4THEDITION GASTU TEORY By G.F.C.RogersandY.R.Mayhew EngineeringThermodynamicsWorkand HeatTransfer(Longman) Thermodynamicand Transport Propertiesof Fluids:SIBy G.F.C.Rogers TheNatureof Engineering (Macmillan) 4THEDITION GAS TURBINE THEORY H U KAT'. It followsthat I)c P04/Pcthenozzleischoking.t ( 2)2 x9923 Ts= Tc=--1T04== 8507 K l' +2333 (_1)1284 Ps=Pc= P04 \P04/Pc= 1.914= 0671bar 100x0671_3 PsRTc0.287x850.7 - 0275 kg/m Cs == (1333x0287x8507x1000)! = 5705 m/s As11 -;:;;= PsCs = 0275x570.50006374 m2s/kg 101 t Thenextexampleinthischaptershowshowthecalculationisperfonnedwhenthenozzleis unchoked. 102 GAS TURBINECYCLESFOR AIRCRAFTPROPULSION Thespecificthrustis IAs Fs= (C5- Ca)T-(Pc - Pa) m = (5705- 2396) + 0006347(0671- 0.265)105 =3309 + 2588=5897 N sjkg FromFig.2.15,withT02 =4868KandT03- To2 = 1200- 4868=7132K, wefindthat thetheoreticalfuelJairratiorequired is00194.Thus theactualfuel/ airratiois 00194 f=--=00198 098 ThespecificfuelconslUnptionistherefore SFC=f= 00198x 3600= 0.121kgjh N Fs5897 Optimizationof theturbojet cycle When considering the design of a turbojet the basic thermodynamic parameters at thedisposalof thedesigner aretheturbineinlet temperatureandthecompressor pressureratio.It iscommonpracticetocarryoutaseriesof designpointcal-culationscoveringa suitable rangeof thesetwovariables,usingfixedpolytropic efficienciesfor the compressor and turbine,and to pIlOtSFC versus specific thrust with turbine inlet temperatureT03and compressor pressure ratio fcas parameters. Suchcalculationsmaybemadeforseveralappropriateflightconditionsof for-wardspeedandaltitude.Typicalresultsapplyingtoa subsoniccruisecondition areshowninFig.3.l2. Theeffectsof turbineinlet temperatureandcompressor pressureratiowillbeconsidered intum. It can be seen that specific thrust isstrongly dependent on the value of To3 , and utilizationof thehighestpossibletemperatureisdesirableinordertokeepthe engineassmallaspossibleforagiventhrust.Ataconstantpressureratio, however, an increase in T03will cause some increase in SFC.This is in contrast to theeffectofT03onshaftpowercycleperformance,whereincreasingT03 improves bothspecific powerandSFC asdiscussedinsection2.4.Nevertheless thegaininspecificthrustwithincreasingtemperatureisinvariablymore importantthanthepenaltyinincreasedSFC,particularlyathighflightspeeds wheresmallenginesizeisessentialtoreduceboth weightanddrag. The effect of increasing the pressure ratio fcisclearly toreducethe SFC.At a fixedvalueof T03,increasingthepressureratioinitiallyresultsinanincreasein specificthrust but eventually leadstoa decrease;andthe optimum pressure ratio formaximum specific thrustincreasesasthe valueof T03isincreased.Evidently theeffectsof pressureratiofollowthe pattern alreadyobserved toholdforshaft powercyclesand need nofurthercomment. SIMPLETURBOJETCYCLE 0.16, SFC

0.14 0.12 0.10 10 15 Ma=O.Ball. =9000 m Compressor pressure ratio 0.08:::__-:::::-__--:::-::-____:-L1___---LI___.--JI 5006007001300 9001000 Specific thrust![Ns/kg] FIG.3.12Typicalturbojet cycleperformance 103 Figure3.12appliesto.aparticularsubsoniccmisecondition.Whensuch calculationsarerepeatedforahighercruisingspeedatthesamealtitudeitis found that in general, for anygiven valuesof fcandTo3, the SFC isincreased and thespecificthrustisreduced.Theseeffectsareduetothecombinationof an increase in inlet momentum dragand anincreasein compressor work consequent upontheriseininlettemperature.Correspondingcurvesfordifferentaltitudes show an increase in specific thrust and a decrease in SFC with increasing altitude, duetothefallintemperatureandtheresultingreductionincompressorwork. Perhapsthemost notableeffectof anincreaseinthedesigncmisespeedisthat theoptimum compressorpressureratioformaximmnspecificthrustisreduced. Thisisbecauseofthelargerramcompressionintheintake.Thehigher temperatureatthecompressorinletandtheneedfora higher jet velocitymake theuseof ahighturbineinlettemperaturedesirable-andindeedessentialfor economicoperation of supersonicaircraft. The thermodynamic optimization of the turbojet cycle cannot be isolated from mechanicaldesignconsiderations,andtheof cycleparametersdepends verymuchonthetypeofaircraft.Whilehighturbinetemperaturesare thermodynamicallydesirabletheymeantheuseof expensivealloysandcooled turbine blades leading toan increase in complexity and cost,or tothe acceptance of a decrease inenginelife.The thermodynamicgains of increased pressure ratio must beconsidered inthelightof increasedweight,complexityandcost dueto theneedformorecompressorandturbinestagesandperhapstheneedfora multispool configuration. Figure 3.13illustrates the relation between performance anddesignconsiderations.Asmallbusinessjet ortrainer,forexample,needsa 104 GASTURBINEc:YCLESFOR AiRCRAFTPROPULSION SFC Business jet SpecHic thrust FIG. 3.13Performanceanddesign(:onsiderations Lifting engine' Long rangesubsonic simple,reliableengineof lowcost:SFCisnot criticalbecauseof thesmallamount of flyingdone,and low pressure ratio turbojet of modest turbme inlettemperaturewouldbesatisfactory.Inrecentyears,however,noise restrictionshaveledtothedispla.cementof theturbojetbytheturbofan.For thebusiness jet,thischangehasalsobeendriven by theneedforlonger range. Anotherexampleof interestwasthedevelopmentof specialized liftingengines for Vertical Take Off and Landing (VTOL), where the prime requirement was for maximum thrustperunit weightand volume,withSFClesscriticalbecauseof theverylowrunningtimes:theserequirementsweremetusinga lowpressure ratiounit witha very high turbineinlet temperature(permissible becauseof the short life required). The compressor pressure ratio was governed by the maximum thatcouldbehandledbyasingle-stageturbine.Thistypeof enginewasnot widelyusedbecauseoftheinereasedaircraftcomplexity,butenginesof exceptionallyhighthrust/weightratiowerebuiltintheearly1960s.Lastly, turbojetsofhighpressureratiowereusedinearlycommercialaircraftand bombersbecauseof theneedforlongrangeandhencelow SFC.Theincreased engine weight wasacceptable because of thelarge reduction in (engine plus fuel) weightfora longrange.Turbojetshavenowbeensupersededbyturbofansfor commercialsubsonicaircraft.Tw:bojetsarenolongersuitableforcommercial supersouic aircraft because of the take-off noise. In future,such aircraft will need anengine with take-off noisecomparabletothat of conventional turbofans.This willrequirethedesignanddevelopmentof variablecycleenginescapableof operatingasturbofansduringtake-offandasturbojets(orverylowbypass turbofans)atsupersoniccruiseconditions.Thesubsonicfuelconsumptionof a supersonic transport is important because a considerable portion of any journey is SIMPLE TURBOJErCYCLE105 flown at subsonic speeds, and it follows that the optimization procedure cannot be carriedoutarounda singlecruisecondition. Variationofthrust and SFC with flight conditions for a givenengine The reader isremindedthatwehavebeendiscussing the resultsof design point cyclecalculations.Curvessuchasthoseof Fig.3.l2donotrepresentwhat happenstotheperformanceof a particularenginewhentheturbineinlettem-perature,forwardspeedor altitudediffer fromthedesign values.Themethodof arriving at such data isdescribed in Chapter 8:here we will merely notesomeof themoreimportantaspectsof thebehaviourof a turbojet. AtdifferentflightconditionsboththethrustandSFCwillvary,duetothe changeinairmassflowwithdensityandthe variation of momentum dragwith forwardspeed.Furthermore,evenif theenginewererunatafixedrotational speed,thecompressorpressureratioandturbineinlettemperaturewillchange withintakeconditions.Typicalvariationsof thrustandSFCwithchangein altitudeandMachnumber,forasimpleturbojetoperatingatitsmaximum rotationalspeed,areshowninFig.3.l4.Itcanbeseenthatthrustdecreases significantly with increasing altitude, due tothe decrease in ambient pressure and density,eventhoughthespecificthrustincreaseswithaltitudeduetothe favourableeffectof thelowerintaketemperature.Specificfuelconsumption, however,showssomeimprovement withincreasingaltitude.Itwillbeshownin Chapter 8 that SFC is dependent upon ambient temperature, but not pressure, and hence its change with altitude is not so marked as that of thrust. It is obvious from the variation in thrust and SFC that thefuelconsumption will be greatly reduced at high altitudes. Reference to Fig. 3.14 shows that with increase of Mach number at a fixed altitude the thrust initially decreases, due toincreasing momentum drag, and then starts to increase due to the beneficial effects of the ram pressure ratio;at supersonicMachnumbersthisincreaseinthrustissubstantial. 15000 5000 Sea-level 3000m 0.15 Z0.10 .c fi ____--___ --- (J)0.05 11 DOOm

3000m OL-_______L____ ____OL-__ o0.20.40.60.8o0.20.40.60.8 Flight Mach numberFlight Mach number FIG.3.14Variationof thrust and SFC withMachnumber andaltitude for typical turbojet 106 GASTURBINECYCLESFOR AIRCRAFT PROPULSION 3.4The turbofan engine Theturbofanenginewasoriginallyconceivedasamethodof improvingthe propulsionefficiencyof thejetenginebyreducingthemeanjetvelocity,par-ticularlyforoperationathighsubsonicspeeds.It wassoonapparent,however, thata lower jet velocityalsoresulted inless jet noise,an importantfactorwhen large numbers of jet propelled aircraft entered commercial service. In the turbofan a portion of the total flowby-passes part of the compressor,combustion turbineandnozzlebeforebeingejectedthroughaseparatenozzleasshownm Fig.3.15. Thus the thrust is made upcomponents, the (orfan) thrustandthehotstreamthrust.Figure3.15showsanengmeWIthseparate exhausts, but it is sometimes desirable to mix the twostreams and eject them as a single jet of reduced velocity. Turbofan engines are usually described in terms of bypass ratio,defined as the ratioof the flowthrough the bypassduct (coldstream) to the flowatentry tothe high-pressure compressor (hot stream).t With the notation of Fig. 3.15 the bypass ratioB isgiven by B=mc mh It immediatelyfollowsthat mBm m=-- mil=-- andm =mc+m" cB+l'B+l For the particular case where both streamsareexpanded toatmospheric pressure inthepropellingnozzles,thenetthrustisgiven by m F= (mcCjc+ mhCjh)- mCa qc

.-._------ .-------- -----------_.----FIG. 3.15Twin-spoolturbofan engine tThe tenns turbofan and bypass engine may both be encountered, often referring to the same engine. Early engines with a small portion of the flow bypassing the combustion (low value ratio)wereinitiallyreferredtoasbypassengines.Asthebypassratio. IStheoptimmn pressure ratio for the bypass stream is reduced and can eventually be prOVIded bf astage.Thetermturbofanwasoriginallyusedforengmesof highbypassratiobutIS mcreasmgly employed forallbypassengines. , .! THETURBOFAN ENGINE107 Thedesign .pointcalculationsfortheturbofanaresimilartothoseforthe turbojet;in view of this,onlythedifferencesincalculation will beoutlined. (a)Overall pressureratioandturbineinlet temperaturearespecifiedasbefore, but itisalsonecessary tospecify the bypassratio(B)andthefanpressure ratio(FPR). (b)From the inlet conditions and FPR, tht: pressure and temperature of the flow leavingthefanandenteringthebypassductcan becalculated.Themass flowdownthebypassductcanbeestablishedfromthetotalflowandthe bypassratio.Thecoldstreamthrustcanthenbecalculatedasforthejet engine,noting that air isthe workingfluid.It isnecessary tocheck whether the fan nozzle is choked or unchoked;if choked the pressure thrust must be calculated. (c)In the two-spool confignration shown in Fig. 3.15 the fan is driven by the LP turbine.Calculationsfor theHPcompressor and turbinearequitestandard, and conditions at entry to the LP turbine can then be found.Considering the workrequirementof thelowpressurerotor, mCpallTo12= '1mmhcpgllTo56 andhence mcpaCpa AIlT056= - X--XIlTo12== (B + 1)X--XUT012mh'1mCpg'1mCpg Thevalueof B mayrangefrom03to8 or more,and itsvalue hasa major effectonthetemperaturedropandpressureratiorequiredfromtheLP turbine.KnowingT05,'11 andIlTo56, theLPturbinepressureratiocanbe foundandconditionsatentrytothehotstreamnozzlecanbeestablished; thecalculation of thehotstream thrustisthenquitestraightforward. (d)If thetwostreamsaremixeditisnecessarytofindtheconditionsafter mixingbymeansof anenthalpyandmomentumbalance;thiswillbe considered following an example on the performance of an enginesimilar to that shown inFig.3.15. EXAMPLE3.2 The followingdata apply toa twin-spool turbofan engine,with the fandriven by theLPturbineandthecompressorbytheHPturbine.Separatecoldandhot nozzlesareused. Overall pressure ratio Fanpressureratio Bypassratiome/mh Turbineinlet temperature Fan,compressorandturbine polytropicefficiency Isentropicefficiency of eachpropellingnozzle Mechanicalefficiencyof eachspool Combustionpressureloss Totalair massflow 250 165 50 1550K 090 095 099 150 bar 215kgls 108GASTURBINECYCLESFOR AIRCRAFTPROPULSION It is required to find the thrust andSFC under sea level static conditions where the ambientpressureandtemperatureare10bar and288K. Thevaluesof (n- l)lnforthepolytropiccompression andexpansion are: forcompression,II- 1 = _1_ (1'- 1)=1 = 0.3175 II.1]ooe l'a09X.J5 .n - 1(1'- 1)09 forexpanslOn,-n- = 1]oot --1'- g = 4= 0225 UnderstaticconditionsTOl= Taand POI= Pasothat,usingthenomenclatureof Fig.3.14, T02=P02yieldsT02= 288X1650.317S= 3376 KG )(n-I)/n T01 01T02- TOl = 3376 - 288= 496 K P03= 250= 15.15 P02165 (p)(n-1)/n T03== 3376x15.150-3175 = 8001K T03- T02= 8001- 3376 = 4625K Thecold nozzleflressureratiois P02=FPR = 1.65 Pa and thecritical pressureratioforthis:nozzleis P02_1_ _I= 1.965 Pc- [1-iGDf(Y-l) -[1- G::) fS. Thus the cold nozzle is not choking, so that pg =Paand the cold thrust Feisgiven simplyby Fe= meCg Thenozzletemperaturedrop,fromequation(3.l2),is T02- Ts= I1j T02 [I -()'-1)h] [( 1)l/3,SJ == 095)( 33761 - T6s= 428K andhence 11 Cg= [2c/To2- Tg)]'=(2)(1005)( 428)(1000)2= 2932mls THETURBOFANENGINE SincethebypassratioB is50, mB215x50 me= B + 1 =6.0= 1792kg/s Fe= 1792X2932= 52532 N Consideringtheworkrequirementof theHProtor, 'cpa1005X4625 T04- Tos= --(T03 - TO?)== 4090 K YimCpg - 099x1148 andfortheLProtor 109 cpa60X1005X496 Tos-- T06= (B + l)--(T02 - T01 ) ==263.2 K 1]mCpg099X1148 Hence ToS= T04- (T04- Tos)= 1550- 4090= 1141.0 T06= Tos- (Tos- T06 ) =J 1410 - 2632=877.8 P06maythenbefoundasfollows. P04(T04) n/(n-1)(1550) 1/0225 Pos== Tos== 1141.0= 3902 Pos(Tos) n/(n-l)(1141.0) 1/0225 -= - =--- =3208 P06T068778 P04= P03-I1Pb =250 X]0 - 150== 235bar P04235 P06=== == 1878bar (P04/Pos)(PoslP06)3902X3208 Thusthehotnozzlepressureratiois P06== 1.878 Pa whilethecriticalpressureratiois ,1 P06=4== 1.914 Pc[11 (0.333)J 0952333 Thisnozzleisalsounchoked,andhence P7 =Pa. T06- T7= 1]j T06 1 - -- [ 1)(Y-I)IYJ 06/Pa = 095X877-8[1- (_I_)*J == 1216 K 1878 C7== [2cp CT06 - == (2)(1148x1216x== 5283mls m215 mh== B + 1 == 6.0 == 3583kg/s Fh== 3583x5283== 18931N 110 GAStlJRBINE CYCLESFOR AIRCRAFTPROPULSION Thusthetotalthrustis Fe+Fh = 52532 + 18931= 11463 Nor 715kN Thefuelflowcanreadilybecalculatedfromtheknowntemperaturesinthe combustorandtheairflowthroughthecombustor,i.e.mhoThecombustion temperatureriseis(1550- 800) = 750Kand thecombustion inlet temperature is800K.From Fig.2.15theidealfuel/airratioisfoundtobe00221andthe actual fuel/airratiois then (0.0221/099) =00223.Hence the fuelflowisgiven by and mf = 00223x3583x 3600 = 28764 kg/h 2876-4 SFC= --= 00403kg/h N 71463 Becauseboth nozzleswereunchoked,thethrustcould beevaluatedwithout calculating the nozzle areas. It isalwaysgood practice, however, tocalculate key pieces of information which may be required for other purposes. In both cases the area can be calculated from continuity, i.e. m =pAC. The density is obtained from p =p/RT,where pandT arethestatic valuesin the plane of the nozzle;forboth nozzles pwill equal Pa'Thefollowingresultsareobtained forthe twostreams: Staticpressure(bar) Statictemperature(K) Density(kg/m3) Mass flow(kg/s) Velocity (mls) Nozzle area (mz) ColdHot 1010 19267498 119104647 17923583 29325283 0513201459 Thecold nozzle area is much larger than the hot one,and Fig.1.12(b) showsthe physicalappearanceof anengineof similar cycleand bypassratio;Fig.1.12(a) showsanengineof lower bypass ratio,around2.5. Thisexampleillustratedthemethodfollowedwhenapropellingnozzleisun-choked,whilethepreviousexampleshowedhowachoked nozzlemay bedealt with. Notethatatstaticconditionsthebypassstreamcontributesapproximately 74percentofthetotalthrust.Ataforwardspeedof60mis,whichis approaching a normal take-off speed, the momentum drag mCa will be 215x60 or12900 N;theram pressureratioandtemperaturerisewillbe negligibleand thusthenet thrust isreduced to58563N.Thedropin thrust duringtake-off is evenmoremarkedforenginesof higherbypassratioandforthisreasonitis preferable toquote turbofan thrusts at a typical take-off speed rather than at static conditions. THE TURBOFANENGINE111 Mixingof hot and cold streams Mixing isessential foran afterburning turbofan when maximum thrust boosting is required, toavoid the need for two reheat combustion systems. In certain cases mixingmayalsobeadvantageousinsubsonictransportapplications,givinga smallbut significant gainin SFC.Weshallpresentasimplemethod of dealing with mixing. in a constant area duct,with nolosses and assuming adiabatic flow. TheductisshownschematicallyinFig.3.16,with thehotandcoldflowsbe-ginningtomixat plane Aand with complete mixingachieved by plane B. Starting from the enthalpy balance, with suffix m denoting the properties of the mixed stream, mccpeToz+ mhcphT06 =mCpmT07wherem=mc+mh Wealsohavethefollowingequationsthatrelatethepropertiesof amixtureof gasestothoseof itsconstituents: mccpc + mhcph (me+ mh) meRe+mhRh Rm(me+mh) From themomentum balance, (mcCc+ P2A2)+ (mhCh+ P6A6)= mC7 + P7A7 If there isno swirlin the jet pipe downstream of plane A,the static pressure will be uniform acrosstheduct,andso pz =P6' Fromcontinuity, m = P7C?A? It isthepressureaftermixing,PO?,thatis. requiredforthecyclecalculation, because this is the stagnation pressure at entry to the propelling nozzle. When this has been found,the calculation for the thrust is thesame asdescribed previously. AB II T02 P02 Iime II T077"IInl_ '06P06II P07nlhII __._ FIG. 3.16Mixing in aconstant area duct 112GASTURBINECYCLESFOR AIRCRAFTPROPULSION Thecalculation ofPo7issimplified if weworkMach numbers. in thehotandcoldstreams:thehotstreamMachnumberISfixed' bytheturbme design,andtypicallyM6willbeabout05.Havingselecteda of M6,the procedureisasfollows.Wewillmalce:useof thestandardrelatIOnsbetween M andstaticandstagnation P andT,equations(8)inAppendixA.2. (a)KnowingM6, T06and P06,' wecandetermine P6andCh: P6andT6giveP6, hence A6followsfromcontinuity.Wenowknow(mhCh + PI06).. (b)With P6 =P2, P2/P02 yields the value of M2With M2, P02and T02known we can nowfindCe and A2sothat(meCe +P2A2)isImown. (c)(mC7 +A7P7)isnowobtainedfromthemomentum balance. (d)A7 =A6 +A2,andm = pC7A7 =C7A7 (e)T. isImownfromtheenthalpybalance,but neither P7nor P07isknown. a valueof M7, andthenfindT7andC7; continuity thengives P7. (f)It is now necessary tocheck that (mC7 +A7P7)is equal tothe value obtained fromthemomentum balancein(c). (g)IterateonM7untilcorrect P7isfound. (h)P07isthenobtainedfromp7,M,. Thefanoutletpressure(Poz)shouldbeonlyslightlyhigherthantheturbine outletpressure(P06)tokeepmixinglossestoaminimum;typically,P02/P06 shouldbeabout105-107. In practice,quitesmallchangesincycle parameters cancausesignificantchangesintheratioPOZ/P06andnegatethebenefitsof mixing.Nohardandfastrulescangivenand thedecisiononwhetlrer touse mixingornotwillalsobeinfluencedbyinstallationandengineweight considerations,combinedwithadetailedinvestigationof thepressurelosses causedbythemixer;Ref.(7)describesanexperimentalinvestigationof mixing losses. Optimizationof theturbofancycle DesiQTIersof turbofanshavefourthermodynamicparametersattheirdisposal; over:1Ipressureratioandturbineinlettemperature(asforthesimpleturbojet), andalsobypassratioandfanpressureratio.Optimizationof tirecycleissome-whatcomplexbut thebasicprinciplesareeasilyunderstood. Let usconsider anengine with the:overall pressure ratioand bypass ratioboth specified.If weselect a value of turbine inlet temperature theenergy input isbecausethecombustionchamberair flowandentrytemperaturearedetermmed bythechosenoperatingconditions.Theremainingvariableisthefanpressure ratioandasa firststepitisnecessarytoconsider the variationof specific tlrrust andspecificfuelconsumption with FPR.Ifwe start with a low value of FPR,the fanthrust willbesmallandthe workextracted fromtheLPturbine willalsobe small;thuslittleenergy will be extracted fromthe hot stream anda large value ?f hotthrustwillresult.AstheFPRisraiseditisevidentthatthefanthrustwill increase and thehot thrust will decrease.A typical variation of specific thrust and THE TURBOFANENGINE 113 Overall pressure ratioandbypassratiofixed SFC

Fs ------ Turbine inlet temperatureincreasing Fanpressureratio FIG.3.17Optimizationof fanpressureratio SFC with FPR,for a range of turbine temperature,isshown in Fig.3.17. It can be seen that for any value of turbine inlet temperature there will be an optimum value of FPR;optimum valuesof FPRforminimum SFC and maximum specific thrust coincide because of the fixedenergy input.Taking the valuesof SFC and specific thrustforeachof thesevaluesof FPRinturn,a curveof SFCagainstspecific thrust may be plotted asshown inFig.3.18(a).Notethat each point on thiscurve is the result of a previous optimization, and isassociated with a particular value of FPRandturbineinlettemperature. The foregoing calculations may be repeated for a series of bypass ratios,still at thesameoverallpressureratio,togiveafamilyof curvesasshowninFig. 3 .18(b).This plot yields the optimum variation of SFC with specific thrust for the SFC Bypassratio fixed Specific thrustFs(a) FIG.3.18TlIrbofanoptimization SFC Increasing bypass ratio Optimum Specific thrustFs(b) / / (/ 114GASTURBINECYCLESFOR AIRCRAFTPROPULSION selectedoverallpressureratioasshownbythedottedenvelopecurve.The procedurecanthenbe repeatedforarangeof overallpressureratio.It willbe clearthattheoptimizationprocedureislengthyandthatalargeamountof detailedcalculationisnecessary.Thequalitativeresultsofsuchaseriesof calculationscanbesununarized asfollows: (a)Increasingbypassratio' improvesSFCattheexpenseofasignificant reductioninspecificthrust. (b)Theoptimumfanpressureratioincreaseswith turbineinlettemperature. (c)Theoptimum fanpressure ratiodecreases with increase of bypass ratio.(At a bypass ratioof about5 theFPRmay be low enough to pennit the use of a single-stagefan.) Thechoiceof cycleparametersisdependentontheaircraftapplication,and both high and low bypassratios have their place.Specific fuelconsumpti?n isof majorimportanceforlongrangesubsonictransportaircraftand the can best be met by using a bypass ratioof 4-6 anda high overall pressureratIO, combined with a high turbine inlet temperature.A higher bypass ratioof 8-9 was chosen fortheGE90,which entered service in1995,representing thefirstmajor change in thisparameter for many years.Military aircraft with a supersonic dash capabilityandarequirementforgoodsubsonicSFCwoulduseamuchlo,:"er bypassratio,perhaps05-1,tokeepthefrontalareadown,andafterburnmg wouldbeusedforsupersonicoperation.Enginescurrentlyunderdevelopment, withsignificantlyhighervaluesof TIT,willpermitoperationuptospeedsof about M1-4withouttheuseof afterburning.Short-haulconnnercialaircraftare notascriticalaslong-haulaircraftregardingSFCandformanyyearsbypass ratiosof around1 were used;modem designs,however,use higher bypass ratios similar to those used in long-haul aircraft.The prime reason forthis change is the significantdecreaseinenginenoiseresultingfromincreasedbypassratio. Another reason forconcentrating on enginesoflow SFC,so that they aresuitable fora widerangeof aircraft,istheescalatingcost of developingnewengines. Figure 3 .18(b) showed that optimizing the SFC required the use of high values of BPRresultinginenginesof lowspecificthrust.Thesecurvesapplyonlyto uninstalledengineperformance,andinstallationeffectsmustbeconsideredfor evaluationina specificaircraft.HighBPRengineswould havealargediameter fanandbothdiameterandweightwouldincreasewithBPR;groundclearance effectswouldcauseincreasesinundercarriagelengthandweight,increasing aircraftweightand thrustrequired. Thecombinationof the nacelle anditssupport system isnormally refened to asthe'pod'and it isinstructive todoa simple analysison theeffect of poddrag. Thepropulsionefficiency willbe modifiedtogive __2_ [net thrust - pod drag] '1p - Cjnetthrust 1+-Ca THETURBOFAN ENGINE Theidealpropulsionefficiency canbeexpressed inadifferentfmIDas 2 11 - -----,---p- 1 +gross thrust momentum drag 2)(momentum drag momentum drag + gross thrust 2)(momentum drag_2 '2xmomentum drag + net thrust- 2 +net thiust momentum drag Thus,includingtheeffectof poddrag 1] =2[net thrust- pod drag] p2 +net thrustnet thrust momentmn drag - r1 - 2net thlUst(net thrust) + momentum dragmomentum drag 115 Theratioof (netthrust/momentumdrag)isdirectlyrelatedtothebypassratio, decreasing with increasein BPR.UsingIX todenote(pod drag/momentum drag), theeffectof pod dragon'1pcan be readilyevaluatedgiving the resultsshownin Fig.3.19.It canbeseenthatathighvaluesof BPRtheeffectof poddragis significantandtheinstallationeffectsmust be carefully evaluatedby theaircraft manufacturer,workinginconjunction withtheenginemanufacturer. It wasmentionedearlierthat,becauseof theirreducedmeanjetvelocity, turbofans produce lessexhaust noise than turbojets.At firstsight it wouldappear thatnoiseconsiderationswoulddemandthehighestpossiblebypassratio, 1.00

Poddrag Momentum drag 0.50 ...,------c.... ____ ------;... HighBPRJets __-'-_--1__ o0.51.52.0Net thrust/momentumdrag FIG.3.19Effect of poddrag Uill propulsiollefficiency 116 GAS TURBINE CYCLESFOR AIRCRAFT PROPULSION resulting in a low jet velocity. Unfortunately, however, as bypass ratio is increased the resulthig high tip speed of the fan leads to a large increase in fan noise. Indeed at approachconditions,with theengineoperatingata low thrust setting, thefan noisepredominates;fannoiseisessentiallyproducedatdiscretefrequencies whichcanbemuch moreirritatingthanthebroad-bandjet noise.Theproblem can be alleviated by acoustic treatmeilt of the intake duct, avoiding the use of inlet guide vanes,and carefulchoiceof axialspacing between the fanrotor and stator blades. Turbofanconfigurations The cycle parameters fora turbofan have a much greater effect on the mechanical design of theengine than in thecaseof the turbojet.Thisis because variation in bypass ratio implies variation in component diametersand rotational speeds,and theconfigurationof enginesof lowandhighbypassratiomaybecompletely different. Some early turbofans were directly developed fromexisting turbojets,and this led to the'aft fan'configuration shown in Fig.3.20.A combined turbine-fan was mounteddownstreamof thegas-generatorturbine.Twomajorproblemsarise withthisconfiguration.Firstly,thebladingoftheturbine-fanunitmustbe designed togiveturbinebladesectionsforthehot streamandcompressor blade sectionsforthecoldstream.Thisobviouslyleadstobladingof highcostand, because the entire blade must be made from the turbine material,of high weight. Theotherproblemisthatofsealingbetweenthetwostreams.Theaft-fan configuration has not been used in a new design for many years, but is a possible contender for ultra high bypass (UHB) engines.In this case, however,the turbine andfansectionswouldprobablybeconnectedbyagearboxandthecomplex blades usedinearlierengineswould not be required. Formoderatebypassandoverallpressureratiosthesimpletwo-spool arrangementof Fig.3.15isadequate.Atveryhighbypassratios,especially when combined with high overall pressure ratios,design problemsarise because thefanrotational speed must be much lower than that of the high-pressure rotor; theimportant limitationson bladetipspeed willbediscussedinChapter 5. Fourdifferentconfigurationswhichmaybeusedtoobtainhighbypassratio and highoverall pressure ratioareshown in Fig.3.21.Theconfigurationof Fig. Fan --------------------------------------FIG. 3.20Aft-fan configuration THETURBOPROPENGINE117 (\ll Twospool(b) Twospool (cl Threespool------------(dl Two-spoolgeared fan FIG.3.21Configurationsforhighbypassratioturbofans ; .21 (a)suffersfromthefactthatthelaterstagesontheLProtorusuallcalled booster stages', contribute little becauseof their low blade speedShY (b)' moreattractive,but requiresa veryhih..cerneISh'hid.. .gpressure ratiofromtheHPcompressor wIC eas tomstabiijty problems referred toinsection1.2.The three-s001 arrangementof FIg:3.2l(c) isin manywaysthemostattractiveconcetwftha ratio each compressor. All of these have been useJ for large Trent(FIg.1.12(bisanexampleof (a)whiletheRolls-Royce .g..IS anexampleof (c).A geared-fanarranementas.. :ssible forsmallerengines,andunitsof thissorthavebee! develope: 1: boPbroPbbackground.Thepowerrequirementsforthefanof a largeturbofan mayeaout60MWandthede.fl'gh. uld'.0a1twelghtgearboxtohandlethis be deSIgnstudIes,however,are beingcarried outfor ultra g.ypassratIO(UHB)engineswithvariablepitchfanswherethelow of the large diameter fan requires a gearbox to using a large rome stages. 3.5The turbopropengine The turboprop engine differs from the shaft power units dealt with in Chater 2 in thatsomeof theusefuloutputappearsasJ' etthrustInthithP.. nectb'.s case,ereforeItISessary0comme shaft power and J' et thrustThibd'' b.I.s caneone m a number of ways,ut m alcasesa knowledgeof theaircraftspeed isinvolved .bedelivered tothe aircraft in theformof thrustower driving a propeller.thrust power (TP) power (SP), propeller effiCIency IJpr and jet thrust F by TP= (SP)IJpr+ FCa 118GAS TURBINECYCLESFOR AIRCRAFT PROPULSION In practicetheshaftpowerwill account fora proportionof theenthalpy dropavailableatthegas-generatorexit,andthrustpoweristhereforelargely dependentonthepropellerefficiencywhichmayvarysignificantlywithflight conditions. It is desirable to find some way of expressing the power so that it can be readily compared with that ofa piston engine, and so that it is not quite so dependent on propeller efficiency.Thesame basicenginemaybe usedinconjunctionwitha varietyof propellersfordifferentapplications,andit istheperformanceof the engineitselfwhichisthenofmostinteresttous.Amoresuitablewayof expressing the power is to quote the equivalent (or effective) power EP defined as TPFea EP=-=SP+-'1pr '1pr '1prnowaffectsonlythesmallerterm.Theequivalentpowerisanarbitrarily defined quantity and it should not be quoted without reference to the flight speed. thatbydefinitiontheEPandSPareequalatstaticconditions,although thereissomebeneficial jet thrust.Allowancemust be madeforthiswhen com-paring engines under take-off conditions.Experimentshave shown that an aver-age propeller produces a thrust of about 85N per kW of power input under static conditions, so that the take-offEP is conventionally taken as SP + (F/8.5) with SP in kW and F in newtons. Turbopropsareusuallyratedonthebasisof equivalentpowerattake-off conditions,andthespecificfuelconsumptionandspecificpowerareoften expressed in termsof thatequivalent power.It isneverthelessdesirable toquote boththeshaft powerand jet thrustavailableatanyconditionof interestandit should be recognizedthattheequivalent power ismerelyauseful,but artificial, concept. In view of the similarity between the turboprop and the shaft power uuits discussed in Chapter 2 it is not necessary to elaborate on cycle requirements.The onlybasicdifferenceisthatthedesignercanchoosetheproportionsof the availableenthalpydropusedtoproduceshaftpowerandjet thrust.Itcanbe shown that there is an optimum division forany given flight speed and altitude;a simple ruleof thumbistodesignsothat the turbineexit pressure isequaltothe inletpressure.Theequivalentpowerisnot particularlysensitiveto turbineexit pressure in thisrange.Turbopropsgenerally operate with thenozzle unchokedanduseasimplestraightthroughtailpiperatherthanaconvergent nozzle. Thecombinedefficiencyof thepowerturbine,propeller,andthenecessary reductiongear,iswellbelowthat of anequivalent propelling nozzle.It follows that 'Iefora turbopropengineis lower than thatof a turbojet or turbofan engine. The turboprop has held its position for speeds up to M 06 because the propulsion efficiency is so much higher than that of the turbojet; the turboprop is widely used inbusinessaircraftandregionalairliners,mostlyatpowerlevelsof500-2000kW.The propeller efficiency, using conventional propeller design methods, decreasesdrasticallyatflightspeedsaboveM06andforthisreasonthe THETURBOPROP ENGINE 119 did not widely used for longer haul aircraft,being superseded yd turofansofeqUIvalentpropulsionefficiency;anexceptionwaslong enurancepatrolaircraft,whichflytothesearchareaat M06andfuen1o"t at much lower flight speedsIn thI.h.I er fi..e ear y eigtIes,however,considerable attention wasocusedonthedeSIgnof propellerswiththeoa1of obt.. of 080 at fli?ht Mach numbers of 08.Th! would give largesavingsin fuelcompared toexistingturbofans .eof thepropeller wasmarkedlydifferentfromconventionaldesis su?ersouic blading and 8-10 blades; these devices were c:ci to them from conventional propellers. Studies showed that propaIrcraftwouldrequiremuchhigherpowersfu.1 expenencedwifuturbdanprevIOUS y b.oprops,anatpowerlevelsinexcessof 8000 kWfue gearox deSl?n becomes difficult.Afurther major problem isthe transruission of propellernOIsetothepassengercabinanditwaswIdel.dh bt1..'yrecogIllZetata sus antiaill noIse over existing turbofan levels would not be acceptable toItprobable that this can be overcome only by the use of 'pusher' b guranons . Wlfufuepropellersmountedbehindfuepassengercabin uroprop aresurveyedin Ref.(8).. Analternativeconfiguration,acomproruisebetweentheturboproandth was. the'unductedfan'(UDF),developedbyGeneral inth: eIghties.In tillStwoc?unter-rotating variable pitchfansweredirectI coupledtocounter-rotatingturbmeswithnostatorselinIintinthdfiY gearboxThh.,ageneeora .escematicarrangementdepictedin Fig.3.22showsa similarity to Stationary support structure ---t-----{jt:-------.-..-.--_._. -----------FIG. 3.22Dnductedfanengine 120 GASTURBINECYCLESFOR AIRCRAFTPROPULSION theaft-fanconfigurationdescribedearlier.Thisrevolutionaryapproachdemon-strated a significant reduction in fuelconsumption in flight tests, but problems with noiseandcabin vibration;airlineswerealsotopIOneer thisnewtechnology,andnotallenginemanufacturerswereconvmced. thatthe gearbox could be eliminated. The UDr concept, however, wasalso to beapromisingteclmologyforthedevelopmentof longrange mIssIles, whereitsexcellentSFCwouldimproverangeforafixedquantItyof fuelon board. Thestatusof advancedturboprop8,prop fansandunductedfansinthemid 1980s wasverysimilar tothat of turbofansin themid1950s.At that time there wasaheatedcontroversyregardingtherelativemeritsofturbopropsand turbofans,withthelatterbeingtheclearwinner.In themid1990stheturbofan has once again emerged as the winner;fuel prices, however, have been stable over a long period, but ifthis were tochange significantly engines having ultra low fuel consumption would beessential. The conventional turboprop will certainly continue todominate the market for smaller aircraftof 10-60 seatswith flightspeedsin therangeof 400-600 kmIh; theseaircraftwould beusedon flightswith adurationof 60-90minuteswitha range of perhaps 400-500km.The turboshaft engine,in which the output power drivesahelicopterrotorisalsoof greatimportanceandisvirtuallyuniversally used because of its low weight and high power.In the helicopter application,free turbineenginesareused.The helicopterrotor isdesignedtooperateatconstant speedbychangingthepitchandthepowerisvariedbychangingthegas generatorspeed.Whileat firstsightthedesignrequirementsof. turboprops. and turboshafts appear identical, the former may be optinuzed for crUIse at an altItude of 6-10 000m while the latter isoptimized for operation at very low altitudes.A numberof enginesareavailablein bothturbopropandturboshaftversions,but invariablyanygivenengineisfoundtobemuchmoresuccessfulinone application thantheother. 3.6Thru.staugmentation If thethrustof anenginehasto increasedabovetheoriginaldesignvalue several alternativesareavailable. Inc:reaseof turbine inlet temperature,for exam-ple,willincrease thespecific thrust and hence the thrust forasize. AlternativelythemassflowthroughtheenginecouldbemcreasedWithout altering thecycle parameters.Both of these methodsimply some redesign of the engine,and eitheror both maybe used touprateanexisting engine. Frequently,however,therewillbearequirementforatemporaryincreasein thrust,e.g.fortalce-off,foraccelerationfromsubsonictosupersonicspeedor duringcombatmanoeuvres;theproblemthenbecomesoneofthrust augmentation.Numerousschemesforthrustaugmentationhave been proposed, but thetwomethodsmost widelyusedareliquidinjectionandafterburning(or reheat). THRUST AUGMENTATION121 inje.cti?nis primarilyusefulforincreasing take-off thrust.Substantial of hqUldrequired,but if theliquid isconsumed during take-off and lll1t1al theweightpenaltyisnotsignificant.Sprayingwaterintothe compressor inlet causesevaporationof the water droplets,resulting inextraction of heat fromtheair;theeffectof thisisequivalent toa dropin compressor inlet 8willshowthatreducingthetemperatureatentrytoa turbOjet mcreasethethrust,duetotheincreaseinpressUI'eratioandmass flow resultmg from the effective increase in rotational speed.In practice a mixture of isused;themethanollowersthefreezirIgpointof water, and ItWill?urn wh.enit reachesthecombustionchamber.Liquidis ll1JecteddIrectlymtothecombustionchamber.Theresulting blockageforcesthecompressortooperateatahigherpressureratiocausing thethrust tomcrease.In both casesthemassof liquid injectedadds tothe useful massflow,but thisisasecondary effect.Liquid irIjectionisnowseldom used irI aircraftengines. . Afterburning,as .thenameimplies,involvesburningadditionalfuelin the jet pIpeasshownmFIg.3.8.In theabsenceof highlystressedrotatingbladesthe allowablefollowingafterburningismuchhigherthantheturbine inlettemperature.Stoichiometriccombustionisdesirableformaximumthrust augmentationfmaltemperaturesof around 2000Kare possible.Figure3.23 showstheT-s fora turbojet withtheadditionof afterburningto 2000K.Thelargemcreasemfuelflowrequiredisevidentfromtherelative inthecombustion chamber andafterburner,and thepenalty in IS heavy.Assunllng that a choked convergent nozzle is used, the jet velOCity WIll to the sonic velocity at the appropriate temperature in the of nozzle,I.e.T7orT5dependingonwhethertheengineisoperated WIthor WIthoutafterburning.Thusthe jet velocitycan befoundfrom(yRT )1/2 withTcgiven either by T06ITc= (y-/- 1)/2 or T04ITc= (y-/- 1)/2.It followsth:t th; a s FIG.3.23Cycleof turbojet withaftel'burnillg 122GASTURBINE CYCLESFOR AIRCRAFTPROPULSION jet velocity isproportional toJTo at inlet to thepropelling nozzle,and that the gross momentum thrust, relative to that of the simple turbojet, will be increased in theratioJ(TorJT04).Forthetemperaturesshownin Fig.3.23thisamountsto J(2000/959)or144.Asan approximation,the increase in fuelwould bein the ratio(2000- 959)+ (1200- 565)withafterburning,to(1200- 565)without afterburning,i.e.264.Thusa '44per centincreaseinthrustisobtainedatthe expense of a 164 per cent increase in fuel flow,and clearly afterburning should be used only for short periods. This might be the thrust augmentation under take-off conditionswherethegrossthrustisequaltothenetthrust.Athighforward speeds,however,thegainismuchgreaterand isoften wellover100percent. This is because for a fixed momentum drag an increase in gross thrust represents aconsiderablygreaterincreaseinnetthrust.TheConcordemakesuseof afterburningfortransonicaccelerationfromM09toM1-4;thesignificant increaseinnetthrustprovidesfasteraccelerationthroughthehighdragregime near M 10, resulting in a reduction in fuelconsumption in spite of the short term increaseinfuelflow.Afterburning offers even greater gainsfor low bypass ratio turbofans,becauseof therelatively lowtemperatureafter mixingof thehot and coldstreamsandthelargerquantitiesof excessairavailableforcombustion; militaryturbofansuseafterburningfor take-off and combat manoeuvring. It isessentialforenginesfittedwithanafterburnertoincorporatea variable area nozzlebecauseof thelargechangeindensityof theflowapproachingthe nozzle resulting from the large change in temperature. Afterburning will normally be brought intooperation when the engine isrunningatitsmaximum rotational speed,correspondingtoitsmaximumunaugmentedthrust.Theafterburner should bedesigned sothat the engine willcontinue tooperate at the same speed when it isin use,and hence thenozzle must pass thesame massflowat a much reduced density.This can be achieved only if a variable nozzle is fitted permitting asignificantincreasein nozzlearea.Notethatthepressurethrustwillalsobe increasedowingtotheenlarged nozzlearea. Thepressurelossintheafterburner canbesignificant.Combustion pressure losses are discussed in Chapter 6, where it is shown that the pressure loss is due to bothfluidfrictionandmomentumchangesresultingfromheataddition.In combustion chambers the former predominates, but in afterburners the losses due tomomentumchangesaremuchmoreimportant.Thetemperatureriseis determinedbytheturbineoutlettemperatureandthefuel/gasratiointhe afterburner,and thepressure lossdue tomomentum changescan bedetermined usingtheRayleighfunctionsandthemethodoutlinedinAppendixA.4.This pressurelossisfoundtobeafunctionof boththetemperatureratioacrossthe afterburner and the Mach number at inlet to the duct.If the inlet Mach number is toohigh,heatreleasecanresult inthedownstreamMachnumber reaching10 and thisplacesan upper limit on theallowableheat release:thephenomenonis referred toasthermal choking.Figure 3.24 shows valuesof the pressure loss due tomomentumchangesandemphasizestheneedforalowMachnumber. Typically, the exit Mach number from the turbine of a jet engine will be about 05 and it is necessary tointroduce adiffuser between the turbineand afterburner to NOMENCLATURE Inlet Mach no. 1.5 Afterburner temperature ratio (TosITo4)FIG.3.24Momentum pressure loss 123 reducetheMach number toabout025-030beforeintroducingtheafterburner fuel. Even when not in use,an afterburner incurs some penalty in pressure loss due thepresence?ftheburnersandflamestabilizingdevices.Another thismethodof thrustaugmentationisthattheveryhighjet velOCItiesfroma degree of afterburning result in a noisy exhaust. The Olympusengmesused onConcordeprovideabout15-20 per centmcrease. mtake-off !hrust,resultinginanexhausttemperatureof around 1400 The mcreased nOIselevel, thougha seriousconcern,issignificantly less than mIght be expected from with military aircraft. It is most unlikely, how.ever, anysupersomc transport will use afterburning for take-off,as a.pnme WIllbe take-off noiselevelscomparable tocurrent subsonic arrcraft;mentionedearlier,thiswillnecessitatethedevelopmentof variable cycleengmes. NOMENCLATURE asonicvelocity Across-sectionalarea Bbypassratio(mJmh) Fnet thrust Fsspecificthrust Kpspecificthrustcoefficient MMachnumber l1e efficiency of energy conversion Iiiintakeefficiency I1j nozzleefficiency 11m mechanicalefficiency THRUST AUGMENTATION125 124 GAS TURBINECYCLESFOR AIRCRAFTPROPULSION Internatiollal Standard Atmosphere 110overallefficiency zpTa I1ppropulsion(Froude ) efficiency [m][bar][K] p/Po [m/s] I1prpropeller efficiency I1rramefficiency 010132528815100003403 1100polytropicefficiency 50009546284909529338-4 Suffixes 1000089882817090753364 criticalcondition,coldstream 1500084562784086383345 c 2000079502752082173325 hhotstream 2500074692719078123306 jjet 3000070122687074233286 mmixed3500065782654070483266 4000061662622066893246 4500057752589063433226 5000054052557060123205 5500050542524056943185 60000-47222492053893165 65000-4408245905096314-4 7000041112427048173123 75000383023950-45493102 8000035652362042923081 8500033152330040473060 9000030802297038133038 95000,28582265035893017 10000026502233033762995 1050002454220003172297-4 11000022702168029782952 11500020982167027552951 12000019402167025462951 12500017932167023542951 13 000016582167021762951 13 500015332167020122951 14000014172167018602951 14500013102167017202951 15000012112167015902951 15500011202167014702951 16000010352167013592951 165000095722167012562951 170000088502167011622951 17 5000081822167010742951 1800000756521670099302951 1850000699521670091822951 1900000646721670084892951 1950000598021670078502951 2000000552921670072582951 DensityatsealevelPo = 12250kg/m3 Extractedfrom:ROGERSG FCandMAYHEWYR Thermodynamicand Transport Propertiesof Fluids(Blackwell 1995) 4 Centrifugalcompressors Veryrapidprogressinthedevelopmentof gasturbineswasmadeduringthe SecondWorldWar,whereattentionwasfocusedonthesimpleturbojetunit. Germaneffortswerebasedontheaxialflowcompressor,butBritishdevelop-mentsused thecentrifugalcompressor-Refs(1)and(2).It wasrecognizedin Britain that development time was critical and much experience had already been gainedonthedesignof smallhigh-speedcentrifugalcompressorsforsuper-chargingreciprocatingengines.Centrifugalcompressorswereusedinearly BritishandAmericanfighteraircraftandalsointheoriginalCometairliners which were the first gas turbine powered civil aircraft in regular service. As power requirementsgrew,however,it becameclear that theaxialflowcompressor was moresuitableforlargeengines.Theresultwasthataveryhighproportionof development funding was diverted totheaxial typeleading totheavailabilityof axialcompressors with anappreciably higher isentropic efficiency than could be achieved by theircentrifugalcounterparts. Bythelatefifties,however,it becameclearthatsmallergasturbineswould have to usecentrifugal compressors,and serious research and development work startedagain.Smallturboprops,turboshaftsandauxiliarypowerunits(APUs) havebeenmadeinverylargenumbersandhavenearlyallusedcentrifugal compressors;notableexamplesincludethePratt and WhitueyCanada PT-6,the Garrett 331and the large stable of APUs built by the latter organization. They are alsousedforthehigh-pressurespoolsinsmallturbofans,seeFig.1.12(a). Centrifugals wereused primarilyfortheirsuitability forhandlingsmallvolume flows,butotheradvantagesincludeashorterlengththananequivalentaxial compressor,betterresistancetoForeignObjectDamage(FOD),lesssuscepti bility to loss of performance by build-up of deposits on the blade surfaces and th ability to operate over a wider range of mass flowat a particular rotational speed Theimportanceofthelatterfeature,inalleviatingproblemsofmatchin operatingconditionswiththoseof theassociatedturbine,willbe madeclear. Chapter8. Apressureratioof around4 : Icanreadilybeobtainedfromasingle-stag compressor made of aluminium alloys, and in section 2.4 it was shown that this i adequateforaheat-exchangecycle when theturbineinlet temperatureisinth PRINCIPLEOFOPERATION127 region of 1000-1200 Ie Many proposals for vehicular gas turbines were based on thisarrangement and manufacturerssuchasLeyland,Ford,GeneralMotorsand Chryslerbuiltdevelopmentengineswhichneverwentintoproduction.The advent of titaniumalloys,permittingmuchhighertipspeeds,combinedwith advancesinaerodynamicsnowpermit pressure ratiosof greater than8 : 1 tobe achievedinasinglestage.Whenhigherpressureratiosarerequired,the centrifugalcompressor may be used in conjunction with an axial flowcompressor (Fig.1.11),orasatwo-stagecentrifugal(Fig.1.10).Eventhoughthelatter arrangement involvesrather complex ducting betweenstages,it isstillregarded as a practical proposition. Reference (2) of Chapter Idescribes the design process leading to the choiceof a twin-spool all-centrifugal compressor for the Pratt and WhitueyCanada PWlOOturbopropwhich entered servicein1984. 4.1Principle of operation The compressor consists essentially of a stationary casing containing a rotatmgimpeller which impartsa high velocity to the air,and anumber of fixed diverging passages in which theair is decelerated with a consequent rise in static pressure.Thelatter process isoneof diffusion,and consequently the part of the compressorcontainingthedivergingpassagesisknownasthedifJit.ser.Figure 4.1(a) isa diagrammatic sketch of a centrifugal compressor.The impeller may be single- or double-sided asin 1(b) or I ( c),but the fundamental theory is thesame for both. The double-sided impeller was required in early aero-engines because of the relatively small flowcapacity of the centrifugal compressor for a given overall diameter.. Airissuckedintotheimpellereyeandwhirledroundat highspeedbythe vanes on the impeller disc. At any point in the flow of air through the impeller, the centripetal acceleration is obtained bya pressure head,sothat the static pressure of theair increasesfromtheeyetothetipof theimpeller.The remainder of the static pressure rise isobtained in the diffuser, where the very high velocity of the air leaving the impeller tipisreduced tosomewhere in the region of the velocity with which the air enters the impeller eye; it should be appreciated that friction in the diffuser willcause some loss in stagnation pressure. The normal practice is to design thecompressor sothat about half thepressure riseoccursin theimpeller and half in thediffuser. It will be appreciated that owing totheaction of the vanesin carrying theair aroundwiththeimpeller,therewillbeaslightlyhigherstaticpressureonthe forwardfaceof avanethanonthetrailingface.Theair willthustend toflow round the edges of the vanes in the clearance space between theimpeller and the casing.Thisnaturallyresultsinalossof efficiency,andtheclearancemustbe keptassmallaspossible.Ashroudattachedtothevanes,Fig.4.1 (d),would eliminatesuch aloss, but the manufacturingdifficultiesare vastly increased and therewouldbeadiscfrictionor'windage'lossassociatedwiththeshroud. 128 Vaneless seace Impeller eye (a) CENTRIFUGAL COMPRESSORS or" diffuser channel Impeller shroud Ij-'M

(b) (c) FIG. 4.1Diagrammatic sketchesof centrifugalcompressors Althoughshroudshavebeenusedonsuperchargers,theyarenotusedon impellersforgasturbines..., Theimpellersof moderncentrifugalcompressorsoperateWIthveryhightIp speeds resulting in very high stress levels. It will be shown the next that backswept curved vanesaredesirableforcompressorsof high pressure ratio,but for many years designers were forced to use radial vanes because of the ten?ency for curved vanes tostraighten out under the action of the considerable centrIfugal forceinvolved,settingupundesirablebendingstressesinthevanes.Modern methodsofstressanalysiscombinedwithstrongermaterials,however,now permit backswept vanestobeusedin high-performancecompressors. 4.2Work doneandpressurerise Sincenoworkisdoneontheairinthediffuser,theenergyabsorbedbythe compressor will bedetermined by theconditions of the air at theinlet and outlet of theimpeller.Figure4.2showsthenomenclatureemployed. WORK DONEANDPRESSURE RISE129 In the, first instance itwill be assUmed that the air enters the impeller eye in the axialdirection;sothat theinitial angular momentumof theair iszero.Theaxial . portion of the vanes must be curved so that the air can pass smoothly into theeye. The angle which the leading edge of a vane makes with the tangential direction exwillbegivenbythedirectionof therelativevelocityof theairatinlet,Vbas shown in Fig.4.2. If theairleavestheimpellertipwithanabsolutevelocityC2,itwillhavea tangential or whirlcomponent Cw2,and a comparativelysmall radialcomponent Cr2Under idealconditionsC2would besuch that the whirl component isequal to the impeller tipspeedU,as shown by the velocity triangle at the top of Fig. 4.2. Due toits inertia, theair trapped between theimpeller vanesis reluctant to move round withtheimpeller,and wehavealreadynotedthatthisresultsina higher staticpressureontheleadingfaceof a vanethanonthetrailingface.Italso prevents theair fromacquiringa whirlvelocity equal tothe impeller speed.This effect is known asslip. How far the whirl velocity at the impeller tip fallsshort of thetipspeeddependslargelyuponthenumberof vanesontheimpeller.The greaterthenumberof vanes,thesmallertheslip,i.e.themorenearlyCw2 approachesU.It isnecessaryindesigutoassumea valuefortheslip factor(1, where (1 is defined as the ratio Cw21U.Various approximate analyses of the flowin animpellerchannelhaveledtoformulaefor(1:theoneappropriatetoradial-vaned impellers which seems to agree best with experiment is that due to Stanitz, Ref.(4): (1= 1- 063n n wheren isthenumber of vanes. Ideal conditions at impeller tip FIG.4.2Nomenclature Velocityrelative toimpeller !C/u. Section througheye atradiusr1130CENTRIFUGALCOMPRESSORS Asexplainedin.anyelementalYtextonappliedtherrnodynamics,the theoreticalltorquewhichmustbeappliedtotheimpellerwillbeequaltothe rateof changeof angularmomentumexperiencedbytheair.Consideringunit massflowof air,thistorqueisgivenby theoretical torque= Cw2r2 If wistheangular velocity,theworkdoneontheairwillbe theoretical workdone= Cw2r2w= Cw2 U Or,introducingtheslipfactor, theoretical work done= eJU2 (4.1 ) (4.2) Forconvenience,inboththechaptersoncompressorsweshalltreatthework doneontheairasa positivequantity. Due tofriction between thecasingand theair carried round bythe vanes,and other losseswhichhavea brakingeffectsuchasdiscfrictionor'windage',the applied torqueandtherefore theactual workinput isgreater thanthistheoretical value.A power inputfactor tjJ can beintroducedtotakeaccountof this,sothat theactualworkdoneontheairbecomes work done= '/leJU2(4.3) If (T03- TOI )isthestagnationtemperatureriseacrossthewholecompressor then,since noenergy isadded in thediffuser,this must beequal tothestagnation temperaturerise(T02- ToI )acrossthealone.It will thereforebeequal tothe temperature equivalent of the work doneon theair given by equation (4.3), namely

T03- TOl= -- (4.4) cp where cpisthe mean specific heat over thistemperature range.Typicalvaluesfor thepowerinputfactorlieintheregionof 1035-104. Sofarwehavemerelyconsideredtheworkwhichmustbeputintothe compressor.If a valuefortheoverall isentropicefficiencylIebeassumed,thenit isknownhowmuchof theworkisusefullyemployedinraisingthepressureof theair.Theoverall stagnationpressureratiofollowsas ==1 + 'Ie03 0] =1 +_',e_'I'__ P (Tol )1'/(1'-1)[n(To - T)J1' /(1'-I)[n,1'eJU2]J'/(1'-1) PO]TOITOICpTOl (4.5) Thedistinctionb":tweenthepowerinputfactorandtheslipfactorshouldbe clearlyunderstood:theyareneitherindependentof oneanothernorof lie.The power input factorrepresentsan increase in the work input,the wholeof which is absorbedinovercomingfrictionallossandthereforedegradedintothennal energy.The fact that theoutlet temperatureis raised by thisloss,and incidentally WORKDONE ANDPRESSURERISE131 by otherfrictionallosses aswell,enablesthemaximumcycletemperaturetobe reached without burning so much fuel,so that asfar as the efficiency of the whole gasturbine unit isconcerned theselossesarenot entirely wasteful.Nevertheless, thiseffect isoutweighed by thefactthat moreturbine work isused indrivingthe compressor andisentropic(i.e.frictionlessadiabatic)compression istheidealat which toainl,It followsthat the power input factor should be as low aspossible, a lowvalueoftjJ implyingsimultaneouslyahighvalueof17e. Itshouldbe appreciated that 11e dependsalsoupon thefrictionlossinthediffuser which does notaffecttheargument uptoequation(4.4).Forthisreasonitisnothelpfulto considertjJ implicitlyaspart of lJe' Theslipfactor,ontheother hand,isa factorlimitingtheworkcapacity of the compressor even under isentropic conditions,and thisquantityshould be asgreat aspossible.ClearlythemorenearlyCw2approachesU,thegreater becomesthe rateatwhichworkcanusefullybeputintoacompressorofgivensize, Unfortunately an increase in the number of vanes, which would increase (J,entails anincreaseinthesolidityof theimpeller eye,i.e.a decrease intheeffective flow area.Additionalfrictionlossesarisebecause,forthesamemassflowor 'throughput', the inlet velocity must be increased.Thus theadditional work input thatcanbeemployedbyincreasingthenumberof vanesmaynotresultinan increaseinthat portion which isusefullyemployed in raisingthepressureof the air;itmayonlyincreasethethennalenergy produced byfrictionresultinginan increaseintjJ andreductioninlie'Asuitablecompromisemustbefound,and present-daypracticeistousethenumberof vaneswhichgiveaslipfactorof about 09,i.e.about19or 21vanes(seealso under heading'Mach nUl1lber in the diffuser',in section4.4). Fromequation(4.5)itwillbeseen that theremainingfactorsinfluencingthe pressure ratiofora given working fluidaretheimpeller tipspeedU,and theinlet temperatureTol . Anyloweringof theinlet temperatureTOlwillclearlyincrease the pressure ratio of thecompressor for a given work input, but it is not a variable under thecontrol of thedesigner,Reference totextsonstrengthof materials will show that thecentrifugalstressesina rotatingdiscareproportionaltothesquare of therimspeed.Forsingle-sidedimpellersof lightalloy,Uislimitedtoabout 460mlsbythemaximumallowablecentrifugalstressesintheimpeller:sucha speed yieldsa pressure ratioof about4 : 1.Higher speedscan be used with more expensivematerialssuchastitaniumandpressureratiosof over8: 1arenow possible.Becauseof theadditionaldiscloading,lowerspeedsmustbeusedfor double-sidedimpellers. EXAMPLE4.1 Thefollowingdataaresuggestedasabasisforthedesignofasingle-sided centrifugalcompressor: powerinputfactortjJslipfactor(J rotationalspeed N 104 09 290rev/s 132 overalldiameterof impeller eyetipdiameter eyerootdiameter air massflowm inletstagnationtemperatureTOIinletstagnationPOl isentropicefficiency11eCENTRIFUGALCOMPRESSORS 05m 03m 015m 9 legis 295K 11bar 078 Requirementsare(a)todeterminethepressureratioof thecompressorandthe power required todrive it assuming that the velocity of theair at inlet is axial;(b) tocalculate the inlet angle of the illlpeller vanes at the root and tip radii of the eye, assuming that the axial inlet velocity is constant across theeye annulus;and Cc) to estimate theaxialdepthof theimpeller channelsat the periphery of the impeller. (a)ImpellertipspeedU=nx05x290=4555mls. Temperatureequivalentof theworkdoneon unit massflowof airis, __IjJ(JU2 _104x09x455.52 _193K T03 TOI - c- 1.005X103-p P03=[1+ 11c(To3- TOI )]V/(V-ll = (1 + 078x193)3.5=4.23 POITOI 295 Powerrequired =mCpCT03- TOI ) =9 x1005x193= 1746 leW (b)Tofindtheinletangleof thevanesitisnecessarytodetenninetheinlet velocity which in thiscaseisaxial,i.e.Cal= CI .Calmust satisfy thecontinuity equationm=PIAtCa],whereAlistheflowareaatinlet.SincethedensityPI dependsuponC],andbothareunlmown,a trialanderrorprocessisrequired. The iterative procedure isnot critically dependent on the initial value assumed fortheaxialvelocity,but clearlyitisdesirabletohavesomerationalbasisfor obtaininganestimatedvalueforstartingtheiteration.Thesimplestwayof obtaining a reasonableestimate of theaxial velocity istocalculate thedensity on thebasisof theImown stagnationtemperatureandpressure;inpracticethiswill givea density that istoohighanda velocity that istoolow.Havingobtained an initial estimate of the axial velocity, the density can be recalculated and thence the actualvelocityfromthecontinuityequation;iftheassumedandcalculated velocitiesdonotagreeitisnecessary toiterateuntilagreementisreached(only thefinaltrialisshownbelow).Notethatitisnormaltodesignforanaxial velocityof about150mis,thisprovidingasuitablecompromisebetweenhigh flowper unitfrontalareaandlowfrictionallossesintheintake. n(032 - 0.152) Annulusareaof impeller eye,A]=4= 0053 m2 Basedonstagnationconditions: POI11x1003 PI::::: RTOI= 0.287x295= 130 kg/m m9 Cal= PIAl= 1.30x0.053=131m/s WORKDONEANDPRESSURERISE SinceCI = Ca],theequivalent dynamictemperatureis C?1312 1.312 -==--=85K 2cp2 x1005x1030201 C2 TI= TOI - _I = 295- 85= 2865K 2cp .POI11 PI= (TodTIy/()'-I)= (295/286.5)35= 0992bar _PI_0992x1003 PI- RTI- 0.287x286.5= 121kg/m CheckonCal: C9_ alPIAl1.21x0.053- 140 m/s Finaltrial: TryCal= CI= 145m/s Equivalent dynamic is Cf1452 1.452 2cp = 2)( 1.005X103=0.201=105K C2. TI= TOI- -21= 295- 105= 2845K cp POI11 PI= (TOI/TIY/(V-I)= = 0968 bar 0968x100_3 PI- RTI- 0.287x284.5- 1185leg/m CheckonCal: C9_ 01 PIAl1185x0.053- 143 m/s 133 Thisisa goodagreementanda furthertrialusingCal= 143mlsisunnecessary becausea smallchangeinC haslittleeffectuponp.Forthisreasonitismore accurate touse the finalvalue143mis, rather than the meanof 145mls(the trial value)and143mls.Thevaneanglescannowbecalculatedasfollows: Peripheralspeed at the impeller eyetipradius =nx03x290= 273m/s andat eye root radius= 1365 m/s r:t. at root = tan-l 143/1365= 46.33 IX at tip= tan-I143/273= 27.65 (c). The shape of the impeller channel between eye and tipis very much a matter of tnal andThetoobtain asuniform a change of flowvelocity up the channelaspOSSIble,aVOldmglocaldecelerationsupthetrailingfaceof thevane 134CENTRlFUGAL COMPRESSORS which might lead to flowseparation. Only tests on the machine can show whether thishasbeenachieved:theflowanalysesalreadyreferredto[Ref.(4)]arefor inviscidflowandarenotsufficientlyrealistic tobeof directuseindesign.To calculatetherequireddepthof theimpellerchannelattheperipherywemust makesomeassumptionsregardingboth theramalcomponentof velocityat the tip,andthedivisionof lossesbetween theimpellerandthediffusersothatthe densitycanbeevaluated.Theradialcomponentof velocitywillberelatively small and can be chosen by the designer;a suitable value isobtained by making it approximatelyequaltotheaxialvelocityatinlet totheeye. Toestimate thedensity at theimpeller tip,thestatic pressure and temperature arefoundbycalculatingtheabsolutevelocityatthispointandusingitin conjunctionwiththestagnationpressurewhichiscalculatedii'omtheassumed lossuptothispoint.Figure4.3may helpthereadertofollowthecalculation. Making the choiceCr2= Cal'we haveCr2 = 143mls Cwz=(JU=09x4555=410m/s ci_ C;2+ C ~ 2_1.432 + 410z - - 0.201= 938 K 2cp2cp Assumingthat half thetotalloss,i.e.o 5(1-. '1e) = 011,occursin theimpeller, theeffectiveefficiencyof compressionfrom POIto POlwillbe089sothat P02=(1 + 089x193)3.5 = 1-5823.5 POl295 T TOl --------FIG.4.3Divisiollof lossbetween impellerallddiffuser WORl( DONEANDPRESSURERlSE Now(P2lpoz)=(T2ITo2l5, andToz=T03= 193 + 295 =488 K,sothat C2 T2== Toz_.-.1.. =488 - 938=3942 K 2cp pz= (394.2)3.5 P02488 Hence,since(P2IpOI) = (P2IP02)(P02IpOl), pz(394.2)3.5 - =1582x-- =235 POI488 pz= 235x11= 258bar pz258x1003 P2 = RTz = 0.287x394.2= 228kg/m 135 The required area of cross-section of flow in the radial direction at the impeller tip ISA = ~ =9_.2 P2CrZ 228x143- 0 0276 m Hencethedepthof impellerchannel 00276 = --= 00176 m or 176em nx05 Thisresultwillbeusedwhendiscussingthedesignof thediffuserinthenext section. Beforeleaving the subject of theimpeller,it is worth noting theeffect of using backswept curved vanes,which wesaidat theend of section 4.1are increasingly beingusedforhigh-perfonnancecompressors.Thevelocitytriangleatthetip sectionfor a backswept impeller isshown in Fig. 4.4,drawn for the ideal case of zeroslipforeaseof nnderstanding.Thecorrespondingtrianglefortheradial-vanedimpellerisshownbydottedlines.Assumingtheradialcomponentof velocitytobethesame,implyingthesamemassflow,itcanbeseenthatthe velocity relative to the tip,V2,is increased while the absolute velocity of the fluid, Cz, is reduced.These changesimply lessstringent diffusion requirements in both theimpeller and diffuser,tending toincreasetheefficiencyof both components. Thebacksweepanglef3 maybeintheregionof 30-40degrees.Thework-absorbing ca:pacity of the rotor isreduced, however,because Cwz islower and the temperaturerisewillbelessthanwouldbeobtainedwiththeramal-vanedim-peller.Thiseffectiscounteredbytheincreasedefficiencyanditmustbere-membered that theultimategoalishighpressure ratioand efficiency rather than hightemperaturerise.Theuseof backsweptvanesalsogivesthecompressor awideroperatingrangeof air-flowatagivenrotationalspeed,whichisim-portant for matchingthecompressor toitsdrivingturbine;this will be discussed inChapter8. 136 \-!iTos= !iTs.Furthermore,sincethepressureratioperstageis small theconstant P03and P3linesarevirtually parallel between3'and 3sothat Y "" x.It followsthatlis hasthesamevalueoneither basis.For thestageof our example,TOI was333K,andhencethestagnation pressureratiois [ 092X20J3.5 Rs=1 += 1252 333 It willberememberedthatitwasnecessarytomakeuseof twoassumed efficienciesatthestartof thedesignprocess:apolytropicefficiencyforthe compressor asa whole, and a stage efficiency.Asa first approximation these were taken to be equaland a value of 090 wasassumed.Theestimated valueof 11s for thethirdstage,i.e.092,isinsufficientagreement,bearinginmindtheuncer-tainties in predicting thesecondary and annulus losses. If similar agreement were tobeobtainedforallthestages,wemightconcludethatthedesignhadbeen conservativeandthat thecompressorshouldhavenodifficultyinachievingthe specified performance. Toobtain an estimate of the overallefficiency it would be necessarytorepeattheforegoingforallthestages.Theproductof thestage pressure ratios would then yield theoverall pressure ratiofrom which theoverali isentropic temperaturerise,and hence theoverallefficiency,could becalculated. Before continuing, it may be helpful tosummarize the main steps in the design proceduredescribedintheprevioussections.Havingmadeappropriate assumptionsabouttheefficiency,tipspeed,axialvelocity,andsoon,itwas possible to size the annulusat inlet and outlet ofthe compressor and calculate the airanglesrequiredforeach stageat themeandiameter.A choice wasthen made of a suitable vortex theory toenable theair angles tobe calculated at various radii from root totip.Throughout thisworkit wasnecessary toensure that linlitations onbladestresses,ratesofdiffusionandMachmnnberwerenotexceeded. Cascadetestdatawereusedtodeterminea bladegeometrywhichwouldgive these air angles,and alsothelift and dragcoefficients fora two-dimensional row of bladesof thisform.Finally,empiricalcorrectionfactorswereevaluatedto enable thesecoefficients to beapplied to theannular row at the mean diameter so thatthestageefficiencyandpressureratiocould beestimated. 5.10O!J;mpressibilityeffeds Overa longperiodof time,thedevelopmentof high-performancegasturbines hasledtotheuseof muchhigherflowratesperunitfrontalareaandbladetip speeds,resultinginhighervelocitiesand Machnumbersincompressors.While earlyunitshadtobedesignedwithsubsonicvelocitiesthroughout,Machnum COMPRESSffiILITY EFFECTS217 bersexceeding unity are.now found in thecompressorsof industrialgasturbines Mach num?ersashigh as15areused in thedesignof fansforturbofansof ratIO.It is not wit?in thescope of this book toGOver transonic design m detall,andthetreatment confined toa brief introductionand provision some. key.references;agam,Itshouldberealizedthatmuchof therelevant mfor:natlOnISof a proprietarynatureandisnotavailableintheopenliterature. cascadetestingisrequiredtoprovideexperimentaldataon compresslbilltyeffects,andinparticulartodeterminethevaluesof theMach numbers,con-espondingtoentryvelocitiesrelativetotheblades,whichbring about cascadep.erformance.Thefirsthighvelocityofinterestisthat correspond:ug to what IS called the'critical'Mach number Me;atentry velocities lower than .his,the performance of thecascadediffersvery little fromthat at low Abovethisvelocity,thelossesbegintoshowa markedincreaseuntila pomtIS reachedwherethelossescompletelycancelthepressureriseandthe blade ceasestobeany useasa diffuser.The corresponding Mach nunlber isthen referredtoasthe'maximum'alMF....v,uem.oratypicalsubsomccompressor cascadeatzerothe valuesof theseMach numbersarein theregionof 07and 0.8.5respe.ctively.A typical variation of fluiddeflection and pressure loss forsubsomcbladmgwasshowninFig.5.25,representingflowatlowMach numbers.ABtheMach increases,twoimportanteffectstalceplace:first, overalllevel.oflosseslllcreasesubstantially,andsecond,therangeof mClden.cefor lossesareacceptableisdrastically reduced.Thismeansthat off-deSIgn performanceof the .compressor may beseriouslyaffected.Figure5.33 showstest results for a subsomc blade section over a rangeof Mach number from 05to08;theusablerangeof incidenceatM 08can beseen tobevery narrow 0.35 0.30- Mach number 0.750.8 0.25 C '" ."0.20 ;;: '" 0 c.>(I) (I) 0.15 0 ....J0.10 0.05 0I -20-15-10-5015 Incidence, degrees FIG. 5.33Effectof Mach !lumber Oillosses 218AXIALFLOWCOMPRESSORS and clearlycompressor bladingofthis typecould not be.used at Mach numbers approachingorexceedingunity.It shouldbepointedoutthat forcompressible flowthedenominator inthelosscoefficient is(POl- Pl) rather thanpVl/2. Compressibilityeffectswillbe mostimportant atthefrontof thecompressor wheretheinlettemperature,andhencetheacousticvelocity,arelowest.The Mach number corresponding tothe velocity relativetothetipof therotor isthe highest encountered and is important both from the viewpoint of shock losses and noise.Thestator Mach number isgenerally highest at thehubradiusbecauseof theincreased whirl velocity normally required togiveconstant work inputatall radiiintherotor.TherotorandstatorMach numberscorrespondingtothefirst stageof thecompressordesignedinsection5.7areshowninFig.5.34,which shows that the flow relative to the rotor issupersonic over a considerable length of theblade.ThestatorMachnumberscanbeseentobesignificantlylower,this beingduetothefactthat theyareunaffected by thebladespeed. Analysisof alargeamountof compressortestsbyNACA[Refs(10,11)] showedthatlossesforsubsonicconditionscorrelatedwellonthebasisof diffusionfactor,asshown inFig.5.8,but weresignificantlyhigher fortransonic conditions. It wasdeduced that thisincrease in lossmust beduetoshock losses, butitwasalsofoundthat thespacingof theblades had aconsiderableeffect;a reduction in solidity (i.e.an increase in pitch/chord ratio)caused a rapid increase inloss.Asimplemethodforpredictingthelosswasdevelopedandcanbe explained with referencetoFig.5.3 5. Apairof doublecirculararc(DCA)bladesareshown,withasupersonic velocityenteringinadirectionalignedwiththeleadingedge.Thesupersonic expansionalongtheuncoveredportionof thesuctionsidecanbeanalysedby meansof thePrandtl-Meyer relationsdiscussed in Appendix A.S,and the Mach number willincreaseastheflowprogressesalongthesuctionsurface.Ashock structureisassumedinwhichtheshockstandsneartheentrancetotheblade passage,striking the suction surface at B,extending in front of the blade at C and 1.2 :;;1.0 Q;.c E ::l c:J:: the incidence will be atitsdesign valueand a high efficiency will be achieved.With theassumptionthat0(1 and/32 areconstant, OFF-DESIGNPERFORMANCE223 where k = tan0( 1+tan/32. This relationship isshown by the dotted straight line in Fig.5.39.It isobviousthatthetemperaturecoefficientincreasesastheflow coefficientdecreasesandreachesavalueof1whenc/> =O. Thetemperature coefficientcanbeinterpretedfromtheshapeof thevelocitydiagram.Recalling that itimmediatelyfollowsthat cpl1TosI1Cw U2U Thus, ift{! = 1,ACw = U and from earlier examination of velocity diagrams it will berecognizedthatthisresultsinexcessivediffusioninthebladepassageand efficiency willdecrease.For satisfactory operation,I1Cw/U,and henceIj!,should bearound03-04. The pressure rise across the stage,I1pos,isdetennined by the temperature rise andtheisentropicefficiency of thestage. POI+ I1pos= 1+ /).pos= (1+ rtsATos)Y/(l'-ll POlPOITO!Expandingbymeansof thebinomialtheorem,assumingthatI1Tos To],we have AposYI1Tos --=--rts--POIY - 1TOJ......... ActualV'N ............ / :::::, Stalled........,.( t-9 I............ mea.nsa larger annulusareaforagivenmassflow.Foranindusnialgasturbmewhen and weightareof littleconsequenceandalowSFC isvital,it wouldbesensibleto ELEMENTARYTHEORY OF AXIALFLOWTURBINE 277 6.0 5.0 60' N:;:,--",f.2 AN,whichitshouldbebyvirtueof thetipleakagelossintherotor blades. Thenextstepsin thedesignare (a)toconsider the three-dimensional nature of the flow in so far as it affects the variationof thegasangleswithradius; (b)toconsiderthebladeshapesnecessarytoachievetherequiredgasangles, andtheeffectof centrifugalandgasbendingstresseson thedesign; (e)tocheck the design by estimating AN and ARfrom the resultsof cascade tests suitablymodified totakeaccount of three-dimensionalflows. 7.2Vortextheory Earlyintheprevioussectionitwaspointedoutthattheshapeof thevelocity trianglesmustvaryfromroottotipof thebladebecausethebladespeedU increaseswithradius.Another reasonisthatthe whirlcomponent in theflowat outlet fromthe nozzlescauses thestaticpressure and temperaturetovaryacross theannulus.Withauniformpressureatinlet,oratleastwithamuchsmaller variation because the whirl component is smaller, it is clear that the pressure drop acrossthenozzlewillvarygivingrisetoacorrespondingvariationinefflux velocity C2Twisted blading designed to takeaccount of the changing gas angles iscalled vortex blading. It hasbeencommonsteamturbinepractice,exceptinlow-pressureblading wherethebladesareverylong,todesignonconditionsatthemeandiameter, keep the blade angles constant from root totip, and assume that no additional loss isincurred by the variation inincidencealong the bladecaused by thechanging gas angles.Comparative tests, Ref.(4), have been conducted on a single-stage gas turbineof radiusratio137,usinginturnbladesof constantangleandvortex blading.Theresultsshowedthatanyimprovementinefficiencyobtainedwith vortexbladingwaswithinthemarginof experimentalerror.Thiscontrastswith similartestsona6-stageaxialcompressor,Ref.(5),whichshowedadistinct 288AXIALANDRADIALFLOW TURBINES ...".improvement from the use of vortex: blading. This was, however, not so much an improvelllent in efficiency(of about 15 per cent)asinthedelay of theonset of surging which of coursedoes not arisein accelerating flow.It appears, therefore, thatsteamturbinedesignershavebeencorrectinnotapplyingvortextheory exceptwhenabsolutelynecessaryattheLPend.Theyhavetoconsiderthe additionalcostof twistedbladesfortheverylargenumberof rowsof blading required,andtheyknowthattheRankinecycleisrelativelyinsensitiveto componentlosses.Conversely,it isnotsurprisingthat thegasturbine'designer, struggling toachievethe highest possible component efficiency,hasconsistently usedsomeformof vortexbladingwhichitisfeltintuitivelymustgivea better performancehowever small. Vortextheory hasbeenoutlinedinsection 5.4 where it wasshown that if the elementsof fluidaretobeinradialequilibrium,anincreaseinstaticpressure fromroottotipisnecessarywheneverthereisa whirlcomponentof velocity. Figure7.8showswhythe gasturbinedesigner cannot talk of impulseor50per centreactionstages.Theproportionof thestagepressureortemperaturedrop which occurs in the rotor must increase from root totip.Although Fig.7.8 refers toa single-stage turbine with axial inlet velocity and noswirlat outlet,the whirl componentatinlet andoutletof a repeatingstagewillbesmallcompared with Cw2:thereactionwillthereforestillincreasefromroottotip,if somewhatless markedly. Freevortexdesign Referringtosection5.6,it wasshown thatif (a)thestagnationenthalpyhoisconstant over theannulus(i.e.dho/dr=O), (b)theaxialvelocityisconstantover theannulus, (c)thewhirl velocityisinversely proportionaltotheradius, thentheconditionforradialequilibriumof thefluidelements,namelyequation (5.13), is satisfied. A stage designed in accordance with (a), (b) and (c) is called a freevortexstage.ApplyingthistothestageinFig.7.8,wecanseethatwith iI ~ I iI .I IPlI~ II .I / Rotor blades Conditions constantConditions constant across annulus nacross annulus if inlet velocity is axialoutlet velocity is axial FIG. 7.8Changes inpressureand velocityacrossthe annulus VORTEXTHEORY289 uniform .inlet conditions to thenozzles then,since no work is done by the gasin thenozzles,homustalsobeconstantover theannulusatoutlet.Thuscondition '(a) isfulfilledin thespace between thenozzlesand rotor blades.Furthermore,if thenozzlesaredesignedtogiveCa2 = constantandCw2r =constant,allthree conditionsarefulfilledandtheconditionforradialequilibriumissatisfiedin plane2.Similarly,if therotorbladesaredesignedsothatCa3 = constantand Cw3r =conStant,it iseasy toshow asfollowsthatcondition (a)will be fulfilled, andthusradialequilibriumwillbeachievedin plane3 also.Writingwforthe angular velocity wehave Ws=U(Cw2 + Cw3 ) =w(Cw2r + Cw3 r)=constant But when the work done per unit massof gas is constant over the annulus,and ho isconstantatinlet,homust beconstant atoutlet also:thuscondition(a)ismet. It isapparent that a freevortex design isone in which the work done per unit massof gasis constant over theannulus,and toobtain the total work output this specific value need only be calculated at one convenient radius and multiplied by themassflow. In contrast, we may note that because the density varies from root to tip at exit from the nozzles and the axial velocity is constant, an integration over the annulus willbenecessaryif thecontinuityequationistobeusedinplane2.Thus, consideringa flowOmthroughanannularelementof radiusrandwidthor, om= P22nrorCa2 m = 2 n ~ a 2Jr!P2rdr r, (7.22) With the radial variation of density detennined from vortex theory, the integration canbeperformedalthoughthealgebraislengthy.Fordetailedcalculationsit wouldbenormaltousea digitalcomputer,permittingreadycalculationof the densityata seriesof radiiand numerical integration of equation (7.22)toobtain the mass flow.For preliminary calculations, however,it issufficiently accurate to take theintensityof massflowat themean diameter as being themeanintensity of mass flow.In other words, the total mass flowisequal to the mass flow per unit area calculated using the density at the mean diameter (P2mCa2)multiplied by the annulus area (A2).This is one reasonwhy it is convenient todesign the turbine on conditionsat meandiameter (aswasdoneinthepreviousexample)andusethe relationswhich willnowbederivedforobtainingthegasanglesatotherradii. Using suffix m todenote quantities at mean diameter,the freevortex variation of nozzleangle(1(2 maybefoundasfollows Cw2r = rCa2 tan(1(2 = constant Ca2 = constant Hence(1(2 atanyradiusrisrelated to(l(2m atthemeanradiusrmby tan(1(2 = C;)2 tan (l(2m (7.23) 290JI.xIALANDRADIAL FLOWTURBINES Similarly,whenthereisswirlatoutletfromthestage, tan 1X3 = C; \ tan 1X3m (7.24) Thegasanglesatinlettotherotorblade,/32,canthenbefoundusingequation (7.1),namely U tan /32 = tanC ~ 2- C-a2(rm)(r)Urn=- tanlX2m- - -r12rm2Ca2 andsimilarly/33 isgiven by (rm)(r)Urntan /33 =- tan 1X3m +- -C r3rm3a3 (7.25) (7.26) Toobtainsomeideaof thefreevortexvariationof gasangleswithradius, equations(7.23}-(7.26)willbeappliedtotheturbinedesignedintheprevious section.Wewillmerelycalculatetheanglesattherootandtip,althoughin practicetheywouldbedeterminedatseveralstationsupthebladetodefinethe twist more precisely. We will at the same time clear up twoloose ends: we have to checkthatthereissomepositivereactionattherootradius,andthattheMach number relativetotherotor bladeatinlet, Mn , isnowherehigher thansay 075. From the velocity triangles at root and tipit will be seen that this Mach number is greatest attherootanditisonlyat thisradiusthatit needbecalculated. Fromthemeandiametercalculationwefoundthat 1X2m = 58.380 /32m = 20.490,1X3m = 100,/33m = 54.960Fromthecalculatedvaluesofhandrmwehaverr = rm - (hI2)andrt = rm+(hI2),andthus (rm)= 1.164, rr2 (rm)= 0.877, r t2 Also Um = Urn =.!.= 1.25 Ca2 Ca3 4>Applyingequations(7.23)-(7.26)weget Tip Root (X25493 6215 1X3t'm)= 1217, \J"r3 (rm)= 0.849 r t3 The variation of gasangles with radius appears asin Fig.7.9, which also includes thevelocitytrianglesatrootandtipdrawntoscale.That Mn = V2IJ(yRT2)is VORTEXTHEORY 70,...--'--.,---__ U) ~30 Cl '" Cl 200--10'--__'--_-' Root MeanTip -Root ---- Tip FIG. 7.9Variationof gasangleswithradius 291 greatest at the root isclear from the velocity triangles:V2is then a maximum,and J(yRT2)isaminimumbecausethetemperaturedropacrossthenozzlesis greatestat therootThatthereissomepositivereactionattherootisalsoclear becauseV3r >V2/". Althoughthereisnoneedliterallytocalculatethedegreeof reaction at theroot,wemustcalculate (Mdr toensurethat thedesignimpliesa safe value.Usingdata fromtheexampleinsection7.1wehave Vz,. = Ca2 sec/32r = 272sec3932= 352 mls C2,. = Ca2 secIXZ,. = 272sec6215= 583mls C ~ , .5832 952'( TZr= Toz- 2cp = 1100 - 2294=x V2r352= 0.58 (Mvz)r= J(yRT2r).1(1333x0287x952x1000) This isa modest valueand certainly fromthis point of view a higher value of the flowcoefficient4> couldsafely havebeenusedinthedesign,perhapsinsteadof introducingswirlatexitfromthestage. Constantnozzleangledesign Asin the caseof theaxialcompressor,it is notessential todesign forfreevortex flow.Conditions other thanconstant CaandCwrmaybe used togive someother formof vortex flow,whichcan still satisfy therequirement forradialequilibrium of the fluidelements. In particular,it may be desirable tomakea constant nozzle angleoneof theconditionsdeterminingthetypeof vortex,toavoidhavingto 292 AXIALANDRADIALFLOWTURBINES manufacture nozzles of varying outlet angle. This, as will now be shown, requires particular variationsof CaandCWoThe vortex flowequation(5.15)statesthat dCadCw dho Considertheflowinthespacebetweenthenozzlesandblades.Asbefore,we assume that the flowis unifonn acrossthe annulus at inlet to the nozzles,andso the stagnation enthalpy at outlet must also be unifonn, i.e.dho/ dr = 0in plane 2. Also,if ()(zistobeconstant wehave C =cot ()(z = constant Cw2dCa2 dCw2--=-- cot()(? drdr-The vortex flowequation thereforebecomes 2dCw2 dCwzCw2 cot()(2T+ CwzT+-r- = 0 (1 +r)dCw2+ Cw20 co()(2 T-,-. = dCw2 .2dr --=-sm ()(,-Cw2L. r Integratingthisgives Corsin'",= constant w_(7.27) (7.28) AndwithconstantG(z, Ca2 oc Cw2sothat the variationof Ca2 must be thesame, namely (7.29) Normallynozzleanglesaregreaterthan60degrees,andquiteagoodapproxi-mation to theflowsatisfYingtheequilibriumcondition isobtained bydesigning withaconstant nozzle angleand constantangular momentum,i.e.()(2 = constant andCw2r =constant.If thisapproximationismadeandtherotorbladesare twisted to give constant angular momentum atoutlet also, then, asfor free vortex flow,theworkoutput perunitmassflowisthesameatallradii.On theother hand,if equation (7.28) were used it would be necessary tointegrate from root to tiptoobtain the work output.Weobserved early in section 7.2that thereislittle difference in efficiency between turbines oflow radius ratio designed with twisted and untwisted blading. It follows that the sort of approximation referred to hereis certainly unlikelytoresultin asignificantdeterioration of perfonnance. Thefree vortexandconstant nozzleangle typesof design donotexhaustthe possibilities.Forexample,oneothertypeof vortexdesignaimstosatisfYthe radialequilibriumconditionandatthesametimemeetaconditionof constant massflowperunitareaatallradii.Thatis,theaxialandwhirlvelocity distributionsarechosensothattheproductP2Ca2isconstantatallradii.The CHOICEOFBLADE PROFILE,PITCHANDCHORD293 FIG.7.HI of t':isapproachcorrectlypointoutthatthesimplevortextheory outlmed msectIOn5.6assumesno radialcomponent of velocity,and yet even if the turbine isdesigned with no flare there must be a radialshift of the streamlines shown in Fig.7.lO.Thisshift isdue to the increase in density from root totip m plane 2.Theass.umption that the radial component iszero would undoubtedly betrueforaturbmeof constantarmulusareaif thestageweredesignedfor constant per unit area.It isargued that theflowis then more likely to behaveasmtended,sothatthegasangleswillmorecloselymatchtheblade angles. Further details can be found in Ref.(2). In view of what has just been said about the dubious benefits of vortex blading for turbinesof modest radius ratioit isverydoubtfulindeedwhethersuchrefinementsaremorethananexercise. 7.3Choiceof blade profile,pitchand dumd So ourexample we have shownhow toestablish thegasanglesat allblade heIghts.The next step is to choose stator and rotor blade shapes whIchWIll thegasincidentupontheleadingedge,anddeflectthegas throughthereqUIredanglewith theminimumloss.An overallbladelosscoeff-icientY (orIl)mustaccount forthefollowingsourcesof frictionloss. (a) withboundarylayergrowthovertheblade profile lossunderadverseconditionsof extremeanglesof mCldenceor highinlet Machnumber). (b)Annulus loss-associated with boundary layer growthon the inner and outer wallsof theannulus. (c)Secondaryflowloss-arisingfromsecondaryflowswhicharealways pr:sentwhenawallboundarylayeristurnedthroughananglebyan adjacentcurvedsurface. (d)Tipclearanceloss-neartherotorbladetipthegasdoesnotfollowthe intended path,failsto contribute itsquota of work output,and interacts with theouter wallbounda.rylayer. TheprofilelosscoefficientYpismeasureddirectlyincascadetestssimilarto described forcompressor blading in section 5.8.Losses(b)and(c)cannot easIly be separated,and they are accounted for by a secondary loss coefficientY". 294 AXIALANDRADIAL FLOWTURBINES Thetipclearancelosscoefficient,whichnormally arises. onlyforrotorblades, will be denoted byYk.Thus the totallosscoefficientY comprisesthe accurately measuredtwo-dimensionallossYp ,plusthethree-dimensionalloss(Ys + Yk) whichmustbededucedfromturbinestagetestresults.Adescriptionof one importantcompilationof suchdatawillbegiveninsection7.4;allthatis necessaryforourpresentpurposeisa knowledgeof thesourcesof loss. Conventionalblading Figure7.11showsaconventionalsteamturbinebladeprofileconstructedfrom circulararcsandstraightlines.Gasturbineshaveuntilrecentlyusedprofiles closely resembling this,although specified by aero foilterminology.Oneexample isshown:the T6base profile which issymmetrical about thecentreline.It hasa thickness/chordratio(t/c)of 01,aleadingedgeradiusof 12percenttanda trailingedgeradiusof 6 per centt.Whenscaleduptoat/c of 02and usedin conjunction with a parabolic camber line having the point of maximum camber a distanceof about40per centcfromtheleadingedge,theT6profileleadstoa blade section similar to that shown but with a radiused trailing edge. In particular, thebackof thebladeafterthethroatisvirtuallystraight.Othershapesusedin BritishpracticehavebeenRAF27andC7baseprofilesonbothcircularand parabolicarccamber lines.Allsuch bladingmaybereferred toasconventional blading. Angleofincidencei "p 6/",," ,-

:::cos-1 (oIs)Pitchs xII 0.025-0.05-0.10-0.15-0.20-'i f--->- Y ylL _0.0154- 0.0199- 0.0274 - 0.0340 - 0.0395..J0.30 - - 0.0472 040-1"';'-0.05(tlL=0.10) 2II 0.50- - 0.0467tr5-'" E '" (,) 0.60-0.70-O.BO-0.90--0.95--0.0370 - 0.0251- 0.0142- 0.0085- 0.0072Leadingedge radius0.121 Trailingedgeradius0.601 Symmetricalabout. /32//33"c "'130.12 ;;: " 0"U) 0.08 U) .Q ' 0..0.04 0 -30'-20'-10'0'10'20'30' Incidencei FIG.7.12Effect on incidenceuponYpIt is inlportant to remember that the velocity triangles yield the gas angles,not thebladeangles.Typicalcascaderesultsshowingtheeffectof incidenceonthe profilelosscoefficientYpforimpulse(A =0and/32 :::: /33)andreactiontype bladingaregiveninFig.7.l2.Evidently,withreactionbladingtheangleof incidencecan vary from-15 to+15without increaseinYp'The picture is not verydifferentevenwhenthree-dimensionallossesaretakenintoaccount.This meansthat a rotqr blade could be designed tohavean inlet angle82equal tosay (/32r- 5)attherootand(/32t +10)atthetiptoreducethetwist required by a vortexdesign.It must be remembered,however,that a substantial margin of safe incidencerangemustbelefttocopewithpart-loadoperatingconditionsof pressureratio,massflowand rotationalspeed. Withregardtotheoutletangle,ithasbeencommonsteamturbinepractice totakethegasangleasbeingequaltothebladeangledefinedbycos-I (opening/pitch).Thistakesaccountof thebendingof theflowasitfillsupthe narrowspacein the wakeof thetrailingedge;thereisno'deviation'in thesense of thatobtainedwithdeceleratingflowinacompressorcascade.Testsongas turbinecascadeshaveshown,however,thatthecos-I (o/s)ruleisanover-correctionforbladesof smalloutletangleoperatingwithlowgasvelocities,i.e forsomerotorblades.Figure7.13showstherelationbetweentherelativegas outlet angle, /33 say,and the blade angle defined by cos-1 (o/s).The relation does notseemtobeaffected by incidence withinthe workingrangeof 15degrees. Thiscurve isapplicable to'straight-backed'conventional bladesoperating with a relativeoutlet Mach number below05.Witha Mach number of unity thecos-1 (o/s)ruleisgoodforalloutletangles,andatMachnumbersintermediate betvveen 05and 10 it can be assumed that [cos-I(o/s) - P3]varies linearly with Machnumber.Reference(3)givesanadditionalcorrectionforbladeswitha curved-backtrailingedge. Notethat untilthe pitchandchord havebeen established itisnot possible to drawa bladesectiontoscale,determinethe'opening',and proceed bytrialand 296 OJOJ" QlQi'5 0 '"co0> (]) > "fdill cc80' 70' 60' 06:..: 50' ' 40' 3LO-'-,COS-1 (a/s) AXIALANDRADIALFLOWTURBINES FIG. 7.13Relationbetweengasandbladeoutletangles error to makeadjustments until the required gasoutlet angle(X2 or 133 is obtained. Furthennore, this process must be carried out at a number of radii from root totip tospecify theshapeof the bladeasa whole.Now the pitch and chord have tobe chosen with due regard to(a)theeffect of the pitch/chord ratio(sic)on the blade losscoefficient,(b)theeffectof chord upontheaspect ratio(hlc),remembering that h has already been determined, (c) the effect of rotor blade chord on the blade stresses,and(d)theeffectof rotorbladepitchuponthestressesatthepointof attachment of theblades totheturbinedisc.Wewillconsider each effect in turn. (a)'Optimum' pitchlchord ratio In section7.4(Fig.7.24)arepresentedcascadedataonprofilelosscoefficients andfromsuchdataitispossibletoobtaintheusefuldesigncurvesinFig. 7.14.Thesecurvessuggest,asmight beexpected,that thegreater thegasdeflec-tionrequired[( G( I + G(2) forastatorbladeand(132 + 133) forarotorblade]the smallermustbethe'optimum'slcratiotocontrolthegasadequately.Thead-jective 'optimum'isin inverted commas because it is an OptimUlll with respect to Ynot to the overall lossY.The true optimum value of sic could be foundonly by a detailedestimateof stageperfonnance(e.g.onthelinesdescribedin section 7.4) forseveral stagedesignsdiffering in sic but otherwise similar. In fact thesic valueisnotverycritical. For the nozzle and rotor blade of our example turbine we have established that IXlm = 00,G(2m = 5838;132m =