Thrust is the force which moves
an aircraft through the air. Thrust is generated by the
How is thrust generated?
Thrust is a mechanical force which is generated through the
reaction of accelerating a mass of gas,
as explained by Newton's third law
A gas or working fluid is accelerated to the
rear and the engine and aircraft
are accelerated in the opposite direction. To accelerate the gas, we
need some kind of propulsion system. We will
discuss the details of various propulsion systems on some other
pages. For right now, let us just think of the propulsion system as
some machine which accelerates a gas.
From Newton's second law of motion, we
can define a force F to be the change in momentum of an object with a
change in time. Momentum is the object's mass m times the
velocity V. So, between two times t1 and t2, the force is given by:
F = ((m * V)2 - (m * V)1) / (t2 - t1)
If we keep the mass constant and just change the velocity
with time we obtain the simple force equation - force equals mass
time acceleration a
F = m * a
If we are dealing with a
keeping track of the mass is relatively easy; the molecules of a solid are
closely bound to each other and a solid retains its shape. But if
we are dealing with a fluid (liquid or gas) and particularly if we
are dealing with a moving fluid, keeping track of the mass gets
tricky. For a moving fluid, the important parameter is the mass
flow rate. Mass flow rate is the amount of mass moving through a
given plane over some amount of time. Its dimensions are mass/time
(kg/sec, slug/sec, ...) and it is equal to the
r times the
V times the
A. Aerodynamicists denote this parameter
as m dot (m with a little dot over the top).
m dot = r * V * A
The "dot" notation is used a lot by mathematicians, scientists, and
engineers as a symbol for "d/dt", which means the variable changes
with a change in time. For example, we can write Newton's second law
F = d(mv)/dt or F = (mv)dot
So "m dot" is not simply the mass of the fluid, but is
the mass flow rate, the mass per unit time.
Since the mass flow rate already contains the time dependence
(mass/time), we can express the change in momentum across the
propulsion device as the change in the mass flow rate times the
velocity. We will denote the exit of the device as station "e" and
the free stream as station "0". Then
F = (m dot * V)e - (m dot * V)0
A units check shows that on the right hand side of the equation:
mass/time * length/time = mass * length / time^2
This is the dimension of a
force. There is an additional effect which we must account for if the
exit pressure p is different from the free stream pressure. The fluid
pressure is related to the momentum of
the gas molecules and acts perpendicular to any boundary which we
impose. If there is a net change of pressure in the flow there is an
additional change in momentum. Across the exit area we may encounter
an additional force term equal to the exit area Ae times the exit
pressure minus the free stream pressure. The general thrust equation
is then given by:
F = (m dot * V)e - (m dot * V)0 + (pe - p0) * Ae
Normally, the magnitude of the pressure-area term is small
relative to the m dot-V terms.
Looking at the thrust equation very carefully, we see that there
are two possible ways to produce high thrust.
One way is
to make the engine flow rate (m dot) as high as possible. As long as the
exit velocity is greater than the free stream, entrance velocity, a
high engine flow will produce high thrust. This is the design
A large amount of air is processed each
second, but the velocity is not changed very much. The other way
to produce high thrust is to make the exit velocity very much greater
than the incoming velocity. This is the design theory behind pure
turbojets and turbojets with
A moderate amount of flow is accelerated to a high velocity in
these engines. If the exit velocity becomes very high, there are
other physical processes which become
important and affect the efficiency of the engine. These effects are
described in detail on other pages at this site.
There is a simplified version of the general thrust equation that
can be used for gas
The nozzle of a turbine engine
is usually designed to make the exit pressure equal to free stream.
In that case, the pressure-area term in the general equation is equal to
is then equal to the exit mass
flow rate times the exit velocity minus the free stream mass flow
rate times the free stream velocity.
F = (m dot * V)e - (m dot * V)0
Since the exit mass
flow rate is nearly equal to the free stream mass flow rate, and the
free stream is all air, we can call the mass flow rate through the
engine the engine airflow rate.
F = (m dot)eng * (Ve - V0)
We can further simplify by absorbing the engine airflow dependence
into a more useful parameter called the
specific thrust. Fs
Specific thrust only depends on the velocity change across the engine.
Fs = F /(m dot)eng = (Ve - V0)
There is a different simplified version of the general thrust equation that
can be used for
Since a rocket carries its own oxygen on board,
the free stream mass flow rate is zero and the second term of the
general equation drops out.
F = (m dot * V)e + (pe - p0) * Ae
We have to include the pressure correction
term since a rocket nozzle produces a fixed exit pressure
which in general is different than free stream pressure.
There is a useful rocket performance parameter called the
specific impulse Isp,
that eliminates the mass flow dependence in the analysis.
Isp = Veq / go
where Veq is the equivalent velocity, which is equal to the
nozzle exit velocity plus the pressure-area term, and g0 is
the gravitational acceleration.
For both rockets and turbojets, the nozzle performs two
important roles. The design of the nozzle determines the
for a given pressure and temperature.
And because of
in the throat of the nozzle, the nozzle design also sets the mass flow rate
through the propulsion system. Therefore, the nozzle design determines the
thrust of the propulsion system as defined on this page.
You can investigate nozzle operation with our
interactive thrust simulator.
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