de Laval nozzle

A de Laval nozzle (or convergent-divergent nozzle, CD nozzle or con-di nozzle) is a tube that is pinched in the middle, making a carefully balanced, asymmetric hourglass shape. It is used to accelerate a hot, pressurized gas passing through it to a higher supersonic speed in the axial (thrust) direction, by converting the heat energy of the flow into kinetic energy. Because of this, the nozzle is 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.^[1]

History

The nozzle was developed by Swedish inventor Gustaf de Laval in 1888 for use on a steam turbine.^[2]^[3]^[4]^[5]

This principle was first used in a rocket engine by Robert Goddard. Very nearly all modern rocket engines that employ hot gas combustion use de Laval nozzles.

Operation

Its operation relies on the different properties of gases flowing at subsonic and supersonic speeds. The speed of a subsonic flow of gas will increase if the pipe carrying it narrows because the mass flow rate is constant. The gas flow through a de Laval nozzle is isentropic (gas entropy is nearly constant). In a subsonic flow the gas is compressible, and sound will propagate through it. At the "throat", where the cross-sectional area is at its minimum, the gas velocity locally becomes sonic (Mach number = 1.0), a condition called choked flow. As the nozzle cross-sectional area increases, the gas begins to expand, and the gas flow increases to supersonic velocities, where a sound wave will not propagate backwards through the gas as viewed in the frame of reference of the nozzle (Mach number > 1.0).

Conditions for operation

A de Laval nozzle will only choke at the throat if the pressure and mass flow through the nozzle is sufficient to reach sonic speeds, otherwise no supersonic flow is achieved, and it will act as a Venturi tube; this requires the entry pressure to the nozzle to be significantly above ambient at all times (equivalently, the stagnation pressure of the jet must be above ambient).

In addition, the pressure of the gas at the exit of the expansion portion of the exhaust of a nozzle must not be too low. Because pressure cannot travel upstream through the supersonic flow, the exit pressure can be significantly below the ambient pressure into which it exhausts, but if it is too far below ambient, then the flow will cease to be supersonic, or the flow will separate within the expansion portion of the nozzle, forming an unstable jet that may "flop" around within the nozzle, producing a lateral thrust and possibly damaging it.

In practice, ambient pressure must be no higher than roughly 2–3 times the pressure in the supersonic gas at the exit for supersonic flow to leave the nozzle.

Analysis of gas flow in de Laval nozzles

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

For simplicity, the gas is assumed to be an ideal gas.
The gas flow is isentropic (i.e., at constant entropy). As a result, the flow is reversible (frictionless and no dissipative losses), and adiabatic (i.e., there is no heat gained or lost).
The gas flow is constant (i.e., steady) during the period of the propellant 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 behaviour is compressible since the flow is at very high velocities (Mach number > 0.3).

Exhaust gas velocity

As the gas enters a nozzle, it is moving at subsonic velocities. As the throat contracts, the gas is forced to accelerate until at the nozzle throat, where the cross-sectional area is the smallest, the axial velocity becomes sonic. From the throat the cross-sectional area then increases, the gas expands and the axial velocity becomes progressively more supersonic.

The linear velocity of the exiting exhaust gases can be calculated using the following equation:^[6]^[7]^[8]

v_{e}={\sqrt {{\frac {TR}{M}}\cdot {\frac {2\gamma }{\gamma -1}}\cdot \left[1-\left({\frac {p_{e}}{p}}\right)^{\frac {\gamma -1}{\gamma }}\right]}},

where:
$v_{e}$	= exhaust velocity at nozzle exit,
$T$	= absolute temperature of inlet gas,
$R$	= universal gas law constant,
$M$	= the gas molecular mass (also known as the molecular weight)
$\gamma$	= ${\frac {c_{p}}{c_{v}}}$ = isentropic expansion factor
	( $c_{p}$ and $c_{v}$ are specific heats of the gas at constant pressure and constant volume respectively),
$p_{e}$	= absolute pressure of exhaust gas at nozzle exit,
$p$	= absolute pressure of inlet gas.

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

1,700 to 2,900 m/s (3,800 to 6,500 mph) for liquid monopropellants,
2,900 to 4,500 m/s (6,500 to 10,100 mph) for liquid bipropellants,
2,100 to 3,200 m/s (4,700 to 7,200 mph) 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 p = 7.0 MPa and exit the rocket exhaust at an absolute pressure p_e = 0.1 MPa; at an absolute temperature of T = 3500 K; with an isentropic expansion factor γ = 1.22 and a molar mass 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.

References

^ C.J. Clarke and B. Carswell (2007). Principles of Astrophysical Fluid Dynamics (1st ed.). Cambridge University Press. p. 226. ISBN 978-0-521-85331-6.
^ British patent 7143 of 1889.
^ Theodore Stevens and Henry M. Hobart (1906). Steam Turbine Engineering. MacMillan Company. pp. 24–27. Available on-line here in Google Books.
^ Robert M. Neilson (1903). The Steam Turbine. Longmans, Green, and Company. pp. 102–103. Available on-line here in Google Books.
^ Garrett Scaife (2000). From Galaxies to Turbines: Science, Technology, and the Parsons Family. Taylor & Francis Group. p. 197. Available on-line here in Google Books.
^ Richard Nakka's Equation 12.
^ Robert Braeuning's Equation 1.22.
^ George P. Sutton (1992). Rocket Propulsion Elements: An Introduction to the Engineering of Rockets (6th ed.). Wiley-Interscience. p. 636. ISBN 0-471-52938-9.

External links

Other applications of nozzle theory Flow of gases and steam through nozzles

[1] C.J. Clarke and B. Carswell (2007). Principles of Astrophysical Fluid Dynamics (1st ed.). Cambridge University Press. p. 226. ISBN 978-0-521-85331-6.

[2] British patent 7143 of 1889.

[3] Theodore Stevens and Henry M. Hobart (1906). Steam Turbine Engineering. MacMillan Company. pp. 24–27. Available on-line here in Google Books.

[4] Robert M. Neilson (1903). The Steam Turbine. Longmans, Green, and Company. pp. 102–103. Available on-line here in Google Books.

[5] Garrett Scaife (2000). From Galaxies to Turbines: Science, Technology, and the Parsons Family. Taylor & Francis Group. p. 197. Available on-line here in Google Books.

[6] Richard Nakka's Equation 12.

[7] Robert Braeuning's Equation 1.22.

[8] George P. Sutton (1992). Rocket Propulsion Elements: An Introduction to the Engineering of Rockets (6th ed.). Wiley-Interscience. p. 636. ISBN 0-471-52938-9.

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