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 the aircraft can fly. 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 range is the maximum range the aircraft can fly when carrying ordnance. 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.

Derivation

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 $W$ of the aircraft at a particular time $t$ is: $W$ = $W\_0\; +\; W\_f$, where $W\_0$ is the zero-fuel mass and $W\_f$ the mass of the fuel (both in kg), the fuel consumption rate per unit time flow $F$ (in kg/s) is equal to $-\backslash frac\; =\; -\backslash frac$. The rate of change of aircraft mass with distance $R$ (in meters) is $\backslash frac=\backslash frac=\; -\; \backslash frac$, where $V$ is the speed (in m/s), so that $\backslash frac=-\backslash frac$ It follows that the range is obtained from the definite integral below, with $t\_1$ and $t\_2$ the start and finish times respectively and $W\_1$ and $W\_2$ the initial and final aircraft masses $R=\; \backslash int\_^\; \backslash frac\; dt\; =\; \backslash int\_^\; -\backslash fracdW\; =\backslash int\_^\backslash fracdW\; \backslash quad\backslash quad\; (1)$

Specific range

The term $\backslash frac$, where $V$ is the speed, and $F$ 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.

Propeller aircraft

With propeller driven propulsion, the level flight speed at a number of airplane weights from the equilibrium condition $P\_a\; =\; P\_r$ has to be noted. To each flight velocity, there corresponds a particular value of propulsive efficiency $\backslash eta\_j$ and specific fuel consumption $c\_p$. The successive engine powers can be found: $P\_=\backslash frac$ The corresponding fuel weight flow rates can be computed now: $F=c\_p\; P\_$ Thrust power, is the speed multiplied by the drag, is obtained from the lift-to-drag ratio: $P\_a=V\backslash fracWg$ ; here ''Wg'' is the weight (force in newtons, if ''W'' is the mass in kilograms); ''g'' is standard gravity (its exact value varies, but it averages 9.81 m/s^{2}).
The range integral, assuming flight at constant lift to drag ratio, becomes
$R=\backslash frac\backslash frac\backslash int\_^\backslash frac$
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:
$R=\backslash frac\; \backslash frac\; \backslash ln\; \backslash frac$

Electric aircraft

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 $W\_/W\_$ $R=E^*\; \backslash frac\; \backslash eta\_\; \backslash frac\; \backslash frac$ where $E^*$ is the energy per mass of the battery (e.g. 150-200 Wh/kg for Li-ion batteries), $\backslash eta\_$ the total efficiency (typically 0.7-0.8 for batteries, motor, gearbox and propeller), $L/D$ lift over drag (typically around 18), and the weight ratio $\backslash frac$ typically around 0.3.https://www.mh-aerotools.de/company/paper_14/09%20-%20Electric%20Flight%20-%20Hepperle%20-%20DLR.pdf

Jet propulsion

The range of jet aircraft can be derived likewise. Now, quasi-steady level flight is assumed. The relationship $D=\backslash fracW$ is used. The thrust can now be written as: $T=D=\backslash fracW$ ; 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. $F=c\_TT=c\_T\backslash fracW$ Using the lift equation, $\backslash frac\backslash rho\; V^2\; S\; C\_L\; =\; W$ where $\backslash rho$ is the air density, and S the wing area. the specific range is found equal to: $\backslash frac=\backslash frac\; \backslash sqrt$ Inserting this into (1) and assuming only $W$ is varying, the range (in meters) becomes: $R=\backslash frac\; \backslash sqrt\backslash int\_^\backslash frac\; dW$ ; here $W$ is again mass. When cruising at a fixed height, a fixed angle of attack and a constant specific fuel consumption, the range becomes: $R=\backslash frac\; \backslash sqrt\; \backslash left(\backslash sqrt-\backslash sqrt\; \backslash right)$ where the compressibility on the aerodynamic characteristics of the airplane are neglected as the flight speed reduces during the flight.

Cruise/climb

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: $V=aM$ where $M$ is the cruise Mach number and $a$ the speed of sound. W is the weight in kilograms (kg). The range equation reduces to: $R=\backslash frac\backslash frac\backslash int\_^\backslash frac$ where $a=\; \backslash sqrt$ ; here $R\_s$is the specific heat constant of air 287.16 $\backslash frac$ (based on aviation standards) and $\backslash gamma\; =\; 7/5\; =\; 1.4$ (derived from $\backslash gamma=\backslash frac$ and $c\_p=c\_v+R\_s$). $c\_p$ and $c\_v$ are the specific heat capacities of air at a constant pressure and constant volume respectively. Or $R=\backslash frac\backslash fracln\backslash frac$, also known as the ''Breguet range equation'' after the French aviation pioneer, Breguet.

See also

* Flight length * Flight distance record * Endurance

References

* G. J. J. Ruijgrok. ''Elements of Airplane Performance''. Delft University Press. {{ISBN|9789065622044.

Prof. Z. S. Spakovszky

' MIT turbines, 2002 * Martinez, Isidoro.

Aircraft propulsion. Range and endurance: Breguet's equation

' page 25. Category:Aeronautics

Derivation

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 $W$ of the aircraft at a particular time $t$ is: $W$ = $W\_0\; +\; W\_f$, where $W\_0$ is the zero-fuel mass and $W\_f$ the mass of the fuel (both in kg), the fuel consumption rate per unit time flow $F$ (in kg/s) is equal to $-\backslash frac\; =\; -\backslash frac$. The rate of change of aircraft mass with distance $R$ (in meters) is $\backslash frac=\backslash frac=\; -\; \backslash frac$, where $V$ is the speed (in m/s), so that $\backslash frac=-\backslash frac$ It follows that the range is obtained from the definite integral below, with $t\_1$ and $t\_2$ the start and finish times respectively and $W\_1$ and $W\_2$ the initial and final aircraft masses $R=\; \backslash int\_^\; \backslash frac\; dt\; =\; \backslash int\_^\; -\backslash fracdW\; =\backslash int\_^\backslash fracdW\; \backslash quad\backslash quad\; (1)$

Specific range

The term $\backslash frac$, where $V$ is the speed, and $F$ 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.

Propeller aircraft

With propeller driven propulsion, the level flight speed at a number of airplane weights from the equilibrium condition $P\_a\; =\; P\_r$ has to be noted. To each flight velocity, there corresponds a particular value of propulsive efficiency $\backslash eta\_j$ and specific fuel consumption $c\_p$. The successive engine powers can be found: $P\_=\backslash frac$ The corresponding fuel weight flow rates can be computed now: $F=c\_p\; P\_$ Thrust power, is the speed multiplied by the drag, is obtained from the lift-to-drag ratio: $P\_a=V\backslash fracWg$ ; here ''Wg'' is the weight (force in newtons, if ''W'' is the mass in kilograms); ''g'' is standard gravity (its exact value varies, but it averages 9.81 m/s

Electric aircraft

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 $W\_/W\_$ $R=E^*\; \backslash frac\; \backslash eta\_\; \backslash frac\; \backslash frac$ where $E^*$ is the energy per mass of the battery (e.g. 150-200 Wh/kg for Li-ion batteries), $\backslash eta\_$ the total efficiency (typically 0.7-0.8 for batteries, motor, gearbox and propeller), $L/D$ lift over drag (typically around 18), and the weight ratio $\backslash frac$ typically around 0.3.https://www.mh-aerotools.de/company/paper_14/09%20-%20Electric%20Flight%20-%20Hepperle%20-%20DLR.pdf

Jet propulsion

The range of jet aircraft can be derived likewise. Now, quasi-steady level flight is assumed. The relationship $D=\backslash fracW$ is used. The thrust can now be written as: $T=D=\backslash fracW$ ; 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. $F=c\_TT=c\_T\backslash fracW$ Using the lift equation, $\backslash frac\backslash rho\; V^2\; S\; C\_L\; =\; W$ where $\backslash rho$ is the air density, and S the wing area. the specific range is found equal to: $\backslash frac=\backslash frac\; \backslash sqrt$ Inserting this into (1) and assuming only $W$ is varying, the range (in meters) becomes: $R=\backslash frac\; \backslash sqrt\backslash int\_^\backslash frac\; dW$ ; here $W$ is again mass. When cruising at a fixed height, a fixed angle of attack and a constant specific fuel consumption, the range becomes: $R=\backslash frac\; \backslash sqrt\; \backslash left(\backslash sqrt-\backslash sqrt\; \backslash right)$ where the compressibility on the aerodynamic characteristics of the airplane are neglected as the flight speed reduces during the flight.

Cruise/climb

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: $V=aM$ where $M$ is the cruise Mach number and $a$ the speed of sound. W is the weight in kilograms (kg). The range equation reduces to: $R=\backslash frac\backslash frac\backslash int\_^\backslash frac$ where $a=\; \backslash sqrt$ ; here $R\_s$is the specific heat constant of air 287.16 $\backslash frac$ (based on aviation standards) and $\backslash gamma\; =\; 7/5\; =\; 1.4$ (derived from $\backslash gamma=\backslash frac$ and $c\_p=c\_v+R\_s$). $c\_p$ and $c\_v$ are the specific heat capacities of air at a constant pressure and constant volume respectively. Or $R=\backslash frac\backslash fracln\backslash frac$, also known as the ''Breguet range equation'' after the French aviation pioneer, Breguet.

See also

* Flight length * Flight distance record * Endurance

References

* G. J. J. Ruijgrok. ''Elements of Airplane Performance''. Delft University Press. {{ISBN|9789065622044.

Prof. Z. S. Spakovszky

' MIT turbines, 2002 * Martinez, Isidoro.

Aircraft propulsion. Range and endurance: Breguet's equation

' page 25. Category:Aeronautics