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.

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 $W$ of the aircraft at a particular time $t$ is:

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

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

$-\frac{dW_f}{dt} = -\frac{dW}{dt}$ .

The rate of change of aircraft weight with distance $R$ (in meters) is

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

where $V$ is the speed (in m/s), so that

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

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 weights

$R= \int_{t_1}^{t_2} \frac{dR}{dt} dt = \int_{W_1}^{W_2} -\frac{V}{F}dW =\int_{W_2}^{W_1}\frac{V}{F}dW$ .

The term $\frac{V}{F}$ 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 $P_a = P_r$ 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:

$P_{br}=\frac{P_a}{\eta_j}$

The corresponding fuel weight flow rates can be computed now:

$F=c_p P_{br}$

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

$P_a=V\frac{C_D}{C_L}W$ ; here W is a force in newtons

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

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

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

Jet propulsion

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

$T=D=\frac{C_D}{C_L}W$ ; 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.

$F=-c_TT=-c_T\frac{C_D}{C_L}W$

Using the lift equation, $\frac{1}{2}\rho V^2 S C_L = W$

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

the specific range is found equal to:

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

Therefore, the range (in meters) becomes:

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

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

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

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:

$V=aM$

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:

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

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

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

References

G. J. J. Ruijgrok. Elements of Airplane Performance. Delft University Press. ISBN 9789065622044.
Prof. Z. S. Spakovszky. Thermodynamics and Propulsion, Chapter 13.3 Aircraft Range: the Breguet Range Equation MIT turbines, 2002
Martinez, Isidoro. Aircraft propulsion. Range and endurance: Breguet's equation page 25.

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