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

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ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS 101

of Secs. 4.2.3 and 4.2.4, and then mark their location carefully so that you can find

them quickly whenever necessary.

The ratio of total (isentropic stagnation) pressures rr and total (adiabatic stag-

nation) enthalpies r are introduced, where

total pressure leaving component i

zri -- (4.4a)

total pressure entering component i

total enthalpy leaving component i

ri -- (4.4b)

total enthalpy entering component i

For the case of the calorically perfect gas, we assume a zero reference value of the

enthalpy at zero absolute temperature. Thus enthalpies are replaced by the specific

heat times the absolute temperature, and ri becomes the ratio of total temperatures

total temperature leaving component i

vi - (4.4b-CPG)

total temperature entering component i

Note should also be taken of the fact that for case of the calorically perfect gas the

relationships of Sec. 1.9 apply to Eqs. (4.4a) and (4.4b).

Moreover, the Jr and v of each component will be identified by a subscript, as

follows:

Subscript Component Station

Afterburner 6A --~ 7

b Burner 3.1 --~ 4

c Compressor 2 ~ 3

High-pressure compressor 2.5 ~ 3

Low-pressure compressor 2 ~ 2.5

d Diffuser or inlet 0 --~ 2

f Fan 2 ~ 13

-- Fan duct 13 --~ 16

m 1 Coolant mixer 1 4 --~ 4.1

m2 Coolant mixer 2 4.4 ~ 4.5

M Mixer 6 -~ 6A

n Exhaust nozzle 7 ~ 9

t Turbine 4 ~ 5

High-pressure turbine 4 --+ 4.5

Low-pressure turbine 4.5 ~ 5

Examples

zrc, rc = compressor total pressure, temperature ratios

zrb, rb = burner total pressure, temperature ratios

102 AIRCRAFT ENGINE DESIGN

Exception. rr and Err are related to adiabatic and isentropic freestream re-

covery, respectively, and are defined by

h,o ho + V~/(2&)

rr "-- -- (4.5a)

ho ho

['to Prto

Er r ~'

-- -- (4.55)

Po go

Thus, freestream total enthalpy hto = horr and freestream total pressure Pro =

Po Err. For the simplifying case of a calorically perfect gas, we have, in accordance

with Sec. 1.9,

rto

v 1

- 1 q- c-=---M~ (4.5a-CPG)

Tr -- TO

Z v

( Y-1 2) ~'--~

Err = Tr r-I =

1 + ----~M;

(4.5b-CPG)

Further exceptions.

It is often desirable to work in terms of design limitations

such as the maximum allowable turbine inlet total temperature, Tt4. The term rx

is thus used and is defined in terms of the enthalpy ratio

ht4

rx -- -- (4.6c)

Similarly, for the afterburner

ht7

r),aB --" -- (4.6d)

For a calorically perfect gas, Eqs. (4.6c) and (4.6d) become

r)~- Cp4Tt4

(4.6c-CPG)

C po To

C p7 Tt7

r)~AB --

(4.6d-CPG)

C po To

Component Er and

r. A complete compilation of total pressure and total

enthalpy ratios follow. Please note in the following that it has been assumed that

Ptl3 = Ptl6, ht13 = htl6

and Pt3 =

Pt3.1, hi3 = ht3.1.

Diffuser (includes ram recovery):

Fan:

Pt2 ht2

Erd = -- rd -- -- 1 (4.7a)

Pro hto

Ptl3

ht13

Erf

Pt2 "~f = ht2 (4.7b)

ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS 103

Low-pressure compressor:

7~cL ~.

et2.5 ht2.5

75cL

P,2 ht2

High-pressure compressor:

2"fcH :

Pt3 ht3

V:cH

Pt2.5 ht2.5

Compressor:

et3

7g c ~- -- ~__ 7~cLYgcH

P,2

~c --

ht3

-- "CcL TcH

ht2

Burner:

et4 hi4

7~b = -- "Cb --

et3 ht3

Coolant mixer 1:

Pt4.1

7"t'ml = ~'ml --

/',4

ht4.1

ht4

High-pressure turbine:

~tH -- --

et4.4 et4.4

-- ~ml--

P,4 Pt4.1

"ftH

ht4.4

ht4.1

Coolant mixer 2:

~m2 -- --

Pt4.5

et4.4

--1

ht4.5

"Cm2 -

ht4.4

Low-pressure turbine:

YrtL =

1'15

Pt4.5

"~tL

ht5

ht4.5

Mixer:

Pt6A

e,6

ht6A

• M --

ht6

Afterburner:

Pt7

2"gAB _

Pt6A

ht7

~AB--

ht6A

Exhaust nozzle:

Pt9

P~7

ht9

-1

ht7

(4.7c)

(4.7d)

(4.7e)

(4.7f)

(4.7g)

(4.7h)

(4.7i)

(4.7j)

(4.7k)

(4.71)

Jr. -- rn -- (4.7m)

104 AIRCRAFT ENGINE DESIGN

For the calorically perfect gas, all of the component r except that of the burner

and afterbumer become total temperature ratios. For example, rc = Tt3/Tt2 and

rtH

= Tt4.n/Tt4.1.

The r for the bumer and afterburner become

C p4 Tt 4

-gb -- (4.7f-CPG)

C p3 Tt3

"gAB-- cp7Tt7 (4.71-CPG)

Cp6A Tt6A

4.2.4 Mass Flow Rates

The mixed-flow turbofan engine with afterbuming, bleed air, and cooling air

is a very complex machine with numerous air and fuel flow rates. The cycle

analysis of this engine includes those mass flow rates that have major importance

in engine performance and, hence, cycle selection. Please note that the mass flow

rate frequently changes between stations as flow is added or removed or fuel is

added (see Fig. 4.2). The symbol rh is used for the mass flow rate with a subscript

to denote the type as follows:

Subscript Description Station

fAB

0--+9

Bleed air 3-3.1

Core airflow through engine 2.5, 3

Cooling air for high-pressure turbine nozzle vane 3-3.1, 4-4.1

Cooling air for remainder of high-pressure turbine 3-3.1, 4.1-4.4

Fan air flow through bypass duct 13, 16

Fuel flow to main burner 3.1-4

Fuel flow to afterbumer 6A-7

Flow rate at numbered station

Mass flow ratios. In engine cycle analysis, it is often most effective to cast

the calculations into dimensionless mass flow ratios. The most useful of these for

the engine of Figs. 4.1 a, 4. lb, and 4.2 include the following:

Bypass ratio (or):

Bleed air fraction (fl):

bypass flow rh F

Ct --' -- (4.8a)

core flow rhc

Cooling air fractions (el and e2):

El-

bleed flow rhb

core flow rhc

82 ~"

(4.8b)

mcooll (4.8c)

rhc

rhc°°12 (4.8d)

rhc

ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS lO5

t • thcfl rhc¢ 2 •

mcEl b.

rhc(fl +el +~ 2 )

rhI[ v~

3.1 • 4 41 high-p ....... turbine 4i4 ,

25~31

,~--~,~--~,~-~ ~----~,~

' i b=:l I iEI I r:= ] I :

rho=rhc+fh ~ = (l+a)rh c

r~= rh c

~, =mc(1-fl-e,-e2)

m 4 = thc(1- fl -ei -e2)(l+ f)

th4. , = rh4. , = rh c {(1 -/3 -e, - e2)(1 + f)

+e,}

m45 =ms = rhc {(1-fl -e,-e2)(1+ f)+e, +e:}

Fig. 4.2 Reference stations--turbine cooling air.

Burner fuel/air ratio (f):

burner fuel flow

/'hf

burner

inlet airflow rh3.1

Mixer bypass ratio (a'):

o/I --

fan air entering mixer rh 16

turbine gas entering mixer

/'h 6

Afterburner fuel/air ratio (faB):

fAB --"

afterburner fuel flow

rhfAB

afterburner inlet airflow rhc + rhF -- rhb

Overall fuel/air ratio (fo):

fo-

total fuel flow

#if -Jv rhfAB

engine inlet airflow thc ~-

?/'/F

Fuel/air ratio at station 4.1:

f4.1=

1-I-f-t-el/(1

--f--E 1 --82)

Fuel/air ratio at station 4.5:

f4.5

l+f-k-(el -k-e2)/(1 --~--E 1 --S2)

Fuel/air ratio at station 6A:

f6A --

f4"5(1 -- /~)

l+ot-fl

(4.8e)

(4.8f)

(4.8g)

(4.8h)

(4.8i)

(4.8j)

(4.8k)

106 AIRCRAFT ENGINE DESIGN

Turbine cooling.

The model of turbine cooling incorporated into the engine

analysis is shown in Fig. 4.2. Cooling air is drawn off at the compressor exit

(station 3). A portion of this cooling air

(rhcoon = rhcei)

is used to cool the high-

pressure turbine nozzle guide vanes. The remainder

(rhcool2 = rhce2)

is used to

cool the high-pressure turbine rotor. For cycle purposes, the cooling airflows

mcooll

and rh~oot2 are modeled as being introduced and fully mixed in coolant mixer 1

and coolant mixer 2, respectively. No total pressure loss is assumed for coolant

mixer 2. No cooling air is included for the low-pressure turbine.

4.2.5 Component Efficiencies

Rotating machinery.

For the rotating machinery components it is usually

convenient to relate their Jr to their r by means of efficiencies, which account

for losses or real effects and are based on experience. Two such efficiencies are

commonly employed, r/corresponding to the overall Jr and r and e corresponding

to an imaginary process in which zr and r are arbitrarily close to one. The latter

efficiency, known as the polytropic efficiency, can be used more broadly because

it represents a level of technology rather than the behavior of a given device (as

given by 0). Table 4.4 gives representative values of the polytropic efficiency for

different levels of technology.

Like Refs. 1 and 2, the subscript i is used to represent the exit state or process of

a 100% efficient or ideal turbomachinery (such as the compressors and turbines).

This ideal exit state or process is used to calculate the overall efficiency (~) from

the polytropic efficiency (e) and the dimensionless zr or 3. A compilation of rela-

tionships for the engine of Figs. 4.1 and 4.2 for variable specific heats are given

next. The reduced pressure at the ideal exit state is used to obtain the corresponding

temperature and enthalpy.

Fan:

ht13

~f=(Prtl3~ ef

"Of

hi2 ' \ Prt2 J '

Low-pressure compressor:

ht2.5 ( ert2.5"~ ecL

rcL- ht2 7rcL ~, Pr,2 ,]

High-pressure compressor:

ht3 ( Prt3 ~ ecH

rd-1- ht2.5 , JrcH = \ P---~t2.5 / '

High-pressure turbine:

ht4.4 ( Prt4.4 ) 1/etH '

rtH - ht4.1' YrtH = ~ P~lt4.1,]

rfi -- 1

Prtl3i = 7gfPrt2,

Of -- --

(4.9a)

rf--1

ert2.5i ~- 7gcLert2, T]cL -- --

"EcL i -- 1

VcL- 1

(4.9b)

ert3i ~-- 7gcHert2.5, l']cH --

ZcHi -- 1

VcH-- 1

(4.9c)

Prt4.4i ~ 7gtHPrt4.1, TltH -- --

I -- TtH

1--rtni

(4.9d)

ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS lO7

Table 4.4 Component polytropic efficiencies and total pressure losses

Component

Level of technology

Figure of merit Type 1 2 3 4

Diffuser

Compressor

Fan

Burner

Turbine

Afterburner

Nozzle

Maximum

Tt4

Maximum Tt 7

A a 0.90

rrd max B b 0.88

C c 0.85

0.80

0.78

rrb 0.90

r/b 0.88

Uncooled 0.80

Cooled

~AB

r/AB

7fn

0.90

0.85

D d 0.95

E e 0.93

F f O.9O

(K) 1110

(°R) 2000

(K) 1390

(°R) 2500

0.95 0.98 0.995

0.93 0.96 0.97

0.90 0.94 0.96

0.84 0.88 0.90

0.82 0.86 0.89

0.92 0.94 0.96

0.94 0.99 0.995

0.85 0.89 0.91

0.83 0.87 0.89

0.92 0.94 0.95

0.91 0.96 0.97

0.97 0.98 0.995

0.96 0.97 0.985

0.93 0.95 0.98

1390 1780 2000

2500 3200 3600

1670 2000 2220

3000 3600 4000

aA = subsonic aircraft with engines in nacelles.

bB = subsonic aircraft with engine(s) in airframe.

cC = supersonic aircraft with engine(s) in airframe.

fixed-area convergent nozzle.

eE =

variable-area convergent nozzle.

fF =

variable-area convergent-divergent nozzle.

gStealth may reduce Jrd max, 7rAB, and zrn.

Note:

The levels of technology can be thought of as representing the technical capability for 20-year

increments in time beginning in 1945. Thus level 3 technology presents typical component design

values for the time period 1985-2005.

Low-pressure turbine:

ht5 ( Prr5 ] t/etL

1--rtL

r/tL -- ht4.5' 7ftL = ~Prt4.5J , Prt5i -D- 7rtLPrt4.5, rhL -- 1 -- rtLi

(4.9e)

A compilation of relationships for the engine of Figs. 4.1 and 4.2 for the case

of constant specific heats (calorically perfect gas) includes the following:

Fan:

yes

( __

Ttl3

Jrf = rtl3 y t rtl3i -- (j~I)'T- ~f =

"Of -- Tt2 ' \-~t2 ,] , "rfi -- Tt2 ' r f - 1

(4.9a-CPG)

108 AIRCRAFT ENGINE DESIGN

Low-pressure compressor:

yecL

Tt2.5

{

rt2.5

"~ Y -1 Tt2.5 i ~,-I

"(cLi -- 1

, = ! I - -- - (ZrcL) 7-, ~d. --

rcL -- Zt2 7t'cL ~-~t2 /] ' "CcLi Tt2 "gcL -- 1

High-pressure compressor:

(4.9b-CPG)

Tt3 ( rt3 ~ ~,-1 Zt3 i ×-1 "VcH i --1

= , = ----(JrcH) 7, 0cH=--

"EcH

Tt2.5

7gcH

Ik Tt2.5 /I

, "gcHi

Tt2.5 rcH

High-pressure turbine:

(4.9c-CPG)

Zt4.4 ( Tt4.4 ~ (Y-iSetH

Zt4.4i ~-,

1 -

rtH

---(n',H) 7, n~//- ~

"CtH

Tt4.1

~t, : Ik ~ ] , ~tHi

T,4.1

_ "Etni

(4.9d-CPG)

Low-pressure turbine:

rt5 (rls~(Y-~)etL

7:tL- Zt4. % , TCtL

~ T--~4.5/] '

Tt5i Lz2

-- "(tL

"gtLi

-- --

(2"gtL) Y

?]tL

-- --

Tt4.5 1 -- "gtLi

(4.9e-CPG)

Combustion components.

For those components in which combustion takes

place, combustion efficiency is used to characterize the degree to which the chem-

ical reactions have gone to completion. The efficiencies are based upon the ratio of

the actual thermal energy rise to the maximum possible thermal energy increase,

as represented by the lower heating value of the fuel

(heR,

see Table 9.2). Thus,

we have the following for a perfect gas with variable specific heats:

Bumer:

lil4ht4 --/h3.1ht3.1

/lb ~--- < 1

(4.10a)

lh f hpg

Afterburner:

#17ht7 -- th6Aht6 A

~AB

= < 1

(4.10b)

rhfaBhpR --

For the case of the calorically perfect gas, these combustion efficiencies are written

as follows:

Burner:

th4cp4Tt4 --

th3.1Cp3.1

Tt3.1

t/b = < 1 (4.10a-CPG)

lf, l fhp R

Afterburner:

l'h7cp7 Tt 7 - l~16ACp6 A Tt6A

OAB

= < 1 (4.10b-CPG)

mfaBheR

ENGINE SELECTION: PARAMETRIC CYCLE ANALYSIS lO9

Power transmission components.

For those components that merely trans-

mit mechanical power by means of shafts, gears, etc., a simple definition of me-

chanical efficiency is used to account for the losses due, for example, to windage,

bearing friction, and seal drag. In such cases

mechanical power output

TIm

(4.11)

mechanical power input

so that rlmn, tlmL, Omen, and t/mPL refer to the high-pressure turbine shaft, low-

pressure turbine shaft, and power takeoffs from the high-pressure shaft and low-

pressure shaft, respectively.

4.2.6 Assumptions

Before proceeding with the analysis, the underlying assumptions to be employed

are summarized as follows:

1) The flow is, on the average, steady.

2) The flow is one-dimensional at the entry and exit of each component and at

each axial station.

3) The fluid behaves as a perfect gas (but not necessarily calorically perfect) with

constant molecular weight across the diffuser, fan, compressor, turbine, nozzle, and

connecting ducts.

4) For the case of variable specific heats, the NASA Glenn thermochemical data

and the Gordon-McBride equilibrium algorithm are used to obtain thermochemical

properties of air and combustion gases at any station (see Chapter 9). For the case

of the calorically perfect gas, Cp and F are assigned one set of values from station

0 through stations 3.1 and 1.6 (denoted as Cpc and Fc), a second set of values from

station 4 through 6 (denoted as Cpt and Ft), a third set of values leaving the mixer

at 6A (denoted as CpM and FM), and a fourth set of values from station 7 through

9 (denoted as CpA8 and FAS).

5) The total pressure ratio of the diffuser or inlet is

Ygd = 7"(dmaxT]Rspec

(4.12a)

where Zrd max is the total pressure ratio caused only by wall friction effects and

OR spec is the ram recovery of military specification MIL-E-5008B (Ref. 5) as

given by

17Rspec

1 for M0 < 1 (4.12b)

OR~ve¢=I--O.O75(Mo--1) 1"35 forl <M0 <5 (4.12c)

800

for5 <M0 (4.12d)

1"JR spec -- M4 +

935

6) The fan and low-pressure compressor are driven by the low-pressure turbine,

which can also provide mechanical power for accessories, ProL.

7) The high-pressure compressor receives air directly from the low-pressure

compressor and is driven by the high-pressure turbine, which can also provide

mechanical power for accessories, Pron.

8) High-pressure bleed air and turbine cooling air are removed between stations

3 and 3.1.

9) The flow in the bypass duct (from station 13 to 16) is isentropic.

110 AIRCRAFT ENGINE DESIGN

10) The effect of cooling on turbine efficiency is accounted for by a reduction

of etH due to

Filcool 1

and

rhcool 2.

11) The fan and core streams mix completely in the mixer, the total pressure

ratio YEM being

7"(M = 7"gMideal

YEMmax

(4.13)

where

YgMideal

is the total pressure ratio across an ideal constant area mixer and

YEMmax is the total pressure ratio due only to wall friction effects.

4.2.7 Engine Performance Analysis

The definitions and assumptions just catalogued will now be used to analyze

the overall and component performance of the engine cycle of Figs. 4.1a, 4.1b,

and 4.2. We would like to emphasize the fact that the following solution process

may be successfully applied to a wide variety of airbreathing engine cycles. We

recommend that you employ the following sequence of steps whenever a new

engine cycle is to be studied.

Uninstalled specific thrust (F/rho).

Equation (4.1) can be rearranged into

the following nondimensional form for the uninstalled specific thrust (F/m0):

rhoao - 1 +--~ ~o - Me + 1 + fo 1 -~u Re Vg/ao Vo

(4.14)

When the nozzle exit area is chosen for ideal expansion and maximum unin-

stalled specific thrust, then P0 = P9, and the last term in the preceding equation

vanishes. Otherwise,

Po/P9

# 1 is a design choice, and Eq. (4.14) shows that the

nondimensional uninstalled specific thrust depends largely upon the velocity ratio

Vg/ao

and the overall static temperature ratio

T9/To,

which are considered next.

Velocity ratio (Vg/ao).

From

ht = h + V2 /2g¢,

using Eq. (4.6b) and ht9 =

ht7

gives

a-o! = M2 \-Voo J = M2° hto ho

rr-1 ~t9

where

ht9/h9

is a function of the nozzle total exit state (t9) and the total to static

pressure ratio (Pt9/P9) as given by

~9 -m- YEr YEd Y75cL YEcH YEb YEtH 7[tL YEM YEAB YEn

(4.16)

For the case of a calorically perfect gas, the velocity ratio is given directly by

/ /

a~/

~--]- 1 - ~t9 -- ~ --i- 1 - (4.15-CPG)

\P,9/