constant R U C 1 ar C w + dr Special case: n=1 m m w = 2 = +

Transcription

1 Thee-diensionl Flo () o f, nlysis ssued to-diensionl flo Good ssuption fo hub-to-tip tio of gete thn.8 (lst copesso stges) Fist stges of oden copesso (fn of high bypss engine) hve hub-to-tip tios of bout.35. Theefoe telines e not pllel to oto xis Flo ust hve dil velocity coponent Due to hil coponent, the sttic pessue in the nnulus inceses ith dius Flo ill djust nd undego dil oveent hnge in dil pessue (nd hence density) distibution cuses chnge in fluid velocity vecto. tong vition in cicufeentil blde speed fo oot to tip Velocity tingles t en dius e not good epesenttion of flo conditions t oot nd tip Dend fo high efficiency equies blde nd vne ngles to tch i ngles closely t ll dii. Theefoe Bldes nd vnes hve to be tisted fo oot to tip to suit chnges in i ngles Methods hve to be deived to define blde nd vne ngles s function of dius dil equilibiu Blnce beteen pessue nd ineti foces cting on fluid eleent oving ound ottionl xis nd long cuved steline. Ineti foces in dil diection ise fo Axil Flo opesso 3

2 Thee-diensionl Flo () dil equilibiu (continued) the centipetl foce ssocited ith cicufeentil flo F ( i ) θ ( ρ dd ) the centipetl foce ssocited ith the flo long the cuved steline ( ii ) ( ρ dd θ ) F cosα cos α the dil coponent of the line cceletion long the steline dt ( iii ) Theefoe, the totl ineti foce is The totl pessue foce poducing this ineti foce is dt ( ρ dd θ ) F sinα sin α ( dd θ ) F I ρ cos α sin α dp F P ( p dp )( d ) d θ pd θ p dt Equting ineti nd pessue foces nd neglecting second ode tes yields coplete nd (fo design puposes) educed dil equilibiu eqution ρ dp dp s ; α s cos α sin α d dt ρ d d d θ Axil Flo opesso 4

3 Thee-diensionl Flo (3) tgntion enthlpy vition ith dius At ny dius ith bsolute velocity : h d dh h dh d h d d ( ) Bsic theodynic eltion Tdsdh-vdp nd dopping second ode tes: dh d T ds d ρ dp d ubstitution of dh/d nd using dil equilibiu eqution leves bsic eqution fo flo in the copesso nnulus: In ost peliiny design clcultions, the dil loss vition, T ds/d, cn be ignoed the finl enegy eqution, efeed to s votex enegy eqution is given by pecil cse: constnt specific ok t ll dii d d dh ds d d T d d d d dh d d d d d tgntion enthlpy (nd tepetue) usully unifo coss nnulus t inlet. Fequent design condition is to keep it unifo (though pogessively incesing) though the copesso: dh d d d d d Axil Flo opesso 5

4 Thee-diensionl Flo (4) pecil cse: fee votex condition onstnt specific ok t ll dii nd constnt xil velocity coss nnulus: d d d d d d d d constnt Inteedite conclusion: The thee conditions of constnt specific ok, constnt xil velocity nd fee votex vition of hil velocity stisfy the dil enegy eqution nd fee votex blding ppes to be cndidte design concept fo dil blde nd vne ngle vitions. Fee votex blding nd its effect on the dil vition of degee of ection Fo constnt xil velocity nd se bsolute velocities t inlet nd exit of stge Λ (tn α tn α Fo fee votex design constnt. Thus constnt Λ / ) Evidently, Λ inceses kedly fo oot to tip nd, bsed on 5% ection t en diete, Λ y be too high t the tip o too lo t the oot. The ltte is countepoductive becuse gete te of diffusion (oe fluid deflection) is equied t slle dii due to loe cicufeentil speeds. Thus, fee votex blding not the best choice fo copessos. Hoeve, it is idely used concept in xil flo tubines nd ill be discussed gin in the ppopite sections. / Axil Flo opesso 6

5 Thee-diensionl Flo (5) Othe possible sets of design conditions hich stisfy constnt stgntion enthlpy: Note: In genel, constnt stgntion enthlpy coss the nnulus is not equieent nd ny cobintion of dil distibutions fo to of the thee independent vibles (stgntion enthlpy, xil velocity, nd hil velocity) cn be chosen. The eining vible hs to be deteined fo the votex enegy eqution. The folloing ssues the nol design conditions, i.e. constnt stgntion enthlpy coss the nnulus, nd bity hil velocity distibutions. onditions fo the dil distibutions of xil velocity nd degee of ection ill be deived. onside the folloing hil velocities t inlet nd t outlet of the oto blde (, b, nd n e constnts nd is the dius tio / : b n n pecil cse: n- b Fee votex fo nd hs to be constnt. The vition of Λ s given befoe nd ith nd defined s functions of nd b, the degee of ection cn be itten Λ pecil cse: n b eiting votex enegy eqution in tes of diensionless b / b d d d b constnt Axil Flo opesso 7

6 Thee-diensionl Flo (6) pecil cse: n (continued) Folloing integtion fo the en dius () to ny othe dius, the dil distibution of t the inlet nd the exit of the oto ed: Note tht cnnot be constnt except t the en dius. Theefoe, the oiginl definition of L hs to be used nd the degee of ection fo this type of blding hs to be deived fo fist pinciples. The ssuption of se inlet nd outlet velocity, 3, cn be etined. Theefoe, T DT nd c c P P Λ ( T T ) ( ) T ( ) ( ) ( ) ( ) Assuing the se xil velocity fo the oto inlet nd outlet t the en dius nd evisiting the hil velocities, the degee of ection fo n, lso efeed to s fist poe design, is given s Axil Flo opesso 8 ( ) [ b ln ] ( ) [ b ln ] 8 b ln b Λ ln, ( Λ ), b c P T /,.

7 Thee-diensionl Flo (7) pecil cse: n Anlogous to the pevious cse nd ith the xil velocity distibutions t oto inlet nd outlet becoe ( ) ( ) ln b ln b b b Agin, constnt xil velocity t en dius is ssued nd the degee of ection fo n, lso efeed to s exponentil blding, is given s Λ, ( Λ b b, ) uy tble: Degee of ection fo n-,,, nd (-Λ ) t the en dius n Λ ( Λ ) ( Λ ) Blding Fee Votex Exponentil ( ln )( Λ ) Fist Poe Axil Flo opesso 9

8 Thee-diensionl Flo (8) Evlution of Λ vitions fo ll thee cses (Λ.5) Fee votex gives the getest, fist poe blding the loest eduction in Λ fo lo vlues of All cses sho liit to belo hich Λ becoes negtive. Negtive Λ iplies eduction in sttic pessue in the oto (i.e. huge loss). Fo fee votex the liit is.77 (Λ.5). Note tht typicl hub-to-tip tios fo fist copesso stges e in the nge of bout A stge of lo hub-to-tip tio ust hve high degee of ection t en dius to povide cceptble conditions t the blde oot section. oents onstnt ection design ith constnt xil velocity, constnt ok input, nd constnt degee of ection, leds to lee tisted bldes. onsequence: dil equilibiu not stisfied nd flo djusts itself beteen blde os. The design hs to llo fo these djustents to void high losses due to lge diffeences beteen blde nd i ngles. Fns of high bypss engines feed diffeent i stes (bypss nd coe) hich y equie diffeent pessue tios. In this cse, the specific ok ill vy ith dius (though it y be constnt in the inne nd oute segent, espectively). Axil Flo opesso

9 opesso Design Pocess Fo equested pessue tio, i ss flo, nd tubine inlet tepetue, the design pocess ill step though the folloing considetions: hoice of ottionl speed nd nnulus diensions Deteintion of nube of stges (bsed on ssued efficiency) lcultion of i ngles fo ech stge t the en dius Deteintion of the vition of i ngles fo oot to tip Investigtion of copessibility effects election of copesso blding (bsed on expeientlly veified ifoil pefonce dt) heck on efficiency peviously ssued Estition off-design pefonce ig testing Itetions on ll of the bove ple clcultions ill be bsed on L nol dy nd opession pessue tion 4.5 Ai ss flo kg/s Tubine inlet tepetue K Axil Flo opesso

10 ottionl peed nd Annulus Diensions () No explicit foul fo ottionl speed. It follos fo ssued tip speed, xil flo velocity, nd hub-to-tip tio t inlet The nnulus e is deteined fo the eqution of continuity, the ss flo equieents, bient conditions, nd n ssued xil flo velocity Fo expeience (popiety dt bse, stndd ok pinciples) Tip speed, t, loe thn 35 /s cceptble (stess) Axil velocity in the nge of 5 /s < < /s (no hil coponent) Hub-tip tio in the nge of.35 < / t <.6 Fo specified nnulus e (ss flo equieent, bient conditions, nd ssuption of ) Fo continuity Tip dius is function of hub-tip tio Fo fixed blde speed (stess liit), the ottionl speed is lso function of hub-tip tio The designe fces viety of potentil solutions nd hs to use on judgeent nd close intection ith othe disciplines to select the ost poising (best?) { ( [ ] ) }. 5 t / πρ ( / t ) Explicit foultion of tip dius t, hich llos to solve fo N bsed on N t / π t [ s ] Axil Flo opesso

11 ottionl peed nd Annulus Diensions () The ctul choice fo / t hs to conside tubine equieents (nube of stges fo equested ok output), eight considetions (lge vlue inceses eight), nd stess considetions (sll vlue leds to high hub stesses). y, / t.5 is good copoise fo the cuent ppliction, then t.6 nd t /s. heck Mch nube eltive to oto tip to pove fesibility of choice. y the xil velocity is 5 /s, then M t.65, hich is tnsonic but not too high. Note: With selection of inlet hub-tip tio nd tip speed, the nnulus t inlet is fixed. Next step: Estite nnulus diensions t copesso exit Fo given copesso pessue tio nd ssuption of polytopic efficiency, sy η c.9, the copesso delivey tepetue cn be estited T ( n ) / n p T p 45.5 K With constnt xil velocity thoughout the copesso (no hil coponent t exit), the sttic tepetue, pessue, density, nd eventully the nnulus e e edily deteined T T 44.3 K c P, γ / ( γ ) T p p b T Finlly, nothe design ssuption hs to be de eltive to the vition of the en diete. If is held constnt, the blde height s ell s oot nd tip dietes e deteined s h.43, t.93,. 49, ρ p 3.3 kg / T 3, A. 44 ρ Axil Flo opesso 3

12 Estition of Nube of tges Tepetue ise though copesso: TT -T 64.5 K tge tepetue ise (constnt velocity) Axil inlet velocity iteion fo xiu diffusion (deceletion of flo) T tn β λ (tn β tn β c Exple: Men blde speed: 66.6 /s P V cos β DeHlle : V β,in V.7; cos Axil velocity: 5 /s ) Wok done fcto: λ.85 T 4 K Nube of stges T / T 6.9 V onclusion: se 7 stges ith n vege tepetue ise of 3.5 K pe stge. Note: Nol designs use soeht loe DT in the fist nd lst stges, sy K, leving highe equieent fo the eining stges! Axil Flo opesso 4

13 tge-by-stge Design () Evlute i ngles fo ech stge t the en dius to check tht estited nube of stges epesent cceptble design lcultions e bsed on oto considetions (no ok tnsfeed in the sttos!) Wok done fcto vies though the copesso tge hnge in hil coponent of velocity: No hil t inlet of fist stge: tn tn tn β β α Velocity tingle fist stge: β β α c P T λ V V Axil Flo opesso 5

14 tge-by-stge Design () Deflection nd diffusion in the oto blde pssges of the fist stge: V cos β β β 8.98 ;. 79 V cos β lculte stge pessue tio nd tepetue ise to deteine conditions of stte t inlet to next stge (note tht vlues of polytopic nd isentopic efficiencies e the se fo fist stge): γ /( γ ) p 3 η T,.36 p 3.49 b p T, Degee of ection (ppoxite vlue; Λ s deived bsed on the ssuption of 3!): Λ.856 high enough to void negtive vlues t oot dius gol: 5% ection stges fo the thid stge onds nd, theefoe, Λ.7, sees to be good choice fo the second stge. Ai ngle t outlet of stto hs to conside the diffusion nd deflection equieents t second stge ( α 3, α, ) ; T T T 38 3,,, K tge tge tepetue ise T 5K; ok-done fcto λ.93 Axil Flo opesso 6

15 tge-by-stge Design (3) Λ T Thus onclusion λ tn β tn β tn β c P tn β tn β tn β β 57.7 nd β 4.9 α.6 α /s 3. 6 ( ) ( tn β ) ( ) ( tn β ) hnge in hil velocity incese fo W 76.9 /s to W.3 /s. Fluid deflection hs incesed fo ε 8.98 o to ε 5.57 o tto exit ngle fist stge: α 3, α,.6 o /s.6756 DeHlle nube fo fist stge stto 3 /.97, i.e. odete nd iplying sll ount of diffusion in the stto, hich is consequence of the high degee of ection. DeHlle nube fo second stge oto ithin liits: Velocity tingle second stge: cos cos β β.7 V V Axil Flo opesso 7

16 tge-by-stge Design (4) Note: Degee of ection is elted to the shpe of the velocity dig: Λ, ;, Λ High degee of ection: keed velocity dig 5% ection: yeticl velocity dig tge outlet pessue nd tepetue γ /( γ ) p 3 η T,.8 p 3,.599 b ; p T, T T T 333 3,,, K tge 3 tge tepetue ise T 5K; ok-done fcto λ.88; Gol: Λ.5 e pocedue s fo stge, ( tn β tn β ).74 ( tn β tn β ) yields nd, leding to dehlle nube of.79, hich is β 5.4 β 8. the lo. epet nlysis ith ) diffeent degee of ection b) educed tepetue ise c) incese in d) incese in Axil Flo opesso 8

17 tge-by-stge Design (5) Eventully: β 5.9, β 8.63, DeHlle T,3 Λ.5 β 4 K, Λ.5,3 α,3 α 3, Whil velocities /s /s tges 4,5, nd 6 tge tepetue ise T 4K; ok-done fcto λ.83; degee of ection Λ.5; se clcultion s befoe tge p [b] T [K] p 3 /p [-] p 3 [b] T 3 [K] Note: Pessue tio deceses ith stge nube! Explin hy. Axil Flo opesso 9

18 tge-by-stge Design (6) tge 7 Inlet conditions: p, b, T,7 49 K opesso delivey pessue: p 3,7 4.5 x. b 4.9 b Pessue tio seventh stge nd tepetue ise equied fo desied pessue tio K.8 η T,7 T, 7 T,7 ) γ /( γ 3 p p p p Fo 5% ection, the i ngles nd the diffusion pete e ( α ); β 8.5 ( α ); DeHlle β oents An i ngle of β is not desible fo the subsequent cobustion pocess. Hoeve, copesso exit guide vne is esily incopoted into the diffuse. The nlysis ssued constnt en diete hich is not lys the best choice if constnt, ppopite vlues fo nd ust be used insted of, e.g. ust be eplced by - constnt OD ould incese the vege blde speed, I.e. the stge tepetue ises ith the potentil of educing the nube of stges constnt OD fi given tepetue ise iplies eduction of hich is beneficil in tes of the eodynic loding (decese of deflection, incese of DeHlle nube) constnt ID y be beneficil fo ultistge industil copessos hee lo cost cn be n issue Axil Flo opesso 3

19 tge-by-stge Design (7) Vition of Ai Angles fo oot to Tip eding ssignent: Axil Flo opesso 3