Rocket engine nozzles

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Figure 1: A de Laval nozzle, showing approximate flow velocity increasing from green to red in the direction of flow

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.

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.

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. The optimal size of a rocket engine nozzle depends on the 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. A number of sophisticated designs have been proposed, such as the plug nozzle, expanding nozzle, 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.

The shape of the nozzle also 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 about the same efficiency as the cone nozzle, but is shorter and hence 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 may be more costly.

Magnetic nozzles have been proposed for some types of propulsion, in which the flow of plasma is directed by magnetic fields instead of walls made of solid matter.

1 Analysis of gas flow in rocket engine nozzles
2 Specific Impulse
3 See also
4 External links
5 References

[edit] 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]

$V_e = \sqrt{\;\frac{T\;R}{M}\cdot\frac{2\;k}{k-1}\cdot\bigg[ 1-(P_e/P)^{(k-1)/k}\bigg]}$

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

[edit] 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]

$F$	$=\,\dot{m}\,V_e + (P_e - P_o)\,A_e$
	$=\,\dot{m}\,\bigg[V_e + \bigg(\frac{P_e - P_o}{\dot{m}}\bigg)A_e\bigg]$
	$=\,\dot{m}\,V_{eq}$

The specific impulse, $I s p$ , 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]

$I_{sp} =\, \frac{F}{\dot{m}\,g_o}\,=\, \frac{\dot{m}\,V_{eq}}{\dot{m}\,g_o}\,=\,\frac{V_{eq}}{g_o}$

where:
$F$	= gross rocket engine thrust, N
$\dot{m}$	= mass flow rate of exhaust gas, kg/s
$V e$	= exhaust gas velocity at nozzle exit, m/s
$P e$	= exhaust gas pressure at nozzle exit, Pa
$P o$	= external ambient pressure, Pa (also known as free stream pressure)
$A e$	= cross-sectional area of nozzle exhaust exit, m²
$V e q$	= equivalent (or effective) exhaust gas velocity at nozzle exit, m/s
$I s p$	= specific impulse, s
$g o$	= Gravitational acceleration at sea level on Earth = 9.807 m/s²