# Laval nozzle

Laval nozzle in section with flow direction of the medium, flow velocity (v), pressure (p) and temperature (T)
Cutaway model of an RD-107 rocket engine

The Laval nozzle (also diverging ) is one of Ernst Körting 1878 for steam ejectors and the Sweden Carl Gustav Patrik de Laval 1883 for the admission of steam turbines with steam independently developed nozzle .

A Laval nozzle is a flow organ in which the cross section first narrows and then widens, the transition from one part to the other taking place continuously . The cross-sectional area is usually circular or elliptical at each point.

Laval nozzles have been used in rocket engines since  V2 and still today . The aim is to accelerate a flowing fluid to supersonic speed without causing strong compression shocks. The speed of sound is reached shortly after the narrowest cross section of the nozzle. The relaxation in the divergent part of the nozzle converts thermal energy into kinetic energy. Furthermore, the largest possible proportions of the outflowing fluid should have a speed running parallel to the axis in order to be more effective in thrust.

## Derivation of the form

{\ displaystyle {\ begin {aligned} c \ cdot {\ frac {\ mathrm {d} c} {\ mathrm {d} x}} & = - {\ frac {1} {\ rho}} \ cdot {\ frac {\ mathrm {d} p} {\ mathrm {d} x}} \\ & = - {\ frac {1} {\ rho}} \ cdot {\ frac {\ mathrm {d} p} {\ mathrm {d} \ rho}} \ cdot {\ frac {\ mathrm {d} \ rho} {\ mathrm {d} x}} \ end {aligned}}}

With

• the flow rate of the fluid${\ displaystyle c}$
• the coordinate in the direction of flow${\ displaystyle x}$
• the pressure ${\ displaystyle p}$
• the density ${\ displaystyle \ rho}$

together with the speed of sound, which depends on the density, results in : ${\ displaystyle a = {\ sqrt {\ frac {\ mathrm {d} p} {\ mathrm {d} \ rho}}}}$

${\ displaystyle c \ cdot {\ frac {\ mathrm {d} c} {\ mathrm {d} x}} = - {\ frac {a ^ {2}} {\ rho}} {\ frac {\ mathrm { d} \ rho} {\ mathrm {d} x}}}$

Inserting the Mach number gives: ${\ displaystyle {\ mathit {Ma}} = {\ frac {c} {a}}}$

${\ displaystyle {\ frac {1} {\ rho}} \ cdot {\ frac {\ mathrm {d} \ rho} {\ mathrm {d} x}} = - {\ mathit {Ma}} ^ {2} \ cdot {\ frac {1} {c}} \ cdot {\ frac {\ mathrm {d} c} {\ mathrm {d} x}} \ qquad (1)}$,

This equation states that the relative density change along the current filament is proportional to the relative speed change with the proportionality factor. It follows from the square proportionality factor that ${\ displaystyle x}$ ${\ displaystyle {\ mathit {Ma}} ^ {2}.}$

• with a subsonic flow ( ) the relative density change is (substantially) smaller than the relative speed change${\ displaystyle {\ mathit {Ma}} <1}$
• with a supersonic flow ( ) the relative density change is (substantially) greater than the relative change in speed.${\ displaystyle {\ mathit {Ma}}> 1}$

Furthermore, the continuity equation has to be considered:

{\ displaystyle {\ begin {alignedat} {2} & \ rho \ cdot c \ cdot A && = \ quad {\ dot {m}} \ quad & = {\ text {const}} \\\ Leftrightarrow \ quad & \ ln \ rho + \ ln c + \ ln A && = \ ln ({\ dot {m}}) & = \ ln ({\ text {const}}) \\\ Leftrightarrow \ quad & {\ frac {\ mathrm {d } \ rho} {\ rho}} + {\ frac {\ mathrm {d} c} {c}} + {\ frac {\ mathrm {d} A} {A}} && = 0 \ end {alignedat}} }

With

• the cross-sectional area ${\ displaystyle A}$
• the mass flow ${\ displaystyle {\ dot {m}}}$

If one differentiates along the stream filament, one obtains

${\ displaystyle \ Rightarrow {\ frac {1} {\ rho}} \ cdot {\ frac {\ mathrm {d} \ rho} {\ mathrm {d} x}} + {\ frac {1} {c}} \ cdot {\ frac {\ mathrm {d} c} {\ mathrm {d} x}} + {\ frac {1} {A}} \ cdot {\ frac {\ mathrm {d} A} {\ mathrm { d} x}} = 0}$

Taking into account equation (1) it follows:

${\ displaystyle {\ frac {1} {c}} \ cdot {\ frac {\ mathrm {d} c} {\ mathrm {d} x}} = {\ frac {1} {{\ mathit {Ma}} ^ {2} -1}} \ cdot {\ frac {1} {A}} \ cdot {\ frac {\ mathrm {d} A} {\ mathrm {d} x}}}$
Laval nozzle

If one takes the cross-sectional area  as given and,  on the other hand, as unknown, the last equation enables the following qualitative discussion of the flow through a nozzle. ${\ displaystyle A (x)}$${\ displaystyle c (x)}$${\ displaystyle {\ mathit {Ma}} (x)}$

If you want to accelerate a flow, the form of the Laval nozzle follows from the last equation: ${\ displaystyle {\ frac {\ mathrm {d} c} {\ mathrm {d} x}}> 0}$

• Inlet with subsonic flow ( )${\ displaystyle {\ mathit {Ma}} <1}$ : must be here , the nozzle must therefore narrow (convergent part)${\ displaystyle {\ frac {\ mathrm {d} A} {\ mathrm {d} x}} <0}$
• shortly after the narrowest cross-section is reached${\ displaystyle {\ mathit {Ma}} = 1}$
• further acceleration in the outlet: it must be here , the nozzle must therefore expand (divergent part).${\ displaystyle {\ mathit {Ma}}> 1}$${\ displaystyle {\ frac {\ mathrm {d} A} {\ mathrm {d} x}}> 0}$

## literature

• Erich Hahne: Technical Thermodynamics. Introduction and application, 5th edition, Oldenbourg Verlag, Munich 2010, ISBN 978-3-486-59231-3 .
• Herbert Oertel jr., Martin Böhle, Thomas Reviol: Fluid mechanics for engineers and natural scientists. 7th edition, Springer Fachmedien, Wiesbaden 2015, ISBN 978-3-658-07785-3 .
• Heinz Schade, Ewald Kunz, Frank Kameier, Christian Oliver Paschereit: Fluid dynamics. 4th edition, Walter de Gruyter GmbH, Berlin 2013, ISBN 978-3-11-029221-3 .