Vertical Sluice Gate Discharge Coefficient

Transcription

1 , Scienceline Pulication Journal of ivil Engineering an Uranism Volume, Issue : 8-4 () ISSN-5-4 Vertical Sluice Gate Discharge oefficient Navi Nasehi Oskui * an Farzin Salmasi M Sc Stuent of hraulic structures, Department of water sciences engineering, Agriculture facult, Tariz Universit, Tariz-Iran Department of water sciences engineering, Agriculture facult, Tariz Universit, Tariz-Iran *orresponing author s navinasehi@gmailcom, salmasi@tarizuacir ABSTRAT: Sluice gates are wiel use for flow control an ischarge measurement in irrigation an rainage channels Their ischarge coefficient epens on geometric an hraulic parameters Errors are inevitale when their values are astracte from empirical curves for a range of reasons incluing the resolution of the graphs, jugments in reaing the values, the nee for their igital values at the computational stages This stu evelops two equations, linear an nonlinear, to etermine ischarge coefficient using imensional analsis an linear an nonlinear regression analsis, for oth free an sumerge flow conitions A total of 5 ata point was generate, which involve ifferent effective hraulic parameters The stu also inclue the results of the past stuies carrie out ifferent investigators concerning sluice gate ischarge coefficient etermination for comparison purposes The performance of the nonlinear equation improves in comparison with the linear equation All numerical computations were carrie out Wolfram Mathematica v6 software Kewors: Discharge coefficient, free flow, multiple regressing, sluice gate, sumerge flow INTRODUTION Discharge can convenientl e measure hraulic structures for controlling ischarge an water epth, as the create a one-to-one relationship etween epth an ischarge Their applications inclue irrigation an rainage canals an overflow spillwas Notal, other ischarge measuring evice are costl, eg Laser Doppler Anemometer Attention to the unerstaning of the performance of weir-tpe flow control structure, eg over-flow an sharp-creste weirs is relativel etter than that of stanar gates The asic knowlege of the hraulic performances of gate structures is even poorer than of the viration of these gates, eg see [] Open channel flow software moelling has ecome stanar esign tools for irrigation canals Over the ears, research was focuse on eveloping numerical schemes for the solution of the shallow-water equations ut sometimes this was at the expense of overlooking the significance of other components, eg the performance of sluice gates, eg see [] The moe of flow associate with gate structures is often complex uner real-time conitions ut their hraulics fall into two regimes of: (i) moular flows when ischarge is inepenent of the tailwater epth; an (ii) sumerge flows when there is epenenc; see Fig This istinction is well estalishe as the sumergence reuces the ischarge through the gates an this is reflecte on the values of ischarge coefficient Figure shows that that sumerge flow occurs when the tailwater epth is greater than the ownstream epth of the hraulic jump, Figure Scheme of a gate operating uner free (a) an sumerge () flow conition A escription of flow equations has een approache theoretical an empirical formulas an graphical approaches Swamee (99) presents two formulas to istinguish moular an sumerge flow conitions ase on Henr s (95) curve Yen et al () presents a theoretical formula an some experimental graphs to etermine maximum allowale To cite this paper: Nasehi Oskui, N an Salmasi, F Vertical Sluice Gate Discharge oefficient J ivil Eng Uran (): 8-4 Journal homepage: 8

2 tailwater epth for free flow (or minimum allowale tailwater epth for sumerge flow) The efine the conition separating free flow an sumerge flow in terms of flow contractions at the gate an erive equations for ischarge coefficient in terms of imensionless ischarge, sumerge water epth, maximum allowale gate opening The also compare their results with other investigators approach an report goo fitness In this stu we ve use Swamee s (99) approach to etermine whether the jump will e free or sumerge Free flow Sumerge flow where = upstream epth, = tailwater epth an = gate opening Discharge Formulation Using the Bernoulli s an continuit equations in sluice gate hraulic jump flow, it is possile to erive the following wiel known expression to calculate the gate ischarge for a rectangular cross section: () () 7 q g Free flow 6 () where q = ischarge per unit with of channel, g = 5 gravit acceleration, = gate opening, = upstream Sumerge epth an = ischarge coefficient epens on 7 7 ifferent parameters such as upstream an tailwater 7 8 epths, gate opening, contraction coefficient of gate an 6 the flow conition 5 Eq () is applicale for oth hraulic conitions Henr (95) use Eq () an evaluate experimentall The outcome of this stu is the wellknown Henr s curve MATERIALS AND METHODS Henerson (966) erive two equations to compute for each flow conition Free flow Sumerge flow c ( ) ( ) (5) c Where /, /, ((/ ) ) ( ) c an c = contraction coefficient The contraction coefficient is efine as the ratio of the water epth at vena contracta, to gate opening ) ( c For sharp-ege vertical sluice gate c varies etween 598 an 6 ase on theoretical reasons [5] Since the contraction coefficient epens on gate opening, shape of the gate lip, upstream water epth, gate tpe, an so forth, it is ver ifficult to know its real value for all operating conitions in practice [7] For practical purposes, selecting c = 6 have an accurate results an man researchers have use this value [6] (4) Another stu of sluice gate ischarge calculation was performe in Rajaratnam an Suramana (967) The expresse the ischarge through a sluice gate as q g( ) (6) c q g( ) (7) A value of 6 was use for c an the analsis of experimental ata inicate that was uniquel relate to / for oth flow conitions For / < this relationship was almost linear with Eq (8) (8) As can e note, Eq (7) makes use of gate sumergence epth, (see Fig ) Because it is ver ifficult to accuratel measure its value (this zone has staning recirculation flows), it must e preicte After certain simplifications the otaine [ 4 4 ] (9) Swamee (99) otaine ischarge coefficient equations for free an sumerge flow, performing nonlinear regression on Henr s (95) curve () flow () First we consier free flow conition Relation among hraulic parameters can e etermine appling the Bernoulli s equation etween sections an, an specific force equation etween sections an in Fig -a onsiering the channel ottom as the atum an neglecting the energ losses at the gate, Bernoulli s an specific force equations iel Eq ()an () q q () g g q q g g () omination of Eq () an () results Eq (4) q q g g f (,, q, ) (4) B selecting ifferent (ut hraulicall feasile) values for, an q, we can solve Eq (4) with respect to Then we can calculate gate opening ( = /6) After that using Eq () an () we coul specif flow conition Now we can calculate ischarge coefficient presente formulas for oth free an sumerge flow All of the computations were carrie out Mathematica v6 software The input ata (, an q) To cite this paper: Nasehi Oskui, N an Salmasi, F Vertical Sluice Gate Discharge oefficient J ivil Eng Uran (): 8-4 Journal homepage: 9

3 generation process is programme to generate ranom real numers etween following efine limits (It s notale that SI sstem of units was use in all over this paper): 5,, 5 q These values are which practicall occurs in irrigation an rainage channels It s notale that the tailwater epth alwas must e less than the upstream epth otherwise the flow irection will e reverse an if these two epths e equal, then there is no flow an ischarge coefficient will e zero At an point three input ata are generating at the same time an are sustituting in Eq (4) Each ata point iels four values for as the roots of Eq (4) in which two of them alwas are with minus sign an are not practicall acceptale values The thir an fourth roots are complex numers for some ata points, so these ata points shoul e eliminate from ata series Use of the thir real roots at calculation process iels negative values for in sumerge conition, so this root is not acceptale too, an onl the natural values of fourth root at an ata point will e use in future steps Now specifing the values of an contraction coefficient, the gate opening coul e calculate In some ata points, an then the gate opening is a ver small value which is certainl not feasile If these ata points e use in following processes (imensional analsis an imensionless numers generation), unusual an igressive values will otain which affects regressing result an ecreases the fitness of fitte relations So ata points with a gate opening less than 5 m are eliminate Initiall generate ata sets of, an q were selecte aout, points,in orer to eset all of the possile situations, ut after ignoring unacceptale ata points, finall 5 ata set were remaine, an it is aout 45 % of initial points 65 ata points of the remaine set are aout the free flow an 455 ata points are aout the sumerge flow conition A part of generate ata points an calculate parameters are presente in Tale At sumerge flow the water epth at the immeiatel ownstream of the gate (section in Fig - ) is So must e use as the piezometric hea at the right sie of the Bernoulli s equation, ut at velocit hea term, is use as flow section epth ecause water flows onl from vena contracta an there is just a stationar circulating flow at the upper part of the vena contracta So Bernoulli s equation alters to Eq (5) q q g g (5) After fining values, sustituting, an q at Eq (5), values will e otaine which are presente in Tale, too For free flow equals to (see Tale ) For Rajaratnam an Suramana (967) metho we / is couln t use Eq (8) to etermine when greater than, so no calculation were one to compute value for this situation This conition is specifie - in Tale R column in Tale correspons the solution of Eq (9) In aition, H, R an S columns in Tale are ischarge coefficients otaine Henerson (96), Rajaratnam an Suramana (967) an Swamee (99), respectivel Dimensional Analsis Effective hraulic parameters are as follows for,,,,,, q g B flow through sluice gate: rearranging theses parameters as imensionless parameters, we have g F(,,, ) (6) q an As can e seen, in aition to which were use in Henr (95) an Swamee (99), there are some other parameters involve in this phenomenon Dimensionless parameter q /g is Froue Numer regaring to gate opening So Eq (6) can e alters to Eq (7) F(,,, ) F r All inepenent parameters in Eq (7) are contriuting in sumerge flow conition an onl two of them (/Fr an /) are contriuting in the free flow conition We can specif F performing experimental stuies for otaining ata or generating these require ata from analsing governing equations an the use of computers As mentione previousl, secon metho was use in this stu At this step multiple regression techniques were use to etermine F regaring to generate an then refine ata series For an flow conition a linear an nonlinear functions are efine an regressing proceure is performe ase on these functions Linear an nonlinear regressing pattern is as following, respectivel 4, i,,, n i i i i Fr i (8) 4 i i i Fr i, i,,, n i (9) where n is numer of ata points an an are unknown parameters that regressing target is to fin them Orinar least square (OLS) metho was use to etermine these parameters Eq (8) an (9) are arrange for sumerge conition an as mentione previousl an in t use in free flow conition RESULTS In this stu the use of Eq (4) an (5), a coe was written in Mathematica v6 to coincie solve of Bernoulli s an specific force equations This coe can omit negative an complex roots of Eq (4) an (5), also it is esigne to generate applicale values for engineering purposes After generation of geometrical an hraulic properties of flow, the flow conition whether free or sumerge was etermine After that, the ischarge coefficient compute Henerson (966), To cite this paper: Nasehi Oskui, N an Salmasi, F Vertical Sluice Gate Discharge oefficient J ivil Eng Uran (): 8-4 Journal homepage:

4 Rajaratnam an Suramana (967), an Swamee (99) approaches an finall a sample of 5 ata points presente in Tale Then using the compute ischarge coefficients, ischarge passes through the gate for each ata points compute Eq () or (6) an (7) Tale presents a part of compute ischarge values three methos an initiall generate values The error introuce each metho is etermine the mean asolute percentage error (MAPE) MAPE is efine as follows n q ˆ i qi MAPE () n q where i i qi is initiall generate ischarge an q ˆi is -calculate values of ischarge ifferent researchers formulas an n is total numer of ata points The MAPE for Henerson (966), Rajaratnam an Suramana (967), an Swamee (99) methos is 67%, 44% an 6% respectivel which have a goo accorance with Sepúlvea et al (9) results Accoring to the efinition of MAPE criteria, the approach of Rajaratnam an Suramana (967) has more accurac to state flow through sluice gates, comparison with two other methos It is notale that the ischarge coefficient compute this metho is approximatel constant value of 59, nevertheless this metho has a high accurac for computation of ischarge rate The onl restriction of this metho is the / < constrain which is satisfie for 45 of 5 ata points Tale A part of generate an calculate parameters Mathematica coe No q R Fr onition H R S Su Su Free Su Free Su Su Su Su Su Su Su Free Su Free Su Su Su Su Su Su Su Su Su Su Su Su Su Free To cite this paper: Nasehi Oskui, N an Salmasi, F Vertical Sluice Gate Discharge oefficient J ivil Eng Uran (): 8-4 Journal homepage:

5 Tale Results of generate ischarge (q) in this stu an comparison with other researchers (q H =Henerson, 966; q R =Rajaratnam an Suramana, 967; q S =Swamee, 99) No q q H q R q S No q q H q R q S Finall F is specifie for oth free an sumerge conitions with the ai of linear an nonlinear regression techniques It shoul e sai, in this proceure S is use ecause Swamee (99) formulas are ase on experimental stuies of flow ehavior through a sluice gate Although, we can use H ecause this metho has low value of MAPE, ut this metho is theoretical an it just has goo accurac comparing with Henr s curve especiall in sumerge for low values of conition So Swamee (99) metho is preferre Regression Analsis Multiple regression analsis was carrie out with ifferent cominations of the imensionless parameters in Eq (7) Several linear an nonlinear multiple regressions were conucte using the Linear an Nonlinear Regressing Package of Mathematica v6 The results for each flow conition are as follows a) Free Flow The fitte linear an nonlinear equations for free flow an their etermination coefficients are given Eq () an () respectivel , R = 54 () Fr , R = 86 () Fr It can e seen from Eq () an () that F r is less important than /, so it can e cancele from regressing proceure to simplifing equations as following , R = 5 (), R = 7894 (4) Determination coefficient of Eq () an (4) are ver close to that of Eq () an () This inicates that the ischarge coefficient has just influence upstream water epth So it is recommene to use Eq () an (4) ecause of their simplicit an ease of application Also, nonlinear equation has more precision compare with linear form, so it is etter to use nonlinear equation Fig epicts variation against / an /Fr an quietl acknowleges the state points In fact, Fig -a is the same Henr s curve The interesting mater aout Fig - is that hsteresis phenomenon exists in ata point tren It means for a constant ischarge rate, is not same for increasing an ecreasing flow rates This ha not aresse other researchers previousl (a) 4 / () /Fr Figure Variation of against (a) / an () /Fr for free flow conition To cite this paper: Nasehi Oskui, N an Salmasi, F Vertical Sluice Gate Discharge oefficient J ivil Eng Uran (): 8-4 Journal homepage:

6 ) Sumerge Flow (a) Fr 6 5 Multiple regression analsis was performe in sumerge flow with ifferent cominations of the imensionless parameters /, /, / an /Fr The perfect fitte equations are given Eq (5) an (6) Fr, R = 747 (5) /, / () 6, R = 8 As can e seen, Eq (9) has less precision with respect to Eq (8), ut the use of this equation we can make a goo jugment aout this stu an other researchers Fig 4 an 5 present preiction in free an sumerge flow conition Eq (4) an (9), respectivel with those of Henr (95) an Swamee (99) It is interesting to note that, Eq (4) an (9) have goo clearance to Henr s curves an sometimes have more conformit compare with Swamee (99) formulas results Also, Eq (4) an (8) or (9) are simpler than Eq () an () (c) 6 (9) () 99 R = 98 (8) As can e seen Eq (8) coul state flow passes through a sumerge sluice gate with a fine precision So it is clear to use nonlinear equation for sumerge conition, too In Fig variation of with /, /, / an /Fr are presente As it can e seen, iscrepanc etween these imensionless parameters an is too high (no tren line can e rawn etween them), ut interaction of these parameters with each other results high accurac in preiction of eg in Eq (6) In orer to comparing the results of this stu with other experimental stuies, another simplifie form of Eq (6) is presente with inepenent parameters of /, / as Eq (9) 6 (7) R = 7 Fr , 748 /, R = 988 (6) Similarl, for sumerge flow we can omit some of these parameters to otain simple equations with approximatel same precision Because four epenant parameters are contriuting in this conition, thus there will e fourteen other cominations of these parameters Among these cominations the simplest an the most accurate linear an nonlinear equations are Eq (7) an (8), respectivel 4 /Fr Figure Variation of against (a) /, () /, (c) / an () /Fr for sumerge flow conition To cite this paper: Nasehi Oskui, N an Salmasi, F Vertical Sluice Gate Discharge oefficient J ivil Eng Uran (): 8-4 Journal homepage:

7 Mean Asolute Percentage Error (%) ONLUSIONS Figure 4 Preiction of in free flow conition with those of Henr (95) an Swamee (99) Stu of free-surface flow uner sluice gate is important to provie a preiction tool for the optimal management of irrigation an rainage channels Flow through the gate ma e free or sumerge epening on the tailwater epth Here, we consiere an alternative to solve the governing equations Our approach is ase on solving Bernoulli s an specific force equations simultaneousl with Wolfram Mathematica v6 software High quantit of ata points (aout 5) in imensionless form was prouce We compare the preictions otaine from numerical simulation an experiments performe on a laorator other researchers Results showe high accurac of present metho in estimation of ischarge coefficient Effect of ifferent parameters on estimation of ischarge coefficient is shown accurate regression equations REFERENES Figure 5 Preiction of in sumerge flow conition with those of Henr (95) an Swamee (99) Finall, using Eq (4) an (8), ischarge coefficient for generate ata points are compute Then using Eq () ischarge rate otaine an MAPE were calculate for these values Fig 6 epicts MAPE values for these formulas an other researcher s formulas to calculate ischarge coefficient As one can see for present stu MAPE equals to 54 % an this emonstrates its conformit with Swamee (99) an improve value than Swamee (99) 5 Roth, A an WH Hager, 999 Unerflow of stanar sluice gate Exp in Fluis, 7: 9-5 Sepúlvea,, M Gómez an J Roellar, 9 Benchmark of ischarge caliration methos for sumerge sluice gates J Irrigation an Drainage Eng, 5(5): Swamee, PK, 99 Sluice-gate ischarge equations J Irrigation an Drainage Eng, 8(): Henr, HR, 95 Discussion of Diffusion of sumerge jets Alertson, ML, YB Dai, RA Jensen an H Rouse Trans ASE 5: Yen, JF, H Lin an T Tsai, Hraulic characteristics an ischarge control of sluice gates J hinese Inst Eng, 4(): 6 Francis M Henerson, 966 Open hannel Flow Macmillan, New York, pp: - 7 Lin, H, JF Yen an T Tsai, Influence of sluice gate contraction coefficient on istinguishing conition J Irrigation an Drainage Eng, 8(4): Rajaratnam, N an K Suramana, 967 Flow equation for the sluice gate J Irrigation an Drainage Eng, 9(): Anonmous, Wolframe Mathematica v6 user s guie 5 5 H R & S S Present Figure 6 omparison of MAPE for otaine in present stu an in Henerson, 966 (H), Rajaratnam an Suramana, 967 (R & S), an Swamee, 99 (S) To cite this paper: Nasehi Oskui, N an Salmasi, F Vertical Sluice Gate Discharge oefficient J ivil Eng Uran (): 8-4 Journal homepage: 4