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de Laval Nozzle Exhaust Gas Velocity Equations and Calculator

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de Laval Nozzle Exhaust Gas Velocity Equations and Calculator

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.

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

The linear velocity of the exiting exhaust gases can be calculated using the following equation

${\displaystyle v_e=\sqrt{\frac{T\cdot <!--\; \cdot \; -->R}{M}\cdot <!--\; \cdot \; -->\frac{2\cdot <!--\; \cdot \; -->\mathrm{\gamma <!--\; \gamma \; -->}}{\mathrm{\gamma <!--\; \gamma \; -->}-<!--\; -\; -->1}\cdot <!--\; \cdot \; -->[1-<!--\; -\; -->{\left(\frac{p_e}{p}\right)}^{\frac{\mathrm{\gamma <!--\; \gamma \; -->}-<!--\; -\; -->1}{\mathrm{\gamma <!--\; \gamma \; -->}}}]}}$

where

v_e = exhaust velocity at nozzle exit (m/s),
T = absolute temperature (K),
R = universal gas law constant ( 8.3144621 j mol^-1 K^-1 )
M =gas molecular mas (kg / mol)
γ = 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) (kg/mol),
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:

1,700 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.

Reference:

• Wikipedia
• Department of Aerospace Engineering, Iowa State University, Shock Waves and De Laval Nozzle

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