Concept

2 Constraint Analysis

2.1 Concept

The design process starts by considering the forces that act on the aircraft, namely, lift, drag, thrust, and weight. This approach will lead to the fortunate dis- covery that several of the leading performance requirements of the Request for Proposal (RFP) can be translated into functional relationships between the mini- mum thrust-to-weight or thrust loading at sea-level takeoff

(TsL/Wro)

and wing loading at takeoff

(Wro/S).

The keys to the development of these relationships, and a typical step in any design process, are reasonable assumptions for the air- craft lift-drag polar and the lapse of the engine thrust with flight altitude and Mach number. It is not necessary that these assumptions be exact, but greater accuracy reduces the need for iteration. It is possible to satisfy these aircraft/engine system requirements as long as the thrust loading at least equals the largest value found at the selected wing loading. Notice that the more detailed aspects of design, such as stability, control, configuration layout, and structures, are set aside for later consideration by aircraft system designers.

An example of the results of a typical constraint analysis is portrayed in Fig. 2.1. Shown there are the minimum

TsL/Wro

as a function of

Wro/S

needed for the following: 1) takeoff from a runway of given length; 2) flight at a given altitude and required speed; 3) turn at a given altitude, speed, and required rate;

and 4) landing without reverse thrust on a runway of given length. Any of the trends that are not familiar will be made clear by the analysis of the next section.

What is important to realize about Fig. 2.1 is that any combination of

TSL/WT"o

and

Wro/S

that falls in the "solution space" shown there automatically meets all of the constraints considered. For better or for worse, there are many acceptable solutions available at this point. It is important to identify which is "best" and why.

It is possible to include many other performance constraints, such as required service ceiling and acceleration time, on the same diagram. By incorporating all known constraints, the range of acceptable loading parameters (that is, the solution space) will be appropriately restricted.

A look at example records of thrust loading vs wing loading at takeoff is quite interesting. Figures 2.2 and 2.3 represent collections of the design points for jet engine powered transport-type and fighter-type aircraft, respectively. The thing that leaps out of these figures is the diversity of design points. The selected design point is very sensitive to the application and the preferences of the designer. Pick out some of your favorite airplanes and see if you can explain their location on the constraint diagram. For example, the low wing loadings of the C-20A and C-21A are probably caused by short takeoff length requirements, whereas the high thrust loading and low wing loading of the YF-22 and MIG-31 are probably caused by requirements for specific combat performance in both the subsonic and supersonic

19

20 AIRCRAFT ENGINE DESIGN

1 8 T H R 1.6 U S 1.4

Landing L

0 A 1.2 D

I N 1.0 G

0.8 TsL/Wro

0.6

20 40 60 80 100 120

WING LOADING W /S _TO

(lbf/ft 2)

Fig. 2.1 Constraint analysis--thrust loading vs wing loading.

0.45

0.40

0.35

TSL'/WTo

^{0 . 3 0}

0.25

0.20 P-3

C-21A

• C o n c o r d e

S - 3 0 767-200

A300-600 C-20A A310-200 ... _ . ~ / L1011

~ . . / A321-200

737-600 • m O • 777-300ER

737-800 ~

7~757.300

777-200 • a eC- 17

c-gA. / - A I ~ Z .Kc-loA

B-52H / / \ "~w--- 747-400

C - 1 4 1 B • / / \ \ 747-300

B-2A KC-135R / A380 • - - • 767-400ER

A330-200 A340-300

KC-135A • C-5B

0,15 , , , , I , , , , I , , , , I , , , ,

50 100 150 200

WFo/S (lbf/ft 2)

1 2

Fig. 2.2 Thrust loading vs wing loading--cargo and passenger aircraft.

B-1E

250

C O N S T R A I N T A N A L Y S I S 21

T /W SL TO

1 . 4 0

1 . 2 0

1 . 0 0

0 . 8 0

0 . 6 0 -

0 . 4 0

i i , i , , , i ' i

YF-23 •

YF-22 SU-27

• AV-8B •

MIG-31 Harrier

Mirage •

4000 • F-150 • MIG-29

F-16

• X-29

Mirage • F-20 F-111F

2000 • JA37

Viggen F-4E F-14B/D ~

KFir-C2 • •

• MIG-25 •

F-106A • F 15E

"l

T-38 • • • F/ 18E/F

F-16XL • Mirage F1

T-45 • F/A-18NB

• A-10 F-117A •

T-37

0 . 2 0 ' I I I ~ J I ~ ~ ~ I i ~

2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

W /S (lbflft 2) TO

Fig. 2.3 Thrust loading vs wing loading--fighter aircraft. 1-2

arenas. Where do you think the location would be for the AAF, supersonic business jet, GRA, and UAV?

2 . 2 D e s i g n T o o l s

A "master equation" for the flight performance of aircraft in terms of takeoff thrust loading (TsL/Wro) and wing loading (Wro/S) can be derived directly from force considerations. We treat the aircraft, shown in Fig. 2.4, as a point mass with a velocity (V) in still air at a flight path angle of 0 to the horizon. The velocity of the air ( - V ) has an angle of attack (AOA) to the wing chord line (WCL). The lift

WCL ~ ~ L

OS w ~ ~ WCL

Fig. 2.4 Forces on aircraft.

22 AIRCRAFT ENGINE DESIGN

(L) and drag (D + R) forces are normal and parallel to this velocity, respectively.

The thrust (T) is at an angle 9 to the wing chord line (usually small). Applying Newton's second law to this aircraft we have the following for the accelerations:

Parallel to V:

W W d V

T cos(AOA + ~o) - W sin0 - (D + R) = --all -- (2.1a) go go dt

Perpendicular to V:

L + T sin(AOA + 9) - W c o s 0 = - - a ± W (2.1b) go

Multiplying Eq. (2.1a) by the velocity (V), we have the following equation in the direction of flight:

{T cos(AOA + ~o) - (D + R)}V = W V sin0 + (i)

Note that for most flight conditions the thrust is very nearly aligned with the direction of flight, so that the angle (AOA + ~o) is small and thus cos(AOA + ~) .~ 1.

This term will therefore be dropped from the ensuing development, but it could be restored if necessary or desirable. Also, multiplying by velocity has transformed a force relationship into a power, or time rate of change of energy, equation. This will have profound consequences in what follows. Since V sin 0 is simply the time rate of change of altitude (h) or

V sin 0 = - - dh (ii)

dt

then combining equations (i) and (ii) and dividing by W gives T - ( D + R ) v = d { h V 2} dze

w = - g

^(2.2a)

where Ze = h + V2/2go represents the sum of instantaneous potential and kinetic energies of the aircraft and is frequently referred to as the "energy height." The energy height may be most easily visualized as the altitude the aircraft would attain if its kinetic energy were completely converted into potential energy. Lines of constant energy height are plotted in Fig. 2.5 vs the altitude-velocity axes. The flight condition (h = 20 kft and V = 1134 fps) marked with a star in Fig. 2.5 corresponds to an energy height of 40 kft. Excess power is required to increase the energy height of the aircraft, and the rate of change is proportional to the amount of excess power.

The left-hand side of Eq. (2.2a) in modern times has become recognized as a dominant property of the aircraft and is called the weight specific excess power

d z e __ d h + (2.2b)

P s - dt dt

CONSTRAINT ANALYSIS 23

A

1

t i

t U

d

e

(ft)

50,000

40,000

30,000

20,000

10,000

i i I i ~ i I i i i i i i i i i i i i i i

0 500 1000 1500

Velocity (fps)

Fig. 2.5 Lines of constant energy height (ze).

This is a powerful grouping for understanding and predicting the dynamics of flight, including both rate of climb ( d h / d t ) and acceleration ( d V / d t ) capabilities.

Ps must have the units of velocity.

If we assume that the installed thrust is given by

T = etTsL (2.3)

where ot is the installed full throttle thrust lapse, which depends on altitude, speed, and whether or not an afterburner is operating, and the instantaneous weight is given by

W = fl WTO (2.4)

where fl depends on how much fuel has been consumed and payload delivered, then Eq. (2.2a) becomes

Wro - ~ \ Tff-~ro + (2.5)

Note: Definition of ot

Particular caution must be exercised in the use of a throughout this textbook because it is intended to be referenced only to the maximum thrust or power available for the prevailing engine configuration and flight condition. For example, an afterbuming engine will have two possible values of ot for any flight condition,

24 AIRCRAFT ENGINE DESIGN

one for mil power and one for max power (see Sec. 1.11E of RFP). You must differentiate carefully between them. Lower values of thrust are always available by simply throttling the fuel flow, thrust, or power.

It is equally important to remember that both T and TSL refer to the "installed"

engine thrust, which is generally less than the "uninstalled" engine thrust that would be produced if the external flow were ideal and created no drag. The dif- ference between them is the additional drag generated on the external surfaces, which is strongly influenced by the presence of the engine and is not included in the aircraft drag model. The additional drag is usually confined to the inlet and exhaust nozzle surfaces, but in unfavorable circumstances can be found anywhere, including adjacent fuselage, wing, and tail surfaces. The subject of "installed" vs

"uninstalled" thrust is dealt with in detail in Chapter 6 and Appendix E.

Now, using the traditional aircraft lift and drag relationships,

L = nW = qCLS (2.6)

where n = load factor = number of g's (g = go) _L to V(n = 1 for straight and level flight even when dV/dt ~ 0),

D ^{= q C D S} (2.7a)

and

R = qCDRS (2.7b)

where D and CD refer to the "clean" or basic aircraft and R and CDk refer to the additional drag caused, for example, by external stores, braking parachutes or flaps, or temporary external hardware. Then

CL-- qS -- q Further, assuming the lift-drag polar relationship,

CD = KIC2L + K2CL + CDO (2.9)

Equations (2.7-2.9) can be combined to yield

D + R = q S { K I ( q f l ~ ° ) 2 + K 2 ( n ~ f l q V ~ ° ) + C D o + C D R } (2.10) Finally, Eq. (2.10) may be substituted into Eq. (2.5) to produce the general form of the "master equation"

q' o) ]

WTO t~ [ flWro L \ q + K2 ^{- -} JC CDO "~- CDR "J- ~- (2.11) It should be clear that Eq. (2.11) will provide the desired relationships between TSL/WTO and Wro/S that become constraint diagram boundaries. It should also be evident that the general form of Eq. (2.11) is such that there is one value of

CONSTRAINT ANALYSIS 25 WTo/S for which TSL / WTO is minimized, as seen in Fig. 2.1. This important fact will be elaborated upon in the example cases that follow.

Note: Lift-drag polar equation

The conventional form of the lift-drag polar equation is 3 CD ~- C n m i n + K'C 2 + K ' ( C L - C L m i n ) 2

where K' is the inviscid drag due to lift (induced drag) and K" is the viscous drag due to lift (skin friction and pressure drag). Expanding and collecting like terms shows that the lift-drag polar equation may also be written

/¢" t I t"~ 2

CD ~-" (K' + K")C2L - (2K't CLmin)CL q- (CDmin ~ .~ ~Lmin]

o r

where

CD = K1C 2 + K2CL + CDO (2.9)

K1 = K' + K"

K2 = -2K"CLmin l¢,'tl t~2 CDO = CD min -'[- *~ ~ L min

Note that the physical interpretation of CDO is the drag coefficient at zero lift. Also, for most high-performance aircraft CL min ~ 0, SO that K2 ,~ 0.

A large number and variety of special cases of Eq. (2.11) will be developed in order both to illustrate its behavior and to provide more specific design tools for constraint analysis. In all of the example cases that follow, it is assumed that the ot of Eq. (2.3), the fl of Eq. (2.4), and the K1, K2, and CDO of Eq. (2.9) are known. If they change significantly over the period of flight being analyzed, either piecewise solution or use of representative working averages should be considered.

2.2.1 Case 1: Constant Altitude~Speed Cruise (Ps = O)

Given: dh/dt = O, d V / d t = 0, n = 1 (L = W), and values of h and V (i.e., q).

Under these conditions Eq. (2.11) becomes

TsL _ fl Ka + K2 + (2.12)

Wro ot q \ - - S - ] f l / q ( W r o / S )

This relationship is quite complex because TsL/Wro grows indefinitely large as WTo/S becomes very large or very small. The location of the minimum for T s J Wvo can be found by differentiating Eq. (2.12) with respect to WTo/S and setting the result equal to zero. This leads to

S -Iminr/W K1

o~. &~ , , ~" ~ "' -e ~11 -~1 ~ ~- + ~ ~+ ~ + ,~ ~ V t~

"* ~l'~ ~ + || Jr-

e~. =~ r~" Cb ~,-~ ..

a.~ ~.-~ g.~ ~ ~~

-=: .~ + n

~~ + ~ ~ ~ ~"

+ ~ o ~ ~"

~1 ~ ~ b~

II

o%1o

~.~.

=

i ! IiI- ~ bO -4-

33 O 33 -I1 --I ITI Z G') Z m E3 m or) Z

CONSTRAINT ANALYSIS 27

I I I I I I I

L = n W I , . - - A W

,~ ~ ~ bank angle : (._W_W ~V 2 ~ : cos-'(1/n)

" Y W

R~

'1

Fig. 2 . 6 F o r c e s o n a i r c r a f t in t u r n .

and