Cmin
2.4 Example Constraint Analysis
The performance requirements of the Air-to-Air Fighter (AAF) aircraft Request for Proposal (RFP) in Chapter 1 that might constrain the permissible aircraft takeoff thrust loading
(TsL/Wro)
and wing loading(Wro/S) are
listed in Table 2.El. In this example, we shall translate each of these performance requirements into a constraint boundary on a diagram of takeoff thrust loading vs wing loading. This will be accomplished with each requirement by developing an equation for the limiting allowable loadings in the formZ {(TsL/ Wro), (Wro/ S)} = 0
Combat So bsonic Cruise Clim b ...~ Escape ~=:~ ~
oovorEx .. ., S =roC
~ ~ ~ . . .
~°~oen~ ~ / ~
I And # ' ~ ! ... ~ ~
t Land / ~ 1 1 J ~ m l ~ Combat Air Patrol
P = • o R' Warm-up
and Takeoff
Fig. 2.12 Instantaneous weight fraction: typical fighter aircraft.
40 AIRCRAFT ENGINE DESIGN
Cruise
3=0.81
fl = 0.73
Loiter f l = 0.95
I Descend F
~ = o . ~ ~=1
and Takeoff
Cruise
Climb
fl = 0.86 ~ I ~ Descend
and Takeoff
Fig. 2.13 Instantaneous weight fraction--typical cargo and passenger aircraft.
Finally, with the help of the constraint diagram so constructed, we shall select initial values of ( T s L / W r o ) and ( W r o / S ) for the A A E
2.4.1 Constraint Boundary Analysis
To proceed, it is necessary to have preliminary estimates for CL max, the lift-drag polar, and engine performance data. These data are based on current technology
Table 2.El Selected AAF specifications
Mission phases and segments Performance requirement
1-2 Takeoff
Acceleration Rotation
6-7 Supersonic penetration
and 8-9 and escape dash
7-8 Combat
Turn 1 Turn2 Acceleration
13-14 Landing
Free roll Braking
Max Mach number RFP para. C. 1
2000 ft PA, 100°E STo = S G ~- S R <_< 1500 ft kTo = 1.2, #tO = 0.05, max power Vro, tR = 3 s, max power
1.5M/30 kft, no afterburning (if possible) 30,000 ft
1.6M, one 360 deg 5g sustained turn, with afterburning
0.9M, two 360 deg 5g sustained turn, with afterburning
0.8 -+ 1.6M, At < 50 s, max power 2000 ft PA, 100°F, SL = SIR + SBR < 1500 ft krD = 1.15, tFR = 3 S, /ZB = 0.18
Drag chute diameter 15.6 ft, deployment <2.5 s 1.8M/40 kft, max power
CONSTRAINT ANALYSIS 41 for fighter-type aircraft and engines. The variation of thrust with Mach number and density is shown in Fig. 2.Elb that depicts the so-called engine thrust lapse of Eqs. (2.54a) and (2.54b) for three typical values of the throttle ratio (TR). In addition, the instantaneous weight fraction (fl) must be estimated for each item of Table 2.El. Reasonable estimates (see Figs. 2.12 and 2.13) will fall between 1.0 at takeoff and 0.5, say, at landing.
The aerodynamic data of Fig. 2.El a provide the initial estimates of CL m a x , K 1 ,
and CDO required by Eq. (2.1 lb) and its descendants. Please note that the AAF is at first assumed to have an uncambered airfoil, for which K2 = 0. The following AAF calculations will therefore use the relevant equations with K2 set equal to zero.
Note that CL max = 2 was arrived at after many values of CL max were examined, such that takeoff would not overconstrain the problem. According to Table 2.1, this requires a fighter with very low wing sweep (improbable, especially in this case with supercruise requirements) or augmented high lift (such as achieved by using vortex lift created by leading edge extensions, etc.). This challenges the aircraft designer to join in the effort by using advanced technology for this requirement.
The installed propulsion data of a low bypass ratio, mixed flow turbofan engine are selected and plotted in Fig. 2.E lb based on Eqs. (2.54a) and (2.54b). A range of TR from 1.0 to 1.08 is shown because the hot day takeoff and supercruise require- ments desire a TR greater than 1.0 (see Appendix D). It is important to have a good first estimate of the installed engine thrust lapse, which is usually obtained from such open literature information as company brochures, technical papers, and textbooks. 1,2 Another very effective method for obtaining engine thrust lapse is simply to run the performance computer portion of the AEDsys program sup- plied with this textbook for an initial guess of the engine design point. That was the origin of the engine thrust lapse equations of Sec. 2.3.2.
The computations and data required to construct each boundary in the complete constraint diagram of the AAF are contained in the Sec. 2.4.3. The Constraint Analysis software in the AEDsys program makes these calculations effortless.
Before determining the constraint for each item of Table 2.El in detail, we will first illustrate the procedure with the takeoff distance and supercruise phases. This will also help us appreciate the role of the engine TR and select a value capable of meeting these two constraints.
Takeoff. For this illustrative case we consider the airplane accelerated by thrust with no resisting forces whatever in the ground roll. The thrust is balanced by drag forces during the constant velocity rotation. Under these conditions our takeoff constraint boundary equation comes from Case 5, Eq. (2.22), solved for SG, and Eq. (2.26), where sTO = SG + SR, or
STO ~--- [ pgofLmaxOlwet(ZsL/WTO) ] I VRCL max
(2.El) which is in the form
a +b v
a)
0.40 0.35 0.30 0.25 K1
0.20 0.15 0.10 0.05 0.00
b) 1.4 T h 1.2
r u
s 1.0 t L 0.8
a
P 0.6
S
e 0.4
0.2 0.0
2
C D = K 1 C L
+Coo,
CLmax= 2.0
I ' ' ' ' r ' ' ' '
C~o...~ / . /
___ / ~ "'~'XK 1
I i n t n I i = n I
0.5 1.0 1.5
Mach Number
/
u
TR = 1.08 1.05 / _
M a x . . / . . . - - . . \
j -. \
1.00 " ,
"%
\
, \
40 kft
Max
Mil
/
s
TR = 1.08
1
0.040 ,,: 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 2.0
1.05 Coo
. 0 " "
0.0 0.50 1.0 1.5 2.0
Mach Number
Fig. 2.El Preliminary AAF data: a) aerodynamic data; b) installed thrust lapse.
42
CONSTRAINT ANALYSIS 43 Table 2.E2 Thrust lapse variation
with throttle r a t i o - A B on
TR ~)lwe t
1.00 0.6885
1.05 0.8400
1.06 0.8703
1.07 0.9006
1.08 0.9309
From item 1 of Table 2.El, Fig. 2.E 1, and the standard atmosphere of Appendix B, we have
kro = 1.2 /3 = 1.0 0 = 1.0796
MT'o = 0.1 CLmax = 2.0 8 = 0.9298 cr = 8 / 0 = 0 . 8 6 1 2 p = 0 . 0 0 2 0 4 7 s l / f t 3 t R = 3 . 0 s go = 32.17 ft/s 2
For the range of TR the corresponding thrust lapses (or) from Eq. (Z54a) are shown in Table 2.E2.
With these values we can evaluate the coefficients a, b, and c in Eq. (2.El), which, for computational convenience, we solve for WT.o/S in order to obtain
where
( - - ~ ) = { - b + ~ } 2 2 a
10.93
a = b---79.57 c = 1 5 0 0 Otwet( TsL/ WTO )
whence we obtain the matrix of results for Wro/S:
(00)M=0.1 = 1.0818 (80)M=0.1 = 0.9363 STO = 1500ft
(2.E2)
TR TsL/Wro 0.4 0.8 1.2 1.6 2.0 2.4
1.00 Wro/S (lbf/ft 2) 27.3 47.9 64.9 79.6 92.5 104 1.05 Wro/S (lbf/ft 2) 32.2 55.7 74.8 91.1 105 118 1.06 Wro/S (lbf/ft z) 33.2 57.2 76.7 93.2 107 120 1.07 Wro/S (lbf/ft 2) 34.1 58.7 78.6 95.33 110 122 1.08 Wro/S(lbf/ft 2) 35.1 60.2 80.4 97.4 112 125 The takeoff constraint boundaries for this range of throttle ratios are shown in Fig. 2.E2. An airplane with thrust and wing loadings in the region to the left of the boundary can take off within the specified constraints, but those to the right cannot.
44 2.4
AIRCRAFT ENGINE DESIGN
2.0
1.6 T / W
SL TO
1.2
0.8
0.4
40 50 60 70 80 90 100 110 120
W /S (lbf/ft 2)
TO
Fig. 2.E2 Constraint diagram for takeoff and supercruise conditions.
Supercruise.
For constant altitude/speed cruise, Case 1, Eq. (2.12), withCoR
= K2 = 0 givesW~ro ,] K l q +
Coo }
fl /q (Wro/ S)
From item 2 of Table 2.El, Fig. 2.El, and the standard atmosphere of Appendix B, we have
0 = 0.7940 (0o)M=1.5 = 1.1513 3 = 0.2975 K1 = 0.27
Coo
= 0.028 /~ = 0.78 q = 991.6 lbf/ft 2(6o)M=1.5 = 1.0921
The supercruise constraint boundary equation is, therefore,
\~oro]
( TsL ~
0.78ot 2.12 x 10 .4+ (Wro/S)I
(2.E3)CONSTRAINT ANALYSIS 45 Table 2.E3 Thrust lapse variation
with throttle ratio-AB off
TR ado,
1.00 0.3278
1.05 0.4359
1.06 0.4576
1.07 0.4792
1.08 0.5008
with a = t~dry as a function of TR from
whence we obtain the matrix of results for TsL/Wro:
TR Wro/S (lbf/ft 2) 20 40 60 1.00 TsL/Wro 4.25 2.14 1.44 1.05 TsL/Wro 3.19 1.61 1.08 1.06 Tsl./Wro 3.04 1.53 1.03 1.07 TsL/Wro 2.90 1.46 0.986 1.08 TsJ Wro 2.80 1.40 0.944
Eq. (2.54b), as shown in Table 2.E3,
80 100 120
1.10 0.898 0.767 0.827 0.675 0.576 0.788 0.643 0.549 0.752 0.614 0.524 0.720 0.588 0.502 As shown in Fig. 2.E2, this constraint boundary places a lower limit on the allowable wing loading and, together with the takeoff constraint boundary, encloses the "solution space" of allowable combinations of thrust and wing loadings that satisfy the two performance requirements considered here. The main consequence of increased throttle ratio, as expected, is to reduce TsL/Wro by sustaining thrust to higher values of M0 and 00 (see Fig. 2.Elb).
The selection of a thrust loading and wing loading from Fig. 2.E2 is a compro- mise of many factors. For a given Wro, a low (Wro/S) value means large wing area while a high value of (TsJ Wro) results in a large thrust requirement. In addition, a low wing loading reduces the airplane riding qualities and range and can increase the aircraft radar cross section. We would prefer, therefore, relatively low thrust and high wing loadings. We might, based on the constraint diagram of Fig. 2.E2, select 1.0 and 64 lbf/ft 2 for our AAF thrust and wing loadings and an engine throttle ratio of 1.07. However, we must take into account all performance requirements as well as the takeoff ground roll drag in constructing the fighter's complete constraint diagram.
When this is done, we employ the AEDsys Constraint Analysis software to obtain and construct Fig. 2.E3, which is the complete constraint diagram we seek in order to make a judicious selection of (TsL/Wro) and (Wro/S) for the AAE Notice that the solution space in the diagram is bounded with constraints formed by the supercruise, 0.9M/5g combat turn, takeoff, and landing RFP requirements.
It is interesting to superimpose the AAF solution space of Fig. 2.E3 on Fig. 2.3, which contains the wing and thrust loadings of 22 fighter-type airplanes, as is done in the composite Fig. 2.E4. We see that none of the contemporary fighters meets all of the RFP requirements for the AAF, thus the need for a new airplane. However, the larger YF-22 and YF-23 have similar thrust and wing loadings, which might be expected because they have similar requirements.
46 AIRCRAFT ENGINE DESIGN 1.6
1.4
1.2 L 0 A 1.0 D
I N 0.8 G
0.6 T / W
SL TO
0.4
20 40 60 80 100 120
WING LOADING W /S (]bf/ft 2)
TO
Fig. 2.E3 The complete preliminary AAF constraint diagram ( T R = 1.07).
2. 4.2 Selection of the Preliminary Air-to-Air Fighter Design Point We are now in a position to make a preliminary selection of the thrust loading and wing loading that will guarantee that all the flight constraints are met. To do this, we will refer repeatedly to the constraint diagram of Fig. 2.E3. You will find that the tools are provided to allow the solution to be iterated and refined as better information becomes available, until a final converged solution is obtained.
As you can see from Fig. 2.E3, the limits imposed by the sustained 1.6M/5g combat tum, the acceleration requirement, and the maximum Mach are, for this aircraft, not important. Please bear in mind, however, that this will not be the case for aircraft designed for other applications and may not even remain true for the AAF if the underlying assumptions change (e.g., if the afterburner is not used during sustained combat turns). For the time being, then, we will concentrate on the four flight conditions that do form the boundaries of the AAF solution space, namely, supercruise, 0.9M/5g combat turn, takeoff, and landing.
At this stage of the design process, the constraint analysis also can be used to bring down the ultimate takeoff weight (and cost) of the AAE On the one hand, lower thrust loadings lead to smaller engines and greater wing loadings lead to smaller wings. On the other hand, as you will see in Sec. 3.2.7, the fuel consumed in supercruise can be reduced by selecting a wing loading closer to the minimum thrust loading, which is evidently to the right of the solution space. This means that we must focus on design points on the right side of the "bucket" formed
CONSTRAINT ANALYSIS 47 1.6
T H R 1.4 U
S T 1.2 L O A 1.0 D
I N 0.8 G
°° I
Ll%o
0.4 - 20
YF-23 • AAF
Mirage 4000 q
0.9M/5g Turn
Takeoff
F- 106A
. - YF-22
MIG-31 AV-8B • SU-27 Harrier
• MIG-29 F- 16 X-29
JA37 •
Viggen F-20
F-4E •
KFir-C2 [ T-38 MIG-25
• •
F - 1 6 X L • F-5E Mirag T-45
F-117A ~ A-10
40 60 80 100
WING LOADING Wo/S (lbf/ft 2)
120
Fig. 2.E4 Composite thrust loading vs wing loading--fighter aircraft and prelimi- nary AAE
near the juncture of the three constraining lines, preferably those near to the region of previous thrust loading experience, which extends to about TSL/WTO = 1.2 according to Fig. 2.E4. Fortunately, the wing loadings in the region are well within the design experience or the industry, as seen in Fig. 2.E4, and high enough to assure good handling qualities) It is equally important to avoid trying to put too fine a point on our choice, lest the movement of any of the constraining lines caused by improved estimates of aircraft and/or engine performance render it useless.
Following this logic, it appears that the preliminary design point