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

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142 AIRCRAFT ENGINE DESIGN

1) The flow areas are constant at stations 4, 4.5, 6, 16, 6A, and 8 dry (afterburner

off).

2) The flow is choked at the high-pressure turbine entrance nozzles (choking

area 4), at the low-pressure turbine entrance nozzles (choking area 4.5), and at the

exhaust nozzle (station 8). Because the exhaust nozzle may unchoke at low throttle

settings and affect the fan operating line (see Fig. 7.E 10), the case of the unchoked

exhaust nozzle (station 8) is also included in this analysis.

3) The component efficiencies

(Of, OcL, OcH, Oh, OtH, rltL, 1TAB, OmL, rlml-l, ~]mPL,

and

rlmeH)

and total pressure ratios (rOb, Zrgmax,

rCaBary,

and rrn) do not change

from their design values.

4) Bleed air and cooling air fractions are constant. Power takeoffs are constant.

5) The air and combustion gases are modeled as perfect gases in thermody-

namic equilibrium, and their properties are based on the NASA Glenn thermo-

chemical data and the Gordon-McBride equilibrium algorithm reference, s

6) The simplifying gas model of a calorically perfect gas is included in the

analysis. It assumes that the gases are calorically perfect upstream and downstream

of the bumer and afterburner and values of gt,

Cpt, FAB, and CpZ B

do not vary with

throttle setting, but included is the variation of g and

due to mixing with bypass

ratio. For this model, fuel-air ratios f and

faB are

ignored when compared with

unity.

7) The exit

area A9

of the exhaust nozzle is adjustable so that the pressure ratio

Po/P9

can be set to a predetermined value.

8) The area at each engine station is constant. However the area of station 8

changes with the afterburner setting to maintain constant pressure at the mixer exit

or nozzle entrance.

9) The diffuser total pressure ratio, 7rd, is given by Eqs. (4.12a-4.12d).

5.2.3 Referencing and the Mass Flow Parameter (MFP)

Two techniques are worthy of discussion at this point to prepare the path for their

efficient and frequent use in the analysis to follow. The first is called referencing

and involves the use of reference point conditions to evaluate constants appearing

in equations for the dependent performance variables. The second exploits the mass

flow parameter, introduced in Sec. 1.9.3, to capitalize on the law of conservation

of mass.

Referencing.

The functional relations for engine cycle analysis are based on

the application of mass, energy, momentum, and entropy considerations to the

one-dimensional steady flow of a perfect gas at an engine reference or off-design

steady state operating point. Thus, if at any off-design point

f(r, zr) = constant

represents a relationship between the two performance variables r and Jr at a steady

state operating point, then the constant can be evaluated at the reference point, so

that

f(r, 7r) =

f(re,

ZrR) = constant

ENGINE SELECTION: PERFORMANCE CYCLE ANALYSIS 143

because f(v, zr) applies to both on-design and off-design points. This technique

for replacing constants with reference point values is used frequently in the per-

formance analysis to follow.

Mass flow parameter.

The mass flow parameter

(MFP)

is defined as the

grouping rn 4~t/Pt

that can be written as

MFP-- p----~ V R Pt V T

The static to total pressure and temperature ratios

(P/Pt, T~ Tt) are

functions of the

total temperature of gas, chemical equilibrium properties, and the Mach number.

Thus the mass flow parameter can be written in the following functional form:

FnVr~t -- M. V/-~ P./~ = MFP(M, Tt, f)

(4.24)

MFP-- ~tA V R P,V T

The subroutine RGCOMP that is used to calculate the compressible flow functions

(P/Pt, T/Tt, and MFP)

is given in Appendix F. For the simplifying case of the

calorically perfect gas, the properties

P/Pt, T/Tt,

and

MFP are

given by

T 1 + Y - (1.1)

T, 2

P ( Y- 1M2) #'

= 1 + 2 (1.2)

y+l

~'~ ~/~-t ~/~( 1) 2(1-)/)

MFP-- Pt~--M

l+Y2- M2 (1.3)

5.2.4 Performance of Turbines with~without Coolant Mixers

The first step in determining the performance of the entire engine at conditions

away from the reference point is to analyze the behavior of the high- and low-

pressure turbines. This is greatly expedited by the fact that they are both deliberately

designed to be choked in their entrance stator airfoil or vane passages, and that

the static pressure downstream of the low-pressure turbine is tied to the mixer

entrance conditions. As you will see in the development that follows, this both

restricts turbines to a very narrow range of operation and furnishes us with a

straightforward method of solution. The remainder of the engine performance

analysis flows directly from this step because the turbines, in conjunction with

the throttle setting (i.e., T,4), provide the power for the fan and compressors and

control the fan and compressor mass flows (see Sec. 5.2.5).

The performance (rrt and rt) of a turbine at off-design is primarily determined

by the efficiency and mass conservation relationships. It is shown in Ref. 1 that

zrtH and VtH

remain constant for an uncooled turbine with constant specific heats

(calorically perfect gas). Similarly, as shown below, these ratios can be considered

constant for a cooled turbine with constant specific heats (calorically perfect gas).

144 AIRCRAFT ENGINE DESIGN

When a cooled turbine with variable specific heats is modeled, it is found that Zrtn

and Ttn

vary only slightly with engine operating condition.

It is also shown in Ref. 1 that

7rtL

and

TtL are

constant in an uncooled low-

pressure turbine with calorically perfect gas for a turbofan engine having choked

separate (unmixed) fan and core streams. For a mixed flow turbofan, on the other

hand, these ratios cannot be considered constant because

~rtL

(and hence rtL) must

modulate at off-design conditions to maintain P6 = P16, the Kutta condition, at

the mixer entrance.

Variable specific heats.

Consider a high-pressure turbine with cooling air

fractions el and e2 modeled as in Fig. 5.lb. Let the nozzle throat stations just

downstream of stations 4 and 4.5 be denoted by 4' and 4.5'. With choked flow at

stations 4 t and 4.5', Eq. (4.24) yields, assuming

Pt4 = Pt4'

and

Pt4.5 = Pt4.5,

l~t4' Pt4.5/Pt4

A4.5,

MFP(M4,, Tt4,

rh4.5' T~-~-t4.5/Tt4 a4,

MFP(M4.5,, Tt4.5,

f4.5)

or,

since Y/'m2 = l,

7rJtH = Pt4.4/Pt4

and

rn4.5,

rh4

then

YgtH

[ 1

/4/4 @ /'hC(E'I +

e2) (el + e2)

= =1+

/h 4 (1 -/5 - el - e2)(1 + f)

(_~l -~_ 82 A I A4, MFP(M4,, Tt4, f)

(1 - fl - el - e2)(1 + f) J A4.5,

MFP(M4.5,,

Tt4.5, f4.5)

(5.1)

where the right-hand side of the equation is essentially constant for the assumptions

of this analysis, namely

An,/A4.5, =

constant, M4, = 1, M4.5, = 1, and constant

bleed and cooling airflow fractions. Since

f4.5 = (4.8j)

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

E2)

then for a specific value of Tt4 and f, Eq. (5.1) can be used to calculate Jrtn for an

assumed value of Tt4.5.

Likewise, for given values of

Tt4, f, rrtn,

and Orn, the resultant turbine exit

temperature (Tt4.5) can be determined as follows:

1) The total enthalpy at state t4.1(ht4.1) follows from that at state

t4(ht4)

and

the total enthalpy ratio of coolant mixer 1 (rml).

~rn l

and

ht4.1

ht4.1 = ht4~-'- = ht4"~ml

nt4

where r,,i is given by

(1 - fl - el - e2)(1

+ f) + elrrrcLrcH/rX

(1 -/3 - el - e2)(1 + f) +

(5.2a)

(4.20a)

f4.1 = (4.8i)

1 + f + el/(1 -- 13

- s1 -

S2)

ENGINE SELECTION: PERFORMANCE CYCLE ANALYSIS 145

2) With state t4.1 known and using Eq. (4.9d), the reduced pressure at the ideal

exit state

t4.4i (Pr

t4.4i)

follows from its value at state t4.1 (Pr t41) and the turbine

total pressure ratio (zrtH).

Pr t4.4i ~-- ~tHPr

t4.1

(4.9d)

3) With

Prt4.4i

known and noting that f4.4 =

f4.1,

then

ht4.4 i

is known using

the subroutine FAIR. The total enthalpy ratio of the ideal turbine (rim) follows

directly using its definition

rtHi = ht4.4i/ ht4.1

(4.9d)

4) By the assumption of constant efficiency of the high-pressure turbine,

Eq. (4.9d) can be solved for the high-pressure turbine enthalpy ratio

(rtn)

~tH

1 - rhn(1

"CtHi)

(5.2b)

5) With the total enthalpy ratio of coolant mixer 2 (rm2) given by

(1 - ~6 -- el - e2)(1 + f) + el +

e2{rrrcHrcL/(rZ'rmlrtit)}

rm2 = (4.20b)

(1 -/3 - el - e2)(1 + f) +

E1 n t-

and f4.5 given by Eq. (4.8j), the total enthalpy at station 4.5 (ht4.5) can be calculated

using

ht4.5 = ht4

hJ 4"1

ht4"4 ht4"----~5 - ht4rml'Ctnrm2

(5.2c)

t'It4 ht4.1 ht4.4

6) With

ht4.5

and f4.5 known,

Tt4.5

follows directly using the subroutine FAIR.

We will refer to this system of equations used to determine

Tt4.5

as Eq. (5.2).

Equations (5.1) and (5.2) give us a system of equations to satisfy that determine

the high-pressure turbine performance

rtH and rctH.

This system of equations is

programmed into the cooled turbine subroutine TURBC outlined in Appendix E

Subroutine TURBC solves this system by first assuming an initial value of

Tt4.5

and using Eq. (5.1) to find ~rtH. Using this value of zrtH, Eq. (5.2) gives a new

value of total temperature at station 4.5 called

Tt4.5,.

This new value is input into

Eq. (5.1) and calculations are repeated until successive values are within 0.01.

Figure 5.2 shows the variation of rrtn with throttle setting

(Tt4)

for a typical mixed

flow turbofan engine. Note that

7rtn

only varies 0.9%, which shows the common

assumption used in the constant specific heat analysis that

zrtrl

= constant is a very

good approximation.

A similar, but simpler, set of equations can be solved for the uncooled low

pressure turbine. Writing Eq. (5.1) for the low-pressure turbine gives

rrtL A4.5, MFP(M4.5,,

Tt4.5,

f4.5)

,V/~/Tt4.5

A5 MFP(M5, Tt5,

f4.5)

Because the flow is assumed to be isentropic from engine station 5 to station 6 and

we know the area at station 6, we use the Mach number and area at engine station

6 for the low-pressure turbine exit and write

YrtL

A4.5'

MFP(M4.5,, Tt4.5,

f4.5)

(5.3)

~¢/~/Tt4.5 A6 MFP(M6, Tt5,

f4.5)

146 AIRCRAFT ENGINE DESIGN

et4.5

e,4

0.34

0.33

0.32

0.31

' I I I I

Fig. 5.2 Variation of high pressure turbine pressure ratio with throttle setting

(Tt4)

for a typical mixed flow turbofan engine.

where

A4.5,/A6

---- constant and M4.5, = 1. Then for a specific value of

Tt4.5,

f4.5,

and M6, Eq. (5.3) can be used to calculate :rtL for an assumed value of

Tts.

Likewise, for given values of Tt4.5, f4.5, 7rtL, ~tL, and Mo, the resultant turbine

exit temperature (Tts) can be determined as follows:

1) Using Eq. (4.9e), the reduced pressure at the ideal exit state

t5i (Pr tsi)

follows

from its value at state t4.5

(Pr

t4.5) and the turbine total pressure ratio (TrtL).

Pr t5i : 3TtL Pr

t4.5

(4.9e)

2) With

t5i

k_rlown,

then

ht5i

is known using the subroutine FAIR. The total

enthalpy ratio of the ideal turbine (~Li) follows directly using its definition:

"~tLi : ht5i / ht4.5

(4.9e)

3) By the assumption of constant efficiency of the low-pressure turbine,

Eq. (4.9e) can be solved for the low-pressure turbine enthalpy ratio (rtD

rtt = 1 - tilL(1

-- "CtLi)

(5.4)

4) Tt5 follows directly using the subroutine FAIR.

We will refer to this system of equations used to determine Tt5 as Eq. (5.4).

Equations (5.3) and (5.4) give us a system of equations to satisfy that determine

the low-pressure turbine performance rrtL and rtL. This system of equations is

programmed into the uncooled turbine subroutine TURB outlined in Appendix E

Subroutine TURB solves this system by first assuming an initial value of

Tts

and

using Eq. (5.3) to find ZrtL. Using this value of :rtt, Eq. (5.4) gives a new value of

0.30 = i i I i

2000 2200 2400 2600 2800 3000 3200

T~ 4 (°R)

ENGINE SELECTION: PERFORMANCE CYCLE ANALYSIS 147

total temperature at station 5 called

Tthn.

This new value is input into Eq. (5.3),

and calculations are repeated until successive values are within 0.01.

Constant specific heats.

In the case of a perfect gas with constant specific

heats [calorically perfect gas (CPG)], the equations that model the performance of

the high-pressure and low-pressure turbines are simplified. The mass flow para-

meter for a CPG is given by Eq. (1.3). Since the

MFP

is the same at stations 4' and

4.5' when

M4, : M4.5' =

1, Y4'

: Y4.5' ~---

~/t, and

R4' = R4.5' -----

gt,

then Eq. (5.1)

becomes

7Z't/4 --{1+

(81q"82)

} aA~4'~ (5.1-CPG)

-- ~ --71

--~-2)(1 + f)

za4.5'

where

A4'

= constant

A4.5,

and where, by the assumptions of Sec. 5.2.2,

(El + E2)

1 + = constant

- 13 - 81 - e2)(1 + f)

and where, combining Eqs. (4.20a) and (4.20b),

rmlrm2 = 1 + (1 -~ -el - e2)(1 +f).] (1 -~ --e-i ~- e-2)(1 +f)

(5.1 b-CPG)

Consider the order of magnitude of the second term in the numerator of

Eq. (5. lb-CPG) compared with unity. Noting that, for the performance operation

of interest,

7JtH ~

0.85 and

r~. -- cptTt4 ~

2 ---> 4

12r12cL12cH Cpc Tt3

the second term in the numerator of Eq. (5. lb-CPG) will be of the order of e or less

and its variation at engine off-design can be considered small in comparison with

unity. The denominator is constant, whence the product

72mllhm2

can be considered

constant and Eq. (5.1-CPG) becomes

712tH

-- constant (5.1c-CPG)

when e is an order of magnitude less than unity.

For a calorically perfect gas, Eq. (5.2b) can be written as

"CtH : 1 -- thrill -- 7r~H Y'-I)/y' }

and, by the assumption of constant efficiency of the high-pressure turbine, can be

rewritten as

1 -- "~tn

(e,-1)/y, -- r/tH = constant (5.2b-CPG)

1 - 7rtH

148 AIRCRAFT ENGINE DESIGN

To satisfy both Eqs. (5. lc-CPG) and (5.2b-CPG), the total pressure ratio and the

total temperature ratio of the high-pressure turbine must be constant, or

2"gtH ~

constant

75tH ~

constant

When the same order of magnitude analysis used with Eq. (5. lb-CPG) is applied

to Eqs. (4.20a) and (4.20b) separately for rml and rm2, the coolant mixer total

temperature ratios are found to be approximately constant, or

rml = constant

Z'm2 ~

constant

For the low-pressure turbine with a CPG and M4.5 = 1, Eq. (5.3) can be written

• y~+l

{ ( 62)/"'"

~rtL A4.y 1 2 ?'t -- 1 M (5.3-CPG)

A6M y-T 1+ 2

where

A4.5, / A 6 =

constant

Likewise, Eq. (5.4) can be written as

(×,-b/×,

rtL = 1 -- OtL{ 1 -- ZrtL I

and, by the assumption of constant efficiency of the low-pressure turbine, can be

rewritten as

1 - "(tL

(×,-1)/×, -- OtL

= constant (5.4-CPG)

- 7rtL

For a given value of the Mach number at station 6 (M6) and the constant values of

the area ratio

(A4.5,/A6)

and the turbine efficiency (rhD, there is only one set of

low-pressure turbine total properties (rtL,

rCtL)

that satisfies both Eqs. (5.3-CPG)

and (5.4-CPG).

5.2.5 Component Performance Analysis

The performance of the mixed flow turbofan engine in Figs. 5.1a and 5.1b

can now be analyzed using the assumptions of Sec. 5.2.2 and the techniques of

Secs. 5.2.3 and 5.2.4. The goal is to obtain the 24 independent equations required

to determine the dependent performance variables of Table 5.1. To underscore the

logic of the iteration solution sequence that follows in Sec. 5.2.6, the required equa-

tions will be developed in the order that each arises in the solution flowcharts of

Figs. 5.3a and 5.3b. The equations to be developed next are represented in turn by

the following 24 functional relationships between the 24 dependent performance

ENGINE SELECTION: PERFORMANCE CYCLE ANALYSIS 149

Calculate new

value of a'

using Newtonian

iteration

Initial values of f,a', M6, M 8 , th o

Fig. 5.3a

Calculate

HPT ~,, rtn, ~:ml, r.,2' T,4.5 Eq. (5-1,2)

LPT ~,L, "r,L, T,s Eq. (5-3,4)

~r s Eq. (5-5a)

~j Eq. (5-5b)

~'~L Eq. (5-6b)

:rL Eq. (5-6d)

"t'cH Eq. (5-7)

~H Eq. (5- 8b)

f Eq. (4-15)

M~6 Eq. (5-9)

a ',e,' Eq. (5-10)

, _ a'°ew-a'l

a er~o~ --

a Eq. (5-11)

' > 0.001 9

Yes

~ error "

Flowchart of iterative solution scheme (Part I).

variables of interest:

Zrtn = fl(Zml, rtn, Tin2, f)

Zml = f3(rcL, rcn, f)

ZrtC = fs(rtL, f)

Tf = f7(rtH, rtL, TcL, rcn, Ol, f, l~lo)

"rcL

= f9(rf)

rcH = fll(rtH, rcL, r.cH, Or, f, fnO)

f = f13(rd,,

ZcH)

Ol t = f l5( Pt6/ Ptl6, Tt6/ Ttl6, m6,

m16)

rM = fl7('gf, "gml,

~TtH, rm2, rtL, O lt)

ZrM = fl9(rM, M6,

M6A, or')

M6 = f21(rM, JrM, M8,

a')

fAB =

f23('~ml,

rtn, rm2, rtL, rM, f)

rtH = f2(zrtu, f)

rm2 =

f4(rcL,

rcm rml, r~n, f)

r,/~ = f6(~r,L, f)

~rl = f8(rD

zr~L = flo(r~D

~r~/ = flz(r~H)

M16 = f14(:rrf,

~rcL, 7rcH, Zrb, 7rtn, ZrtL,

M6)

a = fl6(Ot')

M6A =

fls(M6, MI6, o~ t)

M8 = f20(zrcL, ZrcH, 7rb, zr,n, zrtL, JrM, f)

rho = fz2(ZrcL, zr~/~, a)

M9 = f24(TrcL, zr~H, Zrb, :r,m zrtL, rrM, f)

150

AIRCRAFT ENGINE DESIGN

If M 6 > M 6"ew then

M 6 = M 6 -0.0001

else

M 6 =M 6 +0.002

I Yes

z" M Eq. (5-12)

M6A Eq. (5-13)

~M Eq. (5-14)

M8 Eq. (5-15)

M6,,e w

Eq. (5-16)

.......

=IM6,e~.-M6[

Is M 6 ..... > 0.0005 ?

rh 0 ..... Eq. (5-17)

t~/0new --~/o

th0error =

Is rn o ..... > 0.001 ?

Remainder of Calculations

(Appendix I)

Yes

?nO = mo new

Is an engine control limit (~., ~ Reduce

Tt4 ]

Tt3, Pt3, etc.) exceeded?

I t

Fig. 5.3b Flowchart of iterative solution scheme (Part II).

Please notice that each equation can be solved, in principle, in the order listed for

given initial estimates of the three component performance variables or', M6, and

rn0 (see Sec. 5.2.6). As the solution progresses, these estimates are compared with

their newly computed values and iterated if necessary until satisfactory conver-

gence is obtained.

It is very important to recognize that the first six quantities of the solution se-

quence (i.e.,

rCt~l, rtH,

~'ml, "gm2,

~tL,

and

rtH ) can

be determined by the methods

presented in Sec. 5.2.4. The fuel/air ratiosfand

fAB are

found using Eqs. (4.18)

and (4.19), respectively. Additionally, the fuel/air ratios faA, f4.5, and

f6a

at sta-

tions 4.1, 4.5, and 6A, respectively, are required to solve the system of equations

for variable specific heats. Equations (4.8i), (4.8j), and (4.8k) give the needed

relationships for f4.1, f4.5, and

f6a

in terms of

f, faB, Or, and ft.

ENGINE SELECTION: PERFORMANCE CYCLE ANALYSIS 151

Fan temperature ratio

(rt)

and low-pressure compressor temperature

ratio

(rcL). From the low-pressure spool power balance, we have

rtL = 1--

rr{(rcC - 1) + u(rf - 1)} + (1 +

ot)CToL/rlmPL

rlmnrXrta{(1

-- fl -- el -- 62)(1 + f) + (el +

F-.2/'EtH)'Er'~cLTEcH/~3.}

which can be rearranged to yield

rf=l+

(1--'CtL)rlmL{th~'4CC r:x'rtH+(61"~tH-I-62)'ccL~:cH }rr

(4.22a)

(1 + ot ) PTOL

~r OmPL rhoho

{(rcL -- 1)/(rf -- 1) + Or}

where, from Fig. 4.2,

th4/thc =

(1 -/~ - el - 62)(1 + f)

Because the low-pressure compressor and the fan are on the same shaft, it

is reasonable to assume that the ratio of the enthalpy rise across the fan to the

enthalpy rise across the low-pressure compressor is constant. Using referencing,

we can therefore write

htl3 --ht2 ~f -- 1 (Tf --

1)R

ht2.5

--ht2

"EcL- 1 (Z'cL --

1)R

Thus the fan enthalpy ratio can be written as

~:f = 1+

{ r~t4

"E~"CtH _1¢_

(,Sl-Ct H ..1- e2)-CcLr;cH}

+ or)

PTOL

(1-- rtL)TlmL I'h C r''~ rrrlmPL fnoho

{(ZcL -- 1)R/(rf --

1)R

+ or}

(5.5a)

and the low-pressure compressor enthalpy ratio can be written as

rcL = 1 + (zf -

1)[(ZcL --

1)g/(rf --

1)R] (5.5b)

For a calorically perfect gas, h0 =

cpcTo

in Eq. (5.5a) and Eq. (5.5b) is

unchanged.

Fan pressure ratio (~f) and low-pressure compressor pressure ratio

(~cL),

From the definition of fan efficiency, Eq. (4.9a),

htl3i =

ht2{1 q-

rlf('gf --

1)} (5.6a)

Given

htl3i ,

the subroutine FAIR will give the reduced pressure

Pr tl3i.

Thus the

fan pressure ratio is calculated using Eq. (4.9a) written as

ert

13i

(5.6b)

~ f -- Pr t2

Likewise, from the definition of low-pressure compressor efficiency, Eq. (4.9b),

ht2.fi

ht2{1 +

17cL(ZcL --

1)} (5.6c)