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2.7 Endurance
       The endurance of an airplane is the total time an airplane remains in the air and
 is usually expressed in hours. Assume that most of the time the airplane is in level
fiight; we can determine the endurance ofjet and propeller airplanes as follows.
2.7.1   Endurance of rlet Aircraft
    We have
                                    dW = -cT
                  dr '
              dW
                dt=- T
With T - D and L - W,
 dWW
-  cT W
dt = _(l)E(dWW)
(2.176)
(2.177)
(2.17 8)
or
                                        , = -[w;' (1)E(dWW)                            (2.179)
Assuming that the angle of attack is held constant throughout the flight,
(1) E V.(Y:/)
(2.180)
     tmax=(-,)E,e.(W)  . (2.181)
We observe that the altitude and flight velcocity do not directly affect the en-
durance. However, the altitude has an indirect infiuence through the specific fuel
AIRCRAFT PERFORMANCE
115
consumption, which generally varies with altitude. Therefore, the endurance im-
proves when thejet airplane fiies at that altitude where the specific fuel consumpLion
is low.
   For a given altitude, the endurance is maximum when the airplane angle of
attack is such that E = Em. We know that when E - En, CL = C7, V  :  VR,
and D - Dmin.
2-7.2  Endurance of PropeLferAircraft
or
For propeller aircraft, we have
dW_ cP
dt
dt = _d W
   cP
    _ _/7P dW
  cDV
-(-,) (~) (~) dWW
(2.182)
(2.183)
                                    t = -(-, ) [w:' E   ;7) dWW                         (2.184)
As said before, we assume that the angle of attack is held constant during the
flight so that E is constant. The endurance of a propeller aircrafc depends on
velocity. Here, we consider two options: 1) fiight at constant altitude or 2) flight
with constant velocity.
   For flight with constant altitude, the velocity is allowed to vary to compensate
for the decrease in weight as given by
V:
where  W  is the instantaneous weight. With this, we obtain
t = (-, ) (-,.~~(
(2.185)
(2.186)
From Eq. (2.186) we observe that the endurance of the propeller aircraft depends
explicitly on the altitude. The maximum endurance occurs at sea level when Lhe
116            PERFORMANCE, STABILlfY, DYNAMICS, AND CONTROL
 0.866 VR           VR
 tmax   Rmax     V
Fig. 2.24    'Velocities for maramum range and rmurimum endurance of propeUer air-
craft
aerodynamic parameter (C~/2/CD) is maximum. From Eq. (2.53), we have
 -,.) ax = -,
(2.187)
and this happens when <       :  JT~-jZ or CL =  ,~CZ, CD,i = 3CDO, CD =
4CDO, and V  =  VR];%-C:  V p = 0.866 VR,the velocity for PR,nm inlevelflight.
     For fiight at constant velocity, the endurance is given by
                                       t = ( 7: ) ( VE ) &,,(;~ )                           (2.188)
The maximum endurance occurs when V  :  VR and E - Em. Here, the velocity
VR iS based on initial weight Wo as given by
　
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