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# Range (aeronautics)

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.

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#### Transcription

But as manufacturing techniques evolve and improve there is still one component that even the largest aircraft manufacturers admit is crucial to the future of the industry engines we have ever-more fuel-efficient aircraft and that counts for the airlines obviously for the probability of the airlines which is usually very shallow It is no surprise then that engine industry leaders from GE to Rolls-Royce and Pratt & Whitney are all vying for business from plane manufacturers but the allure of a multi-million dollar contract has also attracted some newcomers looking to propel the industry forward in some rather innovative ways This is Scimitar, a prototype jet engine from UK-based group Reaction Engines Its developers say the experimental apparatus can extend the speed the power and the range of existing systems by up to 5 times the speed of sound all by tackling the biggest barrier to a new generation of ultra-fast aircraft engines: heat The fundamentals of jet engine technology haven't changed much from the original 1930s design air is drawn into the front of the jet where it's compressed mixed with a fuel like high octane kerosene and ignited but in many cases engine temperatures can reach up to 2,000 degrees Celsius something which pushes the structural integrity of the aircraft to its limits Scimitar is designed to change all of that We are in the process of testing a very very important development in aerospace propulsion which is a pre-cooler a device for cooling the air entering the high speed engine so that the engine can continue to operate pretty much as normal This means that we're going to be able to fly at speeds of Mach 5 pretty easily in the future The company is now engaged in a fifty-percent EU-funded project called LAPCAT Long-term Advanced Propulsion Concepts and Technologies a study designed to examine the propulsion concepts and technologies required to create a hypersonic aircraft with a flight range near to 20 thousand kilometers It enables very high speed terrestrial aircraft so for example a aircraft carrying 300 passengers could go from Europe to Australia in about four hours, four and a half hours, we're looking at a revolution in transportation equivilant to the jet engine With partnerships to commercialize the technology underway the program has even given birth to a bigger development a synergistic air-breathing rocket engine or SABRE for short This innovative concept engine promises to propel the ultimate flight a single-stage earth to orbit craft designed for space travel But what if heat and fossil fuels could be removed from an engine's equation altogether? Researchers from MIT are working on developing technology they hope will completely rethink aviation propulsion altogether This is an Ion thruster it might look like something from science fiction and it's not necessarily far from it the thruster operates using a ring of magnets to electrically charge atoms within an engine's combustion chamber propelling the craft forward but charging these atoms requires large amounts of electricity with most organizations considering it unviable for terrestrial aircraft until now In 2013 MIT found ionic thrusters may be a far more efficient source of propulsion than conventional jet engines the MIT team says there's still a lot of work needed to figure out the best way to store the voltage required for ionic thrusters but the hope is one day they might just be turning science fiction into science fact

## 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 weight ${\displaystyle W}$ of the aircraft at a particular time ${\displaystyle t}$ is:

${\displaystyle W}$ = ${\displaystyle W_{0}+W_{f}}$,

where ${\displaystyle W_{0}}$ is the zero-fuel weight and ${\displaystyle W_{f}}$ the weight of the fuel (both in kg), the fuel consumption rate per unit time flow ${\displaystyle F}$ (in kg/s) is equal to

${\displaystyle -{\frac {dW_{f}}{dt}}=-{\frac {dW}{dt}}}$.

The rate of change of aircraft weight with distance ${\displaystyle R}$ (in meters) is

${\displaystyle {\frac {dW}{dR}}={\frac {\frac {dW}{dt}}{\frac {dR}{dt}}}=-{\frac {F}{V}}}$,

where ${\displaystyle V}$ is the speed (in m/s), so that

${\displaystyle {\frac {dR}{dt}}=-{\frac {V}{F}}{\frac {dW}{dt}}}$

It follows that the range is obtained from the definite integral below, with ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ the start and finish times respectively and ${\displaystyle W_{1}}$ and ${\displaystyle W_{2}}$ the initial and final aircraft weights

${\displaystyle R=\int _{t_{1}}^{t_{2}}{\frac {dR}{dt}}dt=\int _{W_{1}}^{W_{2}}-{\frac {V}{F}}dW=\int _{W_{2}}^{W_{1}}{\frac {V}{F}}dW}$.

The term ${\displaystyle {\frac {V}{F}}}$ 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.

### Propeller aircraft

With propeller driven propulsion, the level flight speed at a number of airplane weights from the equilibrium condition ${\displaystyle P_{a}=P_{r}}$ has to be noted. To each flight velocity, there corresponds a particular value of propulsive efficiency ${\displaystyle \eta _{j}}$ and specific fuel consumption ${\displaystyle c_{p}}$. The successive engine powers can be found:

${\displaystyle P_{br}={\frac {P_{a}}{\eta _{j}}}}$

The corresponding fuel weight flow rates can be computed now:

${\displaystyle F=c_{p}P_{br}}$

Thrust power, is the speed multiplied by the drag, is obtained from the lift-to-drag ratio:

${\displaystyle P_{a}=V{\frac {C_{D}}{C_{L}}}W}$ ; here W is a force in newtons

The range integral, assuming flight at constant lift to drag ratio, becomes

${\displaystyle R={\frac {\eta _{j}}{gc_{p}}}{\frac {C_{L}}{C_{D}}}\int _{W_{2}}^{W_{1}}{\frac {dW}{W}}}$ ; 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:

${\displaystyle R={\frac {\eta _{j}}{gc_{p}}}{\frac {C_{L}}{C_{D}}}ln{\frac {W_{1}}{W_{2}}}}$

### Jet propulsion

The range of jet aircraft can be derived likewise. Now, quasi-steady level flight is assumed. The relationship ${\displaystyle D={\frac {C_{D}}{C_{L}}}W}$ is used. The thrust can now be written as:

${\displaystyle T=D={\frac {C_{D}}{C_{L}}}W}$ ; 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.

${\displaystyle F=-c_{T}T=-c_{T}{\frac {C_{D}}{C_{L}}}W}$

Using the lift equation, ${\displaystyle {\frac {1}{2}}\rho V^{2}SC_{L}=W}$

where ${\displaystyle \rho }$ is the air density, and S the wing area.

the specific range is found equal to:

${\displaystyle {\frac {V}{F}}={\frac {1}{c_{T}}}{\sqrt {{\frac {C_{L}}{C_{D}^{2}}}{\frac {2}{\rho SW}}}}}$

Therefore, the range (in meters) becomes:

${\displaystyle R={\frac {1}{c_{T}}}{\sqrt {{\frac {C_{L}}{C_{D}^{2}}}{\frac {2}{g\rho S}}}}\int _{W_{2}}^{W_{1}}{\frac {1}{\sqrt {W}}}dW}$ ; here ${\displaystyle W}$ is again mass.

When cruising at a fixed height, a fixed angle of attack and a constant specific fuel consumption, the range becomes:

${\displaystyle R={\frac {2}{c_{T}}}{\sqrt {{\frac {C_{L}}{C_{D}^{2}}}{\frac {2}{g\rho S}}}}\left({\sqrt {W_{1}}}-{\sqrt {W_{2}}}\right)}$

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 11–20 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:

${\displaystyle V=aM}$

where ${\displaystyle M}$ is the cruise Mach number and ${\displaystyle a}$ the speed of sound. W is the weight in kilograms (kg). The range equation reduces to:

${\displaystyle R={\frac {aM}{gc_{T}}}{\frac {C_{L}}{C_{D}}}\int _{W_{2}}^{W_{1}}{\frac {dW}{W}}}$

where ${\displaystyle a={\sqrt {{\frac {7}{5}}R_{s}T}}}$ ; here ${\displaystyle R_{s}}$is the specific heat constant of air 287.16 ${\displaystyle {\frac {J}{kgK}}}$ (based on aviation standards) and ${\displaystyle \gamma =7/5=1.4}$ (derived from ${\displaystyle \gamma ={\frac {c_{p}}{c_{v}}}}$ and ${\displaystyle c_{p}=c_{v}+R_{s}}$). ${\displaystyle c_{p}}$en ${\displaystyle c_{v}}$ are the specific heat capacities of air at a constant pressure and constant volume.

Or ${\displaystyle R={\frac {aM}{gc_{T}}}{\frac {C_{L}}{C_{D}}}ln{\frac {W_{1}}{W_{2}}}}$, also known as the Breguet range equation after the French aviation pioneer, Breguet.