STAGE DISCHARGE RELATIONSHIPS FOR TRIANGULAR WEIR. Masoud Ghodsian

Transcription

1 STAGE DISCHARGE RELATIONSHIS FOR TRIANGULAR WEIR Masoud Godsian Associate rofessor of Hydraulic Engineering, Tarbiat Modarres University, Teran, Iran Abstract: Te caracteristics of triangular weir are studied under free and submerged condition. Tis paper reports te results of an investigation carried out to establis te stagediscarge relationsip for sarp and broad crested triangular weir. Te stagediscarge relationsips are deduced by te application of te π teorem of te dimensional analysis and te incomplete selfsimilarity teory coupled wit experimental data. Keywords: Triangular weir, sarp crested, broad crested, free, submerged, πteorem, Incomplete selfsimilarity.. INTRODUCTION s are measuring devices tat are used in ydraulic and water resources engineering. Triangular weirs are widely used in te field and laboratory cannels for measurements. Te sarp crested triangular weir is best described as a Vsaped notc symmetrically located in a tin plate, wic is placed perpendicular to te sides and bottom of an open cannel. A broad crested triangular weir is a structure wit a triangular crest above wic te fluid pressure may be considered ydrostatic. In te longitudinal section, it is simillar to te broad crested weir wit orizontal crest []. In natural streams and irrigation canals were a wide range of discarges must be measured, te triangular weir as a number of advantages, wic recommend its use. First, at large s it provides a large breadt so tat te backwater effect is not excessive. Second, at low s, te widt is reduced so tat te sensitivity of te weir is acceptable [2]. Moreover it is easy to make and is inexpensive structure. Te dimensions of triangular weirs depend on te maximum discarge to be measured and te available widt of te approac cannel. revious works on triangular weirs, as compare to rectangular weir, are less. Bos [3] related te discarge coefficient to notc angle for sarp crested triangular weir and to ratio of total ead to weir tickness for broad crested triangular weir. Results of experiments on te variations of discarge coefficient for sarp crested triangular weirs are reported by Montes [4], Smit [5] Brater and King [6]. Te caractristics over broad crested triangular weir wit notc angle 9 degree was investigated by Smit and Liang [7] and itlo and Smit [8]. Teir measurments included bot free and submerged conditions. Exessive studies on broad crested triangular weir wit notc angle 2 deggree were conducted at te agricultural university at Wageningen [9 and ]. Boiten and itlo [] obtained te discarge coefficient of broad crested triangular weir as a function of boundary layer tickness, notc angle and ratio of weir tickness to dept. Boiten [2] developed te grapical representation of discarge coefficient for broad crested triangular weir as a function of dept to weir tickness. Gill [3] obtained te discarge coefficient of broad crested triangular weir as a function of ratio of dept of and weir tickness. Smit [5] performed experiments for free and submerged over broad crested triangulare weirs. In tis paper results of experiments on sarp crested and broad crested triangular weir under free and submerged conditions are reported. Te stagediscarge relationsip is obtained by teoretical analysis based on te application of dimensional analysis and te application of incomplete selfsimilarity teory coupled wit an experimental investigation carried out by using a laboratory flume. 2. THEORY Figs. and 2 sow scematic view of te over a sarp crested and broad crested triangular weirs respectively. Te over a sarp crested International Journal of Civil Engineering, Vol. 2, No., Marc 24

2 and broad crested triangular weir is expressed by (Frenc [2] and, Montes [4]): Sarp crested triangulare weir: Q = Cd 2g tan( θ ) (a) 5 Broad crested triangular weir: Q = Cd 2g tan( θ ) (b) 25 5 in wic Q = discarge over te weir; C d = discarge coefficient; = total ead over te weir in te upstream section; g = acceleration due to gravity; and θ = alf notc angle of te weir. a) b) d) a) b) d) Flow Fig.. Flow over sarp crested weir; a) lan, b) condition; c) Front view; and d) condition L L Fig. 2. Flow over broad crested weir; a) lan, b) condition, c) Front view, and d) condition t t c) c) Te discarge coefficient includes te effects of viscous, turbulence, nonuniform velocity distribution of, type and sape of crest. Te generall equations for sarp crested and broad crested triangular weirs are obtained in foregoing sections of paper. Te stagediscarge relationsip, for a sarp crested triangular weir can be expressed by te following functional relationsip: f(q,,, g, µ, θ) = (2) in wic: = weir eigt; and µ = viscosity of water. Eq. (2), according to te Buckingam πteorem and after neglecting te effect of µ, can be expressed as a non dimensional form as follows: f(π, π 2, π 3 ) = (3) in wic π, π 2 and π 3 = dimensionless groups given by: Q π = (3a).5 g π 2 = (3b) π 3 =θ (3c) Terefore, Eq. (3), after combining π and π 3, for sarp crested triangular weir can be written as: = f (4) in wic: K = Q 2/5 g /5 (tanθ) 2/5. It is obvious tat te condition of incomplete selfsimilarity, as explained by Ferro [4] prevail for te above relationsip. Following te incomplete selfsimilarity approac one obtain te following stagediscarge relationsip for sarp crested triangular weir: b = a (5) in wic: a and b are numerical constant to be determined by using experimental data. For broad crested triangular weir, te stagediscarge relationsip can be expressed as: f(q,, L, g, µ, θ)= (6) in wic: L = tickness of weir, i.e. lengt of weir in te direction of. Using te Buckingam πteorem and after neglecting µ, Eq. (6) leads to te following 2 International Journal of Civil Engineering, Vol. 2, No., Marc 24

3 dimensionless group: = f (7) in wic K = Q 2/5 g /5 (tanθ) 2/5. Te following stagediscarge relationsip derives from incomplete selfsimilarity condition for broad crested triangular weir: d = c (8) in wic c and d are numerical constant to be determined by using experimental data. 3. EXERIMENTS Experiments were conducted in a recirculating flume wit a lengt of 2m, widt of.3m and dept of.45m. Te flume ad a mild steel bed and te glass sides. A mild steel triangular weir of crest tickness L was installed at te mid lengt of flume. In order to obtain different tail water dept, for studying te effect of submergence, a tailgate was provided at te end of flume. For te broad crested weir, te upstream corner was rounded wit a radius equal to. of maximum water dept to avoid separation. A scematic view of te experimental setup is sown in Fig. 3. Te discarge was measured by volumetric metod. In order to ave sufficient accuracy, for eac run, discarge measurement was repeated at least tree times and te average of tem was considered. Flow depts (i.e. upstream dept of over te weir and tail water dept over te weir) were measured at te centerline of cannel wit a point gauge aving an accuracy of ±.mm. Four different notc angles were used (i.e. 3, 6, 9 and 2 degree). Experiments were carried out for various combinations of notc angler, weir eigt and upstream dept for free. For submerged condition, in addition to above variables, te tail water dept was varied to obtain various degree of submergence. Table and Table 2 sow te range of various parameters covered in te present study. 4. ANALYSIS In order to ceck te feasibility of Eq. (5) and Eq. (8) and to obtain te proper form of stagediscarge relationsip for triangular weir, te present experimental data were used. Fig. 4 sows te grapical representation of Eq. (5) for sarp crested triangular weir. It is evident tat te assumed stagediscarge relationsip for sarp crested triangular weir (Eq. 4) is acceptable. Te values of constant a and b in Eq. (5) were obtained as.797 and. respectively. Fig. 3. Scematic view of experimental setup; a) lan and b) Longitudinal section International Journal of Civil Engineering, Vol. 2, No., Marc 24 3

4 Terefore, te proposed stagediscarge relationsip for sarp crested triangular weir under free condition is:. =.797 (9) Introducing submergence coefficient K s, Eq. (9) for submerged condition can be written as:. K =.797K s () in wic K s is te ratio of discarge under free and submerged condition and can be assumed to be a function of ratio of tail water ead to upstream dept of (i.e. t /). Fig. 5 sows te variations of K s wit t / for sarp crested triangular weir wit different notc angle under submerged condition. It is evident tat up to about t /.4, te value of K s is unity, and furter increasing t /, te value of K s decreases. Te line drawn troug te experimental data in Fig. 5 is an average line fitted to te data wit te equation as:.3737 Ks = () Terefore, te general form of stagediscarge relationsip for sarp crested triangular weir is obtained from Eqs. ( and ) as: 25 Q = t.9337 (2) g tanθ Comparing Eq. (2) wit Eq. (a), one obtain te discarge coefficient C d as: Cd =.7378 (3) For te free condition, last terms in Eq. (2) and Eq. (3) drops out and results in to te stagediscarge relationsip and discarge coefficient for tis condition respectively. Fig. 6 sows te grapical representation of Eq. (8) for broad crested triangular weir wit different notc angles. Te values of constant c and d in Eq. (8) were obtained as.692 and. respectively. Notc angle (deg) eigt (cm) 24 2&4 2& & &6.8 Table. Range of parameters studied for sarp crested weir / t/ Q (L/s) Type of No. of data Notc angle (deg) 3 6 eigt (cm) Table 2. Range of parameters studied for broad crested weir Discarge tickness L /L t/ (l/s) (cm) Type of No. of data International Journal of Civil Engineering, Vol. 2, No., Marc 24

5 K/ / Ks t/ Fig. 4. Variations of K/ wit / for sarp crested triangular weir wit different notc angle Fig. 5. Submergence coefficient for sarp crested triangular weir wit different notc angle K/L /L Fig. 6. Variations of K/L wit /L for broad crested triangular weir wit different notc angle K s t/ Fig. 7. Submergence coefficient for broad crested triangular weir wit different notc angle Terefore, te proposed stage discarge relationsip for broad crested triangular weir under free condition is:. =.692 (4) Introducing submergence coefficient K s, one can obtain te following stagediscarge relationsip for broad crested triangular weir under submerged condition:. =.692K s (5) Fig. 7 sows te variations of K s wit t / for broad crested triangular weir for different notc angles. Tis Figure sows tat K s = for t /.6. Te trend of variations of t / is almost similar for broad crested and sarp crested weirs. Te line drawn troug data points in Fig. 7 is an average line fitted to te data and can be expressed by te following equation:.488 Ks = (6) Terefore, te general form of stagediscarge relationsip for broad crested triangular weir is obtained from Eqs. (5 and 6) as: 25 Q = t. (7) g L tanθ L Comparing Eq. (7) wit Eq. (b), it can be sown tat for broad crested triangular weir C d is given by:..25 Cd =.9843 (8) L It is obvious tat for free condition te last terms in Eq. (7) and Eq. (8) drops out to yield te stagediscarge relationsip and discarge coefficient for tis condition respectively. In order to ceck te validity of Eq. (4, 5 and 7) te data collected by Boiten [5] on triangular broad crested weir for free and submerge were used. Te range of Boitens data is given in Table 3. Fig. 8 sows te comparision of K/L versus /L for te data of Boiten[5] and toes obtained by Eq. (4). It is evident tat Eq. (4) give satisfactory results for te data of Boiten [5] International Journal of Civil Engineering, Vol. 2, No., Marc 24 5

6 K/L Boiten(98).8 Equation (4) /L Qc(m^3/s).3.2. Boiten(98) Line of perfect agreement Q a (m^3/s) Fig. 8. Validation of equation (4) wit Boitens (98) data for free condition Fig. 9. Comparison of computed discarge using Eq. (4) and actual discarge for Boitens(98) data under free condition K/L Boiten (98) Equation (5) Qc (m^3/s).5..5 Boiten(98) Line of perfect agreement /L.5..5 Q a (m^3/s) Fig.. Validation of equation (5) wit Boitens (98) data for submerged condition Fig.. Comparison of computed discarge using Eq. (5) and actual discarge for Boitens (98) data under free condition Notc angle (deg) 9 eigt (cm) Table 3. Range of parameters studied by Boiten(98) ticknes L (cm) 45, 9 45, 9 /L , , t/ Discarge (l/s) Type of No. of data under free condition even for noc angle 5 degree. Fig. 9 sows te comparision of computed values of discare Q c [using Eq. (4)] wit actual values of discarge Q a for te data of Boiten[5] under free condition. It is obvious tat Eq. (4) is capable of predicting te discarge over a braod crested triangular weir for free accurately. Figs. and sow te validation of Eqs. (5 and 7) respectively for te data of Boiten [5] under submerged condition. Fig. sows te comparision of K/L versus /L for te data of Boiten[5] and toes obtained by Eq. (5). It is evident tat Eq. (5) give satisfactory results for te te data of Boiten [5] under submerged condition even for noc angle 5 degree. Fig. sows te comparision of computed values of discare Q c using Eq. (7) wit actual values of discarge Q a for te data of Boiten[5] under submerged. It is obvious tat Eq. (7) is also capable of predicting te discarge over a braod crested triangular weir 6 International Journal of Civil Engineering, Vol. 2, No., Marc 24

7 for submerged accurately. Te verification of developed equations for sarp crested triangular weir is possible wen data on tis type of weir is available. 5. CONCLUSION In tis paper te results of an experimental investigation on sarp crested and broad crested triangular weir, under free and submerged condition, are reported. Te analysis sows tat te stagediscarge relationsip for sarp crested and broad crested triangular weirs can be teoretically obtained by applying te Buckingam teorem of dimensional analysis and te incomplete selfsimilarity teory. New equations for stage discarge relationsip and discarge coefficient are developed. REFFERENCES [] Metods of Measurments of liquid in open cannels, (969). Britis standard 368, part 4B, Long Base weirs, Britis standard institution, London, England. [2] Frenc, R. H. (985). Open cannel ydraulics, McGrawHill Book Company. [3] Bos, M.G. (976). "Discarge measurement structures," Int. inst. for land reclamation and improvement, Wageningen, te Neterlands. [4] Montes, J.S. (998). "Hydraulics of open cannel." ASCE ress, NewYork, USA. [5] Smit, C.D. (996). Hydraulic structures, University of Saskatcewan, Canada. [6] Brater, E.F. and King, W. (996). Handbook of ydraulics, MacGraw Hill Book Company. [7] Smit, C.D. and Liang, W.S., (97). "Triangular broad crested weir", J. Irrig. And Drain. Engrg., ASCE, 3(4), [8] itlo, R.H. and Smit, M., (97), Discusion of "Triangular broad crested weir " by Smit, C.D. and Liang, W.S, J. Irrig. and Drain. Engrg., ASCE, 3(3), [9] Agricultural university Wageningen, (976). Aanvullend onderzoek Vvormige lange overlaat, aper 35 of te Laboratory of ydraulic and catment ydrology. [] Agricultural university Wageningen, (98). Boundary layer development on a Vsaped broad crested weir, aper 46 of te Laboratory of ydraulic and catment ydrology. [] Boiten, W. and itlo R.H. (982). "Te V saped broad crested weir." J. Irrig. and Drain. Engrg. ASCE, 42(4), 426. [2] Boiten, W. (984). "Vsaped sort crested weir wit semicircle cross section," J. Irrig. and Drain. Engrg. ASCE, (4), [3] Gill, M.A. (985). "Flow measurement by triangular broad crested weir." J. waterpower and dam construction, [4] Ferro, V. (2). "Simultaneous over and under a gate," J. Irrig. and Drain. Engrg., ASCE, 26(3), 993. [5] Boiten, W. (98). "Te Vsaped broad crested weir, discarge caracteristics," Report on basic researc S7VI, Delft ydraulic laboratory, Wageningen, Te Neterland. International Journal of Civil Engineering, Vol. 2, No., Marc 24 7