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

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540 AIRCRAFT ENGINE DESIGN

Table E.1 Interrelationships between the three primary airbreathing engine

performance measures. (For example, in order to express the specific thrust

F/rho

in terms of the specific fuel consumption S and the overall fuel/air ratio

fo,

read across the

F/~no

row to the S column to find that

F/mo = fo/S.)

Measure

F l rno S rio

F F

fo fohpR

mo rno S Vo Oo

S= fo S Vo

Fl~no

hpRriO

Vo F Vo

rio = foheR #lo

hpRS

riO

providing a method for estimating the levels and trends of typical turbojet engine

performance.

A turbojet is being flown at a velocity of 1000 ft/s and is burning a hydrocarbon

fuel with a heating value of 19,000 Btu/Ibm. The uninstalled specific thrust F/rh0

is 75 lbf-s/lbm and the specific fuel consumption S is 1.0 lbm/(lbf-h). We wish to

determine the remaining engine performance measures.

F 75 x 1.0

fo = -- • S - -- -

0.0208 Ibm fuel/Ibm air from Table E.1

rh0 3600

V0 1000 x 3600

Oo -- -- -- = 0.243 from Table E.1

heRS

19,000 x 778.2 × 1.0

Voo = \moVo +1

1(75×32.174)

-- 1.0208 1000 + 1 = 3.34 fromEq. (E.5)

I T-~

}/{ (V9~2 }

=2 (l+fo) v0-1 (l+fo)\v00,/ -1 =0.464 fromEq.(E.6)

0P --~ V9 = 0.461 [0.65% lower than Eq. (E.6)] from Eq. (E.7)

~Tr~/-- -- 0.524 from Eq. (E.4)

r/p

Internal Thrust

The internal thrust

of an engine is defined as the net axial force exerted on

all of the wetted surfaces inside the engine by the fluid within the engine flowpath

control volume. It is the equal and opposite reaction force to the force exerted

APPENDIX E: ENGINE EFFICIENCY AND THRUST MEASURES541

on that fluid. The sign convention is that

is positive when it acts in the di-

rection of flight (i.e., opposite to the direction of the freestream flow). Because

the fuel and bleed airflows may be assumed to have negligible axial momen-

tum when they cross their respective control volume boundaries, they contribute

nothing to the internal thrust. The impulse function of Sec. 1.9.5 may there-

fore be applied directly to show that the internal thrust is given by the simple

expression

Fi = 19 - 11

(E.8)

The internal engine thrust

may be easily evaluated from quantities provided by

ONX or AEDsys by means of Eq. (E.8).

Uninstalled Thrust

The uninstalled thrust F of an engine is an extremely useful engineering ide-

alization that allows all parties concerned to communicate about this important

property with clarity and precision. It is also the primary thrust measure used in

this textbook. The uninstalled thrust F is defined as the net axial force that would

be produced by an engine immersed in a perfect or inviscid external flow. It is

equal and opposite to the force exerted on all of the fluid influenced by the engine

and is positive when it acts in the direction of flight. The derivation of the relation-

ships governing uninstalled thrust is greatly simplified by the careful selection of

appropriate control volumes.

The control volume for the external airflow is shown in Fig. E. 1. Please note

that 1) the flow is treated as axisymmetric only in order to minimize the algebraic

complexity and increase the transparency of the results; 2) the inner boundary of

the control volume is the streamtube that divides external flow from internal engine

flow (i.e., it consists of the streamtubes approaching and leaving the engine plus

the outer surface of the engine); 3) the axial extent of the control volume is suffi-

ciently large compared with the engine that the static pressure equals atmospheric

pressure and the streamlines are parallel to the centerline over the entire upstream

and downstream boundaries; 4) the upstream frontal area

Acv

of the control vol-

ume is sufficiently large compared with the engine that the static pressure equals

atmospheric pressure over the entire outer boundary, and the outer boundary is a

streamtube; 5) the A and dA of the inner boundary are measured perpendicular to

the centerline (consistent with throughflow area terminology) and dA is taken as

positive when the throughflow area of the internal boundary increases in the axial

direction (see detailed sketch); and 6) the supporting pylon that holds the engine

in place and transmits the uninstalled thrust F via stresses exacted on the control

volume boundary to the vehicle exerts no axial force on the external flow.

Because the external flow is inviscid or isentropic, traditional streamline argu-

ments lead to the conclusion that the freestream velocity and density are constant

along the upstream and downstream boundaries. Consequently, the external flow

experiences no change of axial momentum and the net axial force exerted over

the control volume boundary must be zero. Further, mass conservation leads to

the conclusion that the upstream and downstream frontal areas are equal. Finally,

because the inviscid external flow generates no frictional boundary forces, the

542 AIRCRAFT ENGINE DESIGN

Acv

Control Volume

Streamline • Vo _~__ Boundary ....

I- "1

~-~o ~ ,, .~ v0

I I PYLON I A

A=IA(x)

[ I ,~-- P =,(x) ', ~

~, __.~ ....... ~_~ .__.

_'.-_---.2 .... [_ ........... I ......... ~_...x

.... -,----%.._ ENO,NE J.

. ~-~'~ - ~-'~' ~ ........ oo

| , - ~ 9

,~ Vo '~Vo

Streamline

• go

Fig. E.1

Control

Volume ~ j~

Boundary ~ ~. ~,,~

dA >O

I ~ - ~ ~-I

.............. -CL-..~x

Detail

Control volume for

the engine external airflow.

condition of perfect external flow requires that the axial pressure integral over the

control volume boundary of Fig. E. 1 be zero, or

PoAcv + Po(A~ - Ao) - PoAcv

- P dA = 0 (E.9)

or, since fo dA

= A~ - A0,

then

~(P - Po)dA = (E.10) 0

APPENDIX E: ENGINE EFFICIENCY AND THRUST MEASURES543

I~ PYLON Control Volume

] [ /Boundary

- A(x)

F., I I I P = P(x) x'7

:--I- " ............ 2i

....

Fig. E.2 Control volume for evaluating uninstalled engine thrust.

This interesting and universal result reveals that the axial area average of the

pressure P over the internal boundary must be equal to P0 for perfect external

flows.

The control volume for evaluating the uninstalled thrust is shown in Fig. E.2.

The same rules as those of Fig. E.1 and Sec. 1.9.5 apply, except that the divid-

ing streamtube is the outer boundary for this case. Assuming that the fuel and

bleed airflows contribute no axial momentum, the axial direction control volume

momentum analysis yields

F + PdA = I~ - Io

(E.11)

The nature of the free exhaust jet is such that the ambient static pressure is impressed

essentially over its entire external surface. Thus, applying control volume analysis

separately to the exhaust stream downstream of the exit plane, we find that

I~ = 19 + Po(A~ -

A9) (E.12)

Finally, Eqs. (E. 10-E. 12) and fo dA = A~- A0 are combined to yield the desired

expression for the uninstalled thrust:

F = 19 - I0 - P0(a9 - a0) = --(rh9V9 - rn0V0) -I- A9(P9 - P0) (E.13)

Equation (E.13), which is identical to Eq. (4.1), allows the uninstalled thrust F

to be easily evaluated from quantities provided by ONX or AEDsys. Because the

uninstalled thrust depends only upon flow quantities governed by cycle parameters,

it is an inherent property of the engine cycle, is independent of installation effects,

and is used as the standard of engine performance.

Installed Thrust

Because the external flow is actually viscous and imperfect, there is a down-

stream axial drag on the engine control volume equal and opposite to the upstream

axial drag on the external flow control volume (see Figs. E.1 and E.2). Thus,

the installed thrust T must be equal to the uninstalled thrust F minus this drag.

The drag is due both to frictional stresses acting along the dividing streamtube

(primarily the boundary layer on the outer surface of the engine) and to pressure or

544 AIRCRAFT ENGINE DESIGN

form drag resulting from boundary layer separation (caused either by adverse pres-

sure gradients or by shock-boundary layer interactions). Because the frictional drag

is included in the vehicle account, engine designers are responsible only for the

pressure drag.

The pressure drag is variously referred to as the installation or integration "drag,"

"penalty," or "effect" in the literature. Because the pressure drag obviously depends

on the entire shape of the dividing streamtube, it also depends on the flight condi-

tions and the engine throttle setting (e.g., A0 and A9) and is therefore deservedly

referred to also as the "throttle-dependent drag." Several specific examples of in-

stallation drag are provided in Sec. 6.2. The following material provides a general

introduction to the analysis of installation drag as well as the background necessary

for the design work of this textbook.

The conventional approach to accounting for the installation penalties is to

separate the drags associated with the forward and rearward portions of the engine,

commonly referred to as the inlet and nozzle drags, as in Secs. 6.1 and 6.2. Referring

to Fig. E.1 and choosing an arbitrary but convenient midpoint m at which the

external flow is reasonably parallel to the freestream flow and the static pressure

most nearly equals P0, control volume analysis of the external flow shows that

the momentum loss or drag is equal to the axial pressure integral over the control

volume boundary, or

L °° L 9 L m S;

D = (P - Po)dA = (P - Po)dA = (P - Po)dA + (P - Po)dA

= Dinlet -'}- Dnozzle >_ 0

(E.14)

where the forward or inlet drag is defined as

L °

Dirtier --

(P --

P0)dA > 0 (E.15)

and the rearward or nozzle drag is defined as

L 9

Dnozzte -- (P -

P0)dA > 0 (E.16)

The drag on the forward portion (where dA is positive) arises because boundary

layer separation causes the average value of P to exceed Po, while drag on the

rearward portion (where dA is negative) arises because boundary layer separation

causes the average value of P to be less than P0.

Referring to Fig. E.2, the axial direction control volume analysis for the real

flow [identical to the derivation of Eq. (E.11) for perfect flow] combined with

Eqs. (E.12-E.14) and fo dA = A~ - A0 yields the desired expression for the

installed thrust, namely,

/0 9

T =/9 - Io - Po(A9 - Ao) - (P - Po)dA

= F

- (Dinlet + Dnozzle)

<_ F

(E.17)

APPENDIX E: ENGINE EFFICIENCY AND THRUST MEASURES545

Equation (E. 17) is not only a very practical result, but it is also rich with meaning.

Two of the most important messages are that designers require the means to evaluate

Dinlet

and Dnozzle

in order to derive T from F, and that T is directly dependent

upon installation effects and therefore should not be used as a standard of engine

performance.

Ground Test Determination of Uninstalled Thrust

One practical difficulty with the uninstalled thrust is that there is no way to

measure this idealized quantity directly in a ground test facility. In particular, F is

not equal to the force exerted on the test facility thrust stand, usually referred to

as the scale force Fs. Because the correct determination of the uninstalled thrust

is of paramount importance to aircraft and engine designers and their customers,

the process of deriving F from F, is described next. It is important to note that

because the real engine inlet and exhaust flows are nonuniform, integral methods

must be used in place of the one-dimensional approach. Nevertheless, the integral

impulse function retains its original significance, namely, that the net axial force

exerted on the fluid flowing through a control volume is the difference between

the exit and inlet impulse functions (see Sec. 1.9.5).

Figure E.3 shows the control volume for an engine mounted in a direct-connect

ground test facility, the most common and best characterized type of altitude testing

(Ref. 2). Please note that the engine is isolated from any axial forces exerted by the

test facility bellmouth by means of a labyrinth air seal. First, the measured airflow

conditions at engine station 2 are used to evaluate the integral I2 and then used in

concert with an accurate model of the inlet compression (or acceleration) process

to either correspond to predetermined freestream conditions or to determine the

corresponding freestream conditions. This allows the upstream impulse function I0

TEST CELL

Control

Volume ~ P = P0

~l Boundary k

Facility -"71 [''~ ~"l I

Be.mouth % I'1 ENO E

I',

Inlet

" ---ill ~ ~ II

THRUST

~ ST A~ID

....

V9 Facility

~.. Exhaust

Duct

Fig. E.3 Control volume for evaluating engine uninstalled thrust.

546 AIRCRAFT ENGINE DESIGN

to be evaluated. Next, the facility exhaust system is used to set the static pressure

surrounding the engine and nozzle exit station 9 equal to the freestream static

pressure P0. Finally, the axial direction control volume analysis is used to show

that

19 = 12 -t- Fs +

P0(A9

A2) (E.18)

where 12 and 19 are integrated impulse functions, F~ is the scale force measured by

the thrust stand, and 12 is obtained from station 2 measurements as just described.

Two types of corrections are required before Eq. (E. 18) can provide a sufficiently

accurate value for 19 at a specific test condition. First, test cell calibrations must be

used to reduce or eliminate such deterministic errors in

as test stand and load

cell ambient pressure and temperature effects, aerodynamic and mechanical tares,

test cell static pressure distribution forces (buoyancy), and the consequences of

leakage flows. Second, empirical relationships in the form of influence coefficients

must be applied to adjust 19 for differences between the desired and actual test cell

conditions and engine control settings. An interesting observation is that, because

the state of the flow at station 2 can be quite far from freestream conditions, F~

can be zero or less.

The expression for the uninstalled thrust is obtained by combining Eqs. (E. 13)

and (E. 18) in order to yield the desired relationship

F = F~ + (12 - I0) - P0(A2 - A0) (E.19)

Modem, highly developed, direct-connect ground test facilities use this process

to provide uninstalled thrust measurements with an uncertainty in the range of

+/- 0.5 to 1.0%, depending on the engine configuration and flight conditions.

You may well imagine how much more difficult the determination of F must be in

flight testing. The close control over simulated flight conditions, large amounts and

variety of data, excellent accuracy of results, and reduced risk to personnel make

ground testing the approach overwhelmingly employed for engine development.

References

IOates, G. C.,

The Aerothermodynamics of Gas Turbine and Rocket Propulsion,

3rd ed.,

AIAA Education Series, AIAA, Reston, VA, 1997.

2Smith, R. E., Jr., and Wehofer, S., "Measurement of Engine Thrust in Altitude Ground

Test Facilities," AIAA Paper 82-0572, 1982.

Appendix F

Compressible Flow Functions for Gas

with Variable Specific Heats

Consider a perfect gas with variable specific heats in thermodynamic equilib-

rium. The gas properties are based on the NASA Glenn thermochemical data

and the Gordon-McBride equilibrium algorithm (see Chapter 9). Given the total

temperature (Tt), fuel/air ratio (f), and one of the following four input properties,

the Mach number (M), the isentropic temperature ratio (Tt/T), isentropic pres-

sure ratio (Pt/P), and mass flow parameter (MFP), the subroutine RGCOMPR

determines the other three using the subroutines FAIR and MASSFP. In addition,

the high- and low-turbines have an iterative system of equations (Sec. 5.2.4) that

determine their off-design performance with variable gas properties. Outlines of

the subroutines TURBC and TURB are given here for the high-pressure cooled

turbine and the low-pressure turbine, respectively.

Subroutine RGCOMPR (Item, Tt, f, M, Tt/ T, Pt/ P, MFP)

Inputs: Item, Tt, f, and one of the following: M, Tt/T, Pt/P, andMFP

Outputs: M, Tt/ T, Pt/ P, and MFP

Select Case Item

Case 1---Mach known

MASSFP (Tt, f, M, Tt/T, Pt/P, MFP)

Cases 2 and 3--Tt/T or Pt/P known

FAIR(l, f, Tt , ht, Prt, (at, Cpt, Rt , Yt , at)

If Item = 2 then

T--

T,/T

FAIR (2, f, T, h, Pr , (b , c p , R, y, a)

Else

Pr~

Pr--

t', / e

FAIR (3, f, T, h, Pr, ~b, Cp, R, V, a)

End if

V 2 = 2(ht - h)gc

If V 2 < 0 then

M=0

T = Tt

Else

M--

End If

547

548 AIRCRAFT ENGINE DESIGN

MASSFP

(Tt, f, M, Tt/T, Pt/P, MFP)

Cases 4 and 5--MFP known

If Item = 4 then M = 2 else M = 0.5

AM = 0.1

MASSFP

(Tt, f, M, Tt/T, Pt/P, MFPo)

4 M=M+AM

MASSFP

(Tt, f, M, Tt/T, Pt/P, MFPn)

MFPerror = [MFPn - MFPol

MFPe ....

> 0.00001 then

MFP- MFPn

AM = AM

MFP. - MFPo

MFPo = MFPn

GOTO 4

End If

End Select

Return

Subroutine MASSFP (Tt, f,

M, Tt/ T, Pt/ P, MFP)

Inputs: Case, Tt, f, and M

Outputs:

Tt / T, Pt / P ,

and

MFP

FAIR (1,

f, Tt, ht, Prt, dpt, cpt, Rt, Yt, at)

Mat

1 + {(Yt -- 1)/2} M2

V 2

A h =ht - --

2gc

FAIR (2,

f, T, h, Pr, 4), Cp, R, y, a)

Vn =Ma

V- V.

If V -~ 0 then Ve

....

-- -- else

Verror = V - V.

IVerrorl

> 0.00001 then V = V.; Go to A

Tt/T = --

Pr~

Pt/P = --

M ~T,

MFP-- Pt/~ T

Return

Subroutine TURBC {Tt4, f,

(A4/A4.5), M4,

Ma.5, 7hH, Tta.sn, Tt3, fl, el,

e2;rctH,

r,H, T,4.5 }

Inputs:

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

M4.5,

rhH, Tt4.SR, Tt3, fl, el, e2

Outputs:

rrtl4, TtH,

Tt4.5

FAIR(l,

f, Tt4, ht4, Pro, dPt4, Cpt4,

Rt4, )It4, at4)

MASSFP (Tt4, f, M4, T4,

MFP4)

FAIR(I, 0,

Tt3,

ht3, Prt3, (9t3, Cpt3, Rt3,

Yt3,

at3)

APPENDIX F: COMPRESSIBLE FLOW FUNCTIONS FOR GAS 549

rnf = f(1 -/~

- E1 --

E2)

gt/4

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

E2)

?J/4.1 = (1

+ f)(1 -/3

- E l -- '£2) + E1

/'n4. 5 =

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

f4.1 ----" thf/(#/4.1 -- rhf)

f4.5 = mf/(rh4.5 - rhf)

th4hti -1- ~ l ht3

ht4.1 --

I'll 4 J¢

E 1

FAIR (2, f4.1,

Tt4.1,

hi4.1,

Prt4.1,

~t4.1,

Cpt4.1,

Rt4.i, Yt4.1,

at4.1)

Tt4.5 ~--- Tt4.5 R

MASSFP (Tt4.5, f4.5, M4.5, T4.5,

MFP4.5)

rh4.5

MFP4 A4 T~t4.5

2"ttH = m4 MFP4.5

A4.5"/--V ~

FAIR(I, f4.5, Tt4.5,

ht4.5,

Prt4.5,

~t4.5,

Cpt4.5, gt4.5,

2/t4.5, at4.5)

ert4.4i ---- 7ttH Prt4.1

FAIR

(3, f4.1,

Zt4.4i, ht4.4i, ert4.4i,

~t4.4i,

Cpt4.4i, Rt4.4i, Yt4.4i, at4.4i)

htaA =-- hi4.1 - ot(ht4.1 - ht4.4i)

"Ctn

---- ht4.4/ ht4.1

tha.lht4.4 -1-

E2ht3

ht4.5 =

/~4.1 "~ E2

FAIR

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

ht4.5, Prt4.5,

~bt4.5,

Cpt4.5, Rt4.5,

Yt4.5, at4.5)

if ITt4.5 - Tt4.5.[ > 0.01

then

Tt4.5

Tt4.5n

goto

End if

Return

Subroutine TURB {Tti,

f, ( Ai / Ae), Mi, Me, Ot, TteR; 7rt, rt, Tte}

Inputs:

Tti, f, ( Ai / Ae), Mi, Me, rh, Ttel~

Outputs: ztt, rt, Tte

FAIR

(1,

Tti, hti, Prti, ~ti, Cpti, gti, Yti, ati)

MASSFP (Tti,

f, Mi, Ti, MFPi)

T,e = TteR

1 MASSFP (Tte, f,

Me, Te, MFPe)

FAIR(I,

f, Tte, hte,

erte, dPte, Cpte, Rte, Yte, ate)

MFPi

7Tt -- MFP----~e Ae V Tti

ertei = 7rt erti

FAIR

(3,

f, Ttei,

htei, Prtei, dPtei, Cptei, Rtei, Ytei, atei)