theory and applicaon of gas turbine systemskshollen/gasturbine/lecture_gt_part_iv.pdftheory and...

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