• ### What is the “Thrust” of a gas jet?

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### 1. Background

[GLOSS]Thrust[/GLOSS] (or impulse) is one of the basic properties of a fluid jet. It appears in the equations used to determine mixing, flame length, entrained mass flows, velocities, and species concentrations in a free jet. It is also used to estimate the quantity of fluid recirculating around a confined jet.

### 2. Definition of thrust

[GLOSS]Thrust[/GLOSS] is the sum of two terms:

• the [GLOSS]jet momentum[/GLOSS]
• the static over-pressure force

The general expression for [GLOSS]thrust[/GLOSS] is:

G = ò (s) ( rv2 + P ) dS       (1)

where:

– G is the [GLOSS]thrust[/GLOSS] (N)

r is the fluid density (kg/m3)

– v is the fluid velocity (m/s)

– P is the fluid pressure (N/m2)
– S is the area of the jet nozzle (m2)

For a nozzle producing a uniform velocity at its exit, (1) becomes:

G = Qmvs+ (Ps – Pa ) Ss       (2)

where:
– Qm is the mass flow rate of jet fluid (kg/s)
– vs is the fluid velocity at the nozzle exit (m/s)
– Ps is the absolute fluid pressure at the nozzle exit (N/ m2)
– Pa is the absolute downstream (ambient) pressure (N/ m2)
– Ss is the exit area of the injector (m2)

### 3. Thrust in sub-critical jets

In sub-critical or sub-sonic jets, the pressure at the nozzle exit is identical to that of the surrounding fluid (Ps = Pa), and the static-pressure force is zero. [GLOSS]Thrust[/GLOSS] is then identical to [GLOSS]jet momentum[/GLOSS], and can be written:
G = Qmvs = (ra Qv2/ S s ) * (Ts/ 273)       (3)

where:

ra is the density of the fluid under normal conditions (kg/ m3)

– Qv is the normal volume flow of fluid (m3/s under normal conditions)

– Ss is the effective exit area of the nozzle taking account of [GLOSS]discharge coefficient[/GLOSS] (m2)

– Ts is the absolute fluid temperature at the nozzle exit (K)

Nozzle discharge coefficients will depend on the internal nozzle geometry, and can be expected to vary from 0,5 for a cylindrical profile to 1 for a profile of the type shown in Combustion File 1.

### 4. Thrust in sonic jets

The static pressure term (Ps – Pa )Ss plays an important role in this case. It is not easy to measure the static pressure at the nozzle exit (Ps). By assuming:

• ideal gas

• ratio of specific heats constant

• convergent nozzle

a general expression for the [GLOSS]thrust[/GLOSS] can be written:

G = 2 Ss P0 ( 2 / (g+ 1))1/(g-1) – PaSs       (4)

where:

– P0 is the absolute upstream pressure (N/m2).

Acknowledgement

This Combustion File has been extracted from the “Recueil Des Fiches Techniques” published in the Revue Generale de Thermique.

We are unaware of the original author’s name. However we would like to thank the author and Gaz de France for the use of the information.

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