The calculation of aircraft range requires that we describe the entire "mission"
or flight profile. A typical mission is illustrated below. Altitude is shown as the
vertical coordinate and distance on the horizontal axis. Note that the altitude is
greatly exaggerated: even on a short trip, the maximum altitude is only 1% to 2%
of the distance flown.

The mission profile consists of two portions: the nominal mission and the reserves.
Each of these is divided into several segments.

*Taxi and take-off*

A certain period of time is assumed for taxi and take-off. This time varies depending
on traffic and airport layout, but a period of about 15 minutes is a reasonable average,
used in cost estimates. The take-off segment also includes acceleration to the initial
climb speed.

*Initial Climb and Maneuver*

The initial climb and air maneuvering involves airport-specific noise alleviation
procedures and is constrained by other regulations such as a 250 kt CAS speed limit
below 10,000 ft. in the U.S. and some other countries. This segment also involves
acceleration to the enroute climb speed.

*Climb *

The climb segment of the mission is discussed in the previous section of these notes.
Detailed calculations of time and fuel burned during climb may include several climb
segments flown at different speeds. Climb computations for supersonic aircraft are
especially important, with several subsonic and supersonic segments computed separately.
For very short range missions the optimum cruise altitude is not reached and the
climb may constititute half of the flight.

*Cruise*

One cannot continue climbing for long because as the altitude increases at a given
speed the C_{L} increases. Speeding up would reduce C_{L}, but this
is limited by Mach number constraints or engine power. Thus, there is a best altitude
for cruise and this optimum altitude increases as the aircraft weight decreases (as
fuel is burned). For long range missions, the initial and final cruise altitudes
are quite different since the airplane weight changes substantially.

We could compute the altitude that leads to lowest drag at a given Mach number, but
the optimum altitude is usually a bit lower since it results in higher true speeds,
smaller engines, reduced pressure loads on the fuselage, and more margin against
buffet. Thus, we will consider both initial and final cruise altitudes as design
variables in the aircraft optimization. Except in a few lightly-travelled regions,
variable altitude, or climbing cruise is not practical from a traffic control standpoint.
Thus the true optimum is not generally attainable. In the U.S. ATC rules specify
that aircraft be flown at specific flight altitudes so that the aircraft must cruise
at constant altitude, and request clearance to climb to the next highest available
altitude when sufficient fuel is consumed. This leads to "step cruise"
profiles shown on the previous page, with 1 to 3 steps of 4000 ft in altitude due
to airway requirements. Such stepped profiles lead to reductions in cruise range
by 1%-2% if the altitudes are chosen to be optimal for the weight at the beginning
of the step.

*Descent, Approach, and Landing*

Like the climb segment, the descent is performed according to a specified airspeed
schedule with speed limit restrictions below 10,000 ft and extra fuel associated
with maneuvers on approach.

*Reserves*

Reserve fuel is carried to allow for deviations from the original flight plan, including
a requirement for diversion to an alternate airport when the planned destination
is unavailable. The FAA specifies a minimum amount of reserve fuel as described below,
but many airlines have additional requirements that result in reserves usually being
somewhat higher than the FAA minimums. The FAR's establish different requirements
for domestic and international flights as shown below.

There are also other "reserve" requirements such as those associated with
"ETOPS" (extended twin engine operations). ETOPS rules currently require
that the airplane be capable of flying with one engine inoperative to the nearest
"suitable" airport. Some operators are certified for 180 minute ETOPS.
Some are allowed 120 minutes, some 90, some only 75. Some aren't allowed to fly ETOPS
under any circumstances. (Typically this is an economic decision made by the airline
- not a reflection of relative safety - because of the onerous bookkeeping requirements.)

Domestic Reserves:

1. Climb from sea level to cruise altitude

2. Cruise to alternate airport at best speed and altitude (typ. 250 n.mi.)

3. Descend to sea level

4. Cruise for 45 minutes at long range cruise speed and altitude

International Reserves:

1. Fuel to fly 10% of planned block time at long range cruise speed

2. Climb from sea level to cruise altitude

3. Cruise to alternate

4. Descend to 1500 ft and hold for 30 minutes

5. Descend to sea level

For the purposes of this course, we compute an equivalent still-air range (no
wind) using a simplified mission profile.

The fuel required for warm-up, taxi, take-off, approach, and landing segments is
sometimes taken as a single item called maneuver fuel. For our purposes, we estimate
this as 0.7% of the take-off weight.

The fuel consumed in the climb segment is estimated in the previous section as a
certain percentage of take-off weight above that needed to cruise the same distance
at initial cruise altitude.

The descent segment of the mission requires slightly less fuel than would be required
to cruise the same distance at the final cruise speed and altitude, so in the simplified
computation the cruise extends to the destination airport and the mission is completed
at the final cruise altitude.

The simplified mission is shown in the figure that follows.

In order to compute the cruise range, we estimate the weight at the beginning and
end of the cruise segment:

W_{i} = W_{tow} - .5 W_{maneuver} - W_{climb}

W_{f} = W_{zfw} + W_{reserves} + .5 W_{maneuver }

Where:

W_{maneuver} is estimated (roughly) as 0.7% of the take-off weight

W_{reserves} is estimated even more roughly as 8% of the zero fuel weight

and W_{climb} is estimated from the plot in the climb section of these notes.

The difference between initial and final cruise weights is the amount of fuel available
for cruise. This is related to the cruise range as follows.

The specific range is the distance flown per unit weight of fuel burned, often in
n.mi. / lb. It can be related directly to the engine specific fuel consumption:

Specific Range = V / cT

where V is the true speed, c is the thrust specific fuel consumption, and T is the
thrust.

In level flight (or approximately when the climb angle is very small):

T = D = W / (L/D),

so, Specific Range = V/c L/D 1/W

V/c L/D is sometimes called the range factor. It is related to the aerodynamic (L/D)
and propulsion system (V/c) efficiencies.

The cruise range is then computed by integrating the specific range:

If the airplane is flown at a constant angle of attack (constant C_{L}) and
M_{div} in the isothermal atmosphere (above 36,089 ft) where the speed of
sound is constant, then V, L/D, and c are nearly constant and:

This is known as the Breguet Range Equation. When the altitude variation is such
that L/D, V, or c is not constant, the integral may be evaluated numerically.

When the value of brake power specific fuel consumption is assumed constant (propeller
aircraft), the range equation becomes:

where h is the propeller efficiency and BSFC is the power
specific fuel consumption in consistent units.

*Range / Payload Diagram
*An aircraft does not have a single number that represents its range. Even the
maximum range is subject to interpretation, since the maximum range is generally
not very useful as it is achieved with no payload. To represent the available trade-off
between payload and range, a range-payload diagram may be constructed as shown in
the figure below.

At the maximum payload weight is often constrained by the aircraft structure, which has been designed to handle a certain maximum zero fuel weight. (Sometimes the maximum payload weight is limited by volume, but this is rather rare. It has been noted that the MD-11 would exceed its maximum zero fuel weight if the fuselage were filled with ping pong balls.)

So, the airplane take-off weight can be increased from the zero fuel weight by adding fuel with a corresponding increase in range. This is the initial flat portion of the payload-range diagram.

At some point, the airplane could reach a limit on maximum landing weight. This usually happens only when the required reserve fuel is very large. Usually we can increase the weight until the airplane reaches its maximum take-off weight, with the full payload.

If we want to continue to add fuel (and range) from this point on, we must trade payload for fuel so as not to exceed the maximum take-off weight.

At some point, the fuel tanks will be full. We could increase the range further only by reducing the payload weight and saving on drag with a fixed fuel load. This is the final very steep portion of the payload range diagram.

Usually we are most interested in the range with maximum take-off weight and here we will focus on the range of the aircraft with a full compliment of passengers and baggage. This point is somewhere on the portion of the curve labeled maximum take-off weight, but often at a point considerably lower than that associated with maximum zero fuel weight (since the maximum zero fuel weight may be chosen to accommodate revnue cargo on shorter routes and to provide some growth capability.)