Jet Propulsion for Aerospace Applications (Second Edition) [2 ed.]

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/If PROPULSION FOR ABROSPACB APPl/CAf/ONS

/If PROPUlSION FOR ABROSPACB APPl/CAf/ONS SECON D EDIT ION

WALTER J. HESSE Program Director, N ucleonic Systems, L TV Vought Aeronauiics DirJision Ling-Temco-Voughl, I nc., D~l~

N I CHOLAS V. S. M UMFORD, JR. M anager, Systems Engineering, LTV M ichigan DirJision Ling-Temco-Voughl, I nc., Dallas

I' ITM AN PUDLIS lllNG CORPOR11T ION I\rc av 1·ork • Toro11to • London

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© 1958 and 1964 by

Pil111an Puhlishi11g Corporation. All rights nstrttd.

No part of this book may be reproduced in any forrn u1ilhoul tt•rillen p rrn1issio11 of U1c publishu. M anufaclured in the U niled S tales of 11n1crica. /Jibrary of Co11grcss Catalog Card N ti111~r: 64-18757. 2 . 9 87654 3

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TO BETTIE AND ROSEMARY AND OUR NINE ASSORTED CHILDREN

Preface to the Second Edition

This edition of J et Propulsion has been extensively revised to incorporate advances in the field ,vhich have taken place over the past six years and to include the new topics of nuclear propulsion and space propulsion. At least two developments in recent years graphically illustrate the need for these revisions - the nearly complete conversion of commercial air transportation to jet propulsion and the almost commonplace orbiting of manned satellites. This revision has been based primarily on the experience gained from use of the earlier edition as a textbook for courses in jet propulsion and as a reference book for the design of new aerospace vehicles. The purpose of this new edition is to present a "·ell-balanced treatment of jet propulsion, 'vith emphasis on the principles of operation, for the senior undergraduate or graduate level. The development of all jet propulsion concepts is explained from the standpoint of fundamental engineering laws rather than fron1 a n equation s tandpoint. A working knowledge of calculus, physics, thermodynamics, and aerodynamics or fluid dyna111ics is required to con1prehend the material in the text adequately. The study of jet propulsion is basically an analysis of gases as they flow through various engine components, and therefore the aerothen11odynamics of gas flow systen1s is covered early, in Chapters 3 and 4. The application of gas aerothern1odynan1ics to diffusers and nozzles is presented in Chapter 5, 'vhich has been re,rised to include a discussion of subsonic as 'veil as superso1iic ducts a11d n1ea.ns of predicting the performance of each. Because jet propulsion engines are d esigned to po,,rer aerospace vehicles, en1phasis is placed on the combin ed performance characteristics of engines and aerospace vehicles. In Chapters 11 and 13 the application of turbine e11gines to aircraft is co11sidered, and in Chapters 14 and 17 the applicatio11 of jet propulsio11 e11gi11es t o space vehicles is presented . Chapter 11 also no\v includes a considcratio11 of tl1e tt1rbofa11 e11gine, ''•hich po"·ers many of the ne\v co1n 1nercial air transports; the 11c'v field of VTOL (Vertical Take Off and Landing) e11gi11es ; and means of predicting the perfor111ance of an engine when it is installed in an aircraft. Chapter 12 110,v i11cludcs a 111ethod for d eter111ining the performance of ar1 af terburning e11gi11e. Chapter 13 prese11ts 111eans of analyzing turboshaft as "'ell as turboprop engines. Reductio11 gear trai11s a11d shaft po,,rer absorbers, such as propellers, are also co11sidered. Chapter 14 l1as been revised to i11clude a brief consideration of so1nc possible air-breathi11g engines for propulsion of hypersoruc vehicles. Chapter 15 has bee11 co111pletely re\vrittcn to cover n1ore t.horoughly the vii

... VIII

I

Preface

field of chemical rocket engines. Chapters 16 and 17, on Nuclear Propulsion and Space Propulsion, respectively, have been added . All of the chapters are illustrated with current examples of the ''hard,vare'' involved in each of these engines, and several include tables sun1n1arizing the characteristics of current engines. The scope of the material covered calls for t'vo se1nesters, but a one-semester course can be based on a limited an1ount of n1aterial i11 the text. A13 a guide, it is suggested that a 011e-semester course might be based on Cl1apter 2 and portions of Chapters 1131 1 5 6191 11 I and 15 • Other combinations of material may be equally attractive for a one-semester course. In most i11stances the choice of material \\·ill d epend on 'vhat has been previously included in the curriculun1. The equations contained in this text are not writter1 in terms of a prescribed set of units ; therefore, conversion factors do not appear in the equations. This should cause no difficulty for students; in fact, experience has shown that it causes les.5 difficulty because in nwnerical calculations the student checks his own units and thereby learns the equation symbols as physical quantities rather than as a specific number . Numerous examples are presented throughout the teA-t to show the procedures for applying jet propulsion concepts and equations to numerical calculations. More than t\VO hundred widely selected assignment problems, many with anS'\vers supplied so that the student may check his work, are given a t the end of each chapter. The ans,vers to the example and assignn1ent problems are of slid e-rule a ccuracy. As is the case in the preparation of any textbook, the at1thors of t his one are indebted to many persons 'vho aided in its con1pletion. Special t hanks are due such groups and agencies as the AEC ; the N ASA and its predecessor the NACA, from which reference material has been freely drawn; the Test Pilot Training Division at the Ka,-al ..\.ir Test Center, Patuxent R iver, Maryla11d 1 'vhere this book originated; and propulsion industry companies, which provided illustrative material for our use. In particular \Ve wish to ackno,vledge the generous help of the Pratt (.~ Whitney Aircraft Di\rision of U nited Aircraft Corporation; the General Electric Co111pany ; the Allison Di,-ision of General Motors Corporation; the Rocketdyne Division of N orth A111erican A\-iation. Inc.; the Thiokol Chemical Corporation; and the Aeroj et-Ge11eral Corporation. '\\-e also 'visl1 to express our sincere appreciatio11 to associates at Li11g-Ten1co-' ' ought, Inr .. who have offered encouragement - specifically to l\1r. \V. E. l\Iallett., ,,·ho pro,·ided valuable suggestions after reading the entire book for technical co11tent, and to l\1r. F. T. Esen\vein for a critical revie'v of Chapter 17. \Ve are indebted to ~Ir. John E . Standefer, the principal contributor to the chapter 011 N uclear Propulsion. Thanks arc due Mrs. Nicholas Mun1ford , Jr. for typing all the 111 a 11 uscript re,·isions. We also \vish to ackno\vledge the painstaking efforts of l\Ir. \Villia111 S . Lo,,·e, a i11~t diligent reader of galley and page proofs. \V. J .

N. Dallas, Texas Birmingham, Michigan April, 1964

HESSE

v. s.

l\1UMFORD, JR.

1

Contents

xv

Symbols and Units Chapter 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.1 1 1.1 2 1.13 1.14 1.1 5 1.16

Chapter 2

Review of Principal Definitions, Concepts, and Basic Physical laws

Introduction 'Veight (W) and Mass (M) Units Systems Work (~ )

Energy (E) Pressure and T emperature (p, T ) Specific H eats (cp and cv) The Principal Laws of Engineering Newton's Second Law of Motion The Law or Equation of State The Law of Conservation of Mass The Law of Conservation of Energy Special Case of the Energy La'v - The H eat E quation R estriction of the Energy Law - Entropy and the Second Law Concluding R emarks

1 1 1 2 2 3 3 5

7 7 8 10 12 13 15 20 21

27

Princ iple of Je t Propul sion and Eng ine Classification

I ntroduction The R amj et Engine The Pulse-Jet Engine The Turboj et Engine The Turboprop or Turbosl1aft Engine The Turbofan Engine The Chemical R ocket Engine Space PrOJJulsion Engines The Basic Thrust Equations Thrust Power and Pro1Julsive Efficie11cy 2.11 Thermal and Over-All Efficiencies 2.12 Specific Fuel Consumption and Speci fic I mpulse

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

Ix

27

28 29 30 31 31 32 34 35 38 40 41

x

I

Chapte r 3 3.1 3.2 3 .3 3 .4 3.5

Chapter 4

4.1

4.2 4.3 4.4 4.5 4.6 4.7 4 .8

Chapter 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5 .8

5.9 5.10 5 .11

5.12 5.13

Jet Prop ulsion Aerothermodynamics of Steady, One- Dime n s ional lsentropic Compre ssible Flow Introduction Total or Stagnntion Conditions ApJJlication of I scntroJJic Flo\v to Ideal Nozzles I sentro1>ic Para n1etcr Variation \vith Prrssure Ratio I sentropic Para meter Variation \vitl1 M ach N umber

44 44 48 55

60

Compressible Flow in Constant-Area Ducts and t he Thermodynamics of Shock Waves Introduction Rayleigh Lir1c Conditions F a nno Line Conditions Normal Shock Conditions Oblique Shocks Prandtl-l\1eyer Expansion I sothermal Conditions Summary of Special Flo\\' Processes

67 67 68 75 80

87 91 9-1 96

Diffuser and No:z::z:le Flow with Friction Introduction Subsonic Diffuser Subsonic Duct Total Pressure R ecovery Supersonic Diffuser l\fass-Flo'v Ratio or Area Ratio Ram Drag of Supersonic Inlets l\fodes of Su1>crsonic Diffuser Operation • Other Supersonic Diffuser Performance Paran1etcrs Some Typical Inlet Shock Systems Nozzle Flo\V \vith Friction Nozzle Friction Parameters Nozzle Discharge J ets Nozzle Thrust Equations

10 1 101 101 103 106 111 115 118 120 123 126 129 136 141

•

Chapter 6 6 .1

6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Energy Transfer in Turbo-Machinery Introduction Velocity Diagrams at Entrance and Exit Derivation of Torque, Po,,·cr, and Jlend Equations Derivation of Iden! General Head Eqt1ntion Discussion of Ge11ernl Iirad Equation General Tl1ermodynnn1ic Energy Annl)·sis Compressor Prcsst1rc Function, Xe AIJo,vancc for the Vnrintio11 of Thern1od)·nan1ic Data \Yitl1 Tl'mr>erature Summary of l\ lomcntun1 a rid l~nergy llclations

151 151 151

153 154 156 157

160 161 161

Contents Chapter 7

Centrifugal Compressors

7.1 7.2

Introduction No Prewhirl (Axial Entrance) 7.3 Radial Exit witl1 No Pre\vhirl 7.4 Circulation or Slip in Blade Compartments 7.5 Slip Coefficient, /( = caM/ ua 7.6 The Effect of Blade Shape on P erformance 7.7 The Compressor Diffuser 7.8 Pressure Coeffi cient, 1/1,, 7.9 W eight-Flow Rate 7.10 Compressor P erformance Charts 7.11 Summary of One-Dimensional Centrifugal Compressor Equations

Chapter 8

I ntroduction Principle of Operation and Basic Terms Momentum or Filament Analysis, One-Dimensional Compressible Flow Blade Element or Airfoil Analysis, One-Dimensional Ideal Incompressible Flow 8.5 Blade Element or Airfoil Analysis, Two-Dimensional Flow with Friction 8.6 Three-Dimensional Consideration 8.7 Polytropic or Small-Stage Compressor Efficiency 8.8 Compressor Performance Curves and Compressor Stall 8.9 Twin Spool a nd Variable Stator Compressors 8.10 Supersonic and Transonic Compressor Considerations

' I ntroduction D esign Considerations }1r-S Process and Bu rner Efficiency Flame T emperatures Gas Turbine Engine Fuels Fuel-Control Units

Chapter 10

10.1 10.2 10.3

10.4 10.5 10.6 10.7 10.8

165 166 168 168 169 170 172

173 175 176 178 181

181 184 187 189 192 195 196 200 203

204:

209

Combustion Chambers, Fuels, and Controls

Chapter 9

209 210 215 219 221 225

230

Gas Turbines

Introduction General Thermodynamic Analysis AJ)plication of the Expansion Ratio Function, X, l\1omentum and Combined Analysis Tl1e Axial-Flo'v Gns Turbine Potentialities and Problems of Higl1 Turbine-I nlet Temperature Operation The n1atching Problem Turbine Blade Failures

• XI

165

Axial-Flow Compressors

8.1 8.2 8.3 8.4

9.1 9.2 9.3 9.4 9.5 9.6

I

230 233 236 ~

239 245 249 251

=--•• XII

I

Jet Propulsion

Chapter 11 11.1 11.2 11 .3 11 .4 11 .5 11.6 11 .7 11 .8 11 .9

Introduction Over-All E ngine Analysis Cycle Pressure and T emperature Variation Variation of Engi11e Data v;ith Cl1a11ges in Operating Conditions Corrected Engine Parameters Installed Engine Performance Turbofan E ngines VTOL (Vertical Take-Off and Landing) Engines Engine-Airplane Combination Characteristics

Chapter 12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8

14.1 14.2 14.3 14.4 14.5 14.6

15.1 15.2 15.3 15.4

302 302 307 317 322

323 325 325

328

32.8 331 334 3-14 349 35.S 359

360

374

High Flight Mach Number Air-Breathing Engines

Introduction The Basic Problem Turbojet and Turbofan Engines Ramjet Engine Combination Po\ver Plants Po\ver Plants fo r Hypersonic Applications

Chapter 15

302

Turboprop and Turboshaft Engines

Introduction The Basic Turboprop or Turboshaft Engine Analysis of P erformance of the Turboprop Engine Analysis of P erformance of the Turboshaft Engine Shaft Po\'1er Absorber - the Propeller Fixed and Free Turbine Engines Gearbox Considerations Cycle Improvements

Chapter 14

256 261 271 273 275 283 289 291 293

Thrust Augmentation of the Turboiet and Turbofan Engines

Introduction The Afterburner or Tailpipe-Burner Afterburner P erformance \Vater Inj ection a t Compressor Inlet ''' ater Injection in Combustion Chamber Bleed-Burn or Bleed-Off Cycle Summary of Thrust Augmentation Devices Effect of Humidity on Engine P erformance

Chapter 13

256

Turboiet and Turbofan Engines

374 374 376 ~

394 396

The Chemical Rocket Engine

I ntroduction Liquid-Propellant Rocket Engines Solid-Propellant Rocket Engines Thermodynamic Equations of the Rocket l\Iotor

407 407 408 421 425

Contents

15.5 15.6 15.7 15.8 15.9

Thermodynamics of Combustion Characteristics of Some Liquid Bipropellant S)1 stcms The Rocket l\1otor Cooling Problem Performance Characteristics of Solid-Pro1)ellu.nt Rocket Engines Cl1aracteristics of Some Solid Propellants

Chapter 16

16. l 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10

17.1 17.2 17.3 17 .4 17.5 17.6

437 445 450 454 465

469

Nuclear Propulsion Engines

Introduction Nuclear Energy Nuclear Reactor Characteristics Effects of Gamma and Neutron Radiation on 1\Iaterials Health Physics Shielding Nuclear Turboj et Engines Nuclear Ramj et Engines Nuclear Rocket Engines Systems for Nuclear Auxiliary Po\\'er (SNAP)

Chapter 17

I

469

470 473 478 480 482 484 487 491

496 501

Space Propulsion

501 502

Introduction Electric Propulsion Engines Electric Power Sources Solar and Photon Propulsion The Problem of E scape and Space Travel Comparison of Space Propulsion Engines

518 52'2 524 535

Ans\'\·ers to Selected Problems

545

Appendix A Standard Atmosphere Data

553

Appendix B Compressible I sentropic Flo\v Table - Subsonic

Flo~v

561

Appendix C Compressible Isentropic Flo\v Table - Su1)ersonic Flo"·

563

Appendix D Tabulated Values of the I sentropic Pressure Ratio Function Xe

600

Appendix E Tabulated Values of the Isentropic Expnnsion Ratio Function X. Index

•

•

•

••• XIII

607

Symbols and Units

Convenl:Wnal Units

DefinitU>n

ft2 in.2 in.2 in. 2 ft 2

a a, b

area solid-propellant burning area solid-propellant port area nozzle throat area wetted area constant in burning rate equation major and minor axes of an ellipse

B B B!, B! BHP Btu b bsfc

applied magnetic field reactor-buckling factor geometrical, material-buckling factors brake-horsepo\ver British Tl1ermal Unit blade width brake specific fu el consumption

webers/ m 2 cm-2

c c I Cai, CBl

Cv CF

c,, Cp

c. Cp

c.

Cpz, c11:

CT

cT,c; c. c•

constant total drag coefficient additive drag coefficient lift coefficient volume flow rate absolute velocity in blade passage velocity of light air bleed correction factors for turbine engi11cs duct loss correction factors for turbine engines rocket engine thrust coefficient propeller po,ver coefficient specific l1eat at constant })ressure S})ecific heat at co11stant volu111e molar S})ecific heat at co11stant 1)resst1re, 1ncp molar S})ecific heat at co11stant volurne, ?nc., ))O\ver extractio11 correction factors for turbi11e engi11cs propeller thrust coefficient ambient tern1)erature correction factors for turbine c11gincs nozzle velocity coefficient characteristic velocity

cm-~

HP Btu ft lb/BHP-hr

fV/ sec ft/ sec 3 X 1010 rn1/ sec

Btu/lb 0 R Btu/lb 0 R Btu/ mole 0 R Btu/ n1olc 0 R

ft/ sec

xv •

• XVI

I

Jet Propulsion

Definition

S ymbol

D

radiation intensity

Do D,

additive drag scoop increme11tal drag diameter differential operator of calculus

d d

F Fe Fa Fp

FPR Fm FN FR f f/a

C') -Y) T, J>,

J>1

•

R~

...-· c

"< n

-

rQ

! -

J>

-

-0

20

I

Jet Propulsion

these relations apply to a gas whether moving or stopped, the gns properties (p, v, T) are those which would be obtained by measuring them relative to the fluid. The last column, entropy change, is discussed in the next section. 1. 15

Restriction of the Energy Law- Entropy and the Second Law

Thus far, we have discussed the energy equation upon the bnsis that any one type of energy can be converted into another type. Tl1is is true for all cases except for the conversion of heat into \vork. It is recalled from the concept of the second law of thermodynamics that, of a given quantity of heat, only part can be converted into work; thus, there is a restriction which we must place upon the law of conservation of energy. This restriction should be well understood, because, in all jct propulsion devices, \Ve are attempting to convert heat energy into useful work. The concept of tliis restriction of energy transformation is called the second law of thermodynamics, and an important consequence of this law is the definition of entropy which relates the available and unavailable portions of heat energy. Since, even in an ideal reversible heat engine operating at certain finite tempernr tures, a certain portion of the heat input is unavailable for performing work, the heat input is divided as follo\vs: heat input = available heat

l

+ unavailable heat

where the available heat is that portion of heat that is converted into work for a reversible engine, and the unavailable heat is that portion of heat rejected by the engine. From the Carnot cycle efficiency equation, the proportion of available and unavailable heat can be changed by altering the cycle temperatures. As \Ve know, the concept of heat availability referred to a certain temperature can be generalized for a reversible process by the entropy function, \Vhich is defined as

dS = dQ T

•

(1.49) I

By this equation, we can compute the entropy change of any proc~ \vherein heat is transferred, either by friction or by some external source. From an analysis of the Carnot cy cle, \Ve can deduce that the entropy of a complete system which operates on heat can never decrease, simply because the transfer of heat is al\vays accompanied by some unavailable portion. For the ideal, reversible, adiabatic process, there is no entropy change because there is no transfer of heat. This type of process is usually referred to as isentropic, • which means equal entropy. The entropy change for a given process is usually computed by expressing dQ in terms of the heat equation as

dS

=

dQ T

dU

T

+ p dv T

v dp T

(1.50)

•

.... -

•

•

-•

...

•

• It should be noted that an isentropic or equal entropy process can occur with friction if the gas is cooled at the same rate as friction heat is generated; ho\vevcr, isentropic usuall.)• means reversible adiabatic and will be so used in this text.

.

•

Review of Principal Definitions, Concepts, and Basic Physical laws

I

21

which can be \Vritten as

dS

=

c.dT

T

+ R!!:!!_

(1.51 )

ncr lb-n1ole, "·hat is the change in density?

I'

•

•

I

--

28. A diffuser \vhich has an entrance area of 1.5 sq ft is operated \Yith nn rntrnnce \'elocit)• of 500 mph in standard conditions at 20,000 ft pressure altitude. \\.l1at is the " ·eight flo"· of air in lb per sec? 29. A constant area combustion chamber rcceivrs air and fuel at its entrance. The mi" I tion, so11ic at the section, and supersonic I ~ D ecrease o f p,, increase I after this section. It is also sho"·11 that, if o f T,, o r b o th ..c. I -111 tl1e critical pressure ratio exists i11 a ·-II 0 ~ 0 v1 - ·I ·-· .. 0 3 nozzle, it occurs at the 1ni1limun1 area ·-.... .... u 10. section. Figure 3.6 also specifies the proper equation to use for each pressure ratio 0 0 1 0 2 0 3 0.4 0.5 0.6 0 7 0.8 0.9 1.0 region. The simple continuity equation N ozzle p ressure ral to, P/ Pt G = AV /v ca11 be used in either region, F10. 3.6 Weight-flow rate versus pressure ratio but '"hen G is expressed in terms of presfor different inlet conditions. 1

-

--

Q)

-

I

O ne-Dimensional lsentropic Compressible Flow

53

sure ratio, the appropriate equation specified on the figure should be used. Note that the G/ A versus p/ p, relation is actually represented by a family of curves, each curve applying to a different set of inlet conditions. Let us assume, for example, that curve X, called the basic condition, represents the function for a given set of inlet conditions. for these inlet conditions, we note that there is a maximum flo\v rate which the nozzle can handle, and even though \Ve reduce the nozzle back pressure to zero, we cannot increase the fto,v rate. If, however, 've increase the inlet pressure or decrease the inlet temperature, we shift to a ne'v curve Y, \vhich sho\VS a greater flow rate. Conversely, if we decrease the inlet pressure or increase the temperature, \Ve shift to a lo\ver curve, as depicted by Z. The flow rate can also be changed by varying the nozzle fto,v area. These facts illustrate 'vhy it is necessary to use a variable area nozzle when an afterburner is installed on an engine. When t he afterburner is turned on, a large increase in nozzle inlet temperature occurs, and in order not to reduce the engine thrust by a reduction in air-fto\v rate, the nozzle exit area must be increased so that the engine can operate on about the same air-fl.o'v rate curve. Figure 3. 7 presents a \Vorking chart of the weight-fl.ow rate parameter G'V'F,/ AzPi versus the nozzle pressure ratio for -y = 1.33 and 1.40.

"$:

.0c

I!

u

0.6

p•

'

y = l .40

t;:I ~ 0.5 C)

-E

p•

0.

-.... QI

'

p; = 0.542

0.4

.I

~

~

y = l.33

QI

0.... 0

I

'

"'(..

p-= , 0.528 at f G) = 0.532

I

I

at f (G)

= 0.523

'\

0.3

~ 0.2

-.L

.,

Cl

"

\

0

0.1

'

l

-

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

N o zzle pressure ra tio, p./ Pt

Flo. 3.7 Variation of ideal weightrftow rate paramel.er f (G) with nozzle pressure ratio for 'Y = 1.33 and 1.40.

From this chart, it is relatively simple to find the \veight-flo"· rate of n. 11ozzle from the nozzle pressure ratio and the inlet conditions. A COUSTI C V E LOCI TY

The question arises as to 'vhat physical phenomenon causes the critical condition to exist. Figure 3.8 depicts an apparatus which would give the experimental results of the graph as plotted in Fig. 3.5.

Chamber 1

Chamber 2

Valve C

Infinite reservoir p 1=constant

F10.

3.8

Nozzle Bo"· apparatus.

54

I

Jet Propulsion

In the illustration, an ideal gas is supplied to chamber 1 at a constant pressure p 1• It then expands isentropically in the De Laval nozzle to chamber 2 at a pressure of P2 controlled by the valve C. If the pressure on the do,vnstream side of the valve is zero, it is seen that fully opening or closing valve C ,viii vary the pressure in chamber 2 from near zero to Pt· As shown in Fig. 3.5, the weight flow increases until Pt is equal to, or less than, the critical p1essure p*. Any position change of the valve C causes a disturbance producing a change in pressure, and this disturbance is propagated through the fluid in the form of a longitudinal \vave. This wave transmits the pressure change from valve C to the nozzle; however, ,vhen the fluid velocity is greater than the velocity of the propagated wave, the pressure change of valve C never enters the nozzle, nor does it ever reach the nozzle exit section. Therefore, as soon as the velocity in the nozzle becomes supersonic, any further reduction in back pressure is not transmitted to the nozzle', and as far as the nozzle is concerned, the conditions at the throat section (the throat is the section where supersonic flow commences) become constant and are only a function of the inlet conditions. When this occurs, flow rate is constant for any reduction in back pressure (Fig. 3.5) . The critical conditions occur when the fluid velocity is equal to the a coustic or pressure disturbance velocity. Sound is defined as a series of condensations and rarefactions transmitted at audible frequencies along a longitudinal wave. Since the propagated pressure disturbances described above are of identical type, they have all the characteristics of a sound wave. If the critical conditions are considered, an expression for the velocity of sound may be derived since, at this condition, the fluid velocity is equal to the velocity of sound. Using Eq. (3.8), we can write the equation for the velocity at the critical section as follows:

v• =

2g CpT*

T, - 1 T*

and since

(3.17)

'Y-1

c11

v• =

=

'YR "'( - 1

and

2g"YRT* "'( - 1

T, _ T* or

Pi -p*

'Y

v• = v g-yRT* =

Va

(3.18)

This equation shows us that the critical velocity is the local acoustic velocity. Thus, at the paint where the critical pressure exists, tlie P.f ach number is 11.nity. We shall sho\v that when a nozzle operates 'vith a back pressure less than the critical pressure, the critical condition exists at the minimum urea section of the nozzle. In the De Laval nozzle, this minimun1 cross section is called the n ozzle throat, and in the convergent nozzle, the cross section of minimum area is the exit section. The critical velocity can be expressed in terms of the inlet total temperature because, in an adiabatic process, the total temperature is constant. By Eq. (3.3) , t l1e static temperature at the critical condition is related to tl1e total temperature ns

T' = 1 + -y ; 1 p.ft

T*

One-Dimensional lsentropic Compressible Flow

Since M•

=

I

55

1, we have

T*

=

2 "Y+l T,

Thus, the acoustic velocity in the nozzle can be expressed as (3.19)

THE DE

LA v AL NOZZLE

The critical conditions of nozzle flow are of paramount importance because they directly influence nozzle design. When the nozzle exhaust or back pressure is greater than the critical pressure, the fluid jet leaves in practically straight lines. H ence, for this t)rpe of expansion, all that is needed is a convergent section directing the fluid in the desired direction. The section is convergent because, until the critical pressure is reached, the rate of velocity increase is greater than the rate of specific volume increase. Conversely, when the exhaust, or back pressure, of a given nozzle is lower than the critical pressure, the rate of velocity increase is less than the rate of specific volume increase, as will be proved subsequently. In the latter case, the exhausting jet is.5ues forth while still expanding, and is turbulent in motion if a convergent nozzle is used. Therefore, any further expansion below the critical pressure must occur in a walled rection if a directional jet and high efficiency are to be maintained. This is accomplished by adding a divergent section to obtain a convergent-divergent nozzle (see Fig. 3.3). This type of nozzle enables the designer to utilize supersonic velocities, thereby obtaining more guided kinetic energy than attainable p,= 100 psia, r, = 2000° R, G = l lb/~ec. with a convergent nozzle for pressure drops -r= variable 100 .---,...--,,..----.---.-------. 500 greater than the critical pressure drop. From the foregoing equations, the variation of pressure, velocity, specific '9'olume, and enthalpy can be determined for a given ideal nozzle configuration. Figure 3.9 shows the variation of these quantities plotted versus the length of a typical De Laval nozzle shape. The plot is made for air with inlet conditions of p, = 100 psia, T, = 2000 °R, and G = 1 lb per I 2 3 4 Nozzle length, inches sec. Note that the velocity in the convergent section is subsonic, the velocity at the throat is .. • gt sonic, and the velocity in the divergent section •E '"I ,_ I .!:' 0 is supersonic. If a convergent no2Jzle \Vere used I for these conditions, the maximum velocity in the nozzle would be sonic, since supersonic F10. 3.9 Typical De Laval nozzle showing velocities cannot exist in a convergent nozzle. gas propert:y variation. ~

-

3.4 lsentropic Parameter Variation with Pressure Ratio

Since the critical conditions a re so important in nozzle analysis, it is desirable to develop equations relating the various gas properties to their values at the critical condition. We shall express these relations i11 terms of ratios which will be called

56

I

Jet Propulsion

parameters. Thus, the velocity para meter V ;v• will be t he ratio of t he velocity at any section in the nozzle to the critical section \vhere M = 1. The following pa rameters will be expressed as functions of pressure ratio:

!!_ .§_

a• s•' I

Tc

Pc V M A T p v T~ ' p f, v•, M* , A. , r• , p* ' v• '

p

p* '

and

The results will then be plotted to show the relative variation with nozzle pressu re ratio. Since, in this analysis, we are assuming isentropic steady fto\.v conditions, it is evident that

Ti = -14 = T,

Pt

(3.20)

1

From Eqs. (3. 11) and (3.19), we have

2g'YRT ,

V=

1-

1

'Y -

and

")'- 1 p "Y p,

2g'YRTt

v• =

+

'Y

1

Thus, the velocity parameter is given by

v

+1

'Y 'Y -

v•

1-

I

p -

")'- 1 "Y

(3.21)

Pt

Note that the maximum value of this function occurs when the nozzle back pressure is zero, or p/ p, = 0. Then

v

=

2.66 for 'Y = 1.33

(3.22)

Dividing the general weight flow Eq. (3.13) by the critical \veight flo\v Eq. (3.16), and noting t hat G is constant for steady flo,v, \Ve have

A A*

2

2 'Y - 1

[

(1!.. )

'l/ ")' -

Pt

(1!.. )")'~1] Pt

Mult iplying and dividing by 'Y + 1/ 2, and simplifying, gives 1

2

A A*

')'- 1

+1 'Y + 1 1 'Y

p

-

Pt

1/')'

'Y - 1

(3.23) p

-

Pt

.,.~

'I'

One-Dimensional lsentropic Compressible Flow

I

57

The temperature parameter can be obtained from "Y- 1

"Y- 1 "Y

-

p* Pt

T* = T t

and

"Y

2

= Tt

Thus T T*

'Y

--

+ 1 -p 2

"Y- 1 "Y

(3.24)

Pt

Since the acoustic velocity V 0 is a function of the local tempera ture, its pa rameter (V./ V:) 2 is also given by Eq. (3.24). The volume ratio is given by

r•

v -= v•

1 "Y- 1

1 -y- 1

2

T

X

l / "Y

Pt p

(3.25)

and the density ratio is, of course, given by the reciprocal of Eq. (3.25). The Mach number at any section is given by

.,

"Y-1

1'!2 =

y2

T Tt

2

-

Y'Y RT

1

'Y -

1-

P pI

Employing the isentropic temperature-pressure relation gives

M M -- M*-

"Y-1

2

~

p

., - I

(3.26)

It is also desirable to define a momentum thrust parameter which can be used to evaluate the thrust in a jet propulsion nozzle. In Chapter 1, it was shown that the momentum force or thrust is given by

Fm=

!!...v g

We shall define our momentum thrust parameter on the basis of a unit area, thus

Fm

GV

A

Ag

-= -

From the continuity equation and the perfect gas Ia,v, this equation can be written as

Fm = P'YM2 A

(3.27)

The momentum thrust parameter can be expressed directly in terms of pres.5Ure ratio by employing Eq. (3.26) and the critical pressure ratio equation to gi,l'e

F,,./ A --,,.--

-

F:.;A• -

2 'Y -

'Y

1

+1 2

"Y- 1

"Y

"Y- 1

p pt Pt P

..,

-1

(3.28)

A summary of the isentropic nozzle parameters ns a fu11ction of pressure ratio, as '"ell as equations for the parameter value at the critical section, is given in Table 3.1. It should be noted that these relations apply to any isentropic process with an area change. Therefore, they apply equally \Veil to a diffuser.

mlilDt.lfOl t'! I

lt '-•n1Y1• ",...., ,.. • •• •• • • - · - - ..

-·-a---·· -

TABLE

°'co

3.1 Summary of Isentropic Flow Parameters as a Function of Pressure Ratio

-

-

Critical values in terms of inlet conditions Parameter expression

Value for 'Y = 1.33 and p - Ib/ ft 1

General expression

.-,, '(I-

!!.._ = P•

G*

A A*

p,•

=

S = T, s,• T,*

(

(]!_)

l/ T

p,

1.0

=

0

"O

c --· "'

2 )~ 'Y + 1

0

]!_) ]

T

T-1

F ../ A = ( 2 )(-y+l ;:i [E.(~-;- - l)] F!/ A * 'Y - 1 2 p, p

•

F*"' A*

G* A•

2 )~ T-1 RTI 'Y + 1

G* A. - p I \

'Y + 1 [ 1 - ( ~ \ 'Y - 1 P• \

-

-

g-y (

p,

=

0·524 VT,

lb sec ft

F! lb A* = 0.72?tfti

T 2 ) ;:i = P•'Y 'Y + 1

(

T-1

M=M= 2 M* ' -y-1

[~-;- -1]

M*

p

= 1.0

T

_p_ = E.( 'Y + l);:i

p•

p,

~= ( T*

v=

y•

2

v. )2 =-y-1 ( ]!_) -;v.• 2 p T- 1

'Y+1[1-E.7 ] \ 'Y - 1 P• p

=

)_!_

( 2 Pt 'Y + 1

T-1

~ = t_ = c~)l/T v•

p•

M* = 1.00

p

(

2 )T~l 'Y + 1

T* = T 1 ( 'Y

v• = v•

~ p •-0542 - . Pt ftt

T- 1

! 1)

T* = 0.859 T, 0 R

2g-yRT,

v• =- 44,a V'F:

'Y + 1

= 1 = gp•

1

RT, ( 'Y + 1):;:i p, 2

v•

1

ft

sec

= gp• = 84.7 !! ~ p, lb

::::J

2

One-Dimensional lsentropic Compressible Flow

I

59

The various parameters summarized in Table 3.1 are plotted in Fig. 3.10 for 'Y = 1.33. This.plot shows that t he critical conditions occur at the section of minimum area (the t hroat in the De Laval nozzle) because the value of all parameters at t his minimum area section is equal to 1.0. The plot furt her shows t hat the gas velocity is subsonic in the region where the pressure is above t he cri tical value (subcritical region), and supersonic where the pressure is below the critical value (supercri tical region). Figure 3.10, even t hough based on ideal flow conditions, presents an explanation of some nozzle fio\v phenomena, and the curves, when used in conjunction wit h Table 3. 1, can be used to advantage for the rapid solution of any ideal nozzle flow problem. 10

9 8

7 6

I I

5

I \

4

2 0

p•

v

=p v•

\

. '' ~\

\

'\ \

3

>

\

...

\

....

\

Q)

E 2

0 .... 0

' v"-

•,-.

\

-

'

:.=

'

' " .............~ ... "'~

'

1

Fm/A ..... i..F •/A*_,• ~ m

I

//

•

-- ·~

~

I

(::.)2/

0

I

I

,X ~

v

~

p

••

l{ -;; = p-

....

v

I

~ ....

I

~

F,,./ A ~ Fm*/A• '

0 .2

0.4

0.6

Pressure ra tio, F10 .

I

•

-vv•

-MM• _ 0

I

A A*

~~ ~

-v•

~

!

I

'

Q)

l

p/ p 1

0.8

'Iii

I

.\ 1.0

3.10 Vo.riation of isentropic flow para.meters with pressure ratio for -y = 1.33.

!!_ = 1!.! =.§_ G• pf s•

_J:J _ 10

- Ti - .

Example

A jet propulsion De Laval nozzle is operated \Vith 58.3 lb of air per sec at inlet conditions of T, = 1600°R, p, = 30 psia, and a back pressure of 9.72 psia. Assuming that ~~e nozzle is designed for complete expnnsion to 9.72 psia, "·hat are the isentropic cond1t1ons at the nozzle throat and at the exit section? 'Vhn.t thrust " ·ould be produced if the nozzle were convergent?

60

I

Jet Propulsion

Solution The conditions at t he nozzle throat can be easily calculated by employ ing t he equations from Table 3.1 for 'Y = 1.33. Thus,

A* = G• v"f';

F! = 0.72p, A*

= 1.03 ft 2

0.524 p, "Jl.f* = 1.0

T*

=

lb 3105 W

= 0.542 p, = 16.26 psia v• = 44.3 v'T, = 1770 ft / sec p•

= 0.859

T,

= 1372 °R

The conditions at the nozzle exit section (where p = 9.72 psia for complete expansion) are easily found from t he parameter values from Fig. 3.10 for p / p, = 9.72/ 30 = 0.324. From these parameter v alues, the exit conditions are computed as

A.

A• = 1.12 A. = 1.152 ft2

:::;. =

M• = M. M* = 1

1.40

P; = p

T. T*

T•

v. v•

= 0.881

= 1210°R

1.17

Fo

0.597

= 1.31

p.

= F,,. =

=

4190 lb

9.72

v• = 2320 sec ft

If the nozzle were convergent, the critical conditions would exist at the exit section, and t he t hrust would be given by Eq. (2.4). Thus

Fo

=

G.V.+ A.(p. - Po)= g

or

Fo = 3105 X 1.03

F! + A*(p*

+ 1.03 (16.26 -

- Po)

9.72) X 144 = 4170 lb

I t is seen t hat the t hrust of a n ideal convergent nozzle is approximately the same as the ideal D e Laval nozzle for t he relatively lo\v pressure ratio used in this example.

3 .5

lsentropic Parameter Variation with Mach Number

Thus far we have expressed t he parameters for isent ropic fto,v as a function of pressure ratio, and t heir usefulness \Vas indicated for the solution of nozzle problems. These same parameters can also be expressed as functions of Mach 11umber to provide further usefulness for t he analysis of an ideal nozzle or diffuser . The d eriva tion of the parameters in terms of Mach number can be obtained from the equations of T a ble 3.1. Solve t he Mach nu mber equation in terms of pressure ratio to give

,.

~ = p

1

+ 'Y -

2

1 M'

,._1

and then replace the pressure ratio functions \vhich appear i11 the other equations with this expression, or obtain them from the basic engineering la,vs. Tl1e nrea param ete r will be derived here in differential form from basic considerations because it deserves

One-Dimensional lsentropic Compressible Flow

I

61

special discussion. The other parameters, ho\vever, will not be derived since they follow directly from Table 3.1. The area parameter can be derived from the continuity equa tion, \vhich in differential form is dA

-

A

dT T

dV

dp

- -p - v

(continuity)

The dynamic equation for isentropic flow (vdp = VdV / g) can be expressed as dp

-

p

--yM

=

dV

2

(dynamic)

v

The energy equation can be written as

d~

-(-y - l)M2

=

d;

(energy)

and substituting the dynamic and energy equations into the continuity equation gives dA = dV (M 2

A

V

l)

_

(3.29)

Solving the Mach number definition M = V / yg:ytff' for dT / T, and equating , it to the energy equation above, gives dV

v

-

dM

~------------------

M 1

+ "Y -2

(3.30)

1M2

When Eq. (3.30) is substituted into Eq. (3.29), we obtain the differential area parameter dA

A

(M2

-

l)dM

- --------------------M 1

+ 'Y -2

1

(3.31)

M'

Equation (3.31) warrants discussion because it sho"~ "·by the area is a minimum when M = 1. Remembering that dA / A represe11ts the chnnge in .1 \nth respect to A, \Ve see that when M = 1, (dA / A ) = 0, which nlenns thnt the area is a minimum at M = 1. Thus it is evident that, in a superso11ic 11ozzle, the ~Iach number at the throat (minimum flow area) must be equal to u11it)'· Furthern1ore, the equation also shows that,vhen M > 1 \vhilel\1is i11creasi11g (d1l!/ ltfis + ), dA l.4 is positi,·e, ,,~hich means that the area must increase. Hence, \Ve i1ced n divergent section for supersonic nozzle flo\v. When l\f < 1 \Vhile 1\1 is i11crensi11g, dA / .4 is i1egati,•e, " ·hich means that the area mus t decrease. He11ce, \Ve 11ced a co11,rerge11t sectio11 for subsonic nozzle flo\v. Co11verscly, \Vhen 1\1 is decreasi11g t1S i11 diffuser flo'\', the opposite is true. Equation (3.31) can also be t1sed to sl10'\' t11e ro11ditio11 for constant-area duct flow, subso11ic venturi flo,v, n.nd supersonic ve11turi flo\v. The above remarks for the various fto,v conditions are sun11narized i11 Table 3.2. Figure 3.11 illustrates the flow changes in an isentropic nozzle and diffuser.

I

62

Jet Propulsion

TABLE

dM M

dA A

+or -

0

Mach No. M - 1.0 M

M

M

M

1

+

+

1

M =any value

Type of flow

Physical meaning

I

Critical conditions

Min. area

/

Subsonic nozzle

M increase Area decrease

•,•

Supersonic nozzle

M increase Area increase

/

-

3.2 Mach Number and Area Variation for Isentropic Flow

+

Subsonic di.ffuser

M decrease Area increase

,.'

-

-

M decrease Area decrease

Supersonic di.ffuser

0

0

Min. area or const. area, ll.p = 0

Const. area duct or throat of venturi

-~

•

•

Continuous pressure decrease and velocity increase I

NO~ZLE

Ml

I

M = l.O I

_ ____,~

Flo. 3. 11

Continuous pressure increase and velocity decrease

M >l

M '

= ln

M•

'Y+ l

--;

The Rayleigh line conditions are best represented by a plot on the T - S or h-S plane. Such a plot can readily be made with the application of (S - S*)/ c11 and T / T* parameters. The result ing curve for some constant value of G/ A will appear ~shown by the solid curve in Fig. 4.2. Points a and. b are 'vorthy of investigation because they represent maxim\1m entropy and maximum temperature for the specified value of G/ A . In order to determine the Mach number at these points, it is nec~ry to obtain a relation to give dT/ dS = f (J.f).

72

I

Jet Propulsion 10,---~-r~~-.--~~.--~__,..~~~~~--~-. 9 t--~-t-~~-+-~~~~-+~~4--~~I--~~

8 t--~-+~~f--~-+-~~~~---+--~~I--~~ 7 ,__---t~~~~~~~~--+~~--+~~-+~~-1 6 r-~--t~~-+~~-+-~~+-~~~~-+-~---I

v - -" -p• Fm" / A· - v· - ,..- P

-r•T

0 .5

1.0

1.5

2 .0

2.5

3.0

3.5

Moch number

Flo. 4.1 Variation of Ray leigh lin e parameters with Mach number for

G A G• = A • = l

Shi ft d irection to lower va lue o f

G

A

/

b ., ,. - • -, \ \

• a'

>-a.

I

I I

0 ..J:.

-... c

w

0

0

-- M = l

I I I I

I s· Entro py, F10.

4.2

S

Rayleigh line conditions for one value of G/A.

'Y

= 1.33.

Constant-Area Ducts and Thermodynamics of Shock Waves

I

73

From Eq. (1.52) we have 'Y -

1 dp

(4.22)

p

'Y

From the dynamic equation \vithout friction and from the Mach number [Eq. (4.11)] we have

,I

\

3

\\

>

I . ,

\

...

I

\

Q)

,I

\

'i 2

\

E

e0

\

\

\

0..

I

/-

'' ...

1

~

A v ,,, / ',,,

Fm/A Fm•/A•

~ ............._. ~

.......... \

' ' ' .......... 0

_

0.5

1.0

/. -

/_,,,

v

p• v ==-=-=-~

v·

"F•ino• 0

~-

v

./ -...__

,,,. ,,, ....

~

,,, ,,,· .,..,,,. .,..,. s- s•

.... p -p•

-c

•

p

v•

-T

............. "'-.

0

•

P,

1.5

2.0

2.5

3.0

3.5

Mach number

Flo. 4.7 Va.ria.tion of Fanno line para.meters with Ma.ch number for

'Y =

1.33.

G A T1 -G• -m - -A• - T,. --1

Example

A compressor delivers air to a pipe of 1 ft diameter at p = 50 psia, T = 800°F, and M = 0.4. If the average friction coefficient is .003, find the ma.'tlmum pipe length, the pressure, and temperature at the end section. Assume 'Y = 1.4.

Solution Since Fig. 4.7 is plotted for 'Y = 1.33, it is not applicable to this problem; therefore, it is necessary to calculate the follo\ving parameters for the inlet condition : 4fXma:c D

=

2.31

1

Ti

T*

= 1.16

E; = p

2.69

The maximum length is obtained \Vhen the exit ~1ach number is 1.0. Thus, at the exit section (subscript 2)

=0 2

The pipe length between any t\VO sections is given by D x =4f

4fx,.,,4Z D

1

4fx,.,,4Z D

!

80

I

-

-~~-

•

•

Jet Propulsion

Thus Xtftaz

=

l (0.00 ) (2.31 - O) 4 3

=

192.5 ft

and the pressure and temperature at exit are

P2 = p* = and

p* 50 - Pi = 2.69 Pi

T2 = T* =

=

18.6 psia

800 = 690oR 1.16

Note in the above example that the pressure drop is appreciable, but the temperature drop is small, because the friction energy reappears as heat 'vhicb goes back into the gas. If any additional pipe is added, the inlet condition \vill change so as to decrease the flow rate. Reference 1 also presents tables of the Fanno line parameter as a function of Mach number.

4.4

Normal Shock Conditions

In our discussion of flow relations up to this point, we have seen how the gas properties change in a relatively smooth and slow manner as a function of flow distance. In isentropic flow it was sho\vn that a change in flow area produced uniform changes in the gas pressure, velocity, density, etc.; \vhereas in the constant-area duct flow, '"e saw that continuous changes in the gas properties were brought about by heat addition and/ or fluid friction. All flow, however, is not characterized by this relatively uniform, continuous, and slow change of gas properties. A certain phenomenon may exist in the supersonic regime where such gas conditions as velocity, pressure, and temperature change very rapidly across a discontinuity of the flo\v. This flow discontinuity is called a shock " 'ave; a normal shock \vave when the discontinuity is perpendicular to the flow direction, an oblique shock \Vave \Vhen the discontinuity is at some other angle to the flow direction. Figure 4 .8 is a shadow photograph, popularly kno,vn as a shadowgrapb, of a high-velocity projectile moving at supersonic speeds from right to left. It illustrates a strong curved shock wave, a portion of \vhich is a normal shock (that portion in the region of the nose of the body) and the remainder of \vhich is an oblique shock.

4.8 Shock wave generated by a blunt body sho,ving both normal and oblique portions. (Courtesy U .S . Naval Ordnance Laboratory, Whiu Oak, J\f d.) F10 .

4.9 Schlicren photograph of a normal shock at the mouth of a duct . ( / 22.97°). 1 1 6. Static temperature increases, 7 2 > 7 1. 7. Total energy or total temperature is constant, h,, = h,,, or Tr. = T,, . 8. Entropy increases, S2 > S1. 9. Density increases, P2 > Pt· 10. The normal components of the Mncl1 numbers before and after the shock wave, Af :c and M 11 respectively, obey all the relat.011s of tl1e normal shock equations of the previous section. 11. When w' = 0 (tl1c lo,ver limit cnsc for oblique shocks) or is negative, a' = µ = arc sin (1/ 111 ), and the flo,v expands isenlropically producing l\Inch lines or expansion \Vavcs rather than shock \Vnves (sec next section) . 12. I•lo\V areu. decreases, A2/ A1 = sin (a' - w') / sin a' .

Constant-Area Ducts and Thermodynamics of Shock Waves

I

91

The method for solving an oblique shock problem involves only a few more steps than the normal shock problem, a nd the method is outlined belo\v. 1. Obtain values of M 1, w', and a'. Of the three variables M 1, w', and a', t\vo are known from given conditions of the problem. Find the third from tables (Ref. 1 or 3 cited above), or from Fig. 4.18. 2. Obtain M :a: and M 11 • From Fig. 4.17, M :a: = M 1 sin a'. M 11 is found from the normal shock equations, Table 4.3, Fig. 4.13, or from appropriate tables (Ref. 1 or 3). 3. Obtain M2. From Fig. 4.17, M2 = M 11/sin (a' - w'). 4. Obtain the gas property ratios p2/ p1, T2/ T1, V2/ V1, v2/v1, Pi. / p,,, Pi. / p1, T,. / T, P2/ Pi, and (S2 - S1) / cp across the shock. Replace subscript 2 'vith y and subscript 1 with x; then use the normal shock relations \vith }r[ :a: found in step 2 above; obtain :values from Table 4.3, Fig. 4.13, or from appropriate tables from Ref. 1 or 3.

Example A fuselage-type inlet supersonic diffuser is designed and operated according to Fig. 4.19. M 1 = 1.5 p 1= 10 psia T1 = 450° R

Flo. 4.19

Find the Mach number (M2), pressure oblique shock.

(~)

and temperature (T2) after the

Solution From Fig. 4.18 or the cited tables in Ref. 1 or 3, we find for w' = i deg, and Mi = 1.50 that a' = 50.88 deg. From step 2, !vi :a: = !111 sin a' = 1.5 sin 50.88 = 1.162. From Fig. 4.13orappropriatetables, !If.., = 0.868 ; ])2/ p 1 = 1.403; and T'!/ T1 = 1.103. Thus, JI,{ 2 = it{ u/ sin (a' - w') = 1.25; p,, = 1.403 Pi = 1-1.03 psia; and T2 = 1.103 T1 = 497°R. From the solution of the exa1nplc problen1, it is seen tl1nt the oblique shock problem is only one step further than the norn1al shock problen1. In fuct, most of the ans"·ers to oblique shock problems are calculated on t11e basis of the norn1al shock equations.

4.6

Prandtl-Meyer Expansion

As demonstrated in the previous sectio11, tlie lo'''er linut case for the oblique shock occurs \Vl1en tl1e \Vedge angle w' is zero. For '''edge a1gles that are negati,re, the superso11ic flo\v is urou11d n. corner, (rather tl1a11 into a, cor11er \\·hich creates compression shocks). 1, his produces n.11 expnnsiort process that is approximately isentropic. An idealized flo\v model of tl1is supersonic expansion can be analyzed by the so-called

92

I

Jet Propulsion

Prandtl-Meyer expansion theory, which approximates experimental results. This theory is developed by considering a uniform t'vo-dimensional supersonic stream fto\ving around a corner of infinitesimal deflection d11 as shown in Fig. 4 .20. A The flo\v is from left to right and at the corner, Moch wove the flo,v rapidly expands and changes direction v through an angle d11. The line 0-A originati11g Streamline from the corner is known by several names : Mach µ wave, Mach line, or expansion wave. It can be regarded as an infinitesimally weak oblique shock 0 dv wave. 1.,he angle that this 'vave makes 'vith the incoming flow is called the lVIach angle, µ, and is given by the well-known relation . 1 µ=arc sin M

Flo. 4.20 Supersonic expansion around a corner.

(4.46)

Applying the basic engineering laws to the Mach \vave for the special conditions of a frictionless adiabatic process, and taking as the limit JI = 0 for l\f = 1 when integrating the relation J1 = f (M) yields JI

=

+ 1 tan- I

'Y 'Y -

1

v M + 1

'Y 'Y 1

2 -

1

- tan- 1

v l\!

2 -

1

(4.47)

which is known as the Prandtl-Meyer relation. For the limit conditions specified above, ~he Prandtl-Meyer angle JI is literally the angle through which a supersonic stream is turned to expand from M = 1 to M > 1. The deflection angle, .6J1, for a flo,v "·hich expands from M 1 to M2, is given by the difference between J11 and J12• The PrandtlMeyer relation is derived in a number of \vell-kno,vn texts (see Ref. 4) . Values of J1 are tabulated together \Vi th Mach angle µ as a function of 1lf in the isentropic flow tables of Appendix B . The Mach angle µ may also be obtained from Fig. 4.18 for the case where the \vedge angle is zero. When a supersonic stream flows A Moch around a finite corner, a series of Mach lines or expansion waves emanate from Streamline the corner in a fan-like shape as shown in Fig. 4 .21. The streamlines are parallel to the walls , both ahead of the initial Mach line 0- A and aft of the last Mach line 0- B . 0 In the region AOB, where there are many expansion waves and where the flow expands isentropically, the streamlines are curved. The angle made by the initial Mach line is the Mach angle µ 1 F10 . 4.21 Prandtl-~l cj·cr expansion around a corner. corresponding to the incoming Mach number M 1• The angle made by th~ final Mach li11e is tl1e l\Iach n11gle µ,, corresponding to the final Mach number M 2. The flo,v is deflected through tl1e a11gle .6J1.

I

Constant-Area Ducts and Thermodynamics of Shock Waves

93

The shado,vgraph shown in Fig. 4.15(b) should be examined carefully for several good examples of actual expansion waves emanating from the body trailing edges. Exampl,e

Assume that flow expands according to the Prandtl-Meyer relation and that in Fig. 4.21 Mi = 2.6, ~v = 10 deg. For these conditions, find the angles of the first and final expansion waves and the final Mach number M 2.

Solution From Appendix C or from Fig. 4.lS. for w' = 0, or from Eq. (4.46), the Mach angle µi for M1 = 2.6 is 22.62 deg. The Prandtl-Meyer angle v1 for Mi = 2.6 (from Appendix C) is 41.4 deg. Since the flow deflection angle is 10 deg, v2 for M2 is 51.4 deg. Entering Appendix C with this value gives a value of 3.09 for M 2. The Mach angle µ.2 for this Mach number, the angle of the final expansion \Vave, is found to be 18.88 deg. The maximum expansion for a given flow is obtained 'vhen the final Mach number is infinite. Substituting M = oo into Eq. (4.47), the limiting expansion angle is obtained. For 'Y = 1.4 v =

(V6 -

(4.48)

1) ; = 130.5 deg

It may be seen that the Prandtl-Meyer theory indicates that a supersonic flow can be turned and accelerated around a sharp corner of a very large angle without separation. A further check of theory shows, however, that the maximum deflection angle, v = 130.5 deg, occurs 'vith a final pressure p = 0; under these conditions the theory is no longer valid since the fluid is not a continuum. The actual deflection angles through which a flow can be deflected will be less than the theory predicts. Figure 4.22 presents the variation of the maximum theoretical Prandtl-i\1eyer angle Vmaz with initial Mach number for 'Y = 1.4. 140 130 120 110 100

"' 90 Q) ~ 80 OI

~ 70 >
= 1.0 - 0.075(M o - 1) 1 ' 35, from 1.0 to 5.0 Mach n11mber Pi. Pi.cw.> = M~ 800 + , above 5.0 Mach number p,.

1.0

935

(5.15) (5.16)

The p ressure recovery prescribed by the above equations 2 0. 9 1---T-~~oe---+-----t----1 can be regarded as good optim11m Q. ........ A specific diffuser may ~ 0.8 1----~--+---~--P..,.,....------t----; values. attain recoveries slightly higher MIL-E-50088 than that specified in the equations, but is more likely to llit.\•e ~ 0.6 1------+-~----4--~--t-~---l a lo\ver recovery. Figure 5.14 pre> 0 v sents a. plot of Eq. (5.13), the ~ 0.5 1-------1--~,.-----4----~----1 Norma AJA standard recovery, and Eqs. a shock curve °' 0 .4 1-----+---~+-----t-~---i (5.14), (5.15), and (5.16), the military standard reco\rery. Also 0.3 '-------'-----~------'----"---' sho,,·n is a curve giving the nor4 2 1 3 Moch number, M 0 mal sl1ock losses. F10. 5.14 Variation of Aircraft Industries Association nnd Fron1 l;-ig. 5.14 it n1ay be seen MIL-E-5008B standard recovery factors nnd normal shock that 11ormal shocks are totally recovery factor \vith Mnch number. \111satisfactory for operation above Mach number 1.5. This fact has given sig11ificant impetus to inlet development, as shown by the top curve labeled MIL-E-5008B. The recoveries shown by this curve arc nO\V the standard on which ne\v military turbi11e engine specification performance is based.

Diffuser and Nozzle Flow with Friction

5.5

I

111

Mass-Flow Ratio or Area Ratio

The criterion of diffuser performance discussed thus far has dealt solely with the ram recovery factor. This factor is importa11t, but does not, in itself, dictate the over-all performance of a diffuser. In addition to having a high ram recovery, a good diffuser must have air-handling characteristics \vhicl1 are matched with the engine, as well as lo\v drag and good fto\v stability. For example, if a given installation had an 77,. value of 0.95 for the air \vhich it handled but supplied only 80% of the air required by the engine, \Ve could not call this a good diffuser. The importance of the air-flow matching characteristics can be sho,vn from the area considerations of Fig. 5.15, which is a sketch of a typical subsonic diffuser and a typical ramp-type supersonic diffuser. A 0 =free stream flow area A 1 = flow area at inlet A c= capture area

Ao

,.I"'----~~;;:::; I

0

2

r ------ - - _;; --, A le

Ao

Station

0

O~,e>:.

Station

A,

1

S

. tot1on 2 (comp. inlet)

a' \

~-RomP

F10.

5.15 Typical subsonic and supersonic diffusers.

For a given set of operating flight conditions, the air-flow requirements are fi..'\:ed by the pumping characteristics of the engine. No\v for either diffuser, if A1 is too small to handle the air, the engine must ''suck in'' the lacking amount of air resulting in a decreased ram recovery. If Ai is too large, the diffuser " rill supply more air than the engine can use resulting in excess drag because \Ve must either by-p~ the e.xcess air around the engine or ''spill'' it back out of the inlet. Too much air or too little air is detrimental to diffuser performance. In order to evaluate the air-fio\v matching characteristics of the diffuser and engine, a parameter called the 1nass-flow ratio is defined as (mass-fto"· ratio)

(5.17)

It is pointed out that this ratio is n. pure definition. Eq. (5.17) can be put into a more convenient form by the use of the continuity eqt1atio11 \\rhich, for the diffuser, is ..lo = ..11 = Po VoAo = P1 V1A1

(5.18)

Thus, Eq. (5.17) ca11 be expressed as

Ao - (nren. ratio) JI - Ai Jl1

(5.19)

Equation (5. 19) sho\VS that tl1e defi11ed term, t11c n1nss-flO\\' ratio, is actuall)r the ratio of t11e fio\v areas at station 0 and 1.

- ·112

I

Jet Propulsion

The magnitude of the mass-flo\v ratio or area ratio varies with the operating condition and the type of flow into the diffuser. For a subsonic diffuser \Vhich operates with free stream compression (most typical condition in flight), the mass-flow ratio is less than 1.0. For supersonic flow with a normal shock at the inlet, Ao = Ai; thus, the mass-flow ratio is unity. For supersonic flow with an oblique shock ahead of the lip (see :F'ig. 5.15), Ao > Ai; thus the mass-flow ratio is greater than one. Usually, for a supersonic diffuser, the inlet area used to define mass-flow ratio is based on the capture area A1. (sec Fig. 5.15) \vhich is larger than Ai and A o; thus, for supersonic flow with oblique shock, \vhen based on the capture area, the morufied area ratio Ao/ Ai. and the corresponding mass-flow ratios are equal to or less than one. The advantage of using this area is that mass-flo\v ratio greater than unity cannot exist. \:Vhen based on the inlet capture area A 1., the mass-fio,v ratio .Li/ .L is literally the ratio of the air mass actually swallowed to the maximum amount that could be S\vallowed \vithin the capture area; therefore, the mass-flow ratio can be thought of as JI / .Lmax· The question arises as to how to use the mass-fio'1.' ratio in an applied problem. Suppose that for a given set of opera.ting conditions Po, Af0 , and V 0 , we know the massflow requirements of the engine .Leos. (from engine specifications). For this given set of conditions, we must design the engine inlet area to be of proper size, that is, we want neither too much air nor too little air to come into the diffuser inlet. The problem is not too serious in subsonic diffusers, because, if the engine needs more air than it is receiving, it can send a ''pressure signal'' to the free stream and the stream tube "ill adjust itself to supply the proper amount. Also, it can signal for less air. These conditions are shown belo\v in Fig. 5.16.

Initial condition

Engine needs

more F10.

•

0 1r

Engine needs

less o ir

5.16 Subsonic dilTuser with soveral demands for inlet air.

In supersonic flo\\', when oblique shocks nre formed , the condition is more serious because the ''pressure signals'' from the engine which are sent to advise the free stream to give more or less air cannot get to the free stream, or even to the inlet section, since supersonic velocities exist \Vithin the inlet. Suppose we have an operating condition as shown in Fig. 5.17. Let us e.xamine qualit.ati,·ely what happens to the inlet fiO\\' characteristics \\·hen the engine demands a change in ma.ss-flo\\· rate. If the en/ Normal gine demands n1ore air than sho"·n in the staA1 shock bilized condition of Fig. 5.17, it \\-ill decrease the pressure behind the normal shock and actually make that portion of the diffuser beF10. 5.17 Supersonic ramp-typo diffuser. hind the normal shock act as an e.~ding supersonic nozzle. More shocks \vill occur ''rith a consequent loss of total and static pressure. If the engine demands less air, the pressure behind the normal shock will

Diffuser and Nozzle Flow with Friction

I

113

increase and become greater than the normal shock can support for the given Mi. Therefore, the normal shock ,vill move forward ahead of the inlet, making the system compress air 'vhich goes around the inlet (spillage) by means of the normal shock. The normal shock may even move to the beginning of the ramp and detach the oblique shock. In either event, the inlet system 'vill be penalized by having more drag due to spillage. For this reason we always want the normal shock to be swallowed otherwise, the normal shock compresses air wliich flows outside the propulsive system. Because of the consequences of changed air-flo,v demand on a supersonic diffuser, we are obliged to design our inlet system to do one of t'vo things, namely: (1) Use variable area inlet ; (2) Use fixed area inlet with Ai large enough to handle the maxim11m air-flow requirements of the engine 'vith some system for handling the excess air intake for all other conditions. For normal values of pressure recovery, the required inlet flow area decreases \vith an increase in flight Mach number; therefore, for fixed inlet areas, Ai is designed for low values of Mo (usually about 0.85) . Operation at ~1ach numbers above 0.85 requires a system to handle the excess air. Both of the above schemes are workable ones. The former is efficient but difficult to achieve because of the inherent complexities and structural problems of variable area designs; the latter is less efficient than the former because we must pay for handling the excess air. Ho,vever, it does have the advantage of being simpler. A comparison of various means of handling the excess air is made later. As mentioned before in our original statement of the problem, we want to determine the proper inlet area for given values of Po, Ptf0 , Vo, and .,Lenc· From the continuity equation, we can write ..Lens = ..L1 =..Lo= PiAiV1 = PGAoVo

(5.20)

Since Pt, V1, and Ao are difficult to measure, we must exp~ Eq. (5.20) in a more practical form, using the area ratio as (5.21) Equation (5.21) provides the real application of the definition of the mass-flow or area ratio in that the mass-flo,v of the engine can be e.~pres.5ed in temlS of free stream conditions, the inlet area (both of 'vhich are easily n1ea.sured), and the area ratio (which is easily calculated). Fron1 the geon1etry of the oblique shock figures (see Fig. 5.15) the area ratio can be expressed as sin (a') sin (a' - w')

Thus, the area ratio and mass-flo,,• ratio cnn be angle a.', and the ramp or ,,·edge angle w'.

~'\lculated

(5.22)

from values of the shock

The method of finding the proper inlet area o\•er a range of flight ~Inch numbers can be illustrated from the sketcl1cs in Fig. 5. 18. Sketeh I presents the relntionship bet,vee11 the shock angle a', the ,,·edge angle w', and the ~Iach nun1ber ~11 0 • Figure -l. 18 prese11ts a full-scale plot of the \rariation of a', with ~\f o for various ' 'nlt1es of w'. Fron1 tl1c dnt ~1 of Sketch I and Eq. (5.22), Sketch II can be plotted. Sketch III is obt:1i11ed f ron1 engine specification data. ketch I'\' presents the required ans"·ers, the inlet area required versus flight ~Iach number. The

I

114

Jet Propulsion

11

(I, I

F10 .

5.18

111

IV

Mo

Mo

Curves required to obtain inlet area variation with Jl.f O·

inlet area is calculated from the equation sho\vn on Sketch IV. If a variable inlet a rea design is used, the area should vary according to Sketch IV, and if a fixed inlet area is used, A 1 should be large enough to handle the inlet flow at Mo ~ 0 .85. From the discussion presented in the above section, it is evident that the first two basic requirements of a good supersonic cliffu.ser are : (1) A high recovery factor; (2) The proper mass-flow or area ratio.

Example The supersonic fuselage-type inlet shown in Fig. 4 .19 is operated at the conditions sho\vn there. In addition to the Cha pter 4 data, the following applies: Gen, = 100 lb p er sec; a normal shock exists at the diffuser entrance at Section A1, and the total pressure recovery factor in the section aft of the norma l shock to the diffuser exit is pi. I p,, = 0.97. A sketch of the diffuser with the appropriate data is shown in Fig. 5.19. vo -1562 ft/ sec p 0 = 0.001 865 slugs/ft 3 M 0 = l.S p 0 - 10 psio T0"" 450° R

0

1 or x F10.

2

y

5.19

Calculate A1 required for no-excess flow, Pt ., p, ,, M 11, Pi., and find the over-all total pressure recovery factor Pi. / p, •.

Solution From the solution of the example in Chapter 4 (Section 4.5), \Ve have the follo\ving

data: a'

= 50.88 ° M1 = M z =

1.25

Pi

= 14.03 psia T1 = 497 °R

The area ratio of the diffuser is calculated from Eq. (5.21) to give

Ao sin a' - = - • -( -I A 1 SID a - w' )

sin 50.88° sin 43.88°

=

1 12 ·

From Eq. (5.20)

A = __1 _ 1

oPVo

en""it _

(100) l 952 (32.2) (.001865) (1562) (1.12) - · ft

I

Diffuser and Nozzle Flow with Friction

115

The free stream total pressure is calculated from

Pt. = Po [ 1

+ -y -2

1

2J'Y y- 1 =

Mo

10 X 3.67

=

36.7 psia

and p, , is

Pt,

=

-y-1 2] ' Y Pi [ 1 + M1 y - 1 2

=

14.03 X 2.59

=

36.4 psia

M 11 and Pt./ Pc, = Pc./ Pc. are obtained from normal shock tables or from Fig. 4.13 for Mr = 1.25 to give

M 11 thus

=

Pt. = 0.987 Pt ,

0.813

Pc. = (0.987) X (p,,) = 35.9 psia

Since

p '• is specified as 0.97

pt,,

p ,, is given by Pt. = 0.97 X 35.9

=

34.8 psia

Therefore, the over-all total pressure recovery factor is p '· = 34.8 = 0 95 Pt. 36.7 .

Note that this value corresponds to the MIL-E-5008B recovery factor of about 0.97 (Fig. 5.14) for the same free stream l\1ach number.

5.6

Ram Drag of Supersonic Inlets

In Chapter 2 it was sho,vn that the ram drag of an air-breathing engine is equal to the free stream momentum flux of tl1e enteri11g air; that is, in equation form

GVo ram drag = Fn = g

[See Eq. (2.5))

This definition is generally used to define the net thrust of an engine because of the many differ---• " 'nlo ---ent inlets which are possible on ~~=--C:::..==3~D~r===---:..J different aircraft. In t he sketch in I I I I I Fig. 5.20, the forces acting on the 0 1 e aircraft are sho\vn. From Fig. 5.20 may be seen fc= ~ Ve+ A.(p. - po) that thrust F11 actually acting 011 F11 -=~ v0 the airplane is not equal to the net thrust F N defined i11 Chapter 2. FN = Fc- F11 The difTerc11cc bet,veen these Fp- FN D0 = 'r?lo+D thrusts is the momc11ium flux beF10. 5.20 !\ l a.jor f OT('CS nnti their rt'lations. t\VeCI1 statiOllS Q a11d 1. rfhis differe11ce is called additive drag a11d is defi11cd by the f ollo,,;11g cqt1atio11 :

..

---

-0:'t1

Q

Fc

11 6

I

Jet Propulsion

D. = additive drag = G

(p1 - Po) dA - -

g

Vo= FN - FP =Fi - Fn

(5.23)

From Fig. 5.20, it is evident that D0 \Vill be zero when the e11tcring stream tube is a cylinder (11 / .,II = 1) because axial pressure forces have no surface upon \vhich to act. At the condition of zero additive drag it can thus be seen that no air \Vithin the boundaries of the projected capture area ratio is spilled around the sides of the inlet. It is also evident that for subsonic ftight \Vhen the shock \Vave does not exist, the relative axial pressure acting on the entering stream tube will be different from that shown; in fact, for subsonic fio,v, the net axial pressure force acting on the entering stream tube is zero, making the additive drag zero, except for certain unique cases where very thin inlet lips are employed and the inlet lip suction forces are virtually nonexistent. (See Ref. 11.) Referring again to Fig. 5.20, it should be noted that, from a bookkeeping standpoint, the choice of subtracting additive drag from the net thrust FN, or adding it to the airplane drag D, is purely an arbitrary one. F rom an engineering standpoint, however, additive drag is more logically tied to the engine, because it varies with engine mass-flow requirements. The actual evaluation of additive drag is somewhat difficult because, in general, the location and slope of the entering stream tu be is not precisely known. Furthermore, even when these quantities are known, the determination of the pressure and friction forces that are present on the bounding surfaces of the stream tube are difficult and sometimes impossible to calculate. To illustrate this problem, Fig. 5.21 shows the equation prescribing the value of additive drag for a single conical inlet operating supersonically with a zero angle of attack. In Fig. 5.21, Xis the flow deflection angle, p,. 11 is the effective pressure of the centerbody on the fluid for\vard of station 1, A, is the cross sectional area of the centerbody at station 1, F1• is the axial force of the conical spike on II A, the fluid due to friction, and the other II terms are as used before. Even for this simple shaped inlet operating in the prescribed simple mode, the evalua- 0 1 tion of such terms as p,. 11 and F 1 • become difficult. For these reasons, the In general, Do= F,- Fo; current practice of evaluating additive drag is predominantly by means of model wind-tunnel tests. To facilitate this procedure, an additive drag coeffiF10. 5.21 Additive drng evaluation. cient has been defined as follows:

-

Cd.

_ -

Do = _!_ A1qo A1

(p -qo Po dA

(5.24)

Diffuser and Nozzle Flow with Friction

I

117

The additive drag coefficient is a function of inlet geometry, mode of operation (subcritical or supercritical), free stream Mach number, and i11let mass-flo'v ratio; therefore, the model tests must prescribe these quantities. l•'igure 5.22 sho\vs the relative variation of Cd. \vi th these quantities. 2.4 0

:-g 0.4

·-

~ 1.6

'

0

v

O>

--o...__ _.__ _ _.__ --_ ---

-u 0.8 G)

-:g 0.4

- - - - - shock

..L.,__ _-~-=-:..:a,;

0

0 .2

0.6

0.4

0.8

1.2

e

Normal shock

•

Open nose no rmal shock

G)

-'

50°

oo

G)

Mixed shock

-uG) 0 • 8

M0 =

1.0

·->

~M 0 = 1.35

Figure 5.29 suggests that \Vhen the free stream M ach number, f.lo, is equal t.o or greater than the inlet Mach number, iV/ 1, p,, / p,. is one and there is no loss because the area ratio Ao/ A1 is less than 1.0. Under these conditions, any separation takes place externally, and the ensuing loss is the additive drag discussed previously. '\\'hen the inlet Mach number Mi is equal to 1.0, no further increase in mass Bo"· is po...~ible. Decreasing the outlet pressure results in supersonic fto\v in the divergent portion of the diffuser 'vi th resultant additional pressure losses. This explains the vertical drop-off of the curves. Figure 5.29 presents the sharp lip inlet losses for tl1e inlet only. The figure indicates that these losses ca11 be appreciable for a choked i11let (213 statically). Additional losses will occur in the diffuser do,v11stream of tl1e inlet, and the methods presented earlier for d etermining subsonic diffuser losses ca11 be used for estimating thes e.

5.9

Some Typical Inlet Shock Systems

In addition to the conical and t'vo-din1ensior1nl inlets ,,·}1ich l1n, re been d iscussed, there are a la rge number of feusible sl1ock S.)'Ste111s \vl1icl1 cit11 be t1sed for con1patibility 'vi th aircraft configuration requireme11ts. Of pa. rticula.r note are tl1e as.)·111n1etric t.)·pes, good exam ples of \vhich are sho,vn i11 Ji'igs. 5.30, 5.31, n.11d 5.32. Figure 5.30 is a photograph of the double shock inlet 011 tl1e LTV FS Crusader I ; it n1a.)r be seen that this inlet takes advantage of t11c co1npressio11 field produced by the conical compression shock from the nose cone of the airpla11e.

• •

'

'\ •

F10. 5.30 Photograph of Navy F8 Crusader I airplane sho\ving double-sho('k external compression supersonic inlet. (Courtesy: Ling-Tern.co-Vought, Inc. )

f///!!t •

2

F10. 5.31

Photograph of Navy F8 Crusader III airplane sho,ving three-shock internal-external compression supersonic inlet. (Courtesy Ling-Te111co-l'ought, Inc.)

.,

• •

•

F1a . 5.32 . Phcitograpb of Xav)· )IQ:\I-15.-\ drone sh~" t~g t \\"O---

•

-1

I

'In 0.90 I

......-

Q)

'

I I

126

I

;.

0

~

_

129

I

11- -- -t I

•

0

...

__._.

I

I

,/

I

0 .8

1.0

a.. 1.24

-

'fn= 0.80 i

1 22

1.20 0

0 .2

0.4

0 .6

1.2

Pressure ratio

5.37 Variation of polytropic exponent versus pressure ratio for several values of Tin·

F10.

Nozzle Friction Parameters

In our analysis of isentropic nozzle flow presented in Chapter 3, we developed certain parameters which enabled us to calculate readily the variation of gas properties along a nozzle for specified pressure ratios or Mach numbers. Thus, it is also desirable at this point to develop the same parameters for the case which includes friction. The friction parameters, of course, \vill be more useful because in all actual problems, "·e deal with a nozzle that has friction and certain losses associated 'vith its fio,,r process. Again, the asterisk(*) will designate conditions \Vhere the Mach number is equal to unity. We shall assume adiabatic steady fl.o\v for this analysis; thus, the follo,ving relations are equal to one. G T, (5.39) G* = T,* = l Since, for the assumed conditions, the total temperature remains const.nnt in the nozzle, we can write from Eq. (5. 1)

T, = Ti

I+

'Y -

2

1

2

M1

(5.40)

where subscripts 1 and 2 correspond to any t,,vo sections in the nozzle. Thus, the temperature parameter can be \Vriticn as 1

1

+ 'Y ; + ')' ;

1

llf~ (5.41)

1 JI{:

I

130

Jet Propulsion

or T T*

+1

'Y

2 ( 1 + '>' ;

1

(5.42)

M2)

It is seen that the temperature parameter is the same as its counterpart for isentropic flow and for Fanno line conditions. Furthermore, it is noted that the temperature parameter is independent of the nozzle friction factor. This is true because the friction 'vhich is present in the nozzle appears as heat energy which reheats the gas, thus increasing the temperature. This \Vas pointed out previously and is mentioned here again to sho\v that the temperature ratios are independent of the friction as long as the process is adiabatic. The pressure parameter is obtained by replacing the isentropic temperature ratio of Eq. (5.28) with the isentropic pressure ratio and solving the result for pressure ratio in terms of sections 1 and 2 to give

T2 -

-r-1 'Y

Ti,

-

1

+ 1Jn (5.43)

= -'-----1Jn

When the temperature ratio in Eq. (5.43) is replaced by its function of Mach number and the pressures are related to the critical section, we have the pressure parameter ____1_ _ _ -

P - = p*

'Y

+1

1

77;;

1

+

'Y -

2

1) M2

1

+ 1Jn

'Y

-r-1

2

2

(5.44)

1Jn

Here we note that the pressure parameter is a function of the nozzle efficiency as well as Mach number. The volume parameter and density parameters are obtained from the gas law as f ollo" s: v p* T p* - = - = (5.45) v* p T* p 1

It can be seen that \vhen Eqs. (5.42) and (5.44) are substituted into the above relation, we can express the volume-density parameters as a function of l\1ach number and nozzle efficiency. The velocity parameter is most easily derived from the definition of l\1ach number, which is V = M g-y RT. The velocity parameter can be \vritten as

v

v

M

T T*

and from Eq. (5.42) \Ve have the end result for the velocity parameter

v

V*

=

M

'Y

+1

2(1 +'Y;

1

(5.46)

Jlt 2)

It should be noted here that the velocity parameter is independent of tl1e frictio11 in a nozzle process \vhen expressed as a function of Macl1 number.

Diffuser and Nozzle Flow with Friction

I

131

The area ratio can be expressed as follo,vs from the continuity equation:

A A*

T

(5.47)

r•

Now, when Eqs. (5.42), (5.44), and (5.46) are substituted into the definition of area ratio and the result is simplified, \Ve obtain

., -

1

A A*=

~·hich

1

'I,.

2

M

')' + 1

-r- 1

T]n

1 - -- - - - - 1

1 + "Y

-

2

1 M2

+ T]n

(5.48)

expresses the area parameter as a function of Mach number and nozzle effi-

• c1ency. The momentum thrust parameter can be given from Eq. (3.27) as

Fm/ A Fm*/A •

= 1!_

M2

(5.49)

p*

Thus, when Eq. (5.44) is substituted for p/ p* in the above expression, the moment11m thrust parameter is a function of Mach number and nozzle efficiency. In Chapter 3 it \Vas sho\vn that the critical pressure ratio of a nozzle with isentropic flow is

.,

p* p,,

2

"Y

+1

(isentropic flo,v)

'Y- 1

(5.50)

In like manner , we can derive the function which expresses the critical pres&Ire ratio for nozzle flo'v 'vith friction . When Eq. (5.44) is \vritten for the total pres.5ure at entrance section (or for the static pressure 'vhere the velocity is zero), ,,.e obtain (remembering that /.f for total conditions is zero)

p*

-

p,,

-

-

-

2

- 'Y

+

,

1

-y-1 11,.

(5.51)

l_

The above expression gives us the~ritical pressure rati9tof a flo,,· process ''hich has friction. It is evident that this expression reduces to Eq. (5.50) ,,·he11 the 11ozzle efficiency is equal to unity. A comparison of Eqs. (5.50) a11d (5.5 1) also sho,,·s us that a greater pressure drop is re c1uired 'vhen \VC have friction i11 order to acl1ie,·e a ~Iach number of one. Since in most nozzle problems \Ve meast1re the total prcsst1re at the e11trance section and the static at some otl1er sectio11 1 it is esse11tinl to deri,·e a11 expression which relates these t'vo pressures. The nozzle pressure ratio can be \vritte11 ns

p* )( P p '· pi When Eqs. (5.44) a11d (5.51) a re substituted, sin1plificatio11 yields

(5.52)

.. I

132

Jet Propulsion

.., I

1+-y2

-

1

y-1

- 1 + .,,"

M2

(5.53)

The above expression relates the nozzle pressure ratio to the Mach number and the nozzle efficiency. In a practical problem ,ve usually relate the nozzle exit pressure to the total inlet pressure in order that ,ve may calculate the 11ozzle exit Mach number. All \Ve need to kno'v in order to find the exit Mach number of a 11ozzle is the pressure ratio and an approximate value of the nozzle efficiency. This is a typical problem which must be solved on such 'vork as evaluating the thrust of a jet propulsion nozzle. Figure 5.38 presents a plot of Eq. (5.53) for 'Y = 1.33 and several values of.,," in order that l\1ach numbers at various sections might readily be determined for any value of pressure ratio. With the aid of this plot, one can obtain the Mach number at any section in the nozzle for a specified pressure ratio and nozzle efficiency. If 'Y for a particular problem does not correspond to 1.33, then Eq. (5.53) must be used. Once the Mach number for a given section has been found, the values of the other properties can be determined with the aid of the various parameters for nozzle flow \vith friction, summarized in Table 5.1. Figures 5.38 through 5.41 are given to replace the equations of Table 5.1. The primary objective here is t'vofold: (1) to reduce the tedious calculations involved in 9 8 7

~---ir-----r---r--.--~-....-----.

1.0

-

09

-

0.8

- 0.7

"'

~

a.

.g0 0.6

-

..."'"'

I

l

6

I I

5

I I

I I

-"E 2

-

0 ....

./

0

I/

Q..

Cl

0 .3

1

•

v

0 .2 TJ n = 1.00~r\

0.1

;'

~ I

I

I

--.....

~ -I

-

1.5

2.0

·-

---r •

oL___L__J___L_J____:::t:::~

1.0

- v-v·

T

0.90 = 0 80 -l---

0 .5

I

I

=

0

-

I I

Cl

-

z

I

...

a. 0.4

0

I

0

Cl

N N

I

>

0.5

-

I-

~3

....

":>

,_

--

I

4

-

- -

- -

2.5

3.0

3.5

0

0

0 .5

15

1.0

2.0

25

30

3.5

Moch number

Moch number

5.30 Variation of nozzle temperature and

Fro. 5.38 Variation of Mach number with

F10.

nozzle pressure ratio for several values of nozzle efficiency for 'Y = 1.33.

vclocit)• pnramctcr

''°itl1 ~Inch nun1lx-r for 'Y = 1.33.

I

Diffuser and Nozzle Flow with Friction

T ABLE

I

133

5.1 Summary of Nozzle Friction Parameters

G = T, = 1

G•

A A•

T T•

p•

-p

v• v

= -

1

M

Tc•

'Y + 1 1 2( 1 + 'Y;

-

T

"·

2

'Y+ 1

Af2)

1

')'-1

fJn

1 1+'Y ·- 1 M2 2

Fm/ A _ _P 2 M where Fm•/ A • Fm*/ A* - p•

=

'YP•

..,

-')'p

-

=

.,.

1

--

.., 1 ----- - 1

p

1+'Y - 1 Mt 2

--

v

p* -=-=

+ '1"

')' - 1

1

- 1+

f]n

I

134

Jet Propulsion

the equations when 'Y = 1.33, and (2) to sho\v the relative variation of the gas prop. erties 'vith respect to Mach number. The most typical nozzle problem which ~ill arise can be illustrated as f ollo\vs. Suppose the pressure ratio across a nozzle is specified and the nozzle exit area and adiabatic efficie11cy are also given. With these data, we can evaluate the exit Mach number from Fig. 5.38. Kno,ving the inlet conditions, we can evaluate the exit velocity and exit temperat ure from Fig. 5.39, and the exit thl1.l!t from Fig. 5.41. The exit density or specific volume can be calculated from the exit pressure and temperature. The \Veight-flow rate can be calculated from Eq. (5.34) or, if the flow coefficient Cd is unkno\vn, then the \veight-flow rate can be calculated from the continuity equation, using the actual conditions at the nozzle exit.

Examp'le A De Laval nozzle expands air with a pressure ratio of 0.3 for inlet conditions of 50 psia and 2000°R. If the throat area is 100 sq in. and 1ln = 0.90, find the exit Mach number, temperature, velocity, area, and thrust. Assume complete expansion and 'Y = 1.33.

Solution From Fig. 5.38, we read the exit Mach number to be 1.37. From Fig. 5.39, we read

T,.

-

T*

'Y

+1 2

=

= 0.89

1.165

and

~;

= 1.295

Thus

T*

T.

2000/ 1.165 = 1717 °R and The velocity at the throat is given by =

V*

=

v g-yRT*

= 1717 X 0.89 = 1530°R

= 1980 ft/sec

Thus

V. = V* F rom Fig. 5.40, we have (A,/ A *)

=

A, = A*

= 2560 ft/sec

1.16; thus

A, A*

=

116 sq in

From Figs. 5.38 and 5.41, we have p* Pt.

= 0.508

and

(F'm/A).

(Fm*/A*)

= 1.11

From Table 5.1

Fm• I A* = -yp* = (1.33)(0.508)(50)

=

33.8 psi

Thus

(Fm/ A) (F' */ A*) (F1 m ) • = (Fm*/A *) X m A. = 4360 lb This problem sho,vs ho'v the special nozzle charts can be used to solve rapidl.)' the nozzle problems \vhich involve friction.

I

Diffuser and Nozzle Flow with Friction 9 8

-

I

....8 ..

II '-- o ·/ // '

I

-

5 j

4

'

// I:;-.~

-

-

L/ 1'

I J I I I I

• 3

< .......


l

Po

Flo. 5.44

Ratio of gross thrusts for De Laval and converging nozzles.

F10.

5.45

De Laval nozzle with line jet.

~traight

From Fig. 5.44 it may be seen that the gross thrust lo...'5 from \1sing a con,·erging nozzle is less than 5% belo\v a pressure ratio of 6. Abo,·e that pres.5Ure ratio, the De Laval nozzle does offer an increasing advantage. For turbojet and ramjet engines, the gain is much greater than that sl10,,·11 in Fig. 5.44 since, for these engines. the net thrust FN is the important parameter. A De Laval nozzle ca11 be designed by t\vo-din1e11sio11al theor.)· to gi,·e a strsightline discharge jet as sho,vn in Ji'ig. 5.45 ; ho,,·evcr, De La,·al 11ozzle design does not al,vays attempt to 1naintai11 a straight-li11c jet, but has a slight di,·ergence to the exit jet as sho,vn in Fig. 5.4G.

Shock -

F10.

5.46 Typical De Laval nozzle operating '"itl1 c..xhnust c..xpansion).

pl'('s.5Ure

Cones

less than c..~t pre."5ure (under-

138

I

Jet Propulsion

The angle a is tl1e half angle of divergence, and in most cases it is less ~ban 15 deg. For normal operation, the gases leaving a De Laval nozzle are supersonic, and thus the characteristics of the free jet \vill not differ greatly from that sho,vn in Fig. 5.43. In a rocket-motor free jet, \vhere the exit velocities are greater than 5000 ft per sec, it is not uncommon to have as many as a dozen visible sl1ock cones in the jet. The velocity of tl1e exhaust gases is decreased appreciably tl1rough each shock cone and also through the mixing action of the atmosphere. In a jct propulsion exhaust nozzle, t he losses due to shock in the free jet are of little co11ccrn as far as performance is concerned; ho,vever, 'vhen the kinetic energy of the free jet is to be utilized, it is important to reduce the shock and turbulence losses to a minimum. \Vhen a supersonic nozzle is operated \vitl1 a back pressure that is greater than the design exit pressure (overexpansion), shock losses and separation \Vill occur within the nozzle, resulting in very poor operation. Figure 5.47 shO\VS an apparatus used for testing this phenomenon ; it also gives 0 .492" _! _ the results of the tests as performed ~E==~======:!::====~ ~ by Stodola (Ref. 27). The test appa.ratus consists of a De La.val nozzle • - - - - - - - - - - - _: 0.158"~ ...,. ' >----- 6.142"----~-i- which has in its center a pressure probe. This can be moved longitudi160 A B . -, nally so as to measure the gas pressure . . c 140 , at various sections in the nozzle. For ../ • C· C" these experiments, the inlet pressure v ""' ..."' D 8. 120 was about 148 psia and the back pres. I/ ..D E sure was varied from 4 psia t-0 147 psia w i 100 in several increments (A to L), and for F / ...... 0 each increment the pre&llres along / ~ 80 the nozzle " 'ere measured. The / c u smooth curve of Fig. 5.47 is the free G ~ 0 60 expansion curv·e, it applies only when CD ...:::> the back pressure is less than the - / H design exit pre~ure. As soon as the ...Q.CD=: 40 ' • J back pressure ,,.as raised , the ,·arious ..... ' "' 20 ..0 - K '"\ ./ ~ curves A through L resulted, and in ... I L ... each case there ' 11S a sharp rise in 0 -0.5 0 I pressure. This, of course, is due to the 2 3 4 5 6 No zzle axis, in. occurrc11ce of compression shocks inside the nozzle at the location of the Fie. 5.47 Pressure vnrinlion in n nozzle ,,·ith con1shn.rp prcsst1rc rise. These shock pression shocks nnd loss of contact of jct. (Frou1 /{ef. f7. ) \Y!l.\'CS arc t1st1ally nccon1panied b)r a separat.io11 of the jet fron1 the nozzle '''all \vhich, in some cases, can be qt1itc sc,rcrc. I t shot1ld be 11oted that the fio"· proce.."S after the shock is a diffuser process bccn.t1se, si tlCC tl\e sl\ock is almost normal the velocity after shock is st1bso11ic. I n tl1e presc11ce of n. di,rcrge1\t sectio 11 this bec~mes a diffusion process. ' !he result of tl1e shock~ and se1)n.ratio11 i11 n11 o'rcrcxpn11ded 11ozzle is a significant loss in thrust. T l1e tl1eoret1cal loss ca11 be calct1ln.tcd fro111 t l1c ise11tropic relations of Tables 3.1 and 3.3 and Ec1. (2.4), si11ce tl1c 11ozzlc n.rea ratio defines the exit ~Iach number, 'vhich i11 turn defines the ratio of tl1e exit pressure to the throat p~ure.

•---------1_ /

=N/

.,/

-

-

Cl)

---

,,

'

-

.)

-

~

.

"'

Diffuser and Nozzle Flow with Friction

I

139

Figure 5.48 shows the theoretical loss due to overcxpansion as a function of nozzle area ratio and pressure ratio.

0.9 I - - ---+---

--0 .7 1---

0.3 i - - - - - + - - - - - + - - ---i-1-15.9 = AAe.,, -

-

+ - --

; - -- --i

throot

0 .2

1 = 1.33

0 . 1 1------ i----~----t----t-------t----t-----j

F10.

5.48 Theoretical overexpansion loss for De Laval nozzles.

The test results in Fig. 5.47 sho,v, ho"rever, that the extremely low exit pressures predicted by the isentropic equations do not extend to the nozzle exit. Instead, as discussed above, shocks occur to raise the pressure to the existing back pressure, and flow separation occurs. The result of this separation is beneficial, ho"·e,·er, in that the loss in thrust is significa ntly less than the theory illustrated in Fig. 5.48. The l~ is still substantial, and thus one of the major disadvantages of a De Laval nozzle is its poor performance at pressure ratios belo\v the design pressure ratio. References 28 and 29 give further study of fto\v separation in a D e Laval nozzle. Since jct engines often operate over a '''ide range of 11ozzle pressure ratios, considerable effort has been expended to develop nozzles ''·hich n1aintain the ad\-ant&oaes of the De Laval nozzle at high pressure ratios ,,·ithout the attendant thrust 1Ck.'5e.S at lo\v pressure ratios. T"·o nozzles '"hich have resulted f ron1 this effort are the plug nozzle a11d t he ejector nozzle. The former is particularly attrncti,·e ,,·here short length and lo\v \veigh t are i111portant. The latter has partict1lar application ,,·here air p11mping is required for internal cooling. Figt1re 5.49(a) suggests tl1at the gas expansio11 fron1 tl1e 11ozzle throat follows the Prandtl-l\1eyer expa11sio11. As a rest1lt, the initial flo,,· direction at the throat must be inclined at the Prandtl-Meyer a11glc from the horizo11tal for the final flow to ha,·e an axial direction. T 11c co11tour of the plug is actuully a strean1 li11e of the flo"- and represe11ts the expa11sion experic11ced in t l1e De W\•nl 11ozzle. The isentropic shape sho\vn in Fig. 5.4D(a) I as \vitl1 tl1c De wvn.l 11ozzle, ea11 be desig11ed by t,,·o-dimensional theory. Replacement of the lo,ver part of tl1e isent.ropic plug \Yi th a co11e results in a sig11ifica11t sl1orte11i11g a11d a11 i11sig11ificant Joss ti p to cone angles of 20 to 30 deg. The outer bou11dary of t l1e exl1aust jet is a free st1rface ,,·hich adj usts to fit the existing pressure ratio.

Jet Propulsion

I

140 •

1.8

Flow

... I ~

Throat

11 ...

~~ \\. I

u

-

1.6

- -

t>

. ij

·::::::

1.4

t>

I

Expansion {M och} waves

I \\

Streamlines

0 u

':ii

::> ....

1.2

2

A1hroo1 ~ n(R - r ) A ,-- nR 2

~v

~

-

,,.

I

..., ""

~

•

I

I

v·

Desi;,; pressure ratio

--1 • Experimental plug

/

Conventional '/ - De Laval nozzle

nozzle da ta

.L I-

1.0 2

~

~

Plug n ozz~

c

\\~

I \\\

•

1

cos [(w+8)/2)

....

I 15 20

30 40

60 80 100

I

150 200

1

I

I 400 600

Pressure ratio, chamber lo ambient

(b)

(a)

Fro. 5.49 (a) Schematic plug nozzle. (b) Comparison of theoretical performance of plug and De Laval nozzles. (From Ref . SO.)

Experiments on plug nozzles reported in Refs. 30, 31 , and 32 show tha~ their performance is equivalent to that of De Laval nozzles when operated at design or underexpanded pressure ratios and is even better under conditions of overexpanded pressure ratios due to the lack of shocks and separation. This is shown in Fig. 5.4_9 (~), which is taken from Ref. 30. The most significant problem for the plug nozzle IS its location in the midst of the hot exhaust stream. Such a placement n~itates an adequate cooling system and adds to the design complexity. For pumping purposes, the effect of the primary stream in Fig. 5.50 is to indu~ a secondary flow. The important parameters are the diameter ratio D./ D11, the pnmary nozzle pressure ratio p1./ p0, the secondary to primary pressure ratio p./ pr,, '" the secondary to primary temperature ratio T./ T11, and the length-diameter ratio L/ D 11 . Because of the 'T complexity of the mixing flow and the fact that f or Op Ds al "'.1 , , I aircraft installations L/ D11 is much below the ,. ue ~ . , -~ required for complete mixing, ejectors a.re designed P, - t ~ empirically on the basis of tests. Sufficient tests Ti have now been run to define the performance of a wide range of geometry and pressure ratio \-ariaF10. 5.50 Schematic of typical ejector nozzle. tions. See Refs. 33 and 34 for more detail. For ,~e11· high pressure ratios typical of supersonic flight, a divergent secondary shroud is often used. Reference 34 presents data on such a configuration . All these tests as \veil as actual flight e.xperience sho,,· that ejectors cart be designed to simulate closely the De Laval 11ozzle at very 11igh speeds, ,,;th reaso11able performance at lo\v pressure ratios. Eac11 application requires special tailoring to assure the best compromise among the co11Aicting requireme11ts. Another important characteristic of nozzle discharge jets is noise, principal\)· caused by the turbulent mixi11g of the jct 'vitl1 t11e surrou11di11g air. Tests ,,·ith small air jets and full scale engines, i11cluding o.fterbur11ers, sho\\' that the noise sound po,,·er is proportional to the product of tl1e eighth po\\'er of the jet velocity and the square of the nozzle exit diameter. Because of the public nuisance ''·hich this noise represents, large amounts have been spe11t attempti11g to suppress the noise ,,;thout significantly compromising engine performance.

', · ' T

Diffuser a nd Nozzle Flow with Friction

I

141

Since the jet velocity has such a significant influence on noise, reduction of the jet velocity '\viii have a corresponding quieting effect. The turbofan engine makes such a velocity reduction possible, and the trend to\vard turbofan engines for commercial transport planes is undoubtedly influenced by this consideration. I.,or straight turbojet engines, a reduction in velocity is undesirable since it means a thrus t loss or an increase in sfc . Turbojet noise suppression is accomplished, therefore, by decreasing the effective nozzle exit diameter. This is done by breaking up the exhaus t into a number of small nozzles. The total exhaust area of the small nozzles or slots is equal to the original nozzle exit area. Figure 5.51 illustrates t\vo different approaches to sound suppression in this manner.

(a)

(b)

5.5l(a) General Electric sound suppressor on a CJ805 turbojet cngi.nc in a Gcncrnl D ynamics CV-880 airplane. (Courtesy General Ekcln·c

Flo. 5.51(b) B()('ing sound su pp~r on s Pratt"~ \\'hitne.y JT'J C turbojet engine in a Boeing 707 airplane. (Co11rt~.sy The B oeing Com-

co,npan y.)

pany.)

F10.

Reference 35 notes that t11e principle of the sot1nd st1ppressors illustrated in Fig. 5.51 is based on the marked directio11nlity of jet noise. .-\s a result, the peak noise from a cluster of 11ozzles is less tl1a11 the st1n1 of the peak 11oise from each separately. This is true because 11oise i11 a peak dircctio11 fron1 each 11ozz.I~ is redirected in some differe11t dircctio11 \vl1e11 t.l1c jets fro111 adjnce11t 11ozzles are e11countered. The noise proble111 'vill co11t.i11t1e to be i1r1port.a11t i11 the con1n1ercial field , particularly \vhen supersonic transports ,,·itl1 a.fterbt1r11i11g e11gi11es con1e into use. References 36 and 37 also provide addi t io11al infor111atio11 011 jct 11oise and its suppresfilon.

5. 13

Nozzle Thrust Equations

The thrust equations of a 11ozzle ha,·c l)ec11 briefi~· discussed in Chapte rs 2 and 3. In this sectio11 these equatio11s \Yill be dealt ,,·ith i11 n10~ de tail, and the application of these equations to actual co11vergc11t nozzles ,,·ill be illustrated. The details of the

I

142

Jet Propulsion

---

v

•

v I

-

I

I

I

I

I I

I

I I

I

I

ef

0

5

e

F10 .

5.52 Typical convergent nozzle.

De Laval nozzle thrust equations are discussed in Chapter 15 a s they relate to rocket engines where the De Laval type of nozzle is a neceBBity. The thrust developed by the exhaust nozzle of a jet propulsion device is defined as the gross thrust F 0 . Figure 5.52 shows a typical convergent nozzle with station nomenclature.

I

The stat ions illustrated in Fig. 5.52 are defined as follows: 5 = nozzle inlet section (the turbine outlet section of a turbojet engine) ; gas properties at this section are usually defined in terms of total conditions.

e

=

nozzle exit section.

ef

=

nozzle effective exit section ; that section 'vhere the exhaust gas pressure first equals the surrounding atmospheric pressure.

o = surrounding atmospheric or ambient conditions.

It \Vas shown in Chapter 2 that the gross thrust could be expressed by two equations, t he momentum-pressure equation, and the effective exit equation as

Fa

= G.V. + Ae(P. - Po) g

_ Ge1V., Fa g

(momentUin-pressure)

(5.55)

(effective e.'tlt)

(5.56)

Equation (2.7) shows that the momentum flux terms of the above equation could be expressed in terms of Mach number; thus

Fa = -yp,A.M! Fa

=

+ A,(p, -

Po)

'YPoA.1 (M,,) 2

(moment11m-pressure)

(5.57)

(effective exit)

(5.58)

The identifying names given to the above equations originated from the method of evaluating the thrust. The momentum-pressure equation is based on an evaluation of t hrust at t he actual nozzle exit section; it sums the mon1entum flux force and the pressure force at this section. We must note that the pressure force may not al,,-a)-S exist ; that is, in subcritical flo,v the nozzle exit pressure is equal to t he ambient pressure. Therefore, for tl1is case the pressure force \vill be zero. Tl1e effective exit equation is based on an evaluation of thrust at tl1e effective exit section, the section ,,-here there is no pressure force . 1"'11e effective exit ct1uation merely states the n1omentum flu.~ force a t this section. T o evaluate the thrust of a nozzle by n1eans of the equations cited, it is desirable to express the thrust equation in terms of gas properties " ·hich can be measured. Consideri11g first the effective exit e(1uation, a11d recalling that l\Iach number is a function of pressure ratio [Eq. (3.G)], \Ve 111ay \Vrite

Fa= 2-ypoA et 'Y -

1

'Y- 1 'Y

- 1

(effective exit equation)

(5.59)

'

Diffuser and Nozzle Flow with Friction

I

143

The difficulty in evaluating Eq. (5.59) is the determination of Aet· The normal method of handling this problem is to use the actual exit area of the nozzle together ,vith an experimentally determined thrust coefficient based on Ae. Before reducing the momentum-pressure equations to forms containing gas properties, '''e must recall the difference bet,veen subcritical and supercritical nozzle fio,v. Depending on the fio,v regime, the exit pressure of a convergent nozzle will be either equal to or greater than the ambient pressure ; therefore, the momentum-pressure thrust equation is expressed by t\vo distinct equations, one for each flow regime. For subcritical fto,v, Pe = po. Thus, Eq. (5.57) can be written as 'Y- 1

Fa

p,, Po

= 2'YPoAe 'Y - 1

.., _

1

(momentum-pressure equation for subcritical nozzle flow)

(5 .60)

It should be noted that for subcritical flow, the momentum-pressure equation and the effective exit equation are identical - because in this flow regime, the effective exit section is located at the actual exit section. For supercritical flo\v in a convergent nozzle, Pe > Po and Jfe = 1.0; thus, Eq. (5.57) reduces to

F G = 'YPeAe

+ Ae(Pe -

Po)

or 'Y

Fo

=

Ae Pt.('Y

2

+ 1) 'Y + 1

..,_ 1

- Po

(moment11m-pre...c;sure equation fl ) for supercritical nozzle ow

(S. l) 6

The latter form of Eq. (5.6 1) is obtained from the fact that in supercritical nozzle flow, the critical pressure ratio exists between the inlet condition and the nozzle throat; therefore, in a convergent nozzle the exit pressure may be \vritten as a function of the nozzle inlet pressure. Equations (5.59) to (5.61) are the equations normally used to compute nozzle gross thrust. It is most significant to point out tha.t in all of these equations, the thrust is expressed as a function of pressures, of an area, and of some constants. I t is logical that thrust should be so expressed because the only physical ,,.a)' in ,,·hich a gas can exert a force is by a pressure differe11tial across an area. When applying the thrust equations to real nozzles, it is usuall)· nece._"881)· to modify the equations by the so-called tJir11st correction, factor, Kr. This is defined as [(

= T

Fa (as meast1red by tl1rust stand) F0 (by equatio11 calct1lut.ion)

( . ) 5 62

If it \Vere possible to calculate or measure t11e actt1al values of pressures and areas and use the exact valt1e of specific heat ratio i11 the tl1rt1st equations, then the , ..alue of l( r would be unity. Normally, 110\Yever, it is 11ot po...~ible to do so. Thus, the ,-alue of K r is usually less tl1an 011e. It should be recog11ized that the thrust correction fact.or actually lumps together the nozzle flo,v and velocit)' coefficients into one term; therefore, it \vill be a func tion of pressure ratio. Thus, the norn1al method of presenting the variations of J(r for a give11 11ozzlc is to prese11t it i11 graph as a function of nozzle pressure ratio. Reference 26 prescr1ts st1cl1 plots for t)rpical nozzles. It should be noted that for a give11 nozzle, ]{ r 'vill l1ave differe11t values for the effecti,re thrust equation and the mo111entu1n-pressure equatio11.

I

144

Jet Propulsion

Frequently, it is convenient to express and speak of thrust in terms of a special ' thrust parameter called the corrected gross thrust defined as

Fa

~

=

14.7 p Fa

(5.63)

= corrected gross thrust

where o is the ratio of the ambient pressure to standard sea level pressure. This parameter corrects or refers the thrust to sea level conditions. The corrected parameters will be discussed in detail in a later chapter. Table 5.2 presents a summary of the gross thrust equations which have been discussed in this section. The symbol X, used in this table is the isentropic pressure ratio function discussed in Chapter 3. Since the momentum-pressure gross thrust equation is preferred to the effective exit gross thrust equation by the jet propulsion industry in general, it is recommended that the former equation be used \Vhere possible. Actually, one method is as accurate as the other (see example problem at end of this section); ho,vever, to be consistent with general usage, the momentum-pressure equation will be used as the standard gross thrust equation in this book. In order that the gross thrust of a convergent nozzle may be readily computed, Fig. 5.53 is given. It shows the corrected ideal gross thrust per unit exit area F cl o_-t.. variation with nozzle pressure ratio. 40

I J

36

I j

32

I

. c:

·::0 28

-

)

I

•

' ~

J

..~24

58

I/

(.')

u.: ._'

J

~ 20 E 0...

I

)

I/

7

II

8. 16

-"' 2

0- 12

..c

I

41

-

)

I

I

J J

\)

8

I

J

I

I

j

46

'

38

'· -

I

0 1.0

50

42

I

I'

4

54

1.4

1.8

2.2

2.6

3.0

3.4

3.8

Over-all nozzle pressure ratio, p 13 / p 0 F10.

5.53

Va.ria.tion of thrust pnro.n1eter based on nton1C'ntt1m-thrust equation \vit h over-all nozzle pressure rntio.

Diffuser a nd Nozzle Flow with Friction TABLE

l

)

'Y

1.

5.2 Summary of Convergent Nozzle Gross Thrust Equations

1\IO~ I ENTUi\l-PR ESSURE

(a) Subcritical flow, Jl.f. Fo =


po.

Fo = GV. + A.(p. - Po) g

l

,

Fo

=

•[ A, Pi.('Y

=

14.7 A, [1 .26

=

-yA,p,

+ 1) ( 'Y +2 l )

+ A,(p, -

Po)

J

..,., , - - Po

= A,(l .26p,, - Po)

!

I

~o

~':

- 1]

2. EFFECTIVE EXIT EQUATI ONS

Fo

=

GV.1 g

=

2

-yA,,paM., 'r -1

F;

= ~'Y~·~o [ c ;~ )-:y

,

- 1] =

8.06 A,po(X,).1

Fo = 118.4 A.(X, ),1 0 3. THRUST-CORRECTI ON FACTOR K T

_ F o (as measured by thrust stsnd) F 0 (by equation calculation)

Figure 5.53 is based on the momentt1n1-pressttre thn1st equations and, as indicated, two forms of t11e equation are required for this plot, 011e for subcritical flow, and one for supercritical fto,v. I t should be noted tl1at the ,·ariation of thrust ,,;th pressure ratio is linear in the supercritical fto\Y regin1e. The applicatio11 of this plot for a problem is simple; e11ter the cl1art \Vi th t11e gi,re11 nozzle pressure ratio p,. Po, and obtain the value of /?~/ oA.. Tl1en, calculate the ideal thrust fron1 t he \•alues of o and A •. Example A co11vcrgent nozzle \Vitl1 a11 exit area of 100 i11 .~ is operated at sea le\ 1el (o = 1.0)

\vith a presst1rc ratio p,./ po of 1.5, 2.0, a11d 3.5. Assurnc t11at 'Y = 1.33 and J\ r = 1.0, and cal ct1latc the gross thrust for these three conditions by (a) t11e cffecti,,c exit equation a11d (b) the 1non1entum-pressure • • equation.

146

I

Jet Propulsion

Solution (a) From Table 5.2 the appropriate equation reduces to

,

, Fa

=

Fa 100 A•

= 11,840

)

(X. ,,

From Appendix E 've find that (X.),, l1as the follo\ving values: (X,),, = 0.1058, 0.1877, and 0.3646 for pressure ratios of 1.5, 2.0, and 3.5. Thus F ~ = 1250, 2220, and 4320 lb for the pressure ratios. (b) This part can be ans\vered directly from Fig. 5.53 by entering the graph at pressure ratios of 1.5, 2.0, and 3.5, reading the value of F~/ oAe and multiplying the result by 100. Performi11g this operation yields F ~ = 1250, 2235, and 4875 lb for the pres.mre ratios. DISCUSSION

It is seen that the agreement between the two methods of computing thrust is good except for the case of high nozzle pressure. It is obvious that the answer obtained \vith the effective exit equation is wrong because the actual exit area was used. Had the ideal effective exit area Ao1 been computed and used in part (a), the two answers would have agreed very closely. It is thus apparent that the ideal thrust, when computed by the momentum-pressure equation, will always produce an answer which is closer to the actual thrust than that obtained by the effective exit equation. For this reason, the moment11m-pressure equation is preferred. PROBLEMS

I. A turbojet engine operates at standard 10,000-ft altitude conditions at a flight :'\Isch number of 0.8. If its diffuser ram recovery is 0.85, what is (a) the t-0tal temperature (°R) and total pressure at the diffuser exit? (b) What is the total temperature and t-0ta1 pressure of the free stream? (c) If the air were incompressible, what '"ould be the total pressure at the diffuser exit for no losses? 2. A pilot who is flying at standard 20,000-ft conditions at M = 0.6 notes th!\t the cliffu...c:er exit total pressure is 8.10 psia. \Vhat is tl1e ram recovery of his diffuser, nnd what is the t-0tal temperature at the diffuser exit? (0 R) 3. Sketch the actual and ideal diffuser process on n T-S and n p-t' diagrsn1 and indit"ste all important parts of the diagram. 4. A test pilot operates a turbojet airplane '"itl1 a diffuser ram ttto,·e11· of 0.90 and resds a value of diffuser total pressure of 10 psia nt fligl1t ~Inch 11t1mber of 0.85. At "·hst prcs.5Ure altitude is he operating? If tl1e temperature nt this altitude is st."lndard, "·hat is the tot31 temperature at the diffuser exit? (0 R) 5. Sketch the shape of nn idenl su1)ersonic diffuser nnd point out t\\·o diS..'ld,·antsges of this design for aircraft propulsion. 6. A simple inlet diffuser operates nt sen level at ~f = 2.0 so that it has a normal shock at the entrance and an internal rnn1 recovery p,Jp,¥ of 0.85. Calculate the total p~ure (a) after the normal shock, (b) at the diffuser exit, (c) of the free stream. (d) Discuss the results.

Diffuser and Nozzle Flow with Friction

I

147

7. If the ratio of total exit pressure to free stream pressure is 5.0 for the diffuser of the previous problem when it operates at M = 2.0, \vhat is the ram recovery?

8. Explain \vhy the modified diffuser design of l''ig. 5.11 gives better results than a simple inlet \Vhen operating at supersonic velocities. 9. A fusclage supersonic inlet is designed and operated under conditions shown in the sketch belo\v. In the area at section 1, A 1 is 1.2 ft 2, and the total pressure recovery factor from section y to 2, P1 / p, 11 is 0.90. 1

\ or " I

I

M0 - 2 ..5 : p 0 -10 psio 1

Vo

I I I

fAo

ro-4.50° R

I I

I

~

I

':::5:-0

'?

'

y

I

I

I I I I

I

I

I

I

I I

w'= 10°= Romp angle 2

0 F10.

5.54

Calculate the following : (a) a'; M1; pi; T1;G1; A o/A1;M11 ;p,.;p 11 ; T,.; T.;p ,,; T,,;Fa;C ,.; and Da. (b) Also calculate Ac for the case where the oblique shock just intersects the diffuser lip. IO. Why is it best to "swallow" normal shocks? 11. Calculate the additive drag and the additive drag coefficient for the fuselage inlet of the example problem given in section 5.5. 12. Dra'v the actual and ideal nozzle process on a T-S and p-v plane and indicate the important parts on the diagram. 13. The enthalpy change across an air nozzle is 81 Btu per lb. Calculate (a) the ideal exit velocity, (b) the actual exit velocity for '11" = 0.90, (c) the velocity coefficient. (Unless other,visc specified, use momentum-pressure equation for thrust calculations in the remaining problems.) 14. A convergent nozzle \Vith an exit area of 2 sq ft has tot.sl inlet conditions of 1500°R and 30 JJSia. (a) If the velocity coefficient is 0.95 and the exhaust pre...~ure is 12 psis, cslculate the e.~t velocity (use "Y = 1.33). (b) '''l1at is the nozzle ndiabntic efficiency? (c) \Vhat is the entropy increase per lb of air? (d) ' Vhnt is the value of the polytropic exponent of this proe~? (e) \\' hat thrust is JJroclt1ced for /\. r = .93?

15. Derive, step by step, Eq. (5.54). 16. '''l1at is tl1c criticnl pressure rntio for n co11\'crgi11g nozzll' \Yhich has sn adiabatic efficiency of 0.5, 0.6, 0.7, 0. , 0.9, nnd 1.0? l'lot the results. t~te \Yh)· r11orc pres.5ure drop is nccproxi111ntc ~Ictl1od for the Design of Hub Shroud P rofiles of Centrifugal I 1n1>cllcrs of Givc11 Blnde hn1lc," NACA TN 3399, 1955. 4. Osborne, \V. l\ I., nnd Hamrick, J . T ., " Design nnd T est of ~lixcd Flo\\· I mpellers: !Aerodynamic Design Procedure," NACA R~l E52E85, 1952.

8

8.1

Axial-Flow Compressors

Introduction

During the early development of turbojet engines, it 'vas realized that the centrifugal fto,v compressors 'vould impose certain performance limitations upon the high-thrust engines of the future. Consequently, the axial-fio,v con1pressor de,·elopment program was initiated early in turbojet engine history. This is borne out by the fact that the first all-American turbojet engine '''as the 19A engine, an axial-flow compressor engine designed and constructed by Westingl1ouse (Ref. 1). As mentioned in Chapter 7, the maximum pressure ratios attainable in centrifugal-Ho,,· compressors is about 4: 1 (unless multistaging is employed, resulting in multiple air-turning problems) at an efficiency of about 70 to 80%. The axial-fio\v con1pressor, however, can achieve much higher pressure ratio at a high level of efficiency; thus, ,,-here high pressure ratios are required, it is this type of compressor that must be used. Perhaps the greatest advantage of the axial-flo\v compressor is its high thrust per unit frontal area. In today's engines, the average axial-flo,v type attains a static thrust per 11nit area of about 1500 lb per ft 2 (1000 to 2000), \Vhich is about four t.in1es the an10UI1t of the average centrifugal-flo,v type engine developing about 400 lb per ft~ (350 to 450) (Ref. 2). These t'vo characteristics of the axial-flo,v con1pressor - l1igh presst1re ratios at good efficiency and high thrust per unit frontal area - i11dicate the realn1 of its best application in high-thrust engines for 11igh-speed aircraft. Briefl)', tl1e a.~al-fto,,­ compressor provides large air-handling abilities '''ith a s111all f ro11t.al area, a straightthrough flow system, and higl1 pressure ratios '''ith relati\ el)' high efficie11cies. I ts chief disadvantage is its complexity a11d cost. Iiu11dreds of blades are 11eeded to achieve the pressure ratios required by turbojet e11gi11es. }-;'igure 8.1 sho,,·s the stator and rotor blades of the 12-stage J47 turbojet e11gi11e co111pressor, ,,·}1ich is typical for an axial-flo\v type compressor. The com_plexity C~Jl be visualized from Fig. 8.1. In general, each rO\\' of rotor and stator blades is of different size a.11d acsig11. ~0111pressor blades are usuiill)· 111ade steel, magnesium alloy, aluminum alloy, or tit.a11it1111, and it is not UJ1comn1on for one compressor to have sorne steel blades a11d so111e allo)r blades. T?~ ;,otor ,blag~ must be secured properly to tl1e rotor disc to \Vitl1stand the stresseS" in1posed b.}· high rotation~! ~e.eeds.--pjgure 8.2~ (fron1 Ref. 1) sl10,vs the ''fir tree'' a11d ''bulb root'' metliods of attachi11g blades to the rotor. 1

or

181

., I

182

Jet Propulsion

(a)

(b)

F10.

8.1 Compressor blades of the J47 turbojet engine: (n) stator blades, (b) rotor blades.

Axial-fl.o\v compressors come i11 many sizes and con1pression ratios. The follo"~g cl1aractcristics are take11 from Ref. 2 for tl1e sea level static co11ditio11. The con1p~or sho\vn in li'ig. 8.1 has an air-flo\v of about 100 lb per sec and a compression ratio of 5.5. The JT3D t urbofa11 e11gine illustrated i11 l1'ig. 10.2 of Cl1apte r 10 has an air-Bo"· of 450 lb per sec, a fan pressure ratio of abot1t 1.7, and an over-all con1pression ratio of about 14. l 'igure 8.3 sl10\VS the T63 compressor '''hich has a11 air-Bow of only 3 lb per sec and a compression ratio of about 6. 1

Axial-Flow Compressors

(a) Fro. 8.2

F10 .

8.3

I

183

(b)

Two methods of attaching blades to compressor rotor: (a) fir tree root, {b) bulb root.

Compressor rotor of T63 turboprop engine.

(Courtesy Allison Du:ision., General .UolMs.)

As may be seen, the T63, con1pressor has six a."'tia.l stages follo"·ed b)· a single centrifugal stage. The cuta\\'ay of the engine also sho''~ in Fig. 8.3 sho"-s that. this stage matches tl1e remai11der of the engine \veil by transferring the air radiall)' out around the co1nbustor a11d reduction gears. It is also interesting to note the large

184

I

Jet Propulsion

vaneless diffuser utilized \vith tl1e ce11trifugal impeller. Other small engines such as the Pratt & Whitney T74 also ha\'e a combined axial-centrifugal compressor. Some of the ne'\\'er axial-fto\v compressor designs are even more complex than those illustrated in Figs. 8.1, 8.2, and 8.3. In order to attain pressure ratios of the order of 10 to 15 \vhere, perhaps, 15 or more stages are required, it is generally advantageous to split the compressor into t\vO sections, \vhich can _.JJ.:J.~~:u:::u:~========~..L::..------then rotate at different speeds. Figure 8.4 is a schematic diagram of the comFlo. 8.4 Sketch of the compressor section of a dual pressor section of a dual rotor or, as it rotor turbojet engine. is usually called, a ''t,vin-spool'' turbojet engine. Figure 11.3 in Chapter 11 illustrates a cutaway of the Pratt & Whitney JT4A turbojet engine showing the t\vin-spool compressor with eight stages in the front lowspeed rotor and seven stages in the high-speed rotor. From Ref. 2, the air-flow of this compressor is 265 lb per sec with a compression ratio of about 12. The degree of increased complexity of the dual-rotor over the single-rotor compressor is readily apparent - concentric sl1afts, additional bearings, and more n11merous blades. Another somewhat recent development in axial-flo\v compressors is the utilization of variable stator-vane angles in the inlet guide-vanes and some of the stages. Figure 11.1 illustrates a cuta,:vay of the General Electric CJ805 turbojet engine showing the variable stators in the inlet guide-vanes and the first si.x stages of the seventeen-stage compressor. From Ref. 2, this compressor has an air-flow of 168 lb per sec and a compression ratio of about 13. This feature, "·hich makes the compres&>r operation more versatile \vith regard to air-handling abilities, also increases the complexity of the equipment by a significant amount.

8 .2

Principle of Operation and Basic Terms

The basic pripciple of operation of the axial-flo,v co1npressoi:_ ~ the same as that of lli'e centrifugal compressor, namely, i111parting kinetic e11erg)r to the air b • means of the rotating blades, and the11ce converting tl1e ki11ctic energ)• to a pressure rise. Referring again to li'ig. 8.l (b) , tl1e air enters axially fron1 the right and into the inlet guide-vanes \vhere it is turned t hrougl1 a certain a11glc to in1pu1ge 011 the first rO\\: of rotating blades 'vith tl1c proper angle of attack. The rot.ati11g va11es add kinetic energ)· to the air and increase the pressure slightly, tl1en discl1arge it ,,·ith tl1e proper a11gle to the first ro\v of stator blades \vherc tl1c pressure is ft1rther i11crcased_b,J: difft1sion. The air is then directed to the second rO\\' of rotati11g blades and the proce..."-5 is_!:Cpeated through the remai11ing s tages of the compressor. A con1pressor stage consists of a row of rotatir1g blades follo,,·cd by a ro\v of stator blades. i\Iost con1pressors ha,·e one to three ro\VS of "straighte11er" or "difTuscr" blades i11stalled after the last stage to straighten and slo\v do,vn the air prior to its e11try i11to the con1bustion chan1ber. The straightener vanes are easily identified i11 Fig. 8.l (a) as the last stator stage at the bott om - ide11tification is made from tl1e relative blade angles of the Inst stage with other stages. If the purpose of these latter stator vanes is to provide additional air turbulence

Axial-Flow Compressors

I

185

(as is sometimes necessary to alleviate cornbustion problems) , they are called " mixer " bl~es.

. . The pressure ratio accomplished per stage of compressio11 for subsonic .st~ges IS very modest '''hen compared to 011e stage of a centrifugal compres.sor. This IS ~~ ·, illustrated by Fig. 8.5, '''hich sho,vs t he stage pressure ratio accomplished by a typiC'.s. 1.16 ·, ~"i!i1.~v··~>

•. :..'N·,·.z.~". ..,...• . " ·:·-~ ,._..,...·-~.!L':;s .·'.~~::4~~~\ ·.·.·.\' ; •••• ~· .:••• , \ · ,''

._g 1.12

·-:t~·'(."il'•

,_,.,,,,.,.,..,..

····-·,1·'"'< "~,·

:1r.•.:i.'' ·:,·-·. ~\)!, • t',

'F,,

\

, ·'1

···.•~.;··~' ~·:: ;.., ,,

::...._ ....,....

:> .,,

Gl 1.08 '

~~!"-·.-:.-...... ••• • •• ::·····=:•.','

•. . _.. ; •. -...

;\·~~·:!;~:E:1i

i.....

•:',•1:·.•.

:~\~·:~~:~:~:~·:·~~

OI 0

Vi 1.0 4

·'. ._..,_.,, ·-·-...•,·~,(. ,- 1 ·-· .. •• • .:

•.. .'· ·•• ...-•_..

.·.~·1~·::--:.·

... -••• ·- ·:-

._.,,

. .\·.. .:.,. ;;··.' . -=··

.1:····ll·'· ............. ........ :,,-:•: .·,·.·.·.·.'···'· •••• :\.',••t\''-· .·..... :,. ..... .... ·. ............ ,,,... .. .... •'•\' ·,r····. . .:

·........... ~ ·;·

GI

:_.,.;.:..·..;... ........

.

::·':..;·.--:::-:.:' 1;: •• • ;: .,, ••. , " _ "' _ :-!:! .. " • • t·: "'::: • . . .. • ,\, ,:.......>:' :-:-.:;·:....:· ... '" .:~ .:~~:,.,-.....~.· :·;-~:.-,·'::". •.. ..,. :·.· -~·. ·-·~,· ...~.....";o • .,.. • .-~·!·· . . ··,·.~: ~ ]:":::::::-~·: .'.. ....·--:.•.., ,..~:......;. .:.·.-:.·'...·.•.·.•. ...·.'•...•::. :... -........ ·.:-~·..:. •::::·:::.,~.. . •.•••.••• ,,,. : .·.--;;,·t:,:.. :.·::, .. ::.~•..: ·.~ .. ·r ~e

:ti

It is interesting to exami11e special cases of the combined velocity diagrams to see how the velocity plots appear for different degrees of reaction . Figure 8.8 present..q five such special cases '''ith notes to describe each diagram. Figltre 8.8 presents a case 'vhere both (c~ - ci) and (wi - w~) are negati~ ... which means that the pressure decreases through the rotor. This represents tJ:• action which occurs in a turbine. Axial exit C2 = Co CJ U1

r

U2

Ca=C2

~~

- 6.cu

100'1 reaction C2=C1

'.1 UJ

~Acu

,..j

"

l

OX reaction

:> ..,

W2=W1

1

W1

1

Ca

-

•

U1

L

u~I

/!,.cu

50% reaction W2=C1

w,

)

W1=C2 Ca

c2

UJ~

,>

SOX reodion

/!,.cu

W2 = C 1 Wl =C2

& W2>Wl thus, turbine action or C1>C2

Ca

u2 F10. 8.8

8.4

6. cu

U1 ..

I

pres.sure drop

Special types of vclocit)· diagmn1s, all I'('}sting to constant s.xia.l \-elocit.)'-

Blade Element or Airfoil Incompressible Flow

Analysis,

One-Dimensional

Ideal

This analysis is based 011 forccs ,,·}1icl1 act 011 a11 i11di,·idual blade " ·ith the assun1ption of one-di1ne11sio11ul fto,,· '''ithout f rictio11. I t sl1ould be realized that this t)·pe of analysis is not a realistic 011e because of the si111plified flo,,· assun1ptio11. Ho,,·e,·er, it will provide a11 i11troduction that ,,·ill be used i11 the 11ext. sectio11 ,,·here ,,.e shall co11sider t" 0-dimensio11al flo'v \Vi tl1 friction. Figure 8.9 prese11ts a typical axial-flo''' con1pressor rotor blade ,,·ith the perti11~11t symbols used in a blade eleme11t a11alysis. 1

>

I

190

Jet Propulsion ~I = p itch

o}- - /

F

The new symbols used in the figure are defi11ed as follo,vs :

-~c I

I

/ / f)

I Fa w,,.

Fa

F"' = tangential force on rotor blade,

/~t.:'

/ OS

Fu I

/ ,0 /~

I

I

I

I b F10.

.

1

1

.

--- J_w~ 8.9

1

I

= axial force on rotor blade,

I

- ;Jd

Typical rotor blade with its symbols.

F

= resultant force on rotor blade,

l

= blade length along chord,

t

= pitch or distance between rotor blades,

Wm

= mean or average relative velocity,

ab

=

cd - streamlines of flow path,

t:.r =

Ttip -

=

rhub

blade height or length.

The forces generated by the blade originate from t\vo sources, namely, pres.5Ul'e and momentum. Since the pressure forces along the streamlines ab and cd are equal along any tangential line, there is no net tangential pressure force. The pressure force in the axial direction can be evaluated by \Vriting the energy equation for incompressible ideal flow which, when solved for pressure difference, is

P2 - P1 = !p(w~ - w~)

(8.2)

The axial pressure force is given by the axial area over which the pressure difference acts, thus

Fa

A (p2 - Pt)

=

=

(t:.r)t

2

2

2

p(W1 - W2)

(8.3)

Equation (8.3) can be expressed in several other forms by considering the blade velocity diagram \vhich is sho,vn in Fig. 8.10.

- - - -w, WI - w 2 v v

F10.

8.10

v

+-!

Combined velocity diagram for constant a.xial velocity.

It should be noted that for the condition of co11sta.11t axial velocity, 2

2

2

2

I

I

(8.4) Thus, Eq. (8.3) may be \Vritten in tern1s of any of these velocity square differences. Let us now examine tl1e mon1e11tun1 forces. For consta11t a..xial \•elocit.)·, there is no momentum change and, co11sec1uently, 110 n1on1e11tum force in the a."tial direction; thus, Eq. (8.3) represents the total axial forces. Since the tangential pressure force is zero, the total tangential force is due only to the momentu1n force \vhicl1 can be \vritten as G Fu = (wi. - W2.) = p(~r)tco(Wi. - Wt,,) (8.5) W1 -

g

W2

= W1,. -

tV2,.

= Ct,. -

C1.

Axial-Flow Compressors

I

191

The resuUant force F is the vector sum of the axial and tangential forces, and its magnitude can be expressed as

+ F2

p2 = F2a

(8.6)

u

Substituting E qs. (8.3), (8.4), and (8.5) into the above, \Ve have (8.7)

w!

Since the mean or average velocity squared is equal to the last terms (within the brackets) of Eq. (8. 7), \Ve can \Vrite, for the resultant force, F = p(Ar)t(w1,. -

(8.8)

W2.)Wm

= p(Ar)t(Ac.)wm

The circulation concept can also be used to obtain the net force (lift) on the blade. Recalling that circulation is defined as the line integral of the velocity around a bounding surface, \Ve can evaluate the circulation around the path abed of Fig. 8.9. Since the line integral across ab and de are equal and opposite, \Ve have a

d

I'=

Vdx

= b

Vdx

+

(8.9) c

Hence, the resultant force can be written as

(8.10) It is noted that this equation is of the same form as the l(utta-Joukowsk)· law used in aerodynamics for lift determinations, except that here " ·e use the mean ,·elocity (because of finite distance bet\\1een blades) instead of the inlet or free stream ,·elocit}·. From the relations just given \Ve can no\v \Vrite the expre-'5ion for torque, power, and pressure head. The torque equation for one ro,,· of rotating blades can be ''-ritten as

(8.11) where r = effective radius of force application, n = number of blades in stage. From Eqs. (8.5) and (8.9), the torque equation beco1nes

Tq = p(Ar)reanr (8.12) Since the mass-How rate G/ g can be expressed as p2rr(.ir)c0 and in = 2rr, Eq. (8.12) may be \Vritte11 as

G

nr

G

G

Tq = - g 21r = - g r(tv1 • - tvi• ) = - g rAc.

(8.13)

Note that this equation for torque is ide11tical to tl1e torque expression deri,·ed from the filament n11alysis; 11e11cc, tl1e po,,·er a11d 11end equatio11s deri,·ed b)· this anal)-si.s \Vill also be ide11tical to their cou11tcrparts derived from tl1e filament anal,-sis. These • equatio11s are

p

= Gw1ir = !!_ it 6 Cw 21'"g

11/c =

H:

=

(8.14)

g

W1~ r

2rg

=

tl

Aew g

(8.15)

192

I

Jet Propulsion

It has been sho,vn that the airfoil theory for one-dimensional ideal incompressible flow yields equations of torque, po,ver, and head identical to those obtained from the filament theory. The equation for pressure ratio across a stage "vill, of course, be different because the filament theory yields a compressible eql1ation, 'vhereas an incompressible equatio11 \vill result from the airfoil theory for the assumed fto,v conditions. Even though the results of the t\vo different theories are similar, the uses of the results of these analyses for design purposes are limited because they do not take into account the characteristics of the blades, that is, flo,v variation along the blade length, CL and CD section values of the blade, and so on. In order to account for specific blade characteristics, a t\vo-din1ensional analysis that considers various effects (variation of flo\v along 6.r and t) should be used. Tl1is is discussed in the next section. In our analysis thus far, \Ve have assumed that the fluid follo,vs the blades '''ithout slip. This assumption is valid only for blade length to pitch (solidity) ratios l/ t 'vhich are greater than about 1.2. For l/ t values of less than 1.2, slip becomes appreciable, and relatively large correction factors are needed to modify the one-dimensional equations. It is recalled that in centrifugal compressors, slip \Vas accounted for by a slip coefficient.

8 .5

Blade Element or Airfoil Analysis, Two-Dimensional Flow with Friction

This type of analysis can be made by considering the blade section drag coefficient and lift coefficient. Figure 8.11 presents a typical rotor blade sho,ving the pertinent forces, velocities, and angles. The lift and drag forces are directed perpendicl11ar and parallel to the mean velocity Wm, and they are defined as follo,vs:

L --

CL 2

2 A pWm,Lib

(8.16)

where At1 = 6.rl. Since the torque, po,ver, and head equations are functions of the tangential force Fu, 've must evaluate this force in usable form, that is, e~i>~ it in terms of the lift and drag forces. From Fig. 8.11

F

1J.

=

Du + Lu = D cos flrn

+ L sin fl,,.

\vhich can be expressed, from Eq. (8.16), as

F

u

F10. 8. 11

Rotor blade nomenclature for t\\·crdimensional anal)·sis \Yitl1 friction.

Axial-Flow Compressors

I

193

(8.17)

The tangential force can also be expressed in terms of the momentum flux in th.., tangential direction as

Fu = Q_ (Awu) = p(6r)tc0 Awu = p(Ar)twm sin {3,,.(Awu) g

(8.1 >< l

Combining Eqs. (8. 17) and (8.18) gives

(8.19) A parameter which provides an index of the guiding effect of the blades is called the solidity ratio u, \Vhicl1 is defined as

l

-t

" =

Expressing Eq. (8. 19) in terms of t l1e solidity ratio yields

CDu cot f3m

+ CL" =

(8.20)

2

From the combined velocity diagram sho,v11 in Fig. 8.12, \Ve ha\·e Wm..

cot f3m = --'Ca

and

Ca

Flo. 8.12

Thus, Eq. (8.20) can be written as ,,,1lO'

2gc. [CL

• SlD

Remarks Cannot predict. 6c.. for a gi\""en blade

t~C..

One-dimensional blade elcment. - ideal - incompressible Tv.·o-dimensional blnde clcment. incompressible \Yi th friction

Head Equations

fj.. +CD cos

tl-1

>

1.2

CD and CL n.lues from tests. Can be applied to compressible flow

Axial-Flow Compressors

8.6

I

195

Three-Dimensional Consideration

In our analysis thus far \Ve have assumed no flo,v in t~e radjal ru~~ction and t11at the conditions at one particular blade section arc indicative of cond1t1ons at all sections. Actually, there is a co11siderable difference in the velocity diagram between the blade hub and tip sections, as can be seen from Fig. 8.13. Tip

Mean element

Root or hub

Flo. 8.13

Flo. 8.14

Variation of velocity diagram along blade.

Fluid element in compressor stage.

Since the axial velocity c0 is virtually constant along the blade length, as sho"'ll in Fig. 8.13, it is seen that the blades require a considerable twist (change in angle {3) to insure a uniform angle of attack along the blade length. In an actual design problem, it is necessary to analyze the forces over several sections of the blade. The criterion just mentioned, that is, zero flo,v in the rarual illrection, is an actual goal in design practice. Consequently, the centrifugal force of each air particle must be balanced by an equal and opposite pressure force, " ·hich means that a pres.5U.re gradient must exist in the radial direction 'vith pressure increasing from the blade-root section where centrifugal forces are low to the tip section " ·here centrifugal forces are relatively high. In order to maintain radial equilibrium, free vorte.'I: flow must exist in the compressor stages. The equations \vhich prescribe radial equilibrium flo,, can be developed by considering the fluid eleme11t sho'''n in Fig. 8.1-1. For radial equilibrium, the centrifugal force must be balanced b)· the pres.5Ure force, or 2

.,/lrw = 2

.,//Cu

= A dp

r

(8.26)

For urut depth as shown, Eq. (8.26) can be expressed as 2

Cu

p dr dO -

r

= dO dp

dp

or

i

dr

-=Cu-

r

p

(8.27)

The dynamic equation, 'vhc11 \vritte11 alo11g a streamline (no friction) and using c as the absolu te velocity, is

v dp Cornbir1ing tl1e above

t\VO

+ c2gde =

0 =

~ + c de p

equations gives

2 dr r

+

c de = 0 t Cu

2

(8.28)

196

I

Since c~ = c2

Jet Propulsion -

c!,

we have 2 dr

+_

C2 -

r \vhich, " 'hen integrated for constant

Ca,

rcu

•

c_d_c_ = 0 2

Ca

becomes

(8.29)

= constant

Equation (8.29) specifies the free vortex or irrotat ional fto\v and prescribes the variation of Cu \vith r to maintain radial flo\v equilibrium. It states that Cu must decrease linearly \vith an increase in radius. For a t hree-dimensional flow analysis, Eq. (8.29) is used to simplify the basic differential equations. Reference 4 gives a. more complete description of design procedures for axial-flow compressors.

8.7

Polytropic or Small-Stage Compressor Efficiency

Sometimes in the analysis of multi-stage axial-flow compressors, it is more convenient to use the polytropic or small-stage compressor efficiency 17 11 (not defined to this point) in preference to the adiabatic efficiency. The basic advantage of the polytropic efficiency is its equivalent value for the individual stage efficiencies and the over-all efficiency. To illustrate, if a compressor has n stages, each ha\ring the same efficiency 1 t hen the polytropic efficiency of each stage "-ill be equal to the pol)rtropic efficiency of t he over-all compressor, \vhereas the adiabatic efficienC)' of each stage will differ from the over-all adiabatic efficiency. These facts, together ";th the mathematical derivation and the physical meaning of the polytropic or small-stage efficiency, are discussed below. In this discussion on compressor efficiency definitions, kinetic energy changes "'ill be ignored for the sake of simplicity. If it is desired to account for kinetic energy change, then gas total properties should be used in the follo,,ing equations. Consider a compressor \Vhich is made up of n stages, each having equal ,,a.lues of T1c· Thus across some typical stage (from x toy), \Ve have

.,

-P11 =

1

l'- 1

+ T/c

(8.30)

Across the \Vhole compressor of 1i stages \Ve have

"

-

and

(8.31)

Combining the above cquatio11s gives tlie over-all pressure ratio 1

~ P2

=

1

+ fi e

Ts " - 1 T 2

"'Y

,._1

(8.32)

Solving the above equation for the adiabatic efficiency gives ,._1

fi e =

Ps P2

(Ts T1

ft ")'

- 1 (8.33)

l/n

- 1

•

Axial-Flow Compressors

197

This equation sho,vs that T/c for a given stage is a function ?f the nun:iber of stages · of compression as 'vell as the pressure and temperature ratios. To illustrate the dependence of T/c on the number of compression stages , suppose 've . fix Pa/ P2 at the value of 10 and Ts/ T 2 at the value of 2.035 and then solve for T/c for different nt1mbers of stages. The results are sho,vn in Table 8.4.

•

TABLE

8.4

•'

.•

I

Variation of T/ e with Number of Compression Stages for a Pres.5ure Ratio of 10 and a T emperature Ratio of 2.035

Stages n

1

2

4

6

8

10

100

T/c

0.90

0.916

0.923

0.924

0.925

0.9256

0.9260

From Table 8.4 it is apparent that T/ c changes "ith the number of compres.5ion stages. For the over-all compressor, the single stage value T/c has a \ ralue of 0.90 ; \vhereas, for an eight-stage unit, T/c is 0.925 for each stage. It is also e\r:ident that 1Jc approaches a limiting value as the number of stages is increased indefi.nitelj·. Th.is is the exact definition of the polytropic efficiency, namely T/p =

lim

T/c

as the number of stages n~

co

Evaluating the limit of Eq. (8.33) as n approaches co (or as the pressure ratio approaches 1) by taking the derivative of both numerator and denominator with respect to n gives

T/p =

In Ps - 1 P•

"(

"(

In or, in different form, 've have

(8.34)

Ts Ti

.,.,. Ps -= P2

Ts T2

-r-l

(8.35)

The polytropic efficiency is sometin1es called the s1nall-stage efficiency. The main advantage of this type of efficie11cy is tl1at. its o\rer-all ,-slue for a compressor with any number of stages is equal to the efficie11cy of each stage. This, in effect, is how it is defined. Because of the co11stant pol) tropic efficie11c)r value throughout the nUlchine, the poly tropic efficiency is preferred O\rer t.he adiabatic efficienc)· for n1an)· special types of a11alysis. 111 fact, so111e people prefer it. for all cases. In this te:\.-t, preference has been given to tl1e adiabatic efficie11c)r, a11d for the purpose of teaching the principles of jet propulsion, little is gai11ed b)' using 011c or the other t)·pe of efficienc)·. This is true because, for a given problem, ,,.e can al\Ya)'S con,rert from one t)•pe of efficienc~· to the other. Solving Eqs. (8.32) and (8.35) for the ten1perature ratio and equating gives this relationship: 1

I

198

Jet Propulsion -r-1 ')'

-Pa

-1

P2

11c =

(8.36)

')'- 1 'Y'I•

-Pa

-1

p

Figure 8.15 presents the variation of 77c \vith 11P and pressure ratio. It is noted that for a given compressor stage, 77JJ is always greater than 77,. 1

lJ p-1

0.95 lJp=0.95 I I

0.90

I I

' >-.

u

c .!

·-u

I

lJp=0.90_

0.85

G)

·-u Ci 0.80 ...a

I

TJp=0.85

·-0

"
fi e· They are equal 011ly \Vhcn the process is isent ropic, when both fie and flp are equal to 1.0. An equation of further interest can be obtained by considering the polytropic exponent n of the actual process 2- 3. From Eq. (1.48) an expression for the polytropic exponent can be written as n - 1 In (Ta/ T2) (8.42)

In (pa/ p2)

n

'''here n is the polytropic exponent. A comparison of this equation \vith Eq. (8.34) sho,vs that the follo"ring is true: n-1 -y-1 (8.43)

n

Equation (8.43) can also be deduced from Eq. (8 .35). The above equat ion shows that the polytropic efficiency is merely a modifying factor on the isentropic exponent ( "Y / ("Y - 1)) \vhich produces the actual process or polytropic exponent. This concept shO\\'S us "·here the name polytropic efficiency originated. Once the polytropic efficiency is determined, \Ve can calculate the polyt ropic exponent n, and then use the equations given in Table 1.1 for the polytropic process.

8.8

Compressor Performance Curves and Compressor Stall

The compressor performance curves of an axial-flow compressor are presented in the same manner as they are for a centrifugal compressor. Although it is not shown, application of the dimensional analysis (see Chapter 11 ) to the variables "·hich affect compressor performance results in the groupi11g of t hese variables into the f ollo'\\ing parameters: N

~ For a given value of Reynolds number, \Vhich is of seco11dar)r in1portance, the con1pressor pressure ratio p,,/ p, , is plotted versus corrected air-flo,,· rate G 8,, /~ •• for various values of corrected rp111 N /VO::, \vi t h supcrin1posed cttr\•es of adiabatic efficiency. A typical axial-flo\v cornprcssor perforn1ancc n1ap looks ,·er)· sinular to its counterpart for the ce11i rifugal co111prcssor, and tl1c reader is referred to the one already prese11ted in Fig. 7.14. T l1c re111arks n1ade tl1erc arc nlso applicable to the axial-flo,v compressor. The main difTcrc11cc i11 sl1apcs bct,,·cc11 t 11e f\Y O t)~pes of co111pressor-performa11ce 1naps is t hat tl1c corrected rpn1 li11es 011 the axin.1-flo,,· con1pre ,~ion cl1arts are steeper (greater 11egative slope) tl1n11 t.l1eir cou11tcrpa rts for the centrifugal compressors. Compressor stall of a11 axial-flo\v co111prcssor can be n11al)•zcd fron1 tl1e blade velocity diagrarn as \Veil us f ron1 t l1e pcrfor111n11ce cl1arts. 1;-igttre .17 presents a typical rotor blade \vith its velocity diagran1 for proper angle of attack.

Axia l-Flow Compressors

I

201

It can be seen from this figure that the proJX'r a1tgl E out vapor, cannot, of course, burn becau...~ 0 the fuel-nir ratio is 0. To burn a liquid fuel ~ properly, the fuel must first be ,·aporized so tl1nt the vapor can combine with the Root Radial distance Tip proper an1ount of air to achieve an i.nfiamma.ble mixture. For this reason, thl' fuel F10. 9.5 Turbine inlet temperature profile liminozzles must produce n fuel spray u·hicb tations. (f 'ro1n R ef. B.)

-

\

--,

;I

)(

--

Combustion Chambers, Fuels, and Controls

I

213

ronSst.s of very fine droplets or mist, so that the fuel can readily evaporate and quickly l\1Illbine ";th the air. If the evaporation process does not occur quickly, some of the liquid droplets will be carried down the flame tube \vith consequent incomplete combustion. Figure 9.6 sho'\\'-s a photograph of a \veil-developed fuel spray, the type of spray which will readily evaporate and combine quickly ''ith the air in vapor form.

FIG. 9.6

Photograph of a well-developed fuel spray.

(C-

20

40

60

80

100 120 140

Relative spark energy for ignition

9.9 EfTcct of altitude on spark energy required for ignition. (Courtuy i\ •.4CA.}

F10.

Combustion Chambers, Fuels, and Controls

I

215

After ignition occurs and burning begins, reactants are kept at a high temperature by the heat released from the burning fuel, and the spark energy is no longer required. Sufficient turbulence must be created and maintained in order that combustion be complete, because each fuel molecule requires an exact number of oxygen molecules before complete combustion can occur. For example, 1 octane (CsH1s) molecule requires 12! oxygen molecules to convert completely the carbon .to C01 and the hydrogen to H20. Adequate turbulence insures that each fuel molecule \viii intimately ro.ix 'vith the air and find its proper number of oxygen molecules. Since turbulence is accompanied by a fluid pressure drop, it is essential that only enough turbulence be created to achieve proper mixing, other,vise excess pressure drop will occur with a consequent reduction in over-all engine thrust. Sufficient time must be allocated for the fuel to burn if the combustion process is to be complete. Laminar flame speeds are as lo\v as 5 to 20 ft per sec, and turbulent flame, the type occurring in jet propulsion combustion chambers, are in the order of 60 to 100 ft per sec; thus, the primary combustion must occur in a relatively quiescent zone (see again Fig. 9 .3). If the air-flow velocities are greater than the flame speeds, the flame ";u be blown down the combustio11 chamber and out of the engine to cause a flame-out. The burning time requirement is obviously met by designing the burners to provide a lo\\·-velocity combustion zone where velocities are sufficiently lo\v to maintain the flame within the zone. This can easily be accomplished by using large volume combustion chambers such as those used in furnaces of steam-generating plants, but aircraft engine size precludes extremes in this direction. I n afterburners and in some primary combustion chambers, metal cones or flanges (called "flame holders" and "gutters") are mounted in the burner to maintain a quiescent combustion zone. In accomplishing combustion, particu1.06 larly with respect to the requirement for turbulence, pressure losses occur. Even in 1.05 the absence of turbulence, ho,vever, there is 1.04 a loss in total pressure in the combustion -2 chan1ber because of friction and the momen- ;nu ~"i 1.0 3 "' tum change caused by heating. Figure 9.10 1 02 from Ref. 4 shows the effect of pressure loss in the combustion chamber on the specific fuel consumption of a turbojet engine. The importance of minimizing the pressure loss 2 4 6 10 12 14 may be seen from Fig. 9.10. The methods 8 Combustion pressure loss, l:lP/ P, 1 presented in Chapter 4 can be used to evaluate the total pressure loss in combustion Flo. 9.10 Eff--• ···--~~-1 -~ V'-'h,j'/, - • t • ,--'I/.'. 1;/I Ri11 ::~t· ~-

:

l.. " " . ·- . . ~ y.. Jr,,.,.,,1

1·~·. ·· I. I '/J'. ~ -1' '-i '-+...... t-·ft'~/".l'Yf.. ... -

•

°1'

..

..l.-

· t

~ ·-1 1 ~- • -

·-

I

•

1

_µ

--

.,_ -- .. ,.. • • "'T

•

•

I -- .,___

• •

•

' -·+YH-H+-tt+n~~-1

...._ __ ..,. . 1-:~1-:

•

•

·-

• r..

•'

..... ~

· ·-

· .. . -

........ j• - ..... .,, .... 1.:-::: ~~ ·- -:1 . :.;

!

•

-

j

-·~

f

• '1

t -

. ._ 1--r - t·

l

t ·• 14 1~- t · r ..

•

.- - - - ...

I

- 1

-•

-·-..! • · -

'

,

•

!·-..... _, • - ·- ~ r -

j

-.i

-

•

t

-ti

I .

- •1 • t

I

f

1000

:

I

I

I ' •• f • -+i ~-~--1 f °" "'-1-r--..,·~·---1 t

' -

1- . ~ l- .: 1

.

-t t

t-

•

- -·-

~•...,.._.,

~~------,·~-. ·'"'.'"·---1'

,.

--:~···~ -·•·+-·'t-~--

1 • ... -

l

i

I

I

1500

-

I·•··

t ft

tf

'

•-

I

~

--t

-+,-...,~•--i

2000

2500

Flo. 9.13 Chart for determining the fuel-air ratios in a combustion chamber. (From Ref. 4.) (Based on H . V. = 18,900 Btu per lb; for other fuel-heating values, multiply the fuel-air ratio b)· the factor 18,900/ H . V.)

Figure 9.13 yields a more accurate ans,ver. Entering the chart ''·ith the proper temperatures yields a fuel-air ratio-efficiency factor [(f/ a)11b] of 0.176. Thus, the required ans,ver by chart is

f a This example sho,vs t\vo methods of obtaining a satisfactor)' solution to the fuel-air ratio problem of the combustio11 chan1ber, and for n1ost purposes the t\\O methods presented herein provide adequate results. Ho,,·evcr, if a n1ore accurate ans,,·er is required, then a11 accou11ti11g of tl1e hydroge11-carbo11 ratio of the fuel and the initial fuel temperature is necessary. R eference 6 provides a series of tables, the data of \vhich \Vere computed to account for these as ,,·ell as the other ' 'ariables mentioned before.

Combustion Chambers, Fuels, and Controls

9 .4

I

219

Flame Temperatures

The adiabatic flame temperature, \Vhich is defined as the temperature attained by the products of a chemical reaction through the absorption of all the heat released by the reaction, can be calculated accurately by an analytical procedure. The heat balance equation for a chemical reaction \Vhere all the heat goes into the products of combustion can be 'vritten as T1

H. V .

=

"'T,lir - "'T,h,,

=

T,

(9.5a)

("'T,Ncp dT),,

where N equals the number of pound-moles, and subscripts r and p refer to reactants and products respectively. Other symbols are used as before. This equation states that tl1e heating value of the fuel is equal to the differenci:in enthalpy of the constituents of the reactants and products. When all this hea" energy is absorbed by the products of the reaction as we have assumed, the temper"" tures of the products increase from T 1 to T 1 , the adiabatic flame temperature. To apply this equation \vith accuracy, a calculation of the variation of the specifi,heat values of the products of combustion with temperature must be made. Referencf 7 presents empirical specific heat equations based on spectroscopic data for most gases occurring in the products of combustion. Some of these equations are presented in Table 9.1. These specific heat equations can be applied to solve Eq. (9.5a), but as is evident, the solution is tedious, because to obtain T 1, the method of solution becomes one of trial and error. Equation (9.5a) enables us to solve for the maximum adiabatic flame temperature which accrues from the stoichiometric mixture ratio, as \veil as Bame temperatures for mixtures \vi th excess air. Thus, this equation, together with the specmc heat equations, is the basis for Fig. 9.13 as well as for the cited tables of Ref. 6. Equation (9.5a) also provides the basis for an equation to find the heating value of a fuel at some temperature T 2 \vhen its heating value is kno\\rn at a given temperature Ti. To illustrate this principle, let us re,vrite Eq. (9.5a), assuming that the products and reactants are made up of only t\VO parts, that is (9.5b)

Now the rate of change of heating value 'vith temperatt1re for constant pressure is expressed as a(H. V.)

aT

ahr, p

aT ,,

+

ah,,, aT ,,

ah,,. aT ,,

It is noted that all terms to the right of the equality sign are, b)r definition, the specific heat at constant pressure. Replacing t l1ese ter111s \Vitl1 c 11 values, then separating the variables and integrating gives tl1e desired res\1lt T,

(H. V.)2 - (H. V.)1

=

T,

(cp,

'

+ ep '•

- c,,

•,

-

r._

~ ••

)

dT

(9.6)

•

I

220

Jet Propulsion TABLE

9.1 Specific Heat Equations of Some Gases Range

l\Iax. error

OR

%

540-5000

1.1

5000-9000

0.3

MQ-9000

1.7

540-9000

1.1

54(}-4()()()

0.8

4000-9000

1.4

5-10-5-lOO

1.8

540-0300

0.

36.1 - 2389 + 9.06 x 10 5 VT T T'

5-10-9000

1.7

= 7.92 - .0601 T

400-1100

4.0

Equation

Gas or vapor

=

c,, in Btu/ lb-mole °F

Cp=ll .515-

cal g mole 0 1{ 1

172

VT

0

+ ~

02

.

ll 515 _ 172 • VT

= Cp

Ni

c,,

co

c,,

+ 1530 + T

=

9 47 - 3.47 x 10 3 . T

+

1.16

=

_ 3.29 X 10 3 9.46 T

+

1.07 X 10 6

Cp = 5.76

Cp

5.76

H20

C02

Air

CaH1a

+

0.578 T 1000

+

Cp = 19.86 -

Cp

Cp

= 16 2 _ .

= 9 87 .

Cp

6.53

x

T

x

10'

r2 r2

+ ~·: T +

H2 =

0.05 (T-4000) 1000

20

VT

20 _ 0.33 (T-1000) VT 1000

7

597

v'T 1oa

+ ~ +

1.41

x

10'

T2

" 'here c,,. refers to tl1e specific heats of tl1c products of con1bustion and c.,,.. refers to the specific 11eats of tl1e rencta11ts of combt1stion. For pttrposes of ,,·riting chenucsl reaction equations, air ca11 be considered to be of the follo,,;t1g con1position: \Veigl1t : 1 lb Os volun1e:

+ 3.32 lb N

1 ft 3 0 2

nlolecular ,,·eight :

1

=

-1.32 lb air

+ 3.78 ft 1 1V1 = 111

=

-1.78 ft 1 air

{9.7

28.970 lb per n1ole

..

Combustion Chambers, Fuels, and Controls

9.5

I

221

Gas Turbine Engine Fuels

1''hen_gas tur.Qine engi11es '''ere introduced to the industry, one of the advantages frequently_mentioned as i11here11t to this type of engine \ Vas that practically any type of fuel could be used to operate it. In theory-this is true, hut in practice it is not. An auxiliary comp-onent of-the cngi11e \vhicl1 dictates t he need for lo\v viscosity, clean and homogenous fuels, is the fuel control unit. This uni t, \vhi ch operates at high pressures (some at over 1000 psi), requires pumps, valves and metering cctuipment to be manufactured 'vith very small tolerances so that the fuel qualities just mentioned are mandatory. The over-all selection of a givcn_l_uel is~ased on several .cnnsiderations. The first is ~l--avaiJa:biJity~ the second is factors \vhich \affect engine-per~o~mance (su~h as burner efficier1cy, altitude limits, engi ne rpm limits, carbon depos1t1on, and altitude starting of the main and afterburners) ; and the third is factors of the aircraft fuelS)'Stem (such as vapor and liquid loss, vapor lock, fuel cleanliness, and explosive mixtures in tanks) . When all of t hese factors arc considered, it is readily understood why turbojet engines do not normally operate on "bunker-C" or a similar fuel. ~ 9.14 presents the typical products which a re processed from a barrel of crude oil.

1

Crude

oil

Gasoline 4 0%

Kerosene

6%

Diesel oil

Bun ker fuel s 24%

I 7'X.

Lubricants

3'X.

Others lOX

JP- 3 50%

Flo. 9. 14 Typical product~ fron1 cn1de oil.

J'he small normal yield of kerosene from a bnrrel of crude oil is evident and this is the reason \Vl1y ke~ose11e cn.11not be used i11 turbojet e11gi11es-,,·he11 these are ~111plo~·ed on a large. sen.le, as is the case today. If all gas-lt1rbi11c ('t1gi11rs ,,-ere to start using kerosene, its cost \vould beco111e rapid l)' prohibiti\'C b('CO.t1se ti e111a11d ,,·ould crreatl,· exceed su~ply . The ttse of kcrosc11c 011 a sn1all-scale ba is is r11tirel~· practical: o~ first turboJet-po,vercd aircraft, \vl1i cl1 \Ycrc origi11nll)' lin1it rd i11 11t1111brr, ,,·ere OJX'ratcd on kerose11c (military designation JP-1). Th~ lin1ited yi~ld..ofkCroscnestr 11t1latcd tl1c clcvclop111e11t of a st1itable gas-turbine fuel \Vh1cl1 \vould give a n1ucl1 greater )'ield per barrel of crude oil. ...\s sho,,-n on Fig.

...

222

I

Jet Propulsion

9.14, the fuel developed produced a yield of 503; this fuel is designated as JP-3 and is a mixture of aviation gasoline, kerose11e, and diesel oil. To date five basic fuel types l1ave been developed for military turbine-po\vered aircraft, and t hree types for commercial t urbine-po\vered aircraft. Because of the many different molecular structures \vl1ich a.re possible \vithin the four basic hydrocarbon series that make up aircraft fuels (paraffin, cyclopa.raffin, aromatic, and olefin), it is possible that other fuel types or modifications of existing types \vill be produced , in the future . Table 9.2 from Ref. 8 summari zes the physical characteristics of all of these fuels . Also included is a typical aviation gasoline intended for reciprocating engines but sometimes used in turbine engines \vhen nothing else is available. TABLE

9.2 Specifications for Turbine Engine Fuels Commercial

Military

JP-1 R eid vapor pressure, • psi, max. Reid vapor pressure, • • psi, IDin. Freezing point, °F , max. Specific gravity, max. Specific gravity , min. H eating value (lower) Btu/lb, min . Aromatics, vol. %, max. Olefins, vol. %, max.

Av.gas 115/ 145 JP-3

JP-4

JP-5

JP-6

Type A

Type A-1

Type B

-

7.0

7.0

3.0

-

-

-

-

3.0

-

5.5

5.0

2.0

-

-

-

-

-

-76 0.780 0.739

- 76 0.802 0.751

-55 0.845 0.788

-65 0.840 0.780

-40

-58

-

-76 0.732 0.721

0.830 0.776

0.830 0.776

-60 0.802 0.751

-

18900

18400

18400

18300

18400

18400

18400

18400

25.0 5.0

25.0 5.0

25.0 5 .0

25.0 5.0

20.0

20.0

-

-

20.0 5.0

- 76 0.850

-

-

-

• .,IP-l, kerosene, is cheap in limited quantities, but its suppl)Tis restricted. It hBs a lo\v vapor pressure \Vhich, in operation, means little fuel lOs.5 from fuel boiling, les.5 fire hazard, but difficulty in "air starts" at altitude. l\:erorene is also heavier than other jet fuels; t herefore, more ~nergy can be stored in a gi\·en tank S)"Stem' vol11me. l{erosene has about 103 greater heati11g value than gasoline per l1nit ,·olume. Another factor which is a decided advantage \vitl1 kerosene is its high viscofilt)·. which makes it a better lubricant for tl1e ft1el pumps. Ho\\'ever, it. has been outdated h)r the other fuels and is no longer used . Aviation gasoline, although expe11si\re and po...~s.5ing antiknock characteristics \Vhich are of no value to a turbojet cngi11c, '"as a 11atural fuel to earl)' turbojet aircraft . The fuel \Vas already available at shore buses a11d on carriers ,,·here the jet aircraft \Vere operated at tl1e same time as reciprocati11g engi11e airplanes. \'nere the tv•o types of aircraft \Vere operated togetl1er 1 t.l1e reciprocating t.y1>es ,,·ere usually highperformance airplanes \vl1icl1 required tl1e use of high-perforn1ance fuels. Thus, the a viati on gasoline norn1ally used in jets '"as grade 115/ 145, the n1oot expensi,·e t)i>e· \Vhe11 consideratio11 is give11 to the problenlS of supplying carrier t~k forces located

•

Combustion Chambers, Fuels, and Controls

I

223

far from home ports with t'vo different kinds of fuel from tankers, and to the possible necessity of converting aircraft carriers from a one- to a two-fuel system, the solution of a one-fuel system had much merit. However, \vherever possible, and where the logistic problem permits, efforts have been made in the interest of economy to use inexpensive fuels which are also more suitable for jet aircraft. Gasoline as a jet fuel is different when compared to kerosene. The main difference is clQe to its higher vaPDr pressure 'vhich means easier fuel va ori; ation. This allows for better engine starts at igh altitu-~,-i~0/ schedules wise, if fuel fio\V is increased too ..D :\ , 'F' c; / - ~e} ,.:.'~,o~ / / rapidly, the fuel-air ratio could be~, ' O' 7 a) 0 0 1 ~ . $.0~ ~~ ,,. (.~0 ..,r1-f. come excessively rich and cause loss 0 ~ .v· c,0 0 ~ o"' ~ · .... 0 ~~~ .. ~~~c;, 2 ~ .,c; .. /4, , ' ( ' . r~. of flame due to ''rich blow-out.'' The ~ 0 7 0 · {}'~ / S'0~ 7 ~,o~ it"~ : time for stopping the increase in ,..~ 1 /,.\e"o -~.A -Cl • _,,... ~; ~ > ' c,0 .,_.r fuel-flow rate must also be fixed; 0 u.. // c.;,,eo.. / ,- ,,.,Q /~~"""""·.,....L,.r>'•..,:. .~...... otherwise, the engine could overheat / ,,,.. ... ~..~ "Leon die-out zone" ,,,.. >Y"'~ · ...,_ , ,,....,..:!J.....,....-.,...,..•.,,..,.... ,~ and overspeed. Likewise, a reducr"""·-

..D

High pressure f rlter

-

-

Q)

"'tJ 0

-·-

:::>

u.. '

c 0

E

-........ Q)

0

227

•

Fuel shut-off valve

-

•

-

it

i

Fuel control unit

c

u.. -

CD

:::>

~

F10.

9.16 Typical fuel system of a turbojet-powered airplane.

selector; that is, for a given throttle setting, the rpm 'vill be fixed, even for changes in altitude and flight speed. Suppose the throttle is originally set for 50% rpm and the engine is running under equilibrium conditions (point 1 on Fig. 9.15) . ::\ow suppose a pilot wants to obtain 100% rpm from the engine as quickly as possible. To accomplish this, he jams the t hrottle t o t he full open, or 100% position. The engine, of course, does not respond by immediately going to 100% rpm; the increase in rpm takes time because, in going from 50% to 100% , the kinetic energy of the rotating parts must be increased by a factor of 4, and it takes time to store this additional kinetic energ)in the rotating compressor and turbine. When the throttle is jan1Illed to full open, the fuel flo,v '''ould be somewhere near t he pump-delivery schedule if some mea ns "-ere not applied to keep the fto\v rate lo,ver. Actually, the fuel fio"· increa...~s suddenl.}· to point 2, and then its increase 'vith rpm follo,vs the acceleration schedule. It sh ould be noted that at any rpm, the vertical distance bet,,·een the a cceleration and ste8d~- state schedules represents the fuel available for acceleration of the engine; there.fore. it is desirable to have the accelerat ion schedule as higl1 as nearness to the "rich blow-ou t ., zone \vill permit. When the engine rpm gets 11ear 100% at point 3, the fuel flow has to be reduced to the equilibrium position, point 4. Tl1is reduction in fuel fio,, from point 3 to point 4 is accomplished along the gover11or schedule. The go,·em or, which is normally a flyball '"eight system, senses e11gi11c rpn1 through the centrifugal force of its weight system and throttle positio11 throt1gh a spri11g force. ''lien these t"·o forces are in balance, engine operation is 011 the stcad)r st.ate fu1e. In our illustration, \vhen the throttle is jan1med to ft1ll opc11, tl1c tl1rottle spri11g force is greater than the centrifugal force, and fuel no,v follo,,·s tl1e acceleratio11 schedule w1til the fi,· ball centrifugal force starts to bala11ce tl1c tl1rottlc spri11g force, poit1t 3. The procedure for decrcasi11g e11gine rp111 is sin1ilar to tl1c above descriptio11. ''7hen the throttle is reduced, the fucl flo,,· drops to the deccleratio11 scl1cdule a11d follo,,·s this plot until it approacl1cs tl1e rp1n selected by the tl1rottlc, at ,,·J1icl1 point it follo,,·s the horizontal governor schedule.

..... 228

I

Jet Propulsion

It is emphasized that the schedules sho,vn on Fig. 9.15 apply to one altitude and one flight speed. As the operating altitude or air tcinpcrature is increased, or the for,vard flight speed is decreased, the \vl1ole fuel-control schedule system, 'vith the exception of the pump-delivery schedule, moves d o,vn,vard . This vertical displacement of the schedule system is accomplished by a barometric control un it, normally a gasfilled bello,vs \vhich delivers a spring force to the fuel-cont rol uni t in accordance 'vith variations in pressure altitude, flight speed, and temperature. Some of the ne,ver jet propulsion engines have a much more complicated fuelcontrol system t han the one i11dicated above. These are the variable geometry engines, those equipped \vith variable-area exhaust nozzles, variable-angle inlet guide-vanes, or variable-angle compressor stator-vanes. On engines of this type, t he fuel-control unit, in addition to sensing these variables, must also sense turbine-outlet temperature and control the exhaust-nozzle area or inlet guide-vane area. On units \vith variablearea exhaust nozzles or on engines 'vi th afterburners, an additional scheduling system dictated by nozzle area must be sho,vn vertically at t he 100% rpm line on Fig. 9.15. Fuel-control units in use today are predominantly of the hydro-mechanical type; however, there are some current applications of the electronic or combined electronichydromechanical types. The electronic types have the advantage of being able to sense and process more variables and more complicated variables to easily amplify the signal strength, and in theory to do a better over-all job; however, the big disadvantage of this system up to the present time has been its generally poor reliability. In the fu t ure, this problem 'vill probably be alleviated, but until that occurs, the hydromechanical units \Vill prevail as the predominant type of fuel control. Complex-type engines, such as ones \vith variable exhaust-nozzle areas, are forced to use the more complicated electronic or combination electronic-hydromechanical type fuel-control 11nits. P ROBLEMS

I. A turbojet engine combustion chamber receives air from the compres.50r at a total temperature of 600°F and discharges it.5 gas product.5 to the turbine at 2000°R. If the burner efficiency is 903 and the heating value is 18,900 Btu per lb, calculate the fuel-air ratio by the two methods prescribed herein. 2. What is the fuel-air ratio for the burner of Problem 1 if fuel ha'ing a heating ' -alue of 18,000 Btu per lb is used? 3. Air enters an afterburner at 1200°F and is heated by burning fuel to a t~mperature of 3200°R. If the heating value is 18,900 Btu per lb and the burner efficienc)· is 03, "~hat is the fuel-air ratio? 4. Calculate the fuel-air ratio required by t\ turbojet c11gine nt sea lc,·el, 10,000, 20,000. 30,000, and 40,000 ft for a fligl1t .i\[ach nun1bcr of !if o = 0. , and a turbine inlet tempersture Ti. of 1640°F at each of these altitudes. Assume thnt tl1c con1prcs....~r has a constant rompn.'5sion ratio of 7.0 at each of these altitudes, that t11e fuel-heating ,·nJue is I ,900 Btu per lb, and that the burner and compressor efficiency arc constant nt 1003 . 5. .A combustion cl1nm?er hns n fuel-air ratio of 0.018 '''hen using n fuel of 18,900 Btu per lb heat1ntvnlue and operating at a burner efficiency of 903. If the burner outlet temperature is 1900°R, \vhat is the burner inlet temperature?

Combustion Chambers, Fuels, and Controls

I

229

6. Compute the maximum adiabatic flame temperature (°F) of octane (CJI1s) whlch has a gaseous lower-heating value of 19,256 Btu per lb at 25°C. 7. Compute the adiabatic flame tem1)erature (0 R) of octane (C8H 18) (gaseous lower H . V. = 19,256 Btu per lb at 25°C) \vhen the fuel-air ratio is 0.02 and the initial temperature before combustion is 1000°R. Find the answer by using the specific heat data of Table 9.1 and al.so by the chart of Fig. 9.13. Discuss the results of the two answers thus obtained. 8. Add the afterburner fuel-control schedule to Fig. 9.15. REFERENCES

I. Wilkinson, P. H., A ircraft Engines of the World, P. H . Wilkinson, New York, 1961. 2. Olson, W. T ., "Combustion for Aircraft Engines," presented at Fifth International Aeronautical Conference, June 1955. 3. The Mamba Propeller Turbine, Siddeley-Armstrong Motors, Ltd. 4. Pinkel, B., and Karp, I. M., "A Thermodynamic Study of the Turbojet Engine,'' NACA Report No. 891, 1947. 5. Pinkel, I . I ., and Shames, H., "Analysis of J et-Propulsion-Engine Combustion-Chamber Pressure Losses,'' NACA Report No. 880, 1947. 6. Keenan, J. H ., and K aye, J., Gas Tables, John Wiley and Sons, Inc., 1948. 7. Sweigert, R . L., and Beardsley, l\tI. W., "Empirical Specific Heat Equations Ba...~ upon Spectroscopic Data,'' Bulletin No. 2, The Georgia School of TecbnologJ·. 8. Gartenberg, A., CRC Aviation Handbook, Fuels and Systems, Reli-•.\-Test Corp., 1961. 9. Aviation Week, November 12, 1956, p. 51. IO. Pinns, ~1. L., Olson, W. T., Barnett, H . C., and Breitwieser, R., ''X.i\.C•.\ Resesreh on Slurry Fuels," NACA Report 1388, 1958.

l0

10. 1

Gas Turbines

Introduction

The primary purpose of the gas turbine in a turbojet or turbofan engine is to extract mechanical energy from the hot gases delivered to it by the combustion chamber and to supply shaft po\ver to drive the compressor. An incidental factor is that the turbine must also supply po\ver to the auxiliary equipment, such as fuel pumps, oil pumps, and electrical generators. In turboprop and turboshaft engines, the turbine must also supply power to drive the propeller or helicopter rotor. The turbines in modern jet propulsion engines develop up to 60,000 hp or more. Tbe.se turbines have from one to four stages, depending on the compressor pressure ratio or the shaft po,ver output. A turbine stage, by definition similar to that used for compressors, consists of one ro\v of stator blades f ollo"·ed by a row of rotor blades. One exception to this definition, however, is a velocity-compounded stage called the Curtiss stage, 'vhich consists of four rows of blades. This stage mil be dL~u.ssed later. Figure 10.1 is a photograph of the rotating assembly of the Rolls-Ro)·ce Xene engine, sho,ving a. typical single-stage turbine installation. On this particular engine, the single-stage turbine is directly connected to the main and cooling comp~rs. Figure 7.12, in the chapter on centrifugal compressors, sbo"-s the rotors of a two-stage

1"10. 10.1

R otating assembly of tl1c R olls-llo)·ce Nene engine. 230

(From R tf. 1.)

Gas Turbines

I

231

turbine and the rotating parts driven by this turbine in the Rolls-Royce Dart turboprop engine. In Fig. 7.12 the accessory drive shafts, the propeller-reduction gearing, and the compressor rotors are sho,vn. Figures 11.1 and 11.3 in Chapter 11 sho\V three-stage turbines. The t\VO turbines are different, however. Tl1e JT4A turbine illustrated in Fig. 11 .3 is split into t\vo sections. The first stage drives the seven-stage high-speed compressor rotor . The second and third turbine stages drive the eight-stage lo,v-speed rotor th rough a shaft that runs inside the sl1aft of the higl1-speed compressor. This separation allo\vs the t\VO sections to operate at different speeds to provide the t\vin-spool matching described in Chapter 8. The three stages of tl1e CJ805 turbine sho,vn in Fig. 11. 1, on the other band, all drive the 17-stage compressor, 'vhich incorporates variable stators for matching. Four-stage turbines are illustrated in Figs. 10.2 and 13.1. The schematic dra,,ring of the Pratt & Whitney JT3D turbofan engine in Fig. 10.2 sho,vs a four-stage turbine split in the same manner as the JT4A turbojet engine. The first turbine stage still drives the high-speed compressor. A fourth stage is added to the second section , however. I t provides the additional po,ver for t he t'vo-fan stages at the front of t he low-speed compressor. The four-stage turbine in the General Electric T64 turboshaft engine shown in Fig. 13.1 is also split. In this case, ho,vever, the first t,,.o turbine stages drive the 14-stage compressor, \Vhile the last t'vo turbine stages furnish shaft power. I

•

lnlet a ir .

L.

1

IJ

.. ..

•••••••

r'

F10.

10.2

Schematic drawing of Pratt & "\'\7hitney JT3D turbofan engine, sho\\-ing a four-stage turbine. (Courtesy Prall tf: lVhi l 1iey Division, Uniled .4. ircraft Corp.)

From the above it may be see11 tl1at one tt1rbi11e stage dri,·es as n1an)· as 7 compressor stages. This is true becat1se tl1e tt1rbi11c operatio11 is i11 a fa,·orable "do\\'llhill" pressure gradient; thus, large prcsstire ratios ca11 exist. across 011e turbine stage, \vhereas only small presstire ratios, less t.l1u.11 1.-10, exist i11 ct1rre11t prodt1ction axial-ft 0 ,,compressors. There are t'vo basic types of turbines; t.l1e axial-Ho,,· t)•pe (tl1e t~·pc illt1strnted thus far), and the radial- or ce11trift1gal-flo'v t)•pc . The axial-fto,,· t)·pe has bce11 u..~d almost exclt1sivcly i11 aircr11ft gas-tt1rbi11c c11gi11cs to date a11d ,,·ill be discus.5ed in detail in this cl1a pter. It is \Yortl1,vl1ilc to n1c11tio11 tl1e nlai11 featu res of the radialfto\v type, 110,vever, si11cc they arc t1scd extc11sivel)• i11 sucl1 sn1all tt1rbine-po\\·ered auxiliaries as air-tt1rbi11e motors for aircraft electric po,ver generatio11 and air-conditioning u11its for cockpit or equipn1e11t cooli11g, a11d occasionally in small turboshaft

232

I

Jet Propulsion

engines like the Boei ng T60. Figure 10.3(a) sho\VS a cross-sectional s ketch of a radial. flow turbine; Fig. 10.3(b) is a photograpl1 of the rotor of the ,.f 60 engine sbo\ving the radial turbine on the right. Note that this turbine is like a singleBurner g os es faced centrifugal compressor with reverse Bow. The radial-flow turbine has several advantages over the axial-flow turbine: Ta ilpipe (1) It produces an axial-Bow con cenShaft trated exhaust jet. (2) It extracts more N ozzle energy per stage because of centrifugal effect. (3) For certain a pplications, par(a) ticularly for very small-size engines, it has higher efficiency because of lower leakage losses. (4) It offers relatively simple variable turbine stators for matching turbine and compressor over a wide range of load conditions.

(b) Flo. 10.3

Radial turbine: (a) Schematic dra,,·ing. (b) Photograph of the rotor of a Boeing T60 cngino, radial turbine on the right. (Cou rtesy The Boeing Co11rpany.)

The disadvantages of t l1e radial turbi11c nre: (1) poorer peak efficienc)·, (2) much less compatibility for multistagi11g1 a 11d (3) lo,,·cr inlet te n1pemture capabilit)' bec8tL't? of t he much larger \Vetted area. These cl1nracteristics of tl1e radial turbine help to explain its limited applicatio11s me11tioned abo\'C. Table 10.1 on page 238 sumrnariz('s the one-dimensio11al eqt1ations for radial turbines; a n1ore detailed discus.5ion of this topic is given in Ref. 2.

10.2

233

I

Gas Turbines

General Thermodynamic Analysis

Figure I0.4 depicts the energy balance of a gas turbine. It receives high-temperature high-pressure gas at section 4, extracts energy from it in the form of shaft \vork, and discl1arges the gas at a lo,ver level of pressure and temperature. The energy equation for this machine, when written for no change in potential energy and an adiabatic • process, IS

Shaft work W,

I

2

v. + h, = 2g

I I

I

4

5

Fro. 10.4 Turbine energy balance.

(10. I)

Solving Eq. (IO. I) for the turbine work and expressing the gas energy content at sections 4 a11d 5 in terms of total conditions gives if't = Ji,. - ht, = Cp(T, . - T,,) (I0.2) Total co11ditions arc used in preference to the sum of static enthalpy and kinetic energy, because it is far easier to measure and evaluate t'vo total enthalpies than it is to measure t'vo static enthalpies and two velocities. Equation (I0.2) merely states that the turbine \VOrk is equal to the change in energy content of the gases passing through the turbine. Figure 10.5 sho,vs the actual and ideal turbine process on an li-S diagram.

T or h About 1600° F to 2000° F for • maximum power

TJ, =

Ah 1 Ah' I

t

.6. h,

5'

s Fro. 10.5

Ideal and actual adiabatic turbine ex.pans.ion proce..."S.

Kate that the ideal and actual processes for the tt1rbine are defined in the same manner as they 'vcrc for the compressor, 11an1ely, bet,,·ce11 tl1e san1e t,,.o total pre~ure lines. For the assumption of an adiabatic flo,,· process bet,,·een pressures Pt . and Pt ., '"e can have either the ideal isc11tropic process 4- 5' or a gc11ernl adiabatic process '''ith friction 4- 5. I t is cvide11t tl1at for an adiabatic prOC('SS bet,,·ee11 f \VO pres.5ure lines, maximum turbine 'vork 'vill occur nlo11g a co11sta11t e11trop)· path; therefore, great effort is cxpc11ded to n1i11i111ize tl1e frict.io11 i11 turbi11cs. The turbi11e adiabatic efficiency is defined as Ji, . - It,, (I0.3) Tit = ,

Ah,

•

234

I

Jet Propulsion

The 'vork produced per pound of fluid can be expressed in terms of the ideal path 4-5' as ~ =

I

rttflht

=

I

(

f'/tCp(Tt. - T,.)

10.4)

Since, by definition, the total pressure at points 5 and 5' are equal, the turbine work can be expressed in terms of pressure ratio and inlet temperature as -r-1 'Y

~ = T/tC11 T1.

{10.5)

1-

The convenience of the polytropic or small-stage efficiency rt,, in analyzing compressor performance \Vas poi11ted out in Chapter 8. The polytropic efficiency has a comparable usefulness in turbine performance. Follo,ving a derivation similar to that in Chapter 8 for compressors, it may be sho,vn that, for turbine expansion

1

'n -

n

1

= T/p 'Y I

(10.6)

"'(

where T/p, is the turbine polytropic efficiency or the turbine small stage efficiency. Equation (10.5) can thus be written ,,,1' , -r..:..._1_..,

~ = CpTt. 1 -

PPt.

..,

n-1

= CpTt.

1-

Pt.

..

(10.7)

'· p '· By comparing Eqs. (10.5) and (10.7), the relation bet,veen the adiabatic and polytropic efficiencies is seen to be

1-

Pt. Pt.

,,, ' = - - --'-----

(10.8)

,._1

..,

1-

Pt. Figure 10.6 presents the variation of T/t 'vith '11> and pressure ratio. 1.00 Note that .,,, is always greater than llp1 = l 00 C'" .,,,,,. This is caused by the reheat effect, ~ 0.95 ltpr = 0 .95 v which occurs " ..hen a gi\Ten actual expanc 0 .90 sion is broken into a number of smaller llp1 = 0 .90 v ·expansions, as in a. nlultistage turbine. ~ 0 .85 'tlp1 = 0 .85 v Considemtion of the actual expanfilon lJp1 = 0 .80 ·~ 0 .80 _o proces.5 sho,,·n in Fig. 10.5 as a number 0 0.75 '""O of separate steps "-ill show that the re-< 0 .70 • ' 1 2 3 4 5 6 7 8 9 10 heat effect actually increases the t otsl available isentropic enthalpy drop. T his Pressure ratio, P,4 / P,3 nlso increases the O\Ter-all adiabatic effiF10. 10.6 Variation of adiabatic cfficicnc)' '11 'vith ciency above the stage efficienc:>· as pressure ratio and poly tropic efficien cy '11t, . sho,,rn in Fig. 10.6. Considering either Eq. (10.5) or (10.7), the important factors " ·hich affect turbine \Vork, na1nely, turbi110 ~fficiency .,,,, tt1rbine inlet ten1perature T, ., and turbine pressure ratio Pt ./ Pt, n1ay be see11. An i11crcnsc in a11y of these three factors allo\v the turbine to develop n1ore 'vork per pound of fluid . Onl.)· small gains can be expected to accrue from improven1cnts of turbi.i1e efficiency, since present efficiencies are up near t11e peak of dcvelopme11t, 85% to 90% . Because of gas friction o,·er the I

-

Q)

-

I

I

l

I

I

I

''ill

Gas Turbines

I

235

many turbine blades and the leakage losses over the blade tips, turbines inhe rently have about 103 over-all loss. In spite of the Jact that \Ve are near the point of diminishing returns on the improvement of 'It, and a marked increase of 'It is not possible, development work should continue to minimize turbine losses, because a11y gain of turbine efficiency reflects a further gain in t l1e over-all engine performance. rrhe prospect of operating turbines at higher inlet temperatures is indeed an attractive one to achieve more work per pound of fluid because, ns sho,vn in Eq. (10.5), the turbine work is directly proportional to the absolute temperature of the entering gases. Present-day maximum turbine-inlet operating temperatures are about 1800°1..- . This value, as shown in-chapter 9, is still considerably belo\v the adiabatic flame temperatures of hydrocarbon fuels. It is possible to boost the gas inlet temperature ~ppre­ cia.bly. In fact, this value ca11 be more than doul>le 0

3.2 -

8

3.8

...

2.8 16 Nonafterburning

0 3.4

....

-c

CD ~

... 0

c

Q)

2500 2000

:; 2.6 ;,... 2.2

4

II> 0..

-...

, , ,

:; 1.8

- 1.4

..c

/ 2500

u ~ u

1500

I

2 .0 -

Q)

I

I

I

''

------OF ..' 2500

'

I

I

I I

"' Q)

-

I

-

>

0 1.6 -

No cooling - - - A ir cooling

1500 1.0

1. 1

1.2

1.3

1.4

1.5

Relat ive specific fuel consumption

(a)

1.6

1.7

Turbine - inlet temperature,

,

I

I

0..

OF

·-

0.6 0.9

,

-- --,.._-

2500

I I

_, 2000

'

'2000

CD

1.0

I

...c.

Turbi ne - inlet temperature,

> 0

°'

-

..."'0

4

·...0

-·-c

8

8.. 2 .4

Compressor 8 pressure ratio

-II> 3.0

-II>

Compressor pressure ratio

1.2 1500 0 .8 0 .7

0 .8

0 .9

1.0

1. 1

Relative specific fuel consumption

(b)

10.16 (a) Effect of turbine inlet tempern.tt1ro on turbojet performance a t i\Iach 2.0 in the stratosphere; afterburner gas tem1>ern.ture 3000°F. (b) Effect of turbine inlet ten1perature on tur~ prop engine pcrformancc at sen level static. ( f'ro 111 .t'l.t1C.4 R ,\J E54 Tt.3.) F10.

Figure 10.16(a) sho,vs t11at, for a no11afterbt1r1\i11g turbojet engine at l\Iach 2.0 in the stratospl1crc, i11crcas ing T,. from 1500°F to 2500°F results in an increase of up to 1603 in thrust otttput, 'vith so111e i11crense in sfc. The figure also sho'vs the effect of increas ing T,. for afterburni11g c1\gi11cs (sec Cl1apter 12 for further d iscussion of afterburners) ,v}1ere additio11al fuel is bt1r11ed do,,·11strearn of the turbine to pro,-ide

I

246

Jet Propulsion

thrust augmentation. Here it may be see11 that increasing the turbine inlet temperature not only increases the augmented tl1rust output but also decreases the ..sfc. The explanation of this is that as T t. is increased,~ more of the fuel is burned ahead of the turbine. This improves the cycle efficiency (a11d sfc ) because the combustion, and hence temperature rise, occt1rs at a higl1er pressure level than in the afterburner. Figure 10.16(b) sho\\'S that the same increase i11 Ti. results in an increase in shaft horsepo\ver output for a turboprop at the sea-level static condition comparable to that for the turbojet at l\1ach 2.0. Unlike the turbojet, ho\vever, the sfc of the turboprop engine is significantly decreased along \Vith the increased output. Figure 10.16 also shows the effect of bleeding air from the compressor to cool the turbine blades. While this bleed does decrease the performance improvements, the results still sho\v • • • very impressive gains. These significant gains have stimulated mucl1 research and development. A good part of this \vork has been directed to,vard improving the high-temperature strength characteristics of metals and alloys. From this effort has come a series of cobaltand nickel-based alloys that offer significant 11igh-temperature strength improvements over iron-based alloys. Use of these materials has allowed a slow but steady increase in operating temperature during the last ten years. It is sho\vn in Ref. 10 that turbine inlet temperatures for civil and transport aircraft turbine engines have increased about 200 °F to 1800°F during the last fiv-e years. Fundamental metallurgicafConsiderations . l iffiit the improvements wliich can be -realized through material research only. As a result, an increasing part of the R & D for higher temperatures is being directed toward turbine cooling to allow operation at higher temperatures \vhile maintaining the turbine material temperatures within allowable limits. I t is indicated in Ref. 10 that inlet temperatures approaching 2500 °F may be reached soon for engines po"'·ering civil supersonic transports. -Turbine blades can be cooled by several different methods, but basically, each method utilizes a cooling fluid that passes through t.he blade so as to keep the blade metal within safe operating limits. Figure 10.17 presents the theoretical po,ver gain

-

-

-

300

250 3000 °F -.fi. ~

Liquid-cooled

c 200 0 O>

~3000°F

...

4)

:J 0

gas

150

Q.

...0.

~ ..... 100

2000 °F ~ 2000 ° F

50

0

0.05 Coolant

Flo. 10.17

0.10

0.15

gas flow ratio

Theoretical engine po,ver gain by several methods of cooling blades.

(Courtuy .cV.4.C'.4 .)

Gas Turbines

I

247

(based on cooling effectiveness) of three different types of hollo\v blades plotted against coolant gas-fto,v ratio. It is noted that the liquid-cooled blade produces hy far the best cooling effectiveness - at a 3000°Ji' gas temperature it prodt1ces a po,ver gai11 of 230% \\•ith a coolant gas-flo''' ratio of abot1t 0.055; the fin11ed air-cooled blade produces a po,ver gain of about 195% at coola11t gas-ft;,v ratio of about 0.11. The fully open hollo\v blade produces practically no gains at all. Even thougl1 the Ji(1Ui(l-cooled blade provides much better cooli11g effectiveness, the complication of ha11dli11g a separate cooling liquid in an aircraft applicatio11 and the fact that air-cooled blades can produce appreciable po,ver gains (see fin11ed-blade curve), make utilization of compressor bleed-all: appear to be the best over-all system for lade cooling. The criteria for achieving good cooling effectiveness come directly from t l1e principles of heat transfer of a fluid in a closed duct. To attain high heat-transfer rates in such a system, it is necessary to meet t'''o basic requirements, namely, (1) flo,v the cooling fluid \vith a high Reynolds number, and (2) provide a large surface area for the beat-flo,v path. With these points in mind, it is obvious why Fig. 10.17 sho,vs the cooling effectiveness increasing \Vith coolant gas-fto\v ratio, and why the finned blade is many times better than the open hollo'v blade. The open hollo'v blade doe.s very little cooling, because, in operation, a boundary layer \vhich acts as an excellent insulator to heat transfer, forms o\•er the inner surface of the blade. The insertion of fins or tubes in the blade causes the cooling air to pass over greater surface area \vith high turbulence or a rubbing action, which produces a turbulent boundary layer that readily passes beat. ~other disadvan!_a_ge of th_~.P.e.n 4.2 ho~ow blade is its structural limitation. Type of cooling Withou.t fins or supporting n1embers, the Porous open hollow blade vibrates readily and a 1 3.8 v.rith large magnitude at it§_J:esonaDt fi:equency to prgduce a ''breathing aQtion'' "' 25 fins ::> 3.4 0 wth consequent fatigu.e failure. Other ... 0 Cl. cooling methods, such as filn1-cooling or "s,veat-cooling,'' are available for application to turbine blades. Figure 10.18 (page 71 from R ef. 11) sho,vs the cooling effec·-u 12 fins tiveness of this system as well ns the n1ore ~ 26 4> normal system. 4> > AB sho,vn, the porous blade provides 0 Ci 2.2 a very effective method of cooling. As 0:: a result, considerable effort has been devoted to developing this tech11ology, and Hollow 18 recently Curtiss-\Vright Corporatiort ar1nounced den1onstration of porot1s blades in a turbine rotor applicatior1. 14 Thus far, \VC hitve disctisscd t11e perforn1ance advantages obtained by rooli11g turbi11e blades to i\llo\v e11gi11c opcrntio11 004 012 016 0.08 at turbine-inlet tc111peraturcs l1ighcr tl1nn Coolant flow/ gos flow those used currently. Anotl1er distinct F 1u. ~0. 18 Coolin~ cffectjveness of Yarious t~-pes feature of utilizi11g cooled blades is tl1nt it of ruN:ooled turbine hlndet:. (Fror11 R~J. J J .)

-

248

I

Jet Propulsion

allows for operation at present-day turbine-inlet temperatures \vi th turbines constructed of materials containing a smaller quantity of strategic metals. Like any other system \Vhicl1 presents adva11tages to engine performancei._ there are . also disadvantages incurred by cooling t urbir1e blades. A cursory examination of the • turbine b1ade-cooli11g problem leads one to thi11k that the solution is relatively simple, that is, merely pass some compressor bleed-air througl1 hollo\v turbine blades and the job is done. A more detailed study of the subject, ho,vever, \vill sho\V that the over-all turbine blade-cooling problem is very complex. The basic problems of heat transfer in a duct are made more difficult and more complicated because the cooling air \vithin the blades is accelerated by centrifuga l forces \vhile it absorbs large quantities of heat and the tendency for internal gas choking is prese11t. At a given turbine-inlet temperature, an engine \vith cooled blades suffers a definite performance loss relative to one with uncooled blades (see Fig. 10.16) because the coolant air bled from the compi:essor does not take part in the combustion process, nor can it develop power in the turbine. . I t also requires pump!!lg wg_rk to_force it through the cooling system. Perhaps _t~ \ • ,1, greatest disadvantage ~f turbine blade-cooling is cost due to complexity 1n .fabricat~on. 1 I t has already been pointed out that the simple, open, hollow blades Clo not coof \Vell enough to work; the ones \Vi th fins, inserts, bundles of tubes, and so on, are difficult to manufacture, but do provide adequate cooling. These complex cooled blades must be nlanufactured properly. In addition to providing adequate cooling, they must still withstand the high stresses imposed on them by centrifugal loads. The turbine rotor required for cooled blades is also difficult to manufacture. Figure 10.19 presents a I photograph of a split rotor used in conjunction \vith an experimental blade-cooling µrogram described by l{emp and lVIoseson in Ref. 12.

--

I

I

I I

F10.

10.19 Split turbine rotor sho\ving cooling nir pa.,c;sagcs leading to blades.

(CourtLsy ,\ ".4CA .)

This particular rotor i11corporatcs 54 111acl1ined ' 'a11es to lead tl1e coolant f ron1 the open 11ub sectior1 to cacl1 i11dividual blade. I t is obvious that the fabrication cost of such a rotor is l1igl1. 111 additio11 to tl1e fabrication problems, a rotor suppl)·ing cooling air is furtl1cr cor11plicated by tl1c air-sealing problen1 at the section " ·here the coola11t is brougl1t ir1to tl1e rotor hub.

Gas Turbines

I

2.49

Some advantages and disadvantages of turbine blade-cooling have hecr1 disCUH.'iC, or P13/ P 12

Compresso r Plot

H>,

Turbine Plo t

or Pi. /p,,

N

.fi,

A ir flow, G ~ /6,2 F10.

Air flow, G

•

V0:. /o,.

10.20 T ypical performance plots of compressor and turbine.

The compressor plot, of course, looks familiar because of its close resemblance to the pressure plot of Fig. 8.19. Operation above the surge line is imp08.5ible, and operation near the surge line is critical ; therefore, the operating line of the compressor (a function only of the compressor-fio\v resistance) is fixed below the surge line "-ith a reasonable margin of safety. The turbine plot is quite different from the compressor plot. This is d ue to the fact that critical fio\v can exist in the turbine. As shown for a given turbine rotor rpm, the horsepower output increases with \veight-fiow rate until critical flow exists, and then increases f urtl1er at a constant value of gas-flow rate. Turbine po,ver can i11crease for a given value of '''eight-fto\v rate, even at the critical value, if the turbine over-all pressure ratio is increased by additional stages. ..\.n operating line is also sho\vn on t he turbine diagram. For proper matching to occur, a given set of \veight-.fto\v rate and rpm values on the compressor-operating line n1ust be equal to the same set of values on the turbine diagran1. This matching ma)· be shown analytically by using t he pertinent parameters that describe the perfor mance of t he compressor and turbine. In Chapter 8 it \Vas shO\\'Il that the paramete rs "·hich describe compressor performance are tl1e f ollo\ving: N

VO:: Applying the dimensional analysis developed i11 Cl1apter 11 to the variables affecting turbine performance results in grot1ping these variables i11to tl1e following parameters :

N

VO::

Gas Turbines

I

251

· d gas flo,vs througl1 the compressor and turbine are equal, A ~'" ·ng that t hc air a11 . l . .n=umi .. . t. point tile parameters must have the follo,v1ng re at1onat each equ1l1br1um opera i11g . 1 ships: r;:;T Gv Ot. X Pt. X Pt. X '· 2 (10.28) t ot. Pt. Pt. Ti.

•

or

GV0:: (I'·

1

and

N

_

N

VO:: - VO::

X T,\ 2

(10.29)

T,.

At any equilibriu111 operating point, the t empera ture ratio 1 1./ T 1, 1 the corCompresso r rected speed N / 81 ., and the corrected surge line ~ air - fiO\V G( 8,,/ o,,) determine t h e matching completely. This may be sho,vn as follows: N ; vn;_ and G(ve:J o,, ) determine the compressor pressure ratio p1 ,/ p,, and h ence the compressor power required, HPc. P, 1 G( e,./ 01 . ) and the temp era ture rise in P.; the combustor, which is determined by T,./ T,,, in turn determine the combustor pressure ratio, p,./ p,.. NI [from Eq. (10.29)] and GVo:]o,.) [from Eq. (10.28)] determine the turbine pressure // ratio, p,./ p1., and hence the turbine power / delivered, HP,, 'vhich must equal HPc for • equilibrium. AJ3 a result the matching or • G ,19,; equilibrium operating conditions for como,, pressor and turbine may be sho,vn as lines of constant temperature ratio T,. / T,. on F10. 10.21 .Equilibrium operating conditions for compressor and turbine. the compressor n1ap. This is sho\vn in Fig. 10.21. 1

v

v

v

ve;_

~to

and s.sor

th a

S...'Or

. for Lntil -ate. tir.al An .cur,

10.8

Dlb1

.\' ~

ance hicb

~ting

·te~:

!

Turbine Blade Failures

Probably the most critical part of a gas turbine engine is the turbine rotor blade. As stated before, high ot1tput is a function of turbi11c-inle t t en1perature. Therefore, turbine-inlet temperatures 011 curren t e11gi11es arc 11ear 111axin1un1 nllo,,·able ,·alues for full throttle conditions. Today's production e11gi11c n1a.-xin1um allowable operating temperature of about 1800 °F is made possible b.)• prod\1ci11g the rotor blades of the best high te111perature alloys obtai11able. Si11ce l1igl1 perforn1ance is a ~c criterion of aircraft e11gi11es, a11d si11cc '''e \VOt1ld prefer t o operate nt 3000 °F in stead of 1800 °F , turbi11c rotor blades at full po,,·cr co11ditio11s arc operated ,,·ith a relati,·el)• lo"· factor of safety . l•'ailt1re of rotor l)lades is 11ot a co111111011 occurre11ce; 110,,e,·er , isolated failures do occur. T'''O types of failt1res that ca11 occur are illt1strated in Figs. 10.22 and 10.23, \vhicl1 arc pl1otographs of l\\' O turbi11c rotors 11a,·i11g failures of t,,.o se\·erities. 1'11c pl1otographs clearly sl10\v 110,,· serious a blade failt1re can be. Serious failures occur \vhe11 a blade breaks, beco111cs 111esl1ed ,,·ithi11 the other bl?ldes and ''ipes out

252

I

Jet Propulsion

•

•

Flo. 10.22 Rear view of gas turbine wheel showing failu.re as the result of operation at excessjve temperatures. •

F10.

10.23 View of first-stage turbine rotor sho"·ing blade damage and failure.

the follo,ving ro'v of stator and/ or rotor blades. Broken blades can fl)· through the engine and into the aircraft structure. \~en a .rotor blru.le_gl the last stage failsJ the results are not likely to be serioua becat1se tl1e blade \\'ill usual I)'~ ot1t of_the tailpipe. In a case like that1 the pilot may be a\varc of it only b)r o. slightly rot1gh-run11ing e11gine One interesting operati11g phcnorncnon in a specific turbojet e11gine "'"SS traced as the cause of many turbine blade failures. 111 order to operate the rotor blades a t uniform stress throughout tl1e blade length 1 the n1a11ufacturer of that engine desigi1rd the combustion chamber to produce a. tc n1perature gradient across the blade length high temperature at tl1e blade tip scctio11 \vhero the stress is lo,v, lo,,·er temperature~

I

Gas Turbines

253

progressing toward the blade root (\vith the lo\vest value at the ~oot section~vh~re ·the stress is high~r. This design procedure is optimum because it works the entire biade with an equal factor of safety; therefore, it was not readily unde~tood 'v.hy this particular-type engine experienced more blade failures than other engines 'v?1ch were designed with no temperature gradient across the turbine blade. _Af ~r ~ght investigations the cause was determined. I t \Vas found that the temperature gradient shifted: at high altitudes and high rpm, the high gas temperature shifted to,vard _the root section. Figure 10.24 from Ref. 16 sho\vs some test results of this temperature gradient shift. Seo level 25,000 ft. 35,000 ft. M = 0.7 07 M 1300.----_.!!...M~-~O~--~---.---~~=~-:__~..,.-~r-~~:--~--,1300 1200

1200 u...

0

'

...0

Q)

-

1100

1100

0

1000

~ 1000 E Q)

-

" 'O

I)

900

0 u

·-

"'O

c

-

800

'TR-PP-403, July 1951.

•

11

11 . 1

Turboiet and Turbofan Engines

Introduction

The chapters thus far ha\'e dealt \Vith the individual components of jet propulsion engines, and considerable detail \Vas devoted to each unit because a thorough understanding of each component is necessary to appreciate the performance characteristics of an over-all engine. This chapter \Vill be devoted to the theory, the methods of analysis, and the performance characteristics of turbojet and turbofan engines. Basically, the concepts and equations developed in prior chapters 'viii be collected and applied here as a group to analyze the over-all characteristics of these engines. Before beginning our analysis, it is appropriate to examine photographs of some complete engines. In preceding chapters, many schematic dra\\'ings were used to illustrate fto,v and energy transitions. To a person "'ho has not actually examined a turbojet engine, schematic diagrams tend to leave the imp~ion that jet engines are simple. To help eliminate this impression, Figs. 11.l through 11.7 present photographs to illustrate some actual engine "hard,vare." Figure 11.1 sho,vs a cuta\vay dra,ving of the General Electric CJ805-3 turbojet engine. The characteristics of the engine are given in Table 11.1. The CJ805 engine was developed fron1 the J79 military engine, and it is used to po,ver the General Dynamics CV-880 transport airplane. Figure 11.2 presents an exploded vie'v of the CJ805 sho,,·i11g the major components of the engine. Both Figs. 11.1 and 11.2 sho\v the 17-stage con1pressor, the 10-can combustor, and the 3-stage turbir1e, each of \vhich has been described in previous chapters. Figure 11.1 also sho,vs the sot111d suppressor described in Chapter 5, plus a thrust reverser \vhich produces up to 5,000 pounds of re\·erse thrust. Figure 11 .3 presents a cuta,vay dra,ving of the Pratt r

Burner

Turbine

Nozzle

I

I

0

I

l

2

F

--

•

3

'

.4

Ie

I

5

Flight velocity reference Zero velocity reference

Fio. 11.8

ef

t

Variation of gas properties through a turbojet engine during flight.

11~

•

13l

•

• •

F10. 11 .0 Schematic dingra.n1 of centrifugal turbojet C'nginC' 8ho"·ing thC' balanl't' of a..xial gas load~ \vithin the engine. (Cou rtesy Rolkt-lloycr, Ltd., front Report of 1\ cronautics, Sept. , 19+8.)

.. 264

I

Jet Propulsion

The remainder of the forces, which must be takc 11 on the engine mounts, then becomes

Fe/- 2 = Fencinomount.a

=

G -g (V.1 - V2) - (p2 - Po)A2

(11.3)

If Eqs. (11.1), (11.2), and (1 1.3) are added together, the result is

F1-o

+ F2-1 + Fe/-2 = Q.(/ (Ve/ -

Engine net thrus t,

FN)

Engine mounts,

f,.1-2

t

.,u ...0

u..

Diffuser, f 2 . 1 (lT41/•• - Vo) g g G00 = gas generator air- Bow Gl>P = by-pass air-flow V,1,, = gas generator exhaust velocity

~l>P = {J =

(11.6)

by-pass ratio

00

The h-S Cycle Diagram Figure 11.11 presents a sketch of a tl1rbofan e11gi11c together with its h-S c)·cle diagram. This figu re is also applicable to turbojet e11gines by removing t.he fan and references to it. Note that t11c sa1ne statio11 dcsignatio11 is t1scd h ere as that ch "-8.5 used in the discussio11 on the i11dividual co111po11e11ts ; tl1erefore, the application of equations from previous cl1apters ca11 be dor1e ''rith a n1i11in1un1 of difficult)'. Figun' 11.11 suggests that the diff uscr co11verts the kinetic energ_)' of the i11co1ning air into s ram pressure rise, 0 to 2; the fan co1npresscs t l1c total air-flo,v, 2 to 2.5; the mechanic.al

,,·hi

I

Turbojet and Turbofan Engines

265

compressor further compresses the gas generator air from 2.5 to 3; 11eat energy is added in the combustion chamber at a1Jproximately co11stant pressure from 3 to 4; the high-temperature a11d l1igh-pressure gases are the11 expanded in the turlJine to produce an amount of ''rork equal to the 'vork required by tl1e fan and compressor; and finally, the nozzles convert the enthalpy energy of the by-pass air plus the remaining available enthalpy energy of the gas generator gases into kinetic energy for the production of thrust. Several items should be noted on the h-S diagram: (1) for stable operating conditions and constant mass-flo,v through turbine and compressor, (M, + {J~, = !::i.h,); (2) all pressure lines, \Vi th the exception of Po and p,, are total pressures, and (3) station e is coi11cide11t 'vith station ef when subcritical nozzle-flow exists; it lies above ef \vhen supercritical flow exists. B h

Vo ' '

'

,. '

0

c

DF ''

'

''

v el

''

T

H ''

'

l 2 2.5 3

N '

e'

5

4

'

ef

3

T

~h e

vi _.£.

~ ,1 ~

~

2

.•

ef~g

._s.+.

P..lf1'0

2g

0

P.

e

2.5 AhF

\'1~

5

. ressure, po sphenc P

efF

s

Fro. 11.11 Turbofan cycle diagram on a h-S plane.

The equations and methods used to analyze tl1e over-all e11gine cycle are the same ones discussed in previous chapters, the only difference bei11g that here the con1ponents are regarded as integral parts of an over-all engi11e. Tl1e application of foregoing equations and charts will be illustrated by the solution of t,,.o sa111ple problems. SAMPLE T URBOJ ET PROBLEM

A turbojet engine is operated at fttll po,ver at 36,089 ft at l\1ach 0.9. Under these conditions, the follo,ving component efficiencies apply: 17,.

=

1.0

1Jc

= 0.85

'lb

= 0.96

1Jt

= 0.90

'1"

=

0.90

The air-flo\v Ga = 64.4 lb per sec (2 slt1gs per sec); t l1e compl"('ssor pre..."Sure ratio P1,/ p1 , = 18; the turbine inlet ten1peratt1rc T, . = 2-100 °R , tl1e thrust. correction factor Kr = 0.90 ; and t i\ total pressure loss i11 tl1e co111bustion cha.n1ber t::i.p,/ p,. =

266

I

Jet Propulsion

0.02. The engine utilizes a fuel 'vith a heating value of 18,900 Btu per lb. Calculate the appropriate values of the gas properties at each engine station, the ram drag, gross thrust, net thrust, and thrust horsepo,ver.

Solution The solution of this problem is a step-by-step analysis of t he 'vorking fluid as it passes through each component. Equations and graphs used 11erein are those d eveloped and presented in the previous chapters. Diffuser: From the given data, the free stream conditions are: Jf o = 0.9

T o= 390°R

p0

=

472.7 psfa

From Fig. 5.3, for a ram-recovery factor of

=

Pt. Po

1Jr =

~':

1.69

V o= 873 ft per sec

1.0, \Ve have =

1.16

Thus,

pt.

799 psfa,

=

and

T '·

=

453 °R

Compressor: Since the pressure ratio is 18:1, 've have

= 18pt. =

Pt.

14,380 psfa

The work accomplished per pound of air is given by Eq. (6.21) or (6.26) ; thus, 'frc =

0.24~t,X c

= 0.24

X ~~: X 1.288 = 164.8 Btu per lb

5

The horsepo,ver required to drive the compressor is given by the " 'ork per pound, and the amount of air processed in unit time as

= Ga i//c

HP

= 64.4

164.8 = 14 980 HP 0.707 '

0.707

c

x

The temperature at the compressor-outlet section is given by Eq. (6.23) as

T t,

=

T t.

+

11/c c,, = 453

+

164 8 · = 1140°R 0.24

B urner: From the given data, the pressure a11d tcmperatt1re at the burner outlet are Pt. = 0.98 X Pt, = 14,120, a11d Ti. = 2400 °R . The actual ten1perature difference through the burner, T t. - T,. = 1260 °ll. I4'ron1 Fig. 9 .13, 1Jb

x af

= 0.0194

The actual fuel-air ratio is thus

f = 0.194 = 0.202 a

0.96

The fuel flo\v required to operate tl1e e11gi11c is

G1 = Go [

a

=

1.30 lb per sec

'1 11,rbine: The turbir1e is a11alyzed 011 tl1c basis tl1at t l1e horsepo,,·er or ,,·ork 1

required by the compressor (and fa11) is equal to t11e l1orscpo\ver or ,,·ork deli,·ered l)~ the turbine. 111 ma11y proble1ns \Vl1ere only approxin1atc a11s,,·ers are required, it i:-

Turbojet and Turbofan Engines

I

267

logical to asffilme that the fuel-fio'v rate is equal to the "compressor bleed air" required to supply cabin or cockpit air-co11ditio11i11g for the pilot. For tl1is assumption, the mass-flo"r rate through tl1e compressor is equal to the mass-flo,v rate tl1rough the turbine; thus, some simplification results in tl1e turbi11e and nozzle a11alysis. The aforementioned assumption is actually a good one for i11stalled turbojet ar1d turlJofan engines, because the cabin p ressurization a11d conditioning air-flo\v rate is about equal to the fuel-flo'v rate. In some installations 'vhere "compressor bleed air" is rec1uired to power auxiliary devices in addition to providi11g pilot comfort, the mass fio,v through the turbine is less than that through the co1npressor. Today's engine specifications usually have a maximum limitation to the compressor bleed air of about 53 of the air-flo,v rate. The point of this discussion is to illustrate that \Vhere accurate results are required, mass flo\v through both turbine and compressor must be determined; \vhere approximate but still realistic results are desired, the assumption of equal mass flow is permissible. For accurate results we write

GaAhc = (Ga+ G1)Ah, Thus, with the simplifying assumption

Ahc = Ah, = 164.8 Btu per lb The temperature at the turbine outlet section is given by Eq. (10.2) as

T

'·

= T - 1rt = '·

Cp

164 8 · 2400 0.276

=

1803°R

Note that we now use the c,, value for the hot section of the engine. The pressure at the turbine outlet section is given by Eq. (10.10) as

"fJ/i

= 1/tCpT '·

1

!X fl

Solving for X. gives a value of 0.394, from which the pressure ratio across the turbine can be determined from Appendix E to give

Pt. Pt.

=

3.815

thus

~'i 3 5

Pt. = 7.85 Po N ozzle: In any nozzle problem \Ve must first deter1nine ''·hethcr the fio,,· is subcritical or supercritical. Since the over-all nozzle pressure ratio is l1cre p 1 , / p 0 = 7 .85, we see that the nozzle operates \veil \vitl1in tl1e st1percritical rcgio11. Ol1r 11ext step is, therefore, to determine the gas properties at tl1e nozzle exit sectio11. Fron1 Eq. (5.51) for a nozzle efficiency of 0 .9, \Ve have a11 actual 11ozzlc exit presst1re ratio of 0.5 ; thus, Pt. =

p.

= 3710 psfa

= p• =

0.5 X 3710

=

1855 psfa

At the nozzle inlet section \vhere, for total co11ditions, /II = 0 , perature ratios from Fig. 5.38 or Eq. (5.41) :

Tr•'·

=

1.165

thus

.

T -

r• --

1803 1.165

=

\\'C

ha,·e, for the ten1 -

1550oR

I

268

Jet Propulsion

Since t he exit Mach number is unity, the velocity a t the exit section is given by the d efinition of sonic velocity:

V. =

v•

= v' Y'YRT* = v' 32.2 X 1.33 X 53.3 X 1550 = 1880 ft per sec

The required exit area to handle tl1e e11gine mass-flo,v rate is given in t he continuity equat ion as

A.

= GeRTe

x 53.3 x 1550 = 1 525 ft2

= 64.4

PeV• 1855 X 1880 . Perfarmance Evaluation: The gross thrust from Eq. (2.4) is Fo = G.V. + (p. - po)A. g

=

2 X 1880

+ (1855 -

473) 1.525 = 5870 lb

The gross thrust can also be determined from the gross thrust paramete r in Table 5.2:

,

Fo = c5A.

,

1.26 Pi. - 1 2116 = 18,900 Po

Fa = FoKT = 18,900 X 1.525 X .2235 X 0.9 = 5800 lb The agreeme11t beteeen the t\VO methods is quite reasonable here. The ram drag is given by E q . (2.5) as

FR = GaVo = g

2 X 873

= 1746 lbS

Hence, the net thrust is

FN = Fo - FR = 5870 - 1746

=

4124 lbs

based on the Fo previously computed above. The t hrust horsepower developed by the engine is THP

= FNVo = 4124

550 The engine specific fuel consumption is

X 873 550

= G1 = 1.30 X 3600 =

.t S;C

FN

4124

=

6550 HP

lb 1 . 13 lb-hr

The engine over-all efficiency is given by Eq. (2.18) as

FNVo

1'/o

4124X873 18·8 3 = G1 (H. V.) = 1.3 X 18,900 X 778 =

SAMPLE TURBOFAN PROBLEM

A turbofan engine is operated at full po,ver at 36,089 ft at l\1acl1 0.9 . The engu1e has compo11ent efficiencies equivalc11t to the turboj e t e11gi11e. The 011l)r difference bet,veen t he engines is tl1e total air-flo'v a11d the fo.11 pcrfor1na 11ce. These values are

(Ga)total = 110 lb per sec

fJ

= 1.0

hence,

(Ga) 0 0

=

55 lb per sec

(Ga)F = 55 lb per sec

fan pressure ratio

Pi •.• Pi.

=

2.2

1'/F =

0.85

Turbojet and Turbofan Engines

I

269

Solution The analysis of the turbofan engine is a step-by-step process like the turbojet analysis. Since the basic characteristics of this sample turbofan cycle are identical to the turbojet cycle analyzed above, only those sections 'vhere the turbofan analysis differs from the sample turbojet problem are presented. For easy reference, the basic turbofan cycle parameters are repeated, ho,vever.

Diffuser: (same as turbojet) Po = 472.7 psfa

V o = 873 ft per sec

Mo= 0.9

= 799 psfa

Pi.

T,.

=

T o= 390°R

453°R

Fan: Since the fan-pressure ratio is 2.2:1, \Ve have

Pi •.•

= 2.2 X 799

=

Pi •.• Po

1757 psfa

=

3.72

The fan \Vork per pound of air is

il'F = c,,T 1,XP = 0.24 X 453 X .2527

32.3 Btu per lb

=

0.85

T] c

The fan horsepower is Gpif'F = 55 X 32.3 = 2510 HP 0.707 .707

The temperature at the fan-outlet section is T '···

T '·

=

+

"/l'p

c,,

=

453

+

32 3 · = 588 °R 0.24

Compressor (same as turbojet): P 1• = 14,380 psfa

"fl'c

= 164.8 Btu per lb

The compressor horsepo,,•er is HP = 55 X 164.8 = 12 700 HP c .707 '

Burner (same as turbojet): Pi. = 14,120 psfa

G1

= 55

T '·

X 0.0202

f_

= 2400°R

=

= 0.0202

a 1.11 lb per sec

Turbine: The turbine 'vork equals tl1e fa11 a11d cornpressor ,,·ork, thus: i//,

=

ire

+ {3i//p

= 164.8 + 1 X 32.3

=

197 Btu per lb

The turbi11e outlet ten1peratt1rc is T,. = T '· -

Agair1, usi11g Eq. (10.10),

\VC

"fl',

c,,

= 2400 -

197 = 1685°R 0.276

l1avc "#', = 11,c,,T' · 1

! :y•

270

I

Jet Propulsion

Solving for X., 've obtain X. = 0.496. From Appendix E, X. = 0 .496, Pt./ Pt. = 5.07. Thus, 14,120 Pt. = 5.91 Pt. = .07 = 2790 psfa 5 Po

Primary Nozzle: Again we see that the nozzle is operating supercritically. Therefore p,,,

= p• =

=

1395 psfa

1685 -- 1.165 -- 1445 °R

r•

T •• , --

0.5 X 2790

v ... = v• =

v' 32.2

x 1.33 x 53.3 x 1445 = 1815 ft per sec

and

- 55

A ••• -

x 53.3 x 1445 1395 x 1815

2 - 1·67 ft

Fan N ozzl,e: This nozzle also operates supercritically. Since no combustion products are present, and since the temperature is much lo,ver, is d etermined by Eq. (5.51) for M, = 1.0.

"Y =

1.4. The nozzle pres.5W'e

p* ---=-- = 0.495 Pt •.• P•t = p*

0 .495 X 1757

=

= 870 psfa

From Eq. (5.41)

T t, . T*

I

= "Y + 1 = 1 2 2

.

Thus,

T,,

= T* =

V,, =

588 1.2

=

490°R

v 32.2 X 1.4 X 53.3 X 490

= 1083 ft

per sec

and

A

.,

= 55 X 53.3 X 490 = l 52 f tt

870

x 1083

.

Performance Evaluation: The gross thrust from Eq. (2.4) is

Fo = Ga,,V.,, g

=

55

~2 -~

+ (P•,, -

8 15

Po)A .,, +Ga, v., g

+ (1395 -

473)1.67

= 7110 lb The ram drag is

Fn

= 110 X 873 = 2080 lb 32.2

The net thrust is FN = 4130 lb

+

55

+ (p,,

- Po )A q

~2 -~osa +

(SiO - 473)1 .52

I

Turbojet and Turbofan Engines

271

The thrust horsepower is THP

= 4130

X 873 550

= 6550 HP

The engine specific fuel consumption is •1

= 1.11 X 3600 = 0.965 lb per lb-hr

4130

SJC

The engine over-all efficiency is 110

=

4130 x 873 1.11 X 18,900 X 778

= 223

Table 11 .2 summarizes the results of the t\VO sample problems. TABLE

Item Net thrust, F N, lb sfc, lb/lb-hr Over-all efficiency, 170 Air-flow, lb/sec

11.2

Results of Sample Problems

Turbojet

Turbofan

4124 1.13

4130 0.965

18.83 64.4

110

223

The figures in the table suggest the improvement in sfc and 110 for th~ ~bofan. This advantage again explains the attractiveness of this engine for subsoruc Jet transpo~s. It can also be seen that the air-fio,v of the turbofan is about t'''ice that of the turboJet for the same thrust.

11 .3

Cycle Pressure and Temperature Variation

Table 11.3 presents a summary of the cycle temperatures and pres.5ures as functions of ambient conditions, flight l\1ach number, component efficiencies, and compressor pressure ratio for turbojet and turbofan engines. The equations contained in this table are based on idealized a...~mptions (no change in mass-flow rate through engine, constant value of specific heats in all components); t hus, precise ans,vers cannot be obtained from tl1e equations. The equations a.re, however, extremely useful in that they sho\v ho\v the cycle pressures and temperatures change with the variables in the equations. Several equatio11s ,,·ill be examined in this light to illustrate t heir usefulness. The T,, equation sho,vs, for example, \vhy the tailpipe ten1perature of a tt1rbojet engine decreases with an increase in Macl1 11t1n1ber at a co1istant altitt1de. For a given throttle setting, T,. is approximately fixed (at n1a.xi1uun1 po,,·er, it is fi.xed at the limit value of the turbine blades) as are X e a11d TJ~· Tl1t1s, as :\Iach nun1bcr is increased, T '· increases 'vi th a conseqt1ent reductio11 i11 T, ,. It is stated here that _\~ and 1Jc are approximately fixed; 110\vevcr, for large i11creases i11 fligl1t 2\Iach 11un1ber, say from 0.5 to 2.5, t he opcratir1g poi11t 011 tl1c co111pressor ninp 1110,·es appreriabl)· leftward, because the corrected air-flo'v a11d corrected rp1n nre reduced due to higher T,. values. This effect reduces T,, eve11 n1ore rapidly. The p,. equatio11 is particularly in1porta11t, bccat1se n11 i11crease i11 its ,·alue represents an increase in gross t11rust. Fro1n t11is cquat.io11, it is rendil)· appare11t that higher thrust values per pound of air result \vl1c11 tl1e desig11 variables (compone11t efficiencies, the turbine-inlet te1nperaturc, a11d tl1e con1prcssor-p~ure ratio) are increased.

1

272

I

Jet Prop ulsion TABLE 11.3 Summa ry of Cycle Temperature and Pressure Equations

Turbojet

2)

T,,

=

-y-1 Tt, = Tt, = T o( 1 + Mo 2

Tt,

=

T o( 1 + 'Y - 1 M o 2

2)(1 + -;;: Xe)

Pt, = T/rPo [ 1 + Pt, = rp,, =

f

'Y - 1

2 ]_!_ Mo 'Y -

2 (N )Pt.

1

Pt. ~ Pt,, or usually, Pt. = 0.95pt, Pt, where a

=

TTJrPo [ ( 1

+

'Y - 1

2

2 ) ( Mo 1 -

x e )]_.:!._ T/tO'.TJ -y- l

= cy cle temperature ra tio T,./ T,,

r = compressor-pressure ratio, primarily a function of engine rpm

Turbofan Tt . = T,, = T,. = T o( 1 +

2)

-y- 1 Mo 2

2 )1. x,) Ti •.• = T o( 1 + 2 M o \ 1 + T/F 'Y - 1 2 )( Xe) Tt, T o( 1 + 2 M o 1 + T/e -y-1

=

1 T,, = T o(l + 'Y -2 M!)(a- ~T/e - {JX ' )

=

T,. - Tt, ( !.! + f'/e

f'/F

Pt, = T/rPo [ 1 + Pt•.•

'Y - 1

2

2 ]_!_ Mo

'Y- 1

= r FPt. = f '(N )Pt.

Pt, = rp,, =

f (N )p,,

Pt , ~ Pt, or usually, Pt. Pt, = TT/rPo [ (1 +

= 0.95p,.

2)(1 -

'Y - 1 i\f o 2

)]_.:!._

Xe {J X, -- T/ tO:T/ e T/ tO:T/,

'Y -

where a = cycle temr)ernture ratio T,. / T,, r = compressor-pressure ratio, prirr1nrily a function of engine rpm r, = fan-pressure rutio, primarily n function of engine rpm; a nd fJ = by-pass ratio.

1

{3.--X_,) f'/l'

I

Turbojet and Turbofan Engines

11 .4

273

Variation of Engine Data with Changes in Operating Conditions

The sample problems solved in section 11.2 gave us the results for one operating condition, namely, fixed engine rpm, a fixed altitude, and a fixed flight velocity. In actual operating conditions, each of these quantities varies considerably; therefore, our next logical questio11 concerns the effects of engine rpm, altitude, and fti~ht speed on engine performance. The best way of sho\ving these effects on en~ne performance is to study various plots \vhich present the variation of p~rf?rmance \VI th changes of rpm, speed, and altitude. Figure 11.12 presents the var1at1on ?f th~st, tailpipe temperature, specific fuel-consumption, air-flO\V rate,. and fuel rate .'v1~h engine rpm for a turbojet engi11e at a constant altitude and velocity. The var1at1on for a turbofan engine \Vould be comparable. 8000 .-----.---i.-----r-----r--~-f-.i 800

120

u

0

110

7000 1- -- - l - --

+ --

__ ,~, 700

---I

... ..J:.

100

......... ::Q

«>

6000 1- -- l -- - -

--

uG) 90 '"ii;) 5000 1-

~-

600

4--

500

Ta ilpipe temperature - -

~

«>

a.

·-0

t-

'"'O

80

c: 0

40001--- -1- - - 4- - - •:.it---

-0

_Q

-. !:::

0 0.

--......

4000

:

3000

::> ..t::. C>

........ ......

-

.

40,000 ft 100% rpm • 90% rpm A 80% rpm • . ~

.....

-

'--

2000

--

-

.-'

c

·--0 .

E ;: 1.0

'

I~

_......

/

.........

.....

/

/

~

.... .... I

I

i::-I

-··--

I

11.5 --..

-

L

., -- -

... ~ ... ,.. . ... :• r:..· :

u

~

1000

1.0

-·

I

K.T

0.

I/)

0.5

'

0 .2

/

c

~

0

/

0 u ~

0

/

/

0

l......

z

1 2.0

J'

0 .4 0 .6 0 .8 Mach number

1.0

11.13 Typical turbojet engine characteristic curves- net thrust versus Mach number for several rpms and two altitudes.

F10 .

0

0 .2

I

0.4 0 .6 Mach number

-

. .. ··--I-r

I

.

0 .8

1.0

I

Flo. 11 .14 Typical turbojet en gine characte ristic curves specific fuel con.sum~ tion versus l\Iach number a t se~ rpms and t"·o altitudes.

As altitude is increased, the thrust curves te11d to become flatter (note the appreciable dip in t he sea level curves), and at sea level, the n1a~imum thrust line reaches its static sea level value again at about a Macl1 number of 1.0. This characteristic is typical of a low-compression-ratio engine. The sea-level thrust of engines "·ith higher compressor-pressure ratios decreases continuously '''ith the increase of l\Iach nun1bers, as may be seen in Fig. 11.21. Figure 11.14 sho,vs the effect of engi11e rpm, flight speed, a.11d altitude on specific consumption. It should first be noted tl1at this plot is double scaled , one scale for each altitude sho,vn. At a given Macl111un1bcr and rp111, the sfc is bette r \Yith altitude. especially a t the lo,ver rpm 's. Specific fuel consumptio11 i11crcases '"ith l\Iach number. but, as mentioned i11 Chapter 2, specific fuel co11su111ptio11 is 11ot in itself indicati,·r of the over-all efficie11cy. I-Iigl1 over-all efficiency requires a con1bi11ation of high flight velocity and lo\v sfc, as was sl10,vn i11 Eqs. (2.18) a11d (2.22): 110 =

FVo

Vo

G, (H. V .) = (sfc) (H. V.)

22 (2 · )

From this equation, and from the relatio11 of sf c \vit h l' o noted in Fig. 11.14, it i~ evide11t that tl1e over-all efficiency increases co11ti11uousl)' ,,~ith l\Ia ch number to a

I

Turbojet and Turbofan Engines

value of about 18% at Macl1 1.0 at 40,000 feet. Figure 11.15 presents a plot of maximum thrust-horsepo\ver available (100% rpm) versus l\iach nu111ber for sea level a11d 40,000 ft. These curves come directly fro1n the thrust- versus- l\1ach- number curves. The main point of interest here is that the lines are about linear with flight Mach number. Figures 11.16 through 11.18 present the effect of flight velocity on some additional turbojet engine parameters for operating at 100% rpm in standard sea level air. The curves of these figures are based on the calculated data of an engine which has different static sea level ratings from the one used in the previous figures. Realistic component efficiencies and assumptions were used in the calculations.

275 .......

14,000----.---~-----~-

12,000 1 -- - 4 - Seo level

.

10,000 1--- 4 - - - - +-

- t --++---1

«>

):

&. I>

8000 1---~--t---t:t'---t----1

~

0

.L ~

60001---4---t--+---t--i--~

2 ......

.L

0 "=::::=-_L..-~--.L..:--~~~

0

0 .2

0.4 0 .6 Mo.ch num~r

0 .8

1.0

Fro. 11.15 Typical turbojet engme characteristic curves-ma...ximum thrtLc:t horsepower versus l\1ach number for two altitudes at 1003 rpm.

I 14,000

100

I I

12,000 10,000

90 u

I /

G_Y.-

80

I>

~ 8000

/

"'

:a- 70

_/"'

I

/

~

-

HP,

I

0

6000 -

Pt.+ CD l:!.p,. Pt (Re/) Pt.

+ Cnz Gnz + CPs HP:+ Ga

0

CT To - T o 1td T o 1td (11 .20)

l:!.G1 G1

=

Pt, (Ref>- Pt.+ C' 81 GBz Pt, (Ref) Ga

+

C'p. HP:+ C'T T o - T o 1td T o1td

(11 .21)

[pc, - Pt.1/ Pt. is equivalent to l:!.p1. / P1 . , and Pt. is defined in Eqs. (5.13) through (5.15). Reference 4 requires that these correctio11 factors or their equi\·alent be included in all military engine specifications. It also requires that equi\·alent correction factors be included in the IBNI performance "decks." These requirements apply to all air-brea t hing turbine engines - turbojet, turbofan, turboprop, and turboshaft .

11.7

Turbofan Engines

Figures 10.2 and 11.5 through 11 .7 illustrate four current tt1rbofan er1gines. Table 11 .1 gives the pertinent characteristics for each of t.hese engines, plus several other turbofans. Examination of t hese characteristics reveals t\TO basic areBS in ,,·hich turbofans di ffer from turbojet engi11es. The sf c is lo,Yer a11d the air-flo\,. is higher. More specifically, the higher air-flo'v results i11 a lo,ver thrust per pound of air-flo"· and hence in a Jo,ver equivalent jet velocity, l' .1· Reference to Eq. (2. 12) and Fig. 2.16 sho\\'S that this results in a higher propulsive efficie11cy for the turbofan as compared with the turbojet since tl1e ratio of free strean1 to jet vclocit)' is l1igher. The higher propulsive efficiency ir1 turn increases the over-all efficie11c)r [Eq. (2. 18)] a11d thus decreases the sfc [Eq. (2.22) ). It is tl1is cl1aracteristic - lo,,·er sfc than the turbojet - \vhich has caused the great ir1tercst in, a11d en1phasis 011 1 turbofa11 e11gines. Figures 11 .26 and 11 .27 from Ref. 10 prese11t the effect of several paran1ete rs on turbofan engine perforrnance. Figure 11 .26, \vl1icl1 applies to the sea-}e,•el static condition, sho,vs the effec t of compressor prcsst1re ratio Pi .! Pt, a11d fan b)·-pass ratio on the thrt1st per pou11d of air [F/( Go/ g)] and sfc. 'f\yo different fan pressure ratios are included . In additio11, a curve is sho,v11 for a by-pa...~ ratio of zero, ,..-hich is equivalent to a turbojet e11gi11e \Vitl1 the same con1po11e11t perforn1ance. Figure 11 .27 sl10,vs equivale11t effects for operation at l\1ach 0.9 in the stratosphere, a typical subso11ic aircraft cruise co11ditio11.

...

I

290

Jet Propulsio n

2.1

T, 4 = 2400 °R

1.9

Mo = 0, sea level

1.7

-..D... ' ~

..........

v

Q)

~

.J ~ -..D

1.3

..D

l

I 1.1 u 'ti;

2100

- - Fon pressure ratio• 1.15 - - - Fon pressure ratio -2.20

1.5

Bypass ratio = 0

.,,,.

1700

0 .3

1300 ~

FIG.

6

..D

1.2

~

1. 1

~

10

14

18 22 26 30 34 38 PtJIPn

1700

' :11soo

'

~ I

2

0.9 .

\

1900

Bypass ratio = 0

1.0

Bypass ratio = 0

2100 u ~

..D v

6

2300

- - Fon pressure ratio - l.15 - - - Fon pressure ratio - 2.20

13

I

2.8 2

14 18 22 26 30 34 38 P13/Pr2

T, 4 = 2400°R M 0 = 0 .9, 36,089 feet

1.4

..........

10

--

2.2

11.26 The effects of compressor pressure ratio and fan by-pass ratio on turbofan engine performance at sea level static. (From Ref. 10.) 15

-...

----i-; -~

900 2

---- --

~

0 .7

0.5

... - ......

"" ,''I, ,,----...... .........1.6------ ... ........ ,'~

I

~1~

0 .9

.,---....._l.r:= Bypass ratio = 0

2500

...

-----

1300 -

1~1100

--- ------ -- ------gl.O

, ....------- ---

0.8

1.6

-

---

0.7 0.6

500 2

6

10 14 18 22 26 30 34 38

2

6

10 14

• 18 22 26 30 34 38 Po/P12

FIG.

11.27 The effects of compressor pressure ratio and fan b)·-ps.."'5 ratio on turbofan engine performance at !\1ach 0.9 in the stratosphere. (Frorn Rtf. JO.)

The figu res sho\v t l1at increasi11g compressor pressure and b)--pas.5 ratio both decrease t he sfc, though the effect is s1nall abo,•e pressure ratios of about 15. The figu res also indicate t l1at tl1e 111axin1un1 tluust per pound of air-fio"· occurs at relati,·el~· lo\v compression ratios, bt1t tl1at increasing pressure ratio does not ha'\·e a marked effect. Exami11ation of t l1ese figures together \vitl1 the turbofan characteristics in Table 11 .1 reveals t l1at cu rre11t e11gi11es are in ge11eral agreement ,,;th these trends. Thr by-pass ratios are moderate (less than 1.6) to limit. engine size. The compressor p~­ sure ratios arc n1ediun1 to l1igh (approxi111ately 12 or slightly higher, ~ince these engines are generally derived from mediun1 to higl1 pressure-ratio turbojet engines). Onr e11gi11e i11 Table 11. 1 - the General E lectric X353-5 - deviates significantly from th~ trend. For this e11gi11e, t l1e by-pass ratio is 12.3 a11d the diameter is 76 in. Figure 11 .2.. take11 fron1 Ref. 10, sl10\VS tl1at the X353-5 engine does generall.)1 agree ";th the effect of a large i11crease in by-pass ratio.

I

Turbojet and Turbofan Engines

T,4 = 2400° R M 0 = 0, sea level p ,3/p ,7= 16

10

09

291

2200 ' v Cl.>

-... 0 8 '..r. 0.7 ..t:J

"

O>

1800

-::>

'-..t:J 06

" .. . :E. I

I u

1400 Fon pressure rolro

21~

':; 0 .5

~

0 .4

1000

0.3 0.2 .__...........__.__..___.___.._.....____._--L_..1,__~ 0 2 4 6 8 10

600.___.__._~...___.__..~.._~~-....__,

0

By-poss ratio F10 .

4

2

6

8

10

By-poss ratio

11.28 Effect of large by-pass ratio variation on turbofan engine performance at sea level static. (From Ref. 10.)

11.8

VTOL (Vertical Take-Off and Landing) Engines

The X353-5 engine, illustrated in Fig. 11.29, is not intended for the same type application as the other turbofans discussed abo\re. Figure 11.29 shows that the fan and the gas generator (a General Electric J85 turbojet) are not coaxial or even oriented in the same plane. The fan axis is vertical since it is designed for installation in the airplane \Vings to provide thrust in a vertical direction. This thrust provides the lift for the VTOL mode of operation. The driving po,ver for the fan comes from the hot gas exhaust of the J85 engine. It is directed by means of a diverter valve to drive turbine blades located on the tips of the fan blades. The over-all power plant is designed so that after the vertical take-off and transition, the J85 gas generator exhaust gases can be diverted back to expand through a con-

F10.

11.29

' VTOL fan-in-\\•ing engine. Photograph of GE X353-5 Co1npan y.)

(Cou rtesy T he Gt"IW"al E leclric

..,. 292

I

Jet Propulsion

ventional nozzle for propulsion in fonvard flight. Under this mode of operation, vanes above and belo'v the fans close to seal off the fan and form part of the wing surfaces. The X353-5 is thus a nonconcentric turbofan engine improved for a specific application - VTOL. The X353-5 is one of a number of engine projects which are being pursued in the rapidly developing field of VTOL aircraft. Three other concepts, ,,,hich are also being actively follo,ved for VTOL propulsion, are (1) the light-"·eight lifting turbojet or turbofan, (2) the deflected exhaust turbofan, and (3) the tilting turboprop-propeller combination. Light-,veight lifting engines (high thrust/ "·eight ratios) are installed in a vertical attitude and operate only during takeoff and landing maneuvers. Separate engines are provided in the aircraft for propulsion in forward flight. The Rolls-Royce RB 108 e11gine, listed in Table 11.1, represents the first attempt to design a turbojet strictly for VTOL operation. As may be seen in Table 11.1, the General Electric CJ610 also has a thrust/ '''eight ratio greater thB.n 8:1 . Further development 'vill undoubtedly result in significant increases in the thrust/ weight ratios of lifting engines. Deflected exhaust turbofan engines provide the propulsion for both VTOL and fonvard flight. Figure 11.30 is a photograph of the Bristol-Siddeley Pegasus turbofan engine. The figure shows two pairs of swivelled exhaust nozzles. For VTOL operation. the vertical lift is provided by swivelling the nozzles down"-ard. For Bight propulsion the nozzles are turned aft. Some of the characteristics of the Pegasus engine are listed in Table 11.1.

F10.

11.30 Photograph of Bristol-Siddcle)' Pe~us turbofan engine. (Courtuy Bristol-Si~~ Engines, Ltd.)

A fourth VTOL propulsio11 co11cept is the tilting turboprop-propeller combination. As the name implies, vertical lift for VTOL operation is pro,ided h.}· tilting the whole engine-propeller syste1n vertically. For for\\·ard flight the S)>stem is tilted back to the horizontal. l\ll ore details of this S)rsten1, i11cluding pictures of the General Electric T64 turboshnft engine and the Vought-Hiller-R)'an XC-142 \ 'TOL assault transport. are prese11ted in Chapter 13. Anotl1er VTOL syste1n '"hich has alread)' been developed considerabl.}· is thr helicopter. Turboshaft engines are rapidly co111ing into ,,·ide t1sage for helicopter propulsio11. T l1cse e11gincs are discussed further in Chapter 13. Since VTOL aircraft n1ust have a thrust/ ,,·eight ratio greater than 1 :0, the thrust \Veight ratio of an engine is one of the important characteristics determining its suitability for VTOL application. Engines consume considerable quantities of fuel durin~

I

Turbojet and Turbofa n Engines

I

293

bo,rering, ho,vever, and tl1is fuel must also be carried in the VTOL vehicle. 'l'al)le 11 .8 sho,vs the significance of this hovering fuel. li'or each of the pote11tial V'fO I.J e11gir1cs, three thrust/ ,veight ratios are given : (1) the bare engi11e; (2) t l1e er1gi11e plus five minutes of hovering fuel; and (3) the engi11e plus te11 mi11utes of }1overi11g fuel. Table 11.8 makes it clear that a high thrust/ ,veight ratio is not t}1e only criterio11 for VTOL engines. The high sfc of lifting turbojet engines severely penalizes them if significant hovering times are involved. In order to demonstrate this very clearly, three hypothetical engines representing potential future developments are included. Even \vith a thrust/,~·eight ratio of 20 :1, any significant hovering requireme11t all but disqualifies the lifting engine. In addition, Table 11.8 does not sho\v the complete story for lifting engines since separate engines are also required for propulsion in for,vard flight. TABLE 11.8 Thrust-Weight Ratios of "\' TOL Engines

Engine designation

l\1anufacturer

Bare • engme

Equivalent hovering

sfc lb/hr/lb

CJ610-1 JT12A-8 RB108 CF700-2B X353-5 Pegasus T64 +prop. T58 +rotor Turbojet X Turbojet Y Turbofan Z

Gen. Electric Pratt & ''' hitney Rolls-Royce Gen. Electric Gen. Electric Bristol-Siddeley Gen. Electric Gen. Electric _

-

,

8.03 7.10 8.18 6.83 6.50 6.98 7.17 6.00 15.0 20.0 15.0

0.97 0.90 1.06 0.69 0.34 0.60 0.129 0.050 1.00 1.00 0.80

Engine plus 5 min hovering fuel

Engine plus 10 min hovering fuel

4.87 4.63 4.75 4.90 5.48 5.17 6.65 5.86 6.67 7.50 7.50

3.50 3.44 3.35 3.82 4.78 4.11 6.21 5.72 4.29 4.61 5.00

Table 11.8 sho,vs that the advantage of the 11elicopter rotor for hoveri11g operations stems from its very high thrust/ l1orsepo,ver ratio, \Vl1icl1 i11 tur11 results i11 an equi,·ale11t. low hovering sfc. U1uortunately, the rotor also limits tl1e 1naxin1un1 speed of the helicopter because of aerodynamic consideratio11s relati11g to tl1e retreati11g blades. The other engines in Table 11.8 - the X353, Pegasus, a11d tl1e tilti11g TG-l + propeller - thus represent attempts to provide a better co1npro111isc of t l1e l10\reri11g tin1e a11d maximum speed characteristic of V1"'0L aircraft. Eacl1 of these propulsion systenlS is being investigated for differe11t types of VTOL aircraft. \\7ithout doubt otl1er \ ' TOL power plants 'vill be developed in tl1e future . U11like rnost otl1er applicatio11s, hO\\'C\'er, the VTOL po\ver plant is an integral part of tl1e over-all aircraft. Consequcnll)' , future developments \vill be intimately dcpc11dent 011 tl1c i11tc11ded applicatio11s. 11 .9 Engine- Airplane Combination Characteristics Our discussion thus far has bcc11 devoted prin1arily to pure engine cl1amctcristics, \vithout too much regard to the vehicle i11 ,,·}1ich it is i11st.alled. Since turbojet and

.... 294

Jet Propulsion

I

turbofan engines are built for aircraft propulsion, it is desirable to examine the per. f ormance characteristics of the engine-airplane combination. When we con.sider such a combination, the variables associated with the airplane, such as aircraft weight, angle of attack, and so on, also enter into the over-all functional relationships. Before developing the equations necessary to handle the engine-airplane combination, it is '''ell to examine the real purpose and goal of the level-flight performance evaluation of a turbojet- or turbofan-po,vered airplane. The aircraft must first be flight-tested over the complete range of speed and altitude, sufficient and accurate data must be obtained during these tests, a11d then the data must be reduced and assembled for logical presentation. Fig. 11.31 presents four plots ,vhich represent a typical method of presenting the most important performance characteristics of the airplane equipped \vith a fixed geometry engine. 40,000 -

40,000

90

80 90 95

lOOX

N=80%

.,

Cl>

-

-

- :::>

30

--

Cl>

:::> 0

30

., '

u

""'20

c

Q.

.,-...

10

~

0

11 .31

...... ...s::.

..tl

F10.

40,000 ft

0

Plot 3 Specific range

M

20

0 .... :::> ""'C

c

10

w

0 0

Plot 4 Endurance

M

Four plots sho,ving important level flight performance characteristics of turbojetr- or t\J.rbofan-po,vered aircraft.

The data for the four plots i11 Fig. 11.31 are based upon a given airplane weight and 011 standard conditions at all altitudes. These plots show: (1) variation of flight Macl1 number \Vitl1 standard altitude for various engine rpm's, (2) variation of fuel fio\v \Vith standard altitude for various engine rpm's, (3) specific range (miles per lb fuel) variatio11 \vith flight Mach number and altitude, and (4) endurance (hours) variation witl1 flight Mach number and altitude. It is evident that one can determine the complete level flight performance picture of an aircraft from these plots, but the stipulation is for a given aircraft \Veight and standard altitude conditions. To obtain

' I

Turbojet and Turbofan Engines

I

295

the characteristics at different airplane 'veights, one would require another set of these four plots; or to determine the characteristics at some pressure altitude 'vhere the temperature is not standard, one \vould require still another set of plots. One set of plots is required for every change in weight and/ or altitude deviation from standard values. One wonders about the value of these plots and also ho\v they were obtained in the first place (aircraft \veight varies in flight and performance data are rarely, if ever, obtained on a standard day). The plots are valuable \vithin their limitations, namely, the performance story of an aircraft at a given weight and standard altitude conditions. The method of obtaining these plots depends, of course, upon the technique of applying functional dimensionless parameters to the airplane-engine combination, as was done for the engine in the previous section. COMBINED

p ARA?\-IETER

RELATIONS

From an analysis of the engine alone 've have, from Section 11.5 for a fixed geometry engine,

F

-0 =! =f

G,

ov'7J

(11.22)

N M V(J'

(11.23)

A similar dimensional analysis of the variables which affect the drag of an airplane (see, for example, Ref. 11) yields a drag parameter for the airplane. This is

D [; =

W

{11 .24)

f 5' M

A further analysis of the airplane, that of equating the heat energy input as a function of the airplane drag energy, yields the following fuel-flow parameter:

G,

ov'7J

=! ~

o'

M

(11.25)

We note that in the above equations, aircraft \Veight appears in one of t he parameters. If viscosity is included in the airpla11e dime11sional a11al)rsis, Re)·11olds nt1mber 'vill appear in the equations, just as it 'vould appear in Eqs. (11.22) and (1 1.23) if viscosity \Vere included in the engine a11alysis. As already 11oted, viscosit)·, being a function of temperature, is essentially accot1nted for in a secondary ma11ner; ho,,·e,·er, when it is ignored, the airplane parameters, like tl1e e11gi11e paran1eter, ,,·ill 11ot generalize completely. Thus, lo'v a ltitude data cannot be t1sed to predict accuratel)· the high-altitude characteristics. So, it is necessary to obtni11 perforn1ance data O\rer the complete range of altitudes. Since in steady level flight, net thrust is equal to drag, Eq. (11.22) can be equated with Eq. (11.24), and since the fuel-fto,vs in Eqs. (11.23) a11d (1 1.25) are equivalent, these equations can also be equated. Equati11g eitl1er or botl1 sets of relations sho'\\-s that N W (11.26) M =f 6

vo'

I

296

Jet Propulsion

This equation merely states that the flight Mach number of a turbojet-powered aircraft with fixed geometry engine is a function (for the assumptions made) of the parameters N /VO and W / o only. From this equation ,ve can also write

F

-= f 0

N

M

N

M

vo'

or

F

-0 =!

• w -,;,M

(11.27)

•

(11.28)

and

G,

ov!O

-

=!

ve'

or

G,

ov!O

w =! -,;,M

Level-flight performance-testing of airplanes with fixed geometry engines has almost exclusively been accomplished on the basis of solving experimentally the two follo\ving functional relationships:

M

=!

N

W

ve' {;

and

N

vo'

M

A typical solution of these equations is sketched in Fig. 11.32. -w 6

F10.

11 .32

Graphical solution of the performance relations of the turbojet- or turbofa.n-powered aircraft \Vith a fixed geometry engine.

It is readily apparent that tl1e level-flight performru1ce characteristics cur,·es (the four plots in Fig. 11.31) can be obtained fron1 the t,,.o plots above. In fact, an)' gi,-en number of these sets of four plots correspo11di11g to differe11t airpla11e ,,·eights or nonstandard altitude conditions can be obtained from Fig. 11.32. Aclt1a/ly, all the /creljl:ight performance data are generalized and available 01i the two e11gi1weri11g plots of Fig. 11.32. The four plots of lj"ig. 11.31 are 111erely conve11ie11t plots and are special ca...~s because they are applicable or1ly to a certain specified airpla.11e ,,·eight . Thus, the purpose of level-ftigl1t performance-testing is to obtai11 sufficic11t data so that e11gineering plots (such as Fig. 11.32) may be constructed. • It should be noted that the latter forms of Eqs. (11.27) and ( 11.28) appl)' to &n)' t.)'J>C of engine geometry, fixed or variable, because they come directly fron1 Eqs. ( 11.24) and ( 11 .25) which v.· ere

derived from air-frame considerations only.

I

Turbojet and Turbofan Engines

I

297

It should be noted from the above plots that thrust, \vhich is a very basic parameter, does not appear. For the fixed geometry engines, the performance characteristics are not measured in terms of thrust since tl1rust is very difficult to measure directly. It is, therefore, desirable to present performance characteristics in terms of variables '"hich can be evaluated by the pilot in a cockpit. It is pointed out, ho\vever, that the performance characteristics of the aircraft can be presented in terms of the thrust parameter, and for certain engine types, \vhere the geometry varies, it is advantageous to do so. Thus, plots of the follo,ving parameters can be made. These 'viii also yield the performance story of an aircraft:

w -0 =! °""i'M F

G,

ov'O

=

!

w

°""i'M

(11.29) (11.30)

Reference 12 shows that reasonable results can be obtained using gross thrust and a relatively simple gross thrust meter installation. Figure 11.33 presents the typical variation of the gross thrust parameter with flight Mach number for several values of W / o. These curves represent the thrust-required curves of the basic engineairplane combination since the data for them \Vere determined from the thrust meter in stabilized flight.

Note:

FG values at minimum points are about the same

F10.

11 .33 Typical thrust-required curves for turbojet or turbofnn cngine-aircraf t coml)ination.

Reference 13 sho,vs a satisfactory 111etl1od of dctcr111i11i11g the gross-thn1st a\·ailable curves for an e11gine-airpln11c co1nbi11atio11; tl1t1s, ,,.J1c11 the tl1rt1st-n,·nilable a11d thrust-required curves are combi11ed, tl1c lc\rel-fligl1 t pcrfor111n11ce characteristics are knO\VIl. I t should be recognized tliat tvlic1i variable gco111clry engines arc perfor1na1lr.e-icstcd in conj1tnclion will~ an a1·rcrafl, it 1·s 1riorc ati11anlagcous lo use the thrust parameter function, because lite para1r1cler J?q. ( 11.2G) bcco111es 111ore co11iplcx by an add it io1lal ter1n. For exan1ple, Ec1. (11.26) for a variable geo111ctr.)' c11gine bccon1cs

298

I

Jet Propulsion

Af

f

(11 .31 )

=f

{11 .32)

=

and for a dual rotor e11gine becomes

M

The thrust parameter relationship, even for the variable geometry engine or the dual-rotor engine, is still given by Eq. (11.29), 'vhich, 'vhen derived from airplane . considerations, contains only three terms. DISCUSSION OF

S.R.,

7Jo AND

sfc

One plot of Fig. 11.31 sketches the specific range variation of a turbojet-pov1ered aircraft. I•igure 11.34 presents another plot of the specific range (S. R.) variation sho,ving numbers \vhich are typical of a modern fighter-plane at subsonic speeds. 0.30 r------..------..------~------,

92

0 .25 40,000 ft

I 35,000 ft I81

G)

..... .....0

:>

-0

c

:> 0

0.20

a.

100 85 95 85

30,000 ft

...

G)

a. ~

25,000 ft

-·EG)

0

u

·-:> 0


n._~ for cnginc-inl('t matching. (Fron' 1\' rl CA in engine air fto\v due to the evaporative Tf\7 89~~. ) cooling (\vhicl1 decreases T,, and thus increases N /VO,, and G0 0,,/ o,,) and in e11gine pressure ratio. The i11creasing rnn1 temperature rise \vith i11creusing speed n.llo\VS n greater nmot111t of evaporative cooling, hence, the greater thrust n.ugmentatio11 . In vie'v of the analytical rest1lts i11 Ref. 7, Ref. 8 reports n.11 expcrime11tnl in,·cstigation of \Vater injection in the st1bsonic difTuscr of a fixed co11ical spike inlet operating

-

s. if

~

p

322

I

Jet Propulsion

at Mach 2.5. The experimental results generally confirm the analysis of Ref. 7, but they sho'v that the evaporation efficiency i11 t he subso11ic duct is relatively low ranging from 65% at a \vater-air ratio of 0.01 to 50% at a 'vater-air ratio of about 0.04. These lo'v efficiencies 'vill significantly increase the ,vate r consumption rate. This situation 'viii suggest that one of the disadvantages of 'vatcr injection for supersonic inlet matching is the very high liquid consumption, 'vhich 'vould require very large quantities of \vater for extended high speed operation. If significant thrust augmentation at supersonic speeds is required for short periods, ho,vever, 'vater injection in the subsonic diffuser ahead of the afterburning engine appears promising. It should be noted, ho"·ever, that this system is only applicable at speeds higher than about l\1ach 1.5 when operating at high altitude, since the ram temperature rise must be sufficient to raise T,, well above the freezing point. Performance characteristics of an engine operated 'vith 'vater injection are contained in Refs. 9 and 10. All methods employed to evaluate the water injection compression process are fundamentally the same. In the cited references, use of an air Mollier diagram with a psychrometric chart or a lVIollier diagram containing water vapor-air ratios by \Veight is employed. These charts allo'v one to trace the wet or dry compression process through the machine, and arrive at the compressor outlet conditions. Although the basic evaluation of a typical 'vet compression process is simple with the aid of these charts, the details of the actual solution become tedious when refinements are applied to the problem, that is, \vhen \vater is injected at temperatures other than 519 °R , or air is saturated \vi th \Vater vapor at some fi..xed point in the compressor other than at the inlet or exit section, or when various concentrations of alcohol other than zero are used. Application of these refinements plus the other assumptions made for the problem (for example, the evaporation efficiency) al~YB)"S leave some doubt as to the accuracy of the ans,ver. The best method of evaluating the benefit of \Vater injection for a given engine is actually to conduct tests and experimentally determine the performar1ce changes.

12.5

Water lniection in Combustion Chamber

I t was mentioned previously that the 'vater i11jected i11to an a.""ial-fl.o,,· con1pre...'-."Or tends to be centrifugally separated from the air. To alleviate t.his problen1, 'vater injection into the combustio11 chamber has been developed. By ,,.8.)' of illustration, the 'vater injected into the I>ratt (~ Whitney J57-P-43\V turbojet e11gine goes into both the compressor inlet and tl1c combustion cl1a111ber. Tl1c pri11ciple of operation of this method is best seen on the cornpressor-perfor111a11cc 111ap. 1.-igure 12.20 sho"-s the compressor maps for typical axial- and centriftigal-fto,v e11gines. As stated in the chapters 011 cornpressors a11d as sl10,v11 i11 tl1e figures, the 1/ "\ 18 lines for a11 axial-fto,v compressor arc steeper 11ear t11e st1rge li11e than tl1e 1, r/VB line3 of the centrifugal-flo\v compressor. Poi11t 0 i11 tl1e figures rcprcse11ts tl1c nor111al operating point at 100% rpm. At this operating cot1ditio11, tl1c gas-Ho"' is 11ormall)' chokrd in either the turbine or nozzle or botl1; t11us, t11e 011ly ,,·ny to increase the thru~t output of the engine is to raise tl1c pressure l1 pstrea111 to tl1cse co111ponents. ::\o''· ,,·hen 'vater is i11jected into tl1e con1bt1stio11 cl1n.111bcr, tl1c fto,,· rcsis ta11ce on the con1pressor is increa ed , and si11ce the e11gine rp111 is fixed, t11c con1prcssor operating point shifts up,vard on the consta11t NI Vo line u11til o. stabilized opcmti11g point _\ i.s

i,

Thrust Augmentation of the Turbojet and Turbofan Engines

Air flow, G0 F10.

I

323

Ai r flow, G0

12.20 Compressor performance charts for typical axial and centrifugal fio\\' machines.

reached. This point occurs 11vhen the pressure available from the compressor is equal to the pressure required by the ne11v fio\v system. It is apparent that, if too much flow resistance is added, compressor stall \Vill occur; therefore, it is necessary to maintain close control over the rate of water injected. The increase in flow resistance causes the pressure level of the engine, and consequently the thrust, to increase, but at the same time, the added compressor-fto\v resistance produces a tendency to decrea...c:e thrust by reducing the mass-flow through the compressor. Ho"·ever, since the pres.5Ure is higher at the turbine and nozzle, these components \vill handle n1ore mas.cs, the additional mass being the \Vater injected into tl1e bur11ers. Effectively, "·ater injected into the combustion chamber produces a thrust increase by (1) increasing the compressor-pressure ratio due to its reducing tl1e compressor air-flo''', and (2) increasing the total mass-fl.O\V through turbine and exhaust 11ozzle. The magnitude of thrust increase by tl1is metl1od is e11tirely dependent upon the compressor operating characteristics. As illustrated in Fig. 12.20, tl1e centrifugal-Ho,,· compressor, \vith its characteristic flat co1npressor N / VO li11es, produces ,·e11· little increase in pressure. This n1ethod of augmentatio11 is, tl1erefore, i111practical for engines "'1th flat N /VO lines 11ear the stall line. It is also impractical for engi11cs \vhich ha,·e their normal operating li11e very near the stall li11e. To sho\v the possibilities of \vater i11ject.ion into tl1e co111bustio11 chan1ber, Ref. 5 reports the analysis of a con1prcssor \vl1icl1 l1ad at 1003 rp111n11orn1nl opernti11g pressure ratio of 4.0 a11d a surge li111it pressure ratio of -l.7. 1\ t static sea lc\·el conditions it \Vas found that a11 augme11ted thrust ratio of 1.32 co\1ld be nttni11ed before compressor stall occurred.

12.6

Bleed- Burn or Bleed- Off Cycle

This type of tl1rust aug1ne11tatio11 11as 11ot us )ret bee11 applied i11 scr,rice aircraft; ho\vever, it is 1nentio11ed briefly l1erc because it is o. 111ctl1od for obtnirli11g ver)' lugh augmc11tatio11. The bleed-off cycle is sl10\v11 scl1e111aticully i11 14'ig. 12.21 . As sho\v11, tl1e bleed-off cycle requires tl1e i11stullatio11 of n11 nt1xilinr)' engine consisting of an air-li11c \Vit.11 a sl1ut-off valve, n bur11cr H :, a11d a11 c.xl1nust 11ozzle N t.· In

324

I

Jet Propulsion Water line-

c

D

-

-

T

-

'""" ~

--

F10.

12.21

Turbojet engine equipped for a bleed-off cycle.

normal operation, the shut-off valve to the secondary engine is closed, and the primary engine operates as a normal turbojet engine. In augmented operation, the shut-off valve is opened to allo\v air to pass into both burners. The secondary burner receives that amount of air \vhich is required to cool the normal burner; thus, about 50% of the air is burned stoichiometrically \vi th very high temperatures in H ~ (no turbine blade temperature limitations in JI 2) and expanded in N 2 to produce a high thrust. Since the normal burner no\v receives only about one-half of its air supply, water is injected into the burner to make up for this loss and to provide adequate cooling of the gases prior to their contact \vith the turbine blades. Because the secondary engine operates \vith high pressures and stoichiometric fuel-air ratios, considerable ~-t augmentation results from the bleed-burn cycle. To increa~ the thrust further, "-ater can be injected into the compressor. 1"his is a logical application because a water system is required for the combustion cl1ambers, a11d it is relatively sm1ple to pro,·ide a compressor 'vater-injection system. For this reason, the bleed-off c)·cle norma.ll)· includes \Vater injection to both compressor and burner. Utilizatio11 of a \•ariable-ares exhaust nozzle on the primary engine also resul ts i11 a11 additional thrust increa...~, but to a smaller degree. Figure 12.22 f ron1 Ref. 5 prcse11ts t11e calculated static sea le,·el performance characteristics of tl1e bleed-off cycle for a. typical tL~al (for n1edium pressure ratio) a11d a ccntrifugal-flo,v compressor, eacl1 ,,·itl1 a11d ,,·ithout a \•aria.hie exhaust nozzle area. The figure sho\vs tl1at very l1igl1 at1g1ne11tcd tl1rt1st rntios (thrust ,·alues more than double) accompa11ied by very l1igl1 specific liquid co11sumptio11 are a\•ailable from the bleed-burn cycle. A co11sideratio11 of t.110 turbofa11 c11gi11e ,,·ith a duct heater reveals that it is some\vhat sin1ilar to tl10 blced-bur11 C) cle. The high augn1entation ratio calculated in the sample problem is cvide11cc of tl1is. Although the bleed-off cycle produces appreciably n1ore tl1rust tl1a11 tl1c other metl1ods discus...~d, it also possesses more disadva11tages tl1a11 tl1e otl1ers. Its specific liquid consun1ption is ,·e~· high; its i11stallatio11 rec1uircs \Vater-i11jectio11 syste1ns to tl1e burner and compre.ssor It also requires suitable ducti11g and a11 additional engi11c operatu1g at \ 'el')' high 1

Thrust A ugmentation of the Turbojet and Turbofan Engines

I

325

temperatures; and the engine control, which is required to operate two water-injection systems, two fuel systems, the shut-off valve, and possibly the variable-nozzle eyelids, is considerably more complex. 2.6l...-_j~j__JL--L-..l---l--l--~-+~+-~L:To-tLa~l-sp~e-c~if~ic~I liquid consumption (lb/(hr){lb thrust))

2 .4 l-LLLL...!--!---i--+-+-+-r.:-:1L~:-:-/-:;r-;:~TI

1

2.2 ~i--i--l----..j~~~~---+--= ~ ~~...,.J

12 1

I

1---'-~4---.1---1-~+---+--4----lf----i-'. Ar_/~ j

0

·e2 .0 1---+---'---'--...1......--1-..J-----l~-+71~~~~u-.L_..l.--1--1-1 -..."' Centrifugal-flow engine-£;/ ' --Axial-flow engine :::>

-

8 '

J;

~

~

~

1.8 I----+--+----'--+-----+-

&

~

1.6

~

·~ v ,/

,

9

~ ~-+---t-4fl--M ~l~~·:....'---4-..LJ_J__L_L---'--L-_l....-i

~~

- - - Optimum-area exhaust nozzle l---+-+-++~~J..~ ·~l.l---+---1 - Constant-area exhaust nozzle V .. - • - Constant a ir flow

1.4 1----+--+v-.4.1.q.._--1---+-~

/y-'

I 1.2 1---, - --------

o Stoichiometric fuel-air ratio /). Compressor surge limit o Compressor outlet saturated fo r constant-air-flow case

I

1.0 OL.1/_L__L..!..5_1..__L__;lLO-L__J_J,. lS_L__JL..-::'2L::-0-L---1-;2:1;S:-1--.L-~30 Augmented liquid ratio

0

0 .1

0.2

0.3

0.4

0 .5

Total liquid llow Normol..ngine oir llow

12.22 Variation of augmented thrust ratio " ·ith rntio of total liquid fto,, to norxrutl engine fuel flow (au gmented liquid ratio) and to norn1nl engine nir flo,v for blN'd-ofT method of tn~t sugmcntation for a complete engine. Flight l\Inch number = O; altitude = sea lc\-el. (From Rr_f. 5.) F10.

12.7

Summary of Thrust Augmentation Devices

Table 12.3 presents a summary of the augn1c11tcd thrt1st ratios and specific liquid consumptions obtainable f ron1 typical it1rbojet c11gi11cs ns based 011 t l1c data of Ref. 5. Remarks a.re included i11 the table to sumn1nrize tl1c npplicatio11 n11d linutations of the various thrust augmentntio11 devices.

12.8

Effect of Humidity on Engine Performance

Ilumidity affects turbojet o.11d tt1rbofa11 e11gi11c pcrforn1a11ce bccatlSC the nu~iure of 'vatcr vapor and air l1ns gns properties \Vl1icl1 differ slightl)' fron1 those of dr)· air The primary reason for tl1is difTere11ce is tl1c fnci tl10.t ,,·at er ,·apor is lighter than air. This is evident from their relative n1olecular '''eigl1ts: II!O = 1 and air = 29.0. ..\t

1

326

I

Jet Propulsion TABLE 12.3 Summa.11' of Performance Data of Typical Thrust Augmentation Devic~

l\1ethod

Afterburner

Thrust Ratio and Liquid Consumption

35,000 Feet

Sea Level

Remarks !If

=0

"Af

= 2.0

M

=0

ftf =

2.0

FNo / FN

1.5

3.0

1.5

2.5

slc

2.4

2.4

2.0

2.2

Limited by stoichiometric mixture or therms.I choking. In service use on an extensive basis.

7 \\

Requires separate liquid. I.imater injection FNo / FN 1.4 2.6 1.2 2.0 at compressor ----------------------------------~ ited by air saturation at cominlet sic pressor outlet. In service use 3.2 9.0 2.4 6.0 on limited basis, primarily for thrust restoration at take-off.

Water injection into burner

FNo / FN

1.3

2.4

slc

8.0

15.0

Not practical on engines operating near stall line. I ..imited by compressor stall. In service use on limited basis. Produces greatest t..hru_c;t increase with greatest liquid consumption and greatest comple.xity. Limited by com~r-outlet air saturation and stoichiometric mi~u.re. Kot in ser• vice use.

Bleed-off FNol FN 2.3 (includes HiO -------------injection to slc 11.0 compressor)

a specific humidity of 300 grains of 'vater vapor per pound of cir.)' air (100% relati,-e humidity at 100°F at sea level pressure), R, Cp, and c, are respecti,~el.}· 2.-1, 3.5, and 4 % higher than in dry air conditions. It is sho,vn in Ref. 11 that, even at tl1is 11igl1 a specific humidit.}', the effect on engine performance is only about 1%. U11dcr sta11dnrd conditions at sea le,·el, the specific saturation humidity is about 75 grai11s per lb. Tl1e sa1ne ,·alue is realized on a hot day above 15,000 ft. The a11alysis of Ref. 11 sl10,,·s t11a t tl1e effcct of this n1agnitude of specific humidity is negligible. As a rcst1lt, l1\11nidit)' reall.}' onl)· affects engiI1e performance on hot days at lo\v altitude, a11d tl1c11 b)' 011ly n. relnti\·el)' sn1all amount . Referc11ce 11 gives tl1e complete method for corrccti11g for hu111idit.}' i11 th°'-.~ ~~s 'vl1ere extreme accuracy is desired. PRODLE?llS

I. Sho'v by equations nnd discussion 'vl1y a vnrinblc area no1111lc is ncces5ary for sn aftcrburning engine.

Thrust Augmentation of the Turbojet and Turbofan Engines

I

327

2. The following static sea level cln.ta apply to a n a fterbu rning engine when t he afterbu rner is off : Pt, = 32 psia ; T ,, = T ,, = 1300°F ; Pt, = 32 psia; A., = 312 in.2 ; K = Cd = Cv = 1.0; G1 = 3 lb per sec. From these data calculate F0, Pe, T.,( 0 R ), V 111 G,, sf c, and Ga. When the afterburner is turned on, the follo\ving data. tlJ)ply : Pt. = 33 psia ; T ,, = 1300°F ; T ,, = 3600 °R ; p 1, = 30 })Sia; /{ = Cd = Cv = 1.0 ; and Ga is t he same as for nonafterburner operation. Calculate of a fterburner; a lso F 01 p,, Te( 0 R ), Ve, A ., v~,, Bf c (total). Fuel H .V. = 18,900 Btu per lb.

v.,,

a,

3. Using the data in Problen1 2, find the augmented thrust ra tio by Eqs. 12.5 and 12.8; then compare and discuss the results with the ans\ver found in Prob. 2.

4. R epeat the second portion of Prob. 2 \vl1en K = 0.92; Cd = 0.96; Cv = 0.96. 5. Show by the h-S and compressor-performance chnrts ho'v water inj ection improvef thrust . Also state why the h-S diagram does not .show the thrust increase due to increasec l mass-flow.

6. A certain turbojet engine compressor is required to ha ndle 100 lb of air per sec to obtain

a pressure ratio of 6 to 1 with standard sea level air at its entrance section . Compare the '"ork of compression for a n isentropic process and an isothermal process; then discuss the results.

7. List each fa ctor which produces an increase in thrust for wa ter inject ion. 8. Show how water inj ection into the combustion chamber increases the thrust of a turbojet • engme.

9. State by discussion how the bleed-bum cycle achieves such large qua ntities of augmented thrust. REFERENCES

I. R enas, P. E., Harvey, R . W., Sr., and Jansen, E . T ., "Altitude Starting Characteristics of an Afterburner with Autoignition and Hot-Streak Ignition,'' NACA R:\I E53B02, 1953.

2. Wilkinson, P. H., Aircraft Engines of the World, N ew York : ' '1 ilkinson, 1961. 3. Aviation W eek and Space Technology, l\'farch 16, 1964, pp. 189- 90, 203--1. 4. K eenan, J. H., and Kaye, J., Gas Tables, New York: John " 7iley and Sons, 19-18. 5. Lundin, B. T ., " Theoretical Analysis of Various Thrust Augmentation C)·cles for Turbojet Engines," NACA TN 2083, 1950. 6. " JT3C-26 Turbojet Engine Specification," Pratt & Whitne)' Aircraft Spec. X o. li70, October 9, 1959. 7. Beke, A., "Analytical Investigation of the Effect of Water Inj ection on Supersonic Turboj et-Engine-Inlet l\1atching and Tl1n1st Augn1ent.ntion," NACA TN 3922, 195i. 8. Beke, A., "Experimental Investigation of " 7ater Injection in Subsonic Diffu...c:er of a Conical Inlet Operating at a Free-Strenn1 l\•l ach Number of 2.5," NACA R:\I E56Jl5, 195i. 9. Trout, A. l\1., "Theoretical Turbojet Tl1n1st Augn1cnt.stion b)· EYaporntion of '\at.er During Compression as D etermined by Use of n l\lollicr Diagram,'' NACA TN 2104, 1950. 10. 'Vilcox, E. C., and Trout, A. 1\1., "Annlysis of Thrust Augn1entation of Turbojet Engine by Water Injection at Compressor Inlet Including Charts for Calculating Compression Processes 'vith " ' atcr Injection," NACA Report 1006, 1951. 11. Samuels, J. C., and Gale, B. 1\1., " Effect of Humidity on Performance of Turbojet Engines," NACA TN 2119, 1950.

13

Turboprop and Turboshaft Engines •

13. 1

Introduction

All gas turbine engines 1have three basic components in common - compressor, burner, and turbine. These three components function together to generate hot ga.s. Gas turbine engines are thus primarily hot-gas generators. The method of producing thrust from the high-temperature and high-pressure gas developed by the gas generator determines the type of engine. (1) The turbojet engine expands the hot gas from the gas generator through a nozzle to produce a high-velocity jet and therefore thrust. (2) The turbo!an engine partially expands the hot gas from the gas generator through another turbine which drives the fan; the thrust is produced by the combined expansion of both the by-pass fan air and the hot nozzle gas. (3) The turboprop engine expands the hot gas from the gas generator through another turbine to produce shaft power to drive a propeller; the thrust is produced by the propeller.and a -small -residual expansion of the hot gas through a nozzle. (4) The turboshajt engine completel)· expands the hot gas from the gas generator through another tUl'bine to produce shaft po"·er; the thrust is then produced by the utilization of the shaft po\\·er in a helicopter rotor, a ship propeller, or the 'vbeels of a ground vehicle. This chapter "-ill be devoted to the theory, analysis, and performance characteristics of t1lrboprop and ti1rboshaft engines as they differ from turbojet and turbofan engines. As \\•ith the other engines, it is appropriate to consider some actual turboprop and turboshaft engines and their characteristics before beginning the detailed anal)·sis. Figure 13.1 is a cuta\vay dra,ving of the General Electric T64 turboshaft engine. The principal characteristics of the engine are sun1marized in Table 13.1. The basic elements of the engine are identified in Fig. 13.1. These elen1ents include the 14-.stage compressor (with the first four stages plus the guide vanes incorporating ,-ariable stators); the annular combustor; and the split four-stage tltrbine ('''ith tl1e first t,,.o stages driving the gas generator and the last t'vo fur11isl1ing useful shaft po\ver). Figure 13.1 also suggests that this versio11 of tl1e T64 e11gine has only a shaft at the fron t end, running at the same speed as the po,ver turbi11c (13,600 rpn1). (Four of these engines po\ver the Vought-Hiller-Ryan XC-142A ' ' / STOL cargo a~ult transport, \vhich is depicted in Fig. 13.22.) In Fig. 13.2, the reduction gear box 'vith a reductio11 ratio of 13.4-1:1 (propeller shaft rpm 1160) ca11 be seen offset from the front of the e11gi11e and dri\•en b)r the san1e 328

I

Turboprop and Turboshaft Engines

329

Balanced moment weight blades and buckets High pressure ratio compressor with split steel cosing

Shrouded initial stator stages ---~

External nozzles and igniters

Two-stoge gos generator turbine and two-stage free power turbine with split cosing and stator assembly

Short, small diameter annular combustor Fuel control, pumps, filters and accessory pods grouped externally

Flo. 13.1

Cutaway drawing of a. General Electric T64 turboshaft engine. (Courtesy The Gmeral Elutric Company.)

shaft that is exposed in Fig. 13.1. Several small accessories and a large electrical generator may be noted on the rear face of the gearbox. The T64 turboprop engine bas t"ro versions - reduction gearbox offset-up and offset-do"rn - to allow flexibility with regard to the aircraft installation. A comparison of Figs. 13.1 and 13.2 reveals that primary difference between the turboprop and t urboshaft versions of the T6-1 engine is the reduction gearbox on the turboprop version. In actuality, both types of engines require speed reduction in some form of a gearbox. For the turboprop engine, the gearbox is usually incorporated as an integral part of the engine - "'1th a common lubrication system, for example. For the turboshaft engine, the gearbox is usually separate from the engine - in helicopters for example, '"here the dri\re often turns a right angle in the reduction gearbox, or in the XC-142A V/ STOL transport, where the reduction gearbox is made integral 'vith the propeller. •

••

-..

-.• •

•

~~

-•

.

.I T

F10.

I

-.

• -. •

13.2 Photogrnpl1 of tl1c General Electric T64 turboprop engine. (Cour/c$y The General Ekctric Co11ipany.)

330

I

Jet Propulsion

Figure 13.3 is a photograph of a cuta\vay model of the Canadian Pratt & Whitney PT-6 (T-74) turboprop engi11e. The figure indicates that the compressor has three axial stages and one centrifugal stage mounted on a common shaft and driven by a single-stage turbine. A second free turbi11e stage fur11ishes the useful shaft po,ver to the reduction gearbox. The principal characteristics of the engine are tabulated in Table 13.1.

•·

FIG.

13.3

Photograph of a cutawa)' model of the Canadian Pratt & Whitney PT-6 turboprop engine. (Caurlesy Pratt & Whitney Diuision, United Aircraft Corporation.)

The "bolt-on" reduction gearbox to convert the T63 to a turboprop engine is illustrated in Fig. 13.4. The gearbox reduction ratio is 2.93:1. The compressor, incorporating six axial and one centrifugal stages (see Fig. 8.3), is driven by a t\To-stage turbine. A t\VO-stage free turbine provides the useful shaft power to an offset power take-off (on both the front and rear of the gear case) through a built-in, two-stage reduction gear train (reduction ratio 4.959:1). Another interesting feature of the T63

F10.

13.4 Cutn,vny drn\ving of the Allison T63 turboshnft engine. (Courtesy .-\ llison. Dirision, Grnocl Al otors Corporation.)

Turboprop a nd Tu rboshaft Engines

I

331

engine is the single combustion chamber \vith reverse flo,v through the turbines back to"·ard the compressor. All of the engines illustrated in Figs. 13.1 througl1 13.4 arc "free-turhine" engines. As the figures ''·ill suggest, tl1is means that the po,vcr turlJinc is mechanically independent of the gas generator turbine. Almost all of the ne\v turboprop and turboshaf t engines arc of this type. There arc, ho,vever, a large number of "fixed-turbine" engines in service today.

F10.

13.5 Cross section of a Pratt & Whitney T34 turboprop engine. (Courtay Prall &: TrhiLney Division, United Aircraft Corporation.)

Figure 13.5 shows that, in the "fixed turbine" design, the turbine (three stages in this case) drives both the compressor and the gearbox on a common shaft. It is interesting to note that the T34 reduction gearbox (reduction ratio 11.01 :1) is built into the engi11e in line, requiring an annular compressor inlet. Also of interest are the large compressor air bleed valves for starting and ground operation (see Chapter 8 for "bleed valve" effect) . The Allison T56 turboprop is another example of th.IB t~i>e engine 'vith both commercial and military applications (Lockheed Electra and C-130 transports). Figure 13.6 illustrates an engine mod ification '' 'hich lies bet"·een the "fi.'\:ed., and "free" turbine types. T he figure sho,vs a photograph and schematic dra"ing of the Rolls Royce Tyne turboprop engine. As may be seen, the T)•ne 11as a t,,-in spool compressor. The first turbine stage drives the ni11e-st.age, high-pressure compre~r. The last three turbine stages drive the six-stage lo,,·-pressure con1pressor and furnish useful shaft po,vcr to the reduction gearbox through a separate i11ner dri,·e shaft. :\ s ''ith the T34, the reduction gearbox is in-line '''ith t he e11gine. The projections on top of the engine arc oil coolers for cooli11g the cngi11e a11d gearbox lubrication s~-stems. Table 13.1 is compiled fron1 Refs. 1, 2, and 3. I t i11cludes the primar~· characteristics of all engines discussed abo,,e as \Yell ns a 11un1lx-r of other rontempora~· L" .S. and British turboprop a11d turbosl1aft c11gi11cs. l"ron1 tl1c table, the prin1ar~· difference bet,veen turboprop a11d turbosl1nft e11gincs nll\)' be ... ce11 to be the ,,·eight of the reduction gearbox. Reference 1, \Vl1icl1 is re\·iscd n1111unll~·, gi\·cs further i1uormation on this topic.

13.2

The Basic Turboprop or Turboshaft Engine

T l1e cycle of t l1e l)nsic tt1rboprop or t t1rbosl1uft. e11gi11c is sho,,·n on the h-S diagram in Fig. 13.7. A scl1cn1atic diagram of the c11gi11e togctl1er ,,·ith the station designations is also indicated.

1

TABLE 13.1 Engine D esignation

..,c..> (..)

Characteristics of Some Current U.S. and British Turboprop and Turboshaft Engines

M a nufacturer

T ake-Off SHP

T ake-Off Thrust (lb)

T ake-Off

sfc (lb/ hr/shp)

Air-flow (lb/ sec)

Weight (lb)

Diameter (in.)

Length (in.)

ESHP1

Turboprops T56-A-10W T63-A-1 T60-(520-8) T64-GE-4/ 8 T53-L-7 T55-(LTC4G-3) T34-P-9W T74(PT6A-3) Double Mamba 8 Proteus 765 Eland 504 Dart 10 Tyne 11

Allison Allison Boeing G eneral Electric Lycoming Lycoming Pratt & Whitney Canadian P & W Bristol Siddeley Bristol Siddeley Napier Rolls Royce Rolls Royce

4200 250 550 2770 1100 2445 6950 466 3600 3915 3230 2305 5030

750 36 55 210 125 225 1375 85 730 1260 700 650 1235

2

""°.52 0.71 0.65 0.505 0.65 0.62 "'0.702 0.69 0.65 0.48 0.62 0.63 0.53

33 3.0 5.3 24.5 11 21.5 65 5 42 46 33.5 27 47

1845 147 275 1136 530 795 2870 250 2500 2900 1820 1340 2220

3.0 5.3 12.4 24.5 11 19

128 210 285 713 485 570 870 225 835

27 x 44 19.5 25 29 x 36 23 24 34 18 51x57 40 36 37 43

147 45 69 113 58 59 154 52 103 101 116 100 109

4500 264 572 2850 1150 2535 7500 500 3880 4445 3500 2555 5525

41 43 55 83 48 44 99 49 70

264 520 1303

-

-... 'tD"'CJ

.,,c 0

--·"' 0

:::J

Turbo shaft T63-A3 T72-T-2 T58-GE-8 T64-GE-6 T53-L-9 T55-L-5 JITD12A-3 T74-P-2 Gazelle 512 1

ESI-IP •BHP +

Allison Continental General Electric General Electric Lycoming Lycoming Pratt & Whitney Canadian P & W

250 500 1250 2850 1100 2200 4050

36 50 132 210 130 250

500

85

Napier

1750

280

-

0.71 0.67 0.61 0.495 0.65 0.63 "'0.90 0.70 0.68

49 5 17

19.5 19 16 24 23 24 40 18 34

2890 1152 2300 4050

534 1860

Thrust

2.5 ' n l\tin 11;11 nro buHc

05 a.

.....0

i....--

-

--

"' ~4 0

"':::> 0

I

-£ 3 c a..

·-

40,000

J:

..... 2

-0

-

.....0

1 0 0

0 .1

0.2

0 .3

0.4

0.5

0.6

0.7

1.0

0.9

0.8

Moch number F10.

13.11

Variation of the maximum thrust horsepower of the turboprop engine with altitude and Mach number.

...

..t:

a.. J:

!:::.. _.a

1.4 1.3

- , 1.2

....

c 0

·za. 1.1 E :::> 1.0

"'c

8

I

0.9

Seo level

~ 0.8

-::,....-:::: 1o.000 ~ii. 20.000 ~

·-u 8. 0.7

30,000 40,000

"'

~ 0.6

S~o level

0

~

-

/Ji110.000

0.5

.2 0 .4

20000 -

///, ~o:ooo ./.'. ~ o.ooo-

0

: 0.3

·-C> 0 ·-

0 .2

C>

0

-...e ): 0

a..

~,~

0.1

0

0.1

0.2

0 .3

~ 0.4 0.5 0.6 0.7

0.8

0.9

1.0

Mach number

13.12 Variation of specific fuel consun1ption based on tl1rust horsepo"·cr and variation of tbt ratio of jct to total thrust horSCJJO\vcr ''•ith l\ Inch number and altitude for the turboprop engine s· • mn.x1mum po\vcr.

F10.

Turboprop and Turboshaft Engines

I

343

140

-

-

130

•

'

-

120

.

11 0 100

-

.,... :0 80 - 70 u

90

~ >·

~

-

Seo level

"J.~

0

·< 50

10,000

40

20,000

-

I

I

; ... 60

-

I '

~

-

I

30

30,000

20

I

J0,000

10

I

0

0.2

0.1

0

0.4

0.3

0.5

0 .6

0.9

0 .8

0.7

1.0

Mach number

Flo. 13.13

Variation of air-flow rate with l\1ach number and altitude for the turboprop engine at • maximum po"•er.

0 .70

-:r: 0 ..

u. vi. 0.60 . w -: .........

--

,_

-

•

~

0 .40

0

10,000

•

100 200

I

--........ -

0 .50

,0

l I

~ ...

Cl)

.0

•

I

~

... -:::> • Cl)

' -

I

'

~--

20,000

30,000

I I

40,000

50,000

300 400 500 600

knot~

-

...

0

"J.

8.

-

0

...0

E

z

60,000

N.A.C.A. standard altitude (feet)

Flo. 13.14 Typical variation of l'quivnll'nt shaft horsepo"·cr and specific fuel ronsumption for the turboprop engine.

p

I

344

Jet Propulsion

Figure 13.14 sho,vs typical variations of ESHP and esfc \Vith altitude for different flight speeds for a fixed exit-area typical turboprop engine operated at normal po,,·er. Two items are evident from this figure. First, the turboprop engine, like the turbojet engine, is unsupercharged; therefore, its po,ver falls off as altitude increases. Second, fuel economy is improved \vith an increase of speed and altitude, and at high speed and altitude the specific fuel consumption is very competitive \vith the reciprocating engine.

13.4

Analysis of Performance of the Turboshaft Engine

Although it has been sho\vn that the jet power of a turboprop engine is small relative to the propeller po,ver, it is useful to examine the performance of the turbosl1aft engine separately. Actually the turboshaf t engine is also a basic gas-turbine po,ver plant - an engine consisting only of a compressor, burner, and turbine. AB a result, the follo,ving analysis is applicable to stationary power plants and to automotive and ship propulsion as well as to aircraft and helicopter propulsion. Referring again to Fig. 13.7, and considering the ideal cycle (no losses in compressor, burner, and turbine and constant specific heats), it can be shown that the over-all efficiency of Eq. (2. 15) is equal to

.,

I- 1

'Tio = 'Tia =

(13.21)

1-

Because of the assumptions made in the derivation of Eq. (13.21), air-cycle efficiency.

714 is

k:nov.n as the

Figure 13.15 from Ref. 6 presents a plot of the air-cycle efficiency versus pres.5UI'e ratio for several values of -y. As evident, gas-turbine po,,~er plants must operate at high cycle pressure ratios to achieve good efficiencies. Of interest is the gain in efficiency realized by using gases \vith a high value of 'Y· H elium, for example, with a. 'Y

1 loo' ~\-\e\\u~ 1 ""''·I

0.60

I

0.50

0 .40

/

s:- 0.30

/ I /. h~

0.20

0

v

/

0

0.10

7_ - ,395 . I \ .350

__....... /

~

/ /

~~

-

l .300

-

-::::::. ~-

.,.,...

// / .,, ' / l oad

IW

13.15

r----.... T

~

2 _ Go ~

~

c

-

H 3

i G,

. 2

3

5

6

Pressure ratio, F10.

= 5

I/ 1

I

7

8

9

10

p,3 /p,,

Air cycle efficiency variation of basic gas turbine po"·cr plant. (From Ref. 6.)

Turboprop a nd Turboshaft Engines

I

345

of 1.66, attains a 25% efficiency improvement over air at a pres.'3ure ratio of 6. The practical application of such gases is, of course, limited io closed-cycle operation . As shown by Eq. ( 13.21), the ideal plant efficiency is a function of only the plant pressure ratio. No,,·, ,vhen \Ve examine the actual plant 'vhere losses are present, we find tl1at the over-all efficiency is a function of both the cycle pressure ratio and temperature ratio. An examination of the actual plant, assuming zero pressure loss in the burner and constant values of specific heat, gives the follo,ving equation for the net work output and over-all efficiency:

T,. .,,, - X e + 1

T,,

1]o

(13.22)

'Tic

T, . .,,, - Xe+ 1 T ,, TJc

=

(13.23)

1Ja

T ,. - 1 - ~ T ,, TJc Recognizing that X e and 110 are functions of pressure ratio, it is evident that both '\\·ork output and over-all efficiency can be \vritten in fuctional form as "Ir and TJo =

f

Pt., T, ., T,.,

TJc, .,,,

Pi. Figures 13.16 and 13.17, both from Ref. 6, present the variation of net output and over-all efficiency \vith cycle pressure ratio, machine efficiencies (.,,, and TJc), and turbine inlet temperature for operation in 70°F ambient air. .5

Go

_;,

5

Lood ,-_

!-

c . -

~

..;--

.........

H G,

110

0 Of r ooO 1

,' . 0,.1

.,,,,...

~

'l .... ,or~i J~ / ..,o,,;i' I

80

'~ 70 •

3Q. 60

11

-5 50 -z 40

I/

t

rY//

I>

30 20 ...

0

,

...!..

.. .. 08 g 09 ~ 1.0 ~

-v

.........

v

...... ..0 .75

g_

'

I/)

......

•''

-§

.Q

-

I

' ' 070 ,065 •

5 7 9 Pressure rat io,

'Ir- '" / I o.9s 0.90

-

~ 60

0.85

::>

-z., 0

150

:

-«--__.-- --r 100 ..-: ~ -. I

::::::::. 70

-~

_ ., ::: \ .0

•)

8.

50

u

~

-40 30 20 10

I

(o )

FIG.

80

06

~

'~~h

',

. c

...0

0.7

.,~ +

OQ.L

os!.

90

~

Q •

/ I .......

I/ 0

1/·

"'lj

~/ 1 , -;-:),

8

0 .85

"-6

.r

•

I

..£;

.;,"I

/

/ I

6

...

100

/

·O ~"'~

/

110

0.4

.......

:§/

32

go~.

0 11

p,,/p,,

13

15

I

2

3

-4 5 6 7 8 Pressure rolt0, ~

9

10

{b l

(a) Effect of machine efficiencies on over-all efficicnc)·. (b) K et output versus pte5SUtt ratio for various mnchinc effi cie ncies.

Figure 13. 17 sho,vs the effect of 111acl1inc cfficie11cics on 'lo and 'fl! for a gas-turbine po,,·er plant operated at a typical valt1c of turbi11e i11lct tcn1peraturc. As e,;dent, high mach ine efficie11cies are requi red to provide good o\·er-all characteristics. It is interesting to note that gas-tt1rl)inc pO\Ver pla11ts 1 althot1g}1 UJ\\·a)'S theoretically possible, did f not become practical t1r1til reasonable value of nlncl1i11c cfficiencie and C)·cle pres..."lll't' ratios \Vere n1ade possible. 'I'l1 is 1 t l1e11 1 is t l1c basic reason ''°h)' gas-turbine po"·er plants are relatively ne'". l •'igurc 13. 17 also sho''' t l1at tl1c opt i111un1 pressure ratio for ma~i­ mum efficiency is higl1er t l1a11 t l1at for 1110.xin1u111 \York. Since the opti111un1 press\1re ratio is difTerc11t for 111a..xin1un1 po,,·er and efficienc~-. a gas-turbine po,,·er plant is eitl1er dcsig11ed for ma..ximt1n1 po,,·er, n1a..'\.;mum efficienc~-. or a compron1ise of these itc111s. Of spccinl sig11if1cancc ,,·ith regard to the optimu n, pressure ratio i t11e fnet t l1at tl1e llcst ,,·ay to i11cl'('ase c11gi11e po,,·er or economy is t \1 i11crcasc botl1 pressure ratio and turbi11e i11let te n1perature. It is true that an increw:r

I

Turboprop and Turboshaft Engines

347

in turbine temperature 'viii increase po,ver and efficiency, but the simultaneous i11creasc of both turbine temperature and pressure ratio 'viii produce even greater gains in performance. Because of the importance of optimum pressure ratio, it is 'veil to examine its functional relationship i11 more detail. Reference 6 sho,,·s that, for no pressure drop in the burner and cor1stant values of specific heat, the optimum pressure ratios for maximum 'vork and maximum efficiency are given by the follo,ving equations:

Pt. Pt.

(13.24)

(maximum 'vork) opt

Pt. Pt.

af]c

opt

/.....:(_a_-_l~)

1+

+1-

'1c

....: +_ 1_-____:'1.:::..c_-__.:.. a ...:.:. T] ":...:. 11 t::.!.-)

a....:. TJ.:::.. c

...!.... (

"J

CX71t

(13.25)

(maximum efficiency)

where a = Tt. / T,. the cycle temperature ratio. Figure 13.18 from Ref. 6 presents a logarithmic plot of these equations from which it is possible to determine the optimum pressure ratio for maximum 'vork or efficiency 5

load A

2

T ~~

'H

lJ M = lJ I = 1.0 2 = 70 °F 'r =

_c r ' 3

G1

2ooo r-~~-r~-+----1--1~~1-1-+-1---1-U.

T2 = 70°F

....._.HM--"

1900 t----+--l--h-h-11++.l-J-..J-l-.J~

1800 r----·r---+-1-~ J-i#-1--1-1-f-.l...J.,.J.._J.U

G, 2500

I - C

•

.

'

I

I

~

•

2300 2100 ~ 2000

'

D.

j

e 1900

1800 ~ 1700 1600 ti c 1500 ·-

---

-

--

f

ti

I ~

,_

-

1000

1

j

'

J

j

I

2 3 4 5 6 6 10 Pressure ratio,

(a)

I-.-i. ·-·

-.. .I

~

E

1 ~ 1400 r------t~.f-1f-1-:µ..J'-#-l_:.J -l-J~----~

_o

"

..... J

J

//

~

I

j

1 · ~ r.-t--+ -f--J4-ILI--~

-

j

1100

I

1600

- -

~ 1500 t-----+-~f+--f-f~Y--f-'~J-J-4~~

'- /J

t

-v

J

I

~- ~~ 0 r' 0 l o Jo ~ "> 0 - Q) ~j o· o· ' ~. -I -,,.,

o·

.$J

1200

f

~-

c 1400

j.

J

.

8.

I

-

--

1700 1--- - - + - - - ! . -1-

i

J

I

I

LL

1300

1

'lb = l'JM = any value

llp = O

.;

Perfect gases No pressure drops

2100 r--------.----..,lr---.,r-rl ~~ . ,-,....--.Y__.r~r~~

5

·-...

1.395

~"/

... //

- S' J

20 3040 60 100

p,3/ p,.,

1100 I

1000!'--~~'L-:.'.&...-'-":1-'' J....1.-'-'~l.L......!_!_.!....:....l..l..-_J l

2

3

4 5 6

6 lO 12 15

Optimum pressure ratio, p ,3/Pr2

(b)

Fro. 13.18 (a) Turbine inlet tcmperatttrc versus prc...~ urc ratio for ma.Uinum "·ark. (b) Pressure ratio for ma.ximum efficiency.

I

348

Jet Propulsion

for any value of turbine inlet temperature and machine efficiency. Suppose, for example, that a given plant is to operate at a turbine inlet temperature of 1800°F with .,,, = 1Jc = 0.85. From Fig. 13.18, 've find that the pressure ratio for maximum po,ver is 7.4, and for maximum economy it is 14.0. Figure 13.19, also from Ref. 6, shows the variations of maximum work and maximum efficiency \Vith macl1ine efficiencies for various values of turbine inlet temperature and a cross plot of the optimum pressure ratios required to achieve maximum performance. 5

5

Ti = 530 R

loa d~

80

- 4

cp= 0 .243 'la = TIM = 1.0 H.V = 18,500 Btu/lb

1--t-+--t-+--1r-+-+--+-+-+--+--+-~........... 0.17

0.8

F10. 13.19

T1 .,. 530 R "'( = 1.395

r =1.395

0.9

1.0

c,. = 0 .243

'11 = 'JM =

I0

.

.2_o

o~~l_._~--l.-l....._._~-'-~l....i-....;1~ 0 0.7

0.8

0.9

TJr and 'Ile

'11 and 'J c

(a)

(b)

1.0

(a) Maximum efficiency versus mnclline efficiencies for various t~mperatures. (b) Ma.'timum work versus machine efficiencies for vn.rio\lS temperatures.

Thus far, in the discussion of the bnsic gas-tt1rbine po"·er plant, the pressure I~ in the burner is assumed to be zero. In actt1al practice this is, of course, not true, and burner pressure losses affect both the c11gine po,,•er and econon1y. Figure 13.20 from Ref. 6 presents tl1e effect of tl1is pressure loss on the perforn1ance of a basic gas-turbine power plant. As pointed out earlier, this scctio11 on tl1e turbosl1aft engine or the basic gas-turbine po,ver plant is not directly applicable to the ttirboprop engine, because the ha.sic gssturbine po,ver plant does 11ot t1tilize any exl1aust jet po,,·er. The trends predictro, ho,vever, are applicable, because in the turboprop c11gine, the jet po,ver is small compared to the propeller po,ver. Chapter 11 has suggested that tnilitary specifications require a certain method of performance presentatio11. In additio11, correction factors must be provided to allow determination of the installed performa11ce of the engine. Con1parable information is required for turboprop and turboshaft e11gines (see Ref. 7). Since the correction

......

Turboprop and Turboshaft Engines Ga

= flM = 1.0, Ap = 4' (/>7) T4 =1500 F, T2 = 70 °F

Load~

1= 1.395, cP = 0.243 fJr = 0.85, Jlc= 0 .84

'--IH

G, 28

::t:

I

~

0): 30

.. 20

z

10

'I

--

H.V.= 18,500 Btu/l b '

j

I

_I

I

b , == 0%p,

/

/ / J /, '1 II

'J

0 ' 1 2

Flo. 13.20

- 0.8 ~ 9: ., , - 1.0 :::> ..D u - 1.5 ~ - 2.0 ~ - 3 .0~

I

..D

- 40

0.7 ~ _c

I/

60 co

:::>

/J /

-~0 12 ti> 8 0 4

'-. 50 :::>

E

v .-

16

0

- 0.6

~ oiP'l

~20 ti

·-- ~

- 0 .5 0c ·-0 .

be ._.,,,,,...-_ - '01.P2 bP.-~ ...... b P.""' 20~P.2

.. 24

349

fJb

2

5

I

~, == 1~%p, [>P

-

~

-.......

..., iox '2

75 ... Cl

o~

9

Effect of pressure drop on work and efficiency

Q. ~

-.

- 50 -~

~

·u Cl

~

- 25

3 4 5 6 7 8 Pressure ratio, p,3/P12

): ..D

0.

V>

10 (opcn~ycle

combustion turbine).

factors are essentially similar to those discussed in Chapter 11, the)' are not discussed further here. Reference 8 gives further details on engine specifications. It is felt worthwhile, ho,vever, to sho\v a typical "G and A" perforn1ance for turboprop or turboshaft engines. Figure 13.21 from Ref. 8 presents the seA-le,·el and 25,000foot-altitude performance of the Ge11eral Electric T64 turboshaft e11gine. Figure 13.21 sho,vs that, in additio11 to tl1e shaft po,,·er output. the air-Ho''"' fuel flo,v, and gross thrust of the engine arc also plotted. I t is i11teres ting to note that this engine, although a turboshaft, has a tl1rt1st output. The san1c tl1ing ca11 al8o be seen in Table 13.1. 1~he explanation is t\vofold. First, the exhaust ga..."'€'s must be discharged, and some thrust is produced i11 the procc... ... Scco11d 1 since the prim&r)· difference bet,veen tur})oprop a11d tt1rbosl1nft c11gi11cs i' the redt1ctio11 gear box, the performance sho,vn in Ji'ig. 13.21 is also typical of tl1c T64 turboprop engine.

13.5

Shaft Power Absorber -

the Propeller

It was noted that the po,vcr outpt1t of t.\1rboprop or turboshaft engines can be utilized in many \Vays - aircraft or sl1ip propellers, l1elicoptcr rotors, and groundvehicle transmissio11 systcn1s. Because of recent aircraft propeller de\·elopments, however, particularly witl1 respect to VTOL aircraft, it is '''orth,vhile to consider the

I

350

Jet Propulsion - - - Shaft horsepower

100% ram recovery

- - - -

Fuel flow {lb/ hr)

ICAO standard atmo sphere

-

Gross thrust (lb)

Lo wer heating va lue o f fuel, 18,400 Btu/l b

-

- - - Airflow {lb/ sec)

Alt itude effects 1ncluded Output speed - 13,600 rpm Exhaust area = 450 sq ft

o Roting point Seo level Maximum (10 m in)

3200

M ilitary

1130

Torque limit lb· ft (Q) 13,600 rpm

3200 25,000 ft Maximum

2800 (10 min ) 2400 a. ..s::.

... ...•

~- 2000 i-------

- ---

----">

11,

=100%

~ 0.80

Q..

0.75 0.70 0.65 0.60 1

2

3

4

5

6

7

8

9

10

11

12

13

Pressure ratio F10.

13.29 Ratio of intercooled to normal compressor work versus pressure ratio for three values o! intercooler effectiveness (calculated for a. 5% intercooler pressure loss and 11,, = 0.866).

employed in stationary po,ver plant installatio11s, both for gas-turbine 11nits and compressors in general. From Fig. 13.29 one can also see that an intercooler will increase the over-all engine efficiency, because any po,ver which is saved in the compression work is delivered to the shaft. The net result is that for the same amount of fuel burned, more shaft po,ver is available. Thus far, in aircraft application, the most attractive method of reducing compression work is by means of \vater inj ection. This system has already been discussed in Chapter 12 as applied to turbojet engines. Consequently, there is no need to repeat the discussion here since the benefits derived are con1pressor benefits, and thus are the same in both engine installations. 'Vater injection is, of course, of neces.5it)· a part-time augmenting device, \vhereas intercooling can be used continuously. Perha~ our first intercoolers in gas-turbine engines " ·ill appear in engines "·here other t)rpe5 of heat exchanges are required as a necessity, like the heat exchanges i11 a nuclear po\ver plant \vhich \Viii be required to replace t l1e con1bustio1\ el\an1bcr. The application of intercoolers in aircraft po,vcr pla11ts is not ne,v; they l1a\re 1011g bee11 used in reciprocating aircraft engines ,v}1ere they are e111ployed bet'''een stages of a t,,·o-stage supercharged engine. REHEATER

A reheater is an additio11al combustio11 clln1nber installed bet,,·ecn turbine stagr~ to increase the turbine po,ver, a11cl co11se(1ue11tly, t.J1e over-all sl1aft po"·er. Additional combustion chambers ca11 be added i11 gas-turbine engi11cs bccat1se the prodt1cts of combustion of the first cl1ambcr arc still essc11tiall)' pure air. Rcl1cating is thus comparable to afterbur11i11g but to a 1nucl\ lO\\'er te111peraturc. l•' ig. 12.10 can therefore b

-..."'

Q.

E

::>

·- 1000

--

.L I-

u ··-

v

Cl)

M0 = 0

Q.

-"'::>

500

Cl)

u..

1

0

Altitude

0 80,000

14.2 Thrl1st characteristics of an a.fterburning turbojet engine for application to an air-breathing bcroster system. (Frorn Ref. 4.) F10.

4 3 1 2 Flight Moch number, M 0

Fxo. 14.3 Fuel specific impulse characteristic of an afterburning turbojet engine for application to an air-breathing booster system. (From R ef. 8, an artick by ZipJ.-in and Nucci, courtesy AGARD and P ergamon Prus.)

It may be seen that the fuel specific impulse of an afterburning turbojet in an airbreathing booster application ranges from about 1700 sec at Mach 2.0 to about 1200 sec at Mach 4.0. Comparison of these values 'vith rocket engine specific impulses (see Chapter 15) shows that they are from three to four times higher than the I.P of the best rocket propellants (liquid fluorine and liquid hydrogen) and from four to sL'\': times higher than the l, P of more conventional rocket propellants (liquid oxygen and JP-4). This fact explains the interest in high Mach number air-breathing engines. In the preceding section, certain modifications to allow efficient engine operation over a 'vide speed range 'vere mentioned. One n1ethod of "fi..xing" the turbojet engine so that it may operate effectively at subsonic speeds as well as at high supersonic speeds is to incorporate a variable compression ratio compressor - high pressure ratio at low flight speeds and reduced values of pressure ratio at higher speeds. A variable compression ratio can be achieved in a number of ,,·ays; one method, di...~~~ briefly in Chapter 8, is to alter the geometry of the compressor by literall)· changing the compressor stator blade incidence angles in a nt1n1ber of, or in all st~ves. From Fig. 14.1, it appears that the YJ93 engine incorporates this n1ethod of , ·ar)·ing its compression ratio. For the high-pressure ratios, the blades arc rotated to,,-ard the closed position \vhere more air-turning occurs. Conversely, for lo,,·-pressure ratios, the blades are rotated to,vard the open positio11. In the lin1it Ctk~ " ·hen the blade chords parallel the air fto,v, the engine operation approaches that of a ranijet. In this latter limit case, virtually no pressure rise is produced ru1d i10 con1pressor " ·ork is required . By controlli11g tl1e compressor-blade incide11ce angle to a certain schedule \vith fligh t Macl1 number, it is possible to scl1edt1le the o\·er-all con1pression ratio Pi.I Po, the diffuser pressure ratio tin1es tl1e mecl1n11ical con1pl"{'ssor pressure ratio. That is, as the flight speed is increased n11d tl1c difTt1ser pl"{'ssttre increases, the compressor blades ca11 be "ope11cd" so as to n1ni11tain t l1e d esired ' 'nlue of compressor discharge pressure. Figure 14.4 presents tl1e cycle diagrams 011 an h-S pla11e for the two extremes of operation for an e11gi11e of this type.

High Flight Mach Number Air-Breath ing Engines h -

-

-

Comp. closed (high pressure rol10 for subsonic operation) Comp open (low pressure ratio fo r supersonic operolton)

379

6 / )\ / \ /

/

I

//

\

'

\

/ / / T,~

/

/

D.h,

0

s Flo. 14.4 Thermodynamic cycle diagram of a turbojet engine having variable incidence-angle compressor-stator blades sho\ving subsonic and supersonic flight operation.

As noted on the figure, the solid lines correspond to subsonic flight speed operation (p,./ p,,, tlhc, and tlh, are relatively large), and tl1e dotted lines refer to high supersonic speed operation. The turbine inlet temperature is a....'5umed fixed at the same value for both modes of operation, and the over-all con1pression ratio Pi./ Po is sho"-n to be about the same. As sho\vn, afterburner operation is utilized for the high :\Iach number conditions. The larger increase i11 entropy of the con1pressio11 proceS5 for supersonic flight is caused by shock losses in the difft1ser and the more i11efficie11t compressor operation due to its higher i11let ten1perature. E\re11 thot1gh the a\·ailable ene~· for propulsion tlht.- ef - 6.li2-o does not appear to be significantl)' greater for the supersonic case, '''here greater thrusts are required, it sl1ot1ld be ren1en1bered that the enthalpy changes sl10,vn on the diagram are on a pcr-pot111d basis, a11d tl1at i11 supersonic flight, the air-handling capacity of the e11gine is i11crcased n1arkedl)'. Another method of reduci11g tl1e compressor-prcsst1re ratio at tl1e st1personic flight speeds, mentioned i11 tl1e prcccdi11g sectio11, is to 111odify tl1c basic e11gine operation by employing tl1e so-called ''bypass'' cycle. 1\ scl1cn1t1tic nnd h-S c)·cle diagram of a possible version of t11is type engi11e is sl10\\'t1 i11 Fig. 14.5. The engine sho,vn is basically a t"·i11-spool tt1rbojet e11gine ,,·ith afterburner, plus the distinctive added feature of a bypass dt1ct. 111 st1bsonic flight , tl1e b.)·pass duct door is closed ; all the \VOrki11g fluid passes throt1gl1 tl1c n1ait1 c11gi11e co111ponents, and the engi11e operates as a i1ormal t\vi11-spool tt1rbojet c11gi11e \vitl1ot1t. af terbur11er (refer to solid lines on the li-S pla11c) . 111 higl1 supcrso11ic fligl1t, \vl1ere a redt1ction of mechanical compression ratio is desired, tl1e bypass dt1ct door is n1echanicall)r opened, thus producing a reduction i11 presst1re ratio fron1 t\\'O effects. First, the out~r compressor

380

I

Jet Propulsion

is unloaded so that its pressure ratio is reduced; apd second, the reduction in air-flow through the inner compressor C11 \vith a suitable reduction in its rpm causes its pressure ratio to decrease. Even though a considerable portion of the air is bypassed around the main burner and turbines, the reduced air-fto,v \vhich does pass through the turbine can supply adequate po,ver to drive the compressors as a result of reduced pressure ratio operation and less mass-fio,v through the inner compressor.

C1

0

2

I

a

I

I

I

:

I

1 I

-

D

I I I

3

4

I

I

I

I

11

II 11

I

I

-c - n

b

5

6

e

ef

I

t I

I

I

I

I

I

I

1

I

TlI

H

I

I

I

-J::":==~ ­

N

A .B.

r--~~ --

uct

,

Duct door

h

,.,6

- - Duct door closed for subsonic operation - - -

Duct door open for supersonic operation

// /

//

/

/

/

\ \

\ \

/ /

4 4

/ /

/

•

3

s F10.

14.5

Schemat ic nnd h-S cycle dingrnms of a. possible bj·-pnss turbofan engine suitable for cfTcctive subsonic nnd supersonic operation.

The supersonic cycle operation is indicated on Fig. 14.5 by the dotted lines. The bypassed air is compressed through tl1e outer co1npressor to point a, and then routed and dumped into the afterburner \Vl1crc it joi11s the n1ai11 air-fto,v ,,·hich has pas..~ through the main burner and turbines. As \Vas mcntio11ed i11 Cl1apter 12 in con11ection \vith afterburning turbofan engines, more fuel can be added in the afterburner of this engine than i11 a normal afterburner before stoichion1etric mi~i.ure ratios are attained. because the afterburner receives pure air as \veil as the products of combustion fron1

High Flight Mach Number Air-Breathing Engines

I

381

the main burner. This n1eans that more oxyge11 is available. As \Vas indicated for the variable geometry compressor, t11e turbine inlet temperature is assumed constant for subsonic and supersonic flight. The compression process for the supersonic flight condition has more losses than its subsonic counterpart for the same reasons as those mentioned before. It should be noted that many variations of this type of engine are possible. The bypass ratio (bypass air/ gas ge11erator air) can be adjusted by design to suit special purposes or can be varied in flight by a suitable control system; it is not necessary to use a dual-rotor compressor, since the air could be bypassed from some interstage location of a single-spool compressor, and so on. The pure turbofan engine represents still another method of applying the bypass cycle system to high flight Mach i1umber operation. A turbofan engine designed specifically for high flight speeds is illustrated schematically in Fig. 14.6. It is noted that the turbofan engine for supersonic speeds incorporates only a few compres.5or stages and an afterburner. Contrast the design \Vi th the turbofan engines for subsonic operation discussed in Chapter 12. 1"he design illustrated in Fig. 14.6 bypasses a large portion of the air and provides only a relatively small mechanical pressure rise. Although this type of engine is suitable for supersonic operation, it is not as adaptable as the t\vo types mentioned before for subsonic operation. This turbofan engine has no provisions to increase pres.5ure ratios for subsonic flight; therefore, its economy in this speed range is poor.

~~ssssssssss~~Dv.;,,.,,,,._

-

F10.

-

<


.L

-- ...

I I

I

z

/

/

u

-·

....

---.... ........

~

\ 0

/

/

/

v ·..... .....

"'b

8 0.5

\

' 'o ' ;.J'

....

'

....~ / ~

c

\

/

/ /

' o

/

I

/

''

I

,,"'

;

•

c

' O ;lo

,,, .,,,,,.--- ........ .....

0

I)

393

ER

"....l

-·-·- 0.5 -..."' --

I

v

-"'... :>

-z-

.&: \

~

'' 0 ·..r;

o,

0.4 /

/

\

/

/

/

0 .4

/

/

,, '

1.8

2.0

2.2

2.4

0.3 .....__ _.....__ __....__ _ _....__ ___. 1.8 2.0 2.2 2.4 1.6

Flight Moch number, M 0

Flight Moch number, M 0

14.14 Net thrust coefficient and fuel specific impulse versus flight l\la.ch number for (a) 6..i:ed geometry engine; (b) variable geometry engine. (Fro1n Ref. 5, courtesy Applied Ph ysics Laboratory,

F10.

Johns Hopkins Universit y.)

In an earlier section it 'vas mentioned that turbojet engines are current!)' designed for speeds up to :rvlach 3.0. References 10 and 11 indicate that a i\Iarquardt ramjet engine has po,vered the Lockheed X-7 test vel1icle to flight speeds abo,·e ~Iach 4 at altitudes over 80,000 ft. The ramjet depicted in Fig. 14.8 is reprcsentat.i,·e of the engine used for this flight. Reference 12 notes that Nord A\•iatio11 in France has fio,,·n the ramjet-po,vered Vega vehicle to Mach 4.06 at an altitude of 96 1 500 ft . This flight "-as part of an over-all program to explore ramjet propulsio11 at speeds up to ~Inch 5 and at altitudes up to 115,000 ft. Nord Aviation has also dc,·cloped the '·Griffon," a manned experimental aircraft with a dual-cycle turbojet/ ramjet po,,·er plant which has flo\vn at speeds above Mach 2 at 60,000 ft. Relative to the future, Refs. 3, 9, 13, n.r1d 14 corlSider operation of the con,·entional subsonic combustior1 ramjet completely feasible up to l\lach i . Be)·o11d this point, Refs. 9, 14, 15, 16, and 17 consider applicn.tio11 of superso11ic or detonation combustion ramjets up to Macl1 10. In tl1e cxtre111c, Reis. 181 19, a11d 20 c\·en argue for the application of supcrso11ic con1bt1stio11 (inter11nl a11d external) rn1njets for propulsion to orbital speeds - Macl1 25. li'urtl1er co11sidernt.io11 ,,;11 be gi\•en to the..~ various hypersonic ramj ets (speeds above l'vlacl1 5) i11 a subseque11t section.

I

394

14.5

Jet Propulsion

Combination Power Plants

In our considerations of air-breathing engines suitable for high supersonic flight speeds, we have discussed certain modified turbojet engines and the basic ramjet engine. It has been demonstrated that the turbojet engine, although v.·ell suited to subsonic speeds, has definite limitations in the range of Mach n11mbers 3 to 4, and conversely, that the ramjet engine, although unsatisfactory at lo'v subsonic speeds, is 'vell suited to the higl1 supersonic speed region. As pointed out earlier, these diverse performance characteristics suggest a combination turbojet-ramjet power plant. Considerable theoretical investigations have been made on such propulsion systems. For example, Ref. 21 demonstrates on a theoretical basis that a suitable combination power plant of ramjet and turbojet can provide an integrated supersonic propulsion system capable of overcoming the individual deficiencies of the individual power plants; aircraft speed and ceiling can be increased, and propulsion system weight decreased significantly. Figure 14.15 schematically illustrates one method of combining turbojet and ramjet engines in series. Because of the method of combination, this S)rstem is referred to as a turbo-ramjet engine. By proper positioning of the two sets of movable doors, the power plant can be operated as a pure turbojet engine (with or without afterburner) or a pure ramjet engine. The afterburner can be thought of as a ramjet combustion chamber that receives its compressed working fluid from either the (1) simple fresh air inlet of a conventional ramjet, or (2) the rela.tively comple."\: intake system of rotating parts and a burner, that is, the turbojet engine. Movable door Open for turbo jet operation Shut for ram jet operation

Vo

/

/

''

D

c

T

H

'

Afterburner or ramjet

(', ;

v.

......

Vo

Variable area nozzle

Movable door Shut for turbojet operation Open for romjet operation F10.

14.15 Schematic diagram of the turbo-ramjet engine.

The operation of tl1e engi11e is as follo"·s: for take-off, climb, and maneuvering at subsonic or relatively lo\v supersonic speeds, the engine is operated as a turbojet 'vith afterbur11cr. At higl1 st1pcrsonic speeds, the duct door positions are changed and the engine functions ns a ron1jet, '"itl1 the turbojet engine shut do"~ and all the air entering the auxiliary lo\ver dt1ct. References 3, 20, and 21 disct1ss the turbo-ramjet engine. In general the engine is proposed to operate ns a11 afterburni11g turbojet from take-off to ~Iach 3 to 3.5. .i\t that speed tl1c engine is converted to operatjon as a pure ramjet, and it continues as a ramjet e11gine up to l\Iach 6 to 8. Figure 14.16 fron1 Ref. 3 sho,,·s the performance of a turbo-ran1jet engine as a po,,·er plant for a11 air-breathing booster, comparable to t l1e t u rbojet e11gi11e co11sidered earlier in Fig. 14.2.

High Flight Mach Number Air-Breathing Engines

I

395

This engine also delivers reasonably constant thrust up to Mach 5.0, 'vith a gradual thrust dropoff to about half the sea level static thrust at Mach 6.0 and 90,000 ft. As 'vith the turbojet, the operation is a climbing acceleration along t he engine pressure limit. Figure 14.17, also from Ref. 3, sho,vs that the fuel specific impulse for this trajectory is above 1500 sec up to Mach 3.0, decreasing gradually to about 900 sec at Mach 6.0. 0 2000

2.0

GI

....-

"'-

-E- 1 5 v

~ ·

40,000 feet

GI

r----'----J;;.

~ ?o

-

-"'a.

65,000 feet

:::>

A

.. -

c..ev

·-E 1000 u

~ : 1.0 "--~,0~

-·-

Cl

u

GI

!)

~

1500

a. 500

0.5

"'

90,000 feet

GI

:::>

u..

0

1

2

3

4

5

6

0

1

Flight Moch number, M 0

Flo. 14.16 Thrust characteristic of a turboramjet engine for application to an a ir-breathing booster system. (From Ref. 8 , an article by Ferri, courte.sy AGARD and Perganwn Press.)

3 4 5 2 Flight Moch number, M 0

6

F10. 14.17 Fuel specific impulse characteristic of a turbo-ramjet engine for application to an air-breathing booster 5)-"Stem. (From Ref. S, courtesy AGARD and Perganton Press.)

Because of the wide range of operating speeds and hence pressure ratios, an engine of this type would require variable geometry. Reference 3 proposes a mo\ able plug nozzle which allo,vs very wide variations of both the throat area and the expansion ratio. In addition to the turbo-ramjet, several other po,ver plants can be combined into air-breathing engines suitable for high supersonic flight-speed operation. Thus, such combinations as turbojet-rocket or ramjet-rocket e11gines can be devised, or e\·en a combination of all three basic engines, a turbo-ramjet-rocket. Such types as the....c,e "'ill not be discussed here; ho,vever, they are n1entio11ed because, for a certai11 tactical mission, it may be desirable to use the compleme11tary characteristics of t\,·o po"·er plants other than the ramjet and turbojet. If, as i1oted i11 Sec. 1-1.2, for e."an1ple, it is desirable to make turbojet engine pcrforma11ce almost i11dcpc11dc11t of ftigl1t speed, the turbine can be driven by gases produced from n.11 i11depe11dcnt che111ical con1bustor rather than air from the compressor. An exan1ple of this type of po,,·er pln11t is the air-turbo-rocket engine sho,v11 schematically in li'ig. 14.18. 1

,.,,....

--

< < ---...::_-- < < -......::

R.B. R.B. Air

F10 . 14.18

_
"

--

_...---

liigl1 l\1ach number engine of lite air turbo-rocket t)'J>C.

> 396

I

Jet Propulsion

In this engine, the turbine is driven by the gases 'vhich are generated in the rocket burners R.B. The turbine blade temperatures can thus be constant and independent of flight ~1ach number. The compressed air and turbine exhaust gases combine downstream of the turbine and burn in the afterburner proper. Additional fuel can be added in the afterburner to achieve near adiabatic flame temperatures and high thrusts. A13 evident from the cycle concept, this type of engine is attractive from the standpoint of achieving high thrusts at higl1 Mach numbers as \Veil as at static conditions. A disadvantage, however, is high fuel consumption.

14.6

Power Plants for Hypersonic Applications

In Sec. 14.4, reference is made to a number of ramjet engines \vhich operate at speeds above Mach 5. Somewhat arbitrarily, this speed is considered the dividing line between supersonic and hypersonic flight. In this section, some air-breathing engines will be considered which are potential hypersonic power plants. Reference 13 presents a discussion of a Mach 7 transport, powered by turboj ets up to Mach 3.6 and by t\vo-dimensional, external expansion ramjets from ~lach 3.6 to 7.0. Figure 14.19 from Ref. 13 is a schematic diagram of the ramjet engine sho"ing the various engine components.

e. Station: 0-1 Two·d imensionol supersonic diffuser (supersonic a ir compression) 1-2 Two.dimensional subsonic diffuser (subsonic air compression) 3-5 Fuel addition, combustion, thermal choking (A3 = A.s) 5 - Nozzle throat 8 5 - Flow direction al throat Flow direction at lost nozzle • expansion wove A 1 - Inlet area A. - Exit area e - Final conditions

e. -

14.19 Scbemntic dingrnm of n twodimcnsional, extcrnnl-cxpnnsion rnmjct, sl1owing regions of primnry interest. (/t'rout R ef. 18 , courtesy Applied Ph y:tics Laboratory, The Johns H opkins University.)

F10.

p.Jpo= 1200

1..53

1.00

1000

800 10.70

I,

600

400

1.3 1.2 11

e5 -

1..5616

1..S 1.4

A,/Ao = 0.0752 A2 /Ao= A"J/A 0 =

00829

200

A,/A 2 = 0.907

0

0 .4

0.8 1.2 1.6 2.0 2 .4 2.8 3.2 A. / Ao

14.20 Fuel specific impuL~ of a !\lach-7.0, cxtcmal-c.~pnnsion ramjet engine. (From Rtf. 1S, courtuy Applied Physia Labaralory, Joluu HopkiM c.:nit'tt'Sily.)

F10.

As may be seen the engine, although somc,vhat different looking from the ramjets described earlier, docs have the same basic components - inlet, con1bustor, and noule. In addition the proposed locatio11 of tl1e c11gi11e beneath the ,,·ing also pro,;des about a 12% contributio11 to tl1e lift a.11d l1c11co to tl1c lift/drag ratio of the \•ehicle. This comes from the higl1 pressures acti11g 011 tl10 external expansio11 11ozzle. The open halfnozzlc also helps to take care of tl1c extren1ely high flame t.en1peratures (abo'\·e 5000°R according to Ref. 13) through effective radiation cooling.

I

High Flight Mach Number Air-Breathing Engines

397

Typical performance for such a Mach 7 ramjet engine is presented in Fig. 14.20, which is taken from Ref. 13. The variation of engine performance is sho,vn as a function of the ratio of engine exit to inlet area A ./ Ao and the airflo\v direction at the nozzle throat Or,. The curves are based on a pressure recovery, P1.I P1. = 103, and an equivalence ratio, E .R . = 0.5. For eacl1 value of 06, the nozzle expansion is carried to the point at 'vhich the flo\v direction is parallel to the flight direction, 8, = 0. Since this does not result in complete expansion, the nozzle exit pressure ratio, p,./ po, is indicated along the locus curve. Co11sidering the point of equal i11let and exit areas, A ,/ A o = 1, plus parallel exit flow 0, = 0, it may be seen that 11 equals almost 1000 sec. Based on this type of performance, Ref. 13 sho,vs that a hypersonic transport having a take-off 'veight of 700,000 lb with 418,000 lb of fuel has a range of 3600 nautical miles with a 50,000-lb fuel reserve. The fuel reserve corresponds to a range of 900 additional nautical .miles. The total 3,600-nautical-mile trip takes 89 min. As mentioned earlier, separate turbojet engines are required for takeoff and acceleration to Mach 3.6, and for landing. It is also interesting to not~ that 753 of the fuel and more than half of the total range is covered during the acceleration and climb to the Mach 7.0, 100,000-ft cruise condition. Such performance indicates that further development \viii be required t o achieve sufficient range for a hypersonic transport to be attractive. Reference 20 indicates that such additional development is possible. Figure 14.21 from Ref. 20 sho,,·s the possible range as a function of cruise Mach number for an advanced airplane having a take-off v.'eight of 600,000 lb and carrying 100 passengers. The figure suggests that a range of 6000 nautical miles is possible for a Machi air1.00 0 .80 • oo

A

0.60 ~0.50 "'Cl: 0 .40 0

~

7000

GI

~ v

GI

6000 Acceleration

5000

Ol

95X

"'..."'GI a.

Cru ise

0

c

0

0

f-

~

4000

7

'lKE =

14.21 Range versus cruise Mach number for a. 000,()()(}.lb transport carrying 100 passengers and using Jl">-4 f ucl. (From R ef. fO, an article by Ferri, courtesy AGARD and Perga11wn Press.)

92X

0.10 0 .08 0.07 0 .06 0 .05

Fl ight Mach number, M 0 F10.

1JKE =

~ 0.20

c: 1)-

0 .30

...

Total

·-E

Experimental inlet performance

• 5

6

7

8

9

10

Flight Mach number, M 0

14.22 P crformnnec of h)' personic inlets. (Front R ef. 22, art article by AfcLafferty, courtesy A GA llD and J>crgan1on Pre.ss.) F10.

398

I

Jet Propulsion

plane. A study of geography sho,vs tl1at a 6000-nautical-mile range is of real interest since it is greater than the dista11ce bet,veen most major cities of the world. A weight breakdo,vn for such a Mach 7 aircraft is given in Table 14.2. TAB LE

14.2

Weight Breakdown for Mach 7 Transport

(From Ref. 20, an a.rticle by Ferri, courtesy of AGARD and Pergamon Press.) Per Cent of Gross Weight Payload

24,000 lb

4

Turbojet

30,000 lb

5

Ramjet

24,000 lb

4

Cabin cooling, etc.

24,000 lb

4

Structure

138,000 lb

23

JP-4 Fuel

360,000 lb

60

Gross take-off \veight

600,000 lb

Use of high-energy fuels \vould significantly reduce the weight or increase the range. Realization of such an airplane will take extensive development in all aspects. It was mentioned above that the performance in Fig. 14.20 is based on a pressure recovery p,,/ p,. = 10% . This appears low when compared to the standard recovery of Eq. (5.15), which is 24% at Mach 7. In order to put this inlet recovery int-0 proper perspective, Fig. 14.22 from Ref. 22 is presented. The figure shows some measurements of inlet total pres.5U.I'e recovery as a function of ~1ach number. Also sho'vn on the figure are two lines representing inlet kinetic energy efficiencies 11KE of 92 and 95%. It can be seen that the measured data correspond closely to 11xe = 92%. Reference 22 states that T/KB = 953 is indicati,·e, however, of the recoveries 'vhich may be obtained ''ith further de,•elopment. The kinetic energy efficiency is defined as the ratio of kinetic energy available after diffusion (assuming isentropic re-expansion to ambient pressure) t-0 the kinetic energ)' in the free stream. Hence, using the station designation in Fig. 14.9 or 14.19, the inlet kinetic energy efficiency can be written 11KH

=

v' Vo

t

ht, - ho' = ----h,. - ho

(14.i)

'vhere tl1e prime superscript signifies tl1e condition ''•hich would be achie,•ed h)· isentropic re-expansion to ambient pressure from the diffuser e.xit total pres.5Ul'e. For constant specific heat c,,, Eq. (14.7) may be '''itten "r- 1

rt KB

Pt. -1 = 1 - ~~P...;.:t•:....__ _.....:::;. -y-1 t Mo 2

(14.8)

Figure 14.23 from Ref. 23 s hO\\' S the effect of inlet pres.5Ure reco,·ery on the fue' specific impulse of a ~1ach 7 ramjet engine.

I

High flight Mach Number Air-Breathing Engines

399

0 .6 Equilibrium exponsioo

0.5

p---_....________ frozen expansion (from throat)

0 .4

.,.,, -::>

1.0 0 .92

~ 0.8

0.95

7lo 0.3

'1KE

·-v ··.,v 0 .6

-

0 .2

.,,0.

= 94% 7lb= 95% 7ln= 96% A./A0 = 1.5 71KE

~ 0 .4

-.,

0. 1

>

·-g 0.2

.,er:::

0

0 .05 0 .10 0.15 0 .20 0.25 0.30 Flight Mach number,

Inlet pressure recovery ,p,2/P10

14.23 Effect of inlet recovery on the fuel specific impulse of a Mach 7 ramjet engine. (From Ref. 23, an article by Connors and Obery, courtesy AGARD and Pergamon Press.) FIG.

Mo

14.24 Over-all engine efficiencies for subsonic combustion and both equilil>rium and nonequilibrium nozzle expansion. (From R ef. 17, and article by D ugger, courtesy AGARD and F10.

Pergamon Press.)

The figure indicates that the relative fuel specific impulse for TJKB = 923 is within 3 per cent of the highest reasonably attainable point (for TJKB = 95% as an upper limit to hypersonic inlet performance). To achieve this 33 impro,·ement requires the inlet total pressure recovery to be nearly doubled - 143 to 263. I t would therefore appear that on the basis of fuel efficiency, kinetic energy efficiencies in the range of 90-923 a re reasonable performance goals. Of course, the size of the engine for a given t hrust output 'viii be directly affected by the total pres...c;ure reco,·e~·A second consideration 'vhich is also of importance for h)1Je-rsonic engines is the question of 'vhether or not recombination takes place in the e.xhaust nozzle. ..\t the extremely high combustion temperatures in hypersonic engines (temperatures abo,·e 5000 °R are mentioned in Ref. 13), considerable gas dissociation takes place. Since the process of dis.sociation requires a.11 energy input, considerable ene~· " ·hich should be available for accelerating the gas may be unavailable unles.5 the dissociated ga....~s recombine in the nozzle. Because of the very short tin1e that the gas sta~-s in the nozzle (o,ving to the very high exhaust velocities), tl1ere n1a~· not be time for recombination in the nozzle. Figure 14.24 from R ef. 1i sl10,,·s the pote11tial l--"5 in performance if no recombination takes place i11 the nozzle. The over-all efficiency plotted in Fig. 14.24 is '''ritten '70

11 Vo

= ---=----(H.V.)

v: + 2

(14.9)

g

A comparison of Eqs. ( 14.9) and (2.24) sl10,,·s tl1at t.l1ey are essentiall)• equal. Equation (2.24) is actually a sin1plificatio11 of Eq. (1-1.9). The 1 '~2g term in the de11ominat-0r

I

400

Jet Propulsion

of Eq. (14.9) accounts for the kinetic energy of the fuel owing to its motion through the atmosphere, energy that \Vas furnished by the fuel previously consumed. This term is only of significance at hypersonic speeds. At Mach 7 it only amounts to about 53 of the heating value H .V. of JP fuels. From Fig. 14.24 it can be seen that nonequilibrium expansion in the nozzle, with the exhaust gas constituents "frozen" in the relationship occurring at the nozzle throat, causes a very serious decrease in performance as compared to the equilibrium expansion. Also sho,vn in Fig. 14.24 is an estimate of the "actual" flo,v in the nozzle with some recombination. Even this performance represents a very significant decrease from the equilibrium nozzle fto,v. It is pointed out in Ref. 17 that the curve shown for "actual" flo'v is only a calculated estimate. Performance experiments may show that the actual expansion lies much closer to the equilibrium flow curve. Reference 24 states that the limited experimental data available for hydrogen-air rocket nozzles do fall closer to the equilibrium expansion tl1an to the frozen expansion. The study reported in Ref. 25 states that recombination losses are not expected to be serious up to Mach 9 with free stream dynamic pressures greater than 500 lb per ft 2 • Even so, one of the major uncertainties \vith respect to hypersonic engine performance is in the area of the nozzle. Much research is being directed toward resolution of this uncertainty. For a detailed consideration of the thermodynamics of combustion equilibrium, the student is referred to R efs. 26, 27, and 28. For operation at speeds above Mach 7, Refs. 14, 15, and 17 all indicate that supersonic combustion ramjet (SCRJ) engines offer significant improvement over the conventional subsonic combustion engine. Figure 14.25 from Ref. 15 shows the thermcr dynamic cycle for the SCRJ on the h-S plane. 2800

h

2' M2 < 1

I

p•

2400

M =l

I

2000

I

I

V1600

Normal i shock / / Mi> l I

I

Cl

-

""'

SCRJ

- 1200 Po

21

800 400

0

s 14.25 Thermodynamic cycle of a supc~ sonic combustion ramjet (SCRJ) on nn h-S plane. (Fro111 Ref. 15.) F10.

0

Rocket

2

6

4

8

10

Mo Flo. 14.26 Comps.ri...4'0n of ramjet fuel specific imp\ll"CS. (Froni Ref. 14, an article by Dugger, courle.$y Astronautics.)

Stntion designations nrc compnrnble to those ttscd on Fig. 14.9. Operation ge11erally i11volvcs son1c st1pcrso11ic difft1sio11 from tl1e free strean1, station 0, to the diffuser exit, statio11 2, since this yields better pcrforma11ce according to Ref. 14. As before tl1e decrease i11 total pressure a11d increase in entrop)' are due to the shock los..~.

High Flight Mach Number Air-Breathing Engines

I

401

A Rayleigh line, representing heat addition in a constant-area duct, is constructed through point 2. The lo,ver branch of the Rayleigh line represents supersonic Mach numbers, while the upper section represe11ts subsonic Mach numbers. In either case, the addition of heat results in movement to the choking conditio11, represented by point 4 on Fig. 14.25. Thus, as heat is added to the supersonic stream, beginning at station 2, the temperature and static pressure increase \vhile the Ma.ch number decreases until the limit - thermal choking at station 4 - is reached, provided the stoichiometric fuel-air ratio is not reached first. The combustion process can be stopped at any point before choking. Beyond this point the process is similar to the conventional ramjet, with expansion through a nozzle. Figure 14.25 also shows the difference bet,veen the conventional subsonic combustion ramjet and the SCRJ. I n the former a normal shock before combustion moves point 2' to the upper branch of the Rayleigh line. Heat addition then continues until choking at the same station 4 is reached . The major problem for the SCRJ is the injection of fuel into the supersonic stream without disturbing the flow. A natural variation of the SCRJ is thus the standing wave or detonative ramjet (SWRJ). I n t he SWRJ, fuel is added to the supersoniC' stream ahead of a standing shock (oblique or normal) 'vhich is held in place by the engine geometry. The SCRJ and SWRJ have equivalent performance with equal diffusion and equal heat addition, equivalent to that required for thermal choking. Figure 14.26 from Ref. 14 presents a comparison of the fuel specific impulses of the SCRJ, the SWRJ, and the conventional ramjet. From the figure it may be seen that the SCRJ and SWRJ do show better performance than the conventional ramjet above :rvlach 7. I t can also be seen that airbreathing engines continue to exhibit a significant fuel specific impulse advantage over rocket engines up to Mach 10. In addition to the performance advantages illustrated in Fig. 1-1.26, supersonic combustion also offers some other practical advantages. The pressures and temperatures in the combustion chamber are significantly lo,ver. The lo,,·er pressure should allo'v lighter construction. The lo,ver temperatures should reduce the heat transfer per 11nit area. Since the SCRJ must have a much larger combustion chan1ber to provide sufficient residence time for combustion, the over-all he3t trnnsfer for the SCRJ may not be lo,ver, ho,vever. Air-breathing propulsion beyond Mach 10 and e\'en to orbital \•elocit)· ( ~ faeh 25) is suggested by Refs. 18, 19, and 20. T l1e propulsio11 system for a11 orbital air-breathing booster system is described in Ref. 20 as follo,vs: For take-off and accelerntion to l\1nch 3, turboj et engines are used . Ramjet engines are used for the remainder of tl1e acceleration. Tl1e flo'" is decelerated to sub...~nic speed for the lower range of supersonic flight by nn cfficirnt diffuser. At h)·pcrsonic six-eel, the geometry is maintained constant, and the flo\v bccon1cs suprrsonic in the rombustion zone and increases \vith flight l\ Iacl1 nun1brr. At higl1 :\Isch nun1bt'rs, sup

ca

10 3

5

4

3 5

15 20 10 25 Fl ig ht Mach number, M0

30

14.27 SCRJ burner inlet Mach number. (From Ref. 20, an article by Ferri, courttsy AGARD and Pergamon Press.)

F10 .

Flight Moch number, M0

Flo. 14.28 SCRJ thrust for 100 sq ft of free stream capture area and h:rdrogen fuel (Prom R ef. eo, an article by Fe11 i, rourksy AGARD and Pergamon Preu.)

Figure 14.28 shows possible thrust values for the SCRJ engine as a function of flight Mach number at three different altitudes for 100 ft~ of capture area, based on use of hydrogen for the fuel. Figure 14.29 shows the comparable fuel specific impulse for the thrusts represented in Fig. 14.28. Based on the performance represented in Figs. 14.28 and 14.29, Ref. 20 prop-.~ an orbital air-breathing booster. It has an initial "·eight of 130,000 lb to place a 10,000-lb payload in orbit. Figure 14.30 sho"·s the longitudinal acceleration characteristic for such a booster as a function of flight :r-.racl1 n11n1ber. \Vhen it is remembered that typical values of the ratio of orbital pa)·load to launch \veight for rocket syste1ns are as small as 2 to 3 per cent, the e:\ireme attracti\eness of an air-breathing booster system \vhich orbits a pa)•load equi\•alent to about 8 per cent of itR launch \Veight can be appreciated. Supersonic combustio11 engi11es also incl\1de cxter11al-burning ran1j cts. References 12 and 18 both i11dicatc that \Vork is progressi11g 011 this t)•pe of e11gine. According to R ef. 181 the fuel (probably l1ydroge11) is burned i11 nn area properl)· oriented along the outer skin of t he vehicle so t11at tl1c i11crense i11 static pressure due to supersonic hest addition creates tl1rust a11d lift on t11e vehicle. Since tl1e con1bustion also creates shocks and he11cc drng, 011e of the prin1ary problcn1s is to dctcrnune the optimum configuration to n1nximize thr\1st-n1i11us-drag. Rcfcre11ce 12 i11dicates that the e..xternalbur11i11g ramjet may Sltrpnss the performa11ce of co11ventional ramjets abo,·e ~Iach 10. R eference 18 states t11at it 111ay be possible to orbit a vehicle using the external-burning

-

-

High Flight Mach Nu mber Air-Breathing Engines

I

403

system alone above Mach 3, after take-off and initial climb and acceleration 'vitl1 a turbojet engine. It must be noted that the orbital altitude for either of these airbrcathing, supersonic combustion propulsio11 systems is limited. Figure 14.28 i11dicates that an altitude of about 200,000 ft (40 miles) is probably close to this limit. 3200 1.0

2800 Ol

2400

c·

-e

0.8

·-0

.." v

" 0.6 ·=0.4 ~ v

...:: 2000

u 0 0

1600

100,000 ft 150,000 ft

1200

200,000 ft soo.._~_._~~_.._~__...__~~~---

5

10

15

20

25

30

-

-0 ~

·-

0)

i:

0 -'

02

o_~~-:-=-::-=-::----:-::--=--::-:----::-:--1:::--~~..__ 5000 10,000 15,000 20,000 25,000

Fl ight Moch number, M0

Flight velocity, ft /sec

14.29 SCRJ fuel specific impulse for hydrogen fu el. (From Ref. fO, an article by Ferri,

14.30 Acceleration capability of an orbital air-breatning booster s:ystem. (Froni R ef. fO,

courtesy AGARD and P ergan1on Press.)

an article by Ferri, courtesy AGARD and Perga1non Press.)

F10.

F10.

The preceding discussio11 suggests that many different combinations of hypersonic engines are feasible for air-breathi11g booster systems. These systems include the t1sc of air-breathing propulsion all t he \Vay to orbit, like the systems described above, as \Vell as the use of air-breathing engines for only part of the acceleratio11. ;\ system of the latter type, \vbich appears to hold some promise, is described in Ilefs. 18 and 19. Reference 18 describes the basic operation of the propulsio11 S)'Ste111 as follo,vs: The Aerospace plane, a one-stage booster wl1ich J1as no t11rO\\'a'''ay parts nnd cnn take off and land on conventional airfields, is tl1e tl1eoretical ideal. All other t)•pes of boosters seem to be interim steps leading to this goal. Air-breathing hypersonic engines are tl1e key to one-stage boosters. Eve11 tl1c best chemical rockets of the current variety arc 11ot eflicient e11ot1gl1 to pt1t n one-st.age vehicle into orbit carrying a uscful payload. T l1e nuclear rocket hns the efficienc)·, but it probably " 'ill be used only in space . ... Actually, this is a systen1 of engines, ull possibly bt1r11ing h) drogrn fucl. The first engine is a turbo-ramjet, \vhicl1 operates off of its turbinc-con11)rrssor t111it from takeoff to a l\Iach nu1nbcr of about 3. F ro1n l\ Incl13 to about i\Incl1 8 it ,,·ould ft1nction ns a ramjet, J)O\vering the vehicle \Vhilc nnotl1cr syste111 1>robnbly \Vil! scoo1>i11 nir ''·ith n convcr1tionnl inlet and })USS it througl1 a radiator to liquefy it. Tl1r rcfrigen111t \\'ould be liquid hydrogen. . . . Small-scale e11gi11cs l1u.ve rt111 over n. \ride l\lticl1 11un1bcr range.... During the flight of the rescarcl1 nir1)la11r, nftcr tl1c oxygen-collection s)·sten1 has con11)lctcd its job, tl1e vcl1iclc is po,,·crcd b)' n 11ydroge11-oxygcn rocket. 1

•

404

I

Jet Propulsion

Reference 19 indicates the significance of such a system as follows: The principle of the air-scooping vehicle is based on collecting air at high altitudes, compressing and liquefying it, separating the oxygen from the nitrogen and then burning the oxygen \vith liquid hydrogen inserted in the vehicle before takeoff. Since it takes eight times as much liquid oxygen as liquid hydrogen by \\·eight to supply the proper propellant combination, the vehicle could take off at about haU its upper atmosphere flight weight. T11e weight of the oxygen in the vehicle could be from 80 to 88% of the total propellant \veight after collection. Presumably, this type of vehicle would be able to take off from an ordinary runway, fly to the upper atmosphere, take on a load of oxygen and then act as an orbital tanker and supply station for other vehicles or proceed on its own mission to a soft landing on the moon and return. Not much capability is claimed as a possible vehicle for traveling to other planets, but as a near-space launching platform it appears promising.

A brief comment in Ref. 12 indicates that engines of this type (which use liquid hydrogen to liquefy air entering an inlet and then react the liquid air with the hydrogen for propulsive thrust), are known as LACE, liquid air cycle engines. Reference 29 states that LACE is basically a rocket engine utilizing liquid hydrogen as the fuel and liquid air as the oxidizer. Figure 14.31, from Ref. 29, shows a schematic diagram of LACE.

---------'

-Liquefaction p lant

~---

I

Liquid hydrogen

c-1 ...._IPump I

I

I I I I

I Hypersonic : • I 1 in let I I ~~:-::;:-::;: -· ~----, I -on----- 1 p re< :>- - +-iI _,_c,....oo_l...... ~· -~ :::t er_, denser

-

c

Liq uid nitrogen

ft

-

Pump

-

.....

Seporator

-

Liquid . Cir (enriched}

-

L...,.-r-....,;:chamber

I

I

~--

F 10.

... _ ..,.. _ _ ____ _j

----------- - ~

14.31 Schematic diagram of LACE. (From Ref. !9, courtesy Aviation Week and Spsce Technology.)

The engine operates basically as described above. Apart from the immediate use of the enriched liquid air in the thrust chamber, it ca11 also be stored in tanks for later u...~. I t is hoped that the brief descriptio11 in this chapter has given some idea of the future of air-breathing engines. It is believed that this future offers tremendous development potential. P ROBLEht S

I. Define tho basic ltlgh flight-speed problem of the turbojet engine. 2. I n the bypass turbojet engine, why is the enthalpy drop across the outer turbine appreciably greater than tho enthalpy rise across the outer comprc~r? 3. Sketch the thermodynamic cycle diagrarn of the turbofan engine for both subsonic operation and 11igh supersonic operation. 4. Describe the meaning of each thrust term in the net propulsive thrust equation. [Eq. (14.3)].

I

High Flight Mach Number Air-Breathing Engines

405

5. Discuss the operation of each of the sLx inlets pictured in Fig. 14.10. 6. From the operating data shown with the four ramjet configurations of Fig. 14.12, calculate all the data obtained in Table 14.1 . 7. Devise a possible combination turbojet-rocket engine tl1at is suitable for botl1 subsonic and high supersonic flight and discuss its merits relative to otl1er po,vcr J)lants discussed in this chapter. Show its process on the h-S plane (sec Fig. l .11.18). 8. Repeat Prob. 7 for a combination ramj et-rocket engine.

9. Derive Eq. (14.8). Discuss the significance of 11KB for l1ypersonic engines. REFERENCES

1. 'Vilkinson, P . H., Aircraft Engines of the World, vVilkinson, July 1961. 2. Aviation Week, March 12, 1962, pp. 179-195. 3. Zipkin, M.A., and Nucci, L. M., "Composite Air-Breathing Systems," Fourth AGARD Colloquium - High Mach Number Air-Breathing Engines, Ne\V York : Pergamon Press, 1961 .

4. Kappus, P. G., " R ecoverable Boost Vehicles with Air-Breathing Power Plants,''

AS~lE

Paper No. 59-AV-16, 1959.

5. Bader, F., and Bunt, E. A., " Ramjet Technology - Thermodynamics of Ramj et Flow Processes," The Johns Hopkins University Applied Physics Laboratory, 1960.

6. Miller, P. R ., "The Ramjet and Pulsejet Engines," T est Pilot Training Di,;sion, U.S. Naval Air T est Center, March 1954. 7. Harned, M., "The Application of the Ramjet to Aircraft Propulsion," 1952.

8. Love, B. A., and Oates, C., "Ramjet Performance as

~Iodified

AS~IE

Preprint,

by a FL'\'.ed Area I nlet,"

Chance Vought Report No. 6881, November 1946. 9. Perchonok, E., "Ramjet Trends," Astronautics l\{agazine, April 1959, pp. 40 ff.

10. Lockheed Aircraft Company Annual Report, D ecember 28, 1958. 11. Marquardt Aircraft Company Annual Report, December 1958. 12. A stronautics, December 1961, pp. 48 ff. 13. Pietrangeli, G. J., "An Airbrcathing Mnch 7 Trnnsport," APL Ttrhnirol November- D ecember 1961.

Dt~est,

14. Dugger, G. L., ''A Future for Hypersonic Ramjets," .tl strorwutirs J.llagazi1ie, April 1959, pp. 38 ff. 15. \Veber, R. J ., nnd ~1acl{ay, J . S., "An Anl\lysis of Rnmj ct Engines Using Supersonic Combustion," NACA TN 4386, 1958. 16. Arens, 1\1., "The Performnncc of a Hy1Jcrso11ic Rnn1jrt Using Detonnti\'e Con1bustion," SAE Pa1)er 4 19A, October 1961. 17. Dugger, G. L., "Con1pnrison of Hy1)ersonic Rnn1j ct Engines ·,,·itl1 Subsonic nnd Supersonic Combustion," Fourth AGARD Colloquiu111 - l ligll !i!acll Nu111ber Air-Breathing Engines, Ne\v York : Pcrgnmon Press, 1961.

406

I

Jet Propulsion

18. Butz, J. S., "Aerospace Plane: Answer to Rocketing Costa," Air Force Magazint, l\fay 1962, pp. 42 ff. 19. Booda, L., "Space P lane Grows into a Family of Concepts,'' Aviation Week, June 19, 1961, pp. 54 ff.

20. Ferri, A., "Possible Directions of Future Research in Air-Breathlng Engines," Fourth AGARD Colloquium - High Mach Nurnber Air-Breathing Engines, Ne\v York: Pergamon Press, 1961. 21. Friedman, J., "Ram-jct-Turbojet Power Plants Can Extend Flight Regime,'' Aero Digest, June 1956. 22. McLafferty, G. H., " H ypersonic I nlet Studies at UAC Research Laboratories,'' F -0 .. .. ::>

I

~.=:_,--

I

JI

~;;;;;;.. ·-...;-.;;ii --.....?"-.. .....~'-..j""9-I r-

- •CJI IL

~ -

J

Oxidizer tank

---

-.........._

/ ... - - u____:::__:~--..... . . . ~..........;:.-'

400 psio

500 psia F10.

15.2 Liquid-bipropellant gas pressurization system.

energizing valve, and then the rocket motor is started by opening the bipropellant control valve. The motor can be stopped and restarted at will by closing or opening the latter valve. As evident from the figure, the gas-pressurization system is a relati,·ely simple po'\\·er plant. Its big disadvantage, ho\vever, is the fact that both the oxidizer and fuel tanks are pressurized; consequently, when the tanks are of considerable ,-0I11me (as they must be for long-range applications), the tank ,,·alls, which must ";thstand the high pressure, are excessively thick and heavy. In addition to the increase of propellant tank ,,•eight ";th the duration of engine operation, the inert gas volume, and consequently the inert gas bottle "·eight, mu_c:t also increase. Thus, the major limitation of the gas-pressurization sj·stem is its hea')· '''eight \vhen used for long-duration operation. This limitation makes gas pres::i'1.rization applicable only to short-duration operation. Another gas-pressurization system \vhich is used in sn1all storable liquid propulsion engines utilizes the exl1aust products from a solid-propellant gas generator ( PGG) cartridge as the high-pressure gas. (Figure 15.12, ,,·hich schematicall.}· illustrates a hybrid rocket engine, also sho\vs an SPGG pressurization S)•stem.) PUMP-PRESSURIZATION 8YSTE1'i

The lin1itations of tl1e gns-prcssurizatio11 S)'sten1 to the application of long-range rocket flight stimulated tl1e developn1ent of tl1e pun1p-prcssurization s)·stem. A t)-pical liquid bipropellant pu111p-pressurizntion rocket c11gine is sho,vn in Fig. 15.3. In the pump-pressurization systen1, tl1e liquid oxidizer and fuel are stored in tanks at lo"· pressure (thus the tanks ca11 be n1ade very light) and are fed into the rocket motor at high pressure by the fuel a11d oxidizer p\1n1ps. The po,ver required for dri,;ng the pumps is supplied by a gas turbi11e \vhich, in the illt1stration, is supplied ";th hot ga..~ from the gas generator that also uses the main propellants. In order to keep the gas

The Chemical Rocket Engine

I

411

-

Oxidizer tank

-

Fuel tank

I

'

Gos generator

I •

Pump

I

I

.. , J ... . . '.l

'

-

J

'

Turbine

~---...:--

'

,6.

~

-.

l

V'"

•-

. .; j:::;;-

'

~

-::: ,_.

- ----=-::::: - ~

~

1':..- "'-~ J-'.._-:;;!'

.......

Waste ,

Pump

Fro. 15.3 Typical pump pressurization rocket engine.

temperature lo\V enough to be used in the turbine, the gas generator operates fuel-rich. About 23 of the propellants are used in the gas generator. The e.xhaust from the turbine adds its thrust to that of the main thrust chamber. In addition to this method, \vhich is used on almost all of the ne\V high-thrust engines, the power for the turbopump can be supplied from hot gases bled directly from the thrust ch.amber or from steam and oxygen derived by decomposing liquid hydrogen peroxide b~· a catal.)-st such as calcium or sodium permanganate. The use of the separate h.)·d~aen peroxide system is attractive in rocket engine desig11, because the ga...~s generated sre at a relatively lo\v temperature (about 800 °F); the refore, no problen1 of high turbine blade temperature exists. The use of thrust chamber tap-off is also attracti,·e becau....::e it eliminates several components and 11ence sin1plifies the O\'er-all propulsion s,·stem . • The key to this method is proper selection of the tap--0ff locatio11 to ~·ield lo\\·-ten1perature (about 2000°F) gases '''hich can be used i11 tl1e turbine. Because of the gas turbine, tl1e pun1ps, a11d additional lines that are nece~·. the pump-pressurization system is son1e\vhat n1ore co111plex than the gas-pre~urization system. Ho,vever, in spite of tl1is extra con1plexit.) ' and the additional problen1s involved, the pump-pressurizntion system hns n mnjor nd\'21nt.,'\ge 1 Mmel.)·, light ''-eight for long-duration operatio11, \vl1ich out,,·eigl1s its dis:1d,·nnt.. Cv Cd A, Pc r

THRUST COEFFICIENT

c

Xe

2-y 'Y -

+A. (Pc - Po)

1 + x.

1

(15.11)

p

As \Vas d one for t he weight-flow rate, the t hrust of a rocket motor is frequen tl-'

written in terms of a coefficient, A ,, a nd Pc as

F

=

C, Pc A,

{15.12)

Hence, from t he above and from Eq. (15.10), the thrust coefficient is defined as

F = Ca V . + ~ J!.!. _ ~ PcA t g A, Pc Pc The thrust coefficient may also be written from E q. (15.11) as

CF

C,

=

= X Cv

2

Cd r

'Y

'Y -

Xe

1+

1

x.

+~

At

l!.!. - ~ Pc

Pc

( 15.13)

(15.14)

I t sh ould be noted that \vhen t he nozzle operates ,,·ith con1plete e.'\.- pan.sion, p, = Po a nd t he last term of both E qs. (15.13) and (15.14) ,,·ill be zero. I n the ,·acu11m of space, po = 0. Equation (15. 14) t he11 reduces to 2

(c P) vacuum = ' cd cV r I\

'Y -

'Y

1

x•

1+

_\ .

+ ..-.: I!.!.

Pc

( 15.15)

\Vhere E = A e/ A ,, the nozzle a rea expa11sion ratio. The basic t h rust coefficient C, can t hus be \vritten Cp = (C P)vacu um - E J!! (15.16) Pc Examinatio11 of Eqs. ( 15.14) n11d (15.16) sho,,·s thnt tl1e ideal tl1rust coefficient

c~

=

c,. X Cc1 c,,

is a fUI1Ction Of Oilly '}' 1 71,/ po, filld E. }..igttrc 15.18 p reSCilt t}1e \·nriatiOil Of C ~ \\;th and Po/Pc for t\VO difTcre11t values of -y, 1.20 and 1.30.

E

I

430

-

Jet Propulsion

2.0

Pc/Po = -

•

---

{C;)•ocuum

1.8

1000

--

1.6

soa 3.JJ

200 1.4

/

c,'

100

~ (c;)optimum

1.2

/ 10

1.0

0 .8

I

I

33.3

20

Pc/Po = S

50

0.6'--~--'---'~~..____..___.__.__..__._.._._~__..____._~_....~_.___.__....__..___._~

1

2

3

6

8 10

20

30

40

60

80 100

A./ A, (b) r= 1.30 F10.

15.18 Ideal thrust coefficient

c; versus nozzle area expansion rl\lio and pressure ratio.

The curve labeled Pc/Po = oo is cquivalc11t to t l1e ideal \'acuum thru t coefficient (C~)vncuum· The dotted lines 011 both figures rc1)rcscnt the special ca..~ ,,·here the area ratio E is optimum so that p. = po. In this case Eq. (15.14) reduces to

The Chemical Rocket Engine

x. 1 +x.

2-y

= I'

'Y -

1

I

431

(15.17)

The dotted lines in Fig. 15.18 thus represent Eq. (15.17). Reference 12 should be consulted for detailed thrust coefficient tables. In vie'v of the significant effect of nozzle area expansion ratio E on the thrust coefficient, it is useful to revie'v the relation bet,veen E and the nozzle pressure ratio p,/ pc, which was developed in Chapter 3. Equation (3.23) ca11 be rc\vritten as 2

1 -y- l

(15.18)

l!!.

.!

-y- 1

'Y

-

+1

'Y 'Y -

Pc The relation bet,veen sented in Fig. 15.19.

E

'Y

Pc

1

and Pc/p, for complete expansion and for various -y's is preSince large nozzle expansion ratios tend to result in high nozzle weights and since rocket engines often do operate over a wide range of nozzle pressure ratios, efforts have been directed to'\\·ard development of plug nozzles for use on rocket engines. Figure 5.49 and the accompanying disc11ssion are indicative of the advantages of the plug nozzle for application to rocket engines.

100

80 60 40 20

-

: 0

(h1)c11. - 2(h,)o,

4~!!

I

The Chemical Rocket Engine

439

The excess O>..'Ygen and nitrogen do not enter i11to the reaction and he nce do not affect the heat of reaction. From Table 15.5,

H. V. = -169,300

+ 2( -104,000)

- ( - 32,200) - 2(0)

= - 345, 100 Btu per lb-mole

No'v using Eq. (9.5) , \Ve obtain

H.V.

= =

r,

(

T,-S40

"LN Cp

co

-·

::J

200

Q.

--:"..

8 6

E

·-u

·-v 160

Assuming isentrop ic expans io n with maintenance to 14.69 psia (of Chem . Equil.)

No.

7

~

•

System

P, ps10

340

1. liquid oxygen - liquid hydrogen 0

Cl)

2 . l iquid oxygen - gasoline 3 . l iquid oxygen - ethanol 4. l iq uid o xygen - liquid ammo nia 5. l iquid oxygen - hydrazine 6 . RFNAb - aniline 7 . RFNAb- orthotoluidine 8 . Mixed acid - mono-ethyl aniline 9 . Hydrogen peroxide'- hydrazine 10. Hydrogen peroxided- methanol 11 . Hydrogen peroxided- nitromethane

Q. V')

120 80

40

0

Assuming isenlropic expansion without further reaction lo 14.69 psia

2

6

8

10

12

14

300 300 300 300 300 300 340

300 300 300

16

Oxidizer / fuel weight ratio F10.

15.21

Comparison of calculated values of

I,,, for several liquid-propellant ~-~te~.

impulse and temperature. l\1ixture ratios ,v}1ich yield the n1aximum specific u11puL~ are generally chosen. l1~or example, it is obvious '"h)r tl1c German \ --2 po,,·er plant, '''hich used propellant syste1n 11un1ber 3, '''as operated ,,·ith n. 111i.'Xture ratio of 1.25. One pertinent commc11t regarding the specific in1pt1lsc Ctlr\·cs is the fac t that all prop ellant syst ems sho,vn, ' vith the exception of liquid ox)rgen-h)·drogen, genernll)· lie in t he same regio11. Tl1e liquid oxygc11-l1ydrogc11 S)'Sten1 has appreeiabl~- greater performance potentialities, this bei11g pri111aril)r dt1e t o tl1c high heating ,·alue of hydrogen. It is pointed out, hO\\"C\rer, tl1at becat1sc of tl1e lo,,·er de11sit~· of this s~-s­ tem, it has the lO\\'CSt valt1e of de11sity in1pt1lsc. A fc,v othc1· co111bi11atio11s, 11ot s hO\\' Il i11 the figt1rcs, produce I,,, ,·slues con1parable to that of liq uid oxygc11-h)rdrogc11 (sec Table 15.6). Of particular note is fluorinc-hydroge11, \vl1ich produces l1igl1cr \ 7U.lt1cs of botl1 specific and densi t)· impulse. CI-IARAC'fER ISTI C LENGTH

L*

A11otl1c r parameter '''l1icl1 is tisccl 0111)' i11 co1111C'ctio11 ,,·it 11 liqt1itl-propcllant rocket e11gines is t l1c cl1aractcristic lc11gtl1, \vl1icl1 is dcfi11C'cl n.s

(15.38)

448

I

Jet Propulsio n 6400 6200 6000

5

5800

3 - - Assuming 1sentropic expansion without further react ion to 14 69 psia

5600 5400

4

5200

7 Note a. 0

11 9

15 wt frac. hydrogen 0 85 wt fra c carbon b. 0 15 wt. frac. N 2 0 ~ c. 100 wt. per cent concentrotton

~ 5000

...C>' ~ 4800 ...0

8. 4600

E

~

4400

~ E

4200

...

d

No.

System

p

c

1. l iquid oxygen - l1qu1d hydrogen

_g

4000 u 3800

87 wt per cent concentration

0

2 . Liquid oxygen - gasoline 3 . Liquid oxygen - ethanol

L.. l l l l

4. 5. 6. 7.

3600 3400 3200 3000 12 2800

10

8.

M ixed acid - mono-ethyl an iline 9 . Hydrogen peroxide' - hydrazine 10. Hydrogen peroxided - methanol 11 . Hyd rogen p eroxid ed - nitromethane 12. Hydro g en peroxided-nitr omethone, w ith methanol, 30'X.

6

2600

Liquid oxygen - liquid ammonia Liquid oxygen - hydrazine RFNAb - aniline RFN A b -orthotolu1d1ne

7

2400 2

0

6

10

8

12

14

ps1a

340 300 300 300 300 300 300 340 300 300 300

16

Oxidizer / fue l weight ratio F 10.

15.22 Comparison of calculated values of Te for several liquid-propellant S)·stems.

where (Vol), is the combustion chamber volume up t o t he nozzle t hroat. A corollal')· parameter to L* is the stay time t., 'vhicl1 is t he average t ime spen t by each gas molecule 'vithin the combustion chamber ' ' olume. Thus

t.

= (Vol)c

(15.39)

G Pc

where Pc is the average density of the gases in the com bustion chamber . The nu nimun1 stay time 'vit h good performance thus defi11es t he n1i11imun1 comb\tstion chan1lx-r v olu me, and in t urn t he minimum cha racteristic lengtl1 L •. Si11ce the st.a)' tin1r includes the t ime necessary for vaporiza t ion , decon1positio11, and co111ple te bun1ing of the propellants, it is primarily a fun ction of t l1e propclla11t con1bination. Equation (15.38) can be \Vritten 'Y

t, =

(Vol)c

A,

+1

-y + l -y-l

2

Tc -yg(1545) m

-

(Vol), 1

A,

r

1 g(1545)

T, ni

( 15.40

4

I

The Chemical Rocket Engine No

1.30

0

2 . Liquid oxygen - gasoline 3 . liquid oxygen - ethanol 4 . Liquid oxygen - liquid ammonia 5. Liquid oxygen - hydrazine 6 . RFNAb - aniline 7 . RFNAb- ortho toluidine 8 . M ixed acid - mono ethyl aniline 9 . Hydrogen peroxide< - hydrazine 10 Hydrogen peroxided - methanol

I

''••

l .28

'1

c0

1.26

~

1.24

·-"'c &.

-·-.,"'

,,

I\\\ \

1.22

c

2 6

I\

v

·-Q. ...0

-., -::>> 0

c

I I

1.18

0

-•

I

E

~ 1.16 Q. v

II ~

l .14

l .12

\

I

\

I 19 I I I I I I I l I I

I

8

\

4I \

t)

300

300 300 300 340 300 300

\

1I 'I \\ I I \ I I l I I \

1.20

340 300 300

1

Il \ I I

...0

Pc ps1a

System

1 l iquid oxygen - liquid hydrogen

5

449

I \ ' ... ""

I

I

I

I

N ote: a . 0 .15 wt free. hydrogen 0.85 wt. frac. carbon b . 0.15 wt frac. N 10• c. 100 wt per cent concentration d . 87 wt. per cent concentra tion

I

10

A ssuming isentropic expa nsion w i thout further reaction to 14.69 psia

I

I I I

-- -

1 I

Assuming isentropic expa nsion w ith maintenan ce to 14.69 psia (of Chem. Equil.)

Oxid izer/ fuel weight ratio FIG.

15.23

Comparison of calculated values of mean spcrllnnt C'o111binn tions

Con1binntio11 LOX- kerosene LOX- C,II 6011 LOX- LI-12 H NO r-11yclrocnrbons H NO ,--UD~ lH

F,.-N I-1 ,

L • (in.) 60-100

100-120 50 80-120 60--80 40-60

-

I

450

Jet Propulsion 30

5

28

No.

.

Pc ps10

Sys tem

1. l iquid o xygen - liquid hydro gen 0

26

2. Liquid oxygen - goso line 3. Liquid o xygen - ethano l

24

4 . liquid oxygen - liquid ammo n ia 5. RFNAb - aniline 6. RFNAb- ortho to luid ine

E 22

-·--

340 300

300 300 300 300

2

3

.L

CD

Cl

};

....

20

c

4

~

v

-

41 0

18

E c

5

c

41

~

16

6 Nole: a . 0 .15 wt. f roe. hydrogen 0 .85 wt. froc . carbon b. 0. 15 wt. f roe. N 20 4

14

12

- - - Assuming isentropic expansion without further reactio n to 14.69 psio

10 8.___..__.__.___.__.__.___.__.__.___..____.__...___.____._

0

2

4

6

8

10

12

14

_.___

16

Oxidizer/ fuel weight ratio

F10 . 15.24

15.7

Compa rison of calculated values of mean m olecular \Vcight for se\•eral liquid-propelbn: systcn1s.

The Rocket Motor Cooling Problem

A glance at Fig. 15.22 and Table 15.6 sl1o'''s that t.)·pical combustion chamber temperatures are in the 5 to 7,000 °Ji' range. Tl1csc temperatures arc, of course, n1urh greater than kno,vn metals of today can ''•itl1stand; tl1erefore, an adequate coolin!r system must be employed 011 the rocket motor. 111 prcsc11t-da.)' n1otors, onl)r abot1t 5c--. of tl1e heat liberated i11 the combustio11 cl1amber passes throt1gh the n1otor " ·all . B·?: this 53 amounts to magnitudes of even greater tha11 200 Btu per ft2 sec (a t)·pir·u household furr1ace liberates about 30 Iltu per sec) . A11 O\'er-all a11aly is of the coolir'!! problem is very extensive and is based pri1narily 011 heat transfer relations. Onl)· tl.e major aspects \vill be co11sidered 11ere, \vith a brief description of some current n1eth1 ~ of cooling. The over-all cooling problem is one i11 \vl1icl1 l1igh temperature, high pres.5ure, tt 1 high velocities are involved. Future tre11ds i11dica.te tl1at rocket n1otors must ti t 11 • higher chaml)er pressures i11 order to increase their thrust . Therefore, the rocket n1c · designer must })c concerned \vitl1 (a) any st1ddcn change of temperature; (l1 · · · tended high te1nperatures ; (c) t hern1al d istortio11; and (d) gas radiation and r · -

1

I

The Chemical Rocket Engine

I

451

vection. The primary object of cooling the rocket po\ver pla11t is to keep the inside of the metal \Vall at a temperature belo\v its strength failure point. This is done by six major methods of cooling; namely, 11eavy \Valls, ablation, insulation, evaporative cooling, boundary layer cooling, and regenerative cooling. Method 1, heavy walls, employs a principle of supplyi11g a l1eat reservoir capable of receiving the total amount of heat transn1itted to the \Vall 'vithout raising t he metal temperature to dangerous values. Since t}1e amount of heat transferred through the \Valls is extremely l1igh 1 this method of absorbing the h eat is not satisfactory, unless the rocket motor is to be fired for a very sl1ort period. Method 2, abl,atwn, is really a variation of Method 1 in \vhich t l1e latent heat of evaporation (or sublimation) is used to furnish the heat sink to receive the total amount of heat transmitted to the chamber \valls. Figure 15.5{b) sho,vs an ablative skirt attached to the second-stage engine of the Titan II. Ablative materials can also be used to form almost the entire engine - combustion chamber and nozzle less the throat insert. Method 3, insul,ation, utilizes materials of lo\v conductivity to keep the metal walls belo'v their critical temperatures. Figure 15.13{a) sh0\\1S an example of the primary utilization of this method - the propellant grain itself in internal-burning solid-propellant engines. In addition to propellant, other insulating materials are often used to protect the rocket case, particularly \vith external grain-burning solidpropellant engines. Method 4, evaporative cooling, sometimes called film cooling, is another methoJ for combating the high temperatures in the combustion chamber. I ts principle 01 operation is to form a complete film of fuel 011 the inner surface of the nlotor "·all , its heat of evaporation 'vill then keep the \Vall surface reasonably cool. The German V-2 po"rer plant employed a crude method of evaporative cooling. This ,,.as accomplished by admitting the alcohol in a series of circumferential cooling ports. By permitting approximately 7% of the total alcohol consumption to e11ter tl1rough the cooling ports, temperatures of 5400°R could be \vithstood \vithout da111aging the motor, 'vhich \Vas constructed from plain carbon steel. ~1ethod 5, boundary layer cooling, provides for the stabilizatio11 a11d co11trol of the gas film thickness in contact \vi th the motor a11d nozzle \Valls. I t is a \\'ell-k110,,·n fact that the thermal resistance of high-temperatt1re gas is quite 11igl1. Studies 111nde ''"ith combination film cooling and boundary layer cooling i11dicatc that tl1rec i11sulating characteristics are derived from this method; 11nmely, {I) part of the cooli11g fil111 breaks do,vn on the motor \Valls i11to a layer of carbon about 0.03 i11chcs tl1ick; (2) a liquid la)'er of fuel passes over the carbo11 deposit; and (3) a ' 'apor-i11st1lnting ln)·er forms over the liquid layer. These i11sulating pro1)Crties plus t l1e 11cat of ' 'nporization have made it possible to operate alumi11um co111bt1stio11 chun1bcrs at 600 ~in for several minutes. If some method cot1ld be devised for i11st1ring a certni11 111en11 \'apor layer covering the motor \Val ls, tl1is method of cooli11g '''otild 11a\'e great potc11tinlities . ~Iethod 6, regenerative cooling, is t he most con1n1011 syste111 ernployed todn)·. It appears to be the standard cooling n1ethod tl1n.t is to be t1scd for t l1e 11ext fc,,· )·cars. I t has the advantage t l1at once the cooli11g systc111 l1as bce11 de\relopcd correctl)', the motor cn.11 be operated for exte11dcd bur11i11g periods 'vitl1out dan1age. I"urthermore, rcge11eratively cooled motors cn.11 be n1ade cxtren1cly light per pound of thrust . This method also has a sligl1t efficie11cy adva11tage over the previous n1ethods nlentio11ed.

452

I

Jet Propulsion

Figures 15.7 and 15.8 clearly sho\v the tubes co1IBtituting the combustion chamber and nozzle through ''·hich one of the propellants is pnsscd to make the regenerative cooling systen1. Since tl1is method is predon1i11nntly used i11 current rocket motors, additional details be discussed. It ,,·ill be ,,·ell to analyze one section of a rcgcnerati\'C tube in terms of heat flu.~. This is sho'''n schematically i11 Fig. 15.25. Temperature

''ill

...._

~

-

-

VG 4

c:

-

·-0

Hot

.-

.. --

__ ,

.-

~ :: Tw - -

combustion gases

-..=boundary layer ....,_ ~ :::::: . -~Gas -

~-

·-

Cooling fluid --;)lo~ (liquid fuel)

Flo. 15.25 l\Iotor-,vnll section.

The total amount of l1eat transn1itted tl1rougl1 the ,,·all " ·ill be due to gas radiation and convection as represe11ted i11 the for111t1la I

1

q = qr+ qc

{15.43)

The radiation heat n1ay be represe11icd by

qr

=

uT~ - uT:

=

hr(Ta - Tn·)

(15.-1-1'

'''here " is the Stcfa11-Boltz111ann co11sU\11t. Tl1c co11\·ccti\•e heat n1s:,~ be ,,·ritten in tern1s of the co11vective l1eat tro11sfer cocfficic11t lio as

The l1eat tran~fer to tl1e coola11t is tl1c a111ot111t sl10,,·11 i11 Eq. (15.-13) , a.11d c.s11 l~' reprcsc11tcd by

q=

+

t

The Chemical Rocket Engine

I

453

Substitution from the above equation yields

__ T_w_-_ T_L_ = (lir 1 hw

+ lia)(Ta -

Tw)

(15.47)

+

For the sake of simplicity, and because the radiation heat is small compared to the convection heat, a ne\v heat transfer coefficient, \vhich co11tains the comhiried effect of radiation and convectio11 may be used . Ilence, the follo,ving may be \Vritten (15.48)

q = h'(T a - Tw)

R. H. Sabersky has attempted to calculate the gas film coefficient by applying the la\vs of perfect gases to the equation (15.49)

'vhere CH is a coefficient of heat transfer. In rocket motors, the 'veight rate of gas fio,v is al,vays at a maximum value, and this may be written from Eq. (15.2) as

G = CDAtPc vg r

(15.50)

v'RTc The gas velocity may thus be \vritten as

V a = Gv = CdAtPc vY r v A A 5viRTc

(15.51)

If the combustion chamber section is analyzed, Cd ~ 1. Therefore, substituti11g the above relation into Eq. (15.49) 1 the gas film coefficient bccon1es

ha =

C11CppA tPc Vy I'

(15.52)

A5 .../ RT,pg or

ha =

C11Cppt; A, A5

v gR'l'a

r

(15.53)

It is much more convenient to use the ubovc forn1uln. in rocket n1otor \\'Ork tl1n11 to use Eq. (15.49). Ilo,vever, all the variables 1nt1st be calct1lntc(l 011 tl1c n.."'Stt1111ltio11 of a homogeneous, pe rfect gas flo\ving i11 a straigl1t pipe. 1~t1esc li111itntio11s in1p°'"~ i11acct1racies upon the true case of real gases in a11 uctt1al rocket 111otor, so tl1at t}1(' reLtilts arc sometimes up to 253 in error. This being the case, tl1e dcsig11 is ust1ally \vorke

0 .985

-:>a."' E

-·-

0 0

~

0 .980

0 .975

Fro. 15.32 Effect of propellant temperature on the total impuLc;e of solid-propellant rocket engines. INERT COMPONENTS

After the propellant grain has been consurned, t he remainder of the engine represents the inert components. Figure 15.11 sh o\VS that these ine rt component.s normally consist of the case, the nozzle, the igniter, and any insulation remaining. :\n import.ant parameter for solid-propellant er1gi11es is t he ratio of the propellant ,,·eight to the tot.al rocket-engine \veight, or the propellant-\veight fraction. By definition the propellant '''eight or mass fraction is

lV,,

"' =

where

W ,,

=

propellant \veigh t

W,

=

inert componc11ts \vcight

= -:------

lV,,

+ JJT;

(15.i7)

I

JV"1 = rocket engi11e \Veight

Figure 15.33 from R ef. 24 sl10\VS the relation lJct\vee11 the propellant n1ass f ractio11 and total impulse of solid-propelln.11t rocket e11gi11es. l•~igu re 15.33 \VUS compiled fron1 available rocket c11gi r1c data to sho,,· the correct trend .

...

I

462

Jet Propulsion

Another parameter of interest in solid-propellant rocket engines is the total impulse to rocket-engine \veight ratio, I r/W,.1 • From Eqs. (15.22) and (15.77), it can be sho,vn that

Ir =

W1i1

1•P.

(15.78)

Figure 15.34 presents the impulse-\veight relation as a function of total impulse based on Fig. 15.33 and a specific 1,P value. t .0 220

0 .9

0 .8

6

200

•

v

~

~ 0 .7 ~

0

6

v 0

~ 0 .6

v

v v• "'

v

>

':::? 160

:.v

0.5

~~ 180

l,p = 250 sec.

) ,,.,

v

' v

6

v

0 .4

v

140

v

0 .3 L-...::>~_........__ _--L._ _~_ _.._1i.___ 2 3 5 6 10 10 10' 10 10 lr#-sec

___J

7

10

Flo. 15.33 Propellant mass fraction versus solid-propellant rocket engine total impulse.

1

3

5

7

9

11

13

15

I, -#-secxlO-'

Fto. 15.34 Impulse-weight ratio versus total impulse of solid-propellant rocket • engmes.

I r/WM is a useful parameter for comparing the effectiveness of different propulsion systems. A rapid estimate of the over-all size of a solid-propellant engine is sometimes desirable. Equation (15.79) is an empirical relation, developed in Ref. 23, which gi\·es a good approximation to the over-all engine dimensions.

F 7rD2

4

where

L

C, Pc

-

V"";-1

v"7r

{15.i9)

tan a

C,po

= engine over-all length

D = engine diameter

A,,/ A, = combustion chamber port (initial flo,,· area) to nozzle throat area ratio a = nozzle half-angle

Figure 15.35 from Ref. 24 presc11ts a t;>rpical plot, co11 tructed from Eq. (15.79). sho,,•ing the rclatio11 bet,vcc11 tl1e propellant ,,·eight and the engine length for \-arious diameters. THRUST-VECTOR CONTROL

Rocket e11gi11c applicatio11s frequc11tly rcqt1irc some means of controlling the thrw:t vector of the c11gi11c i11 order to supplc1nent or supply aerod)•namic attitude control

I

The Chemical Rocket Engine

463

2000

F= 5000• @ s.L. Pc= 1000 psia PP= 0 .0633 lb/1 n3 E = opt. @ S.L. AP/A, = 2 .0 ')' = 1.16

J)

...-

,,

~ //

~

1500

if

~

~

--

b.p 0(f

_c

CJ) Q)

-~

~

c:

0

Q)

0. 0

...

1000

Q..

(.../f-1)

vrr ton a

500

75

100

125

150

175

200

225

250

Rocket motor length, L, in. F10.

15.35 Solid-propellant engine length versus propellant weight and engine diameter.

for missiles and space craft. It \Vas pointed out in connection '"ith Fig. 15.-1 that the liquid-propellant engines that comprise the Atlas ICBl\1 are S\\'i\reled or gimbaled t-0 provide attitude control for the missile. Although not specifically mentioned, this is also true of the other liquid-propellant engines discussed in Sec. 15.2. Gin1baling of the con1plete liquid-propellant rocket motor is feasible because the n1otor is reasonabl.)· small. Equivalent thrust vector co11trol is also required for solid-propellant rocket engines in installations such as tl1e l\1inuteman ICBM and Polaris FBl\1 , ,,·hich are illustrated in Fig. 15.15. As can be seen, ho,vever, equivalent gin1bnling of the con1plete engines is not feasible because of their size. O\ving to tl1is reqt1ircn1ent, ho,,·e,·er, methods have been developed for adequately controlling the thrust ,·ector of solid-propellant engines. Figure 15.36 illustrates the pri11cipal types of thrt1st ,·cctor control for solidpropellant engi11es. The follo,ving are tl1e methods \Vl1icl1 11ave citl1er bce11 co11sidered or are being used : (1) jet va11cs; (2) jetavator; (3) S\Yivcli11g 11ozzlc; (4) scro11dar.)· i11jection into the nozzle. Tl1c a.dvar1tagcs n.11d disad\rn.11t.agcs of t' ncl1 ,,·ill 110\\. be co11sidered. Jct va11es, '"hicl1 are rotated to deflect tl1e exl1at1st gfi$CS, are pro~11bl)· the oldest method of tl1rust -vcctor co11trol. Tl1cy ,,·c1'C t1scd i11 tl1c Gcrn1an \"-2 111issile. The adva11tagcs of jct va.11es are large co11trol f orccs, l1igl1 rcspo11sc mte, co111pletc control (pitch, ya.,v, and roll) frorn a si11glc i1ozzlc, a11d a I'Clnt,i\rel.)' si111plc, pro\•c11 s)·stcn1. The disadvantages arc a continual tl1rt1st loss bccat1se of tl1c \'a11rs and se\·ere n1aterial problems \)ccausc of tl1c l1igl1 tcn1pcratt1rc.

464

I

Jet Propulsion To actuator / Cylindrical

Bell crank

Spherical Pivot point

--

/To actuator /

...

1. Jet Vanes

_

2 . Jetavator

Actuator

3. Swiveling Nozzle FIG.

4 . Secondary In jection

15.36 1\.Iethods of thrustrvector control for solid-propellant rocket engine.s.

Jetavators were developed to overcome the disadvantages of jet vanes. There is no thrust loss "·hen vector control is not desired, and the materials selection problem is simplified since contact \\'ith the hot exhaust gases is relatively brief. The disadvantages of this method are as follo'''s: 1. An increase in engine " 'eight because of the requirement for more than one nozzle to achieve complete control 2. Possible overheating of the base of the engine unless it is carefully sealed for protection from base blo,vback 3. A definite thrust loss ''rhen the jetavator deflects the exhaust stream The primary present use of jetavators is their application to the Polaris FB:\I. Although it is not shown in Fig. 15.15, a total of four 11ozzles is incorporated into each engine. This means that jetavator motion is t1nidirectional for each nozzle. The third method involves the use of a S\vi,·eli11g nozzle, ,,·l1ich for the solidpropellant rocket engine is analogous to gin1baling for liquid-propellant engines. The ad\•antages of this method are that tl1crc are no losses in thn1st or in1pulse and thst the control characteristics are li11ear. The disad,·antages ii1clude the necessit)· of sealing against the hot high-presst1re exl1aust gases and the r1eccs.5it)" for sigi1ifica11t actuating loads. As i11 the case of the jetavator, nlore tha11 011e 11ozzle is required to provide complete control. The initial applicatio11 for tl1e S\\~i,,eling n1ethod is on the 1'1inutcman ICBl\f. The fourth method of thrust-,rcctor co11trol i11vol,·es secondar)r injection (SITI.C) into the nozzle. This mctl1od employs tl1c i11jection of a fluid il1to the di,·erge11t portio11

The Chemical Rocket Engine

I

465

of the nozzle, resulting in an oblique shock \vave which deflects the entire exhaust stream a s mall an1ount. Ma11y different ftt1ids can l)e used, ar1d these can \)e either reactive or nonreactive \vith the exhaust gases. Advantages of this system are its light ,,·eight and lack of moving parts. 1"'11e primary problem is the selection of secondary fluids giving high side forces for lo\v injectio11 rates. T\vo fluids being investigated a re freon, \vhicl1 is nonreactive, and N 204, 'vhich is reactive. R ecent ne,vspaper reports indicate that secondary injectio11 is being used i11 connection \vith the development of the large solid-propellant rocket e11gines si1nilar to the one illustrated in l"ig. 15.13.

15 .9

Characteristics of Some Solid Propellants

As mentioned earlier, the solid-propellant grai11 in an engine contains both the oxidizer and the fuel. There are t\vo prir1cipal solid-propellant types: homogeneous and heterogeneous. In the homogeneous type, \vith double-base or colloidal propellants, both the oxidizer and fuel belong to each molecule of propellant. In the heterogeneous or composite propellants, the oxidizer and fuel are separate compounds intimately mixed together. Typical double-base propellants are colloidal mixtures of nitrocellulose and 11itroglycerine such as ballistite and cordite. Typical composite propellants are cons tituted of very finely ground oxidizer crystals (perchlorates or nitrates) dispersed in a fuel matrix (polyesters, asphalt, or rubber) . The follo\ving characteristics are considered important for all solid propellants:

1. High specific impulse, I. p, \vhich requires a high adiabatic combustion temperature T 0 and a lo\v molecular \Veight m 2. High densit)', pp, so that the required propellant quantity can be packaged in minimum volume 3. A lo\v burning-rate exponent n for good combustion pressure stability 4. A reasonable burning rate r at the optimum pressure (Reference 25 sho"·s that r can be varied over a \vide range by modifying the oxidizer mi.xed ";th a given fuel) 5. A lo'v temperature sensitivity coefficient 7rK for small cl1anges in engine performance \vith propellar1t temperature 6. Excellent chemical s tability for long-term s torage \Yithout change in the performance characteristics 7. Very lo'v deterioration or explosion hazard 8. A smokeless exhaust \vith lo\v toxicity to redttce detcctio11 i11 n1ilitar)· applications. 9. Good physical properties for ready e11ginc fabrication nnd n1ggro11rN:i under rough handling 10. Inexpe11sive propel Ia11t constitue11ts a11d CUS)' prepnmtion As might be expected , cacl1 propella11t for111ulatio11 reprc~C'11ts son1ctl1ing of s con1promisc among t hese several cl1aractcristics. 1\ s ,,·itl1 liqt1id propella11ts, 111n11~· ricf considemtion of the thro~· is desirable, ho\\'ever 1 in order to give sonic feeling for reactor characteristics. I

t

.. I

Nuclear Propulsion Engines

475

The difference bet,veen finite and infi11itc cl1ain reactions is the loss of neutrons through leakage from the finite reactor. T l1e ratio of the number of neutrons in successive generations in a fi11ite reactor is k110,vn as tl1e effeclive mulliplicalion factor ke1t, or the criticality factor. Another term frequently used in connection 'vith reactors is the reactivity, which is defined as (kctt - 1)/kctt. kctt is al,vays less than km. This can be expressed as follows (16.8)

kert = km X (total nonleakage probability) For a steadily running reactor kctt is equal to unity. The neutron leakage loss in a finite reactor depends on the size and shape of the reactor. As a result kctt can be d etern1ined from k and a kno,vledge of the diffusion properties of neutrons. Since t he diffusion of neutrons in a moderator is quite similar to the diffusion of heat in a solid conductor, \Ve shall first briefly consider the latter process. For the small element of a solid conductor illustrated in Fig. 16.1, the heat flux q entering or leaving the volume can be expressed from the standard conductive heat transfer equation

z

Qz+ d1

00

q = -KA:

aT ax

(16.9)

I

lclz

--i- ~

ax

I

).---- ----;

- - JC

/

/

; /

F10.

,"cly

A I

16.1 H eat flu."< from an element of a solid conductor.

where

q = the rate of heat flow

K = the thermal conductivity of the conductor

A

=

the conductor area normal to flo'v of heat

T

=

the conductor temperature at any point

The rate of change of heat from the element, \vhich represents the net rate of heat loss from the element, can then be \vritten

dq = -K(dx dy dz)

(16. 10)

The difference bct,veen the heat lost by conduction [Eq. ( 16.10)] and an)• heat generated 'vithin the eleme11t is available for l1eating tl1e co11ductor elen1ent. This can be expressed as follo,vs:

aT pcp(dx dy dz) at =

S (dx dy dz)

+ J((dx dy dz)

(16.11)

or (16.12)

I

476

1 ''

Jet Propulsion

here p =

the conductor density

I

c 11 = the conductor specific heat

S

=

the rate of heat ge11eration per unit volume

Using t he Laplncia11 operator V 2, Eq. (16.12) can also be \Vritten

'!:£ = ot

+ a.V2T p Cp S

(16.13)

'''here a = J(/ (p c 11 ), the thermal diffusivity. Equation (16.13) expresses tl1e rate of change of the temperature in a conductor as a function of the properties of tl1e conductor and any heat sources 'vithin the conductor . Analysis of the flo\v of neutrons i11 a moderator, using the Boltzman transport equation and assuming that the distribution of 11eutrons is isotropic and independent of neutron energy or position, reveals tl1at there is no essential difference bet,veen the diffusion of neutrons and the diffusion of heat. Assume that the element illustrated in Fig. 16.1 is a unit volume from a homogeneous reactor. The rate of change of the neutron density n \Vithin the element is given by the difference bet,veen t he rate of production of neutrons and the rate of loss due to leakage and absorption. This can be expressed as (16.14)

I

I

I

I

where

S

= rate of neutron production

X = transport mean free path l;a =

macroscopic absorption cross-section

cf> = the neutron flux, the product of the neutron density and the average neutron velocity = nv

The transport mean free patl1 Xis the effective average distance each neutron tra\·els bet,veen collisions in the moderator. This is a property of the moderator material. The macroscopic absorptio11 cross-section i;a represe11ts t11e probability per unit volume of an absorption reaction bet\veen a fuel nucleus and an impinging neutron. For steady po,ver reactor operation, the neutro11 de11sity is constant. Equation (16. 14) t hus becomes (16.15)

If 've assume that all 11eutron reactions occt1r at 011ly 011c e11ergy le,•el, ,,.e n~ consider the properties of only one group of neutrons. J\ppl)ri11g tl1e Fcrnu 11eutron-age sJo,vif1g-do\vn model, it is sl10\v11 i11 Refs. 8 a11d 10 tl10.t tl1c sot1rce tern1 i11 Eq. ( 16.1.t) can be \vri t ten

S =

~a

le q, e- 8 lr111

(16. 16)

I

. Nuclear Propulsion Engines

I

477

where k B2

= =

multiplication factor the material buckling factor, a measure of t11c bending of the neutron flux at any point in the reactor ,,.," = Fermi age of neutrons to thermal energy, 'vhich is equal to t11e product of the slo,ving-do'''n time and the average difTusion cocfficie11t (>./3) under the time interval

Substituting Eq. (16.16) into Eq. (16.15) and simplifying, '"c o})tain V2

+ B2q, = O

(16.17)

and 2

k e-B r,. 1 + V B2 = l

(16.18)

where

L2 =

~/3~ 0

=

the diffusion length, a measure of the average distance a thermal neutron travels from its point of formation to its point of absorption

The condition of criticality is that the value of B 2 satisfy both Eq. (16.17) and Eq. (16. 18). Equation (16.18) is also \Vritten as follo\vs: I

- B,

k1//

=

e

k

r,,. (16.19)

00 - - - -

1 + L 2 B~ '''here is the buckling from Eq. (16.17). Examination of these equations shows that the buckling determined from Eq. (16.17) depends only on the size and shape of the reactor. B 0 determined from Eq. (16.17) is therefore called the geometric buckling. B,,., determined from Eq. (16.18), is called the material buckling, since it is determined only by the reactor materials. B,,. = B 0 must apply for a critical reactor since this assumption \Vas made in deriving the equations. Comparison of Eqs. (16.8) and (16.19) reveals that

B:

I

-B, r '"

_e___ = Total nonleakage probability 1 + L 2 B! from the reactor Breaking this down further, it can be sho,vn that ( 1 + L 2B:)- 1 is tl1e no11leakage probability of thermal neutrons, \vhile e-n;r,,. is tl1e nonleakage probability of ncutro11S slo\ving do,vn to thermal energy. In order to illustrate the application of Eqs. ( 16.17) and (16.18), \\'C sl1all co11Sider an example problem. Example

Determine the critical dimensions of a cylinclricul, l101nogc11cot1s l'C'actor co n11>0.."f rr..('mtP at th.P ru'Cf"' =". .:lr) tP-mpP.rn!Ure ~tr~ tbl- lL;;e of hyd n allerate-- oUda.tion proble~ it heccr cn1s. 4. \Vl1nt \vould be the tcn1pcrntt1re increase in n n1nn iust~ntl.)· rccciYing 100 rad " ·holebody rndintion ex1)osurc (dose)? Assume tl1at mnn is cssc11tially \\'Ster.

~

00 TABLE

Unit

Contractor (D eveloper)

16.7 Summary of SNAP Systems Energy Source

Application

Po\\·er Output

-

.-.-,, '-

SNAP-1 SNAP-lA SNAP-2

Martin-Marietta Co. Martin-Marietta Co. Atomics International

Space power Space power Space power source

SNAP-3 SNAP-4

l\1artin-Marietta Co. Atomics International

Navigational satellite power source Underwater or remote terrestrial power source

SNAP-7A SNAP-7B

l\1artin-Marietta Co. Martin-Marietta Co.

SN AP-7C SNAP-7D SNAP-8

Martin-Marietta Co. l\1artin-Marietta Co. Atomics International

SNAP-9A SNAP-lOA

Martin-Marietta Co. Atomics International

Navigational buoy power source Land-based navigational light power source Automatic weather station power Unmanned floating weather station power Space power source for experimental electrical propulsion test Navigational satellite power source Space power

125 EW•

Cc-144 Ce-144 Zirconium - hydride - moderated, N a.K - cooled reactor with mercury vapor t urbogenera.tor conversion equipment Po-210 Zirconium-hydride-moderated, boiling-watercoolcd single-loop turbogenerator conversion equipment Sr-90 Sr-90

S EW 30EW

Sr-90 Sr-90 SNAP-2 type

5EW 30EW 3CH>O EKW

Pu-238 SNAP-2 type thermoelectric converter

12EW 30TW

125EW 3EKW

~1artin -Maric tta

Co. Martin-Marietta Co. Pratt & Whitney ·

• E - electric; T - thermal.

Power source for surveyor spacecraft Space power

Cu-242

Large spaccrpowcr source

liigh-temperature, lithium metal-cooled fast reactor

Cu-242

0

"'C

--·"' c

0

:::J

3.5 EW 6- 15 TMW• 1- 2 EMW

500EW

SNAP-11 SNAP-13 SNAP-50

~

25EW Not available 150-1000 EKW

Nuclear Propulsion Engines

I

499

5. Using Fig. 16.4 and Eq. (16.24), calculate t11e linear absorption coefficient for steel.

6. Construct a graph of specific impulse versus temperature using hydrogen as a rocket propellant.

7. 'Vhat is the radius of a spl1ere containing 106 curies of Ce-144? The density of Ce is 6.77 gm per cm3 and its half-life is 285 days. REFERENCES

I. K eirn, D . J., " The USAF Nuclear Proi)ulsion Programs," Air University Quarterly &Mew, Vol. XI, Nos. 3 and 4, Fall and 'Vintcr 1959, Ilcadquarters, 1'1ax,vell Air Force Base, Alabama.

•

2. "U.S. Nuclear Propulsion Progratn," six articles reprinted from Aviation lVeek and Space Technology, New York : l\1cGra\v-Hill Book Co., 1961 . 3. "Nuclear Propulsion for Aircraft," Nucleonics, Ne\v York : McGra\v-1-Iill Book Co., January 1961, pp. 44-51. 4. Blumberg, B., "Performance of t l1e HTRE 3," Dallas, T exas: P aper presented at Al'lS l\feeting, ~farch 29, 1961.

5. Annual Report to Congress of the Atomic E nergy Commission for 1961, Washington, D.C.: U.S. Government Printing Office, J ant1ary 1962, pp. 155- 156.

6. Savage, ,V. F., "Nuclear Turbojets," Advanced Propulsion Techniques (Proceedings of a T echnical ~1 eeting Sponsored by AGARD in August 1960) Ne" ' Y ork: Pergamon Pres.5, 1961.

7. " 7bite, H. E., Moder-n College Physics, Second Edition, Princeton: D. Van Kostrand Co., Inc., 1953.

8. Glasstone, S., and Sesonke, A., Nuclear Reactor Engineering, Princeton: D . ' 'an Xostrand

Co., Inc., 1955. 9. " Present and P otential Annual Availability of I sotopic Power Fuels," Di,i.sion of Isotopes D evelopment, Atomic Energy Commission, Washington, D .C. : U.S. Go,·ernment Printing Office, April 1962. IO. Stephenson, R ., Nuclear Engineering, Second Edition, New York: ~I cGrnw-Hill Book Co., 1958. 11. Glasstone, S., and Edlund, 1\1. C., Elements of Nt1clear Reactor Theory, Princeton: D . Van Nostrand Co., Inc., 1952. 12. i\1urray, R . L ., I ntroduction to N'1tclear E1tginecring, Englewood Cliffs, X.J.: PrenticeHa ll, 1954. 13. \Veinberg, A. l\1., nnd 'Vigner, E. P., Pliysical Theory of !>lcutro1l Chain Reactors, Chicago: University of Cl1icngo Press, 1958.

14. Proceedings of the Second United 1Valio1ls I11tcr11att'-011al Co11fcrc11cc 01l the Peacrful i·su of Ato11iic Energy, Vol. 23, Geneva : United Nntions, 1958, pp. 156-16-1. 15. f\forgan, K . Z., " H cnltl1 Pl1ysics," Nuclear Engi11ccn'11g llandlxxiJ..·, Hill Book Co., 1958.

~C\V

1-ork:

~IcGrsw­

16. T he Nature of Radioactive Fallout and Its Effects 01l Jllau, Pnrt 1, J oint Congn'SS.ional Committee on Atomic Energy, \ \ 7nsl1ington, D .C.: U .S. Go\'cr11n1rnt Pri11ting Offi(.'{', 1957.

17. Selected Materials on Rap E 60 G) , ideal velocity increment from Fig. 17.31 sec ~ 50 300 is less than 30,000 ft per sec. Obviously, ·->\!>P == higher velocity increments are required ' ij 40 \!>P ::: 200 sec -0 for t he missions shown in Fig. 17.29. Very ~ 30 much higher mass ratios or specific im-0 ~ 20 pulses are thus reqwred to obtain these increments. The electric propulsion en10 gines do provide the higher specific impulses, and this is the primary reason 0 20 40 60 80 100 120 140 160 180 M· for our interest in them. I t was also Moss ratio, M ' F pointed out, however, that electric propulsion engines have very low thrust/ FIG . 17.31 Ideal vehicle velocity increment. weight ratios and thus can only be used in lo\v gravity fields. Chemical rocket engines that are limited in specific impulse to about 420 sec or nuclear engines with specific impulses approaching 1000 sec must therefore be used at least for the initial operation to earth orbit. From Fig. 17.31 it can be seen that the high specific impulse of the nuclear rocket may allo,, single-stage-to-orbit vehicles. Ho,vever, the hazard aspects of the nuclear rocket may restrict its use to second-stage applications and beyo11d, ,,·here the radiation ,,·ill not be the same type of problem. All of this leads to the conclusion that vehicles ,,·ith much higher mass ratios are required for space operations "·ith chemical rocket engines. The manner in \vhich high mass ratios arc obtained is by using a number of stages in a given vehicle. In this \vay the velocity of tl1e final stage is the sum of the ,·elocit)· increments contributed by the individual stages. Thus, for a vehicle ,,·ith n stages, the velocity of the nth stage is

-

~

-

1

( V) veh icl~ = AV 1

+ AV + ' ' ' + AV 2

n

The velocity increment for each stage is determined fron1 Eq. (17.-16) , or from Eq. (17.45).

(17.4i) ideal]~·

EXAMPLE

Neglecting the variation of gravity, determi11c the ideal total ''elocit)' incren1ent for a two-stage vehicle 'vhich has t l1c follo,vi11g characteristics:

Space Propulsion

I

535

First stage: I ," = 300 sec, propellant mass fraction = 0.85 Second stage: I ," = 400 sec, propellant mass fraction = 0.90

= g l ,P ln(mass ratio) =

(A V)nrat 11tBge

(A V)aecond 11t11.ge

= 32.2 X 400 In

(A V)totBl = 18,300

32.2 X 300 In

1

1 - .85

18,300 ft per sec

= 29,700 ft per sec

1 - .9

+ 29,700

1

= 48,000 ft per sec

The equivalent mass ratio of the over-all vehicle, \vith an average / ," = 350 sec, is thus AV

Mass ratio

= e"1 •• = 71

Another parameter \vhicl1 is often of in terest, particularly for electric propulsion engines, is the required engine operating time to achieve a given velocity increment. This parameter can be d erived from Eqs. (17.44) and (17.45) to give

At =

1-

-AV) ( e 111 ·• F

!,"

(17.481 I

Win itial

EXAMPLE

D etermine the operating time of an ion engine to accelerate a vehicle through a velocity increment of 15,000 ft per sec. The ion engine has an J," = 20,000 sec. The thrust/ ,veight ratio of the vehicle is lO- 4 • 1 At =

(

- e

-

15.000 ) 32. 2)(20.000

X 20,000 = 4.6 X 106 sec = 53.3 days 10-4

17.~

Comparison of Space-Propulsion Engines

Having considered the characteristics of chemical, nuclear, and electric rocket engines, it appears useful to conclude \vith a brief comparison of tl1e performance of these engines for several typical space n1issio11s. The n1issio11S ,,·hich ,,;11 be considered include {l ) raising of t l1e 24-hour satellite, (2) a scientific probe to Jupiter, and (3) a n1anned Mars mission. Before considering the specific n1issio11s, it is useft1l to sumn1arize the pertinent characteris tics of all of the propt1lsion syste111s. l;-igt1re 1i .32 docs this, pre_c::enting the relatio11ships amo11g the tl1rt1st / ,,·eigl1t ratio, the specific in1pulse, and the specific po,ver.

24-I-IouR

SATELLITE M1ss10N

We sl1all 110\v consider tl1e first n1issio11, \Yl1icl1 in,·ol,•es the raising of a comn1unications satellite into n 24-11ot1r orbit at a11 nltitl1dc of 22,300 n1iles. The 8500-lb satC'llit~. ir1corporati11g tl1e GO-k\v, 3000-lb SNAI> 8 n\1clenr-t \1rbogcnerator po,,·er suppl~·. ,,·ould first be lau11cl1ed to a lo'v 300-n1ilc orbit \vitl1 a cherr~ical rocket boooter like the _..\ tlas

I

536

Jet Propulsion 6000 Io n rocket 60 kw SNAP-8 po wer supply

10 0

~4000

E 4)

"'C 0 0

v

1 .&:. u

~

>~ 2000

c 0

·-

-1

0... 10

-v

Chem ical rocket

J

Arc-plasma 30 kw SNAP-8 po wer supply

4)

0

4)

g 10- 2 ...0

O>

200

300

Flight time, days

17.33 Payload in twenty-four-hour orbit versus flight time. (Front Ref. f4, courtesy Uniltd Aircraft Corporation.) F10 .

.&:.

'ii 10

100

-3

l

Centaur. From here it 'vould be raised to the 24-hour orbit in the equatorial plane. / Comparisons of the final payload that can be placed in orbit using either chemical (H 2 - 02 with I.P = 420 sec) or electric 2 5 1010-' 10rocket propulsion are given in Refs. 24106 L--~~~~-.:.1..~~...)L_~~;,i_..~~ lO' 101 10s 26. Figure 17 .33 from Ref. 24 shows one 10 5 106 10 3 Exh~ust velocity, ft / sec of these comparisons. From the figure it can be seen t hat significantly more pay10~ 10' load can be ultimately placed in orbit lsp, sec using electric propulsion. In fact, only FIG. 17 .32 Comparison of space propulsion engine characteristics. (From Ref. 8, an article by 'vith electric engines can the SNAP 8 SutUm, courtesy J ournal of Aerospace Sciences.) sources be orbited, since for both the arc plasma and ion rocket-propelled vehicles it is used as the propulsion po,ver source. The smaller 30 k'v SNAP 8 is used ,,;th the arc pla-Sma since it is more nearly optimum for this mission. Since SK AP 8 is designed for a 10,000-hr life (417 days), most of the lifetime is spent in orbit. Further, the electric engine can still be used for small trajectory corrections as required. Figure 17.34 sho,vs in bar-chart form the 'veight breakdo,vns for the three vehicles depicted in Fig. 17.33. The figure clearly sho,,·s the superiority of the ion rocket engine for this mission. It offers 503 more payload, in addition to the larger po,,·er suppl}·. Figure 17 .35 sho,vs an artist's co rice pt of the ion rocket-propelled vel1icle utilizing a SN AP-8 electric po,ver supply plus the optimun1 trajectory to raise the satellite and change inclination to the equatorial plane. JUPITER PROBE MISSION

The second mission to be considered is an u11n1a11ned probe to Jupiter. Reference 8 points out tl1at the energy requireme11ts for sucl1 a 111issio11 arc abot1t 155,000 it per sec, 'vl1icl1 '"ould involve an over-all rnass ratio of n.botrt 200. Rrfcrc11ccs 2-l n11 ..... >M I

c: 0

Ch emical rockets

·;;;ox .~

>

E

Q)

·-:::> CT

w

1 STAGE

I

Configuration

2 STAGES I ,....._3_S_TA-G-ES--.j

14 1

10

STAGES~

100

1000

m·

Mass ratio, _!_ m1

Flo. 17 .42

Comparison of space-mission capabilities for different propulsion syst~ms. (From Ref. 8, an article by Sutton, courtesy J ournal of Aerospace Sciences.) PROBLEMS

I. D erive Eqs. (17.6) and (17.7). 2. D iscuss the relative merits of t11e various electric propulsion engines - electrothermal, electrostatic, l\'I HD. 3. Discuss the areas of application of the various electric po\\'er sources - chemical, nuclear and solar bat teries, radioisoto1Je sources, nuclear and sola r turboelcct ric sources. 4. I s the photon rocket the ultimate jet propulsion system? 5. Calculate the escape velocities at t l1e surface of tl1e moon, J upiter, ~fars, and :\Iercul)·. 6. Calculate the period of tl1e moon's rotation about tl1e Earth as based on Eq. (1 7.38) and compare the answer to that given in T able 17 .3. 7. From the data in T able 17 .3 calculate tl1e energy required to esc:ipc f ron1 the solsr system and com JJare the result \vith tl1e energy reqt1ired to C'SC!lJ>C fro111 the Enrtl1. 8. For an Eartl1 ellitJticu.l orbit of eccentricity 0.201 and a1>ogcc 1000 nules, compute the perigee, and the orbital velocities at perigee and apogee. 9. D erive Eq. (17.48). 10. Determine t he 01>crating t ime of u.n nrc rllnsn1~i 37 l . 4 80G

1 294 12 4 3 1194 1147 1101 1057 1 0 14

4 27 4 .12

3 . 99 3.B5 3 . 72 3.60 3. 47

3 . 35 3 .24 3 .12 3.01 2.87 2 .74 2 . 61 2 . 49

2.38 2.27 2.16 2 . 06

x

102

522 498 4 75 45 2 431 411

p

I 033 1032 1()20 1 025 1 02 1 10 1 7 1 0 13 1 009 100!:> 1 00 1

1 . 04 1 . 01 . 97 . 93 . 89 . 85 . 81 . 78 .74 . 70 . 67 . 64

/ ft l)

3 . 06 3.2 1

3.36 3 . 52 3 . 69 3.87 4 . 06 4 . 26 4 .4 7

"x

(ft 2

10

1

'sec)

263 273

283 2 92 303 314 325

948

337

948

~o

948 948

363 376

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(ft ~>

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t ude f~tl

4 .7

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481

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I 64 I . 09 I . '>4 1 . 99 2 . 04 2 . 10 2 . 15 2 .2 1 2 . 27

74 83 93 1 02 1I I 119 1 28 1 36 145 154 1 62 171 180

1 . 87

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100 000

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600. 3 573 . 3 5 47 . 2 522 . l 498 . 0 4 74 . 7 4 52. 4 4 3 1.2 411 . 0

2 . 779 2 . 662 2.549 2 . 441 2 . 350 2 . 267 2 . 186 2 . 1 07 2 . 031 1 . 957 1 . 886 1 . 8 25 1 . 766 1 . 709 1 . 653 1 . 598 1 . 544 1 . 493 1 . 442 1 . 393 1 .3 4 5 1 . 298 1 . 253 1 . 209 1 . 166 1.124 1 . 084 1 . 044 J .006 . 9686 . 9324 . 8920 . 8523 . 8141 .7773 . 7418 . 7076 .6749 . 6436 .6 139 p

T

(oR )

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378 . 3 378 . 3 378 . 3

Pit X 101

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(lb / ftl)

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1 . 884 l . 796 1 . 713 1 . 634 1 . 558 1 . 486 1 . 41 7 1 .352 1 . 289 1 . 230 J . 1 73

391 . 7 373. 3 3.55.8 339 . 1 323 . 2 308 . 0

5 . 854 5 .583 5 . 325 5 . 078 4 . 843 4 . 619 4 . 4 05 4 .201 4 . 007 3 . 8 22 3 . 645 3 . 4 76 3 . 315 3. 162 3 . 01 5

1 . 118

1 . 06 7 1 . 017 . 970 . 925

. 8il2 . 842 . 8 03 . 730

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. 604 .549 . 500 .4 5 4 . 413 . 37G

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279 . 8 266. 7 25 4 . 1 24 2 .2 230. 8 220 . 0 209. 7 199. 9 190. 5 181 . 5 173 . 0 164 . 9 149 . 8 1 36. 1 1 23 . 6 11 2 . 3 1 02 . 0 92 . 6 84 . l 76. 4 69. 4 63 . 1 57 . 3 52 . 0 47 . 3 4 3 .0 39 . 1 35 . 6 32 . 5 29. 6 27 . 1 24 . 7 22 6

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. 80'l2

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• 7~28 • 74115 • 7401 • 7338 . 7714

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. 8000 • 79(,6

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• 0107 • OOll4 • 0061 • 0037 . 0013

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. fl& • G7 • 68 • '10

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I. 97647 I. 97772 1.9Nll7 I. ll821G I. 118-441

60. 713 60. 002 &l. OllO

. 1577 . IM4 . !Ml . 1&38 . 1&26

4. 1157 4. 702 4. 747 4. 838

1. U&Ml l . ll8879 I. 90097 1. 99313 1. 99627

51. MO 51. 83& 52.020 62. 203 62. 38tl

18. 82 18. 711 18. GO 18. 63 18. &7

4. 884 4. 030 4. 977 &. 025 6. 073

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18. &l 18. 4& 18. 39 18. 33 18. ZT

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• 66lg -t • 6546 -J • 5474 -t

. 61g5 -J • 6128 -t . 6062

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• 7376 - I . 7316 -I . 7265 - 1 . 7194 - · • 7136 - 1

10. 72 10. 81 10, gl 11. 01 11. 11

2. 138C» 2. 131136 2. 14-0&2 2. 14188 2. 14312

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14. 48 14. 44 14. 40 14. 37 14. 33

• 2G38 - · . 2613 - 1 . 258g -· . ~ -I . 2540 -1

• 23J6 • 2327 • 231g • 2310 • 2301

3. 925 3. 1136 3. 945 3. 966 3. 966

• 7076 -I • 7017 - I • 6951> -I • 6902 -· . 6845 -I

11. 21 11. 31 11. 41 11. 61 11. 61

2. 144311 2. 14560 2. 14682 2. 14804 2. 141126

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14. 30 14. 26 14. 22 14. lg 14. 15

• 2516 - 1 . 24113 -1 • 2470 -1 • 2447 -·

• 22113 . 2284 . 2276 . 2267 . 22.fS

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11. 71 11.82 11. 92 12.03 12. 14

2.16046 2. 1611'16 2.16286 2.15404 2.1M22

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4. 028 4. 038 4. 048 4. 059 4. 0llG

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12. 24 12.36 12. 46 12. 67 12. 68

4. 079 4. 0llO 4. 100 4. 110 4. 120

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• 6364 . 113()2

-1

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-1 -I - I

. 4331 . 4329 . 4326

4. 125 4. 141 4. 167 4. 173 4. 189

. 1330 • l31g . 1308 . 1297

. 4633 -1 . 41110 -1 . 4581t -I . 45116 -· . 4544 -1

67. 336 67. 462 67. 683

14. 12 14. 08 14. 05 14. 01 13. 118

. 4324 . 4321 . 4319 . 4316 • 4314

10. 45 19. 54 19. 64 111. 73 lg. 83

4. 624 4. 630 4. 635 4. 640 4, 645

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13. G4 13. 01 13. 88 13. 84 13. 81

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19. 113 2o. 02 2o.12 20. 22 20. 32

4.660 4. &M .{. 660 4. 665 4. 670

4. 285 4. 301 4.318 4. 334 4. 360

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12. 70 12. 00 13. 02 13. 13 13. 26

2. 16216 2. 16320 2. 16442 2. 16M4 2. 111666

68. 333 68. 456 68. 678 68. 700 118. 821

13. 77 13. 74 13. 71 13. 67 13. 64

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. 4207 • 4205 . 4292 . 4200

20. 41 20. 51 20. 61 20. 71 20. 81

4.. 675 4. 680 4. 685 4. 1111() 4. 694

4. 367 4. 383 4. 399 4. 416 4. 432

2. 16776 2. 16880 2. 16006 2. 17106 2. 17214

68. 042 60. 183 60. 302 60. 422

13. 61 13. 68 13. M 13. 61 13. 48

. 42£8 . 4286 • 4283 • 4281 . 42711

20. 91 21. 01 21. 11 21. 20 21. 30

4 699 4. 704 4. 700 4. 713 4. 718

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13. 46 13. 42 13. 38 13. 311 13. 32

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• 5902 -· • 6854 - 1

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4. 131 4. 141 4. 161 4. 1112 4. 172

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• 2180 • 2180 • 2140 • 2120 • 2110

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65.253 65. 38C! 65. 620 116. 652 M . 7&5

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4. 532 4. 540 4. 566

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4. 233 4. 244 4. 2M 4. 2Gf 4. 775

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• 206.1 . 2045 • 2038 . 2030 • 2023

4. 285 4. 205 4. 306 4. 316 4. 326

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16. 21 16. 34 15. 47 16. 61 16. 74

. 1825 -1 • lSOlt -1 . 1792 _, . 177«1 -1 . 1761 -1

. 2016 . 111114 • 11187

•. 336 4. 3411 4. 367 4. 367 4. 377

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16.87 16. 01 16. 16 16. 28 16. 42

2.188708 2. 18116117 2. lll0681 2. 11116.W 2. 11126.12

71. 274 71. 386 71.

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• 1287 , 1276

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. 3673 • 3646 • 3518 • 3491 . 3464

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2. 228424 2. 22Cn06 2. 22111184 2. 230768 2. 231628

76. 91111 76. ()611 76. 1112 76.258 76. 36.1

11. 78 11. 76 11. 73 11. 70 11. 118

. 41117 . 41116 • 4164 • 41113 . 41111

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4. 968 4. 9811 4. m 4. 97" '4, 980

6. t'i01 6. 11216 5. 646 6. 6116 6. 1184

• 18115 . 11181> . 1683 • 11178 • 11172

4. 848 4. 868 4..868 4. 879 4..889

. 3437 -I • 3411 - 1 • 3386 -1 • 33MI -1 . 3333 -1

24. 02 2". 21 24. 41 24. tiO 24. 80

2.23m4 2. 233066 2. ZJ3816 2. 234670 2. 236.121

76. 44g 76. 544 76. 638 76. 732 76.82e

11. 611 11.113 11. 61 11. 68 11. 60

• 41ti0 . 4168 • 4167 . 4165 . 4164

28. 42 28. 64 28. M 28. 77 28. 88

4. 983 4. 1187 4. 900 4. 993 4. 997

6. 703 5. 723 5. 742 fl. 761 6. 781

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4.1199 4. 900 4. gig 4. 930 f. 940

. 3308 • 3282 .3267 . 3233

2. 23ll068 2. 2311Rll 2. 237661 2. 238287 2. 23QO:l)

111. m 77. 013 77. 106 77. 199 77. 291

11. 64 11. 111 11. 49 11. 47 11. 44

. 4162 . 41151 . 4149 . 4148 . 4147

29. 00 :29. 12 :29. 23

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25. 00 25. Ill 26. 40 26.61 26.81

211. 47

6. 000 15. 003 5.007 5.010 5.013

6.800 6.Q) 5.839 5.850 5. 878

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4. 9bO 4. 911() 4.970 f.981 4.991

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26.02 26. 22 26. 43 26. 64 26.86

2. 23974g 2. 240474 2. 2411g15 2. 241g14 2. 242G28

77. 3815 77. 477 77. 608 77. 6tiO 77. 761

11. f2 11. 40 11. 38 11. 36 11. 33

• 41415 • 4144 • 4142 • 4141 • 4140

211. Ml 211. 70 29. 82 29. IM 30. Oii

5. 0111 5.lm 5. 023 5.028 5. ()'Jg

6. 898 15. 918 6. 937 5. 957 5.977

. 6938

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-1 -1

• 11139 • 11134 • 11128 • 11123 . 11118

• 6848 _, • 6803 _, . 6759 -1

• IOf.4 -I • 1005 -· • 1027 -1 . 1019 - · • 1010 -I

• Hl12 . 1t'i07 • HI02 . 16417 • 16G1

. 3003 - I .. 2991 2ll80 -_,· • 2988 _, • 2957 -l

5.001 5. 011 6.0'.ll 5. 032 6.00

• 30llll . 3042 . 3019 • 2996

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77. 841 77. 1131 78. 021 78. 111 78. llll

11. 31 11. 211 11. 211 11. " 11. 22

• 4138 • 4137 • 41311 • 4134 . 4133

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. lla9 ... . 2343 ...

. 2&55 ... . 24M ...

• 2.5e7 - ·

. 2•1e - • • 2.JllO -·

-·

• ZJOS - ·

• 2231

• 2157 -·

-·

. l057 - ·

.. ~ 11167 ... • 1898 - · . 1838 - · . 1783 ... . 1730 ... . 1879 - 1

-·

GM. 7 1181. 1 1009 1037 1oee

. 2134 . 11188 . 1854 . 1730 • 1817

... ... ... ... ...

• ISM

. 1540 . 1493 . 1457 • 1418

1085 ll)C 1154 118• 1215

. 1512 ... . 1415 ... . 1321 ~ . l)C.3 .... • IUlll -t

• ll!ll . 13'5 • 1310 .1278 • 1244

. 1~

l.1195 6.M

1245 1277 131S 1341 1373

... . 1030 .... • 9882 - J • 9113 _,

• 1214 . 1184 . 1155 . 1127 . 1101

6.M 5. 91111 5.llM .\.99$ 6.llM

10 1439 1473 1607 1541

. 8092 _, . 71132 -J . 7~ _, • 8804 -r . M.31 -l

• 1075

-I

. l~

...

. 1028 . 1003

-I -I

6.llM 6.llM l.987 5.987 5.187

1578 1411 11147 Ul83 1719

• 8052

.asss ...

6.WS 5.M

8930 9035 9241

m.1

5. 994 5. 1194 5. 9111 6.IKI•

6.~

8829

700. 9 724. 5 7'8. 4

-P•,

. us:u

7467

8429

810. 7 llJ2. 7 GM. I m .8

Pa,

• 22QJ ...

6.~

7845 8037 8232

659. I

8215 M7. 9 873.8 llOO. l 9211. 7

72!U

7G54

-

5.993 5.9QJ 5. 1193 5. 9114 5. 9111

5.~

e.1811

T1 T1

a. 997

•858.5 _,

•5755

. MSO .51~

-l -l -1

_,

.49M _,

-... '(I>-0

0

-c

--·0 c

l it

:l

... ... .... ...

.... ... .... ... -t -I

.... .... -· -·

• 9905 ...

. 11371 . 9175

-4 -4

. 8978 • 8790

-4

-4

•

. 41142

-J

.~

-l -l

.1427 . S2M . 8087 . 1923 ...

. 4405 _,

a.WT

1758 1793 1831

6.997 6.987

1907

• 4183 . 3974 . 3m

-1

6.987

196.\

.sm ...

6.997

-1

-

lMll

. ne.s

-4 -4 -4 -4

-4

~

Appendix C

I

NOTATIONS FOR TABLES I AND 11

Mor M, local Macl1 number or 1f ach nt1rnber upstream of a normal sl1oclc \vnvo

p p, p Pt

T T,

f3 q -p, A

A.

v

ratio of static pressure to total pressure ratio of static density to total densit.}' ratio of static temperature to total temperature

.JM2-I ratio of dj•namic pressure,

to total pressure

ratio of local cross-sectional area of an iscntropic stream tube to cross-sectional area at tl1c point. where 1'1= I ratio of ]ocal speed to speed of sound nt the point where M = l Prandtl-Meyer angle (angle through \\·hich a supersonic stream is turned to expand from -~= I to M>I), deg

1-f ncl1 angle, sin- 1 ~fncl1

-

~ p V 2,

~'

cl cg

number do,vnstrcnm of a normal shock ..,,-a.,c

static pressure ratio across a normal ·shock ,,-a ,.c static densi t.}· ratio across a, normnl sl1ock ,,-a \C s tatic tempcratt1rc rntio across a normal s l1ock wu,·c total pressure rn tio o.cross a normnl sl1ock ,,.a ,.l. ro.tio of sto.tic prosst1ro t1pstrca.m of a L1ormal shock \VO.Ve to total pressure clo,v11strcan1

599

-

Appendices D and E Tabulated Values of the lsentropic Pressure Ratio Function Xe and the lsentropic Expansion Ratio Function Xe

Appendix D

lsentropic Press ure Ratio Function X e -y-1

Xe = r

'Y

-

1 for "'( = 1.40

For use in cold sections of jet propulsion engines: diffusers and compressors. Appendix E

lsentropic Expansion Ratio Function Xe -y-1

Xe = r

'Y

-

1 for "'( = 1.33

For use in hot sections of jet propulsion engines: burners, turbines, afterburners, and exhaust nozzles.

600

Appendix D Values of x. for

y=

I

601

1.40, where

y- 1

X e= r

Y

-

1

0

1

2

3

4

5

6

7

8

9

0.1

000 276 535 778 004

028 303 560 802 031

057 329 585 826 054

085 355 609 849 077

113 381 634 872 098

140 407 658 895 120

168 433 683 918 142

195 459 706 941 164

222 484 731 964 185

249 510 755 987 207

0.2

228 437 637 829 013

250 458 657 847 031

271 478 676 866 049

292 498 695 88/l 067

313 518 715 903 084

334 538 734 921 102

355 658 763 940 120

376 578 772 958 138

396 598 791 976 155

417 617 810 995 173

2.0 2.1 2.2 2.3 2.4

190 361 527 687 842

208 378 543 703 857

225 395 559 718 872

242 411 575 734 888

260 428 591 750 903

276 445 607 765 918

294 461 623 781 933

310 478 639 796 948

327 494 655 811 963

34-4

2.5 2.6 2.7 2.8 2.9

993 139 282 420 555

001• 153 296 434 566

022• 168 309 448 582

037• 182 324 461 595

052• 197 337 475 609

066• 211 351 488 622

081 • 225 365 502 635

096• 239 379 515 648

110• 253 392 529 662

688 816 942 065 186

700 829 955 078 198

714 841 967 080 210

726 854 980 102 221

739 867 992 114 233

752 878 004• 126 248

765 892 016• 138 257

775 029• 150 269

791 917 041 • 162 280

3.5 3.6 3.7 3.8 3.9

304 419 533 644 753

315 431 544 655 764

323 442 555 666 775

338 453 566 677 785

350 465 577 688 796

362 476 588 698 807

373 487 600 710 817

385 499 611 720 828

396 510 622 731 839

408 522 633

4.0 4.1 4.2 4.3 4.4

860 965 069 170 270

871 976 079 180 280

881 986 089 190 290

892 997 099 200 300

902 001• 110 210 310

913 011• 120 221 320

92-1 021 • 130 230 329

93-1 03s• 140 240 339

944

150 250

955 05g• 160 260

349

35 9

369 465 561 655 747

378 475 570 664 766

388 485 679 673 766

398 494 589 682 775

407 504 599 692 784

417 513 608 701 793

427 523 617 710 802

436

446

"56

532 626 719 811

54~

551

636 _ -, 09 820

645

r

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

3.0 3.1 3.2 3.3 3.4

4.5 4.6 4.7 4.8 4.9

0.0

0.3

0.4

0.5

905

(»8 •

510 671 827 978 125• 268 406

542 674 804 930 052• 174 29.2

742 849

73S 829

-

-

I

602

Jet Propulsion 0

1

2

3

4

5

6

7

8

9

838 928 017 104 190

847 937 026 113 199

856 946 036 121 208

865 955 043 130 216

874 964 052 \39 225

883 973 060 147 233

892 981 069 156 242

901 990 078 165 250

910 999 087 173 259

919 008• 095 182 267

5.6 5.6 6.7 5.8 5.9

275 360 442 524 605

284 369 450 533 613

292 376 459 540 621

301 384 467 549 629

309 393 475 557 637

317 401 484 565 645

326 409 492 573 654

334 418 600 581 662

343 426 608 589 669

351 434 616 597 677

6.0 6.1 6.2 6.3 6.4

685 764 842 919 996

693 772 850 927 003•

701 780 858 935 011 •

709 788 865 942 018•

717 795 873 950 026•

725 803 881 957 034•

733 811 889 965 041 •

740 819 896 973 049•

748 827 904 980 056•

766 835 912 988 064•

071 146 219 293 365

079 153 227 300 372

086 160 234 307 379

094 168 242 314 387

101 176 249 322 394

108 183 256 329 401

116 190 264 336 408

124 198 271 343 415

131 205 278 350 422

138 212 285 358 429

7.0 7.1 7.2 7.3 7.4

436 507 577 647 715

443 514 584 653 722

450 522 591 660 729

458 528 598 668 736

465 535 605 674 743

472 542 612 681 750

479 549 619 688 756

486 557 626 695 763

493 564 633 702 770

500 570 640 708 777

7.5 7.6 7.7 7.8 7.9

784 851 917 984 050

790 858 924 990 056

797 865 931 997 062

804 871 938 004• 069

811 878 945 010• 075

817 884 951 011 • 082

824 891 958 023• 088

831 898 964 030• 095

837 904 971 036• 101

844 911 977 042• 108

8.0 8.1 8.2 8.3 8.4

115 179 243 306 369

121 185 249 312 375

127 192 255 318 381

134 198 262 325 387

140 205 268 331 394

147 211 274 338 400

153 217 280 344 406

160 223 287 350 412

166 230 293 356 419

172 237 300 362 425

8.5 8.6 8.7 8.8 8.9

431 493 554 614 675

437 499 560 620 680

443 505 566 627 687

450 511 572 633 693

456 517 578 638 699

462 523 584 644 705

468 529 590 650 711

474 536 697 657 717

480 542 603 663 723

486

734 793 853 911 969

740 800 858 917 974

747 805 864 923 980

752 811 870 928 986

758 817 876 934 992

764 823 882 940 998

770 829 887 946 003•

776 835 893 952 009 •

782 8-ll 899 957 01 5•

787 847

r

5.0 5.1 6.2 6.3 6.4

6.5 6.6 6.7 6.8 6.9

9.0 9.1 9.2 9.3 9.4

0.6 0.6

0.7

0.8

0.8

-

5t8 608 669 728

905

963 020•

Appendix D

0

r

1

2

9.5 9.6 9.7 9.8 9.9

0.9

10.0 10. l 10.2 10.3 10.4

307 362 416 470 525

10.5 10.6 10. 7 10.8 10.9 11.0 11 .1 11.2 11.3 11 .4 11.5 11.6 11. 7 11.8 11.9 12.0

026 083 140 196 252

578 631 684 736 788

1.0

840 891 943 993 043 094 143 193 242 291 340

032 089 145 201 257 313 367 422 476 530 584 636 689 741 793 845 897 947 998 049 098 148 198 247 295 344

037 095 161 207 263 318 373 427 482 535 589 642 695 746 799

•

6

7

8

9

043 100 157 213 268

049 106 162 218 274

055 112 168 224 279

060 11 7 173 230 285

066 123 179 235 290

072 129 185 240 296

077 134 190 246 301

324 379 433 487 541

329 384 438 492 546

334 389 444 498 551

340 395 449 503 556

345 400 455 508 562

351 406 460 514 567

3.57 411 465 519 673

600 652 705 757 809

605 658 710 762 815

610 663 715 767 820

615 668 720 773 825

620 673 725 778 830

625 678 731 783 83.5

865 917 968 019• 069

871 922 973 024• 073

876 927 978 029• 079

881 932 983 033• 084

886 937 ...88 038• 089

594 647 700 752 804

850 902 953 003• 054

855 907 958 059

860 912 963 013• 064

103 153 203 251 300

109 158 207 257 305

114 163 212 261 310

118 168 218 267 315

123 173 222 271 320

129 178 227 276 325

354

359

363

369

373

349

603

5

3

-

-

I

oos•

I I

I

134 183 232 281 330 378

I 138 188 237 286 334

I 383

604

Jet Propulsion

I

Values of X, for y = 1.33, where y- 1

x, =,. ,, -

,.

1

0

1

2

3

4

5

6

7

8

9

000 239 463 673 871

025 262 484 693 890

049 285 506 713 909

074 308 527 733 928

098 330 648 753 947

122 353 670 773 966

146 375 590 793 985

169 397 611 813 003•

193 419 632 832 022•

216 441

058 237 407 670 726

077 254 424 586 744

095 272 440 602 757

113 289 457 618 772

131 306 473 634 787

149 323 489 649 803

167 340 506 665 817

184 357 522 680 832

202 374 538 695 847

66-i

877 021 161 296 426

891 035 174 309 439

906 049 188 322 452

921 064 202 335 465

935 078 215 348 477

949 092 229 361 490

964 105 242 375 503

978 119 256 387 515

993 133 269 401 528

001• 147 282 413 540

553 675 795 911 024

565 687 807 922 035

578 700 818 934 046

590 711 830 945 057

602 724 841 956 068

614 736 853 967 079

627 747 865 979 090

639 759 876 990 101

651 771 888 001 • 112

663 783 899 013• 123

3.0 3.1 3.2 3.3 3.4

134 241 346 448 548

144 251 356 458 558

155 262 366 468 568

166 273 376 478 578

177 283 387 488 587

188 294 397 498 597

198 304 407 508 607

209 314 418 518 617

220 325 428 528 626

230 335 438 538 636

3.5 3.6 3.7 3.8 3.9

646 741 835 927 017

655 751 844 936 026

665 760 854 945 035

674 770 863 954 044

684 779 872 963 053

694 789 881 972 061

704 798 891 981 070

713 808 900 990 079

723 817 909 999 088

732 826 918

4.0 4.1 4.2 4.3 4.4

105 192 277 361 443

114 201 286 369 451

123 209 294 377 459

132 218 302 386 467

140 226 311 394 475

149 235 319 402 483

158 243 328 410 491

166 252 336 418 500

175 260 344 426 508

183 269

4.5 4.6 4.7 4.8 4.9

523 603 681 758 834

532 611 689 766 841

540 619 697 774 849

548 627 705 781 856

556 635 712 789 864

564 642 720 796 871

571 650 728 804 879

579 658 735 811 886

587 666 743 819 894

1.0 1.1 1.2 1.3 1.4

o.o

1.5 1.6 1.7 1.8 1.9

0.1

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

0.2

0.3

0.4

652

851 040• 219 390 711 862

oos• 097

353

435 516 595 673 751 82 6

901

Appendix E

,.

0

5.0 5.1 5.2 5.3 5.4

0.4 0.5

265 334 401 468 533

5.5 5.6 5.7 5.8 5.9

598 663 726 788 850

6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

908 982 054 125 196

0.6

7.0 7.1 7.2 7.3 7.4

•

7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

0.7

9.0 9.1 9.2 9.3 9.4

0.7

911 972 031 090 149

4

I

605

5

6

7

8

9

976 047 119 189 258

l

2

3

916 989 061 133 203

923 996 068 139 210

931 004• 076 147 217

938 011• 083 154 224

946 018• 090 161 231

952 025• 097 168 238

960 033 104 175 245

967 040 111 182 252

286 354 421 488 553

293 361 428 494 559

299 368 434 501 566

306 374 441 507 573

313 381 448 512 579

320 387 454 520 585

327 394 461 527 592

611 675 739 800 863

618 681 745 807 868

624 688 751 813 874

630 694 757 819 881

637 700 763 825 887

644 707 770 832 894

650 713 776 838 899

656 719 782 844 905

923 983 043 102 160

929 989 049 108 166

935 995 055 114 172

941 002• 061 119 178

947 008• 067 125 184

954 014• 073 131 189

960 019• 079 137 195

965 026• 084 143 200

229 286 343 399 453

235 292 348 404 459

240 297 354 409 464

247 303 359 415 470

252 309 365 420 475

258

315 370 426 481

272 340 408 474 540 605 669 732 794 856 918 978 037 096 155

279 347 41 5 481 546

•

•

•

206 263 320 376 431

212 269 326 382 437

218 275 331 387 442

224 281 337 393 448

486 540 594 648 700

492 546 600 653 705

498 551 605 658 710

503 557 610 663 716

508 562 615 669 721

514 567 621 674 726

519 573 626 679 731

524 579 631 684 737

530 l'\84 637 690 742

535 589 643 695 747

773 824 876 926 976

779 830 881 931 981

784 835 886 936 986

789 840 891 941 991

704

896 946 996

799 850 901 951 001•

041 090 139 187 235

046 095 1-H 192 240

051 100 148 196 244 292

752 804 855 906 956

758 809 860 911 961

763 814 865 916 966

768 819 870 921 971

006 056 104 153 201

011 060 110 158 206

016 066 114 163 211

021 070 119 168 216

026 075 124 172 221

031 080 129 178 225

036

249 297 343 390 436

254 301 348 395 441

259 306 353 399 445

264 310 358 404 449

268 315 362 408 455

274 320 367 413 459

278 325 372 418

284 329 376

287 334 381

422

427

386 -132

464

468

473

4 78

0~5

134 183 230

& J5

339

' I !

I

Index

I

I I I I

• •

I

•

Accelerating grid, 507 Acoustic velocity, 6, 53 Additive drag, 38, 117, 389 Adiabatic process, 17 Aerospace plane, 403 Aerothermodynamics, 44 Aft fan, 32 Afterburner, 302 augmented thrust ratio, 308 combustion temperature rise, 310, 311 cooling, 304

level flight performance of, 294 specific range of, 298 Air-turbo-rocket engine, 395 Alpha particle, 472 Aphelion, 531 Apogee, 521> Arc plasma rocket engine, 503 Plasmadyne l kw, 505 General E lectric 30 kw, 507 Area ratio, of flow, 56, 62, 111, 131, 431 Atmosphere, standard data of, 553 Atomic number (Z) , 470 Avogadro's number, 470

diffuser, 303 Bame holders, 304 ignition, 304 hot streak, 304 spark, 304 nozzle, 304 spray bars, 304 .Afterburning, 302 turbofan engine performance, 312 turbojet engine performance, 311 turbojet engines: J57, 305 J58, 306, 376 J75, 306 J79, 303, 306 J85, 306 y J93, 306, 377 Avon, 306 Gyron Jr., 306 Olympus, 306 Orpheus, 306 A.I.A. pressure recovery, 110 Air, standard atmosphere, 553 Air-breathing boos ter, 377 orbital, 401 turbofan, 381 turbojet, 377 Air-breathing engines, 27 comparison to rocket engines, 407 for high-flight Mach numbers, 374. See also Engines. Air bypass, 117 Air composition, 220 Air spillage, 113 Aircraft nuclear propulsion (ANP) , 469, 485 Air-cycle efficienc}', 344 Airplane, engine combination charo.cteristics, 293 flight forces on, 115, 262

Bernoulli equation, 9 Bet.a particle, 472 Blade, air-cooled, 247 attachment, 181 circulation, 168, 191 compressor, 170, 182 film-cooled, 247 liquid-cooled, 247 rotor, 184 separation, 169 shape, effect, 170 solidity, 192 stator, 184 sweat-cooled, 247 turbine, 230 Bleed-bum, 323 Bleed-off cycle, 323 B omb propulsion system (ORIO:'\), 495 Booster engine, 412 Boundary-la}•er effect, 120 Buckingham's pi theorem, 276 Buckling, geometric, 477 material, 477 Buckling factor, 477 Burner inlet parameter, 217 Burning rate, 454 Burning-rote exponent, 454 Buzz, inlet, 121 Bypass cycle, 379 Bypass engine, 262 Bypass ratio, 32, 264 Capture area, 112 Carnot cycle, 20 Centrifugal compressor, 165

607

'

4

-·

608

I

Index

Centrifugal impeller, 165 Chamber pressure, 427, 445 Characteristic half-life, 472 length, 447, 449 velocity, 433 Charged colloid rocket engine, 513 Chemical batteries, 518 Chemical equilibrium, 440 Chemical rocket engine, 407 Coefficient, additive drag, 117 blade-integrated lift, 352 contraction, 136 Fanning friction, 10, 104 heat transfer, 453 nozzle flow, 128 nozzle velocit)', 127 pov.·er, 352 pressure, 173 of reactivit)•, 491 slip, 169 temperature sensitivit)', 458 thrust, 352, 387, 389, 429 weight flow, 427 Combination po"'cr plants, 394 air-turbo-rocket, 395 ramjet-rocket, 395 turbojet-ramjet (also turbo-ramjet), 394 turbojet-rocket, 395 Combustion chamber, 209 annular type, 209 can type, 209, 210 design considerations, 210 efficiency, 216 flame speeds, 215 flame temperatures, 219 fuel-air ratio equation, 217 heat balance, 219 inner liner, 212 lean die-out, 226 pressure loss, 215 primary zone, 211 process on h-S plane, 216 rich blO\VOUt, 226 rich limit, 211 stoichiometric limit, 211 three T's of, 210 tubular, 209 vaporizer, 213 Combustion index, 454 products of, 219 sta)' time, 448 temperature rise, 217, 3 10, 3 11 thermodynamics, 437 Compressibility factor, 104 Compressible flo,v, 44 Compression, ,,·et, 322 Compressor, nirfoil analysis, 189, 192 anal,ysis, 189 a.~al fl0\\'1 181 blade clement analysis, 189, 192 blade shape effect, 170 bleed valves, 202

ccntrifugal, 165 doublardment, 512 oscillating electron, 512, 514 surface catalyst, 509, 512 Ionization chamber, 507, 509 I rrotational flow 1 106 motion, 168 Isentropic, 17, 441 551 60, 63 Isobaric, 16 Isothermal, 16 conditions, 94 Isotope, 470 Isovolumic, 16 J et propulsion, principle of1 27 J ct noise, 140 J ets, nozzle discharge, 136 l{eppler's laws, 529 J{inetic energ}', 4 Kinetic theory of gases, 5 Law, conservation of energy, 13 conservation of mass, 12 first law of thermody namics, 16 f rec vortex, 172 J{utta-Joukowsky, 191 Nev•ton's second, 8 perfect gas, 10 Saint-Robert's, 454 second of thermod)rnamics, 20 universal gravitation, 525 La"-s of engineering, 9 I{eppler's, 520 Lean die out, 226 Limiting expansion angle, 93 Limits of combustion, 211 Linear absorption coefficient, 482 Liquid air cycle engine (Li\CE), 40-1 Liquid-propellant rocket engi nes, 408 Liquid hydrogen, 415 Lorin engine, 28, 382 See also Ramjet. l\1nch angle, 92 line, 92 number, 6, 54, 68 \Yave, 92 l\1ngneto-h)·drodynan1ic (:\III D) rocket engine, 514 1\Ingneto-plnsn1n rocket engine, 514 :rvinss, 1 l\Inss-c n e r~· relntion, 4il l\1nss-flo,,· ratio, 111 l\1nss nun1brr (1\ ), -liO l\1ntching, 249 1\11-ID pO\Yl'r grnrrator, 522 closed cyrlr, 522 open cycle, 522

61 1

612

I

Index

Missile, ballistic (ICBM), 412, 424 Mixture ratio, 210 Molecular \\·eight, 11 Momentum flux, 36 l\1omentum pressure loss, 309 Monopropellant, 408 Motion, laws of, 7, 22 Motor, rocket, see R ocket engine Multiplication factor, 473 effective, 475 Neutron, 470 density, 476 flux, 476 leakage, 475 radiation, 478 Newton's lav•s of motion, 7, 22 Noise suppression, 141 Non leakage probabilit.Y, 4 77 Normal shock, see Shock Nozzle, afterburner, 304 area, 62, 133, 431 convergent, 48, 136 convergent-divergent, 48, 55, 136 correction factor, 428 DeLaval, 48, 55, 136 discharge coefficient, 427 efficiency, 127 ejector, 139 equilibrium expansion, 400, 445 exit velocity, 49, 127 expansion ratio, 429, 431 flow coefficient, 128 flow with friction, 126 friction parameters, 133 frozen expansion, 400, 445 operation, subcritical, 136 operation, supercritical, 136 overexpanded, 140 plug, 139 ramjet application, 390 recombination, 399, 444 separation, 138 thrust equations, 141 underexpanded, 140 velocity coefficient, 127, 428 Nuclear, nuxiliar:r pov;er, 469, 496 chain reaction, 473 cross section, 474 energ)', 470 fission, 471 fusion, 471 propulsion, 469 radiation effects, 478 ramjet engine, 469, 487 ramjet reactor, 487 Tory, 487 T ory IIA, 489 Tor)' IIC, 490 reactor, 474 epithermnl, 474 fast, 474 heterogeneous, 488

homogeneous, 488 intermediate, 474 tl1ermal, 4 7 4 reactor components, 474 rocket engine, 469, 491 rocket reactor, 491 gas core, 495 l{iwi, 492 NERVA, 494 RIFT, 494 turbogenerator, 519, 535 turboj et engine, 484 direct C)•cle, 485 indirect cycle, 485-86 Nuclide, 470 Oblique shock, see Shock One and one-half stage vehicle, 412 One-dimensional flow, 44 Operating line, 177 Orbit, circular, 527 elliptic, 529 transfer, 531 Orbital period, 528 v elocity, 527, 530 Oscillating electron ion engine, 512, 514 Over-all efficiency, 40, 299, 345, 399 Overexpansion losses, 138 Oxidizer, 446 Parameters, combined engine-airplane, 295 corrected engine, 275 Fanno line, 78 isentropic, 55, 58, 60, 63 nozzle friction, 133 Rayleigh line, 71 thrust, 145 Per cent reaction, 188, 242 P erfect gas equation, 67 Performance curves of: airplane in level flight, 294 compressor, 176, 200, 382 gas turbine po\\·er plant, 345 propellant combinations, 447 propeller, 354 ramjet, 393 tt1rbojet, 284 turbojet " ·ith afterburner, 317 turboprop, 350 Perigee, 529 Perihelion, 531 Photon rocket, 524 Pipe length, m a..ximum, 78 Plu.smn, 503 Plnsmn nccl'l