Aircraft Engines and Gas Turbines
`Jack L. Kcrrebrock
`
`GE-1033.001
`
`

`
`Aircraft Engines and Gas Turbines
`
`Jack L. Kerrebrock
`
`The MIT Press
`Cambridge, Massachusetts, and London, England
`
` '
`
`

`
`Contents
`
`Preface ix
`
`Acknowledgments xi
`
`1 I
`
`ntroduction to Concepts 1
`
`1.1 Thermal Efliciency 1
`1.2 Propulsive Eificiency 2
`1.3 Specific Impulse and Range 3
`1.4 Ramjets 4
`1.5 Turbojets 6
`1.6 Turbofans 7
`1.7 Shaft Engines: Regeneration 8
`1.8 Stationary Gas Turbines: Topping 11
`1.9 Energy Exchange, Mach Number, Reynolds Number
`1.10 Stresses 13
`1.11 Noise 13
`1.12 Thrust and Drag 14
`1.13 Some Engines in Cutaway 17
`Problems 20
`
`2 I
`
`deal Cycle Analysis: Trends 21
`
`2.1 Stagnation Temperature and Pressure 22
`2.2 The Ramjet 22
`2.3 The Turbojet 25
`2.4 The Afterburning Turbojet 28
`2.5 The Turbofan 31
`
`2.6 The Afterburning Turbofan 34
`2.7 The Turboprop 35
`2.8 The Regenerative Gas Turbine 39
`2.9 Gas Turbines for Topping 42
`2.10 The Importance of Turbine Inlet Temperature 43
`Problems 46
`
`3 Q
`
`uantitative Cycle Analysis 48
`
`3.1 Variation in Gas Properties 48
`3.2 Diffuser Pressure Recovery 49
`3.3 Compressor and Turbine Efficiencies 50
`3.4 Burner Efliciency and Pressure Loss 51
`3.5
`Imperfect Expansion Loss 52
`
` —j—j
`
`

`
`

`
`3.6 Heat Exchanger Effectiveness and Pressure Loss 53
`3.7 Turbujet with Losses 54
`3.8 Regenerative Gas Turbine with Losses 57
`3.9 Combined Gas Tu:-bine—Steam Cycles with Losses 61
`3.10 Concluding Comments 64
`References 65
`Problems 65
`
`4 N
`
`onrotating Components 66
`
`4.1 Summary of Gas Dynamics 66
`4.2 Diffusers 78
`4.3 Exhaust Nozzles 93
`4.4 Combustors and Afterburners 99
`References 113
`Problems 114
`
`5 C
`
`ompressors 1 16
`
`5.1 Energy Exchange, Rotor to Fluid 116
`5.2 Compressor Geometry and the Flow Pattern 123
`5.3 Limits on Stage Pressure Ratio and the Compromise with Efficiency
`and Mass Flow 143
`5.4 Stage Performance: Corrected Parameters 149
`5.5 Multistage Compressors 151
`5.6 Stall and Surge 154
`5.7 Centrifugal Compressors 157
`References 163
`Problems 163
`
`6 T
`
`urbines 166
`
`6.1 Turbine Stage Characteristics 167
`6.2 Turbine Blading 172
`6.3 Turbine Cooling 174
`6.4 Turbine Similarity 185
`References 186
`Problems 187
`
`

`
`7 S
`
`tructure of Engine Turbomachinery 189
`
`7,1 Centrifugal Stresses 189
`7.2 Gas Bending Loads on Blades 193
`7.3 Thermal Stresses 195
`7.4 Critical Speeds and Vibration 197
`7.5 Blade Flutter 203
`References 206
`Problems 206
`
`8 C
`
`omponent Matching and Engine Performance 208
`
`8.1 Compressor-Turbine Matching: The Gas Generator 208
`8.2 Gas Generat0r—Nozzle Matching 210
`8.3 Engine-Inlet Matching and Distortion 211
`8.4 Overall Performance 212
`8.5 Control and Acceleration 213
`References 220
`Problems 220
`
`9 A
`
`ircraft Engine Noise 222
`
`9.1 Noise Sources: Unsteady Flow 224
`9.2 Jet Noise 230
`9.3 Turbomachinery Noise 237
`9.4 Noise Measurement and Rules 245
`References 249
`Problems 250
`
`10
`
`Hypersonic Engines 251
`
`10.1 Hypersonic Inlets 253
`10.2 Heat Addition in High-Speed Flow 255
`10.3 Heat Release Due to Chemical Reactions 257
`10.4 Nozzle and Performance 262
`10.5 Cooling 263
`References 264
`Problems 265
`
`

`
`Contents
`
`11
`
`Propulsion Systems Analysis 266
`11.1 Takeoff 266
`11.2 Climb and Acceleration 267
`11.3 Cruise 271
`11.4 Maneuvering 273
`References 273
`Problems 273
`
`Index 275
`
`

`
`Preface
`
`This hook is intended to provide an introduction to the engineering of aircraft
`propulsion systems with the empltnsis on the engine, rather than on the dis-
`ciplines involved in its design. Because of the remarkable advances that have
`occurred since the large-scale introduction of gas turbine power plants into
`military aircraft in the [9505 and into commercial aircraft in the 19605, a
`clear tindcrstainling oi" the characteristics of these devices is needed at the
`undergraduate or early graduate level. Such understanding is essential both
`For entrance to professional work in the industry and for graduate study in
`the field. The understanding of a sophisticated engineering system that in-
`volves the sciences of Fluid mechanics, solid mechanics, chemistry, automatic
`control. and even psychology because of the problem of ait'crai'L noise, also
`has intrinsic value apart from its practical applications. At present, the
`ftindainental inf'ormatin'n required for such understanding is widely dispersed
`in the technical literature and subliterature. The aim of this hook is to draw
`the information together in a unified Form. so that the student can appreciate
`why aircraft propulsion systems have evolved to their present form and can
`thus be better prepared to contribute to their further evolution.
`Automotive and stationary applications ol' gas turbines are growing rapidly.
`They use much the same technology as aircraft gas turbines; indeed they have
`benefited greatly from the aircraft engine developments of the last two de-
`cades. While this book is directed primarily at aircraft engines, the discussions
`of component technology are equally applicable to these other applications.
`Some treatment ol‘ cycles has also been included for automotive and sta-
`tionary cngines in chapters 2 and 3.
`The approach in this hook is to treat the propulsion system at successively
`increasing levels of sophistication, beginning with a phenomenological dis-
`cussion in chapter 1 ol' the processes by which energy is converted from heat
`to meclianical energy to thrust.
`Several types of engines are then discussed in chapter 2 in the framework
`oi’ ideal cycle analysis, wliere the components of an actual engine are repre-
`sented parametricnlly in the analysis without quantitative reference to the
`engine structure. Here the dependence of the engine's performance on com-
`pressor pressure ratio and turbine inlet temperature is established, as well as
`the trends of thrust and spccilic impulse with flight Mach number. The argu-
`ments are repeated more quantitatively in cltapter 3 for a narrower spectrum
`of engines to convey the influence ol‘ nonidenlitics in the engine cycles.
`Chapters 4, 5, 6, and 7 examine the mechanical characteristics required of
`eueli rnajor engine component to achieve the parametric behavior assumed
`in the cycle analysis. At this step the enormous literature of the field must be
`abstracted and interpreted to clarify the important physical limitations and
`trends without submcrging the reader in vast analyses and data correlations.
`Naturally, the presentation is strongly influenced by my own viewpoint. If it
`
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`

`
`errs in detail or by omission at some points, I hope that the overview will
`help the serious student to correct these inadequacies for liimsclll
`Cll3[_'|lEI.' S syntltesizes, from the cortiponent characteristics evolved in
`clinpters 4. 5. and 6, 2: complete gas generator and a complete propulsion
`system. An attempt is made to lreat in a reasonably uniform way the prob-
`Icms 0|‘ engine control, inlet-engine and engine-no’/7.le mulching, and inlet
`distortion, which so strongly dictate the ultimate performance of the system.
`The mechanisms by which aircraft engines produce noise are discussed
`in chapter 9. At its present state of development, this subject is both highly
`mathematical and highly empirical. While the mathematics of this chapter is
`somewhat more advanced than that in other chapters, it should be under-
`standable to a well-prepared college junior or senior. In any case, some care
`has been taken to make the physical arguments independent of the mathe-
`matics.
`
`Since flight at very high Mach numbers leads to complex chemical behavior
`of the air as it passes through the engine, the possibilities for airbreathing
`propulsion at Mach numbers above 6 are discussed separately in chapter 10,
`where the thermochemistry of high-temperature combustion products is
`included.
`
`l deals with some of the simpler techniques of propulsion
`Finally, chapter ]
`systems analysis, the tool used by the prelhninary designel‘ to determine which
`engine should be connnittetl to the lengthy and costly process of development.
`To understand this text, a student neetls good first courses in gasdynamics.
`thermodynamics, and solid mechanics along with the appropriate mathe-
`matics. These subjects will not be reviewed here, but some of the results of
`compressible flow are collected at the beginning of chapter 4.
`,
`Though this book developed from a one-semester undergraduate course in
`aircraft engines at MIT, it contains more inlhrmzition than can reasonably be
`taught in one semester. A good one-semester untlergratluate course in aircraft
`engines might consist of chapters 1 and 2 and the Following selections from
`the remaining chapters:
`Chapter 3: 3.1-3.5, 3.7
`Chapter 4: 4.1, 4.2.2.1—4.2.2.3, 4.3, 4.4.3, 4.4.4
`Chapter 5: 5.1, 5.2.2, 5.2.3, 5.2.5, 5.3, 5.4, 5.5, 5.6
`Chapter 6: 6.1, 6.1.1, 6.2, 6.3, 6.3.1, 6.3.2, 6.4
`Chapter 7: 7.1-7.3
`Chapter 8: 8.1, 8.2, 8.4
`The text in its entirety is suitable for first-year graduate students with no
`prior exposure to aircraft engines.
`
`

`
`Acknowledgments
`
`My understanding of aircraft propulsion systems has benefited from associa-
`tions with many persons—from academe, from the industry, from NACA—
`NASA, and from the armed services——over the last two decades. Edward S.
`Taylor, James E. McCune, Jean [7. Louis. Alnjzy A. Mikol-.1_ic7.ak, and
`Leroy H. Smith have been pt1rl.ica.1larly helpful in t‘ormulatEi1g sr.-me of the
`arguments in chapter 5. Those faniiliar with the Leacltitig I-1I.'It]
`l‘L'SCZ!1'C.i‘l of
`Frank E. Marble of the Calil‘ornia Institute ofTccl.1nology will recognize the
`powerful influence he has had, first as te-.m'.-lier and later as colleague, on me
`and on this book. Special thanks are due to my students at MIT, who
`provided the motivation for writing the book, and to my wife Vickie, who
`helped immeasurably in bringing it to completion.
`
`

`
`

`
`Aircraft Engines and Gas Turbines
`
`

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`

`
`1 I
`
`ntroduction to Concepts
`
`All aircraft engines and gas turbines are heat engines, in which thermal energy
`derived from the combustion of fuel with air (or derived, perhaps, from a
`nuclear reactor) is converted to useful work in one way or another. The
`efficiency of this conversion—the ratio of useful work output to energy input
`by the fuel or reactor—is of major and growing concern.
`When the useful output of the gas turbine is in the form of shaft power
`used to drive a wheeled vehicle, a machine, or an electric generator, the
`cfliciency may usually be characterized by the thermal efiiciency familiar from
`thermodynamics, defined as the fraction of thermal energy input converted to
`mechanical work.
`
`In aircraft propulsion the useful work of the engine is work done in pro-
`pelling the aircraft. It is appropriate then to define a second efficiency, the
`propulsive efficiency, as the ratio of propulsive work to total mechanical
`work. Although analogous efficiencies of utilization can be defined for other
`applications _of gas turbines, the propulsive efiiciency is particularly important
`because it plays a dominant role in determining the configurations of aircraft
`engines. The different types of engines—ramjets, turbojets, turbofans, and
`turboprops—result from optimizing the overall efficiency, which is the product
`of thermal efficiency and propulsive elficiency, for different flight regimes.
`Overall efficiency, however, is not the sole criterion for engine design. In
`aircraft engines weight and size are also important. Cost, while high for
`optimum engines, is important. Recently, takeoff noise has become a major
`problem for commercial aircraft operators, so that noise produced per unit of
`thrust has become an important criterion for engine design.
`In automotive applications cost limits engines to much simpler and less
`eflicient designs than those evolved for aircraft. For stationary applications
`reliability, efliciency, and cost are controlling, while size and Wei ght are much
`less important.
`The purpose of this chapter is to describe the fundamental characteristics
`of gas turbines that control and limit their adaptation to any of these applica-
`tions. Some are thermodynamic, some fluid dynamic, some mechanical.
`
`1.1 Thermal Elficiency
`
`The conversion of thermal energy to mechanical energy is subject to the laws
`of thermodynamics. These laws determine an upper limit on the thermal
`efficiency that depends only on the temperatures at which heat is added to
`and rejected from the working fluid of the engine. Most gas turbines use the
`atmosphere as a heat sink, so that the minimum available heat rejection
`temperature is the atmospheric temperature, denoted by To. The maximum
`available heat addition temperature is in principle limited only by the char-
`acteristics of the combustion process (or nuclear reactor). In practice it may
`
`

`
`Introduction to Concepts
`
`be limited by the temperature capabilities of materials. If this maximum heat
`addition temperature is denoted by Tm, the maximum possible thermal efli-
`ciency is that attained by a Carnot cycle operating between these temperature
`extremes, expressed by
`T
`,,c=1——°.
`Tm
`
`(1.1)
`
`In the stratosphere (between ll and 30 km altitude), T0 is approximately
`217°K. Current aircraft gas turbines have peak temperatures near 1500 °K,
`so that 71, is approximately 0.85. Alltomotivc and stationary gas turbines
`generally have peak temperatures below |3[)0"K, for reasons of cost and
`durability, and they reject heat at about 300".K_. so |'or lhem If‘. is about 0.77.
`Actual engines have tlierinnl clliciencies lower than lhese. For comparison,
`the maximum possible efliciency for steam power plants is near 0.66.
`
`1.2 Propulsive Efficiency
`
`Unlike thermal efliciency the propulsive cfliciency, representing conversion
`between two forms of mechanical energy, is limited only by the laws of
`mechanics and can in principle approach unity, It is defined as
`
`thrust power delivered to vehicle
`17” _ net mechanical power in exhaust'
`
`The numerator is equal to the thrust multiplied by the flight velocity, while
`the denominator is the product of the mass flow and the increase in kinetic
`energy imparted by the engine to the airflow.
`By the conservation of momentum, the force acting on the engine due to
`the flow through it is equal to the time rate of change of the momentum of
`the flow. If the mass flow per unit time is m, the flight velocity is uo, and the
`exhaust velocity is us, the thrust is
`
`F = rr'i(ue — uo),
`
`and the propulsive efficiency is
`
`(1.2)
`
`_ n'1 ue — uO)u0 _ 2u0
`(L3)
`11” _ m(uf/2 4 ué/2)_ue + u0'
`The propulsive efliciency decreases as the ratio of exhaust velocity to flight
`velocity increases. From (1.2), we can see that for a given mass flow and
`flight velocity, the thrust increases with u,/uo. Thus, a definite tradeoff must
`be made between propulsive cfliciency and thrust per unit mass flow. This
`relationship, shown in figure l.l, applies generally to all aircraft engines.
`
`

`
`Introduction to Concepts
`
`IO
`
`F/rhuo
`
`1.1 Propulsive efliciency as a function of thrust per unit of inlet air momentum, with ratio
`of exhaust velocity to flight velocity as parameter.
`
`Increased mass flow in general implies increased engine size and weight, and
`it may also increase drag.
`
`1.3 Specific Impulse and Range
`
`The discussion of engine types in terms of efliciencies links cycle analysis
`and thermodynamics, thus providing an intuitive grasp of propulsion system
`characteristics; but the propulsion system efficiency is usually characterized
`in terms of the specific impulse, defined as the number of units of thrust pro-
`duced per unit of fuel weight flow rate. This quantity enters directly into
`calculations of the fractional weight change of aircraft. It is denoted by 1.
`Suppose an aircraft is in steady, straight, and level flight. The thrust F
`must then equal the drag D. The aerodynamic performance of the airframe
`is characterized by its ratio of lift to drag L/D. Since the lift must equal the
`weight W of the aircraft, F = W/(L/D). Now the weight of the aircraft
`decreases as fuel is consumed; the rate of decrease is d W/dt = —F/I, by the
`definition of 1. Thus
`
`"_W _ _ L
`dt _
`I(L/D)’
`
`and if I and (L/D) are constant in time, the flight duration 1 is given by
`
`t = [(L/D) log
`
`m9._
`
`W,,— W,
`
`

`
`Introduction to Concepts
`
`where W, is the initial (gross) weight and Wf is the weight of fuel consumed.
`It is usual to present this result in terms of range, which is simply the pro-
`duct of the flight duration and the flight velocity uo, so that
`W
`(1.5)
`Range = u0I(L/D) l0g .
`Historically, much effort has gone toward increasing the range of aircraft.
`As a result, the fuel weight has become a substantial fraction of the gross
`weight, and the fraction Wg/( W9 ~ Wf) is considerably larger than unity.
`In this case, structural weight or engine weight affect the range logarithmi-
`cally, while I, uo, and (L/D) affect it directly, so a premium is put on these
`factors. On the other hand, when W9/( Wg — WI) is near unity, engine weight
`becomes as important as specific impulse, since it contributes to Wg.
`The specific impulse can be further related to this discussion of efliciencies
`by noting that the overall propulsion system efficiency is simply 11 =
`Fug/(-—dW/dt)h = Fuo/(F/I)h, where h is the energy content of the fuel. Thus
`the factor uol in (1.5) is simply 1111, the product of the energy content of the
`fuel (in units such as ft-lb per lb or m-kg per kg) and the efficiency with which
`it is used. The value of h for liquid hydrocarbon fuels isabout 4800 km.
`For hydrogen it is 14,300 km, and for methane, 5600 km.
`
`1.4 Ramjets
`
`Ramjets are conceptually the simplest of aircraft engines. Figure 1.2 is a
`schematic cross-sectional diagram of such an engine. Focusing for the present
`on the behavior of the airflow passing through the engine (indicated by the
`dashed inlet and exhaust streamtubes), we see that the schematic depicts only
`the internal functions of the engine. This engine consists of an inlet (diffuser),
`a combustor (burner), and a nozzle. The inlet decreases the flow velocity rel-
`
`inlet
`
`streurrrtub33
`
`fuel , Iii,
`
`diffuser
`
`".1
`
`burner
`
`"‘b»"b
`
`nozzle
`
`1.2 Schematic diagram of ramjet engine.
`
`

`
`Introduction to Concepts
`
`ative to the engine from the flight velocity uo to some smaller value uz. The
`difference in kinetic energies of the air (143/2 ~ u§/2) per unit mass is converted
`to an increase in thermal energy, so that T2 > T0; at the same time, the pres-
`sure increases from p0 to a higher value p2. Fuel is then mixed with the air, and
`the mixture is burned in the combustor. If the velocity uz is small compared
`to the sonic velocity (the Mach number M2 << 1), the combustion occurs at
`nearly constant pressure; the net result is that the thermal energy of the fluid
`increases, and its density decreases. In the nozzle the flow is expanded, ideally
`to the original pressure, with a consequent drop in temperature from T3 to
`T4 and an increase in kinetic energy 14,2,/2 — u§/2. Since T3 is larger than T2,
`the difference in thermal energies between stations 3 and 4 is larger than that
`between stations 2 and 0; therefore, the change in kinetic energy in the nozzle
`is larger than that in the inlet, and u4 is larger than uo. The change in
`momentum 144 — uo per unit mass flow provides the thrust.
`The conversion of thermal energy to mechanical energy is represented
`ide:-ilty by a Brayton cycle, as shown in figure 1.3. This cycle may be thought
`of us :1 sLI|1er|3osit.iun of a number of Carnot cycles, indicated by the small
`rectangles, ericli with a temperature ratio T2/To = T3/T4. Accordingly the
`maximum possible efficiency of the cycle is
`
`773 = 1 — To/Tr
`
`(1-6)
`
`The maximum efficiency can approach the limiting Carnot efficiency nc only
`if T2 approaches T3, that is, if all temperature rise occurs in the inlet rather
`than in the combustor. The thermal efficiency of the ideal ramjet is thus con-
`
`combustor
`
`1.3 Temperature-entropy diagram of Brayton cycle for ramjet, with elementary Carnot
`cycles of which it is composed.
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`Introduction to Concepts
`
`trolled by the inlet compression process, which governs the temperature ratio
`T2/T0. In the ideal case where uz << uo this ratio approaches the stagnation
`to static temperature ratio of the inlet flow, T2/To = 1 + -§(y — 1)M§, so that
`
`%(v — 1)M3
`_
`"” "1+ %(r —1)M3’
`
`(1.7)
`
`where M0 is the flight Mach number and y = cl,/c,, is the ratio of specific
`heats. Thus for M0 < 1, T2/To approaches unity and the thermal efficiency
`of the ramjet becomes small. It becomes a highly efficient engine for MO > 3.
`In the ideal ramjet u4/uo (and hence up) is determined by the combustor
`temperature ratio T3/T2. For a given uo, increasing T3 will increase thrust,
`but reduce propulsive efliciency.
`
`1.5 Turbojets
`
`The poor performance of the ramjet at low Mach number is improved in
`the turbojet engine, shown schematically in figure 1.4, by the addition of a
`compressor. The compressor raises the air pressure and temperature prior to
`combustion and thus improves the cycle efliciency. The ideal Brayton cycle
`for the turbojet is shown by the full lines in figure 1.5. The thermal efficiency
`is now given by
`
`113 = 1 — To/T3.
`
`(1.8)
`
`If, for example, the compressor pressure ratio is 12, corresponding to an ideal
`compressor temperature ratio of 2.03, the ideal thermal efliciency is about 0.5.
`To drive the compressor the turbine must have a temperature drop roughly
`equal to the compressor temperature rise. Because T4 > T3, T4/T5 < T3/T2,
`and it follows that p5 > p2. Thus the combination of compressor, combustor,
`
`mam"
`
`1!. T;
`-I'\.rJL
`‘ T J\:.r— 41...‘.
`nozzle
`diffuser
`compressor turbine
`[
`I
`burner afterburner
`
`1.4 Schematic diagram of turbojct engine.
`
`

`
`Introduction to Concepts
`
`compressor
`inlet
`
`S
`
`1.5 Temperature-entropy diagram for Brayton cycle of turbojet, with afterburning
`modification shown dashed.
`
`and turbine, called a gas generator, produces a rise in pressure as well as in
`temperature of the airflow.
`Materials limitations in-the turbine at present r£:sl.rict the turbine inlet tem-
`perature T_, to values below those CO!‘l’CSpO[‘Il.llI‘lg to a stoichiomctric mixture
`of fuel and air in the comhustor. so that the turbine exhaust gas contains
`considerable residual oxygen. Additional thrust can he obtained by adding
`fuel in an afterbursier. The cycle For this modification is shown dashed in
`figure |.5. Because this Fuel is added at lower pressure (and temperature)
`than the fuel in the primary combustor. it is used less elliciently. The penalty
`at subsonic speeds is so large that afterhurning is used only for short bursts
`ol‘e>ttra thrust. At Mach Ittmtbers ol’ 2.5 or more, the allerburning turbojet
`becomes highly ellieient because the pressure rise associated with diffusion in
`the inlet raises the nozzle pressure ratio to it high value. as in the rainjet.
`The propulsive efliciency ol‘ :1 turbojet is determined in the same way as
`that o!' a ramjet, by the coruhtlstor temperature ratio. For an ideal engine
`on can be made to approach unity by letting T4, approach T1, but the thrust
`for a given engine size (or rnass flow) becomes s.n't:-ill.
`indicated by figure
`l.l. so that in practice this option is not applied.
`
`1.6 Turbofans
`
`A better way to improve the propulsive efficiency of the basic turbojet is
`offered by the turbofan, sketched in figure 1.6. Here, a second turbine is
`
`

`
`Introduction to Concepts
`
`prlmory nozzle
`
`burner
`
`Ian nozzle
`compressor
`
`‘fan
`
`1.6 Schematic diagram of turbofan engine for subsonic flight.
`
`added downstream of the compressor-drive turbine, and the power is used
`to drive a fan that pumps air through a secondary nozzle. By this means a
`portion of the energy of the primary jet is removed, its velocity is reduced,
`and the energy is transferred to the fan airstream. Thus the effective value of
`F/rimo is reduced and 11,, is increased as in figure 1.1; the penalty in engine
`weight is less than would be caused by decreasing T4. The turbofan powers
`most modern subsonic transport aircraft; it has replaced both the turbojet
`and turboprop in this application. Modified to include afterburning in the
`duct airflow or in the mixed primary and fan flows, it also powers many
`high-performance military aircraft and is a major contender for the super-
`sonic transport.
`
`1.7 Shaft Engines: Regeneration
`
`For low—speed flight vehicles, where prohibitive size and weight of duct
`would be required to yield a good propulsive efiiciency with a turbofan, the
`turboshaft engine is used. Here, most of the useful work is extracted from the
`exhaust gas by a turbine; it often rotates on a separate shaft from the gas
`generator, as sketched in figure 1.7, and may drive a propeller, helicopter
`rotor, the wheels of a truck, or any other machine.
`In land or marine applications, where weight and size are not primary
`considerations, regeneration is used to increase the thermal efliciency beyond
`that attainable with a simple Brayton cycle. The regenerator is a heat ex-
`changer that transfers heat from the exhaust gas to the compressor discharge
`air. As sketched in figure 1.8, this transfer can be accomplished with a rotat-
`ing heat storage matrix. The regenerative Brayton cycle is shown in figure
`
`

`
`Introduction to Concepts
`
`compressor
`
`burner
`
`power turbine
`
`gas
`generator
`turbine
`
`1.7 Schematic diagram of shaft turbine.
`
`centrifugal
`compressor
`
`rotating heat—storage
`reg enerut or
`
`1.8 Shaft turbine with centrifugal compressor, and rotating matrix regenerator.
`
`

`
`Introduction to Concepts
`
`combustor
`
`gas generator
`turbine
`
`shaft
`turbine
`
`S
`
`1.9 Temperature-entropy diagram for regenerative Brayton cycle.
`
`4
`
`gas turbine
`cycle
`
`5 c
`
`ideal turbine
`:‘?
`ll.’/actual turbine
`——
`- I
`‘”,aufi\
`Ilz steam cycle
`aL-----,gz-
`elementary Carnot cycle
`
`I1
`
`3
`
`1.10 A gas turbine-steam combined cycle.
`
`1.9 for the ideal situation where the temperature of the exhaust gas is reduced
`to that of the compressor discharge air. Since heat is added in this engine
`only between 3’ and 4, instead of between 3 and 4 as in the simple gas
`turbine, the regenerated engine will have higher efliciency than the simple
`one for the same compressor pressure ratio. In other words, for a fixed
`turbine inlet tcmpcrature T4, an acceptable cflicicncy can be obtained with a
`lower compression ratio in the regenerated than in the nonregenerated engine.
`Since eflicient, high pressure ratio compressors are complex and expensive,
`
`

`
`Introduction to Concepts
`
`regenerators are used in engines for automotive applications where cost is a
`major consideration.
`
`1.8 Stationary Gas Turbines: Topping
`
`The turbine inlet temperature of modern gas turbines is considerably higher
`than the peak steam temperature in steam power plants. Depending upon
`the compression ratio of the gas turbine, the turbine exhaust temperature
`may be high enough to permit eflicient generation of steam using the waste
`heat from the gas turbine. Such an arrangement is referred to as a gas turbine-
`combined cycle power plant. The cycle is shown in figure l.l0. It is capable
`of very high efficiencies when the turbine inlet temperature of the gas turbine
`is high. It’s advantage over the regenerative gas turbine is that the steam
`boiler is easier to manufacture and maintain than the regenerator.
`
`1.9 Energy Exchange, Mach Number, Reynolds Number
`
`Four types of energy excliange have been implicitly involved in the above
`rleseriptions of engines. These are (1) the ciccliangc within a flowing fluid ol’
`kinetic energy for thermal energy or vice vet's-.1; (2) transfer of energy to or
`from .1 lluicl by forces acting on moving blades; (3) the conversion of chemical
`energy to. thermal energy: and ‘{4} the traiisfer of thermal energy from solid
`bodies to flowing fluids.
`The exchange from kinetic energy to thermal energy occurs when the
`momentum of a fluid is changed by pressure forces. The increasing pressure
`compresses the gas and the compression work appears as an increase in
`internal (thermal) energy, according to the first law of thermodynamics.
`The Mach number is defined as the ratio of the flow velocity to the Velocity
`of sound in the fluid M = u/a. When squared, it may be viewed as a measure
`of the ratio of kinetic energy to thermal energy of the fluid. Thus
`
`to — 1)M 2 =
`
`1,42/2
`(1.9)
`OPT.
`It follows that if process (1) is to be important, changes in %(y — 1)M2 that
`are large compared to unity must occur. The ramjet depends entirely on this
`process of energy exchange, and this is the reason it must operate at Mach
`numbers above unity.
`The second process appears in the turbojet, the turbofan, the turboprop,
`and all other devices using fluid dynamic machinery. The air flow over a
`blade in a compressor, for example, exerts a force on the blade. If the blade
`moves in a direction opposite the force, then the blade does work on the air,
`
`

`
`Introduction to Concepts
`
`increasing its mechanical energy. Process (1) may take place at the same
`time, so that the overall change in fluid energy appears partly as kinetic
`energy and partly as thermal energy. Now the force exerted on a body per
`unit area by a fluid is proportional to puz/2, where p is the fluid density and
`u is the velocity, which may be taken to be the same order as the velocity of
`the body. The power delivered to the fluid by the body, per unit area, is
`then of the order of pua/2. Thermal energy of the fluid is convected by the
`body at the rate puc,,T per unit area. Thus the ratio of energy addition by
`the body to convected thermal energy per unit time and area is
`2
`
`pu3/2 = u
`puc,,T
`2c,,T
`
`= %(v - DMZ,
`
`(1-10)
`
`and it can be seen that the Mach number plays the same key role in process
`(2) as in process (1). For the moving blades of the compressor or turbine to
`effectively exchange energy with the air, they should move at a Mach number
`of unity or more.
`Process (3) is so familiar that it requires no elaboration, but process (4)
`requires some discussion. In gas turbines we are concerned primarily with
`convective heat transfer, that is, heat transfer that occurs between a solid
`surface and a fluid because of the motion of the fluid over the surface. The
`thermal effects of the surface on the fluid, like the viscous effects, are con-
`fined to a region near the surface that is thin compared to the characteristic
`length of the surface when the Reynolds number is large. That is, if we con-
`sider the flow over a flat plate of length L, as sketched in figure 1.11, with
`fluid density p, velocity u, and viscosity u, then the viscous effects penetrate
`a distance 6,, of order
`.__~_
`1
`5“
`p
`L
`puL = R—e'
`
`If the fluid has a Prandtl number cpa/k near unity, where /c is the thermal
`conductivity, or if the flow is turbulent, the thermal effect of the plate pene-
`trates a distance 5k z 5,, In most of the Components of a gas turbine, we
`wish to minimize viscous effects; hence we desire large Re and thin. boundary
`layers. But in a regenerator the thermal eflect must penetrate the entire flow,
`so either Re must be small or the ratio of spacing between heat transfer
`
`u
`
`5)"! 5|
`
`T-ifwfi,
`
`L
`
`1.1] The penetration of viscous and thermal effects into a. flowing fluid.
`
`

`
`Introduction to Concepts
`
`surfaces to their flow length must be small, of order 1/Re. In either case the
`result tends to be a bulky and heavy device compared to the compressor
`and turbine. For this reason regenerators are not found in aircraft engines.
`
`1.10 Stresses
`
`Since the speed of snunrl in air is: about 340n1,1scc at normal conditions, the
`hlading 0l' compressors and turbines. slwultl have velocities near 340 m/see or
`more. This |‘l3(|l.ll1‘C1‘l'lCIllL has forcetl the designer of gas turbine ettgities to cope
`with inztte-1'iuls, vibration, and stress problems of E}. very high order. By con~
`trast, the piston speed of a typical “high-speed" gasoline ertgine is only about
`15 m/sec.
`Some appreciation for the problem can be had by considering a prismatic
`bar rotating about an axis at one of its ends, as in figure 1.12, with an angular
`velocity cu. The stress in the bar at any radius, due to centrifugal forces, will
`be
`
`0 = j‘ T pwzr dr = (pcoz/2)[r-f — r2].
`
`V‘
`
`For r << rT, the stress is
`(1.11)
`_
`{
`<7/p = (corT)2/2.
`For [m'—_.
`to be 340 tnfsec. it is ttecessary to have :1 material with the ratio of
`properties crfp of the order of 6 X It)" rniiseez. For steel, with a density 0|‘
`8000 kg}
`"*, this implies a stress oft? : 4.3 x In“ N,’t'I12, close enough to
`the potential limit of the rnateri:-1| that great sopliistieation and care in design
`are required. The problem is compounded in the turbine by the exposure ol‘
`the rapidly rotating turbine hlttcles to hot exhaust gases. This factor, probably
`more than any otlter, has lirnitetl the pet'i'urm:1nce of aircraft gas turbines.
`
`1.11 Noise
`
`Acoustical noise is radiated from regions of fluctuating pressure, which may
`be produced in many ways. There are at least four sources of strong unsteady
`
`axis of rotation
`
`'r
`
`1.12 Bar, rotating about axis th