DISCHARGE MEASUREMENT IN TRAPEZOIDAL LINED CANALS UTILIZING HORIZONTAL AND VERTICAL TRANSITIONS

Transcription

1 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt 63 DISCHARGE MEASUREMENT IN TRAPEZOIDAL LINED CANALS UTILIZING HORIZONTAL AND VERTICAL TRANSITIONS Haan Ibrahim Mohamed Civil Eng. Department, Aiut univerity, Aiut, Egypt. ABSTRACT Mot of previou work for diharge meaurement truture were arried out in anal with retangular ro etion. However, in irrigation ytem mot ommon irrigation anal, epeially lined anal, have trapezoidal ro etion. From pratial view point, it i preferable to ontrut meauring truture without hanging the ide lope of the original anal. Uually, a horizontal ontration i ued a a permanent meauring devie and a vertial hump a temporarily meauring devie. In ontrary to retangular ontration, diret analyti olution of head-diharge relation i not poible a the governing equation are impliit. The olution require tediou method of trial and error. Tabular and graphial method are alo available for olution whih are ubjet to error of double interpolation and error of judgment in reading the graph. Reported herein are expliit equation for head-diharge relation for both horizontal ontration and vertial hump in trapezoidal anal without hanging the ide lope of anal. Both horizontal and vertial tranition in the trapezoidal anal are analyzed by the one dimenional momentum equation. An optimization tool i ued to ompute the ritial depth and ritial width imultaneouly for horizontal ontration, and the ritial height of the hump and ritial depth for vertial tranition. Where the objetive funtion i to adjut the peifi energy at the ontration equal to the peifi energy at the approah hannel aording the ontraint of the Froude number at ontration equal to.0. An expliit approximate equation wa developed for the ritial depth and other equation wa developed for the ritial width in the ae of horizontal ontration. Other equation were developed for omputing the ritial height of hump and the ritial depth over it. A general depth-diharge relationhip wa obtained for both type of tranition. Predited diharge uing thi general relationhip were ompared with experimental data and gave a good agreement. Key word: trapezoidal ro-etion, ontration, vertial hump, diharge meaurement.

2 64 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt INTRODUCTION We are at the point where our water upply i being ritially examined to determine quantity, ue, and wate. Plane mut therefore be formulated for extending the ue of preent upplie. One way to inreae the quantity of water i to find new water oure. Thi i not alway poible and i uually ot prohibitive. Another method i to onerve and equitably ditribute the water preently available. The firt tep in thi proe i to etablih how muh water or flow i available for ue; thu, meauring water in open hannel i ritial toward water onervation. For diharge meaurement in open hannel, for example in irrigation anal, ewage treatment plant or indutrial water upply ytem venturi flume and weir are often ued. Thee devie are deribed in the literature, e.g. Bo [], Barzewki and Jurahek []. The preent knowledge on retangular venturi flume i large. A regard to trapezoidal venturi flume, however, there i pratially a little work exept of Robinon and Chamberlain [3]. Baed upon the tudy of Robinon and Chamberlain [3], the trapezoidal flume have many harateriti whih are lited a follow: () extreme approah ondition eem to have a minor effet upon head-diharge relationhip; () material depoited in the approah did not hange the head-diharge relationhip notieably; (3) a large range of flow an be meaured through the truture with a omparatively mall hange in head; (4) the flume will operate under greater ubmergene than retangular haped one without orretion being neeary to determine the exat diharge; (5) the trapezoidal hape fit the ommon anal etion more loely than a retangular one; (6) ontrution detail uh a tranition and form work are implified. Hanen et al. [4] howed that in pite of the relationhip between head and diharge i not a eaily expreed in the form of an equation a i the retangular haper flume, it ha the advantage that a mall hange in head reult in a omparatively large hange in diharge and thu the enitivity of the flume to hange in diharge i le than that for the retangular one. Hager [5,6]; and Samani& Magallanez [7] howed that a ylinder or a irular one intalled axially in a primati hannel an be ued to meaure the diharge. The purpoe of the preent reearh i to develop the neeary expliit equation for permanent and temporarily omputation of the diharge in trapezoidal hannel by uing horizontal and vertial ontration repetively without hange of the hape of the hannel. GOVERNING EQUATIONS The flow movement through the ontrated flume a hown in Fig. an be defined uing the onventional energy equation and the Froude number ( F e ) relationhip. Auming a uniform veloity ditribution, the peifi energy equation uptream of the ritial flow etion an be written a;

3 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt 65 0.Q E = y ga in whih E i the peifi energy uptream of the ritial flow etion; y i the depth of the water uptream; Q i the flow rate; and A i ro-etional area of flow uptream of the ritial flow etion. Auming a level ontrated flume, and negligible energy lo between uptream and ritial flow etion, the uptream energy will be equal to the energy at the ritial etion, and an be deribed a Q E = E = y () ga in whih E i the energy at the ritial flow etion; y i the ditane from the urfae of the water at the ritial point to the flume floor; Q i the flow rate; and A i the ritial flow ro etion. The water will reah the ritial flow at the mallet ro etion for the anal. Sine ritial flow our with Froude number equal to, the ritial flow equation an be deribed a ga A y = F Q 3 e = (3) in whih A y repreent the derivative of ritial flow ro etion with repet to y. By rearrangement of Eqn., the diharge equation an be written a follow; ( b y y tan ) g( E y ) Q = θ (4) in whih b i the bottom width at the ritial flow etion; y i the ditane from the urfae of the water at the ritial point to the flume floor; Q i the flow rate; and θ i the ide lope angle of flow ro etion in degree. At the ae of retangular ro etion y tanθ = 0 and y = 3E whih lead to the well known formula; Q = (5) b g E

4 66 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt By the ame way, at the ae of vertial tranition hown in Fig., Eqn. 4 an be written a follow; ([ B Z tan ] y y tan ) g( E ( y Z ) Q = θ θ (6) in whih B i the bottom width of the hannel at the uptream flow etion; y i the ritial depth of water over the hump; Q i the flow rate; and Z i the hump height whih produe ritial flow over it. In the ae of trapezoidal ro etion, value of y, b and Z depend on value of B, E and θ. In the following etion, expliit equation for omputing y, b and Z will be formulated. DIMENSIONAL ANALYSIS Normally the diharge Q i determined by meauring the uptream water depth y at a poition of one to three time the maximum water depth uptream of the inlet ditortion, (Fig.,). A phyially pertinent relation between the diharge and the uptream water depth, that mean the type of the rating urve, may be found by dimenional analyi. The non-dimenional relationhip i alo ueful for heking the enitivity of the different parameter whih affet the phenomenon, Keller [8]. The funtional relationhip of the diharge Q in the ae of horizontal ontration may be expreed by: Q = f ( B, b, E, y, θ, g, ρ) (7) and for vertial ontration, Fig. : Q = f ( B, Z, E, y, θ, g, ρ) (8) where B, b are the width of the bottom of the uptream hannel and of the throat, y i the ritial water depth at the throat, E i the peifi energy at uptream hannel, θ i the ide lope angle of the hannel, g i the aeleration due to gravity, ρ i the denity, and Z i the ritial height of the hump. In pratial deign, the effet of the veloity of approah may be entirely diregarded and E i replaed by the water depth in the approah hannel.

5 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt 67 Water urfae Profile Setional Elevation Flow diretion φ Plan Fig. (): Definition keth for the horizontal ontration in trapezoidal hannel W ater u rfae P rofile S e tio n a l E le v a tio n F lo w d ire tio n b Z y P la n Fig. (): Definition keth for the vertial tranition in trapezoidal hannel

6 68 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt Some tranformation lead to the non-dimenional relation for both horizontal and vertial tranition repetively. g Q and Q g B b y = f,,, θ E E E.5. E B Z y = f,,, θ E E E.5. E (9) (0) In the following etion the four group on the right hand ide of Eqn. 9 and 0 will be orrelated in two expliit group. Dimenionally onitent equation have been obtained for ritial depth, ritial width and height of hump in the following etion. Uing thee equation it ha been poible to obtain expliit diharge equation without need to iteration. EXPERIMENTAL DATA The experimental data in thi tudy were arried out by Robion and Chamberlain [3] on a trapezoidal flume with horizontal ontration where diverion and onverion angle φ wa kept ontant and equal to about 0 o. The ide lope angle θ wa hanged three time and the width of the hannel wa hanged two time. The flowing diharge wa in the range from.4 to 57 le. and the water depth in the approah hannel wa in the range from 3.5 to 6 m. Table () how the range of ued variable. Thee data are ued a a boundary ondition in the approah hannel while the ritial width and depth of ontration are omputed uing equation. The ame thing i arried out for vertial hump where the ritial height of hump and ritial flow depth are omputed. An optimization tool i ued to ompute the ritial depth and ritial width imultaneouly for horizontal tranition, and the ritial height of the hump and ritial depth for vertial ontration. Where the objetive funtion i to adjut the peifi energy at the ontration equal to the peifi energy at the approah hannel aording the ontraint that Froude number at ontration equal to.0. Table (): Range of ued experimental data. Flume No. B (m) θ o Q (l) y (m) ~ ~ ~4 4.5 ~ ~ ~ ~ ~6

7 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt 69 RESULTS AND DISCUSSIONS Thi analyi i baed on modular flow, where the ontration i trong enough, the major part of the total uptream energy head E will be onverted into kineti energy to obtain ritial flow at the ontrol etion. Suh a etion with ritial flow i required to make the uptream peifi energy head E independent of downtream ondition. Thi flow type i normally referred to a modular flow or free flow, Boiten [9].. Horizontal Tranition (Contration) The horizontal ontration i ueful in ewage and irrigation tehnique, ine undiolved matter and ediment are le depoited in the uptream reah, and bakwater effet remain mall. Where loal redution of hannel ro etion, reate a drop in the water level over the ontration. Provided thi redution i trong enough, the major part of the total uptream energy head will be onverted into kineti energy to obtain ritial flow at the ontrol etion. Suh a etion with ritial flow i required to make the uptream head independent of downtream ondition. Determination of The Critial Depth From the dimenional analyi (Eqn. 9), the parameter y E i funtion of the uptream width to uptream energy head ratio ( B E ) and the ide lope angle of the ro etion, θ. In order to determine the orrelation between thee parameter, the reult are preented in Fig. 3 in whih y E i plotted againt B E together for the different value of the ide lope angle θ. It i notieable from thi figure, that value of y E ratio are in the range from 0.69 to 0.78 depending on value of the ide lope angle and thi differ from the value of 0.66 in the ae of retangular ontration. Alo, it i readily een that the data align logarithmi urve whih atify the following equation: y E K ln( B E) = K ( R = 0. 95) () in whih K and K are oeffiient. It wa found that K and K due independent of ide lope angle θ and take mean value of and repetively. Then, Eqn. beome; y E 0.053ln( B E ) () =

8 70 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt 0.8 ye θ=60 θ=45 θ= BE Fig. (3): Value of y E veru B E for different value of θ. Determination of The Critial Width Figure 4 how the variation of b E with B E for different value of ide lope angle θ. Apparently, for all value of θ, the data are attered around a urve hown in Fig. 5 whih an be repreented by the following equation: ( B E ) 0.7( B E ) b E 0.04 ( R = 0.98) (3) = 3.5 be θ=60 θ=45 θ= BE Fig. (4): Value of b E veru B E for different value of θ

9 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt b E = 0.04(BE ) 0.73 (BE ) R = be BE Diharge Computation Fig. (5): Correlation equation of b E. The diharge an be etimated uing the developed equation by knowing the hannel bed width and the energy head uptream the ontration whih an be approximated by the water depth. The atual meaured flow rate were ompared with the alulated flow rate uing Eqn. 4, and 3. The reult of the omparion i given in Fig. 6, whih how that the developed equation an predit the meaured flow rate with an error le than 5% Q (predited) unit Q (atual) unit Fig. (6): Predited veru atual diharge

10 7 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt. Vertial Tranition (Hump) If diharge have to be reorded at different point in a hannel ytem (uh a irrigation or ewage analization) only over a limited period, a mobile apparatu would be onvenient. Obviouly, uh an intrument hould not hange the original ro etion and be imply adjutable in the hannel. Mobile hump intallation often may beome ignifiant a hown in Fig.. When the hump reate ritial ondition over it, it an be ued for diharge meaurement. The vertial tranition i more ompliated than the horizontal ontration where the width inreae with height of the hump and thi un-uual ae. Determination of The Critial Height of Hump In Fig. 7, value of Z E are plotted veru B E for eah ide lope angle θ. The data for eah value of θ lutered around a urve whih may be expreed in the form; Z ( B E ) C ( B E ) 3 E = C C ( R = 0.96) (4) in whih C, C and C 3 are oeffiient. It wa found that C, C and C 3 to be independent of ide lope angle θ and take mean value of -.057, 0.37 and 0.67 repetively. Then, Eqn. 4 beome; ( B E ) 0.37( B E ) Z E (5) = ZE θ=60 θ=45 θ= BE Fig. (7): Value of Z E veru B E for different value of θ.

11 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt 73 Determination of The Critial Depth Figure 8 how the relation between the relative ritial depth y E and B E. The data for eah value of θ are grouped around a urve that may be expreed in relation of the form; y E M ln( B E ) = M ( R = 0.90) (6) in whih M and M are oeffiient. It wa found that M and M to be independent of ide lope angle θ and take mean value of and repetively. Then, Eqn. 6 beome; y E ln( B E ) (7) = 0.6 ye θ=60 θ=45 θ= BE Fig. (8): Value of y E veru B E for different value of θ. Diharge Computation Now, the diharge an be etimated uing the developed equation by knowing the hannel bed width and the energy head uptream the ontration whih an be approximated by the water depth. The atual meaured flow rate were ompared with the alulated flow rate uing Eqn. 6, 5 and 7. The reult of the omparion i given in Fig. 9, whih how that the developed equation an predit the meaured flow rate with error le than 5%. The approah preented herein hould be heked with other experimental data, where the effet non-modular flow ondition and onverion and diverion of ontrition angle and effet of head lo through the ontration mut be taken into aount.

12 74 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt Q (predited) unit Q (atual) unit Fig. (9): Predited veru atual diharge. CONCLUSIONS The main onluion drawn from thi tudy an be ummarized a follow:. From the preent analyi, an expliit relationhip ha been developed for the diharge omputation in trapezoidal anal uing either horizontal or vertial tranition.. In ae of horizontal ontration, an equation wa developed for the ritial width and another one for the ritial depth at ontration. 3. For vertial tranition, an equation wa developed for omputing the ritial height of the hump and an equation wa developed for the ritial depth over it. 4. The analyi reveal that the ide lope angle of the hannel ha mall effet on ritial depth at ontration and alo width or height of ontration whih an be negleted. 5. Value of ritial depth at ritial ontration of trapezoidal anal are higher than that at retangular one whih equal to 0.67 with repet to the uptream peifi energy. 6. The predited flow rate baed on the propoed approah were ompared with the orreponding meaured one. The omparion howed a good agreement. NOMENCLATURE A = the ro etional area at uptream hannel A = ro etional area at ritial etion B = width of approah hannel

13 Ninth International Water Tehnology Conferene, IWTC9 005, Sharm El-Sheikh, Egypt 75 E = the energy head at ritial etion E = the energy head at approah hannel F e = Froude number g = gravity of aeleration Q = the flow rate V = uptream mean veloity V = ritial veloity y = the water depth at approah hannel y = the ritial depth at ritial etion Z = height of hump whih produe ritial depth θ = the ide lope angle ρ = the water denity REFERENCES. Bo, M. G., Editor, Diharge meaurement truture, Delft Hydrauli Laboratory, publiation Nr. 6, Barzewki, B., and Jurahek, M., Comparion of rating urve of geometrially imilar venturi flume of different ize, Sympoium on ale effet in modeling hydrauli truture, Elingen, Robinon, A. R., and Chamberlain, A. R., Trapezoidal flume for open hannel flow meaurement, Tran. Am. So. Agr. Eng., Gen. Ed., Vol. 3, No., pp. 0-4, Hanen, V. E.; Iraelen, O. W.; and Stringham, G. E., Irrigation priniple and pratie, John Wiley& Son, New York, Hager, W. H., Modified trapezoidal venturi hannel, Jour. of Irrig. and Drainage Eng., Vol., No. 3, Hager, W. H., Venturi flume of minimum pae requirement, Jour. of Irrig. and Drainage Eng., Vol. 4, No., Samani, Z., and Magallanez, H., Meauring water in trapezoidal anal, Jour. of Irrig. and Drainage Eng., Vol. 9, No., Keller, R. J., Boundary layer ale effet in hydrauli model tudie of diharge meauring flume, Sympoium on ale effet in modeling hydrauli truture, Elingen, Boiten, W., Flow meaurement truture, Flow Meaurement and Intrumentation, 3, pp , 00.