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sive efficiency is independent of flight velocity at any given altitude. With these
assumptions, Eq. (2.157) can be integrated to obtain
R-
( lp)E
e.(;;:,)
(2.15 8)
It is interesting to observe that both altitude and flight velocity do not explicitly
appear in the equation for the range of a propeller aircraft Because the power
developed by piston-props and specific fuel consumption varies with altitude, the
range will actually depend on altitude.
The range of a propeller aircraft assumes a maximum value when fiying at
E - Em and i.s given by
Rmax = (1 )Em e,(~:: )
(2.159)
We know that when E - Em, CL = CZ, tlus gives us the value oflift coefficient
or the angle of attack for R uc. What about the flight velocity for R x? For this
purpose, we consider two options: 1) constant-velocity cruise and 2) constant-
altitude cruise.ln constant-velocit)r cruise, the cruise velocity is taken equal to the
velocity VR, i.e., based on the initial weight Wo as given by
V : 'VR =
(2.160)
In constant-velocit)r cruise, the propeller aircraft wiU gain altitude in a similar
manner to that of the jet aircraft.
In constant-altitude cruise, the flight velocity varies continuously and is equal
to the instantaneous value of VR as given by
V : VR -
where W is the instantaneous weighL
(2.161)
2.6.3 Effectof Windon Range
The expressions derived above for range are for calm air, i.e., we assumed that
the Earth~'atmosphere is stationary. However, in practice this is not always true.
It is a common knowledge that tailwind has a beneficial effect and headwind
has an adverse effect on the range. In the following, we will derive approximate
expressions for the range of a jet airplane in the presence of the wind. A similar
AIRCRAFT PERFORMANCE
111
approach can be used for the propeller airplane, and this is left as an exercise for
the reader.
The velocity with respect to the ground in presence of the wind is given by
~t = V + Vw
(2.162)
where Vw is the wind velocity. The "-" and "+" signs apply for headwind and
tailwind, respectively. For a given air speed V, the headwind reduces the Earth-
related velocity, whereas tailwind increases it.
Then,
dx = ~(V I- Vw)d:
= -(V l: Vw)E (1) dWW
so that the range in the presence of the wind is given by
Rw = -[j"(V -1 Vw)E (l) dWW
(2.163)
(2.164)
As before, we assume that the angle of attack is held constant. Furthermore, as-
suming that the wind velocit}r is constant, we have
Rw = R l: E(l) Vw V.~
(2.165)
In Eq, (2.165), the "-" sign applies for the headwind and "+" sign for the tailwind.
Note that Rw is therangein presence ofthe wind and R is the still air range. Equation
(2.165) applies for both constant-altitude cruise and constant-velocity cruise. As
expected, the headwind reduces the range, and tailwind improves it.
2.6.4 Estimation of Cruise Altitude
The cruise, or the most economical altitude, is that altitude where the overall
range is maximum. For propeller airplanes, it is in the range of 4-7 km and, for
jet airplanes, it is 11-17 km.
An approximate estimation of the cruise altitude can be made if we ignore the
variation of speciiic fuel consumption with altitude. The basic concept of this
method is illustrated in Fig. 2.22 for propeller airplanes and in Fig. 2:53 for jet
airplanes. For propeller airplanes, the power required PR based on Wo is plotted
against the equivalent air speed Ve and, on this graph, the power-available Pa curves
at various altitudes are superposed as shown. The cruise altitude corresponds to
the altitude of that Pa curve, which intersects the PR curve at the point where
Ve = Ve,Rw.. Here, Ve,R iS the equivalent air speed for the maximum range andis
obtained by drawing a tangent from the origin to the PR curve as shown in Fig. 2.22.
For the jet aircraft, the thrust required TR based on initial weight Wo is plottecl
against the equivalent air speed V,t,uand the thrust-available Ta curves at various
altitudes are superposed on this graph as shown in Fig. 2.23. The cruise altitude
;j
{
,;
i
'i!
',i
{
!
112 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Ve for Rmac
Fig. 2.22 Schematic illustration of cruise altitude for propeller aircraft
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