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

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582 AIRCRAFT ENGINE DESIGN

htl3i

= ht2{1 +

~f(Tf -

1)}

ht2.5i =

ht2{1 +

rlcL(rcL --

1)}

ht3i

= h/2.5{1 +

rlcH(rcH --

1)}

FAIR (2, 0,

Ttl3, htl3, Prtl3,

~/13,

Cptl3, gtl3,

Ytl3, all3)

FAIR

(2,

0, Ttl3i,

htl3i, ertl3i,

~t13i,

Cptl3i, Rtl3i,

)"t13i,

atl3i)

FAIR

(2, 0,

Tt2.5,

ht2.5, Prt2.5,

¢t2.5,

Cpt2.5, Rt2.5,

Yt2.5, at2.5)

FAIR

(2, 0,

Ttz.si,

ht2.5i, ert2.5i,

~bt2.5i,

¢pt2.5i, Rt2.5i,

Yt2.5i,

at2.5i)

FAIR

(2, 0, Tt3,

ht3, Prt3,

q~t3,

Cpt3, Rt3,

Yt3,

at3)

FAIR

(2, 0,

Zt3i, ht3i, Prt3i, ~t3i, Cpt3i, et3i, Yt3i, at3i)

yr f = ertl3i / Prt2

TfcL = Prt2.5i / Prt2

zrci~ = Prt3i / Pr,2.5

7g" c ~ 7~cLYrcH

Tc = ~cL"CcH

f temp =

FAIR

(1, f, Tt4,

ht4, Prt4,

~t4,

Cpt4, gt4,

Yt4,

at4)

ht4 - ht3

f--

hpROb -- ht4

if If - fiempl >

0.0001 goto

Ptl9

~- 7"(r ~d Yr f 7rnf

RGCOMP (3,

Tt13, O,

M19,

Ttl9/ To,

PtI9/Po,

MFP19)

M19 > 1 then

M19 = 1

RGCOMP (1,

Tt13, O,

M19, Ttl9/T19,

etl9/ el9, MFP19)

Ptl9/P19

Po/ P19 -- --

Ptl9/Po

Ttl3

T19__ --

TtI9 / TI9

else

Po/ P19 = 1

P,19/P19 = P.9/Po

T,13

TI9--

Ttl9/T0

endif

FAIR

(1, 0, Tl9,

hi9,

Prl9,

q~19, Cpl9, R19, Y19, alg)

(1 --,8 -- ~___~1 -- ~2)(___.~1 +

f)j

Tt4 MFP19 A19

Olnew = T(cLTt'cHTt;b/(Tt'fTtJnf) ~Ttl3 MFP4 A4

if ~ - ot,~e., > 0.001

then

OtR

APPENDIX J: HIGH BYPASS RATIO TURBOFAN ENGINE 583

O[ ~ Olne W

goto 1

endif

Pt9

-- ~-- 7t'r 7~d 7~cL 7~cH ~b ~tH ~tL ~n

RGCOMP (3, Tts, f4.5,349, Tt9/To, Pt9/Po, MFP9)

if M9 > 1 then

M9=l

RGCOMP (1, Tt5, f4.5, M9, Tt9/T9, Pt9/P9, MFP9)

Pt9 / P9

Po/P9 - --

Pt9 / Po

T,5

T9--

Tt9 / T9

else

Po/P9 = 1

Pt9/ P9 = Pt9/ Po

r~5

rg--

r,9 / ro

endif

FAIR (1, f4.5, Tg, h9, Pr9, ~9, Cp9, R9, Y9, a9)

RGCOMPR (1, Tt4, f, M4, TtT, PtP, MFP4)

1 + fR Po(1

"q- Ol)Ygr YCd YgcL 7rJcH

MFP4 Tf~t4R

rhOnew

:/~t0R 1

+-~ {Po(1

+ Ol)Ygr 7~d Y~cL YCcH}R MFP4R . I Tt4

• I~lOnew -- mo

mOerror ~ thOR

if lhOerror > 0.001 then

lfVl 0 = l~Onew

goto 1

endif

V9 = M9a9

El9 = M19a19

fo = f(1 --/3 -- el -- e2)/(1 + or)

V~9

[1 + fo(1 + a) - fl] V9 + a-- - (1 + a)Mo

ao ao

_ ao + [1 +

fo(1 + a) - ~]~o

T9 / To

Po/Pg)

tho gc(1 + or) V9/ao Yo

R19 TI9/To (1 - Po/P19)

+ot--

RO V19/ao YO

584 AIRCRAFT ENGINE DESIGN

S--

F/rho

2gcMo(1 + oOF /(rhoao)

TIp :

{1 +

fo(1 + el) - fl} (V9/ao) 2 + ot (Vl9/ao) 2 -

(1 + a)Mg

1 {[[l+fo(l+oO-fl]vZ+otV29 ]

= - vg

riO : rltHrlp

RGCOMPR (1,

Tto, O,

Mo,

TtT, PtP, MFPo)

Yh o ~C-T t o

AO----

PtoMFPo

100/

ho'cr(rgf -

%NL

[h0rr(rf - 1)JR

100/

hor,.rcL(rcH --

%NH

[horrrcL(rcH-

1)]R

If any of the control limits (7rc,

Tt3, Pt3,

etc.) are exceeded, then reduce Tt4 and

go to 1. A Newtonian iteration scheme is used to rapidly converge on the value of

Tt4

that meets the most constraining control limit.

Example Performance Results

Consider a turbofan engine designed for a Mach number of 0.8 at a standard day

altitude of 30 kft using the variable specific heat (VSH) gas model with a compres-

sor pressure ratio of 30 (7rcL ---- 4, Zr,H ----- 7.5), a fan pressure ratio of 1.5, a bypass

ratio of 8, and the other inputs given here:

= 0.89; 7rdmax

0.99; 0b = 0.995;

Tt4 =

2860°R;

eeL

0.90; zrh = 0.96;

Tim L

0.995;

her

= 18,400 Btu/lbm;

ecH

= 0.90; Jr, = 0.99;

~mH

0.995; fl = 0.03;

etH

0.89;

Ygnf

0.99;

CTOL =

0.00;

rlmeL

----

1.0;

etL

= 0.91; el, e2 = 0.05;

CTOH

= 0.005; and

;lmPH

0.99. The

performance variation of the fan pressure ratio, the compressor pressure ratio, and

the bypass ratio with changes in flight Mach number and altitude are presented in

Figs. J.4-J.6, respectively, for full throttle operation with maximum compressor

pressure ratio of 30 and maximum

Zt4

of 3200°R. These are the same trends

observed in Figs. 4.9, 4.10, and 4.11. Because the component performance curves

of Figs. J.4-J.6 break at about a Mach number of 0.45 at sea level on a standard

day, this engine has a theta break (App. D) and throttle ratio (TR) of about 1.04.

Figure J.7 shows the variation in full throttle uninstalled thrust with changes

in Mach number and altitude for an engine sized for a static sea level thrust of

50,000 lbf. The variation of the uninstalled thrust specific fuel consumption at

full throttle is shown in Fig. J.8. The partial throttle performance at an altitude of

30 kft is presented in Fig. J.9 for selected flight Mach numbers.

APPENDIX J: HIGH BYPASS RATIO TURBOFAN ENGINE 585

1.55

1.50

1.45

1.40

I I I I

40 kft

Sea Level ~

1.35

, , , I , , , I , , , I , , , I ~ , ,

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 3.4 Fan pressure ratio of exam ~le turbofan engine at full throttle (standard day).

I I I

20-40 kft

0 kft

22 , , , I , , I , , , I , , , I , , ,

0.0 0.2 0.4 0.6 0.8 1.0

Fig. J.5 Compressor pressure ratio of example turbofan engine at full throttle

(standard day).

586 AIRCRAFT ENGINE DESIGN

I0.0 , , , ,

9.5

9.0

/ 10 kft_ 4

20 kft

8.0 J~ ' 30 kft -I

7.5

"'~-- ' ' I , , , I , , , I , , , I , , , I

0.0 0.2 0.4 0.6 0.8 1.0

Fig. J.6 Bypass

ratio of example turbofan engine at full throttle (standard

day).

50,000

40,000

30,000

(lbO

20,000

10,000

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

Sea Level

10 kft

~ 20 kft

.------" 30 kfi •

0.0 0.2 0.4 0.6 0.8 1.0

Fig. J.7 Thrust of example turbofan engine at full throttle (standard day).

0.8

0.7

I I I I

10kfi

20 kfl

0.6

(l/h)

0.5

0.4

0.3

APPENDIX J: HIGH BYPASS RATIO TURBOFAN ENGINE 587

0.2

, ~ , I , , , I , , , I , , , I , , ,

0.0 0.2 0.4 0.6 0.8 1.0

Fig. J.8 Thrust specific fuel consumption of example turbofan engine at full throttle

(standard day).

(i/h)

0.80

0.75

0.70

0.65

0.60

0.55

0.50

0.45

0.40

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

.._.._ 0.9 M

0.8M

0.7 M

~ 0.6M

0.5M

0.4 M

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

0 2,000 4,000 6,000 8,000 10,000 12,000 14,000

F (k lb)

Fig. 3.9 Partial throttle performance of example turbofan engine at 30 kft (standard

day).

Appendix K

Turboprop Engine Cycle Analysis

This material is included in case turboprop engine cycles are under considera-

tion. They are strong candidates for any application where high thrust and low fuel

consumption are required and the flight Mach number never exceeds about 0.8.

Consider the turboprop engine shown in Figs. K. 1 a and K.lb. The station numbers

of locations indicated there will be used throughout this appendix and include the

following:

Station Location

3.1

4.1

4.4

4.5

Far upstream or freestream

Inlet or diffuser entry

Inlet or diffuser exit

Compressor entry

Compressor exit

Burner entry

Burner exit

Nozzle vanes entry

Modeled coolant mixer 1 entry

High-pressure turbine entry for

rrtn

definition

Nozzle vanes exit

Coolant mixer 1 exit

High-pressure turbine entry for

"Ctn

definition

High-pressure turbine exit

Modeled coolant mixer 2 entry

Coolant mixer 2 exit

Low-pressure turbine entry

Low-pressure turbine exit

Core exhaust nozzle entry

Core exhaust nozzle exit

The component r, zr, efficiencies, and assumptions of Chapter 4 apply to this

engine with the exception of those referring to the exhaust mixer and afterbumer.

Unlike the engine of Chapters 4 and 5, this engine has neither a fan nor a bypass

airflow. The mass flow ratios of Eqs. (4.8a-4.8e) and (4.8i) apply as does the

turbine cooling model of Chapter 4 shown in Fig. 4.2.

The compressor of this turboprop engine and high-pressure spool power takeoff

are powered by the high-pressure turbine, while the low-pressure turbine provides

589

590 AIRCRAFT ENGINE DESIGN

propeller

bleed air

Fig. K.la Reference stationsmturboprop engine.

mechanical power to both the propeller and the low-pressure spool power takeoff.

The engine's nozzle is modeled as a fixed convergent nozzle.

The following sections outline development of both parametric and performance

cycle analysis equations for this turbofan engine.

Parametric Analysis

The "work interaction coefficient" (C) is introduced for use in analysis of

the work interaction of this engine with the vehicle rather than the thrust. The

dimensionless coefficient is defined by

Total power interaction with vehicle

/ho

(K. 1)

C--

Mass flow of air through core engine /

power extraction

P! Ol PTOH

----P bleed air cooling air #2

nozzle

] coolant

high-pressure spool

high-pressureturbine

low-pressure spool

Fig. K.lb Reference stations--bleed and turbine cooling airflows.

APPENDIX K: TURBOPROP ENGINE CYCLE ANALYSIS 591

The use of such coefficients is commonplace in the turboprop industry because

turboprop designers see themselves as converting engine power into flight power.

Please note that the bypass ratio (a) does not appear in this development because

the propeller is treated as a power exchange device. For the propeller, the total work

interaction with the vehicle is given by

rlpropPprop/rh 0

where

Oprop

is the efficiency

of the power transfer from the propeller to the air and

eprop

is the power transferred

to the propeller. The total power interaction of the propeller is also equal to the

effective thrust of the propeller

(Fprop)

times the velocity of the vehicle (V0) or

Total work interaction of the propeller =

rlpro p

Pprop = Fprop Vo

Thus from Eq. (K.1)

Oprop Pprop Fprop Wo

Cprop- l~loho --

rhoh~

(K.2)

Similarly, the work interaction coefficient for the core engine is defined with respect

to the power transferred to the vehicle (thrust × velocity) or

FcV0

-' ~ (K.3)

The total work interaction coefficient of the turboprop engine is the sum of

Cprop

and

Cc,

CTOTAL = Cprop q- Cc

(K.4)

Thus,

CTOTAL

is related to the total power transferred to the vehicle. The effective

uninstalled thrust (F) of the engine can be found from Eqs. (K. 1) and (K.4) to be

C TOTAL I'hoho

-- Fprop

+ Fc --

(K.5)

The uninstalled specific thrust

(Fc/~no)

of the core is given by

{(1 +fo -fl)~ - M0 + (1 +fo -fl)

r'no gc [ ao

R 9 T9/T 0

(1 -

Po/P9) I

Ro V9 / ao

where

= f (1 - fl - el

-- E2).

Thus, the work interaction coefficient of the core

can be written as

{ R9T9/T°(1-P°/P9)I

= (Yo - 1)Mo (1 + fo - fl)

V9a0 --

Mo + (1

+ fo - fl) R--oo V9/ao Yo

]

(K.6)

The velocity and temperature ratios required in Eq. (K.6) are obtained using the

methods of Chapter 4. We have

592 AIRCRAFT ENGINE DESIGN

where h9 is determined from

ht9

and the pressure ratio

Pt9 (eo'~r~ d

= I --~9/] 2rc yl'b 2"ttH ~tL 7fn

The independent design variables for this engine are the compressor pressure

ratio (ztc) and the turbine total enthalpy ratio (rt). The enthalpy ratio across the

low-pressure turbine is obtained from

Z'tL = "tt / ( rm l T;tH'tm2 )

The enthalpy ratio across the compressor (re) is related to its pressure ratio by

Eq. (4.9c). The enthalpy ratio across the high-pressure turbine (rtH) is obtained

from the power balance of the high-pressure spool, which gives

PTOH

/h4.1(ht4.1 --

ht4.4)OmH = Ihc(ht3 - ht2) 4- --

~mPH

and rearrangement allows the calculation of the high-pressure turbine total enthalpy

ratio

"tr('r c --

1) +

CTOH/JTmPH

rtH

= 1 -- (K.7)

r/mHrX{(1 -- fl -- el -- e2)(1

+ f) + elrrrc/rX}

The enthalpy 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

(ZttH)

and low-pressure turbine

OrtL) are

related to their respective enthalpy

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

given by Eq. (4.18). Thus, the only unknown for solution of Eq. (K.6) is the static

pressure ratio,

Po/P9.

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

and choked. For unchoked flow, the exit static pressure (P9) is equal to the ambient

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

flow will exist when

To <-- 7M=1

then P9 = Po, and the exit Mach number (M9) is determined using the compres-

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

Pt9 > (Pt)

Po e M=I

thus

) P~9/P9

Pt__99 = Pt ,

and

Po/Pg- Pt9/~

M9=l, P9 P" M=I

where

Pt9/Po

is obtained by the product of the ram and component zt for the core

airstream.