The maximal total range is the maximum distance an
aircraft
An aircraft is a vehicle that is able to flight, fly by gaining support from the Atmosphere of Earth, air. It counters the force of gravity by using either Buoyancy, static lift or by using the Lift (force), dynamic lift of an airfoil, or in ...
can fly between
takeoff and
landing.
Powered aircraft 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 enhan ...
energy storage capacity (chemical or electrical) considering both weight and volume limits.
Unpowered aircraft range depends on factors such as cross-country speed and environmental conditions. 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 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 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 b ...
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
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
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
Specific range
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.
Propeller aircraft
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); ''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/s
2).
The range integral, assuming flight at a 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:
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
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.
[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, jet ...
can be derived likewise. Now, quasi-steady level flight is assumed. The relationship
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 ...
can now be written as:
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.
Using the
lift equation,
where
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 express ...
, the specific range is found equal to:
Inserting this into () and assuming only
is varying, the range (in kilometers) becomes:
here
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:
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 (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 we ...
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 = \f ...
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:
where
is the cruise Mach number and
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 we ...
. W is the weight. The range equation reduces to:
where
; here
is the specific heat constant of air (based on aviation standards) and
(derived from
and
).
and
are the specific
heat capacities of air at constant pressure and constant volume respectively.
Or
, 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:
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, ...
s, and
bleed air powered
cabin pressurization systems.
We can account for this by extending the assumed fuel flow formula in a simple way where an "adjusted" virtual
aircraft gross weight
The aircraft gross weight (also known as the all-up weight and abbreviated AUW) is the total aircraft weight at ''any moment'' during the flight or ground operation.
An aircraft's gross weight will decrease during a flight due to fuel and oil co ...
is defined by adding a constant additional "accessory" weight
.
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
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.
where
*
is the
geopotential energy height of the fuel (km)
*
is the overall
propulsive efficiency (
nondimensional
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or ...
)
*
is the aerodynamic efficiency (non-dimensional)
*
is the structural efficiency (non-dimensional)
giving the final form of the theoretical range equation (not including operational factors such as wind and routing)
The geopotential energy height of the fuel is an
intensive property. 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.
for kerosene
jet fuel
Jet fuel or aviation turbine fuel (ATF, also abbreviated avtur) is a type of aviation fuel designed for use in aircraft powered by gas-turbine engines. It is colorless to straw-colored in appearance. The most commonly used fuels for commercial a ...
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 ("ops" for flight operations)
The operational efficiency
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 (TAS, assumed constant), and average
HeadWind (HW) component.
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 st ...
divided by the actual route distance
Off-nominal temperatures may be accounted for with a temperature efficiency factor
(e.g. 99% at 10 deg C above
International Standard Atmosphere (ISA) temperature)
All of the operational efficiency factors may be collected into a single term
See also
*
Flight length
*
Flight distance record
*
Endurance
*
Specific energy
*
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. ...
*
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
*
Fuel fraction,
Mass ratio
*
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 mo ...
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