OVERFLOW IN A CYLINDRICAL PIPE BEND. 1. Introduction. Fig. 1: Overflow in a cylindrical pipe bend

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1 circular cannl, ovrflow, brink pt, pip bn Dtlf AIGNER; Carstn CHERUBIM Institut of Hyraulic Enginring an Appli Hyromcanics, Drsn Univrsity of Tcnology OVERFLOW IN A CYLINDRICAL PIPE BEND T ovrflow of watr in a cylinrical pip bn iscarging wit fr surfac is iscuss. T tortical quation is bas on t critical flow, wr t nrgy a is a minimum, an t momntum quation. T solution is only possibl wit an approximation of t ara of t cross sction of circular cannl. T comparison wit xprimntal rsults for circular wirs an brink pt in circular cannls sows a iffrnc btwn 15% an 19%, wic can b xplain as t influnc of t curvatur of stramlins. 1. Introuction During transition from subcritical to suprcritical flow critical flow appars,.g. at rop points, n stills, rops in t rivr b or narrows. HAGER (1994) xplains ncssary an sufficint circumstancs concrning tis topic. E ü Grnz b() Fig. 1: Ovrflow in a cylinrical pip bn Bcaus t conition of critical flow only appars wn t nrgy lvl is, rlating to t cannl bottom, own to a minimum, it is possibl to st up a tortical rlation btwn t iscarg an minimum nrgy a, or t critical watr pt wit t lp of calculating t xtrm valu of t nrgy quation of t flow. Tis possibility may b us to crat a tortical rlation btwn iscarg an watr pt for various gomtris of t cross sction. A nw rlationsip btwn iscarg an watr lvl of a turn spillway for t rotating pip bn, a nw control mcanism for wast watr cannls (CHERUBIM, 1995), was look for. In litratur it was know only (wit xcption of STAUS) for brink pts an circular wirs abov all as mpirical solution.. Approximation of t ara for circular cross-sction For a calculation on a circlar cross sction approximat valus ar ncssary, in a way wn complt solution ar n. Bcaus of t approximat valus t rsults may sow an rror, but nvrtlss ar asir to anl or work wit. T ral solution compar wit t approximat rsult sows t rror an is minimisation. As t

2 ovrflow quation gnrally ns a mpirical corrction, t application of tis approximat rsult may b possibl. An xampl to calculat t cross sction ara in a partly fill pip is sown in quation (1), wic, spcially in t mil of t watr lvl, sows a rlativly small rror (AIGNER, 1995). 4 7 A π (1) 4. Calculation of critical pt T rlations btwn t minimum of nrgy a, t critical a an t iscarg can b calculat wit t xtrm valu of t nrgy a. E = 0 () T solution of quation () wit quation () v = + = + E g A () g can b riv from: g A = 1 (4) In t cas wn t scon rivation of t quation () is gratr tan zro, t rsult of quation (4) is a minimum. From quation () an (4) follows wit t approximation (1): = (5) 8 EMin Grnz Wit t momntum principl in t n of pip (Fig.) an t approximation of g. (1) t rlation btwn n pt an critical a Grnz or minimum of nrgy a Emin is rcir: Grnz EMin = 0.74 = 0.59 () (7) T iscarg, calculat by quations () to (7), is a function of on of t tr paramtrs, Grnz, EMin : Grnz = 4. =.485 Grnz = 1.8 EMin 49 = π g 4 48 (8) or in a non imnsional form: g.5 Grnz EMin = 1.71 = = 0.44 (9)

3 Corrctly is wit A = f ( α) = ( α sinα ) (1a) 8 α an = f ( α) = sin (1b) in a non imnsional form t quation (9a) for t minimum of nrgy a: g.5 =,5 A = ( α sinα ) Grnz α 51 sin Grnz Grnz (9a) 1 0,8 0, 0,4 0, 0 Equation (9a) Equation (9) 0 0,1 0, 0, 0,4 0,5 0, g,5 figur : T minimum of nrgy a. approximatly q. (9) an xact q. (9a) 4. Comparison wit mpirical solutions CHERUBIM (1995): During t invstigation of t ovrflow of t so call rotating pip bn, a nw control mcanism for wast watr cannls (Fig.1), ovrflow rsults of masurmnts wr analys. CHERUBIM foun t following quation for ovrflow in a rotating pip bn in vrtical position: or non-imnsional: 0,8 1,8 ( ) EÜ = C β (10) g.5 E = () GREVE (194) GREVE analys sarp g circular wirs (Fig.). If t cross sction upstram of t wir is xtnsiv, t pt of watr is narly t nrgy a ( E ). T xponnt of t mpirical quation is nar 1.88 (1.8 for =155mm to 1.89 for =740mm). g.5 = ( ) 1.88 (1) Fig. : Circular wir ( E )

4 4 RAJARATHNAM/MURALIDHAR (1991) Rajaratnam an Muraliar av invstigat t n pt in a cylinrical cannl. Ty foun t following function btwn iscarg an t watr pt at t n of cannl : g.5 Grnz = 1.54 = 0.85 (1) T rlation of n pt an critical a was a rsult of many xprimnts: Grnz = 0.75 (14) E Grnz Fig. : En pt of pip DISKIN, M.H.(19) DISKIN sows t pnncy btwn an or Grnz wit: g Grnz = 1.8 = (15) ADVANI (19) In t iscussion of t papr of SMITH (19) Avani spcifis t rsults of RAJARATHNAM un MURALITHAR an otr autors. Among otr tings sows t rsult of KING wit t following narly non-imnsional form : g = ( in m) (1) SMITH (19) Smit as stimulat a vivi iscussion wit is quations for iscarg masurmnt at t n of pip. H foun quation (17) wit t momntum principl at t n of pips. T non-imnsional form of is quation is: g.5 = 1.4 (17) HAGER (1995) HAGER xplain tis subjct of outflow at t n of pips alray from t bginning in 190 (McAULIFFE), follow by VANLEER (19) an ROHWER (194). Tis masurmnt-mto by VANLEER call California-mto was sai to rac an rror lss tan 5%. RAMPONI (19) in STAUS (197) RAMPONI as invstigat a sarp g circular wir (Fig.). H foun a iffrnt quation for t ovrflow wit a wir cofficint µ as function of t rlativ a ( E ).

5 = µ (18).5 g 1 wit µ = aftr STAUS (197) Summary Starting off wit t bginnings of iscarg masurmnt at t n of pips nowaays t pnncy btwn iscarg an nrgy a, critical a or n pt can b spcifi wit t givn quations. T tortical solution, wic was riv wit t lp of t approximation of t cross sction ara, sows a iffrnc to t mpirical solutions of CHERUBIM, GREVE an otrs. T tortical solution trby giv a 15% to 19% smallr valu tn t mpirical. T multiplication of t rigt an si of quation (9) wit 1.15 sows, wit rrors of 4 or 5%, tat it is possibl to calculat t iscarg as function of n pt, critical a or nrgy a, wn t conitions givn by t iffrnt autors ar takn into consiration. Rfrncs ADVANI, R.M.; Argyropoulos, P.A.; Rajaratnam, N.; Muraliar, D.; Vnnar, J.K. (19): Discussion to Smit (19): Brink pt for a circular cannl. Journal of Hyraulics Division 89 (Mai 19) No.Hy AIGNER, D. (1995): Hyromcanisc Grunlagn r Scwallspülung. Wassrbaulic Mittilungn s Instituts für Wassrbau un Tcnisc Hyromcanik r TU Drsn, 1995 CHERUBIM, C. (1995): Drbogn-yraulisc Grunlagn r Sturung. Wassrbaulic Mittilungn s Instituts für Wassrbau un Tcnisc Hyromcanik r TU Drsn, 1995 GREVE, F.W. (194) Smi-circular wirs calibrat at Puru Univrsity, Enginring Nws-Rcor, Vol.9,No 5, 194 HAGER, W.H.: Ausfluß aus Rorn. Korrsponns Abwassr, (199) S. 184 RAJARATHNAM, N.; MURALIDHAR, D. (194): En pt for circular cannls. Journal of Hyraulics Division. Procing of t Amrican Socity of Civil Enginrs. Vol.90, No.Hy, Marc 194, S.99ff SMITH, C.D.: Brink pt for a circular cannal. Journal of Hyraulic Division. 88 (19) S. 15, an 89 (19) STAUS, A.: Dr Krisübrfall. Wassrwirtscaft un Wassrtcnik, Hft 10/,.Jg. 197 DISKIN, M.H.: Discussion to Brink pt for circular cannals by SMITH (19).. Journal of Hyraulics Division. Procing of t Amrican Socity of Civil Enginrs. Vol.89, No.Hy, Marc 19, S.0ff