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

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530 AIRCRAFT ENGINE DESIGN

so that

where

and

F astd

mo~ gc

--fl{Mo}

(D.14)

fl{M0}

C3"gr -- lygdTrcT--YCt7~ n

(D.15)

2rt 1 Cpt Tt4

C3 -- -- (D. 16)

yc - 1 Ts,d Cpc Oo

where it is important to note that, for the case at hand, fl is a function only of the

instantaneous flight Mach number M0 through M0, Zrr, and rr.

The amazing result of Eqs. (D. 14-D. 16) is that the uninstalled specific thrust be-

havior of this engine collapses into a single line, as shown in Fig. D.4 for the typical

reference point turbojet engine that will be used as the example for the remain-

der of this exposition. Example turbojet engine reference point data are as follows:

Tt4/00 =

3000;ÙObreak = 1.1;hpR = t8,000 Btu/lbm; zrc = 20; r~ = 2.592; Oc =

0.85; r, = 0.7626; ~t

0.2957; rh = 0.91; zra = 0.98; Yc = 1.40; c m = 0.238

mo40

lbf /

~lbm/s)

115

110

105

100

0.0

1.0 1.5 2.0

120

asld

Slope = ---

I I i I

0.5

Fig. D.4

F/(mox/-O)

vs M0 according to Eqs. (D.14-D.16) for the example reference

point uncooled, single-shaft turbojet engine (C3 = 25.57,

¢cacrcccbcrtcr, =

5.451, and

3't = 1.33).

APPENDIX D: ENGINE PERFORMANCE 531

Btu/lbm-°R; Jro = 0.95; Yt = 1.33;

cpt

= 0.276 Btu/lbm-°R; zrn = 0.99; ~/b =

0.95; /']m = 1;]~ = 0;

1 .ql- f ~ 1. Figure D.4 clearly displays the thrust minimum

or "bucket" normally experienced by turbojet engines as they accelerate but not

usually captured by simple models. At first,

F/(rhov/O)

decreases, reflecting the

penalty of the captured freestream momentum, but it then increases as rcr, Tt4, and

cycle thermal efficiency increase. At very small Mach numbers, where 7/" r and rr

are essentially constant, Eq. (D.14) can be differentiated to reveal that the slope

--astd/gc[34.7

lbf/(lmb/s)], as noted in Fig. D.4.

It should also be noted for later use that, using the definition of corrected com-

pressor mass flow rhc2 [cf. Eq. (5.23)], Eq. (D. 14) becomes

astdS2lhc2

F -- -- f~{M0} (D.17)

Continuing to follow Ref. 1, the uninstalled specific fuel consumption of the

engine is given by the expression

S- l~lf _

f _ gc

cpcro ~.- ~r'Cc

(D.18)

F F/rho astd rlbhpR ~v"-Ofi

{M0}

which reduces to

where

S "gr

= C4-- (D.19)

fl{Mo}

gcastd (Cpt 1 Tt4 )

--- rc (D.20)

C4 -- ilbhpR(y c __

\Cpc Tstd O0

The right-hand side of Eq. (D. 19) is again independent of theta and collapses the

uninstalled specific fuel consumption behavior of this engine into a single, almost

straight line, as shown for the example reference point engine in Fig. D.5. The

approximate linear curve fit shown in Fig. D.5 for the results given by Eq. (D. 19)

-- 1.08 + 0.26M0 (D.21)

which compares very favorably with the installed specific fuel consumption cor-

relation for the entire class of turbojet engines presented in Sec. 3.3.2.

Hot Day Flat Rating

An important footnote to these proceedings is to clarify a term often found in the

propulsion literature, namely "hot day flat rating." It is imperative that commercial

and military aircraft engines retain their standard day static thrust on "hot days,"

which generally means up to temperatures in the range of 90-110 °F (32-43 °C),

as dictated by the specific application. Otherwise, some undesirable compromise to

aircraft performance is required, such as leaving payload, fuel, and/or passengers

behind. This is clearly a case of a nonstandard atmosphere.

532 AIRCRAFT ENGINE DESIGN

s/~

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

0.0

, , , i I ' ' ' ' I ' ' ' ' I ' ' ' ' ~

, , , , I , , , , I , J , , I , , , ,

0.5 1.0 1.5 2.0

M o

Fig. D.5

S/x/-O

vs M0 according to Eqs. (D.19) and (D.21) for the example reference

point uncooled, single-shaft turbojet engine [C4 = 3.012 lbm/Ibf-h (85.31 mg/N-s)].

Since Eq. (D.11) shows that the corrected mass flow is constant when 00 is

less than or equal to

OObre~k,

then Eq. (D.17) reveals that the static thrust is also

constant there. Consequently, setting

OObreak

[cf.

Eq. (D.6)] at or above the ratio

of the absolute hot day flat rating temperature to

Tstd

guarantees that the con-

stant static thrust requirement will be met. This is in every way equivalent to

picking a theta break or throttle ratio for the engine that is greater than one, and

leads to the conclusion that

Tt4

is actually somewhat less than Tt

max

for most en-

gines at standard sea level static conditions. For example, if

Tt4ma x

first occurs at

1.1

Tstd

= 1.1(518.7)°R = 570.6 °R (111.9 °F = 299.2 K = 44.4 °C), then the

O0 break

= 1. l

at M0 = 0.

Uncooled Nonafterburning Single-Shaft Turbojet Performance:

break

An entirely analogous and parallel development, with equally appealing results,

is now carried out for the second case of maximum thrust with 0o >

OObreak. The

corresponding situation is that

Tt4 = Ttamax,

Tt, and nt are fixed and the same six

assumptions are made.

Equations (D. 12) and (D. 13) become

F astd

-- --f2{Mo, 0} (D.22)

rho~v/O gc

APPENDIX D: ENGINE PERFORMANCE 533

where

/I ,E 1 }

1 \ v, yc-~

fz{Mo, 0} = --g-

.... Mo

7Cd 712b 2"t't 712n f Tr "Jr- C6/0 .]

(D.23)

and

CptTt4max

(D.24)

C pc Tstd

2vt z'~

(D.25)

yc-1

~---

~/c(1 - vt)vff (D.26)

The results of Eqs. (D.22-D.26) are shown in Fig. D.6 for the example reference

point engine.

F/(gno~f-O)

decreases continuously with increasing Mo because ~c

is decreasing while

Tt4

= rt4max

remains constant. When M0 is small, the slope

-a,td/g~,

for the same reason as in the preceding case, as noted in Fig. D.6.

160

' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' '

F 140

too4-0

120

~lbm/s)

IO0

60 , , ~ ,

I L

, , ~ l , , , , I , , ,

0.0 0.5 1.0 1.5 2.0

Fig. D.6

F/ffhox/O)

vs M0 and O according to Eqs. (D.22-D.26) for the exam-

ple reference point uncooled, single-shaft turbojet engine (Cs = 28.15, C6 = 1.489,

7rd 7rb ~rt ~r, = 0.2725, and ~'t = 1.33).

534 AIRCRAFT ENGINE DESIGN

Continuing, the uninstalled specific fuel consumption of the engine is given by

the expression

S -- #If __ f __ cpcT 0

(v>,

-- "gt Z'c) (D.27)

F F/mo tib her F/rho

which reduces to

where

= F/(rhoVr#)

(D.28)

cpc T~

C 7 --

(D.29)

rib hpR

The results of Eqs. (D.28) and (D.29) are shown in Fig. D.7 for the example

reference point engine. Because of compensating changes in the numerator and

denominator, the right-hand side of Eq. (D.28) is weakly dependent on 0 for the

entire range of standard day values of 0. The linear curve fit of Eq. (D.21) again

shows excellent agreement with the simple model, further supporting the turbojet

correlation of Sec. 3.3.2.

1.7

1.6

1.5

1.4

s/~

1.3

1.2

1.1

1.0

0.0

l''''l''''l

o 0 /o

1~ S / x]-O = l .O8 + O.26 Mo

, , , , I , , , , I i i t

l I i I I

0.5 1.0 1.5 2,0

m 0

Fig. D.7

S/x/O

vs Mo and 0 according to Eqs. (D.28), (D.29), and (D.21) for the

example reference point uncooled, single-shaft turbojet engine (C7 = 0.006998 and

~-~ = 7.378).

APPENDIX D: ENGINE PERFORMANCE 535

110

, i

i I i

, l

i I i i i

' l i ,

~N~ AEDsys MSH J ,,

-~ calculations \ " 0 = 0.7519

100 ~ =-----. r l" ~

¢lUf

~lbm]s) s[mplemod'el s / ~ X

a0 o

0.0 0.5

1.0 1.5 2.0

Fig. D.8 Comparison of

F/(thox/~)

vs M0 and 0 for the simple models versus

AEDSys MSH performance computations for the example reference point turbojet

engine with

OObreak =

1.1.

Figure D.8 compares the results of the simple model calculations of

F/(rho~v/O)

with those of the AEDsys modified specific heat (MSH) performance model com-

putations for the same example reference point turbojet engine. They are very

similar because the two methods solve essentially the same equations. The ma-

jority of the approximately 5% difference is due to the fact that AEDsys accounts

for the added mass flow of fuel that is neglected by the simple models. Similar

agreement is obtained for the quantity

Sfiv/-O.

Based on these results, one may

comfortably conclude that the simple models provide a reasonable representation

of reality.

Above and Beyond

It should be obvious that the groundwork just developed can be extended to

investigate a wide variety of variations on the turbojet theme. On the one hand,

more complex cycles, including bypass or afterburning, can be analyzed using this

approach. We encourage you to explore them on your own. You will find these

journeys stimulating and rewarding.

On the other hand, important questions about individual component operation

can be answered as they arise. For example, the definitions of cycle parameters

and Eq. (D.2) can be used to reveal the behavior of the compressor discharge

total temperature Tt3 as a function of flight condition for the turbojet cycle of

536 AIRCRAFT ENGINE DESIGN

this appendix. This quantity is extremely important from the design standpoint

because it controls the cycle thermal efficiency (see Appendix E) and is limited by

available material capabilities (see Sec. 8.2.3). After some exhilarating algebraic

manipulations, it can be shown that

Tt3/Tstd=03

u~-00Tcmax for00 _< 00bre,k (D.30)

and

Tt3/Tstd = 03 = O0 ~-OObreak(Tc max -- l) for 00 > 00break (D.31)

These remarkably simple expressions reveal that Tt3 is always directly proportional

to 0o, although the slope is less after the theta break than before. The consequences

of this situation are swift and clear. For fighter aircraft that have a large flight

envelope and spend a small fraction of their flight time at their maximum 00, the

compressor discharge temperature is usually less than the maximum allowable

value. For supersonic transport aircraft that cruise at their maximum 00, the com-

pressor discharge temperature is usually at the maximum allowable value. Thus,

the

Tt3

material selection problem can be much more difficult for the transport

aircraft than for the fighter aircraft.

But the fun is not over yet. Equations (D.30) and (D.31) allow the possibility

that T,3

max

will be reached before the design O0break, and the throttle will have to

be retarded. For a typical value of Tt3 max/T~ta of 3.5 and the rcmax = 2.592 and

OObreak

-~-

l. 1 of the example turbojet, Eq. (D.31) shows that 00 must exceed 1.75

(i.e., M0 > 1.94 at 0 = 1) before the compressor discharge materials limitation is

reached. Although there is no conflict for this example case, we have discovered

another boundary that could be placed on Fig. D.1 that must be examined for

every new engine. And once this boundary is reached, another break in all of

the performance parameters will occur. Can you use Eq. (D.31) to determine the

general behavior of r~ and Tt4

once

Tt3

max

is reached?

You will find it worthwhile to repeat this investigation for the compressor dis-

charge pressure, which is also limited by available materials.

And so it goes. We hope that these and the other examples found in this textbook

encourage you to use this framework as thefirst resort when trying to understand

the fundamentals of jet engine operation.

References

J Oates, G. C., The Aerothermodynamics of Gas Turbine and Rocket Propulsion, 3rd ed.,

AIAA Education Series, AIAA, Reston, VA, 1997.

2Mattingly, J. D., Elements of Gas Turbine Propulsion, McGraw-Hill, New York, 1996.

3U. S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, DC,

Oct. 1976.

4"Climatic Information to Determine Design and Test Requirements for Military Equip-

ment," U.S. Dept. of Defense, M1L-STD-210C, Rev. C, Jan. 1997.

s,,Climatic Information to Determine Design and Test Requirements for Military Equip-

ment," U.S. Dept. of Defense, M1L-STD-21OA, Nov. 1958.

6Oates, G. C. (ed.), Aerothermodynamics of Aircraft Engine Components, AIAA Educa-

tion Series, AIAA, Reston, VA, 1985, p. 263.

Appendix E

Aircraft Engine Efficiency and Thrust Measures

The goal of this appendix is to provide a deeper appreciation and understanding

of the efficiency and thrust measures that you will encounter in this textbook and in

practice. The title immediately reveals that there is no single, universal measure of

engine efficiency or thrust that serves all purposes. Engineers and designers have

found it necessary instead to define and use many different efficiency and thrust

measures. Three efficiency measures and four of the most frequently cited thrust

measures will be developed here in detail. All flows are assumed to be steady in

the cyclic time average, propulsion sense.

Please note that the definition of each is plain and unambiguous and that, al-

though this material is based on a single exhaust flow engine configuration, it can

easily be extended to multiple exhaust flow situations. The material that follows has

benefited greatly from Refs. 1 and 2 and repeatedly employs the impulse function

as described in Sec. 1.9.5.

Overall Efficiency

The function of the airbreathing engine, viewed as a thermodynamic cycle,

is to convert the chemical energy stored in the fuel into mechanical energy for

the aerospace system. This leads to a performance measure called overall effi-

ciency that, although always introduced but seldom intensely pursued in the litera-

ture, is particularly revealing and is presented in all AEDsys engine computations

(see Secs. 4.2.7 and 4.2.8). The rate at which the engine makes mechanical energy

available to the aerospace system is known as the thrust power and is given by the

expression

Thrust power =

FVo

(E.1)

where it has been assumed that the thrust is parallel to the direction of flight.

Placing an unambiguous value on the rate at which the chemical reactions make

energy available to the engine cycle requires some thought. The standard practice

in the propulsion and power industry is to represent the actual combustion process

by a fictitious one in which the pressure is constant and there are no heat or work

interactions, namely the heat of reaction or heating value of the fuel

heR,

as defined

in Chapter 9. The rate at which the chemical reactions make energy available to

the engine cycle for the overall fuel flow rate of/nj; is therefore

Chemical energy rate = rn f~, h PR (E.2)

537

538 AIRCRAFT ENGINE DESIGN

An alternative approach would be to add the kinetic energy of the fuel being

consumed (i.e.,

Vo2/2)

to the heat of reaction in order to account for the energy

required to make the fuel available to the engine. We have not used this method

for several reasons. First, on philosophical grounds, one has no choice but to

carry the fuel aloft, and the thermodynamic cycle has no way to capitalize on

this kinetic energy because it vanishes in the aircraft/engine frame of reference.

Second, because the kinetic energy is less than about 1% of the heat of reaction

over the normal operating envelope of turbine engines, it would have a negligible

effect on the numerical results.

Based on the foregoing, the overall efficiency 0o of the airbreathing engine

cycle is defined as

Thrust power F V0

= = -- (E.3)

Overall Efficiency ~7o Chemical energy rate

t:nfoheR

It should be emphasized that the overall efficiency is a direct indicator of how

well the engine uses the energy originally deposited in the fuel tanks or, conversely,

how much fuel must be put onboard in order to provide the propulsive energy

needed for a given mission. Moreover, common-sense application of the second law

of thermodynamics leads to the conclusion that overall efficiency cannot exceed 1;

otherwise the chemical energy of the fuel could be restored and the surplus of

mechanical energy used to create perpetual motion.

Thermal Efficiency and Propulsive Efficiency

It can be very enlightening to further break down the airbreathing engine overall

efficiency into its "grass roots" constituents as follows:

Engine mechanical power Thrust power

r/o =

Chemical energy rate Engine mechanical power

Thermal efficiency r/T/4 Propulsive efficiency 0P

This word equation reveals that our purpose is to follow the energy along its

"food chain" from chemical (in the fuel tank) to mechanical (generated by the

engine) to the aerospace system (thrust power). Provided that the exhaust flow is

perfectly expanded to atmospheric pressure and no bleed air or shaft takeoff power,

the mechanical power generated by the engine manifests itself only as a change in

kinetic energy of the flow, so that

Engine mechanical power = (rno + rhL)-~ - rho-2~ -

and, therefore, that

r'n° {(l + fo)V92 FoX}

=g-7

gc 2 2 F Vo

0° = r/TH0/' =

fohpR __

m0 {(1 + f°)~ 2

V2} 2 (E.4)

where

ITt fo/rrta

was mcorporated.

APPENDIX E: ENGINE EFFICIENCY AND THRUST MEASURES539

Thermodynamic analysis of engine cycles teaches us that thermal efficiency

primarily increases with Tt3/To = rrrc, and therefore with zrc, although Orn can-

not exceed (Ref. 1). This is the driving force behind high-pressure ratio aircraft

engine cycles. Furthermore, because the uninstalled thrust when the exhaust flow

is perfectly matched to atmospheric pressure is merely the change in momentum

flux from entry to exhaust, then

F = --{(rh0 -4-

lhfo)V9 --/h0V0}

m°{(1 +

fo)V9

V0} (E.5)

gc gc

so that the propulsive efficiency portion of Eq. (E.4) becomes

/ }// (v9) }

OP=2 (l+fo)~00-1 (l+fo) ~00 -1 (E.6)

which shows that propulsive efficiency primarily increases as the ratio of the ex-

haust velocity to the freestream velocity decreases. This is the driving force behind

high bypass ratio aircraft engine cycles, which spread the available engine mechan-

ical power across more air in order to reduce this velocity ratio. By noting that

the fuel/air ratio is small compared 1 even for stoichiometric combustion (see Sec.

9.1.2), a reasonable approximation and the most revealing formula for propulsive

efficiency is obtained by using this fact to reduce Eq. (E.6) to

~p _ (E.7)

V9 / Vo --[.-1

This is the most transparent and frequently encountered form and, because ex-

haust velocity must exceed inlet velocity in order to obtain positive thrust, this

approximation for propulsive efficiency never exceeds 1.

The exact value of propulsive efficiency given by Eq. (E.6) is always slightly

larger than the approximate value given by Eq. (E.7) as a result of the presence of the

very small additional fuel mass over which the energy is spread. Close examination

of Eq. (E.6) will reveal, for example, that the exact propulsive efficiency will always

be less than 1 even for arbitrarily large fuel air ratios provided that

V9/Vo

is greater

than 2. You are encouraged to calculate both versions of propulsive efficiency and

draw your own conclusions.

Performance Measure Interrelationships

The conventional definitions may be arranged to yield a complete set of exact

interrelationships between the airbreathing engine performance measures. The

results of these manipulations are presented in Table E. 1.

One important and immediate conclusion that can be drawn from these interre-

lationships is that, for a fixed value of r/o, S is directly proportional to V0 while

F/fno

is inversely proportional to V0. Knowledge of these trends is often a shortcut

to useful conclusions.

Airbreathing Engine Performance Measure Example

It is generally true that specifying the flight speed, the fuel heating value, and

any two performance measures allows all of the others to be calculated. The ex-

ample that follows will demonstrate a typical case of this assertion, as well as