ANALYSIS AND FORMULATION OF HYDRAULIC JUMPS IN SLOPING RECTANGULAR CLOSED CONDUITS

Transcription

1 Seventh Internatinal Water Technlgy Cnference Egypt -3 April ANALYSIS AN ORMULATION O HYRAULIC JUMPS IN SLOPING RECTANGULAR CLOSE CONUITS Abdel-Azim M. Negm Assciate Prfessr, ept. f Water & Water Structures Engineering, aculty f Engineering, Zagazig University, Zagazig, Egypt amnegm85@htmail.cm ABSTRACT Hydraulic jumps frmed in clsed rectangular slping cnduits aree investigated experimentally. Bth psitive and negative slpes are cnsidered. The results are analyzed in terms f the inlet rude number, the bttm slpe and the inflw depth rati. The experiments are cnducted in a labratry flume. Nineteen mdels are tested. Ten mdel fr psitive slpe and anther ten mdels nine negative slpe. The analysis f results indicated that bth the inlet rude number and the bttm slpe have majr effects while the inflw depth rati has a minr effect n the depth rati f the jump at the utlet. Predictin mdel is frmulated using multiple linear regressins. The mdel predictins are cmpared with the measurements as well as with the results f the previusly develped predictin mdels using the same technique.. INTROUCTION Basic infrmatin n different types f hydraulic jumps in bth pen channels and clsed cnduits culd be fund in Rajaratnam []. The hydraulic jump in clsed cnduit ccurs when the depth near the inlet f the cnduit is less than the critical depth and the cnduit utlet is submerged. The hydraulic jump frmed in clsed cnduit belw cntrl gates has been frequently bserved, Rajaratnam []. In clsed cnduits, the initial free surface f supercritical flw changes t a pressurized flw dwnstream frm the jump and the cnjugate depth is cnfined by the cnduit height. The tailwater depth in that case prvides the dwnstream subcritical free surface flw. The jump lcatin in the cnduit is very sensitive t any slight variatin in the initial depth, cnduit height, tailwater depth, r cnduit slpe. Thus it is necessary t investigate the relatinships between the parameters affecting such interesting phenmenn.

2 648 Seventh Internatinal Water Technlgy Cnference Egypt -3 April 003 igure shws a definitin sketch fr the hydraulic jump frmed in slping clsed cnduit. Using the mmentum and cntinuity equatins, a theretical equatin fr the depth rati f the hydraulic jump culd be btained. The direct slutin f such theretical equatin f the hydraulic jumps in slping clsed cnduits is nt pssible withut an extensive experimental wrk t calibrate the equatin as it cntains many nn-theretical parameters. Ezzeldin et al. [3,4] shwed that the dimensinless tailwater depth t /d at the utlet f the clsed cnduit is a functin f the initial rude number, the dimensinless initial depth d / and the cnduit slpe S as: t d = f ( S,, ) () d Where t is the depth f water just dwnstream the utlet f the cnduit d is the initial depth f jump, S is the slpe f the cnduit and is the initial rude number(= Q / Bd gd ) with Q being the discharge, B is the cnduit width and g is the gravitatinal acceleratin. d t S igure. efinitin sketch shwing the jump frmatin inside a clsed cnduit Equatin () with S =0 is valid fr hydraulic jumps frmed in hrizntal cnduits. Such types f Hydraulic jumps were investigated by Lane and Kindsvater [5], Haindl [6] and Ezzeldin [7]. In case f slping clsed cnduits, few studies n the hydraulic jump were carried ut. Kalinske and Rbertsn [8] investigated the air pumping capacity f the jump fr the case f slping cnduits. Smith and Chen [9] investigated the relative height f the hydraulic jump frmed in steeply slping square cnduits withut cnsidering the tailwater depth cnditins. They derived the basic theretical equatin fr the relative height f the hydraulic jump frmed in slping square cnduit using the - mmentum and cntinuity equatins. It culd nt be slved prperly because it cntains many unknwns. Hence, they prvided set f.4 empirical equatins f the frm H j / = a + b, (Hj being the height f jump, is the cnduit height, is the initial rude number and a & b are tw cnstants that depend upn the values f the relative height f cnduit and the slpe f the cnduit).

3 Seventh Internatinal Water Technlgy Cnference Egypt -3 April Recently, Ezzeldin, Negm and Attia [3,4] investigated the hydraulic jumps in rectangular slping clsed cnduit with small slpes. The effect f psitive slpe Ezzeldin et al. [3] and the effect f negative slpe Ezzeldin et al. [4] were cnsidered. Empirical equatins () fr psitive slpe and (3) fr negative slpe were develpd in terms f the initial rude number, the cnduit bttm slpe, S, and the relative initial depth f jump, d /. d.5 d = S 9.53S () t Eq.() has R = and SEE = It is valid fr the fllwing ranges f the input parameters 4 6, 0.0<S 0.0 and 0. d / 0.35 t.5 d = S S (3) d Eq.(3) has R = and SEE = It is valid fr the fllwing ranges f the input parameters 4 6, -0.0 S <0.0 and 0. d / EXPERIMENTAL SETUP AN PROCEURE The experiments were cnducted in a recirculating self cntained tilting glass sided flume in the Hydraulics Labratry f aculty f Engineering, Zagazig University. The flume is 3.0 m lng, 0 cm wide and 3 cm deep. A discharge cntrl valve was used t regulate the flw rate. The bttm slpe was adjusted using a screw jack lcated at the upstream end f the flume while at the dwnstream end; the flume was allwed t rtate freely abut a hinged pivt. The slpe was directly determined using a slpe indicatr. A dwnstream adjustable gate was used t regulate the tailwater surface elevatin. A typical test mdel cnsisted f a clear prespex sheet f 50 cm lng and 0 cm wide fixed hrizntally near the middle f the flume wrking sectin abve the flume bed by an amunt equals the cnduit height,. At the dwnstream end f this hrizntal sheet (at the utlet), anther sheet was fixed abve it vertically t retain the flw at the utlet f the clsed cnduit. The sides f the prespex sheets were fixed t each ther and t the flume sides by clrless silicn rubber t ensure gd fixatin and t prevent leakage frm the sides. The experiments were carried ut mainly by Ezzeldin et al. [3,4] using five different cnduit heights,, f 6, 7, 8, 9 and 0 cm. Ten psitive bttm slpes and anther nine negative bttm slpes, S, were used. The used slpes were 0.00, 0.004, 0.005, , 0.008, 0.0, 0.03, 0.06, 0.08 and 0.0. The slpes were selected based n the flume facility. ive different flw rates ranged frm 34 lit/min t 34 lit/min were used fr each particular cnduit height and bttm slpe. The initial rude number ranged frm 4 t 6. r each cnduit height, the upstream cntrl gate was adjusted t prduce an initial supercritical depth, d, and the dwnstream adjustable gate was adjusted t cntrl the tailwater utlet depth, t, and

4 650 Seventh Internatinal Water Technlgy Cnference Egypt -3 April 003 which enabled the jump t be frmed at a certain fixed lcatin in the cnduit. The discharge was measured using a pre-calibrated rifice meter. epth measurements were taken using a pint gauge with an accuracy f ± 0. mm. r each run, the initial depth f jump, the flw rate and the depth f water just dwnstream the cnduit utlet were measured. 3. RESULTS AN ISCUSSIONS igures a t e present the variatins f t /d with the bttm slpe (frm 0.0 t 0.0) fr different inlet rude number at fixed value f d /. It is clear that the depth rati increases with the increase f the slpe at particular inlet rude number. The lwest value is due t the maximum negative slpe (-0.0) and the maximum value is due t the maximum psitive slpe (0.0). This culd be explained by the fact that, in case f the psitive slpe, the weight cmpnent acts in the directin f the flw while in case f the negative slpe, it acts in a directin ppsite t the flw directin impsing mre resistance t the flw. This in turn results in a reductin f the depth rati and the rate f reductin increase as the negative slpe increases. On the ther hand, the jump depth rati at the utlet f the cnduit increases by the increase f the inlet rude number at fixed value f d / and particular value f the slpe. igures 3a present the variatin f t /d with S fr different values f d / at particular inlet rude number f =93. Clearly, the inlet flw depth rati, d / has a minr effect n the variatin f t /d at fixed. igures 3b present the variatin f t /d with d / fr different values f S at particular f The figure indicates that variatin f t /d is mstly minr with the increase f d /. igure 3c shws the variatins f t /d with fr different S at particular d / f The figure cnfirmed what has been indicated by igure. 3.. Predictin Mdel Eighteen regressin mdels were tested based n the fllwing criteria: - The crrelatin cefficient, r. - The mean relative errr, mre. - The rt mean square errr, rmse. Table shws the parameters invlved in each f the tested mdels as well as the crrespnding test criteria. The mdels were presented in table in an ascending rder accrding t the value f the crrelatin cefficient. rm Table it culd be stated that mdel n. 7 has a gd predictive pwer fr the jump depth rati. This mdel takes the frm: d.5 d = S S (4) t has R = and SEE = It is valid fr the fllwing ranges f the input parameters 4 6, -0.0 S 0.0 and 0. d / 0.35.

5 Seventh Internatinal Water Technlgy Cnference Egypt -3 April t/d t/d t/d t/d t/d (a) d/= S (b) d/= S (c) d/= S (e) d/= S (d) d/= S igure. Variatins f t /d with slpe fr different values f and particular value f d /, (a) d /=0., (b) d /=0.33, (c) d /=0.63, (d) d /=0.30 and (e) d /=

6 65 Seventh Internatinal Water Technlgy Cnference Egypt -3 April 003 t/d t/d (a) = S igure 3a. Typical variatins f t /d with slpe fr different values f d / and particular value f (b) = d/ igure 3b. Typical variatins f t /d with d / fr different values f S and particular value f. d/ S t/d (c) d/= igure 3c. Typical variatins f t /d with fr different values f S and particular value f d /. S

7 Seventh Internatinal Water Technlgy Cnference Egypt -3 April Table Evaluatin f tested regressin mdels in this study Mdel Invlved Parameters r mre rmse max % errr 5 d / S S / S S + S S S / S S / S S + S / S / S / S + S / Cmparisns igure 4a shws the cmparisn between and Eq.() fr psitive slpe while igure 4b cmpare the results f with thse f Eq.(3) fr negative slpe. Clearly gd agreement was achieved. Since, igures 4a and 4b are crwded with the bservatins, it may be better t indicate the predictive pwer f in cmparisn with the experimental results as well as thse f Eqs.() and (3) fr a selected set f data as culd be depicted frm igures 6c fr d /= igures 4c cnfirmed that is capable f predictive the depth rati f the jump at the utlet f the rectangular clsed cnduit fr bth psitive and negative slpes using the same parameters as previusly develped by Ezzeldin et al. [3,4] fr each slpe separately urther Cnsideratins All mdels f table were based n the full data set f size 475 recrds. The cefficient f the last mdel (n. 7) were re-evaluated using 75% f the bservatins while the remaining 5% f bservatins were left fr testing the validity f the

8 654 Seventh Internatinal Water Technlgy Cnference Egypt -3 April 003 mdel. The regressin tlbx f the Neural Cnnectin [0] was used t evaluate the regressin cefficients f the mdel. This mdel (n.8) will have the fllwing frm: t.5 d = S S (5) d Eq.(5) has R = 0.99 and SEE = 0.3. It is valid fr the same ranges as fr. The R fr validatin data is and that f test is Table summarize the same criteria as thse in table fr Eq.(5). Table. Evaluatin f last mdel using 75% f the data Mdel Invlved Parameters r mre rmse max % er data set 8tr S + S / training 8V S + S / validatin 8t + + S + S / test.5 The training data set (75%) was used t evaluate the cefficients f the mdel. The ther 5% was used t verify r testing the validity f the by generating values f the target variable based n the input withut prir infrmatin n the actual utput. The data sets were randmly selected t ensure prper calibratin and crrect verificatin f the mdel. Cmparing the perfrmance criteria fr mdels n.7 and n.8, sme slight imprvement in the perfrmance f mdel n. 8 was btained. r example, the maximum errr was reduced frm.7% t 9.4% while the mre was increased by 0.% t reach.7% instead f.6%. igure 5a shws the cmparisn between predicted t /d and the measured nes fr verificatin data set. igure 5b shws the cmparisn between and Eq.(5). Clearly bth equatins culd be used safely t predict the depth rati f the hydraulic jump at the utlet f the rectangular clsed cnduit. igure 5c shws the variatins f the residuals with the estimated values f t /d fr bth mdels n.7 and n.8. The values f the residuals are very small and are symmetrically distributed arund the line f zer errr with a negligible crrelatin cefficient indicating the validity f bth mdels in the predictin. The values f the mean, standard deviatin and crrelatin cefficient are 0.004, and fr Eq.(5) and are , 0.39 and fr.

9 Seventh Internatinal Water Technlgy Cnference Egypt -3 April (a) Predicted t/d Eq.(), +ve slpe, all slpes Line f equality Actual t/d (b) Predicted t/d Eq.(3), -ve slpe, all slpes Line f equality t/d Actual t/d (c) d/= Eqs.(,3) igure 4. Cmparisn between actual, present mdel fr +ve and ve slpes and previus mdels fr (a) +ve slpe (b) -ve slpe, (c) Present mdel versus previus nes S

10 656 Seventh Internatinal Water Technlgy Cnference Egypt -3 April 003 Predicted t/d (a) Validatin data (5%) Test data (0%) Line f equality Actual t/d Estimated t/d (b) ull data 75% f data Line f equality Actual t/d.0 (c) Residuals 0.0 M7 M Estimated and predicted t/d igure 5. Perfrmance f Eq.(5), (a) cmparisn f measurements with predictin f Eq.(5) fr validatin data, (b) cmparisn between and Eq.(5) with the measurements and (c) distributin f the residuals f Eq.(5) and.

11 Seventh Internatinal Water Technlgy Cnference Egypt -3 April CONCLUSIONS Hydraulic jumps frmed in rectangular clsed cnduit with bth psitive and negative bttm slpes were analyzed based n an extensive experimental investigatin. It was fund that bth the bttm slpe S, and the inlet rude number, have majr effect n the variatins f the jump utlet depth rati while the inflw depth rati d /, is f minr imprtance. The jump depth rati increases with the increase f and increases with the increase f the bttm slpe f the cnduit. The negative bttm slpe prduces values f the jump utlet depth rati which are lwer than thse prduced by the psitive slpe. The values f the hrizntal bttm slpe are between the values f the negative and thse f the psitive slpes. A general predictin mdel was prpsed in the same frm f the previusly develped mdels by Ezzeldin et al. [3,4] fr jumps frmed in cnduits with either psitive r negative slpe. The mdel was calibrated using the full set f data by assigning psitive sign fr psitive slpe and negative sign fr negative slpe. The predictin f the prpsed mdel was cmpared t the previusly develped mdels fr bth psitive and negative slpes. The predicted results agreed well with the experimental bservatins as well as with thse f the previusly develped mdels using the same technique. Als, the mdel was calibrated using 75% f the data and verified with the remaining 5% based n randm selectin. The calibrated mdel with 75% f the bservatins shwed better perfrmance than the mdel that calibrated based n all the data. Hwever, the differences are practically negligible. NOMENCLATURE The fllwing symbls are used in this paper d = initial depth f supercritical flw = d n the figures, = cnduit height; t = depth f flw at the utlet, =t n the figures, S = cnduit slpe; = initial rude number; t /d = dimensinless utlet tailwater depth=t/d n the figures; and d / = dimensinless initial depth r cnduit height rati, =d/. REERENCES [] Rajaratnam, N., Hydraulic Jump, Advances in Hydr-science, V. T. Chw, ed., Vl. 4, Academic Press, New Yrk, 967, pp [] Rajaratnam, N., Hydraulic Jumps in Hrizntal Cnduits, Water Pwer, 7, 965, pp [3] Ezzeldin, M.M., Negm, A.M., and Attia, M.I., "Experimental Investigatin n the Hydraulic Jump in Slping Rectangular Clsed Cnduits", Alex. Eng. Jurnal, aculty f Eng., Alex. Univ., Egypt, Vl. 39, N. 5, Sep. 000, pp [4] Ezzeldin, M.M., Negm, A.M., And Attia, M.I., "Experimental Investigatin n the Hydraulic Jump in Adversely Slping Rectangular Clsed Cnduits, Alex. Eng.

12 658 Seventh Internatinal Water Technlgy Cnference Egypt -3 April 003 Jurnal, aculty f Eng., Alex. Univ., Egypt, Vl. 39, N. 5, Sep. 000, pp [5] Lane, E. W., and Kindsvater, C. E., Hydraulic Jump in Enclsed Cnduits, Eng. News Recrd, 06, 938, pp [6] Haindl, K., Hydraulic Jump in Clsed Cnduits, Prc., Internatinal Assciatin f Hydraulic Research, Lisbn, Vl., 957, pp [7] Ezzeldin, M. M., Characteristics f Hydraulic Jump in Hrizntal Rectangular Cnduits, Ain Shams university, Engineering Bulletin, Vl. 8, N., March 993, pp [8] Kalinske, A. A., and Rbertsn, J. M., Clsed Cnduit lw, Trans. ASCE, 08, 943, pp [9] Smith, C.. and Chen, W., The Hydraulic Jump in a steeply Slping Square Cnduit, Jurnal f Hydraulic Research, Vl. 7, N. 3, 989, pp [0] Neural Cnnectins, Artificial Neural Netwrks Sftware and User Manuals, SPSS/Recgnitin Systems Limited, 998.