theory and applicaon of gas turbine systemskshollen/gasturbine/lecture_gt_part_iv.pdftheory and...
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TheoryandApplica0onofGasTurbineSystems
PartIV:AxialandRadialFlowTurbines
MunichSummerSchoolatUniversityofAppliedSciencesProf.KimA.Shollenberger
Introduc0ontoTurbines
• Twobasictypes:radialflowandaxialflow• Majorityofgasturbinesareaxialflowturbinesbecause
theyaremoreefficientexceptforverylowpowerandmassflowrateapplica0ons
• Radialflowturbinesarewidelyusedinthecryogenicindustryasaturbo-expanderandinturbochargersforreciproca0ngengines
• Aback-to-backradialturbineandcentrifugalcompressorhasthebenefitofaveryshortrigidrotorthatissuitableforcompactapplica0onssuchasauxiliarypowerplantunitsandformobilepowerplants
WindmillorAirTurbineforVeryLowPressure
Schema0cDiagramofflowThroughWindmill
Blades
BasicOpera0onofAxialFlowTurbines
Consistsofaseriesofoneormorestageswhereeachstagecontainsasetoftwoblades:– Rowofsta0onarynozzleblades(orstatorblades,ornozzleguidevanes)thataligntheflow,starttheexpansionprocesstoalowerpressureandtemperaturewhichresultsinahighervelocity
– FollowedbyarowofrotorbladesaWachedtotherota0ngshaXthatextractspowerfromthefurtherexpandinggasthatexitsatlowervelocity
NOTE:Oppositeorderforanaxialflowcompressor
Schema0cofAxialFlowTurbineStageandVelocityTriangles272 AXIAL AND RADIAL FLOW TURBINES
2
Nozzle blades I
FIG. 7.1 Axialliow turbine stage
with an increased velocity C2 at an angle IX2.t The rotor blade inlet angle will be chosen to suit the direction p2 of the gas velocity V2 relative to the blade at inlet. p2 and V2 are found by vectorial subtraction of the blade speed U from the absolute velocity C2. After being deflected, and usually further expanded, in the rotor blade passages, the gas leaves at p 3,T3 with relative velocity V3 at angle P3· Vectorial addition of U yields the magnitude and direction of the gas velocity at exit from the stage, C3 and IX3. IX3 is lmown as the swirl angle.
In a single-stage turbine cl will be axial, i.e. IXJ = 0 and cl =Cal· If on the other hand the stage is typical of many in a multi-stage turbine, C1 and IX1 will probably be equal to C3 and IX3 so that the same blade shapes can be used in successive stages: it is then sometimes called a repeating stage. Because the blade speed U increases with increasing radius, the shape of the velocity triangles varies from root to tip of the blade. We shall assmne in this section that we are tall(ing about conditions at the mean diameter of the annulus, and that this represents an average picture of what happens to the total mass flow m as it passes through the stage. This approach is valid when the ratio of the tip radius to the root radius is low, i.e. for short blades, but for long blades it is essential to accomlt for three-dimensional effects as shown in subsequent sections.
(Cw2 + Cw3) represents the change in whirl (or tangential) component of momentum per unit mass flow which produces the useful torque. The change in axial component (Ca2- Ca3) produces an axial thrust on the rotor which may supplement or offset the pressure thrust arising from the pressure drop (p2 - p 3).
In a gas turbine the net thrust on the turbine rotor will be partially balanced by the thrust on the compressor rotor, so easing the design of the thrust bearing. In what
t In the early days of gas turbines the blade angles were measured from the tangential direction following steam turbine practice. It is now usual to measure angles from the axial direction as for axial compressor blading.
V1 Va,1
Vθ,2
V2
Vrel, 2
Vrel, 1
Va,2
Va,3
Vθ,3
V3
α-fluidvelocityangleβ-rela0vevelocityangleforfluidandblade
AxialFlowthroughTurbineBlades
Sta0centhalpyor
temperature
TheoryofAxialFlowTurbine
• Gasenterstherowofnozzleblades(orstatorbladesornozzleguidevanes)atSec0on(1)
• Flowexpandsthroughthenozzlebladestoalowerpressureandtemperature,butahighervelocityatSec0on(2)
• Throughtherotorbladespowerisextractedfromthefluidwithadecreaseinvelocityandtypicallycon0nuedexpansionun0lSec0on(3)
• BladeanglesarechosentoguidetheflowenteringatSec0on(1)andtransi0oningbetweenstages
AnalysisAssump0ons
• InsinglestageturbinesthefluidvelocityatSec0on(1)ismainlyaxial,thusα1≈0˚andV1 ≈ Va1
• Formul0-stageturbines,inletcondi0onsatSec0on(1)approximatelymatchoutletcondi0onsatStage(3):α1≈α3,V1 ≈ V3;called“repea0ngstage”
• Becausebladespeedincreaseswithradius,velocitytriangleschangefrombladerootto0p,howeverforshortbladesintheradialdirec0onitisreasonabletousecondi0onsatthemeandiameteroftheannulus,rCL = DCL/2 = (Dtip + Droot)/4
AxialFlowTurbineForces
• (Vθ2 + Vθ3)representschangeinangularmomentumperunitmassflowwhichproducestorque
• (Va2 - Va3)producesanaxialthrustontherotorwhichmaysupplementoroffsetthepressurethrustarisingfromthepressuredrop(p2 – p3)
• Tosimplifytheanalysis,assumeconstantaxialflowvelocity(Va ≈ Va2 ≈ Va3)throughrotorswithaflaredannulus(usedtoaccommodatedecreasingdensityasthegasexpands)
VelocityTrianglesforFlowEnteringandExi0ngRotorBlades
U
Vrel,3 V2
V3Vrel,2Va = Va,2 = Va,3
β3
α3
β2
α2
Vθ2 + Vθ3
UVa
= tanα2 − tanβ2 = tanβ3 − tanα3
tanα2 + tanα3 = tanβ2 + tanβ3
AngularMomentumBalanceAngularmomentumequa0onforacontrolvolumeonrotor:Assumingsteadyflow,negligibletorqueduetosurfaceforces(fric0on)withrespecttolargeshaXtorque,andmassisbalancedsotorquescancelout(likeatop):NOTE:TorqueappliedtoshaXequalschangeinangularmomentumofthefluid;nowtheveloci0eshaveoppositedirec0onsenteringandexi0ng,sotheyareaddedtogether
!r ×!F∑ =
∂∂t
!r ×!V ρ dV
CV∫ +!r ×!V ρ!V ⋅ !n dA
CS∫
Tshaft = r2 Vθ 2 + r3 Vθ 3( ) !m where !m = ρ!V ⋅ !n dA
CS∫
PowerExtractedfromtheTurbineShaX
Sha/powerorrateofworkdonebythefluidis:Usingvelocityangle,α,getspecificwork:Fromvelocitydiagramintermsofbladeangle,β:Fromthesteadyflowenergyequa0on:
!Ws, ideal !m =U Va tanα2 + tanα3( )
!Wts !m =U Va tanβ2 + tanβ3( ) = cp T01 −T03s( )
!Ws, ideal ="ω ⋅!Ts =ω rCL Vθ 2 +Vθ 3( ) !m =U Vθ 2 +Vθ 3( ) !m
!Ws, ideal !m =U Va tanβ2 + tanβ3( )
TurbineEfficiency
Applyisentropicefficiencyrela0onshiptoeachcombinednozzleandrotorbladestage:whereηsistheisentropicstageefficiency(ortotal-to-totalstageefficiency).Itispartoftheoverallisentropicturbineefficiency,ηt.
T03T01
=1−ηs 1−p03p01
⎛
⎝⎜
⎞
⎠⎟
k−1( ) k⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
TurbineEfficiencyNotes
• Defini0onofηsdoesnotaccountforexi0ngkine0cenergy(KE);appropriateforthefollowingcases:– Stageisfollowedbyothersinmul0-stageturbine– WhenKEisusedforpropulsion(aircraX)– LaststagewithdiffuserorvolutetorecoverKE
• Some0mesdefinedastotal-to-sta0cisentropicefficiencyforeachstageandwholeturbine;allKEisassumedtobewasted,thusitisgenerallylower
DimensionlessParametersinTurbineDesign
FlowCoefficient,φ = Va /U (typicallylessthan1)BladeLoadingCoefficient(orTemperatureDropCoefficient)–workcapacityofastage;ra0oofextractedenergytobladeKEDegreeofReacDon–frac0onofstageexpansionthatoccursintherotor;typicallydefinedintermsofsta0ctemperaturedrops
ψ =cp T01 −T03( )
U 2 2= 2 φ tanβ2 + tanβ3( )
Λ =T2 −T3( )T1 −T3( )
DegreeofReac0onforIdealRepea0ngStages
ForouranalysiswhereweassumeVa ≈ Va2 ≈ Va3andforrepea0ngstageswhereV1 ≈ V3(KEcancels):Forjustthemovingrotorblades:Combinetoget:
!Wt !m = cp T1 −T3( ) = cp T01 −T03( ) =U Va tanβ2 + tanβ3( )
!Wr !m = cp T2 −T3( ) = 12Vrel,32 −Vrel,2
2( ) = Va2
2tan2 β3 − tan
2 β2( )
Λ =T2 −T3( )T1 −T3( )
=φ2tanβ3 − tanβ2( )
GasAnglesinTermsofφ,Λ,andψ
Combineequa0onstogetthefollowing:
tanβ3 =12 φ
ψ2+ 2 Λ
"
#$
%
&'
tanβ2 =12 φ
ψ2− 2 Λ
⎛
⎝⎜
⎞
⎠⎟
tanα3 = tanβ3 −1φ
tanα2 = tanβ2 +1φ
DegreeofReac0onNotes
• Λ=0correspondstoan“impulsestage”withnoexpansionacrosstheblades;mostefficientforextremelyhighpressurera0os(suchasforsteamturbines)duetoleakagelosses
• Forgasturbineswithmuchlowerpressurera0ostypicallyuseΛ≈50%;thiscorrespondstoexpansionbeingdividedevenlybetweenstatorandrotor
Equa0onsforΛ=50%
Subs0tuteintoequa0onsforrepea0ngstagetoget:
1φ= tanβ3 − tanβ2( )
tanα3 = tanβ3 −1φ= tanβ2
tanα2 = tanβ2 +1φ= tanβ3
tanβ3 =12 φ
ψ2+1
⎛
⎝⎜
⎞
⎠⎟
tanβ2 =12 φ
ψ2−1
⎛
⎝⎜
⎞
⎠⎟
VelocityDiagramforΛ=50%
Resultsinsymmetricveloci0estriangles:
U
V3
Va = Va,2 = Va,3
Vrel,3 V2
Vrel,2
VelocityDiagramforΛ=50%andConstantBladeSpeed
Forlowψandφ:• Lowgasveloci0es,thusreducedfric0onlossesandhigherefficiencies
• Requiresmoresstagesforagivenoverallturbineoutput
U
V2
V3
ψ =2φ=0.6
ψ =1.5φ=0.4
ψ =4φ=1
BladeLoadingversusFlowCoefficientforΛ=50%
File:U:\EES_Files\Turbines\Axial_Flow_Turbine_Tables.EES 7/11/2016 11:20:12 AM Page 2EES Ver. 9.902: #0552: for use only by students and faculty, Mechanical Engineering, Dept. Cal Poly State University
0.0 0.2 0.4 0.6 0.8 1.0 1.20
2
4
6
I = Va / U
\ =
cp 'T 0
/ (U
2 /2)
E3 = 80°
E3 = 70° E3 = 60°
E2 = 10°
E2 = 20°
E2 = 30°
ELEMENTARY THEORY OF AXIAL FLOW TURBINE 277
6.0
N
<l IJI=40, ¢= 1.0
<>" C\1 II u "'" E "' ·u C><J ;;: " 0 IJI= 2.0, ¢= 0.6 " c. E Ll
:::1 1§ "' c.
o:3(= P2) = 1 o' ·'\ (swirl) )
1],=0.94 u E
1.0 IJI= 1.5, ¢= 0.4
Flow coefficient dJ = C8 /U
-1.0 IJI=-2.0
FIG. 7.3 50 per cent reaction designs
design with a low t/1 and low ¢. Certainly in the last stage a low axial veloc'ity and a small swirl angle IX3 are desirable to keep down the losses in the exhaust diffuser. For an aircraft propulsion unit, however, it is important to keep the weight and frontal area to a minimum, and this means using higher values of t/1 and ¢. The most efficient stage design is one which leads to the most efficient power plant for its particular purpose, and strictly speaking the optimum t/1 and ¢ cannot be determined without detailed calculations of the performance of the aircraft as a whole. It would appear from current aircraft practice that the optimum values for 1jJ range from 3 to 5, with ¢ ranging from 0·8 to 1·0. A low swirl angle ( IX3 < 20 degrees) is desirable because swirl increases the losses in the jet pipe and propelling nozzle; to maintain the required high value of t/1 and low value of IX3 it might be necessary to use a degree of reaction somewhat less than 50 per cent. The dotted lines in the velocity diagram for t/1 = 4 indicate what happens when the proportion of the expansion carried out in the rotor is reduced and V3 becomes more equal to V2, while maintaining U, t/1 and ¢ constant.
We will close this section with a worked example showing how a first tentative 'mean-diameter' design may be arrived at. To do this we need some method of accounting for the losses in the blade rows. Two principal parameters are used, based upon temperature drops and pressure drops respectively. These parameters
BladeLoadingversusFlowCoefficientforΛ=50%withStageEfficiencyContoursfromTestData:Ref:Horlock,J.H.,AxialFlowTurbines,BuWerworth,1966.
ψ=
cp ∆
T 0 /
(U2 /2
)
φ= Va/ U
Noteson50%Reac0onDesigns
• Manyassump0onsweremadeinthesecalcula0onsaboutbladeshape,thustheresultsareonlygoodforqualita0veanalysis
• Lowψandφdesigns(whichimplieslowgasveloci0esandhencereducedfric0on)yieldthehigheststageefficiencies
• However,lowψmeansmorestagesforagivenoverallturbineoutputandlowφmeansalargerturbineannularareaforagivenmassflowrate
Noteson50%Reac0onDesigns,cont.
• ForindustrialgasturbineswhensizeandweightarenotveryimportantandalowSFCisvital,typicallydesignforlowψandφ
• ForaircraXpropulsion(whereweightandfrontalareaneedtobeminimalized)usehighervaluesofψandφ(currenttypicalvaluesareψ ≈3to5andφ ≈0.8to1.0)
• Lowswirlangleatexit(α3 <20˚)todecreaselossesinexhaust/diffuserorjetpipe/propellingnozzle;mightrequirelowerdegreeofreac0on
“SmithChart”fromRolls-RoyceTurbineTes0ng
NOTE:Resultsareforzero0pclearance,thushigherthanforactualengine.
Reference:Smith,S.F.,“Asimplecorrela0onofturbineefficiency,”JournaloftheRoyalAeronauDcalSociety,69,1965,467-70.
φ= Va/ U
ψ=
cp ∆
T 0 /
(U2 /2
)
Noteson“SmithChart”
• Usefulpreliminarydesigntoolthatshowscontoursofconstantisentropicefficiencyasafunc0onofbladeloadingcoefficientandflowcoefficient
• Turbinestageshavebeendesignedformanyloca0onsonSmithChart:forexample,foratypicalmul0plestageturbine– highpressurestagescanoperateinregionAand– lowpressurestagescanoperateinregionB
T-sDiagramforaReac0onStageTemperature
Entropy
T01 = T02
1
2
3
01 02
T1
T2
T3
T03
p2
p3
p1
032s
3s3ss
12
3
NozzleRotor
Root
Tip
LossCoefficientsforTurbine
Nozzleblades:Rotorblades:NOTE:λiseasierforuseindesignandYNiseasiertocalculatedfromcascadetestresults.
λN =cp T2 −T2s( )V22 2
OR YN =p01 − p02p02 − p2
λR =cp T3 −T3ss( )Vrel, 32 2
OR YN =p02, rel − p03, relp03,rel − p3
IsentropicStageEfficiency
Defineforasinglenozzleandrotorstage:FromT-sdiagramcanassume:Usethisalongwithpreviousdefini0onstoget:
ηs =T01 −T03T01 −T03s
T03 −T03s ≈ T3 −T3s
ηs = 1+ φ2λR cos
2 β3 + T3 T2( ) λN cos2α2
tanβ3 + tanα2 −1 φ
⎡
⎣⎢
⎤
⎦⎥
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
−1
Example#11
Asingle-stageturbineforasmall,inexpensive,turbojetunitistobedesignedbasedonthespecifica0onsinTable6.DeterminethefollowingassumingVa2 = Va3,V1 = V3,andα1=β1=0˚:a. bladeloadingcoefficient,b. bladeanglesanddegreeofreac0on,c. velocitycomponentsatSec0on2andSec0on3,d. opera0ngcondi0ons(p,Τ,andρ),e. annulusgeometry,andf. rotorlosscoefficientandstageefficiency.
Table6.Single-StageTurbineSpecifica@ons
Massflowrateofworkingfluid, 20kg/sInletstagna0ontemperature,T01 1100KStagna0ontemperaturedrop,T01-T03 145KInletstagna0onpressure,p01 400kPaStagna0onpressurera0o,p01/p03 1.873Rota0onalspeed,N 250rev/sMeanbladespeed,U 340m/sFlowcoefficient,φ 0.80Losscoefficientfornozzle,λN 0.050Swirlangleforoutflow,α3 10˚
!m
NotesonExample#11
Nextstepstoconsiderfordesign:• CheckMachnumberforsupersonicflowissues• Threedimensionalnatureofflowandhowitaffectsvaria0onofgasangleswithradius.
• Bladeshapesnecessarytoachievetherequiredgasangles,andtheeffectofcentrifugalandgasbendingstressesonthedesign.
• Validatedesignes0matesusingcascadetestresultstocalculateλNandλRforcomparison.
Three-DimensionalFlow
Velocitytriangleschangeshapefromrootto0pofblades(increasingradius)innozzleandrotordueto:• Increasingbladevelocitywithradius,U=ωr• Increasingsta0cpressurewithradiusduetocentrifugalforcesarisingfromswirlveloci0es;canresultinradialveloci0es
NOTE:Radialchangescanbeignoredifbladeheight,h,issmallcomparedtomeanradius,rm.(h/rm<30%)
VortexBlading
Useoftwistedbladesthataredesignedtoaccountforchanginggasanglesfrombladerootto0p.• Typicallynotusedforlowpressuresteamturbinesandsomesinglestagegasturbineswherechangesinefficiencyareinsignificant
• Typicallyusedformul0plestagegasturbines(andaxialcompressors)whereevensmallimprovementsinoverallefficiencyareimportantforheavydutyuse
VortexTheory
Derivebasicequa0onsforconserva0onofmomentumandenergytoaccountforradialpropertyvaria0ons.r-MomentumEqua0on(Differen0alForm):Forsteadyflow,constantVa,andVθ >> Vrreducesto:
ρDVrDt
−Vθ2
r⎛
⎝⎜
⎞
⎠⎟=
−∂p∂r
+ ρ gr +µ∇2Vr
Vθ2
r=1ρ∂p∂r radialequilibriumequaDon
VortexTheory,cont.
Forenergyequa0onstagna0onenthalpydefinedas:Fromthermodynamicsandmomentumbalance:
h0 = h+V 2
2= h+ 1
2Va2 +Vθ
2( ) for Vr <<Va orVθ
dh0dr
=dhdr+Va
dVadr
+VθdVθdr
dh = T ds+ dp ρ
dhdr
= T dsdr
+1ρdpdr
= T dsdr
+Vθ2
r
VortexTheory,cont.
Forcaseswhereds/dr≈0(subsonicwithoutshocks):For(1)constantVaand(2)constantspecificworkatallradiiinplanesbetweenbladerows,dh0/dr≈0:CalledfreevortexcondiDonwhichsa0sfiestheradialequilibriumequaDon(withnegligibleradialvelocity).
dh0dr
=Vθ2
r+Va
dVadr
+VθdVθdr
vortexenergyequaDon
dVθdr
= −Vθr
integrate to get: r Vθ = constant
FreeVortexStageDesign
Workdoneperunitmassofgas,,isconstantversusradiusatplanes1,2,and3,thuscalculateWatanyrandaccountforvariabledensityusing:Forini0alcalcula0onscanapproximateusingthefollowing,wheresubscriptmdenotesatmeanradius:
W = !W !m
!W =W Va ρ dAA∫ = 2 π W Va ρ r dr
rr
rt
∫
!W = π Wm Va ρm rt2 − rr
2( )
FreeVortexStageDesign,cont.
Tocalculategasanglevaria0ons:Combinetoget:Usewithearlierequa0onstogetbladeanglevaria0on.
r Vθ = rm Vθ ,m = constant
Va =Vθtanα
=Vθ , mtanαm
= constant
tanα = rmr
⎛
⎝⎜
⎞
⎠⎟ tanαm
Example#12
UsingtheresultsfromExample#11,atbothSec0on2andSec0on3attherootand0pofthebladescalculatethefollowing:a. velocityangles,α,b. bladeangles,β,c. reac0on,Λ,andverifyitisalwaysposi0ve,andd. Machnumber,Ma,andverifysubsonicflow.
GasAnglesVersusDimensionlessRadiusatSec0on2andSec0on3
-20
0
20
40
60
80
-1 -0.5 0 0.5 1
Ang
le (d
egre
es)
r* = (r - rm)/(h/2)
a1a2b2b3
α1α2β1β2
root 0p
VelocityTrianglesforSec0on2andSec0on3(DrawntoScale)
Vrel,2
Vrel,3
NotesonGasAngleandVelocityTriangleVaria0ons
• NeedtoinsureMachnumberislessthanabout0.75toavoidpossibleshocks
• MachnumberatSec0on2ishighestatrootfor:– Rela0vevelocityhigherand– Temperature(andspeedofsound)lower
• Needtoinsurereac0onisposi0veatallradialloca0ons
• Reac0onisposi0veforVrel,3>Vrel,2
ConstantNozzleAngleDesign
Formanufacturingsimplicity,constantnozzlebladeanglescanbeused:• Notessen0altodesignforfreevortexflowtoachieveradialequilibrium(wheretheflowhasnegligibleradialveloci0es)
• Alterna0vely,design0pandrootradiisuchthattheaxialandswirlveloci0eswillnowchangeappropriatelytoachieveradialequilibrium
ConstantNozzleAngleDesign,cont.
Tocalculatevelocitycomponentswithconstantbladeandfluidvelocityangles,canusethefollowing:Subs0tuteintovortexenergyequa0onfordh0/dr≈0:
tanα2 =Vθ ,2Va,2
= constant
dVa,2dr
=dVθ ,2dr
cotα2
0 = Vθ ,22
r+ cot2α2 Vθ ,2
dVθ ,2dr
+Vθ ,2dVθ ,2dr
ConstantNozzleAngleDesign,cont.
Rearrangeandintegratetoget:Subs0tu0ngbackintooriginalequa0onwealsoget:
dVθ ,2Vθ ,2
= −sin2α2−drr
Va,2rsin2α2 = constant
Vθ ,2rsin2α2 = constant
BladeProfile,Pitch,andChordDesign
Selectstatorandrotorbladeshapesthatminimize:• Profileloss–duetoboundarylayergrowth(includessepara0onlossunderadversecondi0onssuchasanextremeangleofincidenceorhighMachnumber)
• Annulusloss–duetoboundarylayergrowthoninnerandouterwallsoftheannulus
• Secondaryflowloss–duetosecondaryflowswhichalwaysoccurwhenawallboundarylayeristurnedthroughananglebyanadjacentcurvedsurface
• Tipclearanceloss–duetoouterwallinterac0ons
TypicalBladeProfile
BladeLossCoefficient
Usetoaccountforoveralllosses;definedas:NOTE:Typicallydefineseparatelyfornozzleandrotor.
Yoverall
blade losscoefficient
! = YPprofilelosses
! + Ysannulus andsecondary
losses
! + Yktip clearance
losses
!
RadialFlowTurbines
• Gasflowwithahightangen0alvelocityisdirectedinwardsandleavestherotorwithassmallawhirlvelocityasprac0cableneartheaxisofrota0on
• Turbinelooksverysimilartocentrifugalcompressor,butwitharingofnozzlevanesreplacingthediffuservanes
• Also,therenormallyisadiffuserattheoutlettoreduceexhaustvelocitytoanegligiblevalue
RadialInflowTurbineSchema0cwithVelocityTriangles
1
23
4
α2
β3
V2
V3 = Va3
Vrel,2
U2
U3
Vrel,3
ForNormalDesignCondi0on:Vrel, 2 ≈ Vr, 2andV3 ≈ Va3
r2 r3
Volute
Diffuser
Nozzlevanes
T-sDiagramforRadialTurbineTemperature
Entropy
T01 = T02
1
2
3
01 02
T1
T2
T3
T03
p2
p3
p1
032s
3s3ss
4pa
4s
NormalDesignCondi0on
FornoswirlatSec0on3(Vθ, 3 ≈ 0),workisgivenby:Includingaperfectdiffuserthisbecomes:whereC0iscalledthe“spou0ngvelocity.”ForidealcaseU2/C0 ≈ 1/√2 (actual values for good performance range from 0.68 to 0.71). Also, given by:
!W !m = cp T01 −T03( ) =Vθ ,2 U2 =U22
!Wd !m = cp T01 −T4,s( ) =C02 / 2
C02
2= cp T0 1−
pap01
⎛
⎝⎜
⎞
⎠⎟
k−1( ) k⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
Example#13
Aradialflowturbinewithaworkoutputof45.9kWaXeraccoun0ngformechanicallossesistobedesignedbasedonthespecifica0onsinTable7.Determinethefollowing:a. turbineisentropicefficiency,b. velocitycomponentsatSec0on2andSec0on3,andc. rotorlosscoefficient.
Table7.RadialTurbineSpecifica@ons
Massflowrateofworkingfluid, 0.322kg/sInletstagna0ontemperature,T01 1000KInletstagna0onpressure,p01 500kPaStagna0onpressurera0o,p01/p03 2.0Rota0onalspeed,N 1000rev/sRotorinlet0pdiameter 12.7cmRotorexit0pdiameter 7.85cmHub-0pra0oatexit 0.30Nozzleeffluxangle,α2 70˚Rotorvaneoutletangle,β3 40˚Losscoefficientfornozzle,λN 0.070
!m