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(2.145) gives
R = -[:' ( /:. ) (-,)
dW
.~
                     = (-,) ( ,/-. XFc ~-
        N          ;-,~]    (2.148)
    Thus, for a ~vteer (yCount of fuel load, the range is maximum when the aero-
dynamic paramer        ~]CD) is a ma                    m be shown that this happens
whenCL =.     737rc orCL _ (,/~./)ZtouZ~/:~ae., _vR Thus.thevelocity
for best constant-altitude range is eqUi              Les the velocity for minimum
drag.
    According to Eq. (2.148), the range improves as altitude increases because the
air density appears in the denonunator.ln general, the thrust available drops with
altitude so that there is an altitude where the overall rangeis maximum. This altitude
is called the most economical or cruise altitude. We will discuss an approximate
method to determine this altitude later-in this section.
  Constant-ve/ocitjr range.  When the cruise velocitjr is held constant, Eq.
(2.145) can be integrated to obtain
R= (V)E V_(~ )
(2.149)
AIRCRAFT PERFORMANCE
109
Equation (2.149) is known as the Breguet range formula. The Breguet range is
maximum when the aircraft flies at that velocity for which E = Em. Recall that
when E -. Em, CL - CZ, the flight velocity is equal to  VR, and the drag in level
flight is minimum.
     The maximum range is given by
                                    ' Rmax = ( V, )E,  f,.(~ )                          (2.150)
where VR iS based on irutial weight Wo and is given by
VR -
(2.151)
During this constant-velocitjt cruise, the aiTplane will steadily gain the altitude
because ofcontinuous decrease in weight. This phenomenon is called cruise climb.
The altitude gained could be substantial for long cruises. Later in this section, we
will discuss an approximate method to estimate the height gained during the cruise
climb.
2.6.2   Range of Propeller Aircraft
   For propeller airplanes, the specific fuel consumption is the amount of fuel
consumed per unit power developed by the engine per unit time so that
                                    w - -cP                          (2.152)
where P is the power developed by the engine. The Systeme International (sD
unit for power is kW. Here, we assume that for a given altitude the specific fuel
consumption is a constant. Then,
Then
dx    V
      (2.153)
dW = -cP
P DV
      (2.154)
   rlp
 dx                  r7 p
dW = -cD                        . (2.155)
Multiply and divide the right-hand side of Eq. (2.155) by  W  and use the relation
L - W to obtain
dx
- = -(g ),7W                      (2.156)
dW   F
The power required is equal to the product of drag D and  relocity  V. The power
 available is the product of the power developed by the engine P and the propulsive
efficiency  7P  For steady level flight condition, the power available must be equal
to power requrred, so that
110              PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
With R = xf - x,, we get
(2.157)
We assume that the angle of attack is held constant at some value tjhroughout the
cruise so that the lift-to-drag ratio E remains constant. Furthermore, we assume
that for a given altitude, r7p - consL In other words, we assume that the propul-
　
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