On this page we summarize the information necessary to do a
preliminary calculation of an aircraft's range
under cruise conditions. On this page, we
are taking a very simple view of aircraft range - for academic
purposes. In reality, calculating the range is a complex problem
because of the large number of variables. An aircraft's flight is not
conducted at a single ground speed but varies from zero at take-off,
to cruise conditions, and back to zero at landing. Extra fuel is
expended in climbing to altitude and in maneuvering the aircraft. The
weight is constantly changing as fuel is burned, so the lift, drag,
and thrust and fuel consumption rate are also continually changing. On
real aircraft, just like with your automobile, there is usually a
fuel reserve and the pilot makes sure to land the plane with fuel
still on board. We are going to neglect all of these effects.
There are certain things that we need to know about our aircraft.
From previous wind
tunnel testing, we need to determine the L/D
and the lift coefficient Cl. We also
need some information about the propulsion
system, specifically, the specific fuel
consumption rate TSFC of the engine.
For the aircraft itself, we need
to know the weight W of the aircraft, the wing
area, A and the fuel load M.
We are free to select a flight altitude,
but this determines the air density r from
our model of the standard atmosphere.
In cruise, the lift L is equal to the
weight W and the thrust F is equal to the drag D.
Using the lift
equation we can solve for the velocity necessary to create enough
lift to equal the weight.
L = W = Cl * .5 * r * V ^2 * A
In this equation, all of the variables are known except the velocity V, so
we solve this equation for V.
V = sqrt[ W / (.5 * Cl * r * A)]
Using the L/D ratio, we can solve for
the drag of the aircraft which is equal to the thrust.
F = D = L / (Cl/Cd) = W / (Cl/Cd)
specific fuel consumption, and fuel load determine the maximum
flight time available to the aircraft.
t max = M / (TSFC * F)
t max = (M * Cl/Cd) / (TSFC * W)
This flight time
multiplied by the aircraft velocity determines the range. R
R = V * t max
R = sqrt[ W / (.5 * Cl * r * A)] * (M * Cl/Cd) / (TSFC * W)
which demonstrates the information found
on this slide is also available. This applet presents problems which you
must solve by using the range equation.
- Beginner's Guide Home Page