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Experimental study on triangular central baffle flume
Flow Measurement and Instrumentation 70 (2019) 101641
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: http://www.elsevier.com/locate/flowmeasinst
Experimental study on triangular central baffle flume M. Bijankhan a, *, V. Ferro b a b
Department of Water Engineering, Imam Khomeini International University (IKIU), P.O. Box 3414916818, Qazvin, Iran Department of Earth and Marine Sciences, University of Palermo, Via Archirafi 20, 90128, Italy
A R T I C L E I N F O
A B S T R A C T
Keywords: Triangular central baffle flume Stage-discharge formula Buckingham theorem Incomplete self-similarity Submergence threshold
In this paper the results of the experiments performed to study the flow through a Triangular Central Baffle Flume (TCBF) are reported. The investigated flume consists of a triangular baffle of the apex angle of 75� with a given base width. The theoretical stage-discharge formula was deduced by applying the Buckingham’s Theorem and incomplete self-similarity hypothesis and was calibrated using the laboratory measurements carried out in this investigation. The proposed stage-discharge formula is characterized by a mean absolute relative error of 7.4% and 72% of the data points are in an error range of �5%. The results indicate that TCBF flume is char acterized by a flow capacity higher than that of a typical central baffle flume. Experimental observations show that the contraction ratio is a key parameter to distinguish between free and submerged flow regimes through a TCBF. Finally, to identify the flow condition, submergence threshold condition was formulated.
1. Introduction Simple hydraulic structures for estimating the flow rate in open channels are practically important to enhance irrigation network man agement and operation. In this regard critical flow concept is used to construct flow measuring devices. Changing the channel dimension, e.g. side contraction or bed level change, is a typical action to establish the critical flow state. Channel contraction can be created by either a local channel width reduction or by inserting a vertical obstacle on the channel bed. The most famous flume constructed by a local channel width thickening is the Venturi Flume. Khafagi and Parshall flumes are types of Venturi channels which are widely used [1,2]. Montana flume is also a kind of Parshall flume without diverging downstream walls [3]. According to throat section the flumes can be classified into LongThroated and Throat-Less (Cut-Throated Flumes, CTF). The term LongThroated stands for the flumes having streamlines running parallel in the contracted section. Any flumes with shorter throat section referred to length is called as Cut-Throat Flume [4]. Based on an extensive laboratory measurement on Cut-Throat Flumes (CTF), Torres and Merkley [5] found that free flow condition is converted to submerged flow regime without a transition zone. Consequently, for both free and submerged flow cases a general stage-discharge formula was developed. Temeepattanapongsa et al. [6] applied Computational Fluid
Dynamic (CFD) to obtain generic free flow rating for Cut-Throat Flumes. They found that CFD simulations could be successfully employed to extend the design principles of CTF devices. Hager [7] designed a modified shape of a Venturi flume by inserting a vertical column in the channel center. Using the critical flow concept, the related stage-discharge formula was developed. Experiments indi cated that this proposed flume can be used to estimate the discharge in the range of �5% of the real value. Hager [7] also proposed the stage-discharge relationships for rectangular, trapezoidal, and U-shaped channels. The flow pattern of a trapezoidal flume constructed by immersing a circular cone into a rectangular channel was studied by Hager [8]. The associated stage discharge formula was developed by accounting for the effect of streamline curvature. Hager [9] also proposed application of a circular mobile flume in circular conduits like sewers and drainage pipes. This flume consists of a cylinder located vertically in a circular channel. Considering that the critical flow state occurs at the flume contraction, Hager [9] established a stage-discharge formula for the mobile circular flume. A flow measurement flume can be constructed by sticking two semicylinders to the walls of a channel or installing a vertical cylindrical column in a channel [10–13] or using two pieces of pipes, one installed vertically inside the other [14,15]. Following the principle of inserting a central body in a channel, Peruginelli and Bonacci [16] proposed to construct a flume, named
* Corresponding author. E-mail address: [email protected] (M. Bijankhan). https://doi.org/10.1016/j.flowmeasinst.2019.101641 Received 21 December 2018; Received in revised form 4 September 2019; Accepted 30 September 2019 Available online 5 October 2019 0955-5986/© 2019 Elsevier Ltd. All rights reserved.
M. Bijankhan and V. Ferro
Flow Measurement and Instrumentation 70 (2019) 101641
1.45. The following stage-discharge relationship was proposed: " � � � �2 #� �1:45 Q L L h 0:1914 þ 0:0861 5 1 ¼ 0:655 B B B 2 2 c Bc g
(2)
Eq. (2) can be applied for 0.2 � L/B � 0.8. A low-cost and more efficient central baffle would be obtained when L tends to zero and α ¼ 75� [20]. In this study, such flume is called as Triangular Central Baffle Flume, TCBF (Fig. 2). Eq. (2), is however not valid for TCBF because L/B ¼ 0 is out of its applicability range. Conse quently, experiments are carried out in this study to establish the stage-discharge formula of the TCBF. 2. Experimental setup The experiments were performed in a 0.5 m wide, 0.6 m high and 12 m long Plexiglas flume located at the hydraulic laboratory of the Water Engineering Department, Imam Khomeini International Univer sity (IKIU), Qazvin, Iran. The flume was supplied by a centrifugal pump connected to a reservoir of the volume of about 15 m3. An electromag netic flow-meter was used for measuring the flow rates of up to 80 l/s with the accuracy of �0.5% of the full scale. Taking into account that the only physical parameter to design TCBF, with α ¼ 75� [20], is the baffle width, b, the experimental runs were performed using four TCBF configurations having b ¼ 12, 20.5, 30.5, and 41.5 cm and a contraction ratio r ¼ Bc/B, in which Bc (¼B b) is throat width, equal to 0.76, 0.59, 0.39 and 0.17, respectively. As shown in Fig. 3, the free flow condition was considered in all experimental runs, i.e. the jump toe was located sufficiently far from the baffle downstream to ensure that the upstream flow condition would not be affected (Fig. 3c). For a given flow rate, Q, the associated upstream flow depth, h, was recorded (see Table 1 of Appendix 1). To identify the maximum permitted tailwater depth allowing free flow condition, hth, the tailwater depth was gradually increased so that the upstream water depth started increasing. A maximum upstream flow depth increase of 2 mm was considered. Detailed experimental data of the observed submergence threshold conditions are listed in Table 2 of Appendix 2.
Fig. 1. Schematic view of central baffle flume.
“Central Baffle Flume” (CBF), inserting a baffle, having a width b and a throat length L, in the middle of a rectangular channel (Fig. 1). Ferro [17], using the Buckingham-Theorem of the dimensional analysis and the self-similarity theory, proposed the following stage-discharge formula of the CBF flow: � �n Q2=3 h ¼ a (1) 1=3 Bc B5=3 c g where, Q is discharge, Bc (¼B b) is throat width, B is channel width, b is baffle width, h is upstream flow depth, g is the acceleration due to gravity, and a and n are empirical parameters that should be obtained using experimental data. Samani [18] discussed the design and calibration of the central cir cular baffle in circular and trapezoidal channel cross sections and pro posed stage-discharge formulas whose applications were verified using field data measurements. Ferro [19], using the field measurements carried out by Samani [18], tested the applicability of the proposed theoretical stage-discharge re lationships for the case of both a flume with two semi-cylindrical glued at the channel walls and a circular flume in which a column pipe is installed. Lotfi Kolavani et al. [20] performed extensive experimental runs to investigate the effect of the throat length, L, for the central baffle flume (Fig. 1). The experiments by Lotfi Kolavani et al. [20] also highlighted that a central baffle flume with an apex angle, α, equal to 75� for up stream guide walls provides suitable hydraulic condition at the flume entrance. While, the existence of downstream guide walls, having a length L, did not affect the stage-discharge relationship. Accordingly, they find that Eq. (1) can be applied taking into account that the coef ficient a depends on the ratio L/B while n is a constant value equal to
Fig. 3. Tested triangular central baffle flume (a) side view of the case with b ¼ 20.5 cm, (b) upstream view of the triangular baffle with b ¼ 41.5 cm, and (c) the hydraulic jump occurred downstream of the triangular baffle with b ¼ 20.5 cm.
Fig. 2. Schematic view of triangular central baffle flume. 2
M. Bijankhan and V. Ferro
Flow Measurement and Instrumentation 70 (2019) 101641
Π3 ¼
h Bc
(5c)
Substituting the dimensionless groups, Eq. (5), into Eq. (4) yields the following dimensionless formula: � � Q h ¼ Γ ;r (6) 5=2 1=2 Bc Bc g where Γ is a functional symbol. For a given value of r, when h=Bc →0 then Q=½B5=2 g1=2 � →0 and c
g1=2 � →∞. Consequently, applying the when h=Bc →∞ then Q=½B5=2 c incomplete self-similarity (ISS) condition [21,22] Eq. (6) becomes: � �n Q h ¼ Γ 1 ðrÞ (7) Bc B5=2 g1=2 c where, Γ1 is a functional symbol. According to the ISS condition, n is a numerical constant that should be obtained by experimental data. For each value of the contraction ratio r, the Γ1 (r) function assumes a constant value m and the stage-discharge formula obtained by Eq. (7), depends only on the upstream head ratio, h/Bc: � �n Q h ¼ m (8) 5=2 1=2 Bc Bc g
0.5 Fig. 4. Experimental pairs of (h/Bc, Q/[B2.5 ]) for TCBF with b ¼ 12, 20.5, c g 30.5, and 41.5 cm.
3. Deducing stage-discharge relationship The free flow hydraulic of the central baffle flume (Fig. 1) can be expressed by the following functional relationship [20]: ϕðh; Q; Bc ; B; L; Le ; g; μÞ ¼ 0
(3)
4. Calibrating and testing the stage-discharge relationship of a TCBF
where ϕ is a functional symbol, h is upstream flow depth, Q is discharge, Bc ¼ B – b is the throat width, B is the approaching channel width, b is the baffle width, L is baffle length, Le is the longitudinal distance of the guide wall, g is acceleration due to gravity and μ is the water viscosity. The stage-discharge equation of TCBF, corresponding to a guide wall entrance angle of α ¼ 75� and L ¼ 0, can be described by the following functional relationship: ϕ1 ðh; Q; Bc ; B; g; μÞ ¼ 0
0.5 Plotting of the experimental pairs of (h/Bc,Q/[B2.5 c g ]) (Fig. 4) revealed that the stage-discharge curves corresponding to different rvalues collapsed on a single curve. Consequently, the following stagedischarge formula was calibrated using all available experimental data (Fig. 4): � �1:5734 Q h ¼ 0:6925 (9) 5=2 1=2 Bc Bc g
(4)
where, ϕ1 is a functional symbol. Considering Bc, μ, and g as reference variables the following dimensionless groups were obtained: Π1 ¼
Q B5=2 g1=2 c
(5a)
Π2 ¼
Bc ¼r B
(5b)
Eq. (9) can be used in the range of 0.17 � r � 0.76 and 0.047 � h/ Bc � 3.67. This stage-discharge equation can be applicable when a free hydraulic jump occurs at the downstream pool. For assuring free flow condition, the flume can be installed at the channel end where a free overfall occurs in the downstream section. The relative errors associated with the discharge values calculated by Eq. (9) were plotted in Fig. 5a versus h/Bc. According to the figure, Eq. (9) can be used to determine the flow through TCBF with the relative
Fig. 5. (a) Relative errors associated with the discharge values calculated by Eq. (9) versus h/Bc b) cumulative frequency distribution of the relative errors. 3
M. Bijankhan and V. Ferro
Flow Measurement and Instrumentation 70 (2019) 101641
errors restricted in the range of �10% and the mean absolute relative error of 4%. As shown in Fig. 5b, the cumulative frequency distribution of the relative errors for 72% of the data points are in the range of �5% de viation from the observed values. Also, relative error distribution co incides with the normal distribution function indicating that a power law regression type is useful to define the stage-discharge formula of TCBF. TCBF is a throat-less flume whose performance can be compared with a central baffle flume with a given baffle length. In this regard, the stage-discharge relationship obtained by Eq. (9) is compared with that of obtained by Eq. (2) (Fig. 6) to know the effect of L/B. As shown in Fig. 6, for a given h/Bc, no significant difference is observed between the stagedischarge curves of the cases L/B ¼ 0 and 0.2 and the case of L/B ¼ 0.8 has the least flow capacity. The flow prediction accuracy of a TCBF is compared in Table 1 with that of proposed by Kolavani et al. [20] for a Central Baffle Flume, CBF, (L/B > 0). As listed in Table 1, flow prediction by a TCBF is more ac curate than a CBF. As a general conclusion, TCBF has the highest flow capacity, the least construction cost and is more accurate than a CBF.
Fig. 6. Comparison between the stage-discharge relationships of TCBF and CBF obtained by Eq. (9) and Eq. (2) respectively. Table 1 The flow prediction accuracy of TCBF and CBF. Flume type
5. Distinguishing condition curve
Percentage of the Data points with error range of
TCBF CBF proposed by Kolavani et al. [20]
�5%
�10%
71.4 65.3
93.8 85.3
The free flow stage-discharge relationship of a given Venturi flume can be applied only when the tailwater submergence is not too high [7], otherwise, the flow may be affected due to tailwater increase. There is a specific downstream flow depth beyond which the discharge through the flume starts decreasing [4]. Willeitner et al. [3] defined the submergence transition as a point when downstream con ditions begin to affect the upstream head readings. Based on a laboratory
Fig. 7. Experimental pairs of (h/Bc,hth/Bc) for the given values of r ¼ Bc/B, a) r ¼ 0.76, b) r ¼ 0.59, c) r ¼ 0.39, d) r ¼ 0.17. 4
M. Bijankhan and V. Ferro
Flow Measurement and Instrumentation 70 (2019) 101641
in which κ and β are empirical coefficients. For a given r value, the plotted pairs (h/Bc,hth/Bc) represents the distinguishing condition curve (Fig. 7). Practically, for a given flow rate and r, any downstream water depth greater than hth would make the flume work under submerged flow condition. When it is suspected to see a submerged flow, a flume with higher r-value can be selected due to its less sensitivity to downstream submergence (Fig. 7). As shown in Fig. 7, the exponent β equates to unity while the coef ficient κ depends on the contraction ratio r. Taking into account that the ISS hypothesis can be applied for the contracted ratio, r, the following equation is proposed to describe κ-values (Fig. 8): κ ¼ 0:9478 r0:3705
Finally, combining Eqs. (12) and (13), the following equation is proposed to formulate the distinguishing condition curve: � � hth h 0:3705 ¼ 0:9478 (14) r Bc Bc
Fig. 8. Calculated κ values in terms of r.
Comparing hth/Bc values obtained by Eq. (14) with the associated experimental values, the mean absolute relative errors of 7.4% is obtained. Calculated hth/Bc values versus the associated observed values are plotted in Fig. 9. This figure demonstrates that the distinguishing con dition curve can be estimated by Eq. (14) within a relative error range of �10%.
study on the flow through a submerged Montana Flume, they indicated that the submergence threshold is reached when the ratio of the tail water depth to the upstream depth is greater than 0.51. Although, the transition between free to submerged flow regimes occurs continuously [5], a free flow stage-discharge formula is useless for a drowned flume. Consequently, the identification of the flow regime is important. In this regard, a distinguishing condition curve can be used. For a given up stream flow depth, distinguishing condition curve represents a maximum tailwater depth increase, hth, while flow condition is still free. Such curve can be described by the following functional relationship: ϕ3 ðhth ; h; Bc ; B; g; μÞ ¼ 0
6. Conclusion Flow characteristics of triangular central baffle flume was experi mentally studied using TCBF devices having contraction ratios of 0.17, 0.39, 0.59, and 0.76. Employing dimensional analysis and incomplete self-similarity concept, TCBF’s stage-discharge formula was developed. A general stage-discharge formula accounting for the effect of the contraction ratio was proposed. Finally, to identify between free and submerged flow conditions, TCBF’s distinguishing condition curve was formulated using the experimental data obtained in this study.
(10)
where, ϕ3 is a functional symbol. Based on the Π-Theorem of the dimensional analysis and using Bc, g and μ as reference variables, Eq. (10) can be written as: � � hth h Bc ¼ ϕ4 ; (11) Bc B Bc where, ϕ4 is a functional symbol. For a specific value of Bc/B, Eq. (11) can be written as: � �β hth h ¼κ Bc Bc
(13)
Acknowledgment The authors would like to gratefully acknowledge the assists of Mrs. Farzaneh Lotfi Kolavani and Mr. Qolamreza Babaei for gathering a part of the experimental data and fabricating the central baffles.
(12)
Notations n, m, κ and β are empirical coefficients B the approaching channel width b baffle width Bc B b g acceleration due to gravity h Upstream depth hth the maximum permitted tailwater depth to allow the free flow condition L baffle length Le the longitudinal distance of the guide wall Q discharge r (Bc/B) contraction ratio ϕ, ϕ2, ϕ3, ϕ4, Γ, and Γ1 functional symbols α apex angle μ water viscosity Appendix 1
Fig. 9. Calculated hth/Bc versus the associated observed values. 5
M. Bijankhan and V. Ferro
Flow Measurement and Instrumentation 70 (2019) 101641
Table 1 Stage-Discharge experimental data of the tested TCBF devices Q(l/s)
h (cm)
b(cm)
Bc(cm)
5.34 11.96 16.26 21.48 27.01 32.17 36.51 40.18 45.04 3.49 5.14 9.40 13.96 19.63 25.19 30.65 5.95 11 16.12 20.49 27.1 34.5 40.35 45.65 0.84 6.92 11.81 16.7 25.4 33 0.85 0.99 7.53 12.5 20.25 27.78 33.58 3.50 7.39 11.94 17.06 26 33.06 1.06 4.36 8.6 13.72 23.75 32.75
4.6 7.5 9.1 11 12.8 14.2 15.5 16.5 17.6 3.5 4.43 6.5 8.6 10.6 12.35 13.9 4 6 7.6 8.94 10.64 12.1 13.74 14.75 2.9 11 15.3 18.9 24.6 29 1.4 1.6 5.6 7.9 10.7 13 14.7 2.9 4.6 6.2 7.8 10.1 11.7 2.2 5.4 8.3 11.1 15.5 18.8
20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 12 12 12 12 12 12 12 12 41.5 41.5 41.5 41.5 41.5 41.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 12 12 12 12 12 12 30.5 30.5 30.5 30.5 30.5 30.5
29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 38 38 38 38 38 38 38 38 8.5 8.5 8.5 8.5 8.5 8.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 38 38 38 38 38 38 19.5 19.5 19.5 19.5 19.5 19.5
Appendix 2
Table 2 Experimental data of the distinguishing condition curve Q(l/s)
h(cm)
b(cm)
Bc(cm)
hth (cm)
8.95 6.64 5.11 5.74 10.14 15.81 19.98 25.31 0.84 6.92 11.81 16.67 25.42 33.00 1.06
12.6 10.4 8.82 9.4 13.72 18.1 21.1 24.5 2.9 11 15.3 18.9 24.6 29 2.2
41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 30.5
8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 19.5
6.8 4.8 4.3 4.8 6.8 8.4 10.6 11.9 1.7 5.4 7.2 9.3 11.8 15.1 1.6 (continued on next page)
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Flow Measurement and Instrumentation 70 (2019) 101641
Table 2 (continued ) Q(l/s)
h(cm)
b(cm)
Bc(cm)
hth (cm)
4.36 8.61 13.72 23.75 32.75 5.14 9.40 13.96 19.63 25.19 30.65 8.56 0.99 7.53 12.53 20.25 27.78 33.58 5.95 11.04 16.12 20.49 27.13 34.47 40.35 45.65 3.50 7.39 11.94 17.06 26.00 33.06
5.4 8.3 11.1 15.5 18.8 4.5 6.6 8.72 10.7 12.4 14 1.4 1.6 5.6 7.9 10.7 13 14.7 4 6 7.6 8.94 10.64 12.1 13.74 14.75 2.9 4.6 6.2 7.8 10.1 11.7
30.5 30.5 30.5 30.5 30.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 12 12 12 12 12 12 12 12 12 12 12 12 12 12
19.5 19.5 19.5 19.5 19.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 38 38 38 38 38 38 38 38 38 38 38 38 38 38
3.2 6.1 7.2 10.3 12.5 3.5 4.9 6.4 7.4 8.9 9.7 1.4 1.8 4.3 6.6 8.7 10.4 11.9 4.5 6.1 7.8 8.3 8.9 9.5 11.3 12.1 2.8 4.3 5.9 7.7 9.1 10.6
References
[12] C. Di Stefano, G. Di Piazza, V. Ferro, Field testing of a simple flume (SMBF) for flow measurement in open channels, J. Irrig. Drain. Eng. 134 (2008) 235–241, https:// doi.org/10.1061/(Asce)0733-9437(2008)134:2(235). [13] F.G. Carollo, C. Di Stefano, V. Ferro, V. Pampalone, New stage-discharge equation for the SMBF flume, J. Irrig. Drain. Eng. 142 (2016) 1–7, https://doi.org/10.1061/ (ASCE)IR.1943-4774.0001005. [14] F. Lotfi Kolavani, M. Bijankhan, C. Di Stefano, V. Ferro, A. Mahdavi Mazdeh, Flow measurement using circular portable flume, Flow Meas. Instrum. 62 (2018), https://doi.org/10.1016/j.flowmeasinst.2018.05.008. [15] Z. Samani, S. Jorat, M. Yousef, Hydraulic characteristics of circular flume, J. Irrig. Drain. Eng. 117 (1991) 558–566. [16] A. Peruginelli, F. Bonacci, Funzionamento libero e sommerso di un misuratore di portata a risalto idraulico, Idrotecnica, 1995, pp. 261–272 (in Italian). [17] V. Ferro, Simple flume with a central baffle, Flow Meas. Instrum. 52 (2016) 53–56. [18] Z. Samani, Three simple flumes for flow measurement in open channels, J. Irrig. Drain. Eng. (2017) 2–5, https://doi.org/10.1061/(ASCE)IR.1943-4774.0001168. [19] V. Ferro, Discussion of “ Three Simple Flumes for Flow Measurement in Open Channels ” by Zohrab Samani, vol. 144, 2018, pp. 8–9. [20] F. Lotfi Kolavani, M. Bijankhan, C. Di Stefano, V. Ferro, A. Mahdavi Mazdeh, Experimental Study of a Central Baffle Flume, J. Irrig. Drain. Eng. Accepted F, 2018. [21] G.I. Barenblatt, Similarity, Self-Similarity and Intermediate Asymptotics, Consultants Bureau, NY, 1979. [22] G.I. Barenblatt, Dimensional Analysis, Gordon & Breach, Science Publishers, Amsterdam, Netherlands, 1987.
[1] R. Parshall, The improved Venturi flume, Trans. ASCE 89 (1926) 841–851. [2] F. Blaisdell, Results of Parshall flume, J. Irrig. Drain. Eng. ASCE 120 (1994) 278–291. [3] R.P. Willeitner, S.L. Barfuss, M.C. Johnson, Montana flume flow corrections under submerged flow, J. Irrig. Drain. Eng. 138 (2012) 685–689, https://doi.org/ 10.1061/(ASCE)IR.1943-4774.0000434. [4] M. Bos, Discharge Measurement Structures, third ed., 1989. [5] A.F. Torres, G.P. Merkley, Cutthroat measurement flume calibration for free and submerged flow using a single equation, J. Irrig. Drain. Eng. 134 (2008) 521–526. [6] S. Temeepattanapongsa, G.P. Merkley, S.L. Barfuss, M. Asce, B.L. Smith, Generic free-flow rating for cutthroat flumes, J. Hydraul. Eng. Eng. 139 (2013) 727–735, https://doi.org/10.1061/(ASCE)HY.1943-7900.0000732. [7] W.H. Hager, Modified venturi channel, J. Irrig. Drain. Eng. 111 (1985) 19–35. [8] W. Hager, Modified trapezoidal Venturi channel, J. Irrig. Drain. Eng. 112 (1986) 225–241. [9] W.H. Hager, Mobile flume for circular channel, J. Irrig. Drain. Eng. 114 (1988) 520–534. [10] Z. Samani, H. Magallanez, Simple flume for flow measurement in open channel, J. Irrig. Drain. Eng. ASCE 126 (2000) 127–129. [11] G. Baiamonte, V. Ferro, Simple flume for flow measurement in sloping open channel, J. Irrig. Drain. Eng. 133 (2007) 71–78.
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Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: http://www.elsevier.com/locate/flowmeasinst
Experimental study on triangular central baffle flume M. Bijankhan a, *, V. Ferro b a b
Department of Water Engineering, Imam Khomeini International University (IKIU), P.O. Box 3414916818, Qazvin, Iran Department of Earth and Marine Sciences, University of Palermo, Via Archirafi 20, 90128, Italy
A R T I C L E I N F O
A B S T R A C T
Keywords: Triangular central baffle flume Stage-discharge formula Buckingham theorem Incomplete self-similarity Submergence threshold
In this paper the results of the experiments performed to study the flow through a Triangular Central Baffle Flume (TCBF) are reported. The investigated flume consists of a triangular baffle of the apex angle of 75� with a given base width. The theoretical stage-discharge formula was deduced by applying the Buckingham’s Theorem and incomplete self-similarity hypothesis and was calibrated using the laboratory measurements carried out in this investigation. The proposed stage-discharge formula is characterized by a mean absolute relative error of 7.4% and 72% of the data points are in an error range of �5%. The results indicate that TCBF flume is char acterized by a flow capacity higher than that of a typical central baffle flume. Experimental observations show that the contraction ratio is a key parameter to distinguish between free and submerged flow regimes through a TCBF. Finally, to identify the flow condition, submergence threshold condition was formulated.
1. Introduction Simple hydraulic structures for estimating the flow rate in open channels are practically important to enhance irrigation network man agement and operation. In this regard critical flow concept is used to construct flow measuring devices. Changing the channel dimension, e.g. side contraction or bed level change, is a typical action to establish the critical flow state. Channel contraction can be created by either a local channel width reduction or by inserting a vertical obstacle on the channel bed. The most famous flume constructed by a local channel width thickening is the Venturi Flume. Khafagi and Parshall flumes are types of Venturi channels which are widely used [1,2]. Montana flume is also a kind of Parshall flume without diverging downstream walls [3]. According to throat section the flumes can be classified into LongThroated and Throat-Less (Cut-Throated Flumes, CTF). The term LongThroated stands for the flumes having streamlines running parallel in the contracted section. Any flumes with shorter throat section referred to length is called as Cut-Throat Flume [4]. Based on an extensive laboratory measurement on Cut-Throat Flumes (CTF), Torres and Merkley [5] found that free flow condition is converted to submerged flow regime without a transition zone. Consequently, for both free and submerged flow cases a general stage-discharge formula was developed. Temeepattanapongsa et al. [6] applied Computational Fluid
Dynamic (CFD) to obtain generic free flow rating for Cut-Throat Flumes. They found that CFD simulations could be successfully employed to extend the design principles of CTF devices. Hager [7] designed a modified shape of a Venturi flume by inserting a vertical column in the channel center. Using the critical flow concept, the related stage-discharge formula was developed. Experiments indi cated that this proposed flume can be used to estimate the discharge in the range of �5% of the real value. Hager [7] also proposed the stage-discharge relationships for rectangular, trapezoidal, and U-shaped channels. The flow pattern of a trapezoidal flume constructed by immersing a circular cone into a rectangular channel was studied by Hager [8]. The associated stage discharge formula was developed by accounting for the effect of streamline curvature. Hager [9] also proposed application of a circular mobile flume in circular conduits like sewers and drainage pipes. This flume consists of a cylinder located vertically in a circular channel. Considering that the critical flow state occurs at the flume contraction, Hager [9] established a stage-discharge formula for the mobile circular flume. A flow measurement flume can be constructed by sticking two semicylinders to the walls of a channel or installing a vertical cylindrical column in a channel [10–13] or using two pieces of pipes, one installed vertically inside the other [14,15]. Following the principle of inserting a central body in a channel, Peruginelli and Bonacci [16] proposed to construct a flume, named
* Corresponding author. E-mail address: [email protected] (M. Bijankhan). https://doi.org/10.1016/j.flowmeasinst.2019.101641 Received 21 December 2018; Received in revised form 4 September 2019; Accepted 30 September 2019 Available online 5 October 2019 0955-5986/© 2019 Elsevier Ltd. All rights reserved.
M. Bijankhan and V. Ferro
Flow Measurement and Instrumentation 70 (2019) 101641
1.45. The following stage-discharge relationship was proposed: " � � � �2 #� �1:45 Q L L h 0:1914 þ 0:0861 5 1 ¼ 0:655 B B B 2 2 c Bc g
(2)
Eq. (2) can be applied for 0.2 � L/B � 0.8. A low-cost and more efficient central baffle would be obtained when L tends to zero and α ¼ 75� [20]. In this study, such flume is called as Triangular Central Baffle Flume, TCBF (Fig. 2). Eq. (2), is however not valid for TCBF because L/B ¼ 0 is out of its applicability range. Conse quently, experiments are carried out in this study to establish the stage-discharge formula of the TCBF. 2. Experimental setup The experiments were performed in a 0.5 m wide, 0.6 m high and 12 m long Plexiglas flume located at the hydraulic laboratory of the Water Engineering Department, Imam Khomeini International Univer sity (IKIU), Qazvin, Iran. The flume was supplied by a centrifugal pump connected to a reservoir of the volume of about 15 m3. An electromag netic flow-meter was used for measuring the flow rates of up to 80 l/s with the accuracy of �0.5% of the full scale. Taking into account that the only physical parameter to design TCBF, with α ¼ 75� [20], is the baffle width, b, the experimental runs were performed using four TCBF configurations having b ¼ 12, 20.5, 30.5, and 41.5 cm and a contraction ratio r ¼ Bc/B, in which Bc (¼B b) is throat width, equal to 0.76, 0.59, 0.39 and 0.17, respectively. As shown in Fig. 3, the free flow condition was considered in all experimental runs, i.e. the jump toe was located sufficiently far from the baffle downstream to ensure that the upstream flow condition would not be affected (Fig. 3c). For a given flow rate, Q, the associated upstream flow depth, h, was recorded (see Table 1 of Appendix 1). To identify the maximum permitted tailwater depth allowing free flow condition, hth, the tailwater depth was gradually increased so that the upstream water depth started increasing. A maximum upstream flow depth increase of 2 mm was considered. Detailed experimental data of the observed submergence threshold conditions are listed in Table 2 of Appendix 2.
Fig. 1. Schematic view of central baffle flume.
“Central Baffle Flume” (CBF), inserting a baffle, having a width b and a throat length L, in the middle of a rectangular channel (Fig. 1). Ferro [17], using the Buckingham-Theorem of the dimensional analysis and the self-similarity theory, proposed the following stage-discharge formula of the CBF flow: � �n Q2=3 h ¼ a (1) 1=3 Bc B5=3 c g where, Q is discharge, Bc (¼B b) is throat width, B is channel width, b is baffle width, h is upstream flow depth, g is the acceleration due to gravity, and a and n are empirical parameters that should be obtained using experimental data. Samani [18] discussed the design and calibration of the central cir cular baffle in circular and trapezoidal channel cross sections and pro posed stage-discharge formulas whose applications were verified using field data measurements. Ferro [19], using the field measurements carried out by Samani [18], tested the applicability of the proposed theoretical stage-discharge re lationships for the case of both a flume with two semi-cylindrical glued at the channel walls and a circular flume in which a column pipe is installed. Lotfi Kolavani et al. [20] performed extensive experimental runs to investigate the effect of the throat length, L, for the central baffle flume (Fig. 1). The experiments by Lotfi Kolavani et al. [20] also highlighted that a central baffle flume with an apex angle, α, equal to 75� for up stream guide walls provides suitable hydraulic condition at the flume entrance. While, the existence of downstream guide walls, having a length L, did not affect the stage-discharge relationship. Accordingly, they find that Eq. (1) can be applied taking into account that the coef ficient a depends on the ratio L/B while n is a constant value equal to
Fig. 3. Tested triangular central baffle flume (a) side view of the case with b ¼ 20.5 cm, (b) upstream view of the triangular baffle with b ¼ 41.5 cm, and (c) the hydraulic jump occurred downstream of the triangular baffle with b ¼ 20.5 cm.
Fig. 2. Schematic view of triangular central baffle flume. 2
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Flow Measurement and Instrumentation 70 (2019) 101641
Π3 ¼
h Bc
(5c)
Substituting the dimensionless groups, Eq. (5), into Eq. (4) yields the following dimensionless formula: � � Q h ¼ Γ ;r (6) 5=2 1=2 Bc Bc g where Γ is a functional symbol. For a given value of r, when h=Bc →0 then Q=½B5=2 g1=2 � →0 and c
g1=2 � →∞. Consequently, applying the when h=Bc →∞ then Q=½B5=2 c incomplete self-similarity (ISS) condition [21,22] Eq. (6) becomes: � �n Q h ¼ Γ 1 ðrÞ (7) Bc B5=2 g1=2 c where, Γ1 is a functional symbol. According to the ISS condition, n is a numerical constant that should be obtained by experimental data. For each value of the contraction ratio r, the Γ1 (r) function assumes a constant value m and the stage-discharge formula obtained by Eq. (7), depends only on the upstream head ratio, h/Bc: � �n Q h ¼ m (8) 5=2 1=2 Bc Bc g
0.5 Fig. 4. Experimental pairs of (h/Bc, Q/[B2.5 ]) for TCBF with b ¼ 12, 20.5, c g 30.5, and 41.5 cm.
3. Deducing stage-discharge relationship The free flow hydraulic of the central baffle flume (Fig. 1) can be expressed by the following functional relationship [20]: ϕðh; Q; Bc ; B; L; Le ; g; μÞ ¼ 0
(3)
4. Calibrating and testing the stage-discharge relationship of a TCBF
where ϕ is a functional symbol, h is upstream flow depth, Q is discharge, Bc ¼ B – b is the throat width, B is the approaching channel width, b is the baffle width, L is baffle length, Le is the longitudinal distance of the guide wall, g is acceleration due to gravity and μ is the water viscosity. The stage-discharge equation of TCBF, corresponding to a guide wall entrance angle of α ¼ 75� and L ¼ 0, can be described by the following functional relationship: ϕ1 ðh; Q; Bc ; B; g; μÞ ¼ 0
0.5 Plotting of the experimental pairs of (h/Bc,Q/[B2.5 c g ]) (Fig. 4) revealed that the stage-discharge curves corresponding to different rvalues collapsed on a single curve. Consequently, the following stagedischarge formula was calibrated using all available experimental data (Fig. 4): � �1:5734 Q h ¼ 0:6925 (9) 5=2 1=2 Bc Bc g
(4)
where, ϕ1 is a functional symbol. Considering Bc, μ, and g as reference variables the following dimensionless groups were obtained: Π1 ¼
Q B5=2 g1=2 c
(5a)
Π2 ¼
Bc ¼r B
(5b)
Eq. (9) can be used in the range of 0.17 � r � 0.76 and 0.047 � h/ Bc � 3.67. This stage-discharge equation can be applicable when a free hydraulic jump occurs at the downstream pool. For assuring free flow condition, the flume can be installed at the channel end where a free overfall occurs in the downstream section. The relative errors associated with the discharge values calculated by Eq. (9) were plotted in Fig. 5a versus h/Bc. According to the figure, Eq. (9) can be used to determine the flow through TCBF with the relative
Fig. 5. (a) Relative errors associated with the discharge values calculated by Eq. (9) versus h/Bc b) cumulative frequency distribution of the relative errors. 3
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Flow Measurement and Instrumentation 70 (2019) 101641
errors restricted in the range of �10% and the mean absolute relative error of 4%. As shown in Fig. 5b, the cumulative frequency distribution of the relative errors for 72% of the data points are in the range of �5% de viation from the observed values. Also, relative error distribution co incides with the normal distribution function indicating that a power law regression type is useful to define the stage-discharge formula of TCBF. TCBF is a throat-less flume whose performance can be compared with a central baffle flume with a given baffle length. In this regard, the stage-discharge relationship obtained by Eq. (9) is compared with that of obtained by Eq. (2) (Fig. 6) to know the effect of L/B. As shown in Fig. 6, for a given h/Bc, no significant difference is observed between the stagedischarge curves of the cases L/B ¼ 0 and 0.2 and the case of L/B ¼ 0.8 has the least flow capacity. The flow prediction accuracy of a TCBF is compared in Table 1 with that of proposed by Kolavani et al. [20] for a Central Baffle Flume, CBF, (L/B > 0). As listed in Table 1, flow prediction by a TCBF is more ac curate than a CBF. As a general conclusion, TCBF has the highest flow capacity, the least construction cost and is more accurate than a CBF.
Fig. 6. Comparison between the stage-discharge relationships of TCBF and CBF obtained by Eq. (9) and Eq. (2) respectively. Table 1 The flow prediction accuracy of TCBF and CBF. Flume type
5. Distinguishing condition curve
Percentage of the Data points with error range of
TCBF CBF proposed by Kolavani et al. [20]
�5%
�10%
71.4 65.3
93.8 85.3
The free flow stage-discharge relationship of a given Venturi flume can be applied only when the tailwater submergence is not too high [7], otherwise, the flow may be affected due to tailwater increase. There is a specific downstream flow depth beyond which the discharge through the flume starts decreasing [4]. Willeitner et al. [3] defined the submergence transition as a point when downstream con ditions begin to affect the upstream head readings. Based on a laboratory
Fig. 7. Experimental pairs of (h/Bc,hth/Bc) for the given values of r ¼ Bc/B, a) r ¼ 0.76, b) r ¼ 0.59, c) r ¼ 0.39, d) r ¼ 0.17. 4
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Flow Measurement and Instrumentation 70 (2019) 101641
in which κ and β are empirical coefficients. For a given r value, the plotted pairs (h/Bc,hth/Bc) represents the distinguishing condition curve (Fig. 7). Practically, for a given flow rate and r, any downstream water depth greater than hth would make the flume work under submerged flow condition. When it is suspected to see a submerged flow, a flume with higher r-value can be selected due to its less sensitivity to downstream submergence (Fig. 7). As shown in Fig. 7, the exponent β equates to unity while the coef ficient κ depends on the contraction ratio r. Taking into account that the ISS hypothesis can be applied for the contracted ratio, r, the following equation is proposed to describe κ-values (Fig. 8): κ ¼ 0:9478 r0:3705
Finally, combining Eqs. (12) and (13), the following equation is proposed to formulate the distinguishing condition curve: � � hth h 0:3705 ¼ 0:9478 (14) r Bc Bc
Fig. 8. Calculated κ values in terms of r.
Comparing hth/Bc values obtained by Eq. (14) with the associated experimental values, the mean absolute relative errors of 7.4% is obtained. Calculated hth/Bc values versus the associated observed values are plotted in Fig. 9. This figure demonstrates that the distinguishing con dition curve can be estimated by Eq. (14) within a relative error range of �10%.
study on the flow through a submerged Montana Flume, they indicated that the submergence threshold is reached when the ratio of the tail water depth to the upstream depth is greater than 0.51. Although, the transition between free to submerged flow regimes occurs continuously [5], a free flow stage-discharge formula is useless for a drowned flume. Consequently, the identification of the flow regime is important. In this regard, a distinguishing condition curve can be used. For a given up stream flow depth, distinguishing condition curve represents a maximum tailwater depth increase, hth, while flow condition is still free. Such curve can be described by the following functional relationship: ϕ3 ðhth ; h; Bc ; B; g; μÞ ¼ 0
6. Conclusion Flow characteristics of triangular central baffle flume was experi mentally studied using TCBF devices having contraction ratios of 0.17, 0.39, 0.59, and 0.76. Employing dimensional analysis and incomplete self-similarity concept, TCBF’s stage-discharge formula was developed. A general stage-discharge formula accounting for the effect of the contraction ratio was proposed. Finally, to identify between free and submerged flow conditions, TCBF’s distinguishing condition curve was formulated using the experimental data obtained in this study.
(10)
where, ϕ3 is a functional symbol. Based on the Π-Theorem of the dimensional analysis and using Bc, g and μ as reference variables, Eq. (10) can be written as: � � hth h Bc ¼ ϕ4 ; (11) Bc B Bc where, ϕ4 is a functional symbol. For a specific value of Bc/B, Eq. (11) can be written as: � �β hth h ¼κ Bc Bc
(13)
Acknowledgment The authors would like to gratefully acknowledge the assists of Mrs. Farzaneh Lotfi Kolavani and Mr. Qolamreza Babaei for gathering a part of the experimental data and fabricating the central baffles.
(12)
Notations n, m, κ and β are empirical coefficients B the approaching channel width b baffle width Bc B b g acceleration due to gravity h Upstream depth hth the maximum permitted tailwater depth to allow the free flow condition L baffle length Le the longitudinal distance of the guide wall Q discharge r (Bc/B) contraction ratio ϕ, ϕ2, ϕ3, ϕ4, Γ, and Γ1 functional symbols α apex angle μ water viscosity Appendix 1
Fig. 9. Calculated hth/Bc versus the associated observed values. 5
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Flow Measurement and Instrumentation 70 (2019) 101641
Table 1 Stage-Discharge experimental data of the tested TCBF devices Q(l/s)
h (cm)
b(cm)
Bc(cm)
5.34 11.96 16.26 21.48 27.01 32.17 36.51 40.18 45.04 3.49 5.14 9.40 13.96 19.63 25.19 30.65 5.95 11 16.12 20.49 27.1 34.5 40.35 45.65 0.84 6.92 11.81 16.7 25.4 33 0.85 0.99 7.53 12.5 20.25 27.78 33.58 3.50 7.39 11.94 17.06 26 33.06 1.06 4.36 8.6 13.72 23.75 32.75
4.6 7.5 9.1 11 12.8 14.2 15.5 16.5 17.6 3.5 4.43 6.5 8.6 10.6 12.35 13.9 4 6 7.6 8.94 10.64 12.1 13.74 14.75 2.9 11 15.3 18.9 24.6 29 1.4 1.6 5.6 7.9 10.7 13 14.7 2.9 4.6 6.2 7.8 10.1 11.7 2.2 5.4 8.3 11.1 15.5 18.8
20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 12 12 12 12 12 12 12 12 41.5 41.5 41.5 41.5 41.5 41.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 12 12 12 12 12 12 30.5 30.5 30.5 30.5 30.5 30.5
29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 38 38 38 38 38 38 38 38 8.5 8.5 8.5 8.5 8.5 8.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 38 38 38 38 38 38 19.5 19.5 19.5 19.5 19.5 19.5
Appendix 2
Table 2 Experimental data of the distinguishing condition curve Q(l/s)
h(cm)
b(cm)
Bc(cm)
hth (cm)
8.95 6.64 5.11 5.74 10.14 15.81 19.98 25.31 0.84 6.92 11.81 16.67 25.42 33.00 1.06
12.6 10.4 8.82 9.4 13.72 18.1 21.1 24.5 2.9 11 15.3 18.9 24.6 29 2.2
41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 41.5 30.5
8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 8.5 19.5
6.8 4.8 4.3 4.8 6.8 8.4 10.6 11.9 1.7 5.4 7.2 9.3 11.8 15.1 1.6 (continued on next page)
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Flow Measurement and Instrumentation 70 (2019) 101641
Table 2 (continued ) Q(l/s)
h(cm)
b(cm)
Bc(cm)
hth (cm)
4.36 8.61 13.72 23.75 32.75 5.14 9.40 13.96 19.63 25.19 30.65 8.56 0.99 7.53 12.53 20.25 27.78 33.58 5.95 11.04 16.12 20.49 27.13 34.47 40.35 45.65 3.50 7.39 11.94 17.06 26.00 33.06
5.4 8.3 11.1 15.5 18.8 4.5 6.6 8.72 10.7 12.4 14 1.4 1.6 5.6 7.9 10.7 13 14.7 4 6 7.6 8.94 10.64 12.1 13.74 14.75 2.9 4.6 6.2 7.8 10.1 11.7
30.5 30.5 30.5 30.5 30.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 20.5 12 12 12 12 12 12 12 12 12 12 12 12 12 12
19.5 19.5 19.5 19.5 19.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 29.5 38 38 38 38 38 38 38 38 38 38 38 38 38 38
3.2 6.1 7.2 10.3 12.5 3.5 4.9 6.4 7.4 8.9 9.7 1.4 1.8 4.3 6.6 8.7 10.4 11.9 4.5 6.1 7.8 8.3 8.9 9.5 11.3 12.1 2.8 4.3 5.9 7.7 9.1 10.6
References
[12] C. Di Stefano, G. Di Piazza, V. Ferro, Field testing of a simple flume (SMBF) for flow measurement in open channels, J. Irrig. Drain. Eng. 134 (2008) 235–241, https:// doi.org/10.1061/(Asce)0733-9437(2008)134:2(235). [13] F.G. Carollo, C. Di Stefano, V. Ferro, V. Pampalone, New stage-discharge equation for the SMBF flume, J. Irrig. Drain. Eng. 142 (2016) 1–7, https://doi.org/10.1061/ (ASCE)IR.1943-4774.0001005. [14] F. Lotfi Kolavani, M. Bijankhan, C. Di Stefano, V. Ferro, A. Mahdavi Mazdeh, Flow measurement using circular portable flume, Flow Meas. Instrum. 62 (2018), https://doi.org/10.1016/j.flowmeasinst.2018.05.008. [15] Z. Samani, S. Jorat, M. Yousef, Hydraulic characteristics of circular flume, J. Irrig. Drain. Eng. 117 (1991) 558–566. [16] A. Peruginelli, F. Bonacci, Funzionamento libero e sommerso di un misuratore di portata a risalto idraulico, Idrotecnica, 1995, pp. 261–272 (in Italian). [17] V. Ferro, Simple flume with a central baffle, Flow Meas. Instrum. 52 (2016) 53–56. [18] Z. Samani, Three simple flumes for flow measurement in open channels, J. Irrig. Drain. Eng. (2017) 2–5, https://doi.org/10.1061/(ASCE)IR.1943-4774.0001168. [19] V. Ferro, Discussion of “ Three Simple Flumes for Flow Measurement in Open Channels ” by Zohrab Samani, vol. 144, 2018, pp. 8–9. [20] F. Lotfi Kolavani, M. Bijankhan, C. Di Stefano, V. Ferro, A. Mahdavi Mazdeh, Experimental Study of a Central Baffle Flume, J. Irrig. Drain. Eng. Accepted F, 2018. [21] G.I. Barenblatt, Similarity, Self-Similarity and Intermediate Asymptotics, Consultants Bureau, NY, 1979. [22] G.I. Barenblatt, Dimensional Analysis, Gordon & Breach, Science Publishers, Amsterdam, Netherlands, 1987.
[1] R. Parshall, The improved Venturi flume, Trans. ASCE 89 (1926) 841–851. [2] F. Blaisdell, Results of Parshall flume, J. Irrig. Drain. Eng. ASCE 120 (1994) 278–291. [3] R.P. Willeitner, S.L. Barfuss, M.C. Johnson, Montana flume flow corrections under submerged flow, J. Irrig. Drain. Eng. 138 (2012) 685–689, https://doi.org/ 10.1061/(ASCE)IR.1943-4774.0000434. [4] M. Bos, Discharge Measurement Structures, third ed., 1989. [5] A.F. Torres, G.P. Merkley, Cutthroat measurement flume calibration for free and submerged flow using a single equation, J. Irrig. Drain. Eng. 134 (2008) 521–526. [6] S. Temeepattanapongsa, G.P. Merkley, S.L. Barfuss, M. Asce, B.L. Smith, Generic free-flow rating for cutthroat flumes, J. Hydraul. Eng. Eng. 139 (2013) 727–735, https://doi.org/10.1061/(ASCE)HY.1943-7900.0000732. [7] W.H. Hager, Modified venturi channel, J. Irrig. Drain. Eng. 111 (1985) 19–35. [8] W. Hager, Modified trapezoidal Venturi channel, J. Irrig. Drain. Eng. 112 (1986) 225–241. [9] W.H. Hager, Mobile flume for circular channel, J. Irrig. Drain. Eng. 114 (1988) 520–534. [10] Z. Samani, H. Magallanez, Simple flume for flow measurement in open channel, J. Irrig. Drain. Eng. ASCE 126 (2000) 127–129. [11] G. Baiamonte, V. Ferro, Simple flume for flow measurement in sloping open channel, J. Irrig. Drain. Eng. 133 (2007) 71–78.
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