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

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DESIGN: INLETS AND EXHAUST NOZZLES 477

and the ideal gross thrust (corresponds to (P9 = P0) expressed as

Fg ideal = ~18i Vs/gc

(10.31)

where Vs is the isentropic exit velocity based on

Pts/Po

and Tts. Please note that

the uninstalled gross thrust differs from the uninstalled thrust of Eq. (4.1) only by

the momentum of the incoming air, which is constant.

For one-dimensional flow of a calorically perfect gas, Eq. (10.30) can be written

/'hzV9 {1 +),--1 1--

Po/P9

} (10.32)

Fgactual-

g¢ 2y

(et9/-P9--) (×-1)/----~ - 1

The uninstalled gross thrust coefficient for one-dimensional flow of a calori-

cally perfect gas can be obtained by substituting Eqs. (10.31) and (10.32) into

Eq. (10.27), giving

1 - ~,sJ y - 1 1 - Po/P9

Cfg=CDCv 1 ~ 1+ -2-{ ~P9J 1

,~-2,8, ( p,9~--C- 1-7~ -- (10.33)

This equation reduces to

Cfg = CDCv

for the case of ideal expansion (P9 =

P9i = PO).

For isentropic flow, P9 =

P9i, et9 = Pt8, Cv

----

1, and

CD = 1.

Equation (10.33) is plotted in Fig. 10.75 for isentropic flow vs the nozzle

area ratio

(A9/As)

for different nozzle pressure ratios

(Pts/Po).

Note that ideal

expansion (P9 ---- P0) gives a gross thrust coefficient of unity and that both un-

derexpansion (P9 > P0) and overexpansion (P9 < P0) reduce the gross thrust

1.00

0.95

Cy8

0.90

0.88

1.0

Fig. 10.75

Flow separation limit

P9/Po

= 0.37

I I I I

2.0 3.0 4.0 5.0

A9/A8

Gross thrust coefficient for one-dimensional isentropic flow (-y = 1.3).

478 AIRCRAFT ENGINE DESIGN

coefficient below unity. Both the uninstalled and installed thrust suffer as a result

of non-ideal expansion.

The extent of overexpansion in nozzles is limited by flow separation resulting

from the interaction of the nozzle boundary layer and the strong oblique shock

waves at the exit of the nozzle. In extreme overexpansion, Summerfield et al) 8

noted that the oblique shock waves moved from the exit lip into the nozzle, the

flow downstream of the shock waves was separated in the vicinity of the wall,

and as a result the wall static pressure downstream of the shock waves was nearly

equal to the ambient pressure (P0). A simple estimate for the maximum allowable

ratio of the pressure just preceding the shock waves

(Ps)

to the ambient pressure

(P0), suggested by Summerfield, is given by

Ps / Po "~

0.37 (10.34)

This flow separation limit can be included in the one-dimensional gross thrust co-

efficient of Eq. (10.33) for isentropic flow by considering the effective exit pressure

(P9 =

P9i)

to be the pressure just preceding the shock wave

(Ps).

Equations (10.33)

and (10.34) were used to obtain the flow separation limit shown on Fig. 10.75. The

design area ratio

(A9/A8)

of convergent-divergent nozzles is selected such that the

nozzle flow does not separate due to overexpansion for most throttle settings. This

is because the increase in gross thrust coefficient associated with flow separation

does not normally offset the accompanying increase in installation loss.

Nozzle pressure ratios are 3-5 in the subsonic cruise speed range of turbofan and

turbojet engines. Typically, a subsonic engine uses a convergent exhaust nozzle.

This is because, in the 3-5 nozzle pressure range, the convergent nozzle gross thrust

(intercept of lines with vertical axis,

A9/A8

= 1) is 1-3% below the peak gross

thrust (P9 = P0) as shown in Fig. 10.75. Consequently, there may be insufficient

gross thrust increase available in going to a convergent-divergent nozzle on a

subsonic cruise turbofan or turbojet engine to pay for the added drag and weight

of such a nozzle. In some applications, this loss in gross thrust coefficient of a

convergent nozzle is too much and a C-D nozzle is used.

A band indicating the range of typical nozzle pressure ratios for turbojets is

shown in Fig. 10.64 as a function of flight Mach number. The pressure ratio

across the nozzle increases rapidly with supersonic flight Mach number. At a Mach

number of 2, the pressure ratio is about 12. At this pressure ratio, the convergent

nozzle gross thrust penalty is about 9% as shown in Fig. 10.76. This figure is a plot

Fg ..... 0.95

Egg ideal

0.90

0.85

0 10 20

P, smo

Fig. 10.76 Ratio of convergent nozzle gross thrust to ideal gross thrust vs pressure

ratio ('7 = 1.3).

DESIGN: INLETS AND EXHAUST NOZZLES 479

of the ratio of the gross thrust in Fig. 10.75 of a convergent nozzle

(A9/A8

= 1) to

the peak thrust (P9 = P0) vs

Pt8/Po.

Substitution of convergent-divergent nozzles

for convergent nozzles provides large thrust gains for supersonic aircraft.

This method of calculating the performance of a one-dimensional nozzle, as

demonstrated in the following example, is incorporated in the NOZZLE program

provided with the AEDsys software package.

10.3.3.3 Example calculation based on one-dimensional flow.

Given:

rh8 = 200 lbm/s

Pt8

= 30 psia Tt8 = 2000°R

A9/A8

= 2.0 y = 1.33 Po = 5 psia

Pt9/Pts

= 0.98 Co = 0.98

Rg c

= 1716 ft2/s2-°R

Find the dimensions of an axisymmetric nozzle,

Cfg, Fg,

and

Cv.

Solution:

From Gas Tables,

MFP(@M8

= 1) = 0.5224, thus A8 = 570.7 in. 2

With

CD =

0.98, thus

A8e =

582.3 in. 2 and r8 = 13.61 in.

Since

A9/A8

= 2.0, then A9 = 1165 in. 2 and r9 = 19.25 in.

9i e9i

A _ A9 _ 2.0 _ 2.041 =¢,

M9i =

2.168 and

Pt9i

A --2 Co-'A8 0.9~

= 0.0990

Thus,

P9i =

(0.0990)(30 psia) = 2.970 psia,

V9i = ~ -~. 1 -- k Pt9i/]

Vgi = ~/(1716)(2000)~2(1 33) {1 - (0.0990) °"33/1'33 } = 3475 ft/s

A _ Pt9

A9 _ 0.98 × 2.0 _ 2.00 ~ M9 2.146 and P9 0.1025

A* Pt8 CDA8

0.98

Pt9i

Thus P9 = (0.1025)(0.98)(30 psia) = 3.014 psia,

V9i

~/(1716)(2000)~2(1 33){1

-(0.1025) 0"33/1'33 } ~-

3456 ft/s

Cv = V9/V9i

= 0.9945

• [i (297]0.33/1.331

1/2[ 0.33 1 -- 5.0/3.014 /

Cfg =

(0.98)(0.9945)| 2-- ~ $ |1 -F (

29.4 ,~0.33/1.33

11-(~--~) " J | 2.66~, -1

C fg =

0.9593

480 AIRCRAFT ENGINE DESIGN

1- 54 in ~, 19.25 in

13.61 in l

__. Ce___nterli___n.e_._._._ .I

Fig. 10.77 Dimensions of example exhaust nozzle (one-dimensional calculation).

Figure 10.71b gives 0 ~ 10 deg for CD = 0.98 and

A9/As

= 2. Likewise,

Fig. 10.72 gives ot ~ 6 deg for

= 0.9945 and

A9/A8

= 2. Thus

Ls ~

54 in.

and the dimensions of the exhaust nozzle are shown in Fig. 10.77. The gross thrust

can be calculated several ways: directly from Eq. (10.30)

Fg = r'n8 V9/gc -}-

(P9 - Po)A9

= (200)(3456)/32.174 + (3.014 - 5.0) 1165 = 19,170 lbf

or from the ideal gross thrust and

Cfg

with

V, = x/(1716)(2000) ~ 1 - \~-~] ] = 3151 ft/s

then

Fgi = m8iVs/gc

= (200/0.98)(3151)/32.174 = 19,990 lbf

Fg = CfgFsi

= (0.9593)(19,990) = 19,170 lbf

10.3.3.4 General thrust performance.

The gross

thrust coefficient (Cfg)

is a measure of the nozzle efficiency and accounts for nozzle losses due to friction,

angularity, expansion, leakage and cooling air throttling.

Cfg

can be expressed in

terms of the other nozzle coefficients as

CvCArh7Vgi/gc + (P9i -

Po)A9

Cf~ = - ACfg

(10.35)

rh7 Vs/ gc

where

Cv =

CA =

l'h 7

V9i =

P9i

A9 =

Velocity coefficient, Eq. (10.24)

Angularity coefficient, Eq. (10.29)

Actual total mass flow rate supplied to the nozzle

Ideal velocity at the nozzle exit based on

A9/As, CD,

and Tt8

Ideal static pressure at the nozzle exit consistent with

V9i

Nozzle physical exit area

Isentropic or fully expanded exit velocity based on

Pts/Po

and Tt8

The sum of the loss in thrust coefficient due to leakage

of gas from the nozzle and the loss due to changes in the

cooling air flow

DESIGN: INLETS AND EXHAUST NOZZLES 481

1.0

Angularity and

Cfgpeakf

~ Expan~sion t fricti°n l°ss

P t8/Po

Fig. 10.78 Baseline gross thrust coefficient.

Basic aerodynamic analysis of nozzle efficiency includes the effects of thrust

losses due to angularity and friction. The thrust losses due to over- or under-

expansion are normally attributed to "off-design" operation. If losses due to leak-

age and cooling are neglected, then the other losses establish a base line gross

thrust coefficient curve for a typical C-D nozzle as shown in Fig. 10.78. The one-

dimensional analysis of the previous section did not include angularity losses,

did include the expansion losses, and estimated frictional losses using Zrd. The

following analysis will include all three of the losses shown in Fig. 10.78.

From Eq. (10.35), the gross thrust coefficient of the base line nozzle (neglecting

leakage and cooling) can be written as

CvC A#17 W9i / gc q- ( e9i -

Po)A9

Cfg

= (10.36)

rnTV,/g~

Cfgpeak , P9i ~" eo and V9i = Vs.

Therefore,

Cfg = CvC A

(10.37)

This method of calculating the performance of an axisymmetric two-dimensional

nozzle, as demonstrated in the following example, is incorporated in the NOZZLE

program provided with the AEDsys software package.

10.3.3.5 Example calculation--axisymmetric two-dimensional nozzle.

Given:

rh8 = 200 lbm/s

Pt8

= 30 psia Tt8 = 2000°R

A9/A8=2.0

y=1.33 P0 =5psia

0 = 10deg a = 10deg

Rgc

= 1716ftZ/sZ-°R

Find the dimensions of an axisymmetric nozzle,

Cfg,

gross thrust, and approxi-

mate

zrn( Pt9 / Pts).

Solution:

From Gas Tables,

MFP(@M8

= 1) = 0.5224, thus A8 = 570.7 in. 2

From Fig. 10.71b,

= 0.98, thus

A8e

= 582.3 in. 2 and r8 = 13.61 in.

Since

A9/A8

= 2.0, then A9 = 1165 in. 2 and r 9

19.25 in.

With r8 = 13.61 in., r9 = 19.25 in., and ot = 10 deg, then Ls = 32.0 in.

The dimensions of the nozzle are shown in Fig. 10.79.

482 AIRCRAFT ENGINE DESIGN

13.61 in

Centerline j

19.25 in

Fig. 10.79 Dimensions of example exhaust nozzle (axisymmetric two.dimensional

calculation).

With a = 10 deg and

A9/A8

= 2.0, Fig. 10.72 gives

= 0.9965, Fig. 10.74

gives

= 0.991, and thus

Cfg

peak = CvCa

0.9875.

9i e9i

AA, -- CoA 8A9

-- 0.982"0 _ 2.041

M9i

= 2.168 and

et9i

Thus, P9 = (0.0990)(30 psia) = 2.970 psia,

V9i = ~ ~ 1 - kPt9i/]

= 0.0990

V9i = ~f(1716)(2000)~/~2(1 33){1

- (0.0990) 0'33/1'33 } =

3475 ft/s

Since

then

I =(,.~) { (~ ~°"~"" I

Vs = ~(1716)(2000) ~ 1 - \3-0] / = 3151

Cfg = Cfgpeakth7V9i/gc + (P9i -

eo)A9

fnTVs/ge

(0.9875)(200)(3475)/32.174 + (2.970 - 5.0)1165

Cfg

= = 0.9683

(200)(3151)/32.174

Fg = C fg peakth 7 V9i / ge ÷ ( P9i -

Po)A9

= (0.9875)(200)(3475)/32.174 + (2.970 - 5.0)1165 = 18,966 lbf

Cv-

/ p,

,p ~(y-1)/y

V 9

.1- ( 9/ t9J

Vgi V 1 - (Pgi/Pts) O'-W×

_ j}

DESIGN: INLETS AND EXHAUST NOZZLES 483

1.05

S/Smin

1.00

F/F ma x

0.95 ~ I

0.5 1.0 1.5 2.0

Po/P9

Fig. 10.80 Effect of

Po/P 9

engine performance.

and

P9 = { 1 - (0.9965) 2 [1 - (0.0990)°"33/133]} 1'33/°'33 = 0.1012

Thus, M9 = 2.154,

A/A*I9

= 2.014 and, finally, from one-dimensional flow,

et9 A/A*[9

2.014

n'n -- -- Co-- -- (0.98) = 0.9867

Pt8 A9/As 2

10.3.3.6 Off-design predicted performance.

The performance cycle

equations developed in Chapter 5, listed in Appendix I, and implemented in

AEDsys can be used to calculate the effects of non-perfect expansion at the nozzle

exit. Figure 10.80 shows the effects of the pressure ratio

Po/P9

on both uninstalled

thrust and uninstalled thrust specific fuel consumption at a specific flight condi-

tion for a mixed-stream turbofan engine without afterburning. These plots show

the trends that are typical for most turbojet and turbofan engines.

10.4 Example Engine Component Design: Inlet and Exhaust Nozzle

10.4.1 Inlet DesignEAAF Engine

The requirements of the AAF inlet are specified in terms of required engine air

mass flow rate by a plot of A0 or A~ vs Mach number and altitude (see Fig. 7.E8)

and in terms of total pressure recovery by military specification MIL-E-5008B

[Eq. (4.12c)]. The type of inlet and its design are mainly determined by the type of

aircraft, the required flight envelope, the range of angle of attack that the inlet will

experience, and the maximum Mach number required. The AAF has a required

maximum Mach number of 1.8 at 40 kft altitude. Since the lift coefficient increases

0.1 for each angle of attack, then the required 5g sustained turns at 0.9M/30 kft

(CL = 0.7 from Sec. 3.4.4) correspond to an angle of attack of about 7 deg and

takeoff (CL

CLmax/k20

1.39 from Sec. 3.4.4) will correspond to an angle of

attack of about 12 deg. A two-dimensional external compression inlet is selected for

this preliminary design of the AAF inlet because of its simplicity, ability to better

match inlet airflow requirements of the AAF engine over the aircraft's mission,

and ease of analysis.

484 AIRCRAFT ENGINE DESIGN

3.8

3.6

3.4

3.2

(ft 2)

3.0

''''l''''l''''l''''l''''l''' 'I''''I''''

Ao i +pec

Oy? y

Margin

2.8

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Fig. 10.El AAF inlet area requirements at 40 kft on cold day.

During preliminary engine sizing in Chapter 6, the sizing point for a fixed capture

area, variable geometry inlet was selected to be 1.56M/40 kfl on a cold day and

resulted in a capture area (A 1 ) of 3.519 ft 2 (see Fig. 6.E3 in Sec. 6.4.5). Analysis

and design of the AAF inlet is now performed at an altitude of 40 kft on a cold

day over the range of Mach numbers of 1.0-1.8. The required engine flow area,

required engine flow area plus 4% safety margin, and required inlet flow area are

plotted in Fig. 10.El and tabulated in Table 10.El for the AAF. The required inlet

flow area is composed of the required engine flow area, a 4% safety margin, and

boundary layer bleed that varies linearly from 0% at 0.8M to 4% at 1.8M.

Figure 10.34 can be used to estimate the number of oblique shocks needed for an

external compression inlet to meet the required total pressure recovery at the Mach

1.8 flight condition. Inspection of this figure shows that at least two oblique shocks

plus a terminating normal shock are required for the AAF's external compression

inlet. Equations (10.9) through (10.11) are the analytical tools needed to design

2-D external compression inlets and are imbedded in the INLET design program.

They help us find the ramp angles of the external compression inlet that will meet

or exceed the required total pressure recovery over the range of supersonic flight

conditions (from MIL-E-5008B).

The AAF inlet needs to have excellent performance at supercruise to minimize

fuel consumption and reasonable performance at Maximum Mach of 1.8. The

DESIGN: INLETS AND EXHAUST NOZZLES

Table 10.El AAF inlet area requirements at 40 kft on cold day

485

Inlet requirements based on engine

performance cycle analysis calculations

Inlet area modified for safety

margin and boundary layer bleed

mo ORspec

Aospec,

ft2

Aospec +

4%,

ft 2

Aotspec,

ft2

1.0 1.0000 2.874 2.989 3.012

1.1 0.9966 2.889 3.005 3.039

1.2 0.9915 2.940 3.058 3.105

1.3 0.9852 3.025 3.146 3.207

1.4 0.9782 3.142 3.268 3.343

1.5 0.9706 3.290 3.422 3.514

1.52 0.9690 3.322 3.455 3.551

1.54 0.9674 3.357 3.491 3.591

1.56 0.9657 3.373 3.508 3.611

1.58 0.9641 3.365 3.500 3.605

1.6 0.9624 3.359 3.493 3.601

1.7 0.9537 3.341 3.475 3.595

1.8 0.9445 3.344 3.478 3.612

ramp angles of the AAF inlet will be designed for performance at supercruise

and the inlet sized to meet the mass flow requirements over the range of flight

Mach numbers. For a two-ramp external compression inlet, the inlet performance

is presented in Fig. 10.E2 as total pressure recovery (OR) contours from the

INLET program for the supercruise Mach number of 1.5. This figure shows that

the best total pressure recovery (OR) of 0.9950 occurs at ramp angles 5.4 ° and

5.1 ° for the first and second ramps

(01, Oz),

respectively. Similarly ramp angles

of 5 ° and 5 ° for 01 and 02 will give a total pressure recovery of 0.9947 and this

combination is selected as for the AAF inlet at Mach 1.5. This provides more

than the required total pressure recovery of 0.971 for this flight condition based on

MIL-E-5008B.

Inlet design, like the design of other engine components, is iterative in nature.

The INLET program helps this process enormously. Thus, the example design of

the AAF inlet begins with the simplest inlet design (fixed inlet) and proceeds to

more complex designs, if needed, until one is found that meets or exceeds the

required performance.

10.4.1.1 Inlet geometry and performance.

Consider a fixed ramp, fixed

area external compression inlet with ramp angles of 5 ° and 5 ° for the first and

second ramps, respectively, as shown to scale in Fig. 10.E3. The location of the

lower lip with respect to the intersection of the oblique shock wave is selected to

be 1 ° further back than the angle of the first oblique shock. This location provides

a small margin for subcritical operation of the inlet (normal shock moving forward

from lip) without ingestion of the low total pressure layer that results when the nor-

mal shock intersects the second oblique shock) 9 The performance of the selected

shock system was analyzed using the INLET program and the results are tabulated

in Table 10.E2 under the heading of Selected Shock System. Since the required

486

(o)

AIRCRAFT ENGINE DESIGN

Fig. 10.E2

I I I I I I i I I

P~/P~~9080.9916

( F~

/0.992.

I I I I I I I I I

0 5 10

Ol (0)

Total pressure recovery of a two-ramp external compression inlet at 1.5M.

~\ ,/

~\ ,A '~~ ,

Fig. 10.E3 Two-ramp external compression inlet design.