The maximal total range is the 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 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. For unpowered aircraft, the maximum flight time is variable, limited by available daylight hours, weather conditions, and pilot endurance. The range can be seen as the cross-country ground speed multiplied by the maximum time t_max. The range equation will derived in this article for propeller and jet aircraft.
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Derivation
The fuel consumption per unit time is: <math>F = \frac{dW_f}{dt}</math> Where <math>W_f</math> is the total fuel load. Since <math>dW_f = -dW</math>, the fuel weight flow rate is related to the weight of the airplane by: <math>F = -\frac{dW}{dt}</math> The rate of change of fuel weight with distance is, therefore: <math>\frac{dW}{dR}=\frac{\frac{dW}{dt}}{\frac{dR}{dt}}=\frac{F}{V}</math> where V is the speed. It follows that the range is obtained from the following definite integral <math>R= \int_{t_1}^{t_2} V dt = \int_{W_1}^{W_2}-\frac{V}{F}dW=\int_{W_2}^{W_1}\frac{V}{F}dW</math> 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 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 <math>P_a = P_r</math> has to be noted. To each flight velocity, there corresponds a particular value of propulsive efficiency <math>\eta_j</math> and specific fuel consumption <math>c_p</math>. The successive engine powers can be found: <math>P_{br}=\frac{P_a}{\eta_j}</math> The corresponding fuel weight flow rates can be computed now: <math>F=c_p P_{br}</math> Thrust power, is the speed multiplied by the drag, is obtained from the lift to drag ratio: <math>P_{br}=V\frac{C_D}{C_L}W</math> The range integral, assuming flight at constant lift to drag ratio, becomes <math>R=\frac{\eta_j}{c_p}\frac{C_L}{C_D}\int_{W_2}^{W_1}\frac{dW}{W}</math> 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: <math>R=\frac{\eta_j}{c_p} \frac{C_L}{C_D} ln \frac{W_1}{W_2}</math>
Jet propulsion
The range of jet aircraft can be derived likewise. Now, quasi-steady level flight is assumed. The relationship <math>D=\frac{C_D}{C_L}W</math> is used. The thrust can now be written as: <math>T=D=\frac{C_D}{C_L}W</math> Jet engines are characteristed by a thrust specific fuel consumption, so that rate of fuel flow is proportional to drag, rather than power. <math>F=-c_TT=-c_T\frac{C_D}{C_L}W</math> Using the lift equation, <math>\frac{1}{2}\rho V^2 S C_L = W</math> where <math>\rho</math> is the air density, and S the wing area. the specific range is found equal to: <math>\frac{V}{F}=\frac{1}{c_T W} \sqrt{\frac{W}{S}\frac{2}{\rho}\frac{C_L}{C_D^2}}</math> Therefore, the range becomes: <math>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</math> When cruising at a fixed height, a fixed angle of attack and a constant specific fuel consumption, the range becomes: <math>R=\frac{2}{c_T} \sqrt{\frac{2}{S \rho} \frac{C_L}{C_D^2}} \left(\sqrt{W_1}-\sqrt{W_2} \right)</math> 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 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: <math>V=aM</math> where M is the cruise Mach number and a the speed of sound. The range equation reduces to: <math>R=\frac{aM}{c_T}\frac{C_L}{C_D}\int_{W_2}^{W_1}\frac{dW}{W}</math> Or <math>R=\frac{aM}{c_T}\frac{C_L}{C_D}ln\frac{W_1}{W_2}</math>, also known as the Breguet range equation.
References
- G.J.J. Ruigrok, Elements of airplane performance, Delft University Press
