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

550 AIRCRAFT ENGINE DESIGN

ht~ = hti - rh(hti - htei)

hte

~t ~ ~t i

FAIR (2,

f, Tten, hte, Prte, ~te, Cpte, Rte, Yte, ate)

[r,e]error = Ir, e - r,e.I

if

[Ttelerror

> 0.01 then

r,e = T,e.

GOTO 1

end if

Return

Appendix G

Constant-Area Mixer Analysis

Consider an ideal (no wall friction) subsonic constant-area mixer (Fig. G. 1) with

perfect gases having variable specific heats entering at stations 6 and 16. From

either the parametric or performance cycle analysis up to the mixer, all gas prop-

erties at stations 6 and 16 are known as well as the following ratios at the mixer

inlet stations:

0/t ~" -- rh6 -- O/

(G.1)

#t16 (1 -/~ -

E 1 --

e2)(1 +

f)

+ 81 -{- E2

Ptl6 27j f

= (G.2)

Pt6 2TcLYgcH~bTt'tHTrtL

htl6 "~r'~ f

= (G.3)

ht6 ~)~TJml Ttn~m2"EtL

The mixer total pressure ratio,

Pt6A/Pt6,

is required for performance calcula-

tions. Toward this end, the energy balance on the mixer gives

th6ht6 "[-

#/16ht16 =

lh6Aht6A

from which the total enthalpy ratio becomes

ht6 A 1 + og'htl6/ht6

rm --' -- -- (G.4)

ht6

1 + or'

An expression for the mixer total pressure ratio in terms of rm and the other prop-

erties at stations 6, 16, and 6A is obtained directly from the mass flow parameter

Splitter

Plate

th6

Control Volume

16 / 6A

rn6A =

(1 +

a ')/~/6

6A

Fig. G.1 Constant-area mixer.

551

552 AIRCRAFT ENGINE DESIGN

(MFP)

defined in Eq. (4.24) as

MFP-:- ~ --

M

p-~ = MFP(M, r,, f) (6.5)

Solving Eq. (6.5) for

Pt

and forming the ratio

Pt6a/Pt6

yields

Pt6A _

(1

+ od)~,'~ ~-~ MFP(M6, Tt6,

f6)

ZrM -- Pt~ --,~ MFP(M6A, Tt6A, f6A)

(6.6)

where

t~'16a/l~ 6

=

1 + od and

ht6a/ht6 =

T M

were used and where

A6

1

-- -- (6.7)

A6A

1

qt_ A16/A6

f6

f6a -- (G.8)

1 +or'

If M6 in Eq. (G.6) is specified, then

M6A

need only be determined to find JrM

because M6 fixes

M16

and, hence

AI6/A6

and

A6/A6A

as

shown by the following.

Using the Kutta condition at the splitter plate end, P6

= P16,

and total to static

pressure ratio

(Pt/P),

gives

(Pt) =(Pt) etl6

T 16 P- 6 ~t6 (G.9)

and using the compressible flow functions yields the Mach number at station 16.

From the mass flow parameter, we can write

AI6 , T/-~tl6

Pt6 MFP(M6,

rt6, f6)

(G.10)

o/

A6 -- V Tt6 Vtl6 MFP(MI6,

Ttl6, f16)

Any one of the three variables M6,

MI6, or AI6/A6

in Eqs. (G.9) and (G. 10) may

be specified and the remaining two determined. Because, generally, the desirable

Mach number range for M6 and

MI6 is

known, a Mach number is specified instead

of

A16/A6

in the analysis.

For the ideal constant-area mixer, application of the momentum equation pro-

vides a solution for

m6a

and, hence, for

ZrMideat.

The momentum equation in terms

of the impulse function (I) is, for the constant-area mixer,

where

Thus

16 -]'- 116 =

16A

(G.11)

I = PA(1 +

yM 2) (G.12)

P6AA6.4(1 + Y6AM 2)

= P6A6(1 +

y6M~)+

PI6A,6(1 + YI6M26)

Using the Kutta condition at the splitter plate end, P6

= P16,

gives

{ A16( 1 }

P6AA6A(1 +

y6AM62A) =

P6A6

(1 +

y6M 2) + -~6 +

y16M26)

(G.13)

Since

APPENDIX G: CONSTANT-AREA MIXER ANALYSIS 553

/........_

rh Rr Rr m l gr

PA = P-- = rh = rh

pV V M y~-R-~cT M v Ygc

then the

(PA)

terms in Eq. (G. 13) can be replaced to yield

if/6 R~ A16(1

#16AR~6A(I"~Y6AM2A)-~-

{ (1 -~- y6M2) --~ -~- Y16M26) ]

M6A V Y6Agc M6 V Y6gc ~6

and solving for M6A yields

~1

"~- Y6AM2A R~6T 6 (1 q- F6M6 2) -t-

A16/A6(1

+

Y16M26)

(G.14)

M6A "~-

~I -~6

M6(1 q-a t)

where the right-hand side is a known constant. This is a nonlinear equation for M6A

that can be solved by functional iteration in combination with the compressible

flow functions. The resulting value of M6A is placed in Eq. (G.6) to give

7~Mideal

and, finally,

71: M = 7r M max YC M ideal

(G.15)

where ZrM

max

is the mixer total pressure ratio caused by wall friction only (no

mixing losses).

Gross Thrust

It is interesting to compare the gross thrust capabilities of mixed and unmixed

streams for the case of correctly expanded ideal exhaust nozzles. The gross thrusts

are, using subscript e for the exhaust nozzle exit stations,

FGu M ~---

fft6Ve 6 "1- fftl6Vel 6

for unmixed streams (G.16)

FGM X =

(/9:/6 "-1-

1Ttl6)Ve6 a

for mixed

streams (G.17)

where

It follows immediately that

FCMx

V2=2gcht(1-~)

(1 +

a')~/ht6A(1 -- h6Ae/ ht6A)

(G. 18)

FGuM

~/ht6(1 -

h6e/ht6) -'}-

a'~/htl6(1 -

hl6e/htl6)

Example Results

Contours for parametric values of

F~Mx/FauM are

plotted in Fig. G.2 for the

following input data to the system of equations listed below:

a' = 2, M6 --- 0.5,

Pe/Pt6

= 0.3,

Tt6 =

1600 R,

f6 =

0.03,

f16 = 0,

Yt'Mmax = 0.97

554 AIRCRAFT ENGINE DESIGN

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

m

-f

f

0.990

0.995

1.000

- S

1.005

1.010

1.015

I I I a I

0.9 1.0 1.1

~,~

/ &

1.2

Fig. G.2 Contours of constant

F~Mx/F~vM.

Although the mixer can actually augment (or increase) the ideal thrust under

some conditions, it is most sensitive to Tt 16/Tt6 and ordinarily results in a reduction.

For typical values of turbofan cycles Ttl6/Tt6~-~0.7 and etl6/Pt6 "~

1.0,

the loss

is negligible with 7rMmax

=

0.97 (see Fig. G.2); however it is about 2% with

7rM max = 0.92.

Summary of Equations--Constant-Area Mixer

Inputs

Outputs

Pe

Ttl6

Ptl6

t

O' , Tt6, M6, , , , f6, f16, 7rMmax

Pt6 Tt6 Pt6

A

16

FGMx

--, ~M, "gM, MI6,

M6A, --

A6 FGuM

Equations

FAIR(l, f6, Tt6, h,6,

Prt6, (~t6, Cpt6, Rt6,

Yt6, at6)

Tt16

Ttl 6 = Zt6

Tt6

FAIR(I, f16,

Ttl6,

htl6,

Prtl6,

~/16,

Cptl6, Rtl6,

Ytl6, atl6)

ht6 Jr- ot'htl6

ht6z

--

1 +or'

APPENDIX G: CONSTANT-AREA MIXER ANALYSIS 555

B

rM = ht6A/ ht6

RGCOMPR (Tt6 , M6, f6, (Tt/T)6, (Pt/P)6, MFP6)

Z 6 ~- Tt6/(Zt/Z)6

etl6

(Pt/P)16 = (Pt/e)6

Pt6

ertl6

Prl6 -- --

(Pt/P)16

FAIR (3, 0, T16,

h16, Prl6, q~16, Cpl6, R16, 2/16, a16)

V16 = ~/2gc(htl6 - hi6)

M16 =-- V16/a16

RGCOMPR (Ttl6, M16, f16, (Zt/T)16, (Pt/P)16, MFP16)

A16 , Z~16t16 et6 MFP6

ol

A6 V Tt6 Ptl6 MFP16

A6 1

A6A 1 -[- A16/A6

A16(1

Constant =`/-~- (1

+ y6 M2) -t- -~6 nt- )/16M26)

¥ ~6 M6(1 + ot t)

Set initial value of Mach number at station 6A = m6a i

RGCOMPR (Tt6A, M6A, f6a, (Tt/ T)6A, ( Pt/ P)6A,

MFP6A)

T6A = Tt6A(Tt/ T)6 A

FAIR (1, f6a, T6A, h6A, Pr6A, qb6A, Cp6A, R6a, Y6A, a6a)

• ~ t 1 -t- Y6AM~2i

M6A -~

V Y6A Constant

If

I M6A

- M6Ai

I > 0.000l,

then

M6A i -~- M6A

and go to B; else continue

, ,, ,---- A6 MFP6

Yg M ideal = (1 ~ ot ) ~/ ~ M "~6 A

YgM ~ 7rMmax 7rMideal

Pr6e = ert6( Pe/ Pt6)

FAIR (3, f6,

T6e,

h6e, er6e, qb6e, Cp6e, R6e, Y6e, a6e)

Prl6e = ertl6( Pe/ Pt6)/( Ptl6/ et6)

FAIR

(3, fl6,

Zl6e, hl6e, Prl6e, ~bl6e, Cpl6e, R16e, Yl6e, al6e)

Pr6Ae • ert6( Pe/ Pt6)Tr M

FAIR

(3,

f6a,

Z6ae, h6Ae, Pr6Ae, ~)6Ae, Cp6ae, R6ae, ~6ae, a6Ae)

FGMx (1 qt_ ott)~/ht6a ( 1 __ h6ae/ ht6a)

FGuM ~/ht6(1 -- h6e/ ht6) + ot'~/htl6(1 - hl6e/ htl6)

Appendix H

Mixed Flow Turbofan Engine Parametric Cycle

Analysis Equations

Appendix H summarizes the complete parametric or design point cycle analysis

equations for the mixed flow, afterbuming, 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. Eqs. (H. 1-16) in the

listing represents the 1 6 independent equations in terms of the 1 6 dependent compo-

nent performance variables in the order l-f, TcL, "CcH, f, V,,1, 75tH, 2"gtH, "~m2, "CtL, 7~tL,

a', VM, M16, M6A, :rrM, andfAB. Note this solution procedure requires no iteration.

Inputs

Flight parameters:

Aircraft system parameters:

Design limitations:

Fuel heating value:

Component figures

of merit:

Design choices:

Outputs

Overall performance:

Component behavior:

Mo, To, Po

fl, CTOL, CTOH

hpR

EI~ E2

Y/'b, :r/'d max, Y/'M max,

7~AB, 7"(n

e f, ecL, ecH, etH, etL

T]b, TIAB, OmL, rlmn, rlmPL, T]mPH

7r :, zrcL, ~rc., 0~, T,4, T~7, M6, Po/ e9

F/tho, S, fo, OR, rlrn, V9/ao, Pt9/ P9

3"gtH ~ Y(tL, 7gM

"gf , TcL, 7:cH~ "gtH, TtL, 7:~,~ ~ZAB

f, fAB

Of, I~cL, T]cH, OtH, l"]tL

M16,

M6A, M9

Equations

FAIR(I, 0, To, h0,

Pro, ~bo, Cpo,

R0, Y0, a0)

Vo = Moao

hto = ho "b Vg

2go

557

558 AIRCRAFT ENGINE DESIGN

FAIR (2, 0, Tto, hto, Prto, ~bto, Cpto, Rio, ?'to, ato)

rr = hto/ ho

7"( r = Prto/ Pro

r/R spec = 1 for Mo < 1

OR spec = 1 - 0.075(Mo - 1) 1'35 for 1 < Mo < 5

800

forl <Mo <5

OR spec -- M4 +

935

7Td ~ 2"fd max

OR spec

ht2 = hto

Pr~2 = Pr, o

1/ef

ertl3 = l'rt2Tf f

FAIR (3, 0, Ttl3,

ht13, Prtl3,

t~tl3,

Cptl3, Rtl3,

Ytl3, atl3)

7~ f = htl3/ ht2

Pr~13i = Prt2Zr y

FAIR (3, 0, Tt13i,

htl3i, Prtl3i, ~tl3i, Cptl3i, gtl3i,

Ytl3i,

atl3i)

htl3i - ht2

fly -- htl3 - ht2

rJ -- 1/ecL

Prt2.5 ~ l-rt2JtcL

FAIR (3, 0, Tt2.5, ht2.5, Prt2.5, 4~t2.5, cpt2.5, Rt2.5, Yt2.5, at2.5)

rcL = ht2.5/ ht2

Prt2.5i = Prt27fcL

FAIR (3, 0, Tt2.5i,

ht2.5i, Prt2.5i, flPt2.5i, Cpt2.5i, gt2.5i,

)/t2.5i,

at2.5i)

ht2.5i - ht2

rlc L --

ht2.5 - ht2

Prt3 r~ - 1/ecH

Frt2.5Jt cH

FAIR (3, 0, Tt3,

ht3, Prt3, (bt3, cpt3, Rt3,

Yt3,

at3)

"CcH = ht3/ ht2.5

Prt3 = Prt2.57rcH

FAIR (3, 0, Tt3i,

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

ht3i - ht2.5

~cH --

ht3 - ht2.5

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

f4g

A FAIR(I,

f4i, Tt4, ht4, Prt4, qbt4, cpt4, Rt4,

Yt4,

at4)

ht4 - ht3

f--

~]bhpR -- ht4

If

If - f4il

> 0.0001, then

f4i = f

and go to A; else continue.

r~. --= ht4/ ho

(H.1)

(H.2)

(H.3)

(H.4)

APPENDIX H: MIXED TURBOFAN ENGINE PARAMETRIC 559

(1 -

fl

-- S 1 --

e2)(1

+ f) + elrrrcLrcH/rX

't-ml

VtH =

1

-

(1 - ,8 - el - 82)(1 + f) +

El

rrZcL(ZcH-

1) + (1 +

ol)CToH/OmPH

~/mnVX{(1 -- fl -- el -- 82)(1

+ f) + elVr'rcL't'cla/rx}

ht4.1 ---- ht4Vml

f

f4.1 ~--

1 + f + el/(1 - fl - el -

e2)

FAIR

(2, faa, Tt4a,

ht4.1, Pn4.1,

~t4.1,

Cpt4.1, gt4.1,

Yt4.1, at4.1)

ht4.4 ---- ht4.1TtH

FAIR

(2, f4.1,

Ttn.a,

ht4.4, Prt4.4,

(~t4.4,

Cpt4.4,

Rt4.a,

~/t4.4, at4.4)

= ( Prt4.4 ~ 1/etH

Prt4.4i "= T(tH ert4.1

FAIR

(3, f4.1,

Tt4.4i,

ht4.4i, Prt4.4i,

t~t4.4i,

Cpt4.4i, et4.4i, Yt4.4i, at4.4i)

ht4.1 - ht4.4

OtH--

ht4.1 - ht4.4i

(1

-

fl - el - s2)(1

+ f) + el + s2{rrrcLrcH/(rXrnlrtH)}

~'m2 =

(1 - fl

- e 1 -

e2)(1 + f) + E1 +

E2

ht4.5 ~ ht4.4~:m2

f

f4,5 ~--"

1

+ f + (el + s2)/(1 -/3 - el - s2)

FAIR

(2,

f4.5, Tt4.5,

ht4.5,

Prt4.5,

t~t4.5,

Cpt4.5,

Rt4.5, ~t4.5,

at4.5)

r,{(rcL - 1) +

ot(rf -

1)} + (1 +

oI)CToL/T/mPL

~tL =

1

-

(H.5)

(H.6)

(H.7)

(H.8)

l']mL ~,k "~tH

{ (

(1-/3-sl-s2)(l+f)+ el+~ rx /

ht5 = ht4.5"CtL

FAIR (2, f4.5,

Tts, hts, Prts, dPtS, CptS,

Rt5,

Yt5,

at5)

( Prt5 "~ 1/etL

Pr, si = rrtH Pr~4.5

FAIR

(3, f4.5,

Ttsi, ht5i, Prt5i, (bt5i, Cpt5i, Rt5i, Yt5i, at5i)

ht4.5 -

ht5

OtL--

ht4.5 - htsi

ht6 =ht5; Zt6 • Zt5; f6 =

f4.5

htl6 =

htl3; Ttl6 = Ttl3;

Prtl6 = Prtl3;

f16 = 0

ot

(1 - fl

- E 1 -

e2)(1 + f) + el + s2

f6A

= f6/(1 + Or')

ht6 + otthtl6

ht6A

--

l+a'

(H.9)

(H.IO)

(H.11)

560 AIRCRAFT ENGINE DESIGN

B

C

75 M = h t6A / h t6

(H. 12)

Ptl6 Yc f

Pt6 7"(cL ~cH Tgb 7~tH 7rtL

RGCOMPR (1, Tt6, M6, f45, (Tt/T)6, (t~t/P)6, MFP6)

T6 = Tt6/(rt/r)6

Pt16

(Pt/P)16 = (Pt/P)6

Pt6

Prt16

Prl6 -- --

(Pt/P)16

FAIR (3, 0, T16, hi6, Prl6, ~b16, Cpl6, R16, )/16, a16)

V16 = x/2gc(ht16 - hi6)

M16 = V16/a16

(H.13)

RGCOMPR (1, Ttl6, MI6, f16, (Tt/T)16, (Pt/P)16, MFP16)

A16 ~ r~tl6 Pt6 MFP6

A---6 ~ ol~ ~6t6 Vtl6 MFP16

A6 1

A6A

1

-k- A16/A6

Constant=, I (I+y6Mz)+A'6 (I+Y,6M 6) } /{M6(I+d)'

V ×6

Set initial value of Mach number at station 6A

=

M6A i

RGCOMPR (Tt6a, M6A, f6z, (Tt/ T)6A, ( Pt/ P)6a, MFP6A)

Z6a = Zt6A(Zt/ Z)6a

FAIR(I, f6a, T6A, h6a, Pr6z, q96A, Cp6A, g6a, Y6A, a6a)

R6~AT~A] 1

+ Y6aM2ai

M6A

(H.14)

V ~6A Constant

If

[M6A

- m6ai

[

> 0.0001,

then

m6a i = m6a

and go to B; else continue

A6 MFP6

~Mideal = (1 -~-O~')~'-M - --

A6A MFP6A

7rM = 7rMmax 7rM ideal

(H.15)

Set initial value of fuel/air ratio at station 7

=

f7i

FAIR (1, f7i, Tt7, ht7, Prt7, qbt7, Cpt7, Rt7, Yt7, at7)

ht7

"()~AB ~ --

ho

(1-- fl -- gl -- e2 ) rXAB -- rX rm] rtnrm2rtL rM (H.16)

faB= 1+f i~-~--~-

hpROaB/ho--TkZB

f7 = U6A + UAB

If [f7 -

f7i[

> 0.0001, then

fTi

---- f7 and go to C;

else continue.

fo=f7

Tt9 = Tt7; ht9 = ht7; Prt9 = Prvt

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

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