A New Velocity Expression for Open Channel and its Application to Lyari River

Transcription

1 World Academy of Sciece, Egieerig ad Techology 008 A New Velociy Expressio for Ope Chael ad is Applicaio o Lyari River Raa Khalid Naeem ad Asif Masoor Absrac I his commuicaio a expressio for mea velociy of wase flow via a ope chael is proposed which is a improveme over Maig formula. The discharges, sorages ad dephs are compued a all locaios of he Lyari river by uilizig proposed expressio. The resuls aaied hrough proposed expressio are i good agreeme wih he observed daa ad beer ha hose acquired usig Maig formula. Keywords Compariso, Deph, Flow, Ope Chael, Proposed Model, Sorage C I. INTRODUCTION HEZY [] developed a equaio which compues mea velociy i a ope chael flow ad his equaio is, V = C RS () where R is he hydraulic radius i meer ad S is he chael slope. I () C is called Chezy s coefficie ad is dimesioal havig dimesios (legh) / per ime. The Chezy s coefficie is deermied by experimes. Maig performed series of experimes ad foud ha depedece o hydraulic radius is acually o as give i (), ad modified (). The modified equaio, called Maig equaio, is V = / R S () i which is Maig s resisace coefficies ad is value depeds o he surface maerial of he Chael s, weed perimeer ad is deermied from experimes. This formula is more accurae he Chezy ad is widely used ow a days. The relaioship bewee Chezy s C ad Maig s is easily show o be 6 C = / R () Afer he appearace of he Maig work, may formulae are proposed for C by researchers. Here we quoe some of hem ad for ohers he reader ca refer o refereces here i []-[6]. Bazi [7], cosidered Chezy C o be a fucio of R bu o of S ad preseed he followig formula for compuaio of C Raa Khalid Naeem, PhD, Professor of Mahemaics, is wih Uiversiy of Karachi, Karachi-7570, Pakisa. ( rkaeeem@gmail.com). Asif Masoor is research scholar Deparme of Mahemaics Uiversiy of Karachi, Karachi-7570, Pakisa (phoe: ; asifmasure@homail.com). 87 C (4) M R i which M is he coefficie of roughess of he chael. Pavlovsky (95), modified C i Maig formula. The modified C is y C = R (5) where y R( 0.0). I depeds upo he roughess coefficie () ad hydraulic radius (R). This formula is widely used i U.S.S.R. Keedy [8] gave a classic empirical equaio correlaig mea velociy V i a regime caal wih verical deph D, measured o he approximaely horizoal siled bed. The mea velociy ivolves criical velociy V o is obaied by V o = 0.55 D 0.64 (6) The hydraulic diagrams give by Keedy are exesively used i Idia for desigig irrigaio caals. These diagrams eable o desig ay umber of chael secio for give slope wih differe bed widh ad deph. I he prese work we propose he followig formula for compuig he mea velociy hrough a ope chael. The formula is V R S C model (7) This formula is a improveme over Maig formula ad C model is a dimesioless umber defied i (). I is worh oig ha i he proposed V, he C model is o-dimesioal while i he earlier works i which Chezy C is modified, are all dimesioal. The formula is employed i compuig V hrough he Lyari river. The compued V is compared wih he observed values ad he values obaied hrough Maig formula. I is see ha he compued values hrough he proposed V (7) are i good agreeme wih he observed values, ad beer ha he values hrough Maig formula. II. MODEL CONSTRUCTION The hydraulic codiio of ope chael flow is a fucio of he chael geomery, discharge, roughess, ad slope The discharge Q hrough cross secioal area A is give by [9], [0] Q AV (8) where V is he mea velociy of flow. 57

2 World Academy of Sciece, Egieerig ad Techology 008 The Lyari river ca be early approximaed as a ope chael wih rapezoidal cross secio. The mea velociy for flow hrough i, employig Maig formula, is give by h W s V (9) zh z W where W, h ad z are river widh, chael flow deph ad side slope respecively. Sice he cross secioal area A is give by A Wh (0) Equaio (8), uilizig (9) ad (0), yields Q maig 5 5 zh z W W h s () Equaio (9) ad () are used for compuig he mea velociy ad discharge hrough Lyari river. The subscrip i () idicaes ha discharge Q uilizes Maig formula. I order o obai dimesioless expressio for C i he proposed formula (7), we se C( C model ) C( R, W ) () The dimesioal aalysis ells ha C ca be expressed i erms of -erm R/W ad herefore, we wrie. Cmodel Cmodel ( R / W ) () May raioal approximaios for C model are aemped ad i is foud ha raio of wo cubic for C model produces very good resuls. The raioal approximaio for C model is C model where x R / W ad a =.09E- b = 8.9E- c =.08E- d = 8.99E-4 e =.7E- f = 9.5E-4 The discharge usig C model is ax bx cx (4) dx ex fx Q model x.0089x.0008x W h s x.007x.00095x W zh z (5) The balace of he surface waer sore, S s, wihi wo river bridges is give by ds s I Q (6) d i which I is he iflow. We ow represes equaios for surface waer sore, S s, ad deph, h which already exiss i lieraure ad used i he prese sudy. The surface waer, S s, is assumed o be a liear fucio of ouflow discharge [] ad is give by Q (7) S s i which is he ravel ime bewee wo bridges uder cosideraio. If L is he disace bewee wo bridges, he L (8) V Equaio (6), uilizig (8), (0) ad (7), yields dh I WhV (9) d LW Equaio (9) describes he flow i erms of rae of chage of he flow deph for a give river secio. O usig a explici forward sep fiie differece approximaio for (9), we obai h LW h I WhV (0) Equaio (6) is solved umerically. The discreizaio of (9) o x-plae (Fig. ) leads o I I Q Q S S () i which I = iflow a ime level I + = iflow a ime level Q = ouflow a ime level Q + = ouflow a ime level S = sorage a ime level S + = sorage a ime level = ime ierval 58

3 World Academy of Sciece, Egieerig ad Techology 008 -axis Time level S + I + Q + Time level S I Q x-axis Fig. Discreizaio of sorage equaio i x-plae The Sorage ( S ) a ay ime +, afer rearragig (0), is give by S S I I Q Q () III. COMPARISON OF COMPUTED AND OBSERVED VALUES Comparisos are made bewee discharges (mea mohly, mea aual); mea mohly sorage, mea mohly deph obaied from Maig V ad proposed V ad observaios a all eleve bridges of he Lyrai river. I is see ha a all locaios of he Lyari river he proposed V produces beer resuls ha he Maig V. There are hiry wo figures of comparisos, ad i here we, for he sake of compleeess, give oly four of hem (Fig. -5). I Fig. (-5), by calculaed flow, we mea usig Maig V. Observe Flow Calculaed Flow Model Flow 5.8 Mea Aual Dischage (cumecs) Moh Fig. Compariso of mea aual discharge. 59

4 World Academy of Sciece, Egieerig ad Techology Observe Flow Calculaed Flow Model Flow Mea Mohly Discharge (cusecs) Mohs Fig. Compariso of mea mohly discharge Observed Flow Deph Model Flow Deph Calculaed Flow Deph Deph(m) Mohs Fig. 4 Compariso of deph of Lyari waerways Model Sorage Calculaed Sorage Observe Sorage Mea Mohly Sorage Moh Fig. 5 Compariso of mea mohly sorage bewee wo cosecuive bridges. 50

5 World Academy of Sciece, Egieerig ad Techology 008 IV. CONCLUSION I he prese work, a improved versio for he model proposed by Maig for compuig mea velociy of ope chael is proposed. These ivolves o-dimesioal C, while i he earlier works i which Chezy C, is modified, are all dimesioal. The discharges, sorages ad dephs are compued a all bridges of Lyari river usig proposed V, ad Maig V. I is observed ha resuls obaied hrough proposed V are good agreeme for he rece daa se ha hose obaied by meas of Maig V. REFERENCES [] A. Chezy, Formule pour rouver la viesse de l eau codui da ue rigole doée, Dossier 847 (MS 95) of he mauscrip collecio of he École Naioaldes Pos e Chaussées, Paris. Reproduced i: Moure, G. (9). Aoie Chézy: hisoire d ue formule d hydraulique. Aales des Pos e Chaussées 6, pp , 9. [] G. Lacey, Sable chaels i alluvium, i Proc. Isiue of Civil Egieers, Lodo, 99, pp [] R. K. Sharma, ad T.K Sharma, Irrigaio ad Draiage. New Dehli: Khaa Publisher, 990, pp [4] P. Diplas, Characerisics of self formed sraigh chaels, Hydraulic Egg. J., vol. 6. o. 5, pp. 7-, 990. [5] S. V. Chiale, Hydraulics of Sable Chaels, Ceral Waer ad Power Commissio, Miisry of Irrigaio ad Power, Idia, 966. [6] C. Y. Be, Chael Flow Resisace: Ceeials of Maig s formula, USA: Waer Resources Publicaio, LLC, Highlads Rach Colorado, , pp , 004. [7] H. E. Bazi, Éuded ue ouvelle formule pour calculer le débi des caaux découvers-mémoire, Aales des pos e chausses, vol. 4, o. 7, pp. 0-70, 897. [8] R. G. Keedy, The provisio of silig i irrigaio caals, i Proc. No:86, Proc. ICE, Vol. 9, Lodo. [9] V. T. Chow, Ope Chael Hydraulics. New York: McGraw-Hill, 959, pp [0] M. C. Poce, Egieerig Hydrology Priciples ad Pracices. Preice Hall: Eglewood Cliff, N.J. 076, pp. -5. [] H. E. Hudso Jr., ad R. Haze, Droughs ad Low Sream Flow, i Hadbook of Applied Hydrology, Va Te Chow, Ed. New York: MaGrow-Hill