For most unpowered aircraft, the maximum flight time is variable, limited by available daylight hours, aircraft design (performance), weather conditions, aircraft potential energy, and pilot endurance. Therefore, the range equation can only exactly calculated and will be derived for propeller and jet aircraft. If the total weight of the aircraft at a particular time is
where is the zero-fuel weight and the weight of the fuel (both in kg), the fuel consumption rate per unit time flow (in kg/s) is equal to
The rate of change of aircraft weight with distance (in meters) is
where is the speed (in m/s), so that
It follows that the range is obtained from the definite integral below, with and the start and finish times respectively and and the initial and final aircraft weights
The term is called the specific range (= range per unit weight of fuel; S.I. units: m/kg). The specific range can now be determined as though the airplane is in quasi steady state flight. Here, a difference between jet and propeller driven aircraft has to be noticed.
With propeller driven propulsion, the level flight speed at a number of airplane weights from the equilibrium condition has to be noted. To each flight velocity, there corresponds a particular value of propulsive efficiency and
specific fuel consumption . The successive engine powers can be found:
The corresponding fuel weight flow rates can be computed now:
Thrust power, is the speed multiplied by the drag, is obtained from the
; here W is a force in newtons
The range integral, assuming flight at constant lift to drag ratio, becomes
; here W is the mass in kilograms, therefore
standard gravity g is added. Its exact value depends on the distance to the centre of gravity of earth, but it averages 9.81 m/s2.
To obtain an
analytic expression for range, it has to be noted that specific range and fuel weight flow rate can be related to the characteristics of the airplane and propulsion system; if these are constant:
The range of jet aircraft can be derived likewise. Now, quasi-steady level flight is assumed. The relationship is used. The thrust can now be written as:
; here W is a force in newtons
Jet engines are characterized by a
thrust specific fuel consumption, so that rate of fuel flow is proportional to drag, rather than power.
where is the air density, and S the wing area.
the specific range is found equal to:
Therefore, the range (in meters) becomes:
; here is again mass.
When cruising at a fixed height, a fixed
angle of attack and a constant specific fuel consumption, the range becomes:
where the compressibility on the aerodynamic characteristics of the airplane are neglected as the flight speed reduces during the flight.
For long range jet operating in the
stratosphere (altitude approximately between 11–20 km), the speed of sound is constant, hence flying at fixed angle of attack and constant
Mach number causes the aircraft to climb, without changing the value of the local speed of sound. In this case:
where is the cruise Mach number and the
speed of sound. W is the weight in kilograms (kg). The range equation reduces to:
is the specific heat constant of air 287.16 (based on aviation standards) and (derived from and ). en are the specific
heat capacities of air at a constant pressure and constant volume.
Or , also known as the Breguet range equation after the French aviation pioneer,