Mattingly J.D., Heiser W.H., Pratt D.T. Aircraft Engine Design

Подождите немного. Документ загружается.

50 AIRCRAFT ENGINE DESIGN

Careful note should be taken of the fact that these hand calculations completely

duplicate the AEDsys Constraint Analysis computations obtained by using the

AAF.AED file. Any minor differences are simply the result of the cumulative

effects of rounding off. Thus we see that one of the motivating forces for carrying

four significant figures is to minimize the impact of these small imperfections on

the overall results.

Mission phase 1-2: Takeoff, no obstacle.

2000 ft PA, IO0°F,

kro

= 1.2,

/ZTo = 0.05, tR = 3 s,

sro = sc + sz¢ <

1500 ft, max power.

From Case 6, Eqs. (2.25) and (2.26) with

STo = so + sR

-- Ta J

where

a m

fl ln{1--~TO/I(

~ TSL

PgO~TO Wro

#TO) "-~To max

b = tlckro~/2fl/(pCLrnax) C = Sro

We estimate ot and fl to be constant at appropriate mean values. A conservative

estimate for ~ro for the AAF is obtained by assuming (CDR --

IzToCL) = 0 and

evaluating CD at

CL = CL max/k2o •

With

fl = 1.0 p = 0.002047 slug/ft 3

max = 2.0

= 1.389

K1 = 0.18

CDO

= 0.014

~TO

= 0.3612

cr = 0.8613 0 = 1.0796 8 = 0.9298

= 1.07

MTO

= 0.1 (00)M=0.1 = 1.0818 (60)M=0.1 = 0.9363

(awet)M=O.1

= 0.9006

STO

= 1500 ft

then

a = -42.061n

whence

{1-

0.2599/[0.9006(W~o ) - 0.051 }

b=79.57 c=1500

TSL/ WTO WTo / S,

lbf/ft 2

0.4 15.5

0.8 46.4

1.2 68.7

1.6 86.9

2.0 103

CONSTRAINT ANALYSIS 51

Mission phases 6-7 and 8-9: Supersonic penetration and escape dash.

1.5M/30 kft, no afterburning.

From Case 1, Eq. (2.12), with Con and Ka = 0

With

then

whence

-- -~ X,q \---~-] + fi/q'(--~TTo/S) ]

/3 = 0.78 0 = 0.7940 0o = 1.1513

3o = 1.0921

= 1.07 adry = 0.4792

Kl = 0.27 CDo = 0.028

3 = 0.2975

q = 991.6 lbf/ft 2

(>)

~TO] = 3.457 x 10 .4 +

57.94

( Wro / S)

WTo / S,

lbf/ft 2

TSL/ WTO

20 2.90

40 1.46

60 0.986

80 0.752

100 0.614

120 0.524

Mission phase 7-8: Combat turn

1. 1.6M/30 kft, one 360 deg 5g sustained

turn, with aflerburning.

From Case 3, Eq. (2.15), with

CDR

and K2 = 0

Wro -- ol [ q \ S ] /3/q~ro/S) J

With

fl = 0.78

30 = 1.2645

Coo

= 0.028

0=0.7940 0o=1.2005

= 1.07

Olwe t

= 0.7829

K1 = 0.288

3 = 0.2975

q = 1128 lbf/ft 2

then

W-~ro] = 4.960 x 10 -3 -- +--

40.34

( Wro / S)

52 AIRCRAFT ENGINE DESIGN

whence

WTo/S, lbff~ 2 TsL/WTo

20 2.12

40 1.21

60 0.970

80 0.901

100 0.899

120 0.931

Mission phase 7-8: Combat turn 2.

0.9M/30 kft, two 360 deg 5g sustained

turns, with afterburning.

From Case 3, Eq. (2.15), with

CDn and K2 = 0

With

then

whence

Wro ~ I q \ S /

fl/q(-~ro/S)

fl = 0.78 0 = 0.7940 0o = 0.9226 8 = 0.2975

80 = 0.5032

= 1.07

Olwe t

= 0.5033 q = 357.0 lbf/ft 2

Coo = 0.016 K1 = 0.18

( TsL ] = 0.01524(--~) + --

11.35

( Wro / S)

WTo/S,

lbf/ft 2

100

120

TSL/ WTO

0.872

0.893

1.10

1.36

1.64

!.92

Mission phase 7-8: Horizontal acceleration.

50 s, max power.

From Case 4, Eq. (2.18a), with CDR and K2 = 0

0.8 --+ 1.6M/30 kft, at At _<

WTO -- "~ fl/q(WTo/S) + g~

We obtain approximate constant values of q,

KI,

CDO,

and a at a mean Mach

number of 1.2.

CONSTRAINT ANALYSIS 53

With

/~ = 0.78

30 = 0.7215

CDO

= 0.028

At = 50 s

then

TsL

WTO

whence

0=0.7940 0o=1.0227 3=0.2975

= 1.07

Olwe t =

0.7216 q = 634.6 lbf/ft 2

Kl =0.216 a=994.8ft/s AM=0.8

= 2.870 x 10-4(--~) -q - --

24.62

( Wro / S)

+ 0.5348

WTo/S, lbf/ft 2 TsL/WTo

20 1.77

40 1.16

60 0.963

80 0.866

100 0.810

120 0.775

Mission phase 13-14: Landing, no reverse thrust.

2000 fl PA, 100°F,

kTD

1.15,

tFR

3 s,/zB = 0.18,

SL = SFR "Iv SB ~

1500 ft, GFE drag chute,

diameter 15.6 ft, deploymemt < 2.5 s.

From Case 7, Eqs. (2.33) and (2.37) with

SL = Srn + sn

where

(W~o)__ = {-b+ ~/-bT+4ac] z~a

a m m

PgO~L

,n {. + + -- --

(--~) TSL ~CLmax 1

b = tFRkrD~/2fl/(pCLmax) C = SL

We assume a drag chute drag coefficient of 1.4 based on the RFP chute area

of 191 ft 2. Estimating the airplane wing area to be about 500 ft 2, the chute drag

coefficent

(Cone)

is 0.5348 referred to the wing area. A conservative estimate of ~L

for the AAF is obtained by evaluating

at 0.8 of the touchdown lift coefficient

(CLmax/k2o)

and

by assuming

(Con - IZBCL) = O.

With

/~ = 0.56 p = 0.002047 slug/ft 3 CLmax = 2.0

CDnc

= 0.5348

CL = 1.210 K1 = 0.18 CD0 = 0.014

= 0.2775

~L=0.8123 a=0

SL=1500ft

54 AIRCRAFT ENGINE DESIGN

then

whence

a=14.47 b=57.06 c= 1500

Wro/S

= 70.6 lbf/ft 2

With no drag chute, the same assumptions and computational methods yield a

wing loading constraint of 50.8 lbf/ft 2.

Maximum Mach number. M = 1.8 at 40,000 ft, max power.

From Case 1, Eq. (2.12), with CDR and K2 = 0

Tsz _ fl K1 -[-

Wro ot q \ ~-/ fl/q(WTo/S)

With

/3 = 0.78 0 = 0.7519 (00)M=1.8 = 1.2391

(~0)M=1.8 = 1.0676

= 1.07

Olwe t

= 0.5575

K1 = 0.324 Coo = 0.028

then

whence

= 0.1858

q = 891.8 lbf/ft 2

(TsL]

(_~) 44.79

~OTOJ = 3.965 X 10 -4

+ (WTo/S)

WTo / S,

lbf/ft 2

TSL/ Wro

20 2.25

40 1.14

60 0.770

80 0.592

100 0.487

120 0.421

References

1"2002 Aerospace Source Book,"

Aviation Week and Space Technology,

14 Jan. 2002.

2Jane's All the World's Aircraft, 2000-2001,

Jane's Yearbooks, Franklin Watts, Inc.,

New York, 2000.

3 Raymer, D. P.,

Aircraft Design: A ConceptualApproach,

3rd ed., AIAA Education Series,

AIAA, Reston, VA, 2000.

4Torenbeek, E.,

Synthesis of Subsonic Airplane Design,

Delft Univ. Press, Delft, the

Netherlands, 1976.

5Jonas, E,

Aircraft Design Lecture Notes,

U.S. Air Force Academy, Colorado Springs,

CO, 1984.

6Mattingly, J. D.,

Elements of Gas Turbine Propulsion,

McGraw-Hill, New York, 1996.

Mission Analysis

3.1 Concept

With preliminary design values of takeoff thrust loading (TsL/Wro) and wing

loading (Wro/S) now in hand, the next step is to establish the scale of the aircraft

via the estimation of gross takeoff weight (Wro). This will be accomplished by

flying the aircraft through its entire mission on paper.

The key fact is that WTo is simply the sum of the payload weight (We), the

empty weight (WE), and the required fuel weight (WF), or

WTO = Wp -t- WE -t- WF

(3.1)

These will now be considered in turn.

Wp is specified in the Request for Proposal (RFP) and comes in two parts. The

first is the expendable payload weight, which is delivered during the trip (WpE),

such as cargo or ammunition. The second is the permanent payload weight, which

is carried the entire mission (Wpp), such as the crew and passengers and their

personal equipment.

WE consists of the basic aircraft structure plus any equipment that is permanently

attached, such as the engines, the avionics, the wheels, and the seats. In short, WE

includes everything except Wp and WF. WE can be estimated as a fraction of

Wro, as shown in Figs. 3.1 and 3.2, which correspond to conventional, lightweight

metal construction. The ratio of WE to Wro has obviously depended on the type of

aircraft and its size, but the range of this parameter is not wide. Because WE/Wro

varies slowly with Wro, an initial value may be obtained from an initial estimate

of WTO and any necessary correction made when WTO becomes more accurately

known.

When one contemplates the enormous range of ages and types of aircraft dis-

played in Figs. 3.1 and 3.2, and the relatively narrow range of WE/WTo, it is

possible to conclude that "practical" aircraft have "natural" empty weight frac-

tions that provide reliable future projections. This observation should increase

your confidence in their validity.

WF represents the fuel gradually consumed during the entire mission. Except

for the instantaneous release of WpE, the aircraft weight decreases at exactly the

same rate at which fuel is burned in the engine. The rate of fuel consumption,

in turn, is simply the product of installed engine thrust (T) and installed engine

thrust specific fuel consumption (TSFC). T can be found from whatever version

of Eq. (2.1) is most convenient, while TSFC depends on the engine cycle, flight

conditions, and throttle setting and must initially be estimated on the basis of

experience.

The fuel consumption analysis has several fortunate benefits. For one thing, it

results in calculations based on relatively little information. For another, it reveals

56 AIRCRAFT ENGINE DESIGN

0.70

0.65

0.60

0.55

0.50

0.45

0.40

0.35

0.30

Fig. 3.1

i i i

C-7A

£ -12A C-123

• ~ •767-200

~o • C-9A

777-200

A310 •

UV-18 •

C-21A A300 A330

0C-140 • •

757-200 • L-1011

S-3 777-300ER

767-400ER A340

',-20A • C-130 • C-5B 747-400

• C-170 C.5A O• ~

Concorde

747-300

C-141B • • KC-10A

B-1B •

KC-135R •

A380

KC-135A •

, I , I , I , I , I , I ,

200 400 600 800 1000 1200

(1,000

lbfs)

Weight fractions of cargo and passenger aircraft, l'z

1400

the "best" way to fly certain legs for minimum fuel usage. Finally, it shows that the

fuel burned during each mission leg is a fraction of the aircraft weight starting the

leg, whence WF becomes a calculable fraction of

Wro.

Most of the analysis below

is devoted to the development of the "weight fraction" equations needed for the

many possible kinds of mission legs.

Substitution of the derived results for

We/Wro

and

Wr/Wro

into Eq. (3.1)

yields an unambiguous value for

Wro

as a function of

Wp,o

and

Wee

(see

0.80

0.75

0.70

0.65

°.6°

0.55

0.50

0.45

o4ol

Fig. 3.2

I I

• X-29A

F-102A

F- 16C/D

F-16A/B

F-104C

YF16

O•

YF17

F-15

F-106A •

F-15C

F-105G

F-14A

F-117A •

F-100D • F-4E •

F-IO1B

F/A-18E

• F-14B

I I I F I I

30 40 50 60 70 80

W (1,000 lbfs)

Weight fractions of fighter aircraft. 12

F-11

100

MISSION ANALYSIS 57

Sec. 3.2.12). Then

TSL

and S are found by multiplying

Wro

by thrust loading

and wing loading, respectively.

which may be rewritten as

3.2 Design Tools

The following material might very well be called the "thermodynamics of flight"

because it deals largely with the way the thrust work of the engine is used by the

aircraft. We expect that it should have a special appeal to engine designers, who

always deal with the conversion of energy from one form to another, but seldom

see aircraft treated in this manner.

We begin by considering the rate at which aircraft weight is diminishing as a

result of the consumption of fuel, namely

dWF

.... TSFC × T

(3.2)

dW T

-- -- TSFC-- dt

(3.3)

W W

Note again that T is the installed thrust and

TSFC

is the installed thrust specific

fuel consumption. Also, since

T T dt Tds

dt -- -- ds --

W Wds WV

then this portion of Eq. (3.3) represents the incremental weight-velocity specific

engine thrust work done as the amount of fuel dWF is consumed. Just as it is

with any other thermodynamic situation, this engine thrust work will be partly

invested in the mechanical energy (potential plus kinetic) of the airplane mass

and partly "dissipated" into the nonmechanical energy of the airplane/atmosphere

system (e.g., wing tip vortices, turbulence, and aerodynamic heating). The ratio

of mechanical energy to "dissipation" will, of course, depend on the type of flight

under consideration, as will be seen in what follows.

The main work of this section will be to determine how Eq. (3.3) can be integrated

in order to obtain "weight fractions"

Wf a/ __. Wl

Wi, itial Wi

for a variety of mission legs of practical interest. From a mathematical point

of view, proper integration of Eq. (3.3) requires a knowledge of the behavior

of the thrust specific fuel consumption and the "instantaneous thrust loading"

{T/W = (ot/~)(TsL/Wro)}

as a function of time along the flight path. Experience

indicates that the integration can be separated into two distinct classes, cor-

responding to Ps > 0 (type A) and Ps = 0 (type B). These will now he considered

in turn.

Instantaneous thrust loading behavior: type A, Ps > O.

When Ps > 0,

some of the thrust work is invested in mechanical energy. Also, it is generally true

that specific information is given regarding the amount of installed thrust applied,

58 AIRCRAFT ENGINE DESIGN

as well as the total changes in altitude (h) and velocity (V) that take place, but

not the time or distance involved. In fact, the usual specification for thrust is the

maximum available for the flight condition, or T = a

TsL.

Examples of type A are

found as Cases 1-4: 1) constant speed climb, 2) horizontal acceleration, 3) climb

and acceleration, and 4) takeoff acceleration.

Progress toward a solution may now be made by using Eq. (2.2a) in the form

7" V dt = 7" ds = d(h +

V2/2go) _ dze

(3.4)

W W 1 -u 1 -u

where

D+R

u -- -- (3.5)

whence, combining Eqs. (3.3) and (3.4),

dW TSFC

-- -- -- d(h +

V2/2go)

(3.6a)

W V(1

-u)

dW TSFC

-- -- -- dze

(3.6b)

W V(1 -u)

The quantity u determines how the total engine thrust work is distributed between

mechanical energy and dissipation. More precisely, Eq. (3.5) shows that u is the

fraction of the engine thrust work that is dissipated, so that (1 - u) must be the

fraction of the engine thrust work invested in mechanical energy. This is further

confirmed by Eq. (3.4), which reveals that (1 - u) equals the change in mechan-

ical energy (W dze) divided by the total engine thrust work

Tds.

Note that when

T = D + R and u = 1, all of the thrust work is dissipated, and the type A analysis

yields no useful results.

The actual integration of Eq. (3.6a) is straightforward and depends only upon

the variation of

{TSFC/V(1

- u)} with altitude and velocity. When this quantity

remains relatively constant over the flight leg, as it frequently does, the result is

~-/Wf- exp {

v~TSFC(V2)}u) ~go

-- ~A h + (3.7a)

-- _ I TSFC |

Wfwi --

exp [ - V(1

- u) AZe I

(3.7b)

where

AZe

is the total change in energy height. Otherwise, the integration can

be accomplished by breaking the leg into several sm~ller intervals and applyng

Eq. (3.7) to each. The overall

Wf/Wi

will then be the product of the results for

the separate intervals.

Equations (3.6) and (3.7) highlight the fact that, within certain limits, potential

energy and kinetic energy can be interchanged or "traded." For example, if there

were no drag (u = 0) and

were constant, as in an unpowered dive or zoom climb,

no fuel would be consumed and dW = 0 or

Wf = Wi.

This, in turn, reveals the

MISSION ANALYSIS 59

forbidden solutions of Eqs. (3.2), (3.6), and (3.7) for which W would increase

during the flight leg and which correspond to T < 0 and

dZe <

0. Lacking special

devices onboard the aircraft that could convert aircraft potential and/or kinetic

energy into "fuel," it must be true for any condition of flight that dW < 0 and

w: <_wi.

fs and the minimum fuel path.

As already mentioned, the fuel consumption

analysis also reveals the "best" way to fly type A legs for minimum fuel usage.

Equations (3.2), (2.2b), and (2.3) may be combined to yield

ot TSFC

dWF =

T x TSFC x dt = TsL dZe

where

T-(D+R)} T

P~= ~ V=~(1-u)V

Defining the fuel consumed specific work

(fs)

Ps TSL dze

"-- --

(3.8)

ot TSFC dWF

then

f2 f Ze2 dz e

WF~ 2 = dWF = TsL

(3.9)

tl Zel

Equation (3.9) shows that the minimum fuel-to-climb flight from

Zel

(hl, Wl) to

Zea (he,

V2) corresponds to a flight path, that produces the maximum thrust work

per unit weight of fuel consumed at each energy height level in the climb, i.e., the

maximum value of fs at each

Ze.

As you can see from Eq. (3.8), the units of fs are

feet.

Constant fs and

contours in the altitude-velocity plane (e.g., see Fig. 3.E6

in Sec. 3.4.4) provide a graphical method for finding the maximum f~ at any

and hence the minimum fuel-to-climb path from Zel to

Ze2.

This is analogous to

using the P~ and ze contours of Fig. 2.E5 to find the minimum time-to-climb path

in conjunction with the time-to-climb equation from Eq. (2.2b).

fl 2 fZe ze2dze

At = dt =

, P,

Because fs and Ps are closely related through Eq. (3.8), it should not be sur-

prising to find that the minimum fuel-to-climb and time-to-climb paths are similar.

This is a fortuitous result because there is little conflict between the alternatives.

Instantaneous thrust loading behavior: type B, Ps

-- 0. When Ps = 0, all

of the thrust work is dissipated. Also, it is generally true that the speed and alti-

tude are essentially constant, and specific information is given regarding the total

amount of time (At or

As/V),

which elapses. Examples of type B are found as

Cases 5-11: 5) constant speed cruise, 6) constant speed turn, 7) best cruise Mach