On this slide, we have collected all of the equations
necessary to calculate the thrust of a rocket engine.
In a
rocket engine,
stored fuel and stored oxidizer
are ignited in a combustion chamber.
The combustion produces great amounts of exhaust gas at high
temperature
and
pressure.
The hot exhaust is passed through a
nozzle
which accelerates the flow.
Thrust
is produced according to Newton's
third law
of motion.

The amount of thrust produced by the rocket depends
on the mass flow rate through the engine, the exit
velocity of the exhaust, and the pressure at the nozzle
exit. All of these variables depend
on the design of the nozzle.
The smallest cross-sectional area of the nozzle is called the
throat of the nozzle. The hot exhaust flow is
choked
at the throat, which means that the
Mach number
is equal to 1.0 in the throat and the
mass flow ratem dot
is determined by the throat area.

where A* is the area of the throat, pt is the total
pressure in the combustion chamber, Tt is the total temperature
in the combustion chamber, gam is the ratio of
specific heats of the exhaust, and
R is the
gas constant.

The
area ratio
from the throat
to the exit Ae sets the
exit Mach number:

Solving for the exit Mach number when we know the exit area ratio is quite difficult.
But, we can use a computer program to iteratively solve the equation. Here's
a Java program that solves for the Mach number when you specify the area ratio:

By default, the program Input Variable is the
Mach number
of the flow. Since the area ratio depends only on the Mach number and
ratio of specific heats, the program can calculate the value of the
area ratio and display the results on the right side of the output
variables. You can also have the program solve for the Mach number
that produces a desired value of area ratio.
Using the choice button labeled Input Variable,
select "Area Ratio - A/A*".
Next to the selection, you then type in a value for A/A*.
When you hit the red COMPUTE button,
the output values change. The area ratio is double valued;
for the same area ratio, there is a subsonic
and a supersonic solution. The choice button at the right top selects
the solution that is presented.

If you are an experienced user of this calculator, you can use a
sleek version
of the program which loads faster on your computer and does not include these instructions.
You can also download your own copy of the program to run off-line by clicking on this button:

We can determine
the exit pressure pe and exit temperature Te from the
isentropic relations
at the nozzle exit:

pe / pt = [1 + Me^2 * (g-1)/2]^-[g/(g-1)]

Te / Tt = [1 + Me^2 * (g-1)/2]^-1

Knowing Te we can use the equation for the
speed of sound
and the definition of the
Mach number
to calculate the exit velocity Ve:

Ve = Me * sqrt (gam * R * Te)

We now have all the information necessary to determine
the thrust of a rocket.
The exit pressure is
only equal to free stream pressure at some design condition.
We must, therefore, use the longer version of the generalized
thrust equation
to describe the thrust of the system.
If the free stream pressure is given by p0, the
rocket thrust equation
is given by:

F = m dot * Ve + (pe - p0) * Ae

You can explore the design and operation of a rocket nozzle with
our interactive
nozzle simulator
program which runs on your browser.

The thrust equation shown above works for both
liquid rocket
and
solid rocket engines.
There is also an efficiency parameter called the
specific impulse
which works for both types of rockets and greatly simplifies
the performance analysis for rockets.