Rocket engine nozzle

Rocket engine nozzle

Rockets use nozzles to accelerate hot exhaust to produce thrust as described by Newton’s third law of motion. The amount of thrust which is produced by the engine depends on the mass flow rate through the engine, the exit velocity of the flow, and the pressure at the exit of the engine. The values of these three flow variables are all determined by the design of the nozzle.

A nozzle is a device, which is shaped into tube through which hot gases flow. Rockets  nozzle designs typically consist of a fixed convergent section followed by a fixed divergent section. The nozzle configuration is then called a convergent-divergent, or CD, nozzle. In a CD nozzle, the hot exhaust leaves the combustion chamber and converges down to the minimum area, or throat, of the nozzle.

The throat size is chosen to choke the flow and set the mass flow rate through the system. The flow in the throat is sonic which means the Mach number is equal to one in the middle of the throat. Downstream of the throat, the geometry diverges, and the flow is isentropically expanded to a supersonic Mach number that depends on the area ratio of the exit to the throat, Mach number is greater than 1. 

Expansion of a supersonic flow causes that static pressure and temperature both decreases from the throat to the exit. The amount of the expansion determines the exit pressure and temperature. The exit temperature determines the exit speed of sound, which determines the exit velocity. The exit velocity, pressure, and mass flow through the nozzle determine the amount of thrust produced by the nozzle.

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.

A de Laval nozzle, showing approximate flow velocity increasing from green to red in the direction of flow; source: wiki

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. 

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.

Density flow in a nozzle; source: wiki

Vacuum use

For nozzles that are used in vacuum or at very high altitude, it is impossible to match the ambient pressure. Nozzles with larger area ratio are usually more efficient. However, a very long nozzle has significant mass, which is disadvantage. Additionally, as the temperature of the gas in the nozzle decreases, some components of the exhaust gases, such as water vapor from the combustion process, may condense or even freeze. This needs to be avoided.

A de Laval nozzle

A de Laval nozzle (or convergent-divergent nozzle, CD nozzle or con-di nozzle) is a tube which is pinched in the middle, making a carefully balanced, asymmetric hourglass shape. It is used to accelerate a compressible fluid to supersonic speeds in the axial (thrust) direction, by converting the thermal energy of the flow into kinetic energy. De Laval nozzles are widely used in some types of steam turbines and rocket engine nozzles. It also sees use in supersonic jet engines.

Similar flow properties have been applied to jet streams within astrophysics.

Diagram of a de Laval nozzle, showing approximate flow velocity (v), together with the effect on temperature (T) and pressure (p); source: wiki

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 pinches, the gas is forced to accelerate at the nozzle throat, where the cross-sectional area is the least, the linear velocity becomes sonic, M=1. 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:

𝑇, absolute temperature of gas at inlet (K)

𝑅 ≈ 8314.5 J/kmol·K, universal gas law constant

𝑀, molecular mass or weight of gas (kg/kmol)

𝛾 = 𝑐p/𝑐v, isentropic expansion factor

𝑐p, specific heat capacity, under constant pressure, of gas

𝑐v, specific heat capacity, under constant volume, of gas

𝑣e, velocity of gas at the nozzle exit plane (m/s)

𝑝e, absolute pressure of gas at the nozzle exit plane (Pa)

𝑝, absolute pressure of gas at inlet (Pa)

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

1.7 to 2.9 km/s for liquid monopropellants

2.9 to 4.5 km/s for liquid bipropellants

2.1 to 3.2 km/s for solid propellants

As a note of interest, ve 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.

Nozzle Types; source: Расчёт сопел современных ракетных двигателей / Хабр (habr.com)

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. A gas or working fluid is accelerated out 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:

the term in brackets is known as equivalent velocity,

The specific impulse 𝐼sp is the ratio of the thrust produced to the weight flow of the propellants.

where:

𝐹, gross thrust of rocket engine (N)

, mass flow rate of gas (kg/s)

𝑣e, velocity of gas at nozzle exhaust (m/s)

𝑝e, pressure of gas at nozzle exhaust (Pa)

𝑝o, external ambient, or free stream, pressure (Pa)

𝐴e, cross-sectional area of nozzle exhaust (m2)

𝑣eq, equivalent (or effective) velocity of gas at nozzle exhaust (m/s)

𝐼sp, specific impulse (s)

𝑔o, standard gravity (at sea level on Earth); approximately 9.807 m/s2

For a perfectly expanded nozzle case, where 𝑝e = po, the formula becomes

Optimal expansion

Figure above: Nozzles can be (top to bottom):

• underexpanded 
• ambient 
• overexpanded 
• grossly overexpanded.

If a nozzle is under- or overexpanded, then loss of efficiency occurs relative to an ideal nozzle. Grossly overexpanded nozzles have improved efficiency relative to an underexpanded nozzle (though are still less efficient than a nozzle with the ideal expansion ratio), however the exhaust jet is unstable.

Optimal shape

The ratio of the area of the pinched 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 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 has a ~15° cone half-angle, which is about 98% efficient. Smaller angles give very little higher efficiency. Larger angles give lower efficiency.

More complex shapes are used, such as bell nozzles or parabolic shapes. These give about 1% higher efficiency than the cone nozzle and can be shorter and lighter. 

A shorter bell shape is typically used, which gives better overall performance due to its much lower weight, shorter length, lower drag losses, and only very marginally lower exhaust speed.