Mechanics and Thermodynamics of Propulsion (1st Edition)

1

PHI LIP G. H ILL Deparlmem of Mechanical Engineering Massachusetts [nstitute a[Technology

CARL R. PETERSON Research and Development , Ingersoll-Rand Company Bedminster, New Jersey

MECHANICS AND THERMODYNAMICS OF PROPULSION

Menlo Park, California

ADr;>ISON.WESLEY PUBLISHING COMPANY Reading, Massachusetts Londoll' Amsterdam . 0011 Mills, Ontario' Sydney

This boo k is in the

ADDISON-WESLEY SERIES IN AEROSPACE SCIENCE HOWARD

W.

EMMONS,

S. S.

PENNER,

Consultillg Editors

Preface Th is boo k is based on an introductory co urse in propulsion which has been given for several years to students in aerona utical and mechanica l engineering at Massachuse tts Institute of Technology. It deals mainly with the propulsion of Th ird print ing, November 1970 Copyright ® 1965 Philippines Copyrighf 1965 ADDISON-WESLEY PUBLISHING COMPANY, INC.

Printed ill the United 5{(aes oj America ALL RIGHTS RESERVED. TH IS BOOK, OR PARTS THERWF' , MAY NOT ijE REPRODUCED IN ANY FOR""I

WITHOUT WRITTEN PERMISS ION OF THE PUBLISHERS

Library oj Congress Catalog Card No. 65-10408 ISBN 0-201-02838-7 RST-MA-898765 43

aircraft and space vehicles. The objective of the book is to show how basic pri nciples may be used to predict the behavior of propulsion devices. There are, of course, many methods of propulsion, but the underlying physical principles are relatively few. Often there is a large gap between theoretical studies and their useful app lication to actual engineering pr061ems. This book does not pretend to close such a gap, but it does stress th e app lication of th e student's theoreti cal knowledge to severa l real devices, poi nt ing toward the engineerin g objectives of prop ulsion. Ex perience has show n that the app lication of fundamenta ls to these problems can reward the student not only with a valuable appreciation of their practica l sign ificance but also wi t h a deeper understa nding of the principles Ihemselves. To this end , the book attem pts to show how basic physical laws bot h describe and limi t the perfor mance of particular devices. With the present trend toward increasingly basic engineering curricula there rema ins a definite need for at least one co urse in which a va riety of fundame ntal ideas are brought together and made to interact so that their engineering implications can be clearly seen. Pro pulsion is a n ad mirable subject for this purpose, since it significan tly involves so many di sci plines. This text may therefore serve a more general purpose than simply conveying knowledge of propulsion devices to the student. In general, the most sa tisfacto ry performance of a given propulsio n engine is determined not only by the laws of mechanics and thermody na mics, bUl a lso, to a large extent, by the behavior of materials. The mai n emphasis in this book is on the former, though ma te ri al limitati ons a re identified and discussed. The mathematical portions of the tex t assume a know ledge of calculus. In addition , a n introductory course in vector calculus is des irab le, but not essential. An attempt has been made to minim ize mathematical complication so that physic al principles can be more readil y di scerned , and t he proble m sect ions at the ends of iii

iv

PREFACE

the chapters are designed to give the reader an idea of the practical applications of these principles. Throughout the text much attention has been given to the "working fluid " of the propulsion engine. Generally it is considered to be a continuum, since the principal interest lies in its macroscopic behavior. Two exceptions to this rule occur. First, in the study ofpJasma, it is necessary to examine microscopic processes in order to understand and predict the macroscopic behavior. Second, in the study of ion or electron engines, it is often necessary to discuss the behavior of discrete particles. The control volume method of analysis, whose usefulness for fluid flow has been demonstrated so thoroughly by Shapiro, has been used extensively. One-d imensional approximations are frequently employed, but certain two-dimensional phenomena like heat transfer and other boundary layer behavior are also discussed. Basically, two classes of propulsion devices are considered : (1) air-breat hing engines, and (2) rockets.

Air-breathing engines have been under very intensive development over the past 25 years. They are particularly inst ructive examples of engi neering achievement, because of both the magni tude and the success of the development effort. It is most illuminating to see the power and the limitatio ns of theoretical methods and to observe the mixture of art and science in thei r evolution. Although it has now approached some degree of completion, development continues toward better performance and new applications, so that air-breathing engines will remain a source of challenging engineering problems for some time to come. Rock et engines, by comparison, are currently under much more act ive development. M uch is now known about chem ical rockets, but nuclear and electrica l rockets are only entering the experimental stage. The latter are nonetheless discussed in this text, since current developments indicate that nuclear and electrical phenomena will be important in future propulsion engines. There is need for an introductory discussion of th,e application of these phenomena to propulsion. Part I is concerned mainly with a review of those topics in thermodynamics, combustion, and fluid mecha nics which are relevant to propulsion engines. One of the purposes of th is review is to introduce necessary concepts and laws in the notatio n used in su bsequen t chapters. In Chapter 2 the thermodynamics o f equil ibrium combust ion reactions are discussed in some detai l because of their relevance to chemical rockets especially. Since propulsion systems generally deal with accelerating fluid s, the gas dynamics of isentropic expansion are reviewed in Chapter 3, a long with the modifying effects of friction at solid boundaries, and shocks. In many engines, e.g. , the turbojet, the viscous boundary layer exercises a controll ing effect on the performance of the engine; hence the question of boundary layer behavior is discussed in some detail in Chapter 4. Convect ive heat transfer is also briefly reviewed, chiefly because of its relevance to the cooling of rocket motors. Electrostatic and electromagnetic acceleration are reviewed in Chapter 5 because they are centrally important in the rising genera ti on of propulsion engines

PREFACE

v

for space flig ht. We also give an elementary description of the behavior of charged particle beams and conducting fluid s, in order to aid understanding of releva nt phenomena, though it is not entirely clear at present which phenomena will be of greatest significance in working electrical-thrust engines. Part 2 is confined to ai r- breathing engines. Chapter 6 demonstrates that the performance of any air-breathing engine can be pred icted from the laws of thermodynamics, given the maximum permissible temperature and the appropriate efficiencies of engine com ponents, e.g., the compressor. Cha pter 7 deals with t he components of the ramjet (diffuser, combustor, and nozzle) and shows how their efficiencies depend principally on aerodynamic phenomena. The essential difference between the ramjet and the turbojet is simply the turbomachinery, which is the subj ect of Chapters 8 and 9. Fortuna tely the turbojet is so highly developed lhal several interesting examples of act ual engine configurat ions and performance can be cited by way of illustration. Part 3 is confi ned to rockets. Chapter 10 presents an elementary treatment of rocket vehicle performance in order to show the significance of specific impulse and other variables, and also to ind icate the relat ive advantages of low- and highth rust engines for various missions. The trajectories of space veh icles are briefly disc ussed in order to show what they req uire of the propulsion system. Chemica l rock ets are the su bject of Chapte rs I I, 12, 13 and 14, wh ich ta ke up a series of pertinent top ics. The combustio n process, the gas dynamics of the nozzie, and the cooling of the wall are all exami ned in some deta il , as well as auxiliary equipment such as the tur bopump. Chapter 15 is concerned with the nuclear rocket, in which a nuclear reactor is used to heat a gas such as hydrogen to high temperature before it expands in th e nozzle. The limi tat ions of this engine and th e phenomena th at govern its performance are briefly discussed. Chapter 16 focuses on electrostatic and elect romagnetic rockets, although these are in ea rly stages of development. Co nsiderable experimental work has been done on the components of the electrostat ic rocket, and it appears relatively promising for future applicati on, provided suitable electrica l energy conversion mac hinery can be developed. The electroth ermal and three kinds of possible electromagnetic rockets are briefly desc ri bed. This text may be used fo r either a one- or two-term course in propulsion. For a two-term sequence, the first term could be used fo r either air- breathing or rocket engines. A o ne-te rm course could be based o n Chapters 6, 10, and II , with selected portions of Parts 2 and 3 and appropriate review materia l. The aut hors wo uld li ke to reco rcl their indebted ness to Professor Edward S. Taylor, who has been our teacher, critic, and friend during our years or associa tion in the M. l.T. Gas Tu rb ine Laboratory. The text has been written largely as a result of his encou ragement, and it may be that his influence will be recogni zable in some of the more worthwhile portions. We are certainly grateful to him for writing the introductory chapter. Throughout the text, there are a number of references to the work of Professors J. H. Keenan and Ascher H. Shapiro. Their expositions of thermodynamics

vi

PREFACE

and fluid mechanics were so enlightening to us as studen ts that we could not help drawing heavily from them in presenting t he subject of propulsio n. The preparation of ea rl ier editions of classroom notes upon which the book is based was supported in part by a grant made to the Massachusetts Institute of Technology by the Ford Foundation for th e pu rpose of aiding in the improvement of engineering education. This support is gratefully acknow ledged. The authors wou ld also like to ac knowledge the di li gent and efficient work o f Mrs. Madelyn Euv ra rd and Mrs. Rose Tedmon in piecing together many fragments and revisions of the text. Cambridge, Massachusetts October 1964

P.G .H. C.R.P.

Contents PART I

Introduction and Review 01 Fundamental Sciences

Chapter 1

PROPULSION

1- 1 1-2 ]-3 Chapter 2

2- 1 2- 2

2- 3 2- 4

Cha pter 3

3-1 3-2

3-3 3-4 3-5 3-6

Theory of propulsion Development of propul sive devices Future possibilities .

I 3 5

MECHANICS AND THERMODYNAMICS OF FLUID FLOW

Int roduction. The laws of mechan ics and thermody nam ics fo r fluid flows Mass conservation and the continuity equation. Newton's seco nd law and the momentum eq uation. The first law of thermodynamics and the energy eq uation. The second law of thermodynamics and the entropy equation Thermodynamics of gases . Equilibrium combustion thermodyn amics; chemical reactions Mixtures of gases. Chemical transformation. Composition of products

7 8

]5 20

STEADY ONE-DIMENSIONAL FLOW OF A PERFECT GAS

Introduction. Genera l one-dimensional flo w of a perfect gas Stagnation state. Mach num ber Isentropic flow . Frictionless co nsta nt-area flow wit h stagnation temperatu re change Consta nt-a rea flow with fr iction

Shocks

43 43

47 49 53 57

Normal shocks. Oblique shocks Chapter 4

4- 1 4-2 4- 3 4-4 4-5

BOUNDARY LAYER MECHANICS AND HEAT TRANSFER

The viscous boundary laye r The bounda ry layer equations. La minar boundary layer solutions The turbu lent boundary layer . Boundary layer heat tra nsfer vii

64 71 76

82 91

viii

CONTENTS

CONTENTS

Chapter 5

5-1 5-2

5- 3 5-4 5-5

5- 6 5-7

ELECTROSTATICS, MAGNETO STATICS, AND PLASMAS

Introduction . Basic equation s of electrostatics and magnetostatics Motion within electric and magnetic fie lds Plas mas Conducti on in plasmas in the absence of magnetic fields Conduction and momentum exchange in plasmas with magnetic fields One.dimen sional steady flow of a plasma Constant·area , constanHemperature acceleration. Magnetic Reynolds number. Plasma boundaries

PART 2

Air.Breathing Engines

Chapter 6

THERMODYNAMICS OF AIRCRAFT JET ENGINES

6- 1 6-2

6-3 6-4

6-5 Chapter 7

7- 1

7- 2 7- 3 7-4

7-5 Chapter 8

8- 1 8- 2

8- 3 8-4 8- 5 8- 6

Introduction. Thrust and efficiency The thrust equation. Engine performance . Aircra ft range. Take· off thrust. Specific fuel consumpt ion The ramjet The ideal ramjet. The effect of aerodynamic losses Gas turbine engi nes . The turbojet. The turbofan engine. Turboprop and turboshaft engines Typical engine performance

Chapter

102 102 106

107 112 117

123

139 140

149 157

176

AEROTHERMODYNAMICS OF INLETS, COMBUSTORS, AND NOZZLES

Introduction. Subsonic inlets and diffusers Flow patterns . Internal flow. External flow. Diffuser performance Supersonic inlets Reverse-nozzle diffuser. The starting problem. External deceleration Combustors . Combustion limits and Harne speed. Ram jet combustors and after· burners . Turbojet combustion cha mbers . Ana lysis . Alt itude per· formance Exha ust nozzles .

190 191

201 215

232

JET ENGINE TURBOMACHINES, AXIAL COMPRESSORS

Introduction . Angular momentum Single·stage axial compressors Press ure-rise limitations. Cascade aerodynami cs . Efficiency. Offdesign pe rformance. Degree of reaction. Solidity. Radia l varia tions Mult istage axial compressors . Instabilities and unsteady flow Surge. Rotating stall Aerodynamics of starting

238

239 244

9

272 275

JET ENGINE TURBOMACHINES, CENTRIFUGAL COMPRESSORS AND AXIAL TURBINES

9- 1 Int roduction. 281 9- 2 The centrifuga l compressor 281 ImpeUer and inducer. Difruser • Performan ce. Compa rison of centrifugal and ax ia l compresso rs 9- 3 The axial turbine 294 Stage dynamics . Radial variations . Deviation . Efficiency. Tur· bine performance 9- 4 Turbine and compressor matching 307 9- 5 Stresses in turbines and compressors. 310 PART 3

Rockets

Chapter 10

PERFORMANCE OF ROCKET VEHICLES

10- 1 Introduct ion. 10- 2 Static performance Thr ust. Specific impulse 10-3 Vehicle accele rat ion Gra vity . Drag. Single·stage so unding rocket. Burning time 10-4 Chemical rockets Single-stage rockets . Multistage roc kets 10- 5 Electrical rocket vehicles 10- 6 Space missions . High thrust. Low thru st Chapter 11

Chapter 12

320 322 327

335 338

353 354

CHEMICAL ROCKETS, PROPELLANTS AND COMBUSTION

Introduction Equilibrium composition Liquid propellants . Solid propellants Combustion chambers Liquid·propellant combusti on chambers . Solid· propellant tion chambers 12- 6 Combustion instabilities

12- 1 12- 2 12- 3 12- 4 12- 5

Chapter 13

319

CHEMICAL ROCKETS, AN INTRODUCTION

11- 1 Characteristics of chemical rockets 11-2 Analysis of an ideal rocket

13- J 270

ix

365 365 368 372 375 COIn bus·

392

CHEMICAL ROCKETS, EXPANSION IN NOZZLES

Rocket exhaust nozzles 397 Nozzle shape. The effect of fri ction and heat transfer. The errect of back pressure on nozzle flow. Plug and expansion-deflection nozzles. The effect of propellant property variations all nozzle flow. The effect of solid or liquid particles in the flow

x

CONTENTS

Chapter 14

CHEMICAL ROCKETS, THRUST CHAMBERS

14-1 Introduction. 14-2 Rocket heat transfer Regenerative cooling. Heat sinks. Convective film coefficients. Radiative heat -transfer 14-3 Rocket construction Liquid-propellant rockets: regenerati ve cooling systems. Solidpropellant rockets: casings. Solid-propellant rockets: nozzle~ 14-4 LIquid-propellant pressurization . Gas pressurization. Turbopumps • Centrifugal pumps. Cavitation. Turbines 14-5 Selection of combustion pressure. Liquid-propellant engines. Solid-propellant engines

14-6 .Ignition . 14-7 Thrust vectoring Chapter 15

1

t.

439

445

Introduction and Revie""

460

01 Fundamental Sciences

461 462

NUCLEAR ROCKETS

15-1 15-2 15-3 15-4

Introduction. Nuclear reactions as energy sources Nuclear reactors The solid-core nuclear rocket . Heat transfer to the rocket propellant. Rocket performance 15-5 Advanced nuclear rocket concepts

Chapt.r 16

PART

425 425

468 469 475 480 486

ELECTRICAL ROCKET PROPULSION

16-1 Introduction. 16-2 Electrostatic propellant acceleration . 16-3 Sources of charged particles . Bombardment ionization. Contact ionization

490 491 495

16-4 The plane diode

506

16-5 Beam optics. Exhaust neutralization

511

16-6 The arejet

518

16-7 Stea9Y crossed-field accelerators 16-8 Pulsed-plasma accelerators 16- 9 Traveling-wave accelerators

522

527 528

Append;x I

OPTIMIZATION OF MULTISTAGE ROCKETS

535

Appendix II

TABLES.

539

Appendix III

LIST OF SYMBOLS

545

Answers to Selected Problems

549

Index

553

.-

Kitty Hawk, North Carolina, 1903: man's first plane flight. The Wright brothers' famous flying machine was the first plane to lift a man into the air. Orville Wright took the first airplane ride, lying face downward on Ihe lower wing or Ihe biplane. His ride, which was only a rew feet above the ground, lasted 12 seconds and covered 120 reet before running into the ground. Photograph courtesy the Library of Congress and United Airlines.

1 Propulsion*

1-1

THEORY OF PROPULSION

The verb "propel" means to dr ive or push forward or onward. Clearly th is defin ition implies the existence OWl body. a force (drive or push). a nd a prefem;d di rection (forward or onward). .tccording to Newton's third law of motion, forces occur only in equal and opposlle pa irs, and the existence 01 a force on a body therefore entails a reaction. This reaction cannot be on the body itself, since a pair of equal and opposite forces on the same body wi ll not "drive or push forward or onward." Therefore one or more additional bodies are esse.u.tial...tQ.. propulsion. For a self-propelled vehicle like an automobile, the second body is Ihe earth aself, and the act of propulsion entails a transfer of momentum between the automobile and the earth. In cases of particular interest in this text, the second body is usually a fluid med ium. In the instance of a propeller-driven a irplane, this fluid is the surrounding air, while in the case of a rocket it is a fluid carried along with, and discharged from, the rocket itself. In a jet ajmlane. the major part of the fluid used in propulsion is the surroundin atmos here but the fuel, which ..Qlay canslltu e 0...'> e PAQOl! sive fluid , is carried along and discharged as In a rocket. If ; consider an airplane in st raight level flight at uniform speed, it is clear that the horizontal component of the resultant force acting on the airplane must be zero, in order to satisfy Newton 's second law of mo tion. We know that to propel the airplane it is necessary to overcome the drag (frictional and induced) of the fluid on the body. If this is done by means of a propeller, the fo rce of the propeller on the airplane must be sufficient to cancel the drag forces . .}he propeller can produce this fore onl b transferrin momentum to the strea m orair which passes througlLit. The drag forces also transfer the momentum to the surroun mg air; and indeed, the horizontal momentum change due to the propeller thrust and the horizontal momentum change due to drag forces must be equal and opposite, I

* An introduction by Edward S. Taylor, Professor of Flight Propulsion, Department of Ae ronautics and Astronautics, Massac hu sett s Institute of Technology. 1

p

1-1

1-2

whet her we are speaking of the momentum of the airplane or of the fluid medium surrounding it. Thus, although the air is disturbed by the passage of such an

1-2

airplane, there is no net c hange in its horizontal momentum before and after the

The history of propulsion illustrates the interaction between economic or military need and technological advances. Although da Vinci's sketches of screw propellers date from the fifteenth (or possibly early sixteenth) century, the need for such a device did not arise until the development of the steam engine provided an appropriate dri ving mechanism . The first successful steamboats were driven by paddle wheels, but the paddle wheel must operate at the interface of two media (air and water) and is therefore clearly unsuitable for submarines, lighter-than-air ships, and airplanes. (It might also be said to be unsuitable for any ship when the sea is ro ugh and the in terface is shifting a bouL) The first successful submarine, the first airs hip, and the first airplane all had screw propellers. The airplane propeller still serves well when the speed of the aircraft is less than 350 mi les per hour. When man began to want to fly at higher speeds (originally for military reasons), a lighter and smaller power plant was essential. The jet engine answered thi need, while at the same time avoiding the difficulties encountered by propellers f operatin g at supersonic tip velocit ies. It is interesting to note that there was no incentive for developing a jet engine until the art of airplane design had advanced enough to make high-speed flight practical. As a matter of fact, the jet is inappropriate to low-speed flight, and the propeller provided an essential first step without which we would almost certainly not have developed airplanes at al l. An interesting history of the development of aircraft engines is to be found in Reference 4. The jet engine is a good example of the solution of a technical problem-that of high-speed flight-which required that many conditions exist simultaneously. First the need had to exist; then there had to be sufficient mastery of the art of

2

PROPULSION

passage. If we consider an accelerating rocket (assumed, for the purpose of simplifying the argument, to be in gravity-free flight outside the atmosphere), the total momentum of the rocket and the material which it has discharged must remain constant. It is inconvenient to keep track of the momentum of the propellant after it leaves the rocket, and we shall see (Chapter 2) that in order to calculate acceleration of the rocket, it is necessary only to observe the momentum of the jet leaving the rocket at the nozzle discharge. Clearly the principles of propulsion are merely applications of Newton's laws of motion. Long before Newton announced his laws of motion (1687) which now make the principles of propulsion comprehensible to us, devices for elTeeling propulsion were known and used. The Chinese had roc kets as early as the twelfth century, and screw propellers are visible in some of Leonardo da Vi nci's dra wings. About 200 B.C., the Egyptian philosopher Hero invented a nd constructed a turbine which clearly shows the principle of jet propulsion. In pro pulsion, as in most other technological adva nces, the art preceded the science. However, it is highly improbable that man could ever have produced supersonic ai rcraft or rockets capable of pUlling vehicles in orbit aro und the eart h withou t a very good theory of propulsion. By "theory" we mean a method of generalizing the results of experiments so that we can, with some confidence, predict the results of new ex periments.

A good theory enables one to see the main points without getting confused by details, and makes it possible to perform limit analysis; e.g., to determine what could be done if the machinery could be made without losses. F urthermore, a theory facilitates communication between workers. Moreover, even poor theory

may stimulate thought and experiment which will lead to new ways of accom plishing things. A good theory can prevent wasting elTort on unworkable devices. Many erroneous ideas of propulsion by jets were prevalent as late as the 1920's. It was well known at this time that a fi reboat di recting its streams of water at a fire would gradua lly move away from the fire. Eminent professors of engineerin g

ha ve been heard to scoff at this observation, say in g that "you can't push on a

rope." This statement, of course, reflects a profound ignorance of the problem, which has nothing to do with whether the streams fro m the fire nozzle ever reach any target. Goddard, at about this time, found it necessary to demonstrate that his rockets had thrust even when fired in a vac uum. T his points up the prevalence of the same fallacy as ex isted in the fireboat problem. It apparently was difficult for people to conce ive of something propelling itself without some med ium or body to push on. The id ea of the device pushing on a med ium which was being discarded was

DEVELOPMENT OF PROPULSIVE DEVICES

3

DEVELOPMENT OF PROPULSIVE DEVICES

airplane design, including aerodynamics, structure, and control; and fin ally there

had to be the knowledge of turbomachinery design and the availability of materials capable of withstanding high temperatures. These are only some of the major hurdles. Of course the solution of the problem also depended on a sophisticated manufact uring technique-forging, casting, co ld-forming, welding, and machining - and on the availability of countless "minor" but essential items: bearings, seals, lUbricants, pum ps, control devices, etc. In any development as complex as the jet engine, special circumstances are bound to arise which require changes and advancement of the technology in these "minor" items. However, without a rea-

sonable state of development in these as well as the major items, the development period would be likely to stretch out until it became clear that the project had better be postponed until the necessary supporting technology was able to catch up. John Barber patented a gas turbine in 179 1. There is little doubt that Barber had insufficient knowledge of basic mechanics, and certa inly insufficient knowledge

understood by only a few. Now it is a part of the knowledge of most if not all

of fluid mechanics to develop a successful gas turbine; but even if he had had all the fundamental knowledge we have today, his job would still have been impossible unt il the general technological level had risen enormously. Materials and manu-

newly graduated mechanical and aeronautical engineers.

factu ring techniques were not adequate even to begin this task until well over a

p

4

1-2

PROPULSION

hundred years later. Furthermore, had he been able to build a gas turbine, there would have been no economically feasib le use for it. . History is full of such examples of inventions made too soon. They emphaSIZe the importance of the engineering task of determin ing when a need eXIsts and when

supporting technology, both in theory and practice, are sufficiently advanced to begin a new deve lopment. To determine these facts involves a knowledge or at least an estimate of what extensions of theory and what refinements of practIce are essential to its success.

The first successful aircraft propulsion device was Henry Gifford's steamengine, propeller-driven di ri gible balloon, which went from Paris to Trappes in 1851 at an average speed of about seven miles per hour. However, for an aIrplane power plant, the necessary water supply (or condenser) and boiler would be so unreasonably heavy that even if the steam engine weighed n~thi n g, o.n ly very P?or airplane performance would be possible. The successful aIr plane, III a practIcal sense, had to wait for the development of the internal combustIOn engIne, whIch eli minated the necessity for either a wate r supply (or condenser) or a bolier. At

FUTURE POSSIBILITIES .

1-3

5

While the German V-2 rocket, with a 2000-pound warhead and a range of about a hundred miles, had considerable value as a morale-bu il der for the Germans, the damage done by these rockets was minor and the accuracy was poor compared with the effort invo lved. However, with the advent of nuclear devices, the potent ia l explosive power of a warhead of given size was mu ltiplied by an eno rmous factor, and the effectiveness of a long-range missile with even poor accuracy became

enormous. T his, then , was the motivating force behind the development of rocket propulsion ; it was inevi ta ble that billions of dollars would be poured into the development of such dev ices. Except for the guidance devices (where enormous advantages were yet to be obtained by subsequent improvements), the art of rocket ry was reasonably ready and the science had been ready for some time, with the exception of one factor: the reentry of a rocket a t high speed. This problem was not sufficiently understood until much later, but practically speaking it proved to be less difficult than was feared. Thus in the forties the need was evident and the tec hnology and science existed. With reasonab le luck and not-unreasonable improvement in the technology, the job could be done.

the time of the Wright brothers' experi ments, available internal combustion

engines were still too heavy, but they did exist, a nd the time was ri pe fo r their development to a reaso nable engine weight per horsepower. The Wnght brothers had to take on this problem [1 J, along with the design and constructIOn of the first powered airplanes to demonstrate controlled flight. . Langley, who was faCIng the same problem at the same time, found an extraord inary englllcer

In

Manley [2],

1-3

FUTURE POSSIBILITIES

At the present writing (1964) we see successful chemical rockets, and the need, if we are to travel far in the solar system, for vastly improved means of propulsion.

The possibili ties now appear to be: (I) Improvement in chemical rockets, which undoubtedly will take place, but

who handled this pa rt of the problem with outsta nding success. The possibility of propulsion by means of a jet had been well known [3J for many years before it was usefully applied by von Ohain (first flight August 27, 1939) and Whittle (first fli ght May 15, 1941). Before jet propulsion could become feasible, it was essential tha t ai rplanes be des igned and built for high speed, and also it was necessary that the art of design and construction of high- temperature

(2) Nuclear rockets . These hinge on our ability to make nuclear reactors which will operate at very muc h higher temperatures than they do at present. (3) Some form of electrical propU lsion. Th is appears to req uire better understanding of the properties and behavior of high-temperature ionized gases, and

turbines and efficient compressors be advanced cons iderably to meet the stringent

considera ble improvement in the means for producing electric power in space.

may be insufficient to an swer the needs for ex ploring the solar system.

requirements (primarily light weight combined with high en1ciency) of a useful

What route we shall take from here is unclear. It may be that improvement in

aircraft engine. The development of the sophist icated rocket is an illustration of a similar series

conven tiona l chemica l rockets wi ll offer the least expensive alternati ve. If we ca n judge from history, it may even be that we shall wait for new scientific discoveries J new materials, and new improvement s in the technology of ma nu factur ing before the task can be accomplished. In any case we must rem ember that a continuing

of events. The Chinese seem to have used the rocket mostly for fireworks, although there is some indication of military use.

In the early days of artillery, acc uracy

was so poor that even a crude unguided rocket could compete, and about the beginning of the nineteenth century the British officer Congreve was instrumental in develop in g rockets as a subs titute for ar tillery. There is no doubt that these rockets were useful in frightening cavalry horses, and perhaps were not much less accurate than the artillery of the time. The "rocket's red glare" of the national anthem refers to Congreve rockets in the War of 1812. Subseq uent ly artillery was greatly improved and military rockets became obsolete. Rockets, in order to be effective, would have to be guided. The necessary understand ino of con trol problem s came about in the 1930's, and th e requisite development ~f th is technology came soon after.

strong desire to do the job will be essential to its accomplishment.

F 6

PROPULSION

References 1. MEYER, R. B., JR., Annual Report of the Smithsonian Institutio n, 1961 2. "A Description of the Manley Engine," Soc . oj Automotive Engineers, 1942 3. SCHLATFER, R. , and S. D. HERON, De velopment oj Aircraft Engines m,ld FI~els. Cam-

2

bn'dge , Mass ( .: Graduate School of Business Administration, Harvard Umverslty, . . I1950.

4. T AYLOR, C. F., "Aircraft Propulsion." Ann ual Report of the Sm ithsonIan nshtu-

Mechanics and

tion, 1962, pages 245-298

Thermodynamics of Fluid Flo""

2-1

INTRODUCTION

An understanding of fluid mechanics and thermodynamics is perhaps the most important prerequisite for the study of propulsion. In nea rly all the propulsion methods discussed in this volume, thrust is developed by imparting momentum to fluid streams. Most of the methods involve thermal effects in one way or another. T he purpose of this chapter is to set forth a concise statement of the principles of mass, momentum, energy, and entropy in a for m which is suitable to the treatment of fluid streams. We shall ass ume that the student is th oroughly fa miliar with the first and second laws of thermodynamics and their corollaries as applied to ord inary nonreacting continuum substances in equilibrium. Hence, after careful definition of the concepts of system and control volume, these laws will merely be stated in fo rms which the reader will find useful in the remai nder of the text. The thermodynamics of equilibrium chemical reactions will then be treated in some detail, because of its relevance to combustors in propulsion engines. Considerable literature is ava ilab le as background material for the subject matter of both this chapter and Chapter 3. This summary largely follows the material of Keenan [I] and Shapiro [2]. T he concepts of syslem and cOlllrol volume, as applied in the following pages and throughout the text, have very specific meanings. The purpose of these concepts is the specification either of a defin ite collection of material or of a region in space which is to be analyzed. System . A system is a collect ion of matter of fixed identity. It may be considered enclosed by an invisible, massless, flex ible surface through which no matter can pass. The boundary of the system may change position , size, and shape. 7

8

MECHANICS AND THERMODYNAMICS OF FLU ID FLOW

2-2

COlltrol vO/llme. A control volume is a region of co nstant shape and size which is fixed in space relative to the observer. One can imagine an invisible, massless, rigid envelope (the control surface) enclosing the control volume. The control surface offers no resistance to the passage of mass and is arb itrarily located . Analyses to be deve loped later will show that the most suitable location of a control surface in any particular problem is a matter of convenience. Most problems may be treated by either system or control volume analyses. However, the proper choice often leads to substantial simplification. For example, most fluid-flow problems, and especially steady flows, are much more conve niently handled in terms of an appr opria tely specified control volume. Continuum. In the follow ing discussions the fluids und er consideration will be treated as continua. T hat is, their atomic st ru ct ure will be ignored and they will be considered as capable of being subd ivided into infinitesimal pieces of identical structure. In this way it is legitimate to speak of the properties of a continu ul11e.g., density, pressure, velocity- as point p roperties. 2-2

THE LAWS OF MECHANICS AND THERMODYNAMICS FOR FLUID FLOWS

Mass conservation and the continuity equation

The law of mass conservation applied to a system simply requ ires that the mass of the system remain constant regardl ess of its size or shape, the number of pieces into wh ich it is d ivided, or the length of the time interval during wh ich it is observed (for " nonrelativistic" system velocities). Here, as in most cases, the physical law is stated as it applies to a system. This is inconvenient for the stationary observer of flui d flow. Such an observer is unable to continuously measure the behavior of a moving fluid particle (a small system) and must, therefore, content himself with measurements taken at specific points in the space through whi ch the fluid moves . The basic law must then be expressed in terms of the required relat ionships between fluid properties a t various stationary points; Le., at poi nts with in a control volume. Consider a general control vol ume defined within a region of flui d now as shown in F ig. 2- 1. The conse rv ation of mass simply requires that the rate of change of mass stored within the control vol ume be equal to the net inflow of mass: (2-1 ) where m c\, is the mass within the control vol um e and 1i1 is the indicated mass flow rate. These terms can be expressed in more convenient ana lytical fo rms. Consider the mass flow through a small surface element dA on the control surface. If the local velocity is u and a unit vector normal to dA is n (outward poi nting) as in Fig. 2-1, the ele mental mass flow rate is din ~ plul cos adA,

2-2

LAWS OF MECHAN ICS AND THERMODYNAM ICS

9

Fluid flow through a control volume.

FIG . 2- 1.

where p is the local density and a is the angle between uand n, the product lui cos a being positive for outflow and negative for inflow. The mass withi n the control volume me\' can be fo und by the integration lI1,v

~

(

Joy

p d't),

where dU is a volume element and the integral extends thro ugho ut the con trol vol ume. F ina lly, writing the product lui cos a as the usual dot product u . n, the cOllfillllily equatioll may be wri tten

'IIi· Co

j'

P d't)

cv

+ )cs ( pu · n dA ~

0,

(2- 2)

where t he seco nd integral exte nds over the en tire contro l surface. Newto n's second law and the momentum equation

For a system of mass

117

acted on by a force F, Newton's law requires tha t

F

d

~ K -I (mu), I
I"M + vN.

(2- 35)

At the same time thi s reaction is taking place, there is ge nerally some tendency for the reverse reaction, I"M vN --> "A f3B

+

+

A,B,M,N',

I

/

"PJI

"

\

\ Pure M \

P = P.I+Pfj+P.lI+PS

\

I

\

l

T = constant

"

\\

I

II Pure B

Pure N /

jJ/j ~,

I

\

+ 9H zO + 2.50 z.

In this case, the composition is easily determined by requ iring that the number of atoms of each element taking part in th e reaction be constant. Actuatly, at high temperatures this expression may be a conside rable oversi mplification, because the compound s CO 2 , H 2 0 , and O 2 dissociate to some extent into free atoms and radica ls. A m ore general sta tement is

"-

/

/

/

\

C S HI 8

/-- ---- -- - "

29

/

"',........

/

......

FIG. 2- 11.

_------

....-//

Reaction chamber for derivation of equilibrium constants.

to occur, especially at high temperatures. Equilib rium is that state at which bot h forward a nd reverse reactions are occurring at equal ra les. When eq uilibrium obtains, the concentrations of A, B, M , and N are constant. Using the seco nd law of thermodynamics, we can express the conditions for equili brium in algebraic form. Suppose that an equil ibrium mixt ure resulting from the combi nation of A and B is contained in a chamber which communicates through semipermeable membranes with each individual cons tituent substa nce in a pure form, as in Fig. 2- 11. The entire apparatus is surrounded by a constant-temperature bath. A semipermeable membrane is a hypothetical surface through which onc substa nce may pass freely, though all others are preve nted from passi ng. Such membranes are only approxi mated by certain real substances, but they are nevertheless useful ana lytical concepts. For example, in Fig. 2-11, substance A may pass freely into or out of the mixture at cylinder A, and the pressure of th c pure A wilt be equa l to the partial pressure of A in the mixture, whereas B, M, and N cannot pass through this membrane. Similarly, the membranes of cy linders B, M, and N adm it only substances B, M, and N respective ly. The amount of any particular substance within the mixture chamber can be cont rolled by simply pushing or pull in g on the piston in the cylinder containing the pure substa nce. This control process is reversible if done slowly. Starti ng with an equilibrium mixture resulti ng from the combination of A and B in any propo rtion , consider the introduction of additiona l small amounts of A and B through the semipe rmeable membranes into the mixture chamber, in the stoichiometric ratio kaA + k(3B, where k is somc small number. Once in co ntact Within the main cham ber, the A and B tend to react, forming M and N. There are two ways to restore the contcnts of the ma in chamber to their origina l condition. One is to wi thdraw from the chamber kaA + kf3B. The othe r is to with-

30

MECHANICS AND THERMODYNAMICS OF FLUID FLOW

d raw kp. M

2-4

+

- a R Inp., - (3 R Inp B

reached equilibrium, its composition will be restored to its former value. As a result of the reaction

+

k{3 8

->

kp.M

+

~

(d

q) ~

T

k vN ,

":p;''. P PAP H

T

K,- p(CO)p(H ,O) p(CO,)p(H ,)

-

dSnli~tu re = O. ch alllbc r

dS.\(

+kI'S.lI,

dS e

~ k{3SB ,

dS ,y

+ kVSN.

p(NO)p( H 2) [peN))' 'p(H,o)

K _ [p(H,)I'p(O,j r Ip(H 20 )),

In add ition, the partial pressure of each constituent does not change and hence the pre~s ure in each cylinder does not change. Since the temperature is also constant in each cyl inder. the entropy pel' mole of each pure substance must be constant and the change in entropy within the cy linders may be written -kaSA.

K- - ~ )

KG_ l p( H,) )' ' p(O':!) p(H,o)

K _ p(C0 2) ,- p(CO)[p(O, )) ' ,

{3SB - I'Su - vS.y ~ -

~p ~ fi() --r T.

(2- 36)

The entropy of a perfect gas per mole can be obtained by integrating Eq. (2-17) with the result: 7'

-

-

S = -Rlnp+

r

ciT )T\cPT +

p(NO) [piN ,)!' 'lp(O, ))' ,

K - - peN)' I

p eNt)

10 32 10

10'_8

10 - 1

10·2.11 - - -

I

§ 10- Cl

IG

o

.§ 10- '

.~ =s

.t ~

.~

1\

,

10 1

£

1O- !1

"'

~

lOS

\

10.1

10- 13 10

l ;j

:;--"LLL-,-,LL--1_

-L---L---.l

2000 4000 Temperature, oK (a)

-I R npl.

where the subscript 1 refers to an arbitrary datum state. Thus. for a given datum

\

0

10- 11

+

p(CO)

s - p(C)[p(O,)I ' ,

~

or " -SA

K _

[p( H, )J ' 2

o

~ 10- 3

where S is the entropy per mole of pure substance. The total entropy change is equal to thc sum of the changes in the mixture chamber and in the cylinders: k H il I' dS ~ - kaS., - k {3S n + k p.S.l1 + kvSx T

(2- 37)

p( H,o)

.

K,-

dSA

~ K ,,(T).

K, - p( H, )p(O)

For perfect gases it may be shown that H il l' is a funct ion of tempe rature only. Thus for a given , value of k the change in entropy of the system is a function of temperature only : dS ~ kf(T). Since the temperature, volume, mass, and composition of the mixture do not change, there will be no change of state of the system in the mi xture chamber. Thus

~ g(T),

The term Kp is called the equilibrium constant for th is mixture in terms of parLial pressures. Figure 2- 12 gives typical values of the equilibriulll constant Kp. Several general statements should be made regarding equilibrium constants before their use is demonstrated in a detailed example.

kH 1l 1' .

re v

+ p.71lnp.l1 + vRlnp x

whe re g(T) is another!unction of temperature only. Adding the logarithmic terms and wri ting g(T) ~ R In Kp(T) leads to the foll owing relationship between the partial pressures :

there will be a heat interaction of magnitude kH rt p with the constant-temperature bath surrounding the apparatus, since the reaction occurs at the constant temperature T of the bath. In the limit th is heat interaction will be reversible. Then, for a system composed of all the gases in the mixture and in thc cy li nders, dS

31

state, Eq. (2- 36) can be written

k vN. In both cases the total number of atoms of each element in the chamber will have returned to its original value. After the mi xture has again

k~A

EQUILIBRIUM COMBUSTION THERMODYNAMICS

2-4

F,G. 2- 12.

atmospheres.

I?

lO- .,

6000

Equilibrium constants [8] in terms

t)

~K

l, ~

~I

---

I

1000 2000 3000 4000 5000 6000 Temperature, oK (b)

of partial pressures.

Pressure

in

32

MECHANICS AND THERMODYNAMICS OF FLU ID FLOW

2- 4

First, it may be noted that the partial pressures of the prod uct appear in the numerator and those of the reactants in the denominator, each with an exponent taken from Eq. (2-35). Qualitatively, then, one can expect a large Kp for subreactions which approach completion a nd small Kp for those th at do not. The number of constituents considered in this deriva tion is four, but the result can easily be extended to include any number of reactants a nd products. Second, the exponents are the stoichiometric coefficients. For example, consider the relationship between ca rbon d ioxide, carbon monoxide, and oxygen which would be of interes t in our octane~combustion example:

co, -->

CO

+

EQUILIBRIUM COMBUSTION THERMODYNAMICS

2-4

to form x moles of Hand x / 2 moles of O 2 , accord ing to the reaction

xH,O

-->

The relative concentrations of the constituents of the mi xture may be tabulated as shown below. Moles presen t at Mole fraction Xi eq uilibrium Constituents

~02'

p(CO)[P(02)] 1/2 = K p (C0 2 ) p'

H2

x

O2

2:

K II

= XII, ( xo, ) X

where Kp is the equilibrium constant based on partial pressures. Note that Kn, whi le mo re convenient in calculati ng mole fractions, is a function of two variables, pressure and temperature, and is thus inconvenient to tab ulate. In Eq. (2-39), Pm must be in the units for which Kp is tab ula ted. The actual use of these concepts ca n best be demonstrated by an example which contai ns a considerably simplified description of an actua l chemical reaction. ~ 1?,.sec

One mole of H 0 is raised in temperature to 6000°F, and it may be assumed 2 that it dissociates to a n equili brium mixture of H 2 0, H 2• and O 2 at a total pressure of 1 atm according to H 20 ~ H2 + !O 2· The equilibrium constant for this react ion a t 60000 P is Kp = 0.23 atml / 2. The concentration of t he mixture may be determined as follows. During the d issociation, x moles o f H 20 break down

- _ K -

-1/2

pPm

.

Substituting the equilibrium concentrat io ns, we obtain

x

) (

+ x/ 2

x/ 2

I I

1+ (2-39)

1/ '

I[:P

(2- 38)

cr+f3-P.-"K K n - Pm P'

Example 1

1+ x / 2

The relativc conce ntrations may be determined as follows:

(I

Since, according to the G ibbs-Dalton Law, Xi = p;/p,", it may be shown that

1+ x / 2 x 1+ x / 2 X 2( 1 + x / 2)

x

Total

than partial press ures : =

I -x

x

In this case the equilibrium equation is

Kn

X

+ 2: 0,.

xH,

H 2O

It may be noted that these e"ponents a re not directly related to n, c, or f of Eq. (2-34) for the overall rcaction . TI1is equilibrium equation provides one of the six relationships necessary to the determin at ion of product composition after t he reaction (2-34). In a similar way we can state five other equilibrium requirements to govern t he relat ive pa rtia l pressures o f the products. T hird , since the object of this calculation is to determine composition, it is convenient to define an equilibri um constant K n , based on mole fractions rather

33

+ x/2

)

II'

0.23.

x x/2

T;e quant ity x may then be determined by trial and e rror and the mole fracti ons H ,0,0" and H , at equilibrium obtained from the above tabulation TI he foregoi ng calculation oversimplifies t he dissociati on. Strictly spe~king the foI owmg three dlssociat'IOn reactIOns . ' are also of importa nce at 6000°F :

°

O2

-->

20,

H,

-->

2H.

Determination of the equilibrium concentrations of H 0 H 0 0 d H en req Ulre th . I 2, 2, 2, ,an th kno I s e Slmu taneous solution of three eq uilibrium co nditions wit h wn va ues of the three equilibr ium constants for the a bove react ions. Exam ple 2 (from Reference [3])

Consider a complete set of six products for a hydrogen-oxygen reactio n :

AH2

+ B0 2 --> ilH,oH,O + 110,_,0, + I1 H, H ,+ 1100 + I1 HH + /loaOH. (2-40)

34

MECHANICS AN D THERMODYNAMICS OF FLUID FLOW

2-4

Since two equations can be obtained from the conservat ion of hydrogen and

oxygen atoms, four equilibrium relationships are required. The following method is due to Penner [3], who has shown that it is conven ient

srn,

to define yet another equi librium constant based on number of moles rather than mole fract ions. T he use of this equili brium constant results in simplified calculations, and it is introd uced here because the technique has value beyond this example. Using again the typical reaction (2- 35),

EQUILIBRIUM COMBUSTION THERMODYNAM ICS

2-4

35

and the corresponding relationships in terms of moles may be writ ten from Eq. (2-41):

-no - , 1/ 2

st n3

110I-f =

1/ 2

1/ 2

n0 2 1111 2

n0 2

A procedure for obtaini ng a solution for the six un kn owns is now out lined.

STEP I: From the equilibrium equations write the number of moles of each nelV species appearing in the product in te rms of the reactants and the proper equilibrium constant. In terms of sr n :

(2- 41 )

no (2-45)

and usi ng Eq. (2-25),

(2- 42)

nOlI

STEP 2: Use the equilibrium relationships (2- 45) to elimi nate the number of moles

Strictly speaking, the use of this equilibrium constant introduces another unknown, 11, along with an additional equation which simply states that 11 is the total number of moles in the actual product mixture (not the subreacti o n). However, the simplification attained by th is approach lies in the fact that 11 can be rather easi ly approximated, and in the fact that the results of the calculation are not strongly dependent on II. Thus, as illustrated in this example, II is at first assumed known and later checked. An iterative procedure can be used if greater accuracy is desired . Returning to the example in Eq. (2-40), the two equations resulting from conservation of atoms are :

+ 2nH, + nII + nOH ~ 2A, nll ,o + 2no, + no + nOlI ~ 2B,

For hydrogen,

NH ~ 2nH,0

(2-43)

For oxygen,

No ~

(2-44)

where NH and No are the known num bers of hydrogen and oxygen atoms, respectively. The equilibrium information available is of the following form:

of new species in one of the atom-conservat ion stituting Eq. (2-45) in (2-43), we have

-->

STEP 3. Use this expression to eli minate no, from Eqs. (2-45).

no

=

(2-47)

nOH

dln411H2

0, 112

PlI z

Si'''211 ~1,2) . + Jr1l3nl/22

(NH - 211H, 2hl'n411H2

PH ,

Thus, sub-

(2- 46)

"

t02

~quat ions .

STEP 4: Use these expressions along with Eq. (2-46) to eliminate all but the rema ini ng atom conservation equation (2- 44).

1111 ,

from

-

POT-[ 1/2 1/2

jJo zPn 2 pl-r 20

112

PII zPO z

No·

(2- 48)

36

MECHANICS AND THERMODYNAMICS OF FLUID FLOW

Equation (2- 48) can now be solved for

"H. if the various ,\1'" are known.

2- 4

2-4

The

Table 2-2

EQUILIBRIUM COMBUSTION THERMODYNAMICS

37

Enthalpies and Heats of Formation for a Given Product Composition

,fil n's may be eva lu ated from Eg. (2-42) for k~own values of K p , Pm, and n. The

total number of moles in the mixture is actually unknown, but it can be rather easily approximated , and the results are not very sensitive to II anyway. The act ual product compos ition is not Car from that resulLing from "complete" react ion (unless the temperature is very high) so that Il is approximately equal to the number of moles resulting from "complete" reaction. For thi s example, it is

AH2

+

BO, -> A H 2 0

+

2BH,O if there is excess H ,. Hence

Il

+ (A

-

n b.7i

Qj,

kca l/gm-mole

nQj

"

kca ljgm-mole

2 I

0 0

0 0 0

0 0

0 0 0

0.317 0.1008 0.05399 0.1090 0.2333 1.5119

25.693 28.263 16.028 15.897 25.907 36.920

8.145 2.849 0.865 1.733 6.044 55.819 75.455

0 0 59.162 52.092 10.063 -57.802

0 0 3.194 5.678 2.348 -75. 11 8 -63.898

Hz 0, Products

(2B - A)O,

H2

if there is excess O 2 , or

/J.7i,

Reactants

2B)H,

can be readily approximated in terms of the know n

reactant quantities. For a numerica l example. consid er the equi lib rium compos iti on of2H 2 + O 2 at 35000 K (centigrade absolute) and 300 psia. Then NH ~ 4 and No ~ 2. For

0, 0 H OH H, O

complete reaction, the number of moles of products would be 2, and we write 2H2

+

Adding these gives us the tota l number of moles (which was assumed to be 2.2):

O 2 -> 2H,O.

However, since the temperature is rather high, II is assumed to be 2.2. Assumptions of this nature come from exper ience in this type of calcu latio n ; but as we shall see, iterations are easily made and the ass umption is not cr itica L From the

known Kp (pressure in atmosphe res in this case) and pressure (20.42 at m), and the assumed 11, the new equilibrium terms can be calculated . Using Eq. (2-42), ,\1',,1 ~

Kpl C;;,t I12~ 0.17003 , ,\1',,3 ~

K p3 =

1.3046,

,\1',,2 ~ ,\1'" , ~

K P4

Kp, (P~n) -I/2 ~ 0.19362, (~;n)' /2 ~ 15.020.

These values are then used wit h various assumed values of

IlH2

to ca lculate the

left-hand side of Eq. (2- 48). The correct value is of course No ~ 2. T rial results are tabulated below. Calculated No from Eg. (2-48) 2.2561 2.9549 2.0224 1.9917

0.2500 0.3400 0.3000 0.3200

By linear in terpolation between the last two figures,

flH2

was found to be

0.3170. Then Eqs. (2-46) and (2-47) give nH z

Jln

0.3170, 0.1090,

0.1008, 0.2333,

"0

0.05399, 1.5119.

11H z

Il ~

2.326.

The next step is to recalculate "If .. elc., assuming It

~

2.326. When we do this

we find no apprec iable change in -composition. Hence eve n an inexperienced person cou ld probably complete the calculat ion in two, or at mos t three, iterations.

This solution method is not limited to reactions of only two reactants. The generalization of the method for additional reactants is discussed in Reference [3]. In order to take 2 moles of H, and I mole of O 2 from 3000 K and transform them at constant pressure to the above composition at 3500o K, it would be necessary to accompa ny the reacti on by a heat transfer Q which can be determi ned by Eg. (2- 33). If we use Table 2-1 to com pute the heat of reaction we must let TJ ~ 300°K. Since the inlet temperature is also 3000 K in this exa:nple H H - 0 ' III RJ . Table 2-2 shows how the terms (Hp , - HpJ) and H RPJ are com puted from enthalpy-temperature information (e.g. Fig. 2-5), heats of for ma tion (Table 2-2) , and tlelgiven ' .. compositions. 111 the tab le, 11 is th e number of moles and b.h the ent~alpy per mole of eacl~ const ituent above the datum temperature Tj . As before, QJ s the heat of for matIO n of each constituent, per mole. From Table 2-2 the heat of rea c t'Jon H RPj IS . - 63 .8 98 kcaljgm-m ole of oxygen reactant A lso' the enthalpy d'~ (H ' . Th I erence 1' 2 - H PJ ) IS 75.455 kcal/gm-m ole of oxygen reacta nt erefore, from Eq. (2-33), the heat transfer necessary to take two moles of H' and One mole of O at 300 0 K and transform them to the product composit ion o} 2 Table 2- 2 at 3500 K is: 0

Q

(H p2

-

H/2) -

(Hill -

75.455 - 0 - 63.898

~

H HJ)

+

H UPJ

11.6 kcal/gm-mole·02.

38

MECHANICS AND THERMODYNAMICS OF FLUID FLOW

2-4

EQUILIBRIUM COMBUSTION THERMODYNAMICS

2- 4

P .. oblerns

References 1. KEENAN, J. H. , Thermodynamics. New Yo rk: John Wiley & So ns, 1941 H" The Dynamics and Thermodynamics of Compressible Fluid Flow, Volume l. New Yo rk: The Ronald Press Company, 1953; C hapters 1- 8 and 16 3. PENNER , S. 5., Chemistry Problems ill Jet Propulsion. New York: Pergamon Press, 2. S HAPI RO, ASCHER

195 1 4. KEENAN, J. H. and J. KAYE, Gas Tables. New York: John Wi ley & Sons, 1957 5. ROSSINI, F. D., et al. , "Selected Values of Properties of Hydrocarbons." Circula r No. 500, National Bureau of Standards, Washington, D. c., Feb. 1, 1952 6. "Selected Values of Chemical Thermodynam ic Properties." Series III, National Bureau of Standards, Washington, D. c., April I, 1954 7. ROSSINI, F. D., el al. , "Selected Values of Properties of Hydroca rbo ns." Circular No. 461, National Bureau of Standards, Washington, D. c., Nov. 1947 8. MARTINEZ, J. S. and G. W. E L VERUM, JR " Memo randum No. 20- 121, "A Method of Calculating the Performance of Liquid·Pro pellant Systems Containing the Species C, H, 0, N, F, and One Other Halogen, with Tables Required; Thermochemical Proper· tics to 6000o K. " Jet Propu lsion Laboratory, California Institute of Technology, Pasa· dena, Californ ia, Dec. 6, 1955 9. RICE, H. E., "Performance Calculations of RFNA·Aniline and Three·Component Systems Using Hyd rogen." Memorandum No. 9- 2, Jet Propulsion Laboratory, Pasa· dena , California , Ju ne 30, 1947 10. " Heat of Fo rmation of Hydrazine, Un symmetrical Dimethylhyd razi ne, and Monomethylhydrazine." Inorganic Research and Development Dept., FMC Corporation, Princeto n, N. J. , May 14, 1959 11. JOHNSTON, S. A., "A Short·Cut Method for Calculating the Performance of F uels Containing C, H, 0, and N with HN0 3, Oz, or NH-l-N03 ." Progress Report No. 20-202 . Jet Propu lsion Laboratory, Pasadena, California, November 9, 1953 12. CLARK, C. E., "Hydrazine," first ed itio n. Technical Bulletin No. 123.1, Olin Mathieson C hemical Corporation, Balt imore, Maryland, 1953 13. CARTER, J. M., and M. PROTTEAU, "Thermochemical Calculations on the Whi te Fumi ng N itric Acid and JP·3 Propellant Co mbi nation." Research Technical Memora n· dum No. 58. Aerojet.Genera l Engineering Corporation, Azusa, Califo rnia, April S, 1950 14. PERRY, J. H., Chemical Engilleer's Handbook, third edition. New York: McGraw-

39

1. A ?e rfe~t gas of molecular weight M = 20 and specific heat ratio Y = 1.2 expands adia batIcally from a pressure of 1000 psia, temperature 5000 o R, and velocity 1000 fps to a pressure of 14.7 psia. (a) If the fina l temperature is 26000 R a nd the fina l velocity negligible, how much wo rk has been done by the gas during expansion? (b) If.no work i~ done by or on this gas during expa nsion to 14.7 psia, what is the ma ximum pOSSible velocity at the end of the process? (c) If any ad iabatic expansion process to the same final pressu re is permissi ble is it possible for the exit temperat ure to be 300 0 R? State your reasoning. ' 2. An inventor suggests ~ propulsion device shown sc hematically in the figure. Would s~c h a scheme be. feaSible for steady operat ion over lo ng periods of time? D isc uss rigoro usly and bn efly.

Eleclric

I

generalOr

Q

Fluid circuit s

Propel lant

storagc

Nuclea r

tank

~-"--'===--- Cot11ro l ~--_ _ _---.,("""'-~

PROBLEM 2

Rockct Il oule

Hi ll, 1950 15. GAY DON, A. G., Dissociation Energies and Spectra of Diatomic Molecules, second ed ition. London: Chapman and Hall, 1953 PROBLEM 3 3. A strut of cho rd . I d· . except ~ tl '11 CIS P ace In a n Incompressible flow wh ich is everywhere uniform far dOw~\ le I usftrlated effe~t ~f the strut. Ve locity measurements at a certa in plane . . .s ream 0 t lC strut Indica te the presence of a wake as shown T I I·t , . le ve OCI y d IstnbutlOn ac ross the wa k" e IS app rox imately a cos ine function; that is, Il

U

1 -

(J

(1 +

cos 27bry)

for

b -2h < y-

One-dimensional flow throu gh a control volume.

L

r (Ir + 1~2)-

)es

]f the fl ow a rea, heat tra nsfer, work, shea r stress, and body force are known, changes in the sta te of lh c fluid stream may be obtained from th ese four equat ions. ]n genera l, solutions cannot be expressed in simple algebraic form. There are a numbe r of important special cases, however, which can be in tegrated in closed fo rm . Before we proceed to these special cases, let us int rod uce two concepts useful fo r all cases: the stag nat ion state and the Mach num ber.

X

I -:-_---~

"wc:::e:r FIG.

For steady fl ow the ene rgy req ui reme nt [from Eq. (2-8)] is

dA

A+Tdx (X

I

1

p

45

where liz is the mass flow rat e puA (a constant) and q and 1\1 a re the heat and work tran sfer per unit mass. If the fluid ca n be assumed a perfect gas (as it will be in this chapter), the equa· ti on of state is p = pRT. (3-4)

d (J's

%

GENERA L ONE· DIMENSIONAL FLOW OF A PERFECT GAS

3-2

=

=

is the wall shear stress and c is th e duct circ um ference.

2

(3- 5)

COIl -

plIA dll

plI!!'!. ' dx

/1

+ "2 .

(3 - 2)

The constant of in tegratio n, fro, is cal led the stagnat ion e nlhalpy . For any ad iabat ic fl ow viewed from a reference frame in which no wor k occ urs the stagna tion enthalpy is co nstant. Note parti cularly that this co nclusion h old~ for both reve rsible and irreversible flows. U nlik e ent ha lpy, the press ure reached When there is ad iabatic zero-wor k deceleration of a fluid is decrea sed by irre~ersible ~rocesses . This is ill~s trated in Fig. 3-2, an enthalpy-en tr o~ diagram Or a typIcal gas. The stagnation state corresponding to state is ® a nd the stagnation press ure is POI. which is the pressure atta ined by the flui d afte r it has

CD

46

STEADY ONE-DIM ENSIONAL FLO W OF A PERFECT GAS

3-2

3-3

ISENTROPIC FLOW

47

where a is the local speed of sound in the fluid. The speed of sound is the speed of propagatIOn of very small pressure disturbances. For a perfect gas it is given by [ I]

IIi

flu]

(3- 8)

2

3-3 pz

p,

M.any a~ tual processes, such as flows in nozzles and diffusers, are ideally isentrop'c. It 15, therefore, worthwhile to study the isentropic flow of a perfect gas in the absence of work and body forces . The simple results obtained with the constan t specific heat ~pproximation are useful even for large temperature changes, so long as approprIate average va lues of cp and)' are used . For constant specific heat, Eq. (3-5) may be written

Entropy

FIG. 3-2.

D efinition of stagna tion state.

been stop ped isentropically. If the flow is brought to rcst irreversibly, as at @ (l he increased ent ropy is a measure of the irreversibility), the stagnation enthalpy is the same as before but the pressure attained is less than Po l' For any process, slIch as to..0), the stagnat ion state gene rall y undergoes a change li ke the one from @ to (Q;). The decrease in pressure from Po 1 to P02 is, in the absence of work, an indication of the irreversibility of the process. If the flu id is a perfect gas, the stagnat ion tel11peralUre is related to the stagnation enthalpy through the

CD

re lation

'I'

ho - h =

ISENTROPIC FLOW

f OcI' dT.

!.-~ = l+!/ T

(3- 9)

2cI'T

which, with the definition of Mach number, becomes To = I

T

+ ~M' 2'

(3- 10)

Equation (3-6) may then be wri tten

7'

The stagnation temperature To of the fluid is that temperature which would be reached upon adiabatic, zero-work deceleration. For a perfect gas with constant specific heats the stagnation temperature and pressure are related [using Eq . (2-19)] by

l Po _ ( 'Y - 1 2)I' /n - ) - I +-- M .

P

2

From the perfect gas la w and Eqs. (3- 10) and (3- 11),

(3- 6)

Po

-=

P

since the deceleration process is, by definition, isentropic. Other properties of the fluid at the stagnation state may be determined from Po, To . and the equation of state . The stagnation conditions may be regarded as local fluid properties. Aside frolll analytical convenience, the definition of the stagnation state is useful experimentally since To and Po are relat ively easily measured. It is usually much more convenien t to measure the stagnation tempera ture To than the temperature T. Mach number

A very conven ient variable in compressible-flow problems is the Mach number M, defined as (3- 7) M = !!.. a

(3- 11 )

(I + )' -- 1 M-")

2

1/("(- 1)

.

(3- 12)

For allY flow of a perfect gas these can be used to relate the local conditions (T,p~ p) t? the local stagna tion conditions To, Po , Po, since this relationship is by definff /oll IsentropIC whatever the actual flow considered may be. The mass flow per unit area is liz A = pli.

Using Eqs. (3-7), (3-8), and (3-10), we may express the velocity as !I

= M

J--

0fFo- -

I

+

-y ;

I M' .

48

3-3

STEADY ONE-DIMENSIONAL FLOW OF A PERFECT GAS

10

f II

f~

-

,,-

~-

I2

"-

"-

/

""

I" 'I, I~~ I i

I

A

A*

I I

\

1\ ~ \ .1 \

0.02

I

0.1

0.2

2

0.4

l Po

\

\

4

Mach number, M

FIG. 3-3.

2)J'H ')/2(7-1) .

(3- 15)

1"0 3-4

,

One-dimensional isentropic flow of a perfect gas.

FRICTIONLESS CONSTANT-AREA FLOW WITH STAGNATION TEMPERATURE CHANGE

Anot her special case for which the basic equations may be solved in closed form

I

I

! I

0.0 I

r'

-

a

is frictionless flow in a constant -area duct in which stagnation enthalpy change occurs. Four processes of interest in which the stagnation temperature of a moving stream changes are:

\ I p\:-r" - I

Po

+

perfect gas of y ~ 1.4. Tabulations for other Y are given by Shapiro [I], Keenan and Kaye [2], and others.

_\

.1

,

(3- 14)

appropr iate 'Y. F igure 3-3 gives the one-d imensiona l isentropic function s for a

,

0.04

1' +1

For a given isentropic flow (given 'Y, R, Po, To , 111), it is clear that A* is a constant so tha t it may be used as in Eq . (3- 15) to normalize the actual flow area A. Eq uations (3-10), (3-1 1), and (3- 12) relate the fl uid properties to the Mach number M and Eq. (3- 15) shows how the Mach number depends on the flow area. Expressing fl ow variables in terms of Mach number is a matter of great convenience, since the various flow functions can be plotted or tabulated as a funct ion of M for the

I

I

.

) (1' + 1) / 2(1' - 1)

I [ -2- ( I+ y -I M M y I 2

- ~ -

\\

v RTo

(2 - -

Combining Eqs. (3-13) and (3- 14), we have

"

\\

I

0.1

±

'-

I

A*

I

49

1 with an asterisk, the maximum flow per unit area is, from th Po - ~ -- ...n

V

I\"

I

I

0.2

'

I

,

/

~

I "

I

I

0.4

--

,

,/

/

CONSTANT-AREA FLOW, STAGNATION TEMPERATURE CHANGE

of the flow at M Eq. (3-13),

.f--L-

fA I A'

I-

4

3-4

,

I. Combustion, 2. Evaporation or condensation of liquid drops traveling with stream,

I

3. Flux of electric current through a fluid of finite conductivity (Joule heating), and 4. Wall heat Iransfer.

\0 l' = 1.4

(After Shapiro [11 ·)

Then, using this expression with Eq. (3- 12), we may write the mass flow rate as (3-13)

For a given fluid (y, R) and inlet state (Po , T o), it may readily be shown that the mass fl ow per Unt't area ,'s maximum at M = I . If we indicate those properties

In any real flow, frictional effects are always present, especially near solid boundaries. As we shall discuss in Chapter 4, wall heat transfer and wall friction are in fact so closely related that it is not rea list ic to discuss the forme r with out at the same time talk ing about the latter. Nevertheless, the study of frictionless flow in a consta nt-area duct in which a To c hange occurs illustra tes some important features of the real flows in which the fi rst three effects above are present. Combustion and evaporat ion or condensation processes may result in variations of mo lecu lar we ight an d gas consta nts, but for th e sake of simplicity these may be considered negl igib le. The stagnation-entha lpy cha nge in the processes can be determined fr om I:1h o = q - w,

/

L

3-4

STEADY ONE-DIMENSIONAL FLOW OF A PERFECT GAS

50

. For th is case d 1\I are net heat and work transfers . per unit fmass Eof fluid where q an (3 I) (3 2) and . ·t momentum and energy equations are rom qs. the contlOUI y, ' (3-3), respectively: 1

+

dp

Momentum:

dp

Energy:

dll o = dh

-pu dll,

+

II

and, using the above relation between static pressure rat io and Mach number, T r--;

To

dll.

T Ol

The first two of these can be readily integrated to

P _ p,

in which the subscript also be expressed as

pll = PIll!! = - (pII)(U - Ill) = -(P,II,)(II - II,),

sign ifies an initial condition. The second of these may ,

P - PI = PI III - pll

2

. duc·mg the Mach number , we can transform this to By mtro

p _ I + 1'M ; . I + 1' M'

p; Further, by use 0 f Eq.

(3 -11) we can derive the stagnation pressure ratio from ,

this equation, with the result that

(

l' _

I + -

.fJll.. POI

Th

I + 1'Mi) ( I + "'1 M 2

I

,)''''-1)

- M(3- 16)

2 I +

l' ;

1 M;

the change in stagnation pressure is directly related to the Mach number.

;~e dependence of Mach number on stagnation enthalpy (or stagna l10n tem-

perature) can in turn be obtained as follows : Using the perfect-gas law, T Tl

I' P,

=

P-; p '

or invo king the continuity condition, pll = PIll " it may be seen that T

r-; Now if th e Mach num ber reI at shown that

·lon M = II/V"'IRT is introduced , it can e;1sily be T

51

=

[I ++ 'Y M' (M)J2 'Y M ;

I

M~

.

From Eq. (3- 10) and this expression, we can show that the stagnati on temperature ra tio is governed by

II

=

CONSTANT-AREA FLO W, STAGNATION TEMPERATURE CHANGE

,

du = 0,

Continuity:

P

-

3-4

1 +1'M;

[I

=

M - 2 1+ 1'- 2- 1 M-' ) + 1'M' (M)] ( I + M; .

~~

(3- 17)

Thus the Mach number depends uniquely on the rali o of stagnati on temperatures and the ini tial Mach number. It is desirable in this case, as in the isentropic case, to simpl ify these relationships by the choice of a convenient reference state. Since stagnation conditions are not. co nsta nt, the stagnation state is not suitable for this purpose. However, the state corresponding to unity Mac h number is suita ble si nce, as Eqs. (3-16) and (3-17) show, condit ions the re are constant fo r a given flow (i.e. , for a given Po" To" M I). Again using an asterisk to signify properties a t M = I , it can be shown from the equations above that

T T* To T~ ..P....

(1 +1')' , I + 1'M " M- , 2(1' + I)M 2 (I +

l' ;

1 M2)

(3- 21 )

( I + 1'M" F

1+ 1'

P*

I + 1'M2

Po

(_ 2 l' + I

Po*

(3-20)

rn-I) (

(3- 22) I + l' ) (

I + 1'M"

'Y _ I ?)'Y !('Y - 1 ) M. + 2I

(3-23)

In this way the fluid properties may be presented as a fu nction of a single argument, the local Mach number. Equations (3- 20) through (3-23) are shown grap hically in Fig. 3-4 for the case l' = 1.4. G iven particular entrance conditions, To 1, Po 1, M 1, the exit conditions after a given change in stagnation temperature may be obtained as follows: The value M, fixes the val ue of T orlT I; and th~s the va lue TJ, since T O l is know n. The exit state is then fixed by T02 /TJ, detenmned by or Then M "p z/p* ,Poz/p~ are all fixed by the value of T oz/TI; (for a given value of 1').

• 52

3-4

STEADY ONE-DIMENSIONAL FLOW OF A PERFECT GAS

3-5 4.00

/

/

I

, ,::::,

l-

0.60

t-

~,

-

\

Ij 0.40

Momentum:

T

\ I '

~

+ u du o

~ dh

For flow in long pipes it will be shown (Fig. 4-14) that the wall shear stresses can be correlated as follows:

p'

//

°

Energy:

\

l/

0. 10 0.08

Continuity:

To

kTZ

/

0.20

CONSTANT-AREA FLOW WITH FRICTION

respectively,

p~ /

- - - - - - 1- - -l-

f1.00 0.80

/

p"

1--

53

Another solution of Eqs. (3- 1) through (3- 4) can be obtained for constant-area adiabatic flow with friction but no body force or work. In this case, from Eqs. (3-1 ), (3- 2), and (3-3) the continuity, momen tum , and energy relations are,

/

I

2.00

CONSTANT-AREA FLOW WITH FRICTION

3-5

TO

or

0.0 6 0.04

0.0 2

0.0

,

diameter.

-

For Mach numbers greater than unity, the Mach number is also a

significant variable. The elTects of roughness (or at least of variations in roughness) are usually fairly un important for surfaces which have a reasonably smooth finish.

II); 0.04

FIG. 3-4.

,

in which C, is the skin friction coefficient To / (pu z/ 2), Re is the Reynolds number puD/ !" ~ is the average height of surface roughness elements, and D is the duct

, 0.1

0.2 0.4 0.6 0.8 Mach number, M

2

3

"( = 1.4

This point is discussed in more detail in Chapter 4. T he way in which friction alTects the Mach number may be shown as follows. First the conservation equations are transformed (by ihtroducing the perfect gas law and the definition of Mach number) to read:

Frictionless flow of a perfect gas in a constant-area duct with stagna tion

It may be seen from Fig. 3-4 that inc reasing the stagnat ion tempera ture drives the Mach number toward un ity whether the flow is supersonic o r subsonic. After the M ac h numbe r has approached unity in a give n duct, further increase in stagnation enthalpy is possible only if the initi al conditions cha nge. The flow may be

said to be thermally choked . It is int eres tin g to note the variations in sta gna ti o n press ure as the stream is subjected to energy transfer. The stagnation pressure alwa ys drops when energy is added to the stream and rises when energy is transferred from the stream. Thus, whether the flow is subsonic o r supersonic. there may be s ignificant JOSS of stagnation press ure due to combustion in a moving s tream. Conversely, cooling tends to increa se the stagnat ion pressure.


Continuity:

temperature change. (Afte r Shapiro [1].)

T

2 u-z.

2

0,

= " M 2 (4 C/ dX) _ 2 D

Momentum:

Energy :

~ dll

o ~ dT

T

+

,,- l lIdll RT'

- ,,-

Since

M

the n dM 2

M2

II

V"Y RT '

dll , RT

II

54

STEADY ONE-DIMENSIONAL FLOW OF A PERFECT GAS

3-5

3-5

CONSTANT- AREA FLOW W ITH FRICTION

55

/00

And now, if we use the con tinuity eq uation to eliminate du 2 / U 2 , it follows that

60 40 20

Also, combin ing the momentum and energy eq uations, we can easily show that

\

\

Po

~, I

dp = - 1'M' (4 C dX) P 2 D j

+

0

_1'_ dT 'Y- I T

6 4

Combining these last two expressions yields

dM' = M'

-M ' Y

(4C dX) D

2

,

,, ~

1/

f

I ; /

1

T T'

which, for constant stagnatio n temperat u re [us ing Eq. (3- 10)], becomest 0.6

4C U

0.4

/ DII

(3- 24) 0.2

Equation (3- 24) shows that dM' > 0 for M < I and dM' < 0 for M > I. That is, frict ion a/ways changes the Mach number toward unity. For constant Cj, Eq . (3- 24) ca n be in tegrated between the limi ts M = M and M I to yield

V\ 1\

/ \

I

0.1

~\

P'

0.06 U 4Cf D

0.04

(3- 25) 0.02

in which L* is the le ngth of d uct necessary to cha nge t he Mach n umbcr of the flow from M to un ity. Co nsider a duct of given cross-sectional area and variable length. If the inle t stale, mass flow rate, and average skin fr iction coefficient are fixed, there is a maximum length of duct which can transm it the flow. Since the Mach number is unity at the ex haust plane in that case, the length is des ignated L* and the flow may be said to be fric ti on-choked. F rom Eq . (3-25), we can see that at any poin t I in the duct, the variable Cf L* / D de pends o nl y on M 1 and 1'. Since D is constant and C, is assumed cons tant , then at some other poin t a distance x (x < L*) downstrea m from point I ,

0.01 O.t

0.2

0.4 2 Mach number, M

4

/0 1 = 1.4

FIG. 3-5. Ad iabatic flow of a perfect gas in a consta nt-area duct with friction. (After Shapiro [\].) by lIsing the definition of Mach number and the continu ity and energy re lations are : T

1' + 1

(3- 26)

T*

p

•

(3- 27)

From this we can determine M z . Other relationships which may be derived [I] Po

t Fo r noncirc ular ducts, the hydraulic diameter mation in place of the diameter D .

DII

=

4A / c can be used as an approxi-

Po*

(3-28)

56

3-5

STEADY ONE-DIMENSIONAL FLOW OF A PERFECT GAS

1.7

1.5

I-

0.9 1'1

1)::,..

!,.?

_}-I~

1-p-

---

1

f

--

35~

(min)\

Of

o

3-6

I

2

V IL

3.0

~ t'--- ---l\~ '"\ ~'" ~ "~ r----- '?i--- -

~

-

i

MaChD7IV !/

t-

O

~~~ = ~ t-t-w~~~ ""---

t---

0

20 0

20

1.0

0

2. 0

"

"LL:=

I4

j

3.5

3.0

/15V'

I. 5

/

Initial Mach numbe r, M1

FIG. 3- 10. Shock angle ve rsus inlet Mach number and tu rning angle [1]. Curve~ above dashed li ne hold for M , < I and curves below ho ld for M z > 1. (After ShapIro[!].)

... ~

1.0

0.9

~:-("~

c 0.8

~

-"0 8

~ 0.7

~~

~

------ Gr

t------

~ ~" ~ ""::::: I ~L\, , I'---- 30~

,!2

ee 0.6 o

~

~ O. 5

Normal 4

s~:ck , ~ ~> ~

3

~-...-:: "-35"--

~

O. 2 y~

o.1

1.0

1.5

2.0 2. 5 Initial Mach' nurnber, M\

3.0

I I

V

1.,4

3.5

FIG. 3- 11. Stagnation pressure ratio ver sus inlet Mac h number , with turn ing angle as pa ramete r [I]. Curves above dashed line hold for M z > 1 a nd cu rves below hold for M , < 1. (After Shapiro [11.)

shoc~

1.0

1.5

/rV

5mlx or

-~--,

o.5

/

7V 1 7 ~ ~~~H'\~,:= __

Normal

-tw--

I-- f-..-

V

// V/ 2~ / / V/ f/ / ;;;/ / V

~\ t~1\

\

50 10 0

~~,

I.

V

/

V 1/

V

1/

11VI 0/1

Mach wa ve

Deflection angle 0= 00

5~~V V

I/ V lr

u

~

2.5

2.0

1.5

Dd1ection angle

0

.c

,~

10

1/ / 1/ VV VV

2. 5

E

----r:::: ~ ~

30

~,

61

I I

:

2.0 2.5 Initial Mnch !lumber, M\

: 3.0

3.5 )' = 1.4

FIG. 3- 12 . Exit Mach number versus inlet Mach number, with turning ang le as parame ter [1]. Curves above dash ed line cor respond to small (J and curves below corn. spond to large~. (After Shapiro [I].)

be determ ined from M

In

T 2 / T, and the condi tion

and M 1 and the d ownstream Mach num ber from M 211, lilt

= "2 /-

Alternatively, the above four eq ua ti ons, alo ng with the equa tion of sta te, can be reduced to a set of four independ e nt equati o ns relati ng the va riables M I , M 2, ",II'" p ,;p 1, fT , and 0 [I]. If any two variables are give n, for exam ple M 1 and the turni ng angle 0, then all others are de term ined and the downstream (or fina l) conditions can be founel in terms of th e upstream conditi o ns. The solution of these equations from Shapi ro [IJ is given in Figs. 3- 10, 3- 11 ancl 3- 12 fo r th e case f = l .4 and fo r various v:.1 1ucs of o. Axisym metric oblique shocks arc di scussed by Shapiro [3], who presents a comp lete solution for the case of th e conical shoc k as we ll as methods of treatment of the general axisymmet ri c shock problem .

62

STEADY ONE-DIMENSIONAL FLOW OF A PERFECT GAS

3- 6

References 1. SHAPI RO, ASCHER H., Tire Dynamics and Thermodynamics 0/ Compressible Flllid Flow Volume I. New York : The Rona ld Press Company, 1953; Chapters 1-8, 16 2. 'KEENAN , J. H., and J. KAYE, Gas Tables. New York: John Wiley & Sons, 1957 3. SHA PIRO, ASCHER H., op. cit., Volume H, Chapter 17

3-6

SHOCKS

63

5. A Mach 2 flow passes th rough an oblique shock as show n, a nd deflects 10°. A second obliq ue shock reflects from the solid wall. Wha t is the pressure ratio across the twoshock system? It may be assu med that there is no boundary layer nea r the wall; i.e., the flow is uniform in each of the regions bou nded by the shocks.

M =2

10° • ~

Refl ec ted

oblique shock ~ PROBLEM

Problems 1. Show that for a pure substance the stagnatio n pressure cannot increase for an adiabatic zero-wo rk process. During what physica l processes might it increase for a zero-wor k process? 2. Show that, for a perfect gas, the Mach number at the nose of the Fanno and Rayleigh lines is indeed unity. The nose is defined by ds/(rr = O. 3. A perfect fluid expa nds in a frictionless nozzle fr om stagnation conditions Po = 600 psia, T o = SOOooR , to ambien t pressure of 15 psia . If the expansion is isentropic, determ ine the followin g conditions a t the fin al pressure : (l) velocity, (2) Mach number, (3) temperatu re, a nd (4) area pe r un it ma ss flo w. How does the fi nal flow a rea compare with the th roat area for a give n mass flo w? The specific hea t ra tio 'Y is 1.4 and the molecul ar wei ght M is 30.

MI~ ~ f-:o,.--:=----- ----,=_ T

P:~

L----

2

M, T, PI

PROBLEM 4

4. Air fl ows through a cy lindrica l combustion c hamber of diameter 1 ft a nd lengt h 10 ft. StationsQ) andQ) are the inlet and ou tlet, res~cti velY .. The inlet stagna ti on temperatu re is SOooR and thel in let stagn at ion pressure IS 200 P SIa. (a) H the skin friction coefficient is C[ = 0.0040 (a pproximately uniform) and no comb ustion ta kes pla ce, wha t will the inlet Mach number be if it is subsonic and the exit Mac h number is un ity? What are t,he values of stat ic and stagnat ion pressures a t the exit? (b) If combustion takes place and the heat of reaction is 300 Btu per pound of mixture, neglecting fri ctional effects, and if the inlet Mach numbe r is 0.25, what is the exit Mach number? (c) If, in a simi lar problem, the effects of combust ion an d skin fri ction on ~he flow are o f the sa me order of magn itude, how might thei r simu lta neo us effects be estimated ? The flui d may be cons idered a perfect gas with 'Y = 1.4 a nd M = 29.

5

6. Air ente rs a constant-a rea duct at Mach 3 a nd stagnation cond itions 13000 R a nd 185 psia. In the duct it undergoes a frictionless energy-tra nsfer process such that the exit Mach number is unity. Consider two cases : (a) normal shock at in let to the duct, a nd (b) shockfree supersonic heating. Determine the stagnat ion temperat ure and press ure at the exit in each case. Is there any reaso n why the total energy transfer should d iffer (or be equal) in the two cases? 7. A uniform mixture of very sma l1 solid particles and a perfect gas expands adiabatica lly fro m a give n cha mber tempera ture through a given pressure ratio in a nozzle. The particles a re so small th at they may be considered to trave l with the loca l gas speed. Since the density of the solid material is much higher tha n the gas phase, the total volume of the pa rticles is negligible . The rat io of solid to gas flow ra tes is Ji., where o < J1. < 1. Consider two cases : (a) no heat tra nsfer between solid a nd gaseous phases, and (b) the solid particles have the loca l gas temperature at all points in the flow. Qualitatively, how does the presence of the so lid pa rticles affect the ve loc ity at the end of the expansion? Show that fo r both cases the mi xture is equivalent to the adia ba tic expansion of a homogeneous gas of different molec ular weight M and speci fic heat rat io 'Y . 8. A 30-ft3 rocket- propellan t chamber is filled with combustion gases at 60000 R and 1000 psia at the instant combustion ceases. If the throat area of the nozzle is 1 ft 2 esti mate the time it takes for the chamber pressure to drop to 100 ps ia. Assume tha~ at the insta nt combust i ~ ceases the chamber propella nt supply stops completely, a nd also that l' ~ 1.2 and M ~ 20. 9.

(a) Compare the work of compression pcr unit mass of ai r for both reversible ~~i~batic a nd reversible isothermal compress ion through a press ure ratio of 10 with InIttal conditio ns SOQoR and 14.7 psia. Is the importa nce of specific hea t variation large? Why should cool ing d ur ing compression red uce the wo rk ? . (b) In a n effort to provide a continuous cooling process during compression wate r IS sprayed int? a n airstrea m ente ri ng the axia l compressor of a gas turb ine ~ngine. The compressIOn may be taken to be reversible an d the evaporatio n rate is ap proximately.dw/ dT = k, where HI is the wa ter-a ir ma ss ratio , T is the mixture temperatu re, and k IS a constant. If w« I , the process may be simplified by consider ing the sale effect of the evapo ration to be e nergy trans ferred from the gas phase . Show how the fi na l temperat ure and the work of compression depend o n k.

4-1

THE VISCOUS BOUNDARY LAYER 2

4

"\

supercritica;/

1,,\~,SUb:itical

Boundary Layer

-2

Mechanics and

-3

Heal Transfer

4-1

65

THE VISCOUS BOUNDARY LAYER

Historically the development of boundary layer theory fo~lowed the development of mathematical solutions for flows assumed to be nonvlscou~. In th~ latter half of the nineteenth century-after Euler had formulated the ba~lc equatlOns of motion-Helmholtz, Kelvin, Lamb, Rayleigh, and other~ ap~hed them t.o a number of physical situations. By neglecting the effects of VISCOSIty they obtaIned elegant mathematical descriptions of various flow field~. .I~ was argued that a fluid like air, for example, which is so rarefied as to be InvlSlble, must ~ave negligible viscosity. However, with some exceptions, the so-called perfect-flUId theory disagreed strongly with the results of experiment.

(a) Perfect fluid

(b) Actual fluid, very low velocity

(c) Actual fluid, low velocity

FIG. 4-1. Flow patterns about a cylinder. One particular example, which came to. be kn~wn as d'Alembert's .paradox, attracted a great deal of attention. As an illustratIOn.' th~ flow of a flUId over a cylinder as predicted by nonviscous theory is shown In Fig. 4-:-1 (a). The theory predicts that the velocity and pressure fields are both s~mmetflcal about a plane norma I t 0 th e ups t re am velo city vectors ' Thus . the flUid , can . .exert no net forces (lift or drag) on the cylinder. Moreover , .th~s c?ncluslOn IS Independent o.f the shape of the body; if the fluid is truly invlscld, It can be shown mathematically 64

\ o

)(

I

I /

r---

r! - --

~.../ '/ 2

- - - Perfect-fluidtheo ry prediction -

-

- Measured results

FIG. 4-2. Pressure distribution on a cylinder. (After Flachsbart [11].) that for any body shape the net integrated pressure force upon the body will be zero. Of course, this conclusion directly contradicts experience. Everyone knows that it takes a finite force to hold a body immersed in a moving stream or to propel it through a stationary mass of fluid; hence the paradox. Observations of the actual streamline pattern around the cylinder reveal something like Fig. 4-1 (b), which suggests that the streamline symmetry is grossly disturbed. On the rear side of the cylinder the fluid seems almost stagnant when the stream velocity is quite low. For higher speeds, fairly violent oscillations take place, which affect the entire streamline pattern near the cylinder. Under certain conditions it is possible to observe a periodic shedding of vortices alternately from the top and bottom sides, as suggested by Fig. 4-1 (c). This downstream flow pattern has been called a vortex street. The drag of the actual fluid on the cylinder is largely due to the asymmetry of the pressure distribution on its surface. Figure 4-2 shows measured pressure distributions on a cylinder and , for comparison, the prediction of perfect-fluid theory. The symbol p stands for fluid density and the pressure and velocity far upstream are denoted by poo and uoo . It can be seen that two experimental pressure distributions may be measured (the difference between the two will be discussed later) and that both differ greatly from the theoretical result. The fact that the average pressure on the cylinder is much higher on the upstream hemi-surface explains most of the drag. In addition to this, the viscous force of the fluid on the cylinder imposes a drag force which, in this particular case, is considerably smaller than the pressure drag. For " streamlined" bodies, which have much less pressure drag, the viscous force is relatively more important. For quite a period the discrepancy between existing theory and physical results was unexplained, and a marked rift developed between the theoreticians, who continued to expound the perfect-fluid theory, and the engineering-oriented workers like Bernoulli and Hagen , who actively developed the empirical body of knowledge generally called hydraulics . The equations of motion which do take viscosity into account-the Navier-Stokes equations- were formulated but were

66

BOUNDARY LAYER MECHANICS AND HEAT TRANSFER

4-1

so very complicated that they could not, in general , be solved. It appeared exceedingly difficult to improve mathematically on the perfect-fluid theory except in a few very special cases. Then Prandtl (in 1904) made a major contribution to the solution of the problem by introducing the concept of the viscous boundary layer. . He showed that even for fluids of vanishingly small viscosity there is a thin regIOn near the wall in which viscous effects cannot be neglected. Since the velocity of the fluid particles on the wall is zero, there may be, under certain conditions, a region of large velocity gradient next to it. Then, even though the viscositY.M may be s.malI,. the shear stress on a fluid layer, T = McaU/ ay), may be large In that regIOn since au/ ay is large. This region is called the viscous boundary layer. ~ bound.ary layer of thickness 0 is indicated in Fig. 4-3. Outside t~e laye.r, the velocity gradient au/ay will be so small that viscous shear may be qUIte unlmportant, so that the fluid behaves as though it actually had zero viscosity.

FIG. 4-3. Boundary layer velocity distribution. Thus Prandtl showed that the flow field may be broken into two parts; a thin viscous zone near the waIl, and an outer zone where the nonviscous theory is adequate. This suggested the major simplification o~ the ~avier-Stokes e~uati~ns which is the basis of boundary layer theory. For fairly hIgh-speed flow, In WhICh the boundary layer is so thin that the pressure gradient normal to the wall is negligible, Prandtl and others showed how to treat the boundary lay~r mathematically for special cases, many of which have been solved in the past SIX~y y~ars. Given that a boundary layer exists on a body, it is easy to see qualitatively why the case of vanishing viscosity is fundamentally different from the case of zero viscosity . Figure 4- 2 shows that for zero viscosity a particle on ~h~ su:face streamline of the cylinder rises in pressure to the stagnation value when It Impinges upon the cylinder (8 = 0). Then the pressure falls to a minimu~ (~nd the velocity rises to a maximum) at the position 8 = 7r/2. From that pOint It travel~ to the rearward stagnation point (e = 7r), having just enough momentum to chmb the "pressure hill" and arrive at the stagnation point. . However small the viscosity, the viscous case is fundamentally different. The fluid near the wall has been slowed by viscous action. When a particle in the boundary layer reaches e = 7r/ 2 it will have reduced momentu~ so th~t only a small pressure rise will stop its forward motion, or actually send It movmg back-

THE VISCOUS BOUNDARY LAYER

4-1

--

/ / /

67

Compressor cascade

FIG. 4-4.

Subsonic diffuser

Rocket nozzle

Some practical devices in which separation may occur.

ward. For this reason the particle can no longer follow the contour of the wall. The accumulation of stagnant fluid on the back part of the cylinder deflects the outer streamlines, as suggested in Fig. 4-1 Cb). This in turn greatly modifies the wall pressure distribution, as shown in Fig. 4-2. When the streamlines in the vicinity of the wall cease to follow it, they are said to separate. This accumulation of stagnant or near-stagnant fluid and the resultant gross distortion of the streamlines means that the boundary layer behavior can have an important influence on overall flow behavior even though the quantity of fluid directly affected by viscosity is a small fraction of the total flow. Three separated flows of practical significance are shown in Fig. 4-4. Both the compressor cascade and the subsonic diffuser are designed so that the mainstream velocity decreases in the flow direction; hence the flow is against an adverse pressure gradient. If the pressure rise is too great the boundary layer will separate, creating regions of near-stagnant fluid as shown. Within rocket nozzles the pressure gradient is usually favorable, so that separation need not be a problem. However, under certain off-design conditions shocks can occur within rocket nozzles and the sudden pressure rise across the shock can cause separation. Separation and its consequences in these devices will be discussed in the following chapters where appropriate. Our purpose in this chapter is to establish the nature and cause of separation and to indicate how one might predict its occurrence (although this is seldom possible with purely theoretical methods). Separation is, of course, only one phase of boundary layer behavior. We shall be interested also in the behavior of non-separated boundary layers, since they are of importance to heat transfer and other phenomena. The classic picture of separation is illustrated in Fig. 4-5, which shows a series of velocity profiles in a boundary layer as it approaches and passes the separation point. As the pressure rises the free-stream velocity U falls, as the Bernoulli equation would predict. However, the effect of a given change in pressure is greater on the slow-moving fluid near the wall than on the free-stream fluid.

68

4-1

BOUNDARY LAYER MECHANICS AND HEAT TRANSFER

4- 1

THE VISCOUS BOUNDARY LAYER

69

cussed SUbsequently.) With these assumptions , the momentum equation requires that

1)1-______ u y

j

J

./

7

!

J

~

/

~/ ~/~~~ ~/

]

/ Streamlin e -Z-Which Ie ft ~___ the wall Recirculating flow ~~

-7

f--- x

=

-u du (boundary layer),

where, in keeping with boundary layer assumptions, the free stream and the boundary layer experience the same pressure change. Equating the two expressions for dp and defining the ratio a = u/ U, there results

J

/

p

p

---1

J

dp

dp = _ U dU (free stream),

da = _

d~ (~ -

pU2

(

2 )

.

a

Hence if the pressure rises (dp > 0), a decreases and the boun.dary layer fluid is decelerated more than the free stream. Noting that -dp/ pU 2 is simply dU/ U, this expression can be integrated (for constant density), yielding

Separation point

FIG. 4-5.

A simplified picture of the development of separation.

I -

The fluid near the wall can be slowed to a stagnant state and then be made to flow backward if the adverse pressure gradient continues. The separation point is generally defined as that point where a streamline very near the wall leaves the wall. In two-dimensional flow this corresponds to the point at which the velocity profile has zero slope at the wall. That is ,

(au) ay

=

y=O

°

ai

where the subscript 1 refers to an initial condition. The thickness 0 of the boundary layer can be found from the equation for continuity which requires that

or, again for constant density, OIl_Ul •

(two-dimensional separation).

OIU

Figure 4-5 suggests that, past the separation point, there is a region of recirculating flow. In practice this region is highly unstable. It may be quite difficult to determine analytically, since once separation occurs there is a strong interaction between the body of stagnant fluid and the free stream. Often there results an adjustment of the free-stream geometry such that the adverse pressure gradient is reduced to practically zero, in which case there is no force to drive the recirculating fluid. The effects of pressure gradient on the shape of the boundary layer, which are shown in Fig. 4-5, U may be explained in a simplified manner as follows . Let the boundary layer be replaced by a hypothetical discontinuous flow shown in Fig. 4- 6, in u which the continuous variation in velocity normal J to the wall is replaced by two step changes. Fur~ ther in order to focus on the effect of pressure gradient, let us for the moment neglect shear FIG. 4-6. Discontinuous stresses. (The effects of shear stresses will be dis- boundary layer. ~

~

~

~

T

The two quantities a and 0 are plotted against U/ U l in Fig. 4-7 for the particular case al = l It may be seen that this boundary layer separates (a -70, 0 -7 U) ) when the free-stream velocity has decreased by less than 14%. With this picture of the relatively large effect of a given pressure rise on the more slowly moving fluid, it would appear that any pressure rise applied to a real boundary layer would cause separation since, if the velocity goes continuously to zero at the wall, there will always be some fluid with insufficient momentum to negotiate the pressure rise. However, this tendency toward separation is resisted ~p to a point, by visc?us stress. That is, ~he slowly moving fluid can be "dragged'; hrough a pressure flSe by the faster flUid farther from the wall. An important ~etho~ of reducing. the separation tendency is to enhance this "drag" force by InCreasmg the effective shear stress through turbulent motion . The mention of turbulent motion introduces another important aspect of boundary layer behavior. In 1883 Reynolds showed that under certain conditions dye particles injected into a pipe flow moved smoothly along streamlines without late~al fluctuations . In the same pipe at higher velocities, this smooth laminar motLOn broke down into turbulent motion consisting of rapid random fluctua-

a 4-1

BOUNDARY LAYER MECHANICS AND HEAT TRANSFER

70

1.0,----- --

t -/

5 4

\ \

\_1

I

o0.7

( 0.8

0.3

....

0.2

-

=0

pDD. j..L

0.1

1.0

V

00

u

V

VI

FIG. 4- 7. "Growth" of step boundary layer near separation for a 1 = t·

FIG. 4-8. Comparison of laminar and turbulent velocity profiles.

tions superposed on the mean motion. After the flow had become t~rbulent the dye lines rapidly mixed with the surrounding fluid, the macroscoplC tu.rbul~nt mixing process being at least an order of magnitude faster than molecular diffusIOn (the only agency for mixing in laminar flow). . . The same transition behavior takes place in the boundary layer, resulting In a significant reduction in its tendency to sep~rate. The reaso~ for .this is that the turbulent fluctuations constitute a mechanism for transportmg high momentum from the outer part of the layer to the region near the wall.. The effect is ~s if there were an increased shear stress or, as will be shown, an Increased coefficlent of viscosity. This tends to raise the wall shear stress, but it also means that the turbulent boundary fluid can climb higher up the " pressure hill" than the fluid in the laminar layer. A comparison of typical turbulent and laminar boundary layer velocity profiles is indicated in Fig. 4-8. The fact that the turbulent layer tends to have much higher momentum (and shear stress) near the wall gives it an increased resistance to sep.arati.o~. As a method of specifying the resistance of a boundary layer to separatIOn, 1t 1S useful to define a pressure coefficient C p as f::.p

Cp

71

geometry is completely specified (far from the entrance) by pipe diameter D and wall roughness. The pertinent fluid properties are density p , viscosity j..L,and the average velocity D. These variables can be arranged in a dimensionless form known as the Reynolds number, Re, so that Re

0 0.9

THE BOUNDARY LAYER EQUATIONS

y Ii

ex

ii

',ii i

--------,

0.4

fa -

-

0.5

4-2

=

lxPU2'

where U is the free-stream velocity at the point where the pressure begins to rise . and f::.p is the total increase in pressure up to the point in question. For an approximate comparison of cases in which the adverse pressu~e gradIent is not abrupt (i.e., where the shear stress has time to act) it may be saId that the laminar boundary layer will support a pressure rise given by 0.15 < C p < 0.2, whereas the turbulent boundary layer can tolerate without separation a pressure rise corresponding to 0.4 < C p < 0.8, depending on the Reynolds number and the upstream pressure distribution of the flow. . .' The onset of turbulence, or transition as it is called, IS dependent In a given device on fluid properties and velocity, the general or upstream turbulence level, and the geometry of the particular device. In the case of pipe flow, for example,

Reynolds found that for "smooth" pipes (commercial pipes and tubing are usually "smooth") and very quiet inlet conditions, the transition process could in fact be correlated with a critical Reynolds number below which the flow was laminar and above which it was turbulent. More generally it is found that transition in any series of geometrically similar devices can be correlated with the Reynolds number, where D is replaced by some characteristic length Land D by some characteristic velocity U. Thus Re

=

pUL. j..L

Obviously, a critical value for Re can have meaning only within the geometry for which it was defined . With these facts in mind, the explanation of the two pressure distributions observed on the cylinder of Fig. 4-2 becomes straightforward. For reasonably low velocities (below a "critical" Reynolds number) the boundary layer remains laminar and separates readily on the back of the cylinder' before much pressure rise has taken place. At the critical Reynolds number, the boundary layer becomes turbulent and is able to flow considerably further around the back of the cylinder before separation . As it does so it rises in pressure, thus reducing the discrepancy between inviscid and actual pressure distributions. Since the drag on a blunt body such as a cylinder is primarily a pressure force (as opposed to skin friction), the onset of turbulent flow in this case is actually accompanied by decreased drag. In most practical fluid machines to be discussed in this book the bound ary layers a.re turbulent, and so long as this is true the influence of Reynolds-number variations on overall performance is usually minor. However, under extreme conditions, as for exampl e at very high altitudes, the Reynolds number can become low enough that portions of the flow revert to laminar flow. There usually follows a rather substantial reduction in efficiency of performance, due to the increased tendency toward separation.

4-2

THE BOUNDARY LAYER EQUATIONS

T~e general equations for the flow of Newtonian viscous fluids are called the

Nav~er-Stokes e~uations [1,2].

Because of their mathematical complexity, it is pOSSIble to obtam exact solutions of them for only a few physical situations. I many cases, .howeve:, it is legitimate to make simplifying approximations an~ thereby obtaIn solutions whIch have a useful range of validity. The boundary

72

4-2

BOUNDARY LAYER MECHANICS AND HEAT TRANSFER

4-2

THE BOUNDARY LAYER EQUATIONS

73

y

Table 4-1

u

_ _ _+-__TX_+ ..

aTx

Surface

ay dy

Mass flux

(}),---- -----,Cll I I

I I I ~ I

I~ I

I I p

pV

I I

4-9. Small control volume for development of boundary layer equations. FIG.

~ x

+-__

_ _+-1

Control volume

+ aTy d

p( u + ~ dX)

+ fJ! dx ax

-----J(D

layer equations are much more tractable than the Navier-Stokes equations. They may be derived by neglecting certain terms in the Navier-Stokes equations which can be shown to be quite unimportant in boundary layer flows. The method is developed by Schlichting [I J in careful detail. In the following discussion , equations will be developed less rigorously, but more directly, for the special case of the steady, two-dimensional, incompressible boundary layer. Shapiro [3J uses this approach to develop equations for compressible flow, and includes a good discussion of the range of validity of the simplifications. The object in view is to lay a foundation for a quantitative description of both laminar and turbulent boundary layers, with particular emphasis on skin friction, separation, and heat transfer. Although the solutions of the boundary layer equations which are available are seldom directly applicable to the complex flows in real machines, they at least illustrate the forces at work and the behavior typical of the boundary layer. Consider a very small control volume within a boundary layer, as shown in Fig. 4-9. Velocities in the x- and y-directions are u and v, respectively, while the shear stresses in these directions are Tx and Ty • To apply the continuity and momentum equations to this control volume, we shall need to know the mass and momentum flux across each surface and the force (shear and pressure) acting on each surface. The outward fluxes of mass and x-momentum are shown in Table 4-1. Since the flow is taken to be two-dimensional, no fluid crosses the surfaces parallel to the xy-plane. The continuity equation for steady, two-dimensional, incompressible flow requires that p (u

or

+ :~ dX) dy

-

+ p (v+ ~~ dY) ~':!. + av = o.

pu dy ax

ay

dx -

+ ~: dX) dy

CW)

p (u

@@

-pudy

COO)

p (v

@)(D

-pvdx

+ :; dY)

(u + ~: dX)

2

dy

dx

p (v

+ :;dY)

(u

+ ~;dY) dx

-pvu dx

The momentum equation in the x-direction, using Table 4-1 to obtain the net outward momentum flux, is (applying Eq. (2-4»

L Fx =

p (u

+ ~~ dX) 2 dy -

2

pu dy

+ p (v + :; dY )

(u

+ ~; dY ) dx -

puv dx.

Expanding and disregarding second-order terms, we have

L Fx = p (2U axau + u ayav + v ayau) dx dy. Using the continuity equation, we may simplify this to

L Fx

= p (u au

ax

+ v au) dx dy. ay

~ince only pressure and shear forces act on the control surface, the force summatiOn may be written, in accordance with Fig. 4-9, as

p pdy - (p + aax dX) dy + (Tx + aTX) ay dy dx (-

Tx dx

~~ + ~?) dxdy.

Thus the momentum equation may be written

X

(4-1)

p

_pu 2 dy

p (u aau

pu dx = 0

Flux of x-momentum

+

~p. + aTayx .

u v aa ) = _ y ax

Th~ shear ~tress T developed in a Newtonian fluid is proportional to the rate of sheanng straIn. For the particular case of the boundary layer, in which there is a

74

4-2

BOUNDARY LAYER MECHANICS AND HEAT TRANSFER

large velocity gradient close to the wall, the shear stress is very nearly equal to

4-2

THE BOUNDARY LAYER EQUATIONS

Thus this equation may be written

f

au ay

f..t-'

h

o

where f..t is the coefficient of viscosity. Using this definition, the momentum equation for the x-direction can be written

75

(u

au ax

h

au + v oy

_

dV) U dx dy

f

=

0 /I

2 a u ay2 dy.

Integrating the right-hand term and noting that the wall shear stress is given by TO

=

f..t

(au) , ay y=o

while au/ oy is zero outside the boundary layer, we can write (for constant As a consequence of the thinness of the boundary layer, pressure changes normal to the wall may be neglected inside the layer, so that it may be assumed that p is a function of x only. Then the momentum equation may be written

h

u ou / o ( ax

+ v au

v dV) d

_

oy

dx

= _

y

/I)

TO.

P

I ntegrating the second term in the integral by parts, we have (4--2)

where /I =::0 f..t/ p is the kinematic viscosity. The momentum equation as derived here assumes a fluid of constant viscosity and constant density. Although these conditions may be satisfied within many low-speed laminar boundary layers, we shall see that the "effective" viscosity in turbulent flow may vary strongly through the boundary layer. Further, p and f..t could vary as the result of temperature variation within the boundary layer, due either to high heat-transfer rates or, in compressible flow, to the variation of static temperature in response to velocity variation. In such cases a complete set of equations must include at least an energy equation and an equation of state. However, even in cases not requiring the simultaneous solution of an energy equation, Eqs. (4-1) and (4-2) can present considerable difficulties in the solution for u and v (where p is assumed a known function of x). It is desirable, indeed necessary in most cases, to utilize approximate methods for solving the boundary layer equations. An extremely useful approximation, the momentum integral method, has been developed and widely applied. The object of the method is to find a solution which will satisfy the integral of the momentutn equation across the boundary layer even though it may fail to satisfy the equation at particular points. The integration of the momentum equation (4-2) may proceed as follows:

foh(u auax + v ayau) dy = fh 0

1 dp

-

P dx dy +

fh 0

lJ

a2u ay2 dy,

where h is an undefined distance from the wall outside the boundary layer. the free stream, I dp UdU. dx dx

P

In

h hVaudY f o oy

vul

=

j'h u~1!.dY.

_ 0

From the continuity equation, v at y

= h is given by

h

v

/

ay

0

av -dy o ay

h

au

= - /0 -dy. ax

Thus

{ h au v ay dy = -

} 0

v

fl< au 0 ax dy

+

fh 0

au u ax dy.

The momentum equation may then be written

l< fo

(2U ax au

-

v au

ax

-

v dV) dx

dy

=

TO

p

or, rearranging,

fo h

a~ [u(V -

u)] dy

+ ~~ fo h (U

- u) dy =

+ ~o.

(4--3)

Both the integrands are zero outside the boundary layer, so h can be indefinitely large. . Equation (4-3) is arranged in this particular form because of the physical sig?Ificance (and the common usage) of the individual integrals. In the second Integral, (U - u) is the mass flux defect (actually I/p times the defect) that occurs as the result of the deceleration of the boundary layer fluid. In the case of flow within a duct, for example, if there is a mass flux defect near the walls there must be an increased mass flux within the free stream. This effect on the free stream would be the same if there were no boundary layer but the duct walls were dis-

76

BOUNDARY LAYER MECHANICS AND HEAT TRANSFER

4-3

4-3

lAMINAR BOUNDARY lAYER SOLUTIONS I. 0

placed inward an amount 0* so that Vo*

=

10'"

(V -

0.8

u

U 0.4

0*

=

faoo

(I -

~)

e = faoo ~ (I - i ) dy.

(4-4a) (4-4b)

I

R

1.0

=

Ux v

o 1.08 X 10 5 o 3.64 XIO·; /:,. 7.28x I05

2.0

3.0

4.0

5.0

5.646.0

7.0

q=y!!Z.

~~ Velocity distribution in the laminar boundary layer on a flat plate at zero InCidence, as measured by Nikuradse [IJ . ~IG. 4-10.

The simplest flow is that over a flat plate where dpl dx is zero. In this case, Eqs. (4-1) and (4-2) reduce to

With these definitions, the momentum integral equation, Eq. (4-3), may be written

u au

ax

+ v au ay

=

with the boundary conditions

or

(4-5) As it stands this equation applies to both laminar and turbulent boundary layers. The solution, say for a given Vex), requires knowledge of the relationship between 0* and e, which may be found if an approximate velocity profile shape is assumed (0* 1 e is often called the shape factor H), and the skin friction TO is known. Note that shear stress is evaluated only at the wall where, even in turbulent flow, it is given by M(aul ay), since turbulent fluctuations must go to zero at the wall. 4-3

i

I

o o

.

dy,

/

V

( 00

ve=}o u(V-u)dy. Here e is called the momentum thickness of the bounda ry layer. The two terms are then defined as follows (for incompressibldow):

/1'"

0.6

0.2

~

/ 11

u) dy.

With respect to mass flux then, the boundary layer has the effect of making the thickness 0* unavailable for free-stream flow ; 0* is called the displacement thickness of the boundary layer. The first integral can be treated in a similar ma nner. Note first that since the limits of integration are not functions of x, the partial differentiation al ax can be carried outside the integral where it becomes d(dx. In this case we interpret u as l i p times the actual mass flu x, while (V - II) is the momentum defect per unit mass of the boundary layer fluid . Hence the integral is the momentum defect in the boundary layer which, as above, could be accounted for by the loss of free-stream thickness e such that 2

77

y = 0:

u

y= 00:

u = V = constant.

v

=

=

0,

A stream function t/; may be defined such that v

_ at/;.

=

ax

The convenience of this form is that it automatically satisfies the continuity equation so that the above two equations reduce to

LAMINAR BOUNDARY LAYER SOLUT:ONS

A variety of exactt solutions of Eqs. (4-1) and (4-2) are available, two of which will be mentioned here .

(4-6) This equation may be considerably simplified by substituting

t Exact in the sense that they do not neglect anI of the terms in the boundary layer equations. Most solutions are, however, expressed in terms of series expansions.

1/

= y ".,! V /vx

and

if;

= vvxU /(1/) ,