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Diagram of a de Laval nozzle, showing approximate flow velocity increasing from green to red

Gas flow through nozzles can be greatly accelerated by using properly shaped nozzles such as a convergent-divergent nozzle also known as a De Laval nozzle. Such nozzles are used in modern rocket engines to accelerate the combustion gases, from burning propellants, so that the exhaust gases exiting the nozzles are at supersonic velocities.

Analysis of gas flow in rocket engine nozzles

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

• For simplicity, the combustion gas is assumed to be an ideal gas.
• The gas flow is isentropic (i.e., at constant entropy), frictionless, and adiabatic (i.e., there is little or no heat gained or lost)
• The gas flow is constant (i.e., steady) during the period of the propellent 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 behavior is compressible since the flow is at very high velocities.

Maximum exhaust gas velocity

As the combustion gas enters the rocket nozzle, it is traveling at subsonic velocities. At the nozzle throat, where the cross-sectional area is the smallest, the linear velocity becomes sonic. As the cross-sectional area then increases, the gas expands and the linear velocity becomes supersonic. This improves the thrust of the rocket because the higher velocities increase the the thrust.

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

$V_{e}={\sqrt {\;{\frac {T\;R}{M}}\cdot {\frac {2\;k}{k-1}}\cdot {\bigg [}(1-(P_{e}/P)^{k/(k-1)}{\bigg ]}}}$

where:
Ve	= Exhaust velocity at nozzle exit, m/s
T	= absolute temperature of inlet gas, K
R	= Universal gas law constant = 8,314.5 (N·m) / (kgmol·K)
M	= the gas molar mass, kg/kgmol    (also known as the molecular weight)
k	= cp/cv = isentropic expansion factor
cp	= specific heat of the gas at constant pressure
cv	= specific heat of the gas at constant volume
Pe	= absolute pressure of exhaust gas at nozzle exit, Pa
P	= absolute pressure of inlet gas, Pa

http://members.aol.com/ricnakk/th_nozz.html equation 12 http://www.braeunig.us/space/propuls.htm#intro equation 2.22