A rocket engine uses a nozzle to accelerate hot exhaust to produce thrust as described by Newton's third law of motion. The amount of thrust produced by the engine depends on the mass flow rate through the engine, the exit velocity of the flow, and the pressure at the exit of the engine. The value of these three flow variables are all determined by the rocket nozzle design.

- (M^2) * dV / V = dr / r

Now we use the conservation of momentum equation:

p / r = R * T

Combining this equation for the change in pressure with the momentum equation we obtain:

V / (a^2) * dV = - dr / r

where gam is the ratio of specific heats. This is Equation #10 on the page which contains the derivation of the isentropic flow relations We can use algebra on this equation to obtain:

gam * R * T is the square of the speed of sound a:

dp = gam * p / r * dr

Most full scale rockets use the rocket exhaust nozzle as part of the stability and control system. The nozzle can be pivoted, or gimbaled, to re-direct the thrust vector. The thrust force can then be used to maneuver the rocket in flight.

where R is the gas constant and T is temperature, to get:

tells us that for M > 1, the change in density is much greater than the change in velocity. To conserve both mass and momentum in a supersonic flow, the velocity increases and the density decreases as the area is increased.

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On this slide we derive the equations which explain and describe why a supersonic flow accelerates in the divergent section of the nozzle while a subsonic flow decelerates in a divergent duct. We begin with the conservation of mass equation:

where mdot is the mass flow rate, r is the gas density, V is the gas velocity, and A is the cross-sectional flow area. If we differentiate this equation, we obtain:

A nozzle is a relatively simple device, just a specially shaped tube through which hot gases flow. Rockets typically use a fixed convergent section followed by a fixed divergent section for the design of the nozzle. This nozzle configuration is called a convergent-divergent, or CD, nozzle. In a CD rocket nozzle, the hot exhaust leaves the combustion chamber and converges down to the minimum area, or throat, of the nozzle. The throat size is chosen to choke the flow and set the mass flow rate through the system. The flow in the throat is sonic which means the Mach number is equal to one in the throat. Downstream of the throat, the geometry diverges and the flow is isentropically expanded to a supersonic Mach number that depends on the area ratio of the exit to the throat. The expansion of a supersonic flow causes the static pressure and temperature to decrease from the throat to the exit, so the amount of the expansion also determines the exit pressure and temperature. The exit temperature determines the exit speed of sound, which determines the exit velocity. The exit velocity, pressure, and mass flow through the nozzle determines the amount of thrust produced by the nozzle.

- (M^2) * dV / V + dV / V + dA / A = 0

divide by (r * V * A) to get:

r * V * dV = - dp

This equation tells us how the velocity V changes when the area A changes, and the results depend on the Mach number M of the flow. If the flow is subsonic then (M < 1) and the term multiplying the velocity change is positive (1 - M^2 > 0). Then an increase in the area (dA > 0 ) produces a negative increase (decrease) in the velocity (dV < 0). For our CD nozzle, if the flow in the throat is subsonic, the flow downstream of the throat will decelerate and stay subsonic. So if the converging section is too large and does not choke the flow in the throat, the exit velocity is very slow and doesn't produce much thrust. On the other hand, if the converging section is small enough so that the flow chokes in the throat, then a slight increase in area causes the flow to go supersonic. For a supersonic flow (M > 1) the term multiplying velocity change is negative (1 - M^2 < 0). Then an increase in the area (dA > 0) produces an increase in the velocity (dV > 0). This is exactly the opposite of what happens subsonically. Why the big difference? Because in supersonic (compressible) flows, both the density and the velocity are changing as we change the area in order to conserve mass. For subsonic (incompressible) flows, the density remains fairly constant, so the increase in area produces a decrease in velocity to conserve mass. But in supersonic flows, there are two changes; the velocity and the density. The equation:

- (M^2) * dV / V = dr / r

using the definition of the Mach number M = V / a. Now we substitute this value of (dr /r) into the mass flow equation to get:

and an isentropic flow relation:

dr / r + dV / V + dA / A = 0