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

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572 AIRCRAFT ENGINE DESIGN

the high- and low-pressure spools resulting in Eqs. (4.21a) and (4.22a). The tem-

perature ratios across the two cooling air mixing processes (rml and rm2) are given

by Eqs. (4.20a) and (4.20b). The pressure ratios across the high-pressure turbine

(Trtn)

and low-pressure turbine (zrtD are related to their respective temperature

ratio and polytropic efficiency by Eqs. (4.9d) and (4.9e). The fuel-air ratio (f) is

given by Eq. (4.18). Thus, the only unknowns for solution of Eq. (J. 1) are the static

pressure ratios,

Po/P9

and

Po/P19.

Two flow regimes exist for flow through a convergent nozzle, unchoked and

choked. For unchoked flow, the exit static pressure

(Pe)

is equal to the ambient

pressure (P0), and the exit Mach number is less than or equal to one. Unchoked

flow will exist when

-- <

Po "P M=I

Then

= P0 and the exit Mach number

(Me)

is determined using the compressible

flow functions. On the other hand, choked flow will exist when

-- >

PO- P- M=I

and then

Me = 1, Pte/ Pe

= (Pt/ P)M=I,

and

P~e/Pe

Po/Pe -- --

P,e / Po

where

Pte/Po

is obtained by the product of the ram and component zr for the

respective airstream.

The thrust specific fuel consumption is given by Eq. (4.31), which can be ex-

pressed as

S - m

fo f(1

-fl-el

-e2)

F/rho

(1 +

a)F/mo

All of the equations needed for on-design cycle analysis of this turbofan engine

have now been identified. The following section presents these equations in the

order of solution.

Summary of High Bypass Ratio Turbofan Engine Parametric Cycle

Analysis Equations

This section presents the complete parametric cycle analysis equations for non-

afterburning, separate exhaust flow, two-spool turbofan engine with bleed, turbine

cooling, power extraction, and convergent nozzles. For convenience in computer

programming, the required computer inputs and principal computer outputs are

given. These are followed by the cycle equations in their order of solution.

APPENDIX J: HIGH BYPASS RATIO TURBOFAN ENGINE 573

Inputs

Flight parameters:

Aircraft system parameters:

Design limitations:

Fuel heating value:

Component figures

of merit:

Design choices:

Outputs

Overall performance:

Component behavior:

Equations

Mo, To, eo

fl, CTOL, CTOH

hpR

El, E2

Ygb, Tfdmax, 27:n, YT:nf

e f, ecL, ecH, etH, etL

T]b, I"]rnL, TImH, TImPL, T]mPH

Y'g f , 7TcL , 7fcH , 0[, Tt4

F/rho, S, fo, tip,

17TH,

V9/ao, V19/ao

7¢tH, YgtL

Tf, "CcL , 15cH , "CtH , "gtL, 75~

Of, YIcL, OcH, ~tH, TltL

M9, Pt9 / P9, P9 / Po, T9 / To

M19,

etl9/ Pl9, PI9/ Po, T19/ To

FAIR (1, O, To,

ho, Pro,

¢o,

cpo,

Ro, YO, ao)

Vo = Moao

hto = ho + V2

2gc

FAIR (2, 0,

Tto, hto, Prto, Ckto, cpto, Rto, Yto, ato)

rr = hto/ ho

7rr. = Prto/ Pro

~r=l forMo<l

/~r

= 1 --

0.075(Mo

- 1) 1'35

for Mo > 1

7"( d = YfdmaxTlr

ht2 = hto

ern = Pr, o

r'J 1/ef

ertl3 = l-rt22Tf

FAIR

(3, O, Tt13,

htl3,

Prtl3,

~btl3,

Cptl3,

Rtl3, Ytl3, atl3)

~:f = htl3/ ht2

ertl3i = ert27"g f

FAIR

(3, 0,

Tt13i,

htl3i, Prtl3i,

~btl3i,

Cptl3i, Rtl3i,

Ytl3i,

atl3i)

htl3i - ht2

~Tf -- htl3 _ ht2

574 AIRCRAFT ENGINE DESIGN

-- l/ecL

Prt2.5 -m- rrt2JtcL

FAIR (3, 0, Tt:.5,

ht2.5, Prt2.5,

~bt2.5,

Cpt2.5,

Rt2.5, Yt2.5, a/2.5)

"CcL =. ht2.5/ ht2

ert2.5i = Prt2 7rcL

FAIR (3, 0,

Ttz.5i, hta.5i, Prtz.si, c~t2.si, CptzSi, Rt2.5i,

~/t2.5i,

at2.5i)

ht2.5i -- ht2

OcL--

ht2.5 - ht2

Prt3 = err2.5 7tcl/HeCH

FAIR (3, 0,

Tt3, ht3, Prt3,

~bt3,

Cpt3, Rt3, Yt3, at3)

~cH = ht3/ ht2.5

Prt3i = Prt2.5 7rcH

FAIR (3, 0, Tt3i,

ht3i, ert3i, ~t3i, Cpt3i, Rt3i, Yt3i, at3i)

ht3i - ht2.5

tic H --

ht3 - ht2.5

Set initial value of fuel/air ratio at station 4 =

f4i

FAIR (1,

f4i, Tt4, ht4, Prt4, ~t4, cpt4, Rt4, Yt4, at4)

ht4 -- ht3

f--

rlbhpR -- ht4

If - f4il

> 0.0001, then

fgi

= f and go to A, else continue.

~X = ht4/ ho

(1 - fl -- el - e2)(1

+ f) + elrrrcLrcn/rx

"Cml .~.

(1--fl-el-e2)(l+f)+el

rrrcL(rc#--

I)+ (1

+ ot)Crol4/rhnen

rtn = 1 -

rh.nrx{(1 - fl - el - e2)(1

+ f) + elrrrcLZctt/r;,}

ht4.1 = ht4rml

4,1

1 + f + el/(1 -/3 - e~ -

e2)

FAIR (2, f4.1, Tt4.1,

ht4.1, Prt4.1,

q~t4.1,

Cpt4.1, Rt4.1,

)zt4.1, at4.1)

ht4.4 = ht4.1T, tH

FAIR (2, f4A,

Tt4.4, ht4.4, Prt4.4,

~bt4.4,

Cpt4.4, Rt4.4,

Yt4.4, at4.4)

~ ( Prt4.4 ) l/etH

PT4.1/

Prt4.4i -~- 7rtH Prt4.1

FAIR (3, f4.1, Tt4.4i,

ht4.4i, Prt4.4i, qJt4.4i, Cpt4.4i, Rt4.4i, Yt4.4i, at4.4i)

ht4.1 - ht4.4

rhH -- ht4.1 - ht4.4i

(1 - 13 - el - e2)(1

+ f) + el +

E2{rr75cLrcH/(rXT~mlIStH)}

rm2 =-

(1 - 13 -- el -- e2)(1 + f) + el + e2

APPENDIX J: HIGH BYPASS RATIO TURBOFAN ENGINE 575

ht4.5 = ht4.4"~m2

f4.5

1 + f + (el

e2)/(1 --/~ -- el -

e2)

FAIR (2, f4.5,

Tt4.5, ht4.5, Prt4.5,

~bt4.5,

Cpt4.5, Rt4.5, Yt4.5,

at4.5)

r,{(rcL - 1) +

ct(rf -

1)} + (1 +

et)CroL/rlmPL

75tL ~ 1 --

rlm14rX rtH

{(1 --/3 -- e] -- e2)(1

+ f) + (el +

S2/rtH)~rrcL'CcH/75Z}

ht5 = ht4.5"CtL

FAIR (2, f4.5, Tt4.5,

ht4.5, Prt4.5,

~t4.5,

Cpt4.5, Rt4.5,

Yt4.5, at4.5)

( Prt5 ~ l/etL

Pr, si = rrtL Pr,4 5

FAIR

(3, f4.5,

Ttsi,

ht5i, Prt5i, ~bt5i, Cpt5i, Rt5i, Yt5i, at5i)

ht4.5 - ht5

~tL--

h/4.5 -

ht5i

ht9 =ht5; Tt9=Tt5; Prt9 = Prts;

f9= f4.5

M9=l

RGCOMPR (1, Tt9, f9, M9,

(Tt/T)9, (Pt/P)9, MFP9)

P,9

-- ~ 7"( r 7~ d 7t'cL 7lJcH 2T b 7"(tH 7t'tL 2"f n

If Pt9 (P')

-- > then

p0- p" 9

Tt9

T9--

(Tt/T)9

FAIR

(1, fg, T9,

h9,

Pr9, ~9, Cp9,

R9, 2/9, a9)

Po ( Pt / P )9

P9 Pt9 / Po

Else

Prt9

er9-- m

Pt9 / Po

FAIR

(3, f9, Tg,

h9,

Pr9,

~9, Cp9, R9, )/9, a9)

--~1

End

V9 = ~/2gc(ht9 -

h9)

M9 = V9/a9

htl9

= htl3;

Ttl9 =

Ttl3;

Prtl9 = Prtl3;

f19 ~--- 0

M19 = 1

RGCOMPR (1,

Tti9,

f19,

M19,

(Tt/T)19, (Pt/ P)19,

MFP19)

576 AIRCRAFT ENGINE DESIGN

Ptl9

Y~'r ~d Yg f Y(nf

IfP'19 > (Pt) then

po- p9

rtl9

rl9----

(rt/r)19

FAIR

(1, f19, T19,

hi9,

Prl9,

~b19,

Cpl9,

R19, Y19, a19)

Po ( Pt / P )9

P9 P,9 / eo

Else

Prtl9

erl9 --

PtI9/Po

FAIR (3,

fl9, 1"19,

hi9, Prl9, ~b19,

Cpl9,

RI9, Y19, a19)

End if

W19 =

~/2gc(htl9 - h19)

M19

V19/a19

= f(1 -/~ - el - e2)/(1 + or)

[1 +

f o (1 + or) - fl ]

V9 "-[- ol

F ao

fno gc(1 + a)

S-

Flmo

W19

- (1 +

a)Mo

~ r91ro

PolP9)

fo(1

+ ~) - ~] V91ao Yo

+otm

R19

T19/To

Po/PI9)

Ro V19/ao

)to

2gcMo(1 + ot)F/(rhoao)

\ ao / \ ao /

1 {[ [l+f°(l+~)-fl]V2+°tV29 ]

= G -f ¥ - vg

Example Parametric Analysis Results

Consider a turbofan engine with a fan pressure ratio

(7/'f)

of 1.4 to be designed for

a flight Mach number of 0.8 at standard altitude of 40 kft using the variable specific

APPENDIX J: HIGH BYPASS RATIO TURBOFAN ENGINE 577

0.85 I i i i '

0.80

0.75

~= 1£

0.70

S a~ 1

(l/h)

0.65

0.60 2~

0.55 , , , I , , , I , , , I , , , I , , ,

6 8 10 12 14 16

F / rh 0 { lbf/(lbm/sec) }

Fig. J.2 Specific performance of high bypass turbofan engines.

heat (VSH) gas model with the inputs listed here:

ef =

0.89; ~dmax

0.99;

~b = 0.995;

Tt4 =

2600°R;

eel =

0.90; ~rb = 0.96;

~mL =

0.995;

hpe =

18,400

Btuflbm;

ecH =

0.90; ~Zn = 0.99; ~mH = 0.995; el = 0.04, s2 = 0.03; etH = 0.89;

~Z,y = 0.99;

OmPL =

1.00;

~mPH :

0.99;

etL =

0.91; fl = 0.03;

CrOL =

0.00; and

CTOH =

0.01.

Figure J.2 presents the parametric results for variation in the remaining two

design variables, Zrc and or. These results were obtained using the equations just

listed for :rcL = 4 and values 10 < :re < 30 and 6 < ot < 15. As can be seen

in this figure, increasing either the bypass ratio or the compressor pressure ratio

generally reduces the specific fuel consumption (see Appendix E), while increasing

the bypass ratio naturally reduces the specific thrust. Also, for the bypass ratio

range 12 < ot < 14, increasing the bypass ratio no longer reduces the specific fuel

consumption because the bypass and core velocities are so disparate. Consequently,

for each compressor pressure ratio, there is a bypass ratio that gives the minimum

thrust specific fuel consumption. This is commonly referred to as the optimum

bypass ratio (or*). A performance curve for this optimum bypass ratio is also

plotted in Fig. J.2. The value of the optimum bypass ratio for this engine is plotted

vs the compressor pressure ratio in Fig. J.3.

Performance Analysis

The performance of a selected design point high bypass turbofan engine of

the type shown in Figs. J.la and J.lb is desired at off-design flight conditions

and throttle settings. In this off-design problem, there are 19 dependent and 4

independent variables as shown in Table J. 1.

The assumptions of Sec. 5.2.2 apply, except those referring to the exhaust

mixer and afterburner, and the exit nozzles have fixed areas. As a result of these

14.0

Ililiillllllliillll

13.8

13.6

a'*

13.4

13.2

13.0

Fig. J.3

578 AIRCRAFT ENGINE DESIGN

15 20 25

Compressor Pressure Ratio (~)

Optimum bypass ratio of the high bypass turbofan engines.

Table J.1 Engine performance variables

Component

Variable

Independent Constant or known Dependent

Engine Mo, To, Po fl mo,

Diffuser red =

f(Mo)

Fan t// re f, rf

Low-pressure compressor

O, z. reeL, rcL

High-pressure compressor tT, H zr,.n, r,w

Burner

Zt4 reh f

Coolant mixer 1 el rml

High-pressure turbine 0,/f re,n, rm

Coolant mixer 2 e2, rein2 r.,2

Low-pressure turbine

~hc retc , rtL

Core nozzle

7r,, M9, P9/ Po

Fan nozzle re~,f

M19,

P19/Po

Total number 4 19

APPENDIX J: HIGH BYPASS RATIO TURBOFAN ENGINE 579

assumptions, Eqs. (5.1) and (5.2) apply to the high-pressure turbine. Since the

flow is adiabatic between stations 5 and 9, then

A6MFP6 = rcnA9MFP9,

and

Eq. (5.3) can be rewritten as

YrtL

A4,5

MFP(M4.5, Tt4.5,

f4.5)

(J.2)

~/~/ Tt4.5 7rna9 MFP(M9,

Tt5, f4.5)

In addition, Eqs. (5.4-5.8), and (5.17) also apply to this engine. These equations

allow solution for 14 of the dependent variables, and new relationships are re-

quired for the bypass ratio (or) and the four exhaust nozzle dependent variables

(M9, M19,

P9/Po, and P19/Po).

An expression for the engine bypass ratio (or)

follows directly from its definition written in terms of properties at stations 4 and

19. We write

rhc

where

Ptl9A19 MFP19 = Po Yrr 7rd 7rf 7rnf A19 MFP19

I~l F

and

Pt4A4 . Po rCr red reeL ZrcH Zrb A4 MFP4

th4 =

rhC(1 -- fl -- el -- e2)(1

f) =

-~4t4MFP4 =

Thus the engine bypass ratio can be written as

(1 - fi - el - e2)(1 +

f)/Tt4 MFP19

A19

= (J.3)

WcL TrcH ~b/(Tf f Jrnf) V Ttl3 MFP4 A4

The pressure ratio

Po/Pe

depends on whether or not a convergent nozzle is

choked. This is the same criterion that applied to the parametric cycle analysis of

this engine. Two flow regimes exist for flow through a convergent nozzle, unchoked

and choked. For unchoked flow, the exit static pressure (Pe) is equal to the ambient

pressure (Po), and the exit Mach number is less than or equal to one. Unchoked

flow will exist when

ete< (et)

Po T M=I

Then P0 =/De and the exit Mach number

(Me)

is determined using the compressible

flow functions. On the other hand, choked flow will exist when

ere (et)

-- >

P0-

PM=I

and then Me = 1,

PtdPe = (Pt/P)M=I, and

Pte/Pe

Po/Pe -- --

P,e/ Po

where

Pte/Po

is obtained by the product of the ram and component 7r for the

respective airstream.

580 AIRCRAFT ENGINE DESIGN

Summary of High Bypass Ratio Turbofan Engine Performance Cycle

Analysis Equations

This section presents the complete performance cycle analysis equations for

nonafterburning, separate exhaust flow, two-spool turbofan engine with bleed,

turbine cooling, and power extraction. For convenience in computer programming,

the required computer inputs and principal computer outputs are given. These are

followed by the cycle equations in their order of solution.

Inputs

Performance choices:

Flight parameters:

Throttle setting:

Design constants:

Jr:

Others:

Reference condition:

Flight parameters:

Component behavior:

Others:

Engine control limits:

Mo, To, Po

Tt4

7r d max , Yg b , 7?~ n , Y['nf

l~ f, l~cL, 17cH, l~tH, OtL, Ob, TIroL, l~mH, T]mPL, ?~mPH

A4, A4.5, A8, AI8,

,8, el, e2, heR, PrOL, PrOH

MOR, TOR,

POR

JTJR, 7t'cLR, 7gcHR, T(tHR, 7gtLR

"EfR, "~cLR, Z'cHR, Z'tHR, 75tLR

rmlR, rm2R, fR, M9R, M19R, OtR

Fk, thoR, SR

7Ccmax, Tt3max, Pt3max, %NL, %Nn

Outputs

Overall performance:

Component behavior:

F, tho, S, fo, ~lP, Oft4, V9/ao, V19/ao, ot

et9/ P9, P9/ Po, T9/ To, Ptl9/ el9, PI9/ Po, TI9/ To

7r f, 7fcL , 7rcH , 7~tH, 7rtL

r f, TcL, rcH, rtH, "~tL, "(~.

rml, "~'m2, f, M9, MI9

Equations

Preliminary computations:

FAIR (1,

O, To, ho, Pro, ~Po Cpo, Ro,

)to, ao)

Vo = mo ao

hto = ho + V~

2gc

FAIR (2, 0, Tto,

hto, Prto, C~to, cmo, Rto, Yto, aro)

"Or = hto/ ho

APPENDIX J: HIGH BYPASS RATIO TURBOFAN ENGINE 581

Y~'r =

Prto/ Pro

OR spec ~-- 1

0l¢ we~

= 1 - 0.075(Mo - 1) 1'35

2"gd : YfdmaxOR spec

ht2 = hto

ert2 = Pr, o

Set initial values:

forMo < 1

forl <Mo <5

7rJtH : 9TtHR, YgtL : TftLR, ~'tH ~--- "gtHR, "gtL : "~tLR

7g f ~ 7gfR , 7gcL : YgcLR, 7~cH : 7gcHR

Tf : TfR, TcL ~ "CcLR, "~cH : "~cHR

15ml "~- TmlR, Tm2 = Tm2R,

= fR

M4 = 1, M4.5 = 1,

M9=M9R , M19=M19R

FAIR

(1,

f, Tt4,

ht4, Prt4,

~bt4,

Cpt4, et4,

Yt4,

at4)

ht4.5 = ht4 rml TtH Tin2

f4.5 = f(1 --/3 -- el -- e2)/(1 --/3)

FAIR (2, f4.5,

Tt4.si,

ht4.5, Prt4.5 ¢t4.5,

cpt4.5, Rt4.5

Yt4.5, at4.5)

ht5 : ht4.5 "gtL

FAIR (2, f4.5,

Ttsi, hts, Prts, fbts, cpts, Rts, Yts, ats)

ht3 = hto'gcL'~cH

FAIR (2, 0,

Tt3, ht3, Prt3,

¢t3,

Cpt3, Rt3, Yt3, at3)

FAIR (1,

f, Tt4, ht4, Prt4,

¢t4,

Cpt4,

Rt4, Yt4,

at4)

f4.5 = f(1 -- 13 -- el -- e2)/(1 --/3)

TURBC (Tt4, f, A4/A4.5, M4,

M4.5,

OtH, Tt4.si, Tt3, fl, ~1, e2, YrtH, rtH,

Tt4.5)

TURB

(Tt4.5, f4.5,

A4.5/(A9~n),

M4.5, M9,

OtL, Tt5i, ~tL, 15tL Tt5)

"Ok

ht4/ ho

rf = l + { (1-- VtL)OmL [ (1- /3-el -e2)(l + (el vtn + e2)vcL'ccn

(lq'-ot) ProL}/

TrOmpg

/noho

{(TcL --

1)R/('Cf --

1)R + a}

(rcL- 1)R

"gcL

~-"

1 + (rf - 1)~f Z 1)R

= 1 + (1 --

VtH)OmH[(1 --/3 -- el --

e2)(1 + f) VX

rcH

[

Vr VcL

(1 + Ot )

PrOH

Vr ~cL OmPH l~loho

ht2 = hto

Prt2 = Prto

htl 3 = ht2"c f

ht2.5 =- htzZ'cL

ht3 = ht2.5~cH

-~- SI TcH}