Runnr Francis Turbin Th shap th blads a Francis runnr is cmplx. Th xact shap dpnds n its spciic spd. It is bvius rm th quatin spciic spd (Eq.5.8) that highr spciic spd mans lwr had. This rquirs that th runnr shuld admit a cmparativly larg quantity watr r a givn pwr utput and at th sam tim th vlcity discharg at runnr utlt shuld b small t avid cavitatin. In a purly radial lw runnr, as dvlpd by Jams B. Francis, th bulk lw is in th radial dirctin. T b mr clar, th lw is tangntial and radial at th inlt but is ntirly radial with a ngligibl tangntial cmpnnt at th utlt. Th lw, undr th situatin, has t mak a 90 turn atr passing thrugh th rtr r its inlt t th drat tub. Sinc th lw ara (ara prpndicular t th radial dirctin) is small, thr is a limit t th capacity this typ runnr in kping a lw xit vlcity. This lads t th dsign a mixd lw runnr whr watr is turnd rm a radial t an axial dirctin in th rtr itsl. At th utlt this typ runnr, th lw is mstly axial with ngligibl radial and tangntial cmpnnts. Bcaus a larg discharg ara (ara prpndicular t th axial dirctin), this typ runnr can pass a larg amunt watr with a lw xit vlcity rm th runnr. Th blads r a ractin turbin ar always s shapd that th tangntial r whirling cmpnnt vlcity at th utlt bcms zr ( w 0). This is mad t kp th kintic nrgy at utlt a minimum. Figur 5.0 shws th vlcity triangls at inlt and utlt a typical blad a Francis turbin. Usually th lw vlcity (vlcity prpndicular t th tangntial dirctin) rmains cnstant thrughut, i.. and is qual t that at th inlt t th drat tub. Th Eulr s quatin r turbin [Eq.(5.)] in this cas rducs t E / m U w (5.7) Whr, is th nrgy transr t th rtr pr unit mass th luid. Frm th inlt vlcity triangl shwn in Fig. 5.0. w ct (5.8a) and U ( ct ct ) (5.8b) Substituting th valus Eq. (5.7), w hav w and U rm Eqs. (5.8a) and (5.8b) rspctivly int ct (ct ct ) (5.9)
Th lss kintic nrgy pr unit mass bcms qual t /. Thrr nglcting rictin, th blad icincy bcms b ( / ) ct (ct ct ) ct (ct ct ) sinc, b can b writtn as b ct(ct ct) Th chang in prssur nrgy th luid in th rtr can b und ut by subtracting th chang in its kintic nrgy rm th ttal nrgy rlasd. Thrr, w can writ r th dgr ractin. R ct [sinc ct α] using th xprssin rm Eq. (5.9), w hav ct R (ct ct ) (5.30) Th inlt blad angl a Francis runnr varis 45 0 and th guid van angl angl rm 0 40. Th rati blad width t th diamtr runnr B/D, at blad inlt, dpnds upn th rquird spciic spd and varis rm /0 t /3. Exprssin r spciic spd. Th dimnsinal spciic spd a turbin, as givn by Eq. (5.8), can b writtn as
Ns T NP H / 5 / 4 Pwr gnratd P r a turbin can b xprssd in trms availabl had H and hydraulic icincy as Hnc, it bcms Again, N U / D, Substituting h U rm Eq. (5.8b), P Q g H h / 3 / 4 Ns T N( Q g h ) H (5.3) N (ct ct ) (5.3) D Availabl had H quals th had dlivrd by th turbin plus th had lst at th xit. Thus, ( / ) sinc ( / ) with th hlp Eq. (5.9), it bcms ct (ct ct ) r H [ ct (ct ct)] g (5.33) Substituting th valus H and N rm Eqs (5.33) and (5.3) rspctivly int th xprssin N givn by Eq. (5.3), w gt, s T
N s T 3 / 4 g 5 / 4 ( h Q) / / 3 / 4 (ct ct )[ ct (ct ct ) D Flw vlcity at inlt can b substitutd rm th quatin cntinuity as Q D B Whr B is th width th runnr at its inlt Finally, th xprssin r N st bcms, Ns T 3 / 4 5 / 4 / B / g ( h ) ( ) (ct ct ) D [ 3/ 4 ct (ct ct )) (5.34) Fr a Francis turbin, th variatins gmtrical paramtrs lik, B / D hav bn dscribd arlir. Ths variatins cvr a rang spciic spd btwn 50 and 400. Highr spciic spd crrspnds t a lwr had. This rquirs that runnr shuld admit a cmparativly larg quantity watr. Fr a runnr givn diamtr, th maximum lw rat is achivd whn th lw is paralll t th axis. Such a machin is knwn as axial lw ractin turbin. Such a turbin was irst dsignd by an Austrian Enginr, iktr Kaplan and is thrr namd atr him as Kaplan turbin. EXERCISES 5.5 A Francis turbin has a whl diamtr. m at th ntranc and 0.6 m at th 90 5 xit. Th blad angl at th ntranc is and th guid van angl is. Th watr at th xit lavs th blads withut any tangntial vlcity. Th availabl had is 30 m and th radial cmpnnt lw vlcity is cnstant. What wuld b th spd whl in rpm and blad angl at xit? Nglct rictin. (Ans.68 rpm, 8. ) 5.6 In a vrtical shat inward lw ractin turbin, th sum th prssur and kintic had at ntranc t th spiral casing is 0m and th vrtical distanc btwn this sctin and th tail rac lvl is 3 m. Th priphral vlcity th runnr at ntry is 30m/s, th radial vlcity watr is cnstant at 9 m/s and discharg rm th runnr is withut swirl. Th stimatd hydraulic lsss ar (a) btwn turbin ntranc and xit rm th guid vans 4.8 m (b) in th runnr 8.8 m (c) in th drat
tub 0.79 m (d) kintic had rjctd t th tail rac 0.46 m. Calculat th guid van angl and th runnr blad angl at inlt and th prssur hads at ntry t and xit rm th runnr. (Ans. 4.8, 59.,47.34m,-5.88m) 5.8 Th llwing data rr t an lbw typ drat tub: Ara circular inlt = 5m 6m Ara rctangular utlt = lcity watr at inlt t drat tub. =0m/s Th rictinal had lss in th drat tub quals t 0% th inlt vlcity had. Elvatin inlt plan abv tail rac lvl = 0.6m Dtrmin: (a) acuum r ngativ had at inlt (b) Pwr thrwn away in tail rac (Ans.4.95m vac,578 KW) 5.9 Shw that whn van angl at inlt a Francis turbin is 90 and th vlcity lw is cnstant, th hydraulic icincy is givn by / ( + th guid blad angl. tan ), whr 5. A cnical typ drat tub attachd t a Francis turbin has an inlt diamtr 3 m and its ara at utlt is 0 m. Th vlcity watr at inlt, which is 5 abv tail rac lvl, is 5 m/s. Assuming th lss in drat tub quals t 50% vlcity had at utlt, ind (a) th prssur had at th tp th drat tub (b) th ttal had at th tp th drat tub taking tail rac lvl as datum (c) pwr lst in drat tub. is (Ans.6.03m vac, 0.4m, 0.08m)