overall vehicle performance can always 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 liquefy or even freeze. This is highly undesirable and needs to be avoided.

Magnetic nozzles have been proposed for some types of propulsion (for example 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 can be at millions of degrees kelvin. But there are often thermal problems in the coils, particularly if superconducting coils are used to form the throat and expansion fields.

Analysis of gas flow in rocket engine nozzles

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

• For simplicity, the combustion gas is assumed to be an ideal gas.
• The gas flow is isentropic (i.e., at constant entropy), frictionless, and adiabatic (i.e., there is little or no heat gained or lost)
• The gas flow is constant (i.e., steady) during the period of the propellent burn.
• The gas flow is along a straight line from gas inlet to exhaust gas exit (i.e., along the nozzle's axis of symmetry)
• The gas flow behavior is compressible since the flow is at very high velocities.

As the combustion gas enters the rocket nozzle, it is travelling at subsonic velocities. As the throat contracts down the gas is forced to accelerate until at the nozzle throat, where the cross-sectional area is the smallest, 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 ^[1][2][3]

where:
	=  Exhaust velocity at nozzle exit, m/s
	=  absolute temperature of inlet gas, K
	=  Universal gas law constant = 8314.5 J/(kmol·K)
	=  the gas molecular mass, kg/kmol    (also known as the molecular weight)
	=  = isentropic expansion factor
	=  specific heat of the gas at constant pressure
	=  specific heat of the gas at constant volume
	=  absolute pressure of exhaust gas at nozzle exit, Pa
	=  absolute pressure of inlet gas, 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 k = 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.

Specific Impulse

Thrust is the force which moves a rocket through the air, and through 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:^[1][2][4][5]

The specific impulse, , is the ratio of the amount of thrust produced to the weight flow of the propellants. It is a measure of the fuel efficiency of a rocket engine. It can be obtained from:^[6]

where:
	=   gross rocket engine thrust, N
	=   mass flow rate of exhaust gas, kg/s
	=   exhaust gas velocity at nozzle exit, m/s
	=   exhaust gas pressure at nozzle exit, Pa
	=   external ambient pressure, Pa     (also known as free stream pressure)
	=   cross-sectional area of nozzle exhaust exit, m²
	=   equivalent (or effective) exhaust gas velocity at nozzle exit, m/s
	=   specific impulse, s
	=   Gravitational acceleration at sea level on Earth = 9.807 m/s²

In certain cases, where equals , then:

In cases where this may not be the case since for a rocket nozzle is proportional to , then it is possible to define a constant quantity which is the vacuum for any given engine thus:

and hence:

which is simply the vacuum thrust minus the force of the ambient atmospheric pressure acting over the exit plane.

Essentially then, for rocket nozzles, the ambient pressure acting over the engine largely cancels but effectively acts over the exit plane of the rocket engine in a rearward direction, while the exhaust jet generates forward thrust.

Aerodynamic back-pressure and optimum expansion

As the gas travels down the expansion part of the nozzle the pressure and temperature decreases and the speed of the gas increases.

The supersonic nature of the exhaust jet means that the pressure of the exhaust can be significantly different to ambient pressure- the outside air is unable to equalise the pressure upstream due to the very high jet velocity. Therefore, for supersonic nozzles, it is actually possible for the pressure of the gas exiting the nozzle to go significantly below or very greatly above ambient pressure.

In addition, as the rocket engine starts up or throttles, the chamber pressure varies and this generates different levels of efficiency. At low chamber pressures the engine is almost inevitably going to be under-expanded.

Optimum shape

The ratio of the area of the narrowest part of the nozzle to the exit plane area is mainly what determines how efficiently the expansion of the exhaust gases is converted into linear velocity; the exhaust velocity and therefore the thrust of the rocket engine, although the gas properties have an effect as well.

The shape of the nozzle also modestly affects how efficiently the expansion of the exhaust gases is converted into linear motion. The simplest nozzle shape is a ~12 degree internal angle cone, which is about 97% efficient. Smaller angles give very slightly higher efficiency, larger angles give lower efficiency.

More complex shapes of revolution are frequently used such as parabolic shapes. This gives perhaps 1% higher efficiency than the cone nozzle, and is shorter and lighter. These shapes are widely used on launch vehicles and other rockets where weight is at a premium. They are of course, harder to fabricate, so are typically more costly.

There is also a theoretical optimum nozzle shape for maximum exhaust speed, however, a shorter, suboptimal bell shape is typically used due to its much lower weight, shorter length and only very marginally lower exhaust speed.

Advanced designs

A number of more sophisticated designs have been proposed, such as the plug nozzle, expanding nozzle, Stepped nozzles and the aerospike nozzle each of which adapt in some way to changing ambient pressure. There is also a SERN (Single Expansion Ramp Nozzle), a linear expansion nozzle where the gas pressure transfers work only on one side and which could be described as a single-sided aerospike nozzle.

References

1. ^ Richard Nakka's Equation 12
2. ^ Robert Braeuning's Equation 2.22
3. ^ Sutton, George P. (1992). Rocket Propulsion Elements: An Introduction to the Engineering of Rockets, 6th Edition, Wiley-Interscience, 636. ISBN 0471529389. 
4. ^ NASA: Rocket thrust
5. ^ NASA: Rocket thrust summary
6. ^ NASA:Rocket specific impulse
7. ^ Huzel, D. K. and Huang, D. H. (1971). NASA SP-125, Design of Liquid Propellant Rocket Engines, 2nd Edition, NASA. 

See also

• Choked flow - when the gas speeds reaches the speed of sound in the gas
• de Laval nozzle - a convergent-divergent nozzle designed to give supersonic speeds
• Dual-thrust rocket motors
• Jet engine - engines propelled by jets (including rockets)
• Merlin (rocket engine)
• Multistage rocket
• Pulse jet engine
• Pulsed rocket motor
• Rocket - rocket vehicles
• Rocket engines - used to propel rocket vehicles
• SERN, Single-expansion ramp nozzle - a non-axisymmetric aerospike
• Shock diamonds - these visible bands form in the exhaust of rocket engines
• Solid rocket
• Spacecraft propulsion
• Specific impulse - a measure of exhaust speed
• Staged combustion cycle (rocket) - a type of rocket engine

Heat engine types/configurations using thermodynamic cycles (Power cycles)
Stroke number and stroke parting
Crower six stroke Four-stroke cycle Scuderi Split Cycle Engine One-stroke cycle Six stroke engine Two-stroke cycle
Different work volume types (incl. Pistonless rotary engine)
Britalus Rotary Engine Combustion chamber Controlled Combustion Engine Jet engine Orbital engine Piston engine Quasiturbine Rocket engine Swing-piston engine Toroidal engine Trochilic engine Tschudi engine Twingle engine Wankel engine
Different work volume ports and main forms of
Cylinder head porting D slide valve Four-stroke cycle engine valves Manifold Multi-valve Piston valve Poppet valve Rocket engine nozzles Sleeve valve
Different pistons layouts
Bourke engine Delta engine Double acting/differential cylinder Opposed piston engine Radial engine Rotary engine Single cylinder engine Stelzer engine Straight engine
Different main rotational motion mechanisms or arc to (or even almost) pistons back-and-forth.
Sector straight-line linkage [1] Connecting rod Coomber Rotary Engine [2] Crank Substitute Engine [3] Crankshaft Evans linkage [4] Cam Parallel motion Peaucellier-Lipkin linkage Piston rod QRMC Stirling/HydraLink [5] Revolving Cylinder Engine [6] Rhombic drive Scotch yoke Swashplate Swashplate engine Watt's linkage
One-way stop-and-go like rotational motion mechanisms to rotation.
Toroidal engine Trochilic engine

External links

• NASA web site
• NASA Space Vehicle Design Criteria, Liquid Rocket Engine Nozzles
• NASA's "Beginners' Guide to Rockets"
• The Aerospike Engine
• Richard Nakka's Experimental Rocketry Web Site
• "Rocket Propulsion" on Robert Braeuning's Web Site

rocket engine is a reaction engine that takes all its reaction mass from within tankage and forms it into a high speed jet, thereby obtaining thrust in accordance with Newton's third law.
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de Laval nozzle (or convergent-divergent nozzle, CD nozzle or con-di nozzle) is a tube that is pinched in the middle, making an hourglass-shape. It is used as a means of accelerating the flow of a gas passing through it to a supersonic speed.
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Combustion or burning is a complex sequence of exothermic chemical reactions between a fuel and an oxidant accompanied by the production of heat or both heat and light in the form of either a glow or flames.
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A propellant is a material that is used to move an object by applying a motive force. This may or may not involve a chemical reaction. It may be a gas, liquid, plasma, or, before the chemical reaction, a solid.
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hypersonic speeds are speeds that are highly supersonic. In the 1970s, the term generally came to refer to speeds of Mach 5 (5 times the speed of sound) and above. The hypersonic regime is a subset of the supersonic regime.
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Robert Hutchings Goddard, Ph.D. (October 5, 1882 – August 10, 1945), U.S. professor and scientist, was a pioneer of controlled, liquid-fueled rocketry. He launched the world's first liquid-fueled rocket on March 16, 1926.
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Walter Thiel (March 2 1910 - August 17 1943) was a German engineer who largely designed the rocket engine that powered the V-2 missile.

Thiel was born and grew up in Breslau.
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Function single stage ballistic missile (area bombing)
Manufacturer Mittelwerk GmbH (development by Army Research Center Peenemünde)
Unit cost 100,000 RM January 1944, 50,000 RM March 1945^[1]
Entered service 1944
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Space Shuttle

Space Shuttle Atlantis on the launch pad prior to the STS-115 mission.
Fact sheet
Function Manned partially re-usable launch and reentry system
Manufacturer United Space Alliance:
Thiokol/Boeing (SRBs)
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A nozzle is a mechanical device designed to control the characteristics of a fluid flow as it exits (or enters) an enclosed chamber or pipe.

A nozzle is often a pipe or tube of varying cross sectional area, and it can be used to direct or modify the flow of a fluid (liquid
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The Variable Specific Impulse Magnetoplasma Rocket (VASIMR^TM) is an electro-thermal thruster for spacecraft propulsion. It uses radio waves and magnetic fields to ionize and accelerate a propellant.
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plasma is typically an ionized gas. Plasma is considered to be a distinct state of matter, apart from gases, because of its unique properties. "Ionized" refers to presence of one or more free electrons, which are not bound to an atom or molecule.
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magnetic field is a field that permeates space and which exerts a magnetic force on moving electric charges and magnetic dipoles. Magnetic fields surround electric currents, magnetic dipoles, and changing electric fields.
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An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. Additionally, the constituent atoms or molecules undergo perfectly elastic collisions with the walls of the container.
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In thermodynamics, an isentropic process (iso = "same" (Greek); entropy = "disorder") is one during which the entropy of the system remains constant.

Background

Second law of thermodynamics states that,

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Ice melting - a classic example of entropy increasing^[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice.
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adiabatic process or an isocaloric process is a thermodynamic process in which no heat is transferred to or from the working fluid. The term "adiabatic" literally means impassable (from a dia bainein), corresponding here to an absence of heat transfer.
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A propellant is a material that is used to move an object by applying a motive force. This may or may not involve a chemical reaction. It may be a gas, liquid, plasma, or, before the chemical reaction, a solid.
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Please help recruit one or [ improve this article] yourself. See the talk page for details.
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velocity is defined as the rate of change of position. It is a vector physical quantity, both speed and direction are required to define it. In the SI (metric) system, it is measured in meters per second (m/s). The scalar absolute value (magnitude) of velocity is speed.
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Mach number (Ma) (pronounced: [mɑːk], [mɑx], [mæk], see IPA) is a dimensionless measure of relative speed.
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supersonic. Speeds greater than 5 times the speed of sound are sometimes referred to as hypersonic. Speeds where only some parts of the air around an object (such as the ends of rotor blades) reach supersonic speeds are labelled transonic (typically somewhere between Mach 0.
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trillion fold).]]

Temperature is a physical property of a system that underlies the common notions of hot and cold; something that is hotter generally has the greater temperature. Temperature is one of the principal parameters of thermodynamics.
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The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another.
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molecular mass (abbreviated Mr) of a substance, formerly also called molecular weight and abbreviated as MW, is the mass of one molecule of that substance, relative to the unified atomic mass unit u (equal to 1/12 the mass of one atom of carbon-12).
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The heat capacity ratio or adiabatic index, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). It is sometimes also known as the isentropic expansion factor and ratio of specific heats
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Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the temperature of a unit quantity of a substance by a certain temperature interval.
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Pressure (symbol: p) is the force per unit area applied on a surface in a direction perpendicular to that surface.

Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.
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The pascal (symbol: Pa) is the SI derived unit of pressure or stress (also: Young's modulus and tensile strength). It is a measure of perpendicular force per unit area i.e. equivalent to one newton per square meter or one Joule per cubic meter.
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Monopropellants are chemicals or mixtures of chemicals which can be stored in a single container with some degree of safety. While stable under defined storage conditions, they react very rapidly under certain other conditions to produce a large volume of energetic (hot) gasses for
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