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

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with

AIRCRAFT ENGINE DESIGN

WpE2 = 657 lbf

/~ = 0.7550

WTo

= 24,0001bf

then

Wro - Wp~

- 0.9637 and

Wro

For mission phase 7-8 we have

-I 0.8734

fl = 0.7276

= 0.7276

Mission phase 8-9: Escape dash.

1.5M/30 kft, As89 = 25 n miles, no

afterbuming.

The computational procedure is identical with that of segment G. Here the

distance is smaller (25 n miles vs 92.99 n miles) and the fraction of takeoff weight

(fl) is less (0.7276 vs 0.8880), making CL smaller (0.04696 vs 0.05731) and

CD/CL

greater (0.6090 vs 0.5041). The net effect is a higher weight fraction for

this leg (0.9795 vs 0.9382) and less fuel consumed, as

-I 0.9795

fl = 0.7127

Mission phase 9-10: Minimum time climb.

1.5M/30 kft --~

BCM/BCA,

mil

power.

In this leg kinetic energy is exchanged for potential energy as the airplane

climbs from 30,000 ft to the Case 7 48,000 ft

(BCA)

and the Mach number is

reduced from 1.5 to 0.9 (BCM). Using the initial and final values of h and V in the

following listing, we find that the energy height diminishes by 4800 ft. We assume,

therefore, in accordance with the RFP, a constant energy height maneuver for the

weight fraction calculation of this phase.

From Case 11, Eq. (3.42), with

CDR = 0

]--I = exp {-(C,-q-CeM)~/CO(~L)At

]

9 10

with (assuming the vertical speed is 0.5 of

Vmia) "

A h V Zmia

At -- 0.5Vmi~a

2g0

Zei -- havg

MISSION ANALYSIS 91

and

Then,

= 0.7941

= 1492fffs

30,000fl

Zei =

64,600h

hf =48,000h

= 0.7519

Vf=

871.2fffs

z~ = 59,800h

AZe

= --4800h

ha~=39,000h

Vm~

= 1284fffs

A t = 28.04 s

Omm

= 0.7519

Mmid

= 1.326

$mid=

0.1949

CLmia

= 0.08986

Klmid

= 0.2387

K2mid = 0

C oomid =

0.028

Comia

= 0.2993

(Co/CL)mia

= 0.3330

C1 + C2Mmid

= 1.298 h -1

As = 5.131 n miles

= 0.9971

9 10

fl = 0.7106

This is a conservative estimate in as much

as AZe =-

-4800 ft < 0.

Mission phase 10-11: Subsonic cruise climb. BCM/BCA,

As10_ll =

150 n miles, mil power.

Refer to the outbound subsonic cruise climb leg data, mission phase 3-4. The

weight fraction of this leg is lower (0.9698 vs 0.9736) due only to a higher

cruise distance (144.9 vs 127.0 n miles). The starting

BCA

is higher (48,000 ft

vs 42,000 ft) because of the airplane's lower weight (0.7106

Wro

vs 0.9584

Wro).

Then,

-'[= 0.9698

10 11

fl = 0.6891

Mission phase 11-12: Descend.

BCM/BCA ---* Mtoiter/lO lift.

--I = 1.O

11 12

~ = 0.6891

Mission phase 12-13: Loiter.

Mzoit, r/ l O

kft, 20 min., mil power.

Refer to data for the combat air patrol loiter leg, mission phase 5-6. Compared to

that leg the weight fraction of this leg is about the same (0.9677 vs 0.9675) because

the offsetting effects of the dimensionless temperature 0 (0.9312 vs 0.7941) and

Mach number (0.3940 vs 0.6970). Because of the lower vehicle weight and altitude

92 AIRCRAFT ENGINE DESIGN

ofthisleg, theloiterMach numberismuchlessthanthatofthe

CAPleg. Then,

= 0.9677

12 13

fl =

0.6668

Mission phase 13-14: Descend and

land.

Mloiter/lO

kft --> 2000 ft PA,

100°F.

=I.0

13 14

fl = 0.6668

The results of all of the AAF weight fraction computations in the preceding are

summarized in Table 3.E3 and Fig.

3.E3.

Minimum fuel-to-climb path.

Although the AAF is not required by the RFP

to fly a minimum fuel climb, this section is included to show the application of the

material on minimum fuel path included as a note in Sec. 3.2 and for comparison

with the minimum time path of Fig. 3.E2. Values of fuel consumed specific work

(fi) were calculated for the AAF at military power using Eq. (3.8) for the following

data:

TSL/WTO :

1.25

WTo/S

= 64 lbf/ft 2 /3 = 0.97

q -}-

C2M :

0.9 + 0.3M

The results of these calculations are presented in Fig. 3.E6 as contours of constant

fuel consumed specific work (fi) plotted in the velocity-altitude space with con-

tours of constant energy height

(Ze).

The minimum fuel-to-climb path from energy

60-

50-

40--

Alt

30-

(kft)

20-

fs (min = 0, max = 2800, incr

200 kft) Standard Day Atmosphere

10-

I I I I I I I~

100 500

Fig. 3.E6

i i I I I I / I i i /

• I i i i l I I I i

1000 1500

Velocity

(fl/sec)

Minimum fuel-to-climb path on

chart.

2000

W/S = 64

"TAV - 1.25

_Beta 0.97

g's

- 1.00

Tmx

= 1.0

"TR = 1.07

-AB

off

Engine

TSFC #2

"Aircraft

Min

fuel

MISSION ANALYSIS 93

height

Zel(hl,

V1)

to energy height

Ze2(h2,

V2)

corresponds to a path having a max-

imum value of fs at each Ze. The resulting minimum fuel-to-climb path from a Ze

of 12 kft at sea level (where 111 = 879 ft/s) to a Ze of 95 kft at 57 kft altitude (where

V2 = 1564 ft/s) is also shown on Fig. 3.E6.

Based on the comparison of Figs. 3.E2 and 3.E6, the following comments can

be made:

1) The contours Ps = 0 and f~ = 0 are the same. This follows from the definition

of f~ given in Eq. (3.8).

2) The minimum fuel-to-climb path is similar but different from the minimum

time-to-climb path for evident reasons [see Eq. (3.8)].

3) The minimum fuel-to-climb path stays subsonic to a higher altitude (about

37 kft) than that of the minimum time-to-climb (about 15 kft).

You may find it useful to know that the weight fraction for a type A flight path

can be expressed conveniently in terms of Tslff Wvo, fs, Ze, and/3 by dividing

Eq. (3.9) by the aircraft weight at the start W 1 (= •

WTO),

noting that

WFI_2 =

W1 - W2, and using the definition of 1-[1 2( = W2/WO. Making these substitutions

gives

Ze2

w 12 _1_1-I_ raze

W1 1 2 fl Wro fs

Zel

The resulting equation for the aircraft weight fraction 1-[1 2 is

TSL f Ze2 dZe

I-I=l fl~o Z

1 2

Zel

Application of this equation to the subsonic portion of the minimum fuel-to-

climb path of Fig. 3.E6 (Zel = 12 kft, Ze2 = 52 kft) where fs is approximately 2800

kft gives the following estimate for the weight fraction:

-I 0.9815

References

1"2002 Aerospace Source Book:' Aviation Week and Space Technology, 14 Jan. 2002.

2jane's All the World Aircraft, 2000-2001, Jane's Yearbooks, Franklin Watts, Inc., New

York, 2000.

3Raymer, D. P., Aircraft Design: A Conceptual Approach, 3rd ed., AIAA Education

Series, AIAA, Reston, VA, 2000.

4jonas, F., Aircraft Design Lecture Notes, U.S. Air Force Academy, Colorado Springs,

CO, 1984.

Engine Selection: Parametric Cycle Analysis

4.1 Concept

The preparations are now complete to begin the design of the specific engine

and airframe that can best perform the prescribed mission. Engine and airframe

designers must now part company in order to concentrate intensely on perfecting

their own product. They will come together again and share information only after

they know a good deal more about their machines.

As engine designers, the required installed sea level static thrust (TsD of the

engine(s) is now known, as well as the assumed behavior of thrust and specific

fuel consumption with altitude and Mach number, depending on whether or not an

afterburner is operating. Much good work remains to be done before the right

engine is known in enough detail to decide how well the task can really be ac-

complished. Most of that work is based on a comprehensive, efficient method of

jet engine cycle analysis pioneered by Frank Marble of the California Institute of

Technology and perfected by Gordon Oates of the University of Washington as

presented in Ref. 1 and by Jack D. Mattingly of Seattle University as presented in

Ref. 2.

The object of parametric cycle analysis is to obtain estimates of the perfor-

mance parameters (primarily specific thrust and thrust specific fuel consumption)

in terms of design limitations (such as maximum allowable turbine temperature

and attainable component efficiencies), the flight conditions (the ambient pres-

sure, temperature, and Mach number), and design choices (such as compressor

pressure ratio, fan pressure ratio, bypass ratio, and theta break). The comparative

simplicity of the aerothermodynamic analysis is achieved by treating each stream

as the one-dimensional flow of a perfect gas, by representing nonideal component

behavior through realistic efficiencies, and by using a compact and intuitively ap-

pealing set of nomenclature. Please note that the inclusion of nonideal (or real)

component behavior to the degree desired leads to a close imitation of nature, even

for very complex configurations.

Engine design starts with parametric cycle analysis, which presumes that all

design choices are still under control and that the size of the engine has yet to be

chosen. The mental image usually attached to this stage of development is that of

a "rubber" engine, whose size is arbitrary and performance is largely known in

terms of ratios or "specific" terms. Therefore, each complete set of design choices

results in a piece of hardware with its own geometry and operating characteristics.

This stage of design is usually referred to in the literature as "on-design" or

"design point" cycle analysis. Our experience has taught us that the term parametric

cycle analysis is preferable because it is always difficult and often impossible to

identify a single point that dictates the design. The final design of an engine

is based on its performance over the entire aircraft mission, and the winner is

96 AIRCRAFT ENGINE DESIGN

chosen because of its balanced behavior over the whole flight spectrum and seldom

(if ever) matches any one of the mission flight conditions. Consequently, we have

learned that it best to refer to the operating point at which all design quantities are

known as the "reference point."

Then, why bother with parametric cycle analysis? There are at least three good

reasons.

First,

engine performance analysis at conditions away from the reference

point, usually referred to in the literature as "off-design" cycle analysis, cannot start

until the reference point and the size of the engine have been chosen by some means.

Second,

parametric analysis is much less tedious and time consuming than per-

formance analysis, often providing mathematical optima, that can be directly ex-

ploited.

Third,

and most important, identifying the combinations of design choices

that provide the best performance at each mission flight condition reveals trends

that illuminate the way to the best solution. A reasonable expectation after the com-

pletion of the parametric cycle analysis is that the most promising type of cycle has

been identified and the possible range of each design choice has been bracketed.

To provide the most potent tools for parametric cycle analysis, relationships

will be developed in this chapter for one of the most complex and flexible engine

cycles known: the mixed flow, afterburning, cooled, two-spool turbofan engine

with bleed air and power extraction. This is done because the behavior of many

other cycles can be obtained merely by setting some design choices to trivial values

and because this cycle is needed for the AAF example used in this textbook. Be

aware that, even though this cycle is not included there, the development closely

parallels that in Chapter 7 of Ref. 2. The resulting equations can be solved in

many ways, but the most convenient should be to simply use the ONX computer

program provided with this textbook. If you wish, you may choose to build your

familiarity with and your confidence in this computer program by calculating the

design point performance of some simpler cycles. Otherwise, you will have ample

opportunity to compare your results with those that go with the following example.

Believing it serves a useful purpose, introductory material from Ref. 2 is re-

produced in the derivations, particularly the definitions and notation. From this

point on, the nomenclature is shifted entirely to the propulsion nomenclature of

the Table of Symbols and Ref. 2.

4.2 Design Tools

The uninstalled engine thrust (F) and the uninstalled thrust specific fuel con-

sumption (S) are the primary measures of the engine's overall performance. The

uninstalled thrust (see Appendix E) for a single exhaust stream engine can be

written as

F = --(th9V 9

th0V0) + A9(P 9

P0) (4.1)

where station 0 is far upstream of the engine and station 9 is the engine exit. The

term gc appears in Eq. (4.1) as a reminder that the units must be watched carefully.

In this particular case, for example, if the units of rh are pounds mass/second and

V are feet/second, then

g¢

must be Newton's constant of 32.174 lbm-ft/(lbf-s 2) if

F is to be measured in pounds force. If the units of rn were slug/s instead, then

g~ would be unity. Because incorrect dimensions are responsible for more errors

than any other thing, careful bookkeeping and double-checking are a worthwhile

ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS

Table 4.1 Typical F/mo and S values

Compressor Fan

Engine pressure pressure Bypass Tt7,

I',4, F/tho, S,

type ratio (zrc) ratio (zrf) ratio (or) °R °R lbf/lbm/s 1/h

Turbojet 10-20

no A/B

Turbojet 10-20

with A/B

Turbofan 20-30

low u

no A/B

Turbofan 8-30

low u 10-30

with A/B

Turbofan 30--40

high ot

no A/B

2-4 0.2-1

3600

2000 54-58 1.0-1.1

3000 93-96 1.3-1.4

2000 94-101 2.0-2.2

3000 115-119 1.7-1.8

2000 23-47 0.85-1.0

3000 53-84 0.96-1.5

2-4 0.2-1 3600 2000 75-98 2.1-2.7

3000 102-116 1.7-2.0

1.4-1.6 5-7.5 2000 5.5-12 0.76--0.97

1.4--4 5-10 3000 13-27 0.67-1.03

investment of time. The uninstalled thrust specific fuel consumption (S) is given by

S -- /~f -[- FhfAB (4.2)

where

(rhf + rhfAB)

is the total fuel flow rate to the main bumer and afterburner

of an engine. Since one pound mass of fuel at (or near) the surface of the Earth

weighs one pound force in the British Engineering system, the thrust specific fuel

consumption may be regarded as pounds weight of fuel per hour per pound force

of thrust, and is traditionally reported in units of 1/h.

Parametric cycle analysis employs the uninstalled thrust per unit mass flow of

captured airflow

(F/rho)

or specific thrust and the uninstalled thrust specific fuel

consumption (S) as the performance measures of primary interest. Representative

values of these parameters for several common but different engine types are given

for guidance in Table 4.1.

4.2.1 Station Numbering

Consider the generalized engine shown in Figs. 4.1a and 4.lb. The station

numbers of the locations indicated there are in accordance with Aerospace

bleed air

Fig. 4.1a Reference stations--mixed-flow turbofan engine.

98 AIRCRAFT ENGINE DESIGN

2 13

1 are h

I ~om~ ~om,

2.5

power extraction

PTOL PTOH

ID bleed air cooling air #2 Ib

| cooling air #1

3 /3.1 ,4 "~ 4 1 high-p ....... turbine 4;4 ,~ 4.5

jrL[

l u~or[ I :Oxen: I I rotor [~ [rotor] [ mixer2 I

high-pressure spool ]

low-pressure spool

Fig. 4.1b Reference stations--bleed and turbine cooling airflows.

Recommended Practice (ARP) 755A (Ref. 3) and will be used throughout this

textbook, and include:

Station Location

2.5

3.1

4.1

4.4

4.5

Far upstream or freestream

Inlet or diffuser entry

Inlet or diffuser exit, fan entry

Fan exit

Low-pressure compressor exit

High-pressure compressor entry

High-pressure compressor exit

Burner entry

Burner exit

Nozzle vanes entry

Modeled coolant mixer 1 entry

High-pressure turbine entry for zrtn definition

Nozzle vanes exit

Coolant mixer 1 exit

High-pressure turbine entry for rtH definition

High-pressure turbine exit

Modeled coolant mixer entry

Coolant mixer 2 exit

Low-pressure turbine entry

Low-pressure turbine exit

Core stream mixer entry

Fan bypass stream mixer entry

Mixer exit

Afterburner entry

Afterburner exit

Exhaust nozzle entry

Exhaust nozzle throat

Exhaust nozzle exit

ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS 99

4.2.2 Gas Model

The air and combustion gases are modeled as perfect gases in thermodynamic

equilibrium, and their properties are based on the NASA Glenn thermochemical

data and the Gordon-McBride equilibrium algorithm (see Ref. 7). Thus, this gas

model includes the variation of specific heat at constant pressure

with tem-

perature. In addition, the classical perfect gas model with constant specific heats

(also know as a calorically perfect gas) is also included as a limiting case in this

analysis.

For the perfect gas with variable specific heats, the required additional thermo-

dynamic definitions (Refs. 1, 2, 4) are

h -- Cp dT

(4.3a)

fr~ aT

(p -- C p --T-

(4.3b)

Pr--exp

(~--~) (4.3c)

where ~b is the temperature dependent portion of entropy (s) and Pr (called the

reduced pressure) is the pressure variation corresponding to the temperature change

for an isentropic process (s = constant). The differential equation for entropy is

the Gibbs equation T ds = dh - v dP, which becomes

T ds = dh - R dP/P

for a

perfect gas (Ref. 4). Integration of this latter equation using the definition of q~ gives

the following relationship for the finite change in entropy between states 1 and 2:

$2 -- S1 -~- ~b2 -- (~1 --

R In/°2 (4.3d)

For an isentropic process, Eq. (4.3d) reduces to

~2 -- ~1 =

R In P2 (4.3e)

Using Eq. (4.3c) we can express the pressure ratio for an isentropic process in

terms of the reduced pressure Pr as

P2 nst

= ~-,-1 (4.3f)

We note that for the simplifying case of a calorically perfect gas (CPG) or

constant specific heats for the finite change between states 1 and 2

a) Eqs. (4.3a), (4.3b), and (4.3d) become

h2 - hi = cp(T2 -

T1) (4.3a-CPG)

(92 -- ~1 = Cp

In T2 (4.3b-CPG)

T2 _ R In P2 (4.3d-CPG)

s 2 - S l = C p

In T1 P1

100

AIRCRAFT ENGINE DESIGN

Table 4.2 Calling nomenclature for subroutine FAIR

Symbol Knowns Unknowns

FAIR(l, f, T, h, Pr, 4), Cp, R, y,a) f, T h, P~, (]), Cp, R, y,a

FAIR(2, f, T,h, Pr, c]),Cp, R, y,a) f,h T, Pr, 4),Cp, R, y,a

FAIR(3, f, T,h, P~,c]),Cp, R, y,a) f, Pr

T,h, qb, Cp,

R, y,a

FAIR(4, f, T,h, P,.,qS, Cp, R, y,a) f, q5 T,h, Pr, Cp, R, y,a

b) Eq. (4.3f) becomes the familiar isentropic relationship

(-~l)s=const=(~l) Y-zzT-I (4.3f-CPG)

The effect of variable gas properties can be easily included in a computer analysis

of gas turbine engine cycles, as follows. In addition to the ideal gas equation, two

subroutines are needed: 1) a subroutine that can calculate the thermodynamic state

of the gas given the fuel/air ratio f and one temperature dependent property; and

2) a subroutine that can calculate the compressible gas relationships Tt/T, Pt/P,

and mass flow parameter (MFP) given the Mach number (M), total temperature

(Tt), and fuel/air ratio (f).

The subroutine FAIR was developed to calculate the temperature dependent

properties given the fuel/air ratio f and one of the following properties: T, h,

Pr, or ~p. Table 4.2 gives the calling nomenclature for the subroutine FAIR. The

subroutine also determines the specific heat Cp, the gas constant R, the ratio of

specific heats V, and the speed of sound a.

The compressible flow subroutine RGCOMP was developed to calculate the

compressible flow relationships for this gas model with variable specific heats.

Table 4.3 gives the calling nomenclature for this subroutine. The equations used

in RGCOMP are given in Appendix F.

4.2.3 Total Property Ratios

You will find it very important to be able to swiftly identify and thoroughly

grasp the physical meaning of the quantities that are about to be defined in order

to understand and manipulate the cycle analysis results. We therefore recommend

that you spend enough time to completely familiarize yourself with the material

Table 4.3 Calling nomenclature for subroutine RGCOMP

Symbol Knowns Unknowns

RGCOMP(1, T,, f, M, T,/T, Pt/P, MFP)

RGCOMP(2, Tt, f, M, TilT, P,/P, MFP)

RGCOMP(3, Tt, f, M, Tt/ T, Pt/ P, MFP)

RGCOMP(4, T,, f, M, T,/T, P,/P, MFP)

RGCOMP(5, Tt, f, M, Tt/ T, P,/ P, MFP)

Tr, f,M Tr/T, Pt/P, MFP

Tr, f, Tt/T M, Pt/P, MFP

T,,f, Pt/P M, Tt/T, MFP

T,,f, MFP T,/T, Pt/P,M < I

T,,f, MFP Tt/T, Pt/P,M > 1