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<math>\frac{dW}{dR}=\frac{\frac{dW}{dt}}{\frac{dR}{dt}}= - \frac{F}{V}</math>, |
<math>\frac{dW}{dR}=\frac{\frac{dW}{dt}}{\frac{dR}{dt}}= - \frac{F}{V}</math>, |
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where <math>V</math> is the speed |
where <math>V</math> is the speed), so that |
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<math>\frac{dR}{dt}=-\frac{V}{F}{\frac{dW}{dt}}</math> |
<math>\frac{dR}{dt}=-\frac{V}{F}{\frac{dW}{dt}}</math> |
The maximal total range is the maximum distance an aircraft can fly between takeoff and landing, as limited by fuel capacity in powered aircraft, or cross-country speed and environmental conditions in unpowered aircraft. The range can be seen as the cross-country ground speed multiplied by the maximum time in the air. The fuel time limit for powered aircraft is fixed by the fuel load and rate of consumption. When all fuel is consumed, the engines stop and the aircraft will lose its propulsion.
Ferry range means the maximum range that a aircraft engaged in ferry flying can achieve. This usually means maximum fuel load, optionally with extra fuel tanks and minimum equipment. It refers to transport of aircraft without any passengers or cargo.
Combat radius is a related measure based on the maximum distance a warplane can travel from its base of operations, accomplish some objective, and return to its original airfield with minimal reserves.
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 be calculated exactly for powered aircraft. It will be derived for both propeller and jet aircraft. If the total mass of the aircraft at a particular time
is:
=
,
where is the zero-fuel mass and
the mass of the fuel, the fuel consumption rate per unit time flow
is equal to
.
The rate of change of aircraft mass with distance (in meters) is
,
where is the speed), 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 masses
The term , where
is the speed, and
is the fuel consumption rate, is called the specific range (= range per unit mass 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 lift-to-drag ratio:
; here Wg is the weight (force in newtons, if W is the mass in kilograms); gisstandard gravity (its exact value varies, but it averages 9.81 m/s2).
The range integral, assuming flight at constant lift to drag ratio, becomes
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:
An electric aircraft with battery power only will have the same mass at takeoff and landing. The logarithmic term with weight ratios is replaced by the direct ratio between
where is the energy per mass of the battery (e.g. 150-200 Wh/kg for Li-ion batteries),
the total efficiency (typically 0.7-0.8 for batteries, motor, gearbox and propeller),
lift over drag (typically around 18), and the weight ratio
typically around 0.3.[1]
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.
Using the lift equation,
where is the air density, and S the wing area.
the specific range is found equal to:
Inserting this into (1) and assuming only is varying, 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 and 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:
where ; here
is the specific heat constant of air 287.16
(based on aviation standards) and
(derived from
and
).
and
are the specific heat capacities of air at a constant pressure and constant volume respectively.
Or, also known as the Breguet range equation after the French aviation pioneer, Breguet.
It is possible to improve the accuracy of the Breguet range equation by recognizing the limitations of the conventionally used relationships for fuel flow:
In the Breguet range equation, it is assumed that the thrust specific fuel consumption is constant as the aircraft weight decreases. This is generally not a good approximation because a significant portion (e.g. up to 10%) of the fuel flow does not produce thrust and is instead required for engine "accessories" such as hydraulic pumps, electrical generators, and bleed air powered pressurization systems.
We can account for this by extending the assumed fuel flow formula in a simple way where an "adjusted" virtual aircraft weight is defined by adding a constant additional "accessory" weight
.
Here, the thrust specific fuel consumption has been adjusted down and the virtual aircraft weight has been adjusted up to maintain the proper fuel flow while making the adjusted thrust specific fuel consumption truly constant (not a function of virtual weight).
Then, the modified Breguet range equation becomes
The above equation combines the energy characteristics of the fuel with the efficiency of the jet engine. It is often useful to separate these terms. Doing so completes the nondimensionalization of the range equation into fundamental design disciplines of aeronautics.
whereis the geopotential energy height of the fuel (km)
is the overall engine efficiency (nondimensional)
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is the aerodynamic efficiency (nondimensional)
![]()
is the structural efficiency (nondimensional)
![]()
giving the final form
The geopotential energy height of kerosene jet fuel is 4400 km or 11% of the Earth's circumference. A physical interpretation is the height that a quantity of fuel could lift itself in the Earth's (assumed constant) gravity field by converting its chemical energy into potential energy. Assuming an overall engine efficiency of 40%, a lift-to-drag ratio of 18:1, and a structural efficiency of 50%, the cruise range would be