User:Milton Beychok/Sandbox

Velocity of combustion gases exiting the rocket engine nozzle exhaust

The purpose of a propelling nozzle in a rocket engine is to expand and accelerate the combustion product gases from burning propellants so that the gases are exhausted from the nozzle exit at supersonic [Velocity|velocities]]. Most usually, rocket propelling nozzles are of the de Laval type as depicted in the adjacent Figure 1.

The analysis of gas flow exit velocity from de Laval 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 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 behavior is that of a compressible fluid (i.e., gas at high velocity).

As the combustion gas enters the rocket nozzle, it is traveling at subsonic velocities. As the throat contracts down the gas is forced to accelerate by the venturi effect until at the nozzle throat, where the cross-sectional area is the smallest, the linear velocity becomes sonic (i.e., attains the speed of sound). From the throat the cross-sectional area then increases, the gas expands and the linear velocity becomes progressively more supersonic.

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

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

where:
$V_{e}$	=  Exhaust velocity at nozzle exit, m/s
$T$	=  absolute temperature of inlet gas, K
$R$	=  Universal gas law constant = 8314.5 J/(kmol·K)
$M$	=  the gas molecular mass, kg/kmol    (also known as the molecular weight)
$k$	=  $c_{p}/c_{v}$ = isentropic expansion factor
$c_{p}$	=  specific heat of the gas at constant pressure
$c_{v}$	=  specific heat of the gas at constant volume
$P_{e}$	=  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:

• 1.7 to 2.9 km/s (3800 to 6500 mi/h) for liquid monopropellants
• 2.9 to 4.5 km/s (6500 to 10100 mi/h) for liquid bipropellants
• 2.1 to 3.2 km/s (4700 to 7200 mi/h) 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 of P = 7.0 MPa and exit the rocket exhaust at an absolute pressure of P_e = 0.1 MPa; at an absolute temperature of T = 3500 K; with an isentropic expansion factor of k = 1.22 and a molar mass of 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.