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Vmin        VR        Vmac
            b) Jet aircraft
Fig.2.10 Levelflightsolutions.
:I;g
Il;~
86                 PERFORMANCE, STABILlfY, DYNAMICS, AND CONTROL
of Eqs. (2.61) and (2.62). Thus, spending the same amount of fuel and developing
the same magrutude of thmst or the power, an airplane can fly at either of these
two velocities. At the low or minimum velocity  V-y."' the thrust or power available
is essentially used to overcome induced drag, whereas, at high or maximum speed
Vmvc, it is mostly used to balance the zero-lift drag.
2.4-1   Analytical Solutions of Level Flight for Propeller Airplanes
   Because the power available and power required are the basic quantities for
propeller aircraft, we rewrite Eq. (2.62) as foii~:ws:
TV -DV :0
 Pa - PR ~ O
For an ideal propeller airjplane, whose power developed in kilowatts
propulsive efficiency r7P  are independent of fiight ve7ocity,
Pa = k'rlp P(kW)
where k' -- 1000 is the conversion constant from kW to Nm/s. Then,
k'r7pP(kW)- ~ZpSCDOV3 _ 2kW  =0
        pSV
 (2.79)
 (2.80)
P(kW) and
(2.81)
(2.82)
The above equation is of the form  V4 + a V + b = O, which has no closed-form
 analytical solution.ln other words, we have to solve the above level flight equation
numerically or graphically even for this simplified case where we have assumed
 that the power developed by the reciprocating engine and propulsive efficiency are
 independent of the fiight velocity. Therefore, for propeller aircraft,itis a common
practice to obtain graphical or numerical solutions to deternune the two speeds in
levelflight 14    and 14na
    To ,o.struct a level flight envelope, we have to obtain this type of graphical
 or numerical solutions at several altitudes. However, this task can be simplified if
 we use the equivalent air speed Ve. which is related to the true air speed by the
following relation:
Ve = V.~
where o  = P/Po is the density ratio and Po is the air density at, sea level.
     7Vith this we can rewrite Eq. (2.67) as follows:
D = ~Po Ve2SCDO +
2k W2
,OO \re2 S
(2.83)
(2.84)
The advantage of introducing the equivalent air speed is obvious. We have only
one drag curye given by Eq. (2.84) that holds for all altitudes. Now, on this drag
curve, let us superpose the thrust~available curves at various altitudes as shown
in Fig. 2.11. The thrust available drops as the altitude increases because of a falJ
in air density. The thrust-available curve intersects the drag curve at two points
designated as Ve,nun cl11(1 Ve,max. At these two intersection points, the level fiight
AIRCRAFT PERFORMANCE
Ve,mi ,,h = 0
          Ve
olutt
ing
Vt ,nnai,h= O
Fig. 2.11     Variahon of maxinmm and mirumum speeds in level flight.
87
Eqs. (2.61) and (2.62) are identically satisfied, and, therefore, each point is a level
fiight solution.
      The schematic variation of Ve.rm i and  Ve,miax with altitude is shown in Fig. 2.12a.
Ifwe plot the true air speeds corresponding to Ve,   n and Ve,max, we obtain the level
fiight envelope as shown in Fig. 2.12b. It is interesting to observe that the true air
speed corresponding to high-speed solution increases initially with altitude and
then begins to decrease, whereas the true air speed corresponding to the low-
speed solution increases monotonically with altitude. At a certain altitude, the two
solutions merge, and we have only one level fiight solution. This altitude is c&lled
the absolute ceiling ofthe airplane. Note that the absolute ceiling is also the altitude
where the thrust-available curve is tangential to the thrust-required (drag) curve.
In other words, at absolute ceiling, the thrust available has dropped so much that
the level flight is possible only at one speed. This speed happens to be the speed at
which the drag is minimum. Also, at absolute ceiling, the rate of climb will be zero
as we will see later. Beyond the absolute ceiling, steady level flight is not possible
because thrust available is not sufficient to balance the aerodynamic drag.
    The velocity corresponding to the high speed solution increases initially with
altitude because at altitudes close to the sea level, the drop in thrust available is very
　
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