Range (aeronautics)

The maximal total range is the distance an aircraft can fly between takeoff and landing as limited by its fuel capacity. This time limit is hence controlled by the fuel load. When all fuel is consumed, the engines stop and the aircraft will lose its propulsion. The range can be seen as the ground speed multiplied by the maximum time t_max. The range equation will derived in this article for propellor and jet aircraft.

Derivation

The fuel consumption per unit time is:

$F={\frac {dW_{f}}{dt}}$

Where $W_{f}$ is the total fuel load. Since $dW_{f}=-dW$, the fuel weigtht flow rate is related to the weight of the airplane by:

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

The rate of change of fuel weight with distance is, therefore:

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

where V is the speed.

It follows that the range is obtained from the following definite integral

$R=\int _{t_{1}}^{t_{2}}Vdt=\int _{W_{1}}^{W_{2}}-{\frac {V}{F}}dW=\int _{W_{2}}^{W_{1}}{\frac {V}{F}}dW$

the term V/F is called the specific range (=range per unit weight of fuel). The specific range can now be determined as though the airplane is in quasi steady state flight. Here, a difference between jet and propellor driven aircraft has to be noticed.

Propellor aircraft

With propellor 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 han 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:

$P_{br}=V{\frac {C_{D}}{C_{L}}}W$

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

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

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:

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

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:

$T=D={\frac {C_{D}}{C_{L}}}W$

Jet engines are characteristed by a thrust specific fuel consumption, so that rate of fuel flow is proportional to drag, rather than power.

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

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

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

the specific range is found equal to:

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

Therefore, the range becomes:

$R=\int _{W_{2}}^{W_{1}}{\frac {1}{c_{T}W}}{\sqrt {{\frac {W}{S}}{\frac {2}{\rho }}{\frac {C_{L}}{C_{D}^{2}}}}}dW$

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

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

where the compressibility on the aerodynamic characteristics of the airplane are negelected as the flight speed reduces during the flight.

Cruise/Climb

For long range jet operating in the stratosphere, the speed of sound is constant, hence flying at 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. The range equation reduces to:

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

Or $R={\frac {aM}{c_{T}}}{\frac {C_{L}}{C_{D}}}ln{\frac {W_{1}}{W_{2}}}$

References

• G.J.J. Ruigrok, Elements of airplane performance, Delft University Press