Development of a computer model for long-throated flumes based on manning equation and different side slopes of trapezoidal channels

Transcription

1 Iran Agrultural Researh (08) 37() Development of a omputer model for long-throated flumes based on mannng equaton and dfferent sde slopes of trapezodal hannels M. Mahbod * and Sh. Zand-Parsa Department of Water Engneerng, College of Agrultural, Jahrom Unversty, Jahrom, I. R. Iran Department of Water Engneerng, College of Agrultural, Shraz Unversty, Shraz I. R. Iran. * Correspondng Author: mahbod00@yahoo.om ARTICLE INFO Artle hstory: Reeved 6 July 06 Aepted 5 Aprl 07 Avalable onlne 9 September 08 Keywords: Long-throated-flumes Mannng's equaton Wn Flume model LOTF model head-dsharge table ABSTRACT- Measurement of rrgaton water n agrultural areas s neessary, espeally, n ard and sem-ard regons. Long-throated flumes have been used to measure rrgaton water and are known to be nexpensve deves wth hgh performanes. In these strutures, head-dsharge tables are predted theoretally wth no need to albrate. In ths study, the model of LOTF (LOng Throated Flumes) was developed n VB. net programmng language, n whh, frton head loss and h-q (h s head at measurng staton and Q s dsharge) tables were predted by usng the Mannng's equaton. The predted tables of h-q by the LOTF model were perfetly smlar to the results of Wn Flume and HEC-RAS model whh are unversally used for these flumes and open hannel flow, respetvely. Calulated h-q tables n trapezodal flumes wth two dfferent sde slopes of 0 and, 0 and, and 0 and 4 were omputed by the LOTF and HEC-RAS models and were the same as those predted by the WnFlume model wth equal average of these sde slopes of hannel. At a water depth, n trapezodal setons wth dfferent and equal average sde slopes, the resulted h-q tables n the three models were the same. INTRODUCTION Most ommonly used strutures for measurng flow n open hannels operate by produng rtal flow or flow at rtal depth through a ontrol seton. Under these flow ondtons, the dsharge through the rtal seton s a funton of the seton dmensons and energy at the upstream (energy at measurng staton). Flumes suh as parshall flumes (Parshall, 96), utthroat flumes (Skogerboe et al., 967) and longthroated flumes (Replogle, 975) are examples of rtal-flow deves. The long-throated flumes and wers provde ost-effetve, pratal and flexble apabltes for measurng dsharge n open hannel systems as used n rrgaton hannels and these strutures are generally very aurate when operated under unsubmerged flow ondtons (Keller, 04). An mportant omponent of modern broad-rested wers and long-throated flumes s the streamlned onvergng transton. Older broad-rested wers had ether no transton (just an abrupt rased sll) or a rounded leadng edge that stll allowed some flow separaton and streamlne urvature to our n the throat. The haratersts of the transton nfluened the flow at the rtal seton, and thus these deves stll reled on empral albratons developed through laboratory or feld testng (Akers et al., 978; Bos et al., 984). It beame possble to develop theoretal albratons when t was realzed that a sutably gradual transton would smplfy the flow ondton at the rtal seton (Wahl et al., 005). Prmary advantages of long-throated flumes nlude (Wahl et al., 000, 005; Sahu et al., 0; Guan et al., 04): Ratng table unertanty of ±% or better n the omputed dsharge. Choe of throat shapes allows a wde range of dsharges to be measured wth good preson. Mnmal head loss needed to mantan rtal flow ondtons n the throat of the flume. Ablty to make feld modfatons and perform omputer albratons usng as-bult dmensons. Eonomal to onstrut, and adaptable to a varety of exstng anal onfguratons.

2 Mahbod and Zand-Parsa / Iran Agrultural Researh (08) 37() Vatankhah and Mahdav (0) proposed a smplfed proedure for desgnng long-throated flumes and wers and dsharge omputaton n these strutures. Raza et al. (007) tested the performane of loally fabrated portable long-throated flumes. Ther study showed that the results are aeptable and portable longthroated flumes are relable and an easly be desgned. Van Den Bosh (004) ntrodued flow measurement wth a long-throated flume n the rehabltaton of rrgaton shemes n the western part of Kosovo. He suggested that the flow measurement strutures for small rrgaton shemes should have an easy and sold onstruton, low mantenane needs, aurate and quk readablty of gauges, and a water level drop as small as possble. For analytal albratng of long throated flumes, Replogle (975) developed the prnples for predton of h-q at measurng staton (h s head and Q s dsharge) tables. However, development of a h-q table for a gven flume s umbersome and tme-onsumng. Hene, omputer programs have greatly faltated the analyss of the strutures. The frst software was prepared by Replogle (975) wth the purpose of developng ratng tables only for trapezodal and omplex trapezodal shapes and named FLUME. Later on, newer versons of the software have been developed (FLUME.0 (Bos et al., 984); FLUME.0 (Clemmens et al., 987), FLUME 3.0 (Clemmens et al., 993). Eventually, the last verson of the software wth the name of WnFlume beame avalable n 00 (Clemmens et al., 00). The hydraul theory used n WnFlume was smlar to the FLUME 3.0 program. The model of WnFlume was wrtten entrely n the Vsual Bas 4.0 programmng language. The program has lots of advantages lke: drawng the profle of water surfae flow, flume desgnng, determnng lmts of dsharge for optmal measurements aordng to downstream and upstream dmensons of hannel. In WnFlume, frton head loss from gage seton to ontrol seton s alulated by usng the boundary layer drag oeffent method (Clemmens et al., 00). Furthermore, n WnFlume, sde slopes for the trapezodal seton should be dental. The HEC-RAS software omputes water surfae profles for all hannel ross setons (U.S. Army Corps of Engneers., 005) and unversally used for open hannel flow. Keller (04) oupled HEC-RAS wth WnFlume for desgnng long throated flumes. In longthroated flumes, beause of produng rtal flow n the ontrol seton, the HEC-RAS model ould be used for preparng h-q tables n all hannel shapes. Jhan (03) and Jowhar and Jhan (0) showed that HEC- RAS ould predt the flow ratng table of the flow over wers wth reasonable auray. They showed that HEC-RAS predted the value of water surfae profle wth the absolute error between 0.45 and.5% wth average of.45 % of real water surfae heght. The ontrbutons of ths study are to propose a model (LOTF) to alulate h table based on frton head losses by Mannng's equaton and onsderng trapezodal hannels wth unequal sde slopes. Fnally, the results of the proposed model were ompared wth WnFlume and valdated wth HEC-RAS software. MATERIALS AND METHODS The hydraul theory for predtng dsharge through long-throated flume has resulted from over a entury of development. Fg. llustrates the Plan, lengthwse profle and a ross seton of a long throated-flume. Aordng to Fg., long-throated flumes are dvded nto fve parts as follows:. approah hannel,. onvergng transton, 3. throat, 4. dvergng transton and 5. talwater hannel. Gage s set up at the upstream of flume (Fg. ). Passng through the gagng staton, flow enters the onvergng hannel wth nreased veloty. Passng through onvergng hannel, the flow enters the throat and rtal flow ours at 0.75 of the throat length. Fnally, water flows to tal water hannel after leavng the dvergng hannel. Wn Flume Theory The ontrol seton s the approxmate loaton of rtal flow wthn the throat of the flume. It s not neessary to know presely where ths ours, beause the developed head-flow rate relatonshp s expressed n terms of the head upstream. Wth referene to Fgure., applaton of the energy equaton yelds: = + () where and are water depth and veloty at the rtal seton (on the sll), and s the water energy at the gaugng seton. To proeed further, the shape of the ontrol seton must be known. For a retangular ross-seton, the propertes of rtal flow are suh that: + = 3 = 3 where q s the flow rate per unt wdth wthn the ontrol seton and g s aeleraton due to gravty. () Substtuton of Equaton () nto Equaton () and expandng yelds: = 3 from whh: = 3 3 (3) (4) In terms of the wdth of the ontrol seton, b, Equaton (4) s wrtten as: = (5) 3 3

3 Mahbod and Zand-Parsa / Iran Agrultural Researh (08) 37() Fg.. Plan, profle, and ross seton of long-throated flumes where Q s the total flow rate. The development of Equaton (5) has assumed deal flow ondtons n partular, that there s no energy loss between the loaton of the upstream head, H, and the rtal ontrol. Seondly, Q s expressed as a funton of H, the total energy level, whereas t s muh more useful to express Q as a funton of the measured upstream head, h. These are taken nto aount by ntrodung a dsharge oeffent, Cd, and a veloty oeffent, Cv suh that = 3 3 h (6) It was shown by Bos et al. (98) that and are gven by: = (7) h = + h (8) Equatons (5) to (8) an be generalzed for nonretangular ross-setons one the relatonshp between the rtal depth, y, and the upstream energy level, s known. These equatons, or ther equvalent for non- retangular ross-setons, represent the omputatonal heart of the WnFlume program (Keller, 04). LOTF Model Theory Water veloty n the rtal seton s a funton of energy head at upstream. Therefore, the objetve s to fnd out the heght of water surfae at the gagng staton for a partular dsharge. The alulaton proess for a wde varety of dsharges s tme-onsumng; so, the software (LOTF, LOng Throated Flumes) s prepared. The nput parameters for ths model are Bed wdth of throat and the hannel, sde slopes, maxmum and mnmum dsharges, Mannng's oeffent, length of throat, onvergng and dvergng hannel lengths, and dstane from the gagng staton to onvergng transton. At frst, rtal depth should be alulated n the rtal seton. Froud number under rtal ondton s ; therefore, gven the dmensons of throat and flow dsharge, rtal ross seton area (A ) an be alulated as follows: α Q B (9) ga 3 = Crtal depth an be estmated by the Newton- Raphson method by defnng F as n Equaton (0) and settng t equal to zero.

4 Mahbod and Zand-Parsa / Iran Agrultural Researh (08) 37() F α Q B = ga 3 (0) The dstane between the rtal seton and the gagng staton s dvded nto several elements of length X ( s the number of nodes at the begnnng of eah element). At the frst pont (rtal seton), depth and veloty of the flow are alulated usng Equaton (0). Energy gradent between the two nodes (S f ) an be alulated usng the alulated depth at the - th node and an assumed depth for the next node ( ) (.e.; y - ) from the followng equaton: () S = f where n n R v ( 4 / 3 ) s Mannng oeffent and v and R are mean veloty and hydraul radus, respetvely, between the two nodes. Therefore, the head loss between the - and nodes an be obtaned from Equaton (): H =S f X () where H Χ. So, the value of s head loss along energy at th node s alulated usng Equaton (3): α v (3) H = y + + g EL where y s water depth, and EL s bed elevaton of th node. The value of Y s omputed by the Newton- Raphson method by defnng F e as n Equaton (4) and settng t equal to zero. (4) F e =H -H - + H where H - s the value of energy at the -th node. After alulatng the value of Y, the value of Y + s omputed n a smlar manner. Ths proess wll ontnue untl the water depth n gagng staton s alulated. Fnally, for eah dsharge value, there s a value for water depth n gagng staton. Aordng to the values of mnmum, maxmum and nrement of dsharge (as defned by the user), the values of the dsharges are defnable and water heghts at gagng staton are alulated. Eventually, a ratng table s obtaned for a flume wth spef dmensons and spef lmts of dsharge. If the value of energy at the rtal seton beomes less than the values of energy at the downstream of flume plus eddy losses at the end of dvergng transton, the omputaton proess wll stop. The energy n downstream of the flume (talwater energy) an be alulated based on the proedures by Bos and Rennk (98) and eddy loss through dvergng hannel an be obtaned as follows (Bos and Rennk, 98): ( v v ) (5) H d = ξ g (6) log[ 4.59 artan( )].65 ξ = m.74 where H d s head loss of dvergng transton, v and v are veloty n talwater hannel and rtal veloty n throat, respetvely, ξ s an expanson energy head loss oeffent and m s the slope of bed of dvergng transton. RESULTS AND DISCUSSION In the LOTF model, the ratng tables (h-q) of long throated flumes are predted theoretally. Channel sde slopes (z,z ), bed wdth (b), sll heght (p), length of approahng hannel (L e ), throat (L), onvergng transton (L d ), the values of mnmum and maxmum dsharges (Q mn, Q max ) and also the value of dsharge nrement are neessary for runnng the model. After runnng the LOTF model wth assumed nput parameters, the ratng table would be obtaned. In trapezodal hannels, for dental values of the sde slopes, the results of LOTF and WnFlume models were ompared. The values of dmensons of hannels and used flumes for models nputs are shown n Table. For sde slopes of 0, and and dsharge values rangng from 0. to 50 m 3 s -, ratng tables were generated. The predted dsharges are smlar n HEC- RAS, WnFlume and LOTF models n all the seleted setons. The results of predted dsharges wth the same values of h n gagng staton are ompared by WnFlume and LOTF models n Fg.. When ross-seton of hannel has two dfferent sde slopes, the value of water surfae wdth and ross setonal area are alulated by the followng equatons: B =b+y (z +z ) (7) A = by + ( z + z ) y (8) where Z and z are sde slopes of hannel. For z dental hannel sde-slopes ( + z z = ), the water surfae wdth and ross-setonal area an be obtaned from by + zy and b + zy, respetvely, and they are equal to equatons (7) and (8), respetvely. Consderng the value of Froud number ( Q B ), equal 3 ga to one for rtal ondtons, and the dfferent sde slopes (z,z ) or equal to mean of them ( z ), the urve of dsharge-rtal depth (Q-y ) s the same. When a mean value of sde slopes s used, the wetted permeter and hydraul radus are smaller and greater than alulated values usng two dfferent values of sde slopes, respetvely. So, based on Equaton (), when a mean value of sde slopes s used, smaller value of head loss s omputed.

5 Mahbod and Zand-Parsa / Iran Agrultural Researh (08) 37() Table. The nputs of flume dmensons used n LOTF, HEC-RAS and WnFlume models. z b p L Ld L Q mn Q mn No m m m m 3 s In Fg. 3, the head losses for dfferent sde slopes z =0, z = and ther mean value (Z=) ( z = ) are ompared assumng dfferent Mannng's n oeffents (0.04, 0.08, 0.0). As shown n Fgure. 3, the head losses, even for hgh roughness values for both ases of sde slopes are very small and the dfferenes are neglgble. In Fg. 3, the head losses for dfferent sde slopes z =0, z = and ther mean value ( z = ) are ompared assumng dfferent Mannng's n oeffents (0.04, 0.08, 0.0). As shown n Fgure. 3, the head losses, even for hgh roughness values for both ases of sde slopes are very small and the dfferenes are neglgble. For a spef water level n gagng staton, the LOTF model wth dfferent sde slopes and WnFlume wth mean value of them provde smlar dsharges (Fg. 4). Ths onfrms the fat that n a trapezod hannel wth dfferent sde slopes, ther mean value an be used wthout any error. When the dmensons of seleted flumes and hannels n Table are used n HEC-RAS model, the predted h-q tables are smlar to Wn Flume model (Fg. 5). So, for all hannel setons, the h- Q tables ould be predted by HEC-RAS model, as well Fg. 3. Comparson of frton head loss n long-throated flumes from the gagng staton to rtal seton (H f ) n both ondtons of two dfferent hannel sde slopes and ther mean value n three Mannng's oeffents of flume No. 6 n Table.

6 Mahbod and Zand-Parsa / Iran Agrultural Researh (08) 37() CONCLUSIONS Fg. 4. Comparson of predted dsharges from LOTF model (Q LOTF ) wth assumpton of two dfferent values for sde hannel slopes (z equals zero and z equals, and 4) and WnFlume model (Q Wn ) for the same water level n gage staton. The results of ths researh are summarzed as follows: The h-q urves resulted from the LOTF model wth frton loss alulated from Mannng's equaton and those from WnFlume model wth frton loss alulated from boundary layer drag oeffent were the same. The h-q urve n a hannel wth two dfferent sde slopes ould be obtaned f the mean value of sde slopes s used. The h-q urves for all hannel setons an also be predted by usng HEC-RAS model. Fnally, further researh s needed to ompare the results of the LOTF model wth expermental data n hydraul laboratores for hannels wth dfferent mannng oeffent and dmensons. Fg. 5. Comparson of predted dsharges from HEC-RAS (Q He ) and WnFlume models wth the same water level n gage staton n lsted flumes of Table. REFERENCES Akers, P., Whte, W. R., Perkns, J. A., & Harrson, A. J. M. (978). Wers and Flumes for Flow Measurement. New York, John Wley & Sons. Bos, M., & Rennk G. Y. (98). Head loss over longthroated flumes. Journal of Irrgaton and Dranage Dvson, 07, Bos, M. G., Replogle, J. A., & Clemmens, A. J. (984). Flow Measurng Flumes for Open Channel Systems. New York, John Wly and Sons. Clemmens, A. J., Replogle, J. A., & Bos, M. G. (987). FLUME : A omputer model for estmatng flow rates through long-throated measurng flumes,, Agrultural Researh Serve, U.S. Department of Agrulture, U.S. Government Prntng Offe, Washngton, DC: ASR-57. Clemmens, A. J., Wahl, T. L. Bos, M. G., & Replogle, J. A. (00). Water measurement wth flumes and wers. Wagenngen: ILRI Publaton 58. Clemmmens, A. J. Bos, M. G., & Replogle, J. A. (993). Flumes: Desgn and albraton of long-throated measurng Flumes. Verson 3.0. Wagenngen: ILRI, Publaton 54. Guan, G., Lu, T., Wang, C., Chen, H., & Yao, X. (04). Effet of sde ontraton on long-throated flume albraton. NongyeGonghengXuebao/Transatons of Chnese Soety of Agrultural Engneerng, 30(3), -9. Jhan, M. Q. (03). Laboratory and HEC-RAS smulaton of a sngle step wer. ARPN Journal of Engneerng and Appled Senes, 8, Jowhar, R. M., & Jhan M. Q. (0). Comparson of onedmensonal HEC-RAS wth two-dmensonal ADH for flow over trapezodal profle wers. Caspan Journal of Appled Senes Researh,, -. Keller, R. (04) Physal model testng and valdaton of large long-throated flumes. World Envronmental and Water Resoures Congress: Water Wthout Borders, 8, 3-3. Parshall, R. L. (96). The mproved venture flumes. Transaton of Ameran Soety Cvl Engneers, 89, Raza, A., Latf, M., & Nab, G. (007). Fabraton and evaluaton of a portable long-throated flume. Irrgaton and Dranage, 56, Replogle, J. A. (975). Crtal flow flumes wth omplex ross-seton. In: Irrgaton and Dranage n an age of ompetton for resoures, Spealty Conferene Proeedngs, Ameran Soety of Cvl Engneers (pp ). Logan, Utah, USA. Sahu, M., Khatua, K. K., & Mahapatra, S. S. (0). A neural network approah for predton of dsharge n straght: ompound open hannel flow. Flow Measurement and Instrumentaton,,

7 Mahbod and Zand-Parsa / Iran Agrultural Researh (08) 37() Skogerboe, G. V., Hyatt, M. L., Anderson, R. K., & Eggleston, K. O. (967). Desgn and Calbraton of Submerged Open Channel Flow Measurement Strutures: Part 3, Cutthroat Flumes. Rep.WG3-4 Utah Water Researh Laboratory, Utah State Unversty, Logan U.S. Army Corps of Engneers. (005). HEC-RAS User Manual. Hydrolog Engneerng Center, Davs CA. Van Den Bosh, B. E. (004). Feld experene on the ntroduton of flow measurement wth flumes n Kosovo. Irrgaton and Dranage Systems, 8, Vatankhah, A. R., & Mahdav, A. (0). Smplfed proedure for desgn of long-throated flumes and wer. Flow Measurement and Instrumentaton, 6, Wahl, T. L., Clemmens, A. J., Replogle, J. A., & Bos, M. G. (005). Smplfed desgn of flumes and wers. Irrgaton and Dranage, 54(), 3 47 Wahl, T. L., Replogle, J. A., Wahln, B. T., & Hggs, J. A. (000). New developments n desgn and applaton of long-throated flumes. Proeedngs of the 000 Jont Conferene on Water Resoure Engneerng and Water Resoures Plannng and Management. Mnneapols, Mnnesota. USA.

8 Mahbod and Zand-Parsa / Iran Agrultural Researh (08) 37() ()37 (397) داه راز. *... * (LOng Throated Flumes) LOTF LOTF. WnFlume HEC-RAS. (40) (0) LOTF. WnFlume : 395 /5/5 : 396//6: 397 /6/8 : : WnFlume LOTF