حد (طیرانیات)

کسی ہوائی جہاز میں ایندھن کے خاتمے تک سفر کرنے کی گنجایش۔

ماخوذ

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 exactly calculated and will be derived for propeller and jet aircraft. If the total weight of the aircraft at a particular time $t$ is

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

where $W_{e}$ is the zero-fuel weight and $W_{f}$ the weight of the fuel, the fuel consumption rate per unit time $F$ is equal to

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

The rate of change of aircraft weight with distance $R$ is

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

where $V$ is the speed, 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). 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.

دھکیلو پنکھا طیارہ

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:

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

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:

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

جیٹ دھکیل

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 characterized 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 neglected as the flight speed reduces during the flight.

دریا گردی/بلند ہونا

For long range jet operating in the stratosphere, the speed of sound is constant, hence flying at fixed angle of attack and constant ماک 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 $R={\frac {aM}{c_{T}}}{\frac {C_{L}}{C_{D}}}ln{\frac {W_{1}}{W_{2}}}$, also known as the Breguet range equation after the French aviation pioneer, Breguet.

مزید دیکھیے

• پرواز کی لمبائی

حوالہ جات

• 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.