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Fig.2.18   Specific excess power curves at various altitudes.
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      To evaluate the integralin Eq. (2.136), we need to draw the Ps  curves at var:ious
altitudes. The procedure of drawing Ps curves is as follows:
       1) Given the data on Pa and PR at a number of altitudes, plot the specific excess-
power Ps curves at selected altitudes to cover the range hi to hf , say hi - hs
as shown in the upper sketch of Fig. 2.18. Note that h - const for each of these
curves. Then, draw horizontallines along which Ps - const,-say Psl and Ps2 as
shown, and read the values of altitude and velocity corresponding to every point
of intersection.
     2) Cross plot the altitude vs veloc,ity with P\* a parameter as shown in the lower
sketch of Fig.  2.18. Note that for each of these curves,  Ps _. const.  Draw as many
curves as possible to cover the range Ps - O - P,max. Note that the curve for which
P\. = 0 represents the level flight envelope.
         The problem ofm.inimum time to climb from a given initial altitude to the desired
final altitude is ofinterest to both commercial and militar}r aircraft. A commercial
airliner needs to clear the terminal airspace quickly to reduce terminal area con-
gestion and reach the cruise altitude as early as possible so that the fuel consumed
102              PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
H
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n.
Fig. 2.19    Minimum hme to climb for a subsonic turcraft
st.
during climb is kept to a minimum. Also, this problem is of special interest to
interceptor/fighter aircraft because minimum time to climb is an important mea-
sure of air superiority. In the following, we will discuss the application of energy
climb method to the problem of minimum time to climb for typical subsonic and
supersoruc aircraft.
      Minimum time to c/imb forsubsonic aircraft.     From Eq. (2.136), we observe
that the time to climbis minimum when the aircraft, at each energy state he, follows
the path along which the specific excess power is locally maximum.
   Typically, the Ps curves for a subsonic aircraft are continuous and smooth as
shown in Fig. 2.19. According to the steady climb method, the aircraft will fiy
such that R/C is a local maximum at each altitude. The locus of this path is given
by the curve AB. However, the.minimum time obtained by this method will be in
error if the path AB is not verticalin h-V  plane because it would violate the basic
assumption that the flight velocity is constant all along the flight path AB.
   The minimum time according to the energy height method is obtained when
the aircraft follows the path AB', which is the locus of all those points on Ps
curves that are locally tangential to he = const curves. Along AB',  the value of
P.\. is a local maximum for each he -.const curve. For subsonic aircraft, the curve
AB' is usually quite close to AB so that the minimum time given by the steady-
state method is nearly equal to that given by the more accurate energy height
method.
    Minimum time to ckmb for supersonic aircraft.   Here, we will discuss an
application of the energy climb method for a supersonic aircraft of the 1960s and
1970s because these aircraft had limited excess-thrust capability in the transonic
H
AIRCRAFT PERFORMANCE
103
Fig. 2.20    Schematic illustration of mirumum time to climb for a supersomc aircraft
with limited excess-thrust capability.
and supersonic regions. Because of this, the flight path for rrunimum time to climb
assumes some interesting var:iations as discussed in the following sections.
  The excess-power curves for a typical supersonic aircraft of the 1960s and
 1970s are schematically shown in Fig. 2.20.1n the subsonic region, Ps curves are
well defined, smooth, and open. However, because of limited thrust capability,
discontinuities existin the transonic and supersonic regions where Ps curves form
closed contours. Also, Ps curves may not exist at lower altitudes for transonic and
supersonic Mach numbers.
        The basic principle in the energy height methodis that the aircraft always follows
the locus of highe:t possible values of Ps without decreasing its ener;y height.
With this criterion, the mirumum time to climb starts in the subsonic region as in
the case of a steady climb and follows the path AB until point B, where a certain
　
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