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The maximal total range is the maximum distance an
aircraft An aircraft is a vehicle that is able to fly by gaining support from the air. It counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines ...
can fly between
takeoff Takeoff is the phase of flight in which an aerospace vehicle leaves the ground and becomes airborne. For aircraft traveling vertically, this is known as liftoff. For aircraft that take off horizontally, this usually involves starting with a t ...
and
landing Landing is the last part of a flight, where a flying animal, aircraft, or spacecraft returns to the ground. When the flying object returns to water, the process is called alighting, although it is commonly called "landing", "touchdown" or ...
.
Powered aircraft A powered aircraft is an aircraft that uses onboard propulsion with mechanical power generated by an aircraft engine of some kind. Aircraft propulsion nearly always uses either a type of propeller, or a form of jet propulsion. Other potential ...
range is limited by the
aviation fuel Aviation fuels are petroleum-based fuels, or petroleum and synthetic fuel blends, used to power aircraft. They have more stringent requirements than fuels used for ground use, such as heating and road transport, and contain additives to enhanc ...
energy storage capacity (chemical or electrical) considering both weight and volume limits.
Unpowered aircraft Unpowered aircraft can remain airborne for a significant period of time without onboard propulsion. They can be classified as fixed-wing gliders, lighter-than-air balloons and tethered kites. This requires a trajectory that is not merely a verti ...
range depends on factors such as cross-country speed and environmental conditions. The range can be seen as the cross-country
ground speed Ground speed is the horizontal speed of an aircraft relative to the Earth’s surface. It is vital for accurate navigation that the pilot has an estimate of the ground speed that will be achieved during each leg of a flight. An aircraft diving ve ...
multiplied by the maximum time in the air. The fuel time limit for powered aircraft is fixed by the available fuel (considering reserve fuel requirements) and rate of consumption. Some aircraft can gain energy while airborne through the environment (e.g. collecting solar energy or through rising air currents from mechanical or thermal lifting) or from in-flight refueling. These aircraft could theoretically have an infinite range. Ferry range means the maximum range that an aircraft engaged in
ferry flying Ferry flying is the flying of aircraft for the purpose of returning to base, delivery to a customer, moving from one base of operations to another or moving to or from a maintenance facility for maintenance, repair, and operations. A commercial ...
can achieve. This usually means maximum
fuel A fuel is any material that can be made to react with other substances so that it releases energy as thermal energy or to be used for work. The concept was originally applied solely to those materials capable of releasing chemical energy but ...
load, optionally with extra fuel tanks and minimum equipment. It refers to the 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.


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, the fuel consumption rate per unit time flow F is equal to -\frac = -\frac. The rate of change of aircraft mass with distance R is \frac = \frac = - \frac, where V is the speed), so that \frac=-\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


Specific range

The term \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 \eta_j and specific fuel consumption c_p. The successive engine powers can be found: P_ = \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 In aerodynamics, the lift-to-drag ratio (or L/D ratio) is the lift generated by an aerodynamic body such as an aerofoil or aircraft, divided by the aerodynamic drag caused by moving through air. It describes the aerodynamic efficiency under gi ...
: P_a = V\frac Wg; here ''Wg'' is the weight (force in newtons, if ''W'' is the mass in kilograms); ''g'' is
standard gravity The standard acceleration due to gravity (or standard acceleration of free fall), sometimes abbreviated as standard gravity, usually denoted by or , is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. ...
(its exact value varies, but it averages 9.81 m/s2). The range integral, assuming flight at a constant lift to drag ratio, becomes R = \frac \frac \int_^\frac To obtain an
analytic expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
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=\frac \frac \ln \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_\text/W_\text R=E^* \frac \eta_\text \frac \frac where E^* is the energy per mass of the battery (e.g. 150-200 Wh/kg for Li-ion batteries), \eta_\text 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 / 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 A jet aircraft (or simply jet) is an aircraft (nearly always a fixed-wing aircraft) propelled by jet engines. Whereas the engines in propeller-powered aircraft generally achieve their maximum efficiency at much lower speeds and altitudes, je ...
can be derived likewise. Now, quasi-steady level flight is assumed. The relationship D = \fracW is used. The
thrust Thrust is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that syst ...
can now be written as: T = D = \fracW ; here ''W'' is a force in newtons Jet engines are characterized by a
thrust specific fuel consumption Thrust-specific fuel consumption (TSFC) is the fuel efficiency of an engine design with respect to thrust output. TSFC may also be thought of as fuel consumption (grams/second) per unit of thrust (newtons, or N), hence ''thrust-specific''. This fi ...
, so that rate of fuel flow is proportional to drag, rather than power. F = c_T T = c_T\fracW Using the
lift Lift or LIFT may refer to: Physical devices * Elevator, or lift, a device used for raising and lowering people or goods ** Paternoster lift, a type of lift using a continuous chain of cars which do not stop ** Patient lift, or Hoyer lift, mobil ...
equation, \frac\rho V^2 S C_L = W where \rho is the air density, and S the
wing area A wing is a type of fin that produces lift while moving through air or some other fluid. Accordingly, wings have streamlined cross-sections that are subject to aerodynamic forces and act as airfoils. A wing's aerodynamic efficiency is expres ...
, the specific range is found equal to: \frac = \frac \sqrt Inserting this into () and assuming only W is varying, the range (in kilometers) becomes: R = \frac \sqrt\int_^\frac dW ; here W is again mass. When cruising at a fixed height, a fixed
angle of attack In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference line on a body (often the chord line of an airfoil) and the vector representing the relative motion between the body and the fluid through which it is m ...
and a constant specific fuel consumption, the range becomes: R = \frac \sqrt \left(\sqrt-\sqrt \right) where the compressibility on the aerodynamic characteristics of the airplane are neglected as the flight speed reduces during the flight.


Cruise/climb (Breguet range equation)

For jet aircraft operating in the
stratosphere The stratosphere () is the second layer of the atmosphere of the Earth, located above the troposphere and below the mesosphere. The stratosphere is an atmospheric layer composed of stratified temperature layers, with the warm layers of air ...
(altitude approximately between 11 and 20 km), the
speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
is approximately constant, hence flying at a fixed angle of attack and constant
Mach number Mach number (M or Ma) (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Moravian physicist and philosopher Ernst Mach. : \mathrm = \frac ...
requires the aircraft to climb (as weight decreases due to fuel burn), without changing the value of the local speed of sound. In this case: V = a M where M is the cruise Mach number and a the
speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
. W is the weight. The range equation reduces to: R=\frac\frac\int_^\frac where a= \sqrt ; here R_sis the specific heat constant of air (based on aviation standards) and \gamma = 7/5 = 1.4 (derived from \gamma = \frac and c_p = c_v+R_s). c_p and c_v are the specific heat capacities of air at constant pressure and constant volume respectively. Or R = \frac \frac \ln\frac, also known as the ''Breguet range equation'' after the French aviation pioneer, Breguet.


Modified Breguet range equation

It is possible to improve the accuracy of the Breguet range equation by recognizing the limitations of the conventionally used relationships for fuel flow: F = c_T T = c_T \frac W In the Breguet range equation, it is assumed that the
thrust specific fuel consumption Thrust-specific fuel consumption (TSFC) is the fuel efficiency of an engine design with respect to thrust output. TSFC may also be thought of as fuel consumption (grams/second) per unit of thrust (newtons, or N), hence ''thrust-specific''. This fi ...
is constant as the aircraft weight decreases. This is generally not a good approximation because a significant portion (e.g. 5% to 10%) of the fuel flow does not produce thrust and is instead required for engine "accessories" such as hydraulic pumps,
electrical generator In electricity generation, a generator is a device that converts motive power (mechanical energy) or fuel-based power (chemical energy) into electric power for use in an external circuit. Sources of mechanical energy include steam turbines, gas ...
s, and
bleed air Bleed air is compressed air taken from the compressor stage of a gas turbine upstream of its fuel-burning sections. Automatic air supply and cabin pressure controller (ASCPCs) valves bleed air from high or low stage engine compressor sections. Lo ...
powered
cabin pressurization Cabin pressurization is a process in which conditioned air is pumped into the cabin of an aircraft or spacecraft in order to create a safe and comfortable environment for passengers and crew flying at high altitudes. For aircraft, this air is u ...
systems. We can account for this by extending the assumed fuel flow formula in a simple way where an "adjusted" virtual aircraft gross weight \widehat is defined by adding a constant additional "accessory" weight W_\text. \widehat = W + W_\text F = \widehat _T \frac \widehat Here, the
thrust specific fuel consumption Thrust-specific fuel consumption (TSFC) is the fuel efficiency of an engine design with respect to thrust output. TSFC may also be thought of as fuel consumption (grams/second) per unit of thrust (newtons, or N), hence ''thrust-specific''. This fi ...
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 R = \frac\frac \ln\frac 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 Nondimensionalization is the partial or full removal of dimensional analysis, physical dimensions from an mathematical equation, equation involving physical quantity, physical quantities by a suitable substitution of variables. This technique can ...
of the range equation into fundamental design disciplines of
aeronautics Aeronautics is the science or art involved with the study, design, and manufacturing of air flight–capable machines, and the techniques of operating aircraft and rockets within the atmosphere. The British Royal Aeronautical Society identifies ...
. R=Z_f\frac\frac\ln\frac where * Z_f is the geopotential energy height of the fuel (km) * \frac is the overall
propulsive efficiency In aerospace engineering, concerning aircraft, rocket and spacecraft design, overall propulsion system efficiency \eta is the efficiency with which the energy contained in a vehicle's fuel is converted into kinetic energy of the vehicle, to accelera ...
( nondimensional) \eta_\text * \frac is the aerodynamic efficiency (non-dimensional) \eta_\text * \ln\frac is the structural efficiency (non-dimensional) \eta_\text giving the final form of the theoretical range equation (not including operational factors such as wind and routing) R = Z_f \eta_\text \eta_\text \eta_\text The geopotential energy height of the fuel is an
intensive property Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...
. A physical interpretation is a height that a quantity of fuel could lift itself in the Earth's gravity field (assumed constant) by converting its chemical energy into potential energy. Z_f for kerosene jet fuel is or 11% of the Earth's polar circumference. As an example, with 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


Operational Considerations

The range equation may be further extended to consider operational factors by including an
operational efficiency In a business context, operational efficiency is a measurement of resource allocation and can be defined as the ratio between an output gained from the business and an input to run a business operation. When improving operational efficiency, the ou ...
("ops" for flight operations) R = Z_f \eta_\text \eta_\text \eta_\text \eta_\text The operational efficiency \eta_ may be expressed as the product of individual operational efficiency terms. For example, average wind may be accounted for using the relationship between average
GroundSpeed Ground speed is the horizontal speed of an aircraft relative to the Earth’s surface. It is vital for accurate navigation that the pilot has an estimate of the ground speed that will be achieved during each leg of a flight. An aircraft diving ve ...
(GS),
True AirSpeed The true airspeed (TAS; also KTAS, for ''knots true airspeed'') of an aircraft is the speed of the aircraft relative to the air mass through which it is flying. The true airspeed is important information for accurate navigation of an aircraft. Tr ...
(TAS, assumed constant), and average
HeadWind A tailwind is a wind that blows in the direction of travel of an object, while a headwind blows against the direction of travel. A tailwind increases the object's speed and reduces the time required to reach its destination, while a headwind has ...
(HW) component. \eta_\text = \frac=\frac Routing efficiency may be defined as the
great-circle distance The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a ...
divided by the actual route distance \eta_\text = \frac Off-nominal temperatures may be accounted for with a temperature efficiency factor \eta_\text (e.g. 99% at 10 deg C above
International Standard Atmosphere The International Standard Atmosphere (ISA) is a static atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes or elevations. It has been established to provide a ...
(ISA) temperature) All of the operational efficiency factors may be collected into a single term \eta_\text = \eta_\text \eta_\text \eta_\text \cdots


See also

*
Flight length In aviation, the flight length refers to the distance of a flight. Commercial flights are often categorized into long-, medium- or short-haul by commercial airlines based on flight length, although there is no international standard definition and ...
*
Flight distance record This list of flight distance records contains only those set without any mid-air refueling. Non-commercial powered aircraft Commercial aircraft Shortest distance The Loganair Westray to Papa Westray route and its return flight make up the ...
*
Endurance Endurance (also related to sufferance, resilience, constitution, fortitude, and hardiness) is the ability of an organism to exert itself and remain active for a long period of time, as well as its ability to resist, withstand, recover from a ...
*
Specific energy Specific energy or massic energy is energy per unit mass. It is also sometimes called gravimetric energy density, which is not to be confused with energy density, which is defined as energy per unit volume. It is used to quantify, for example, sto ...
*
Geopotential height Geopotential height or geopotential altitude is a vertical coordinate referenced to Earth's mean sea level, an adjustment to geometric height (altitude above mean sea level) that accounts for the variation of gravity with latitude and altitude. Thu ...
* Energy conversion efficiency *
Jet engine performance The behavior of a jet engine and its effect both on the aircraft and the environment is categorized into different engineering areas or disciplines. The understanding of how a particular fuel flow produces a definite amount of thrust at a partic ...
*
Lift-to-drag ratio In aerodynamics, the lift-to-drag ratio (or L/D ratio) is the lift generated by an aerodynamic body such as an aerofoil or aircraft, divided by the aerodynamic drag caused by moving through air. It describes the aerodynamic efficiency under gi ...
*
Fuel fraction In aerospace engineering, an aircraft's fuel fraction, fuel weight fraction, or a spacecraft's propellant fraction, is the weight of the fuel or propellant divided by the gross take-off weight of the craft (including propellant): :\ \zeta = \fra ...
,
Mass ratio In aerospace engineering, mass ratio is a measure of the efficiency of a rocket. It describes how much more massive the vehicle is with propellant than without; that is, the ratio of the rocket's ''wet mass'' (vehicle plus contents plus propellan ...
*
Newton's laws of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...


References

* G. J. J. Ruijgrok. ''Elements of Airplane Performance''. Delft University Press. .
Prof. Z. S. Spakovszky


' MIT turbines, 2002 * Martinez, Isidoro. http://imartinez.etsiae.upm.es/~isidoro/bk3/c17/Aircraft%20propulsion.pdf#page29 Aircraft propulsion. Range and endurance: Breguet's equation page 25. * Marchman, James, III. (2021)
Aerodynamics and Aircraft Performance.
Blacksburg: VA: University Libraries at Virginia Tech. CC BY 4.0. * David W. Anderson & Scott Eberhardt. ''Understanding Flight, Second Edition'' 2010 McGraw-Hill. (eBook) {{ISBN, 9780071626965 (print) Aeronautics Aircraft performance