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Flow measurement using circular portable flume
Flow Measurement and Instrumentation 62 (2018) 76–83
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Flow measurement using circular portable flume a
a,⁎
b
T
c
F. Lotfi Kolavani , M. Bijankhan , C. Di Stefano , V. Ferro , A. Mahdavi Mazdeh
a,d
a
Water Science and Engineering Dept., Imam Khomeini International University, Qazvin, Iran Dipartimento di Scienze Agrarie, Alimentari e Forestali, Università degli Studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy Dipartimento di Scienze della Terra e del Mare, Università degli Studi di Palermo, Via Archirafi, 90123 Palermo, Italy d Hydrogeology Dept., Institute of Geology, Mineralogy & Geophysics, Ruhr University, Bochum, Germany b c
A R T I C LE I N FO
A B S T R A C T
Keywords: Flow measurement Portable flume Circular flume Buckingham's theorem Stage-discharge curve Drainage network
The circular portable flume is a simple device to measure discharge in circular drainage networks. Since the unit can be easily installed and removed, it is helpful in water distribution measurement and management. First in this paper the available studies are reviewed for highlighting the effect of both the contraction ratio and the flume slope on the stage-discharge relationship. Then the Buckingham's Theorem of the dimensional analysis and the self-similarity theory are used to deduce the stage-discharge curve of the circular flume. The new theoretical stage-discharge equation is calibrated by the literature available experimental data and those obtained in this experimental investigation for a wide range of the contraction ratios. The developed analysis suggested that the contraction ratio, which was neglected so far from the functional relationship, would significantly affect the stage-discharge curve. Finally, the effect of the flume longitudinal slope on the stage-discharge equation is tested by the experimental data obtained in this study and some other available literature experimental data.
1. Introduction Population growth and fast industrialization make water scarcity, water quality and delivery costs urgent problems worldwide. Therefore, flow measurement devices become a key tool for a reliable management. An old management adage “You can’t manage what you don’t measure” could easily show the growing role of flow measurements in many water distribution systems especially irrigation networks. In this regard, simple portable flumes which are low cost and characterized by accurate discharge measurements, are really helpful for reliable flow measurements [1]. Changing in bed elevation or in channel width are generally used to construct inexpensive and simple flumes to measure the discharge in open channels [2]. Such physical changes of the channel cross section would generate critical flow near to the flume cross-section and supercritical flow at the downstream end. Consequently, the critical flow concept can be used to deduce the stage-discharge relationship of the flow measuring flumes. Hager [2] investigated theoretically the flow through flumes with a central cylinder baffle and, using the critical flow condition, formulated the head-discharge relationships of a flume with rectangular, trapezoidal and Ushaped channel cross sections. Hager [2] also performed some experiments in a rectangular channel (0.3 m wide and approximately 5.5 m long) to investigate the flow behavior of a flume with a central cylindrical baffle.
⁎
Hager [3] proposed a mobile circular cross-section Venturi unit with a central cylindrical baffle useful for flow measuring in sewers and drainage pipes. This proposed unit can be easily installed in circular channels for short-time use. This device, as shown in Fig. 1, is a circular flume with a cylinder installed in the middle and, according to Hager [3], such flume can be used for a large discharge variation, i.e. from 1 to 150 L s−1. Employing the critical flow concept Hager [3] proposed the following theoretical stage-discharge relationship:
Ao3/2 QT = D5/2 g1/2 (dAo / dy )1/2
where QT is the theoretical discharge, D is the pipe diameter, A0 = A/D2, A is the contracted cross-sectional area, y = h/D and h is the upstream flow depth. The quantities Ao and dAo/dy have to be computed by the following equations [3]:
Ao =
4 3/2 1 4 2 1 3 ⎛1 − y − y y ⎞ − ⎛δ y − δ ⎞ 3 4 25 ⎠ ⎝ 12 ⎠ ⎝
dAo 5 3 = y1/2 ⎛2 − y − y 2 ⎞ − δ dy 6 4 ⎠ ⎝
Corresponding author. E-mail addresses: lotfi[email protected] (F. Lotfi Kolavani), [email protected] (M. Bijankhan), [email protected] (C. Di Stefano), [email protected] (V. Ferro), [email protected] (A. Mahdavi Mazdeh). https://doi.org/10.1016/j.flowmeasinst.2018.05.008 Received 3 March 2018; Received in revised form 10 May 2018; Accepted 11 May 2018 0955-5986/ © 2018 Elsevier Ltd. All rights reserved.
(1)
(2)
(3)
Flow Measurement and Instrumentation 62 (2018) 76–83
F. Lotfi Kolavani et al.
Nomenclature
upstream flow depth the theoretical discharge (D − d) / D h/D H/D dimensionless groups functional symbols d/D numerical constant water viscosity
h QT r y Y Π1, Π2, Π3 φ, φ1, and φ2 δ ε μ
A contracted cross-sectional area A0 A/D2 ao, a1, b0, b1, b2, m, n empirical coefficients d cylinder diameter D pipe diameter Dc D−d g acceleration due to gravity H critical total upstream head
Fig. 1. Schematic view of the mobile circular flume a) plan view, b) front view.
where δ = d/D and d is the cylinder diameter (Fig. 1b). According to Hager [3] < < the effects of the streamline slope and curvature may become important for critical flow > > affecting the stagedischarge equation. Therefore, the theoretical discharge values, QT, calculated by Eqs. (1)–(3) can be different from the corresponding measured ones. Fig. 2 shows the comparison between the values QT / D5/2 g1/2 calculated by Eqs. (1)–(3) and those measured by Hager [3] for a circular flume having an internal diameter D equal to 0.488 m and a cylinder diameter d of 0.1103 m. Fig. 2 demonstrates that the theoretical curve obtained by Eq. (1), for each h/D value, systematically overestimates the measured discharge values. For taking into account the effect of the streamline curvature, Hager [3] proposed the following equation to calculate the discharge Q through the circular flume:
Q = (m + n Y ) QT
and the incomplete self-similarity theory, Ferro [5] theoretically indicated that following equation could be used as the stage-discharge relationship of the circular flume
Q Dc5/2 g1/2
h = ao ⎛ ⎞ ⎝ Dc ⎠ ⎜
a1
⎟
(8)
in which ao and a1 are two coefficients and Dc = D − d. Using Samani et al. [4] data for four different values of the contraction ratio r = (D − d) / D varying in a small range (0.632–0.7), Ferro [5] concluded that Eq. (8) is independent of the contraction ratio. The estimated values of ao = 0.4416 and a1 = 2.3052 allow to deduce the following stage-discharge relationship:
Q = 0.4416 g1/2 (D − d)0.1948 h2.3052
(9)
Ontkean and Healy [6] investigated the effect of the circular flume slope on the stage-discharge relationship of a corrugated circular flume.
(4)
where m and n are numerical constants which assume the following values:
m = 1 and n = 0 for Y < 0.073
(5)
m = 0.985 and n = 0.205 for 0.073 < Y < 1
(6)
and Y = H/D is the critical total upstream head H normalized by the diameter D. which has to be calculated by the following relationship:
Y=y+
1 Ao 2 dAo / dy
(7)
Samani et al. [4] investigated the effect of different flume diameters on the hydraulic characteristics of the circular flume and proposed to calculate the correction factor Q/QT by Eq. (4) with m = 1.057 and n = 0.2266. For high discharge values, Samani et al. [4] indicated experimentally that the effect of the sloping installation of the flume is negligible for the longitudinal slopes equal to ± 1% and ± 2%. They also indicated that the flume bed slope affects the discharge estimation significantly for the discharges of less than 0.128l/s. Applying the Buckingham's Theorem of the dimensional analysis
Fig. 2. Comparison between the QT / D5/2 g1/2 values calculated by Eqs. (1)–(3) and those measured by Hager [3] for a circular flume of known geometric characteristics. 77
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cases, free flow condition was established using a tailgate installed at the downstream end. The discharge and upstream head were varied from 1.15 to 150 L s−1 and 5.34–43.16 cm, respectively. Samani et al. [4] used three experimental circular flumes of the diameters of 0.201, 0.25, and 0.45 m and one very large field flume of the diameter of 1.346 m with cylinder diameters of 0.0603, 0.081, 0.155, and 0.4953 m, respectively. The flumes allowed to investigate the effect of the diameter on the stage-discharge formula. The upstream head and discharge varied from 3.5 to 73.66 cm and from 0.38 to 493.5 L s−1, respectively. Free flow condition was observed for all cases and critical flow occurred at the cylinder section. Ontkean and Healy [6] investigated the effect of sloping installation of the circular flume. They considered the slope range of +1% to +7%, with a flume diameter of 0.908 m. The vertical cylinder was made with the diameter of 0.272 m and it was installed 0.788 m far from the flume inlet. The flume was fabricated using corrugated pipe having a length of 1.824 m. The experiments were performed with the discharge values ranging from 0.11 to 107.25 L s−1.
These experiments were carried out using five slope flume values (0%, 1%, 3%, 5%, and 7%) and discharge values ranging from 0.13 to 107.3 l s−1. Ontkean and Healy [6] stated that the stage-discharge formula would be affected by the flume slope. Rashwan and Idress [7] used a circular main pipe of the length of 4 m, with three vertical cylinders of the diameters of 10, 8, and 7.1 cm, i.e. 0.6 ≤Dc / D≤ 0.716. Based on the specific energy concept, they proposed three discharge coefficient formulas for d / D= 0.4, 0.32, and 0.284 characterized by the relative errors of 7.9%, 5.3% and 5.7%, respectively. More recently, Samani [1] indicated that simple circular, trapezoidal, and rectangular flumes, can be used easily in the field as accurate water measurement devices. For the circular flume, Samani [1] proposed to apply Eq. (8) with a0 and a1 values which are practically equal to those estimated by Ferro [5,8]. For testing the applicability of Eq. (8), Samani [1] used field scale flumes constructed in irrigation canals and reported the comparison between calculated and measured discharges for the circular flume. Ferro [8] verified the agreement of the stage-discharge relationship obtained by Eq. (9) and field data measured by Samani [1]. According to this literature review, the available stage-discharge formulas were obtained using experimental data with Dc / D ranging from 0.6 to 0.774. In this paper some experimental runs are carried out to extend the range of measurements by considering Dc / D equal to 0.342 and 0.421. Then, the performance of the previously proposed stage-discharge formulas was evaluated using the available literature data and those obtained in this investigation. Buckingham's Theorem of the dimensional analysis and the self-similarity theory are used to deduce a stage-discharge relationship of the circular flume. This analysis highlighted that the parameter Dc / D significantly affects the stagedischarge curves. Finally, using the experimental data collected both by Ontkean and Healy [6] and those obtained in this investigation, the effect of the flume slope on the stage-discharge equation was studied.
2.2. Experimental setup The experimental setup used in this study is located at the hydraulic laboratory of the Water Sciences and Engineering Department of Imam Khomeini International University, IKIU, Qazvin, Iran. The experimental installation consisted of a 12 m smooth bed pipe with the inside diameter of 0.19 m (Fig. 3a). Circular flume was installed at the downstream end of the experimental channel. Two vertical cylinders of the diameters of 0.11 m and 0.125 m were installed vertically inside the main pipe. Flow depth was recorded at a distance of 0.15 m upstream the cylinder (Fig. 3b). A magnetic flow meter with an accuracy of ± 0.5% of the full-scale was installed at the flume entrance pipe to measure the flow rate. Note that, in all experiments the free surface flow condition was established. Both positive and negative longitudinal slopes were tested. The detailed characteristics of the experiments are listed in Table 1. Hydraulic characteristics of the available experimental for horizontal bed flume are listed in Table 2. According to this table, in the previous studies the minimum tested Dc / D value was 0.632. The experimental data obtained in this study cover Dc / D values smaller than 0.632 (0.342 and 0.421).
2. Material and methods 2.1. Laboratory and field data sets The experimental data of Hager [3], Samani et al. [4], and Ontkean and Healy [6] are used in this investigation. In order to extend the applicability range of the proposed formulas, in this work, new experimental program was conducted for the contraction ratio values equal to 0.342 and 0.42. Furthermore, the effect of the flume slope was investigated experimentally for both positive and negative longitudinal slopes. Hager [3] performed experiments using a circular flume of 1 m long, and diameter of 0.488 m. He installed the circular flume in a U-shaped channel with the length of 8.5 m. Three cylinders of diameters 0.1105, 0.14, and 0.1595 m were used. For the smaller cylinder diameter, Hager [3] observed near critical flow condition involving surface waves. In all
2.3. The stage-discharge relationship In order to describe the flow through a horizontal circular flume the following functional relationship can be used [5,7]:
φ (h, Q, D, Dc , μ, g ) = 0
(10)
where, φ is a functional symbol, h is the upstream flow depth, Q is the discharge, D is the pipe diameter, d is the circular column diameter, Dc = D − d, μ is the water viscosity, and g is the acceleration due to gravity.
Fig. 3. Experimental setup a) cross sectional view, b) upstream water measuring location. 78
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Table 1 Characteristics of the experiments performed in this investigation. D (m)
d (m)
Flume slope (%)
h (cm)
Q (l/s)
0.19
0.11
0.19
0.125
0 +0.2 +1.11 −0.5 −0.85 −1.24 0 +0.187 +0.6 +1.375 −0.4625 −1.25 −1.75
7.33–16.9 10.3–16.7 8.5–17.24 8.2–16.75 7.7–16.85 7.4–17.55 8.93–16.5 8.3–17 9.33–16.7 8.5–16.8 9.7–16.64 11.5–17.1 9.9–16.9
1.2–8.8 2.69–8.03 1.63–8.88 1.4–8.56 1.3–8.25 1.12–8.22 1.33–6.16 1.03–6.41 1.64–6.94 1.26–6.45 1.94–6.24 2.83–6.62 1.76–6.46
Table 2 Hydraulic characteristics of the experimental data of different studies for the horizontal flume. Study
Dc/D
h/Dc
Q/[Dc2.5g0.5]
Fig. 5. Comparison between measured Π1 values and calculated by Eq. (15).
Hager [3] Samani et al. [4] Ontkean and Healy [6] Current study
0.673–0.774 0.632–0.7 0.7 0.342–0.421
0.15–1.28 0.25–1.31 0.014–0.494 0.916–2.54
0.005–0.775 0.016–0.735 0.0001–0.118 0.21–1.825
Q h = ϕ1 ⎛ , r ⎞ g1/2 Dc 5/2 D ⎝ c ⎠ ⎜
(11)
in which Π1, Π2, Π3 are dimensionless groups. Considering g, μ, and Dc as independent variables, the following dimensionless groups were obtained [5,7]:
Π1 =
Q g1/2 Dc 5/2
(12a)
Π2 =
D 1 = Dc r
(12b)
Π3 =
h Dc
(13)
where φ1 is a functional symbol. The self-similarity concept can be used to deduce the mathematical shape of Eq. (13) [9,10]. The self-similarity solutions of a physical problem are searched for boundary conditions, where the non-dimensional parameters tend to zero or infinity. For a given Π-group, when the function f tends to a finite limit different from zero, the phenomenon is not influenced by Πn and it can be dropped from the functional relationship Π1 = φ1 (Π2, Π3, ........., Πn − 1) , in which φ1 is a functional symbol. For such a case, the self-similarity is named complete self-similarity (CSS) in a given Πn dimensionless group. When the function φ1 has a limit equal to 0 or ∞, the incomplete self-similarity (ISS) in the parameter Πn occurs In this last case, the phenomenon is expressed by the following functional relationship [9,10]:
Applying the Π-Theorem of the dimensional analysis the following dimensionless functional relationship can be written:
f (Π1, Π2, Π3) = 0
⎟
Π1 = Πnε
φ2 (Π2, Π3, ........., Πn − 1)
(14)
in which φ2 is a functional symbol and ε is a numerical constant, that should be obtained using the experimental data. For a given r value, when h/ Dc → 0 then Q/(g1/2 Dc 5/2) → 0 . Taking into account that when r increases then Π1 increases too and assuming the ISS for h/Dc and r, Eq. (13) can be rewritten as follows:
(12c)
in which Dc / D is the contraction ratio r. Rearranging Eq. (11) the following non-dimensional functional relationship is obtained:
Fig. 4. Experimental stage-discharge curves obtained (a) using the data points of this study and those reported by Hager [3] and Samani et al. [4] and (b) using the data points of Ontkean and Healy [6]. 79
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Fig. 8. Π1 values in terms of h/Dc for different slope installations.
Fig. 6. Cumulative frequency of the relative errors associated with different methods.
Table 3 Mean absolute relative errors and percentages of the data points with the relative errors of less than ± 10% and ± 15% for different stage-discharge equations. Method
Mean absolute relative error
Ferro [5] Eq. (9) Ferro [5] recalibrated Eq. (8) with ao = 0.4187 and a1 = 2.1299 Eq. (15)
Percentages of the data points with the relative errors less than or equal to ± 10%
± 15%
22.2 21.6
40.4 28.4
45.9 41.2
8.8
64.2
84.4
Fig. 9. Comparison between measured Π1 values and those calculated by Eq. (17). Table 4 Mean absolute relative errors and percentages of the data points with the relative errors of less than ± 10% and ± 15% for Eqs. (16) and (17). Method
Eq. (16) Eq. (17)
Mean absolute relative error
11.6 11.1
Percentages of the data points with the relative errors less than or equal to ± 10%
± 15%
69.2 70.2
79.8 79.8
3. Results 3.1. Recalibrating Ferro's equation Eq. (8), originally obtained by Ferro [5] is referred to Ferro's equation in this study. As indicated in Fig. 4, employing the experimental data of Hager [3] and Samani et al. [4] falling in the range 0.63 < r < 0.77, and the data points obtained in this study for r = 0.342 and 0.421, the stage-discharge relationship, Eq. (8), was recalibrated and the coefficients a0 = 0.4187 and a1 = 2.1299 were obtained. The experimental data of Ontkean and Healy [6], for a corrugated circular flume with zero longitudinal slope, were also plotted in Fig. 4b. Using these experimental data, the stage-discharge relationship by
Fig. 7. Mean absolute relative error versus contraction ratio of different methods. b
g1/2
Q h 1 = bo ⎛ ⎞ r b2 5/2 Dc ⎝ Dc ⎠ ⎜
⎟
(15)
in which bo, b1, and b2 are coefficients that should be obtained experimentally.
80
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Fig. 10. Observed and calculated values of Π1 versus h/Dc along with the associated relative errors for the case of r = 0.421 with positive and negative bed slopes.
current study were estimated by Eq. (15) with the relative error range of ± 10%. The same comparison shows that 40% and 28.4% of the data points which is calculated by the original and recalibrated Ferro's [5] methods respectively are in the relative error range of ± 10%. The mean absolute relative error associated both to recalibrated Ferro's [5] method, and current study were 21.59%, and 8.8% respectively. In summary, the developed analysis based on the experimental data of Hager [3], Samani et al. [4] and the measurements carried out in this study indicated that the contraction ratio would affect significantly the stage-discharge formula and it should not be neglected from the functional relationship. To better identify the effect of the contraction ratio on the stagedischarge formula, the mean absolute relative error associated with each value of the contraction ratio was calculated and the results were plotted in Fig. 7. The figure shows clearly that both Ferro's method, Eq. (9), and its recalibrated formula, Eq. (4) with a0 = 0.4187 and a1 = 2.1299, are not sufficiently accurate for the smallest values of the contraction ratios. However, as indicated in Fig. 7, the stage-discharge formula developed in this study, i.e. Eq. (15), for which the effect of the contraction ratio is considered, can be used accurately in the range of 0.342 ≤ Dc / D ≤ 0.774.
Ferro [5] was recalibrated and the coefficients a0 = 0.4442 and a1 = 1.9671 were obtained. According to this figure the overall trend of the corrugated circular flume, i.e. the experimental data of Ontkean and Healy [6], is different from the data points of the smooth bed circular flumes. 3.2. Effect of Dc/D on the stage-discharge formula In the previous studies [1,3–5], the experimental data of Hager [3] and Samani et al. [4], characterized by 0.632 ≤ Dc / D ≤ 0.774 were considered to find the stage-discharge formulas. Consequently, due to a narrow range of the contraction ratio, the effect of Dc / D could not be distinguished, and therefore the stage-discharge relationship of the circular flume was considered to be independent of the contraction ratio. Eq. (15) proposed in this study, can be used to consider the effect of the parameter Dc / D. Using the data by Hager [3], Samani et al. [4], and those obtained in this study, bo = 0.8195, b1 = 2.3908 and b2 = 1.3349 were obtained. Note that Eq. (15) is applicable for smooth bed circular flumes and in the wide range 0.342 ≤ Dc / D ≤ 0.774. Fig. 5 shows the comparison between measured Π1 values and those calculated by Eq. (15). The cumulative frequency distributions of the relative errors associated with different design methods were calculated and plotted in Fig. 6. For a given relative error value, the cumulative frequency indicates the percentage of the data points with a relative error less than or equal to the selected value. According to this figure, detailed error analyses of different stage-discharge formulas were extracted and listed in Table 3. Fig. 6 and Table 3 show that, 64.2% of the data points of the
3.3. Effect of the slope on the stage-discharge equation The effects of bed roughness and bed slope are not included in Eq. (10). Consequently, the constant parameters of b0, b1, and b2 appeared in Eq. (15) may differ from that of being smooth and level. In the following sections the effects of corrugated pipes and non-level conditions are investigated. 81
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Fig. 11. Observed and calculated values of Π1 versus h/Dc along with the associated relative errors for the case of r = 0.342 with positive and negative bed slopes.
is obtained:
Table 5 Mean absolute relative errors associated with Eq. (15) for different flume bed slopes. Dc/D
Slope (%)
Mean absolute relative error (%)
0.421
0.2 1.11 −0.5 −0.85 −1.24 0.187 0.6 1.375 −0.46 −1.25 −1.75
2.83 2.78 4.87 2.87 8.62 8.27 6.43 8.6 4.3 5.25 2.07
0.342
2.1229
g1/2
⎟
s−0.03287
(17)
3.3.2. Smooth bed circular flume The experimental data obtained in this study were employed to evaluate the effect of both positive and negative flume bed slopes on the stage-discharge relationship of the circular flume. In this regard the applicability of Eq. (15) was examined to determine the discharge through the non-level circular flume. Observed and calculated Π1-values were plotted versus h / Dc in Figs. 10 and 11 for r = 0.421 and r = 0.342 respectively. Although for r = 0.342 with positive bed slopes Eq. (15) underestimates the discharge values, this equation can be used for the non-level circular flumes with acceptable accuracy. To better identify the effect of the longitudinal slopes on the stage-
2.1226
⎜
⎜
in which s = longitudinal slope expressed in percentage (%). Comparison between measured Π1 values and those calculated by Eq. (17) is plotted in Fig. 9. Also, Mean absolute relative errors and percentages of the data points with the relative errors of less than ± 10% and ± 15% for Eqs. (16) and (17) are listed in Table 4. According to Table 4 it was revealed that the mean absolute relative error of Eq. (17) is slightly less than Eq. (16). Consequently, no significant improvement was observed by introducing the flume slope in the stage-discharge formula. Note that the experimental data of Ontkean and Healy [6] are characterized by very limited range of Π1-values, i.e. Π1 < 0.12., and, consequently, the effect of the flume slope for wider ranges of Π1 should be investigated.
3.3.1. Corrugated circular flume The experimental data collected by Ontkean and Healy [6] were used to study the effect of the slope installation on the corrugated circular flume. Using the data points with the flume slopes of +1%, +3%, +5%, and +7%, for Dc / D = 0.7, the pairs (Π1, h / Dc) were plotted in Fig. 8 and the following stage-discharge formula was obtained:
Q h = 0.5062 ⎛ ⎞ g1/2 Dc 5/2 D ⎝ c⎠
Q h = 0.4523 ⎛ ⎞ Dc 5/2 ⎝ Dc ⎠
⎟
(16)
Using the experimental data of Ontkean and Healy [6], and taking into account the slope effect on the stage discharge relationship, the following equation (by employing LINEST function in Microsoft Excel) 82
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literature, is more accurate than the equation proposed by Ferro [5] and that recently applied by Samani [1] for the circular flumes. Using of the experimental data collected by Ontkean and Healy [6] allowed to demonstrate the effect of the flume slope on the stage-discharge equation for the corrugated pipes. No significant improvement was observed by introducing the bed slope in the stage-discharge formula. Experimental data obtained in this study were employed to study the effect of both positive and negative longitudinal slopes on the stagedischarge formula of the smooth bed circular flume. Accordingly, it was suggested to use the proposed stage-discharge formula only within the bed slope range of ± 1%. However, more experimental studies should be carried out to illustrate the effects of the flume slope and bed roughness on the proposed stage-discharge equation.
discharge curve, mean absolute relative errors associated with Eq. (15) were calculated for different longitudinal slopes and the results were listed in Table 5. The maximum value of the mean absolute relative error associated with the positive longitudinal slopes was observed for Dc / D= 0.342 and s = 1.375%. For the negative longitudinal slope, however, the highest mean absolute relative error was observed for Dc / D = 0.421 and s = −1.24%. Accordingly, Eq. (15) is suggested to be used only for the longitudinal bed slopes, within the range of ± 1%. According to the results of this study, in order to measure the flow rates in the field, the following points should be considered: 1. The flume must work in a free flow condition. It is advisable to install the flume at the end of the canal/pipe where a free overfall occurs downstream the measuring flume; 2. It is strongly suggested to construct the circular flume according to the following advises: a) The vertical pipe should be installed at a distance of 0.5 m from the downstream end. b) A distant of at least 1.5 m of the pipe having no flinches, bends, and any other obstacles should be considered at the flume inlet.
Acknowledgements This work was supported by the Center for International Scientific Studies & Collaboration, CISSC under the Contract number of 1663. The authors appreciate also the assists of Mr. Qolamreza Babaei, the hydraulic lab supervisor, for his efforts to construct the experimental setup.
Note that the circular flumes tested in this study are constructed based on the above dimensions.
References 3. Non-level conditions with the bed slopes out of the range of ± 1% should be avoided; 4. For the smooth beds Eq. (15) can be used in the range of 0.342 ≤ Dc / D ≤ 0.774 with the mean absolute relative error of less than 8.8%; 5. For the corrugated pipes, Eq. (17) can be used only for Dc / D = 0.7 with the mean absolute relative error of 11.06%. Experiments should be carried out to extend the applicability range of the circular flumes for the corrugated pipes.
[1] Z. Samani, Three simple flumes for flow measurement in open channels, J. Irrig. Drain. Eng. 2 (2017) 2–5. [2] W.H. Hager, Modified venturi channel, J. Irrig. Drain. Eng. 111 (1) (1985) 19–35. [3] W.H. Hager, Mobile flume for circular channel, J. Irrig. Drain. Eng. 114 (3) (1988) 520–534. [4] Z. Samani, S. Jorat, M. Yousef, Hydraulic characteristics of circular flume, J. Irrig. Drain. Eng. ASCE 117 (4) (1991) 558–556. [5] V. Ferro, Discussion of “simple flume for flow measurement in open channel” by Zohrab Samani and Henry Magallanez, J. Irrig. Drain. Eng. ASCE 128 (2) (2002) 129–132. [6] G.R. Ontkean, L.H. Healy, Impact of non – level operation of a circular flume on discharge measurements, in: Proceedings of the CSBE/SCGAB 2015 Annual Conference, Edmonton, Alberta, 2015. [7] I.M.H. Rashwan, M.I. Idress, Evaluation efficiency for mobile as discharge measurement device for partially filled circular channel, Ain Shams Eng. J. 4 (2013) 199–206. [8] V. Ferro, Discussion of “three simple flumes for flow measurement in open channels” by Zohrab Samani, J. Irrig. Drain. Eng. ASCE (2018) (In press). [9] G.I. Barenblatt, Similarity, Self-Similarity and Intermediate Asymptotics, Consultans Bureau, New York, 1979. [10] G.I. Barenblatt, Dimensional Analysis, Gordon and Breach Science, Amsterdam, The Netherlands, 1987.
4. Conclusion The developed analysis carried out by the Buckingham's Theorem of dimensional analysis and self-similarity theory showed that the contraction ratio would significantly affect the stage-discharge formula of a circular flume. New experimental program was conducted in this study to investigate the effect of both contraction ratio and longitudinal bed slope on the performance of the circular flume. The theoretically deduced stage-discharge relationship calibrated by the experimental data obtained in this study and those available in the
83
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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Flow measurement using circular portable flume a
a,⁎
b
T
c
F. Lotfi Kolavani , M. Bijankhan , C. Di Stefano , V. Ferro , A. Mahdavi Mazdeh
a,d
a
Water Science and Engineering Dept., Imam Khomeini International University, Qazvin, Iran Dipartimento di Scienze Agrarie, Alimentari e Forestali, Università degli Studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy Dipartimento di Scienze della Terra e del Mare, Università degli Studi di Palermo, Via Archirafi, 90123 Palermo, Italy d Hydrogeology Dept., Institute of Geology, Mineralogy & Geophysics, Ruhr University, Bochum, Germany b c
A R T I C LE I N FO
A B S T R A C T
Keywords: Flow measurement Portable flume Circular flume Buckingham's theorem Stage-discharge curve Drainage network
The circular portable flume is a simple device to measure discharge in circular drainage networks. Since the unit can be easily installed and removed, it is helpful in water distribution measurement and management. First in this paper the available studies are reviewed for highlighting the effect of both the contraction ratio and the flume slope on the stage-discharge relationship. Then the Buckingham's Theorem of the dimensional analysis and the self-similarity theory are used to deduce the stage-discharge curve of the circular flume. The new theoretical stage-discharge equation is calibrated by the literature available experimental data and those obtained in this experimental investigation for a wide range of the contraction ratios. The developed analysis suggested that the contraction ratio, which was neglected so far from the functional relationship, would significantly affect the stage-discharge curve. Finally, the effect of the flume longitudinal slope on the stage-discharge equation is tested by the experimental data obtained in this study and some other available literature experimental data.
1. Introduction Population growth and fast industrialization make water scarcity, water quality and delivery costs urgent problems worldwide. Therefore, flow measurement devices become a key tool for a reliable management. An old management adage “You can’t manage what you don’t measure” could easily show the growing role of flow measurements in many water distribution systems especially irrigation networks. In this regard, simple portable flumes which are low cost and characterized by accurate discharge measurements, are really helpful for reliable flow measurements [1]. Changing in bed elevation or in channel width are generally used to construct inexpensive and simple flumes to measure the discharge in open channels [2]. Such physical changes of the channel cross section would generate critical flow near to the flume cross-section and supercritical flow at the downstream end. Consequently, the critical flow concept can be used to deduce the stage-discharge relationship of the flow measuring flumes. Hager [2] investigated theoretically the flow through flumes with a central cylinder baffle and, using the critical flow condition, formulated the head-discharge relationships of a flume with rectangular, trapezoidal and Ushaped channel cross sections. Hager [2] also performed some experiments in a rectangular channel (0.3 m wide and approximately 5.5 m long) to investigate the flow behavior of a flume with a central cylindrical baffle.
⁎
Hager [3] proposed a mobile circular cross-section Venturi unit with a central cylindrical baffle useful for flow measuring in sewers and drainage pipes. This proposed unit can be easily installed in circular channels for short-time use. This device, as shown in Fig. 1, is a circular flume with a cylinder installed in the middle and, according to Hager [3], such flume can be used for a large discharge variation, i.e. from 1 to 150 L s−1. Employing the critical flow concept Hager [3] proposed the following theoretical stage-discharge relationship:
Ao3/2 QT = D5/2 g1/2 (dAo / dy )1/2
where QT is the theoretical discharge, D is the pipe diameter, A0 = A/D2, A is the contracted cross-sectional area, y = h/D and h is the upstream flow depth. The quantities Ao and dAo/dy have to be computed by the following equations [3]:
Ao =
4 3/2 1 4 2 1 3 ⎛1 − y − y y ⎞ − ⎛δ y − δ ⎞ 3 4 25 ⎠ ⎝ 12 ⎠ ⎝
dAo 5 3 = y1/2 ⎛2 − y − y 2 ⎞ − δ dy 6 4 ⎠ ⎝
Corresponding author. E-mail addresses: lotfi[email protected] (F. Lotfi Kolavani), [email protected] (M. Bijankhan), [email protected] (C. Di Stefano), [email protected] (V. Ferro), [email protected] (A. Mahdavi Mazdeh). https://doi.org/10.1016/j.flowmeasinst.2018.05.008 Received 3 March 2018; Received in revised form 10 May 2018; Accepted 11 May 2018 0955-5986/ © 2018 Elsevier Ltd. All rights reserved.
(1)
(2)
(3)
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Nomenclature
upstream flow depth the theoretical discharge (D − d) / D h/D H/D dimensionless groups functional symbols d/D numerical constant water viscosity
h QT r y Y Π1, Π2, Π3 φ, φ1, and φ2 δ ε μ
A contracted cross-sectional area A0 A/D2 ao, a1, b0, b1, b2, m, n empirical coefficients d cylinder diameter D pipe diameter Dc D−d g acceleration due to gravity H critical total upstream head
Fig. 1. Schematic view of the mobile circular flume a) plan view, b) front view.
where δ = d/D and d is the cylinder diameter (Fig. 1b). According to Hager [3] < < the effects of the streamline slope and curvature may become important for critical flow > > affecting the stagedischarge equation. Therefore, the theoretical discharge values, QT, calculated by Eqs. (1)–(3) can be different from the corresponding measured ones. Fig. 2 shows the comparison between the values QT / D5/2 g1/2 calculated by Eqs. (1)–(3) and those measured by Hager [3] for a circular flume having an internal diameter D equal to 0.488 m and a cylinder diameter d of 0.1103 m. Fig. 2 demonstrates that the theoretical curve obtained by Eq. (1), for each h/D value, systematically overestimates the measured discharge values. For taking into account the effect of the streamline curvature, Hager [3] proposed the following equation to calculate the discharge Q through the circular flume:
Q = (m + n Y ) QT
and the incomplete self-similarity theory, Ferro [5] theoretically indicated that following equation could be used as the stage-discharge relationship of the circular flume
Q Dc5/2 g1/2
h = ao ⎛ ⎞ ⎝ Dc ⎠ ⎜
a1
⎟
(8)
in which ao and a1 are two coefficients and Dc = D − d. Using Samani et al. [4] data for four different values of the contraction ratio r = (D − d) / D varying in a small range (0.632–0.7), Ferro [5] concluded that Eq. (8) is independent of the contraction ratio. The estimated values of ao = 0.4416 and a1 = 2.3052 allow to deduce the following stage-discharge relationship:
Q = 0.4416 g1/2 (D − d)0.1948 h2.3052
(9)
Ontkean and Healy [6] investigated the effect of the circular flume slope on the stage-discharge relationship of a corrugated circular flume.
(4)
where m and n are numerical constants which assume the following values:
m = 1 and n = 0 for Y < 0.073
(5)
m = 0.985 and n = 0.205 for 0.073 < Y < 1
(6)
and Y = H/D is the critical total upstream head H normalized by the diameter D. which has to be calculated by the following relationship:
Y=y+
1 Ao 2 dAo / dy
(7)
Samani et al. [4] investigated the effect of different flume diameters on the hydraulic characteristics of the circular flume and proposed to calculate the correction factor Q/QT by Eq. (4) with m = 1.057 and n = 0.2266. For high discharge values, Samani et al. [4] indicated experimentally that the effect of the sloping installation of the flume is negligible for the longitudinal slopes equal to ± 1% and ± 2%. They also indicated that the flume bed slope affects the discharge estimation significantly for the discharges of less than 0.128l/s. Applying the Buckingham's Theorem of the dimensional analysis
Fig. 2. Comparison between the QT / D5/2 g1/2 values calculated by Eqs. (1)–(3) and those measured by Hager [3] for a circular flume of known geometric characteristics. 77
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cases, free flow condition was established using a tailgate installed at the downstream end. The discharge and upstream head were varied from 1.15 to 150 L s−1 and 5.34–43.16 cm, respectively. Samani et al. [4] used three experimental circular flumes of the diameters of 0.201, 0.25, and 0.45 m and one very large field flume of the diameter of 1.346 m with cylinder diameters of 0.0603, 0.081, 0.155, and 0.4953 m, respectively. The flumes allowed to investigate the effect of the diameter on the stage-discharge formula. The upstream head and discharge varied from 3.5 to 73.66 cm and from 0.38 to 493.5 L s−1, respectively. Free flow condition was observed for all cases and critical flow occurred at the cylinder section. Ontkean and Healy [6] investigated the effect of sloping installation of the circular flume. They considered the slope range of +1% to +7%, with a flume diameter of 0.908 m. The vertical cylinder was made with the diameter of 0.272 m and it was installed 0.788 m far from the flume inlet. The flume was fabricated using corrugated pipe having a length of 1.824 m. The experiments were performed with the discharge values ranging from 0.11 to 107.25 L s−1.
These experiments were carried out using five slope flume values (0%, 1%, 3%, 5%, and 7%) and discharge values ranging from 0.13 to 107.3 l s−1. Ontkean and Healy [6] stated that the stage-discharge formula would be affected by the flume slope. Rashwan and Idress [7] used a circular main pipe of the length of 4 m, with three vertical cylinders of the diameters of 10, 8, and 7.1 cm, i.e. 0.6 ≤Dc / D≤ 0.716. Based on the specific energy concept, they proposed three discharge coefficient formulas for d / D= 0.4, 0.32, and 0.284 characterized by the relative errors of 7.9%, 5.3% and 5.7%, respectively. More recently, Samani [1] indicated that simple circular, trapezoidal, and rectangular flumes, can be used easily in the field as accurate water measurement devices. For the circular flume, Samani [1] proposed to apply Eq. (8) with a0 and a1 values which are practically equal to those estimated by Ferro [5,8]. For testing the applicability of Eq. (8), Samani [1] used field scale flumes constructed in irrigation canals and reported the comparison between calculated and measured discharges for the circular flume. Ferro [8] verified the agreement of the stage-discharge relationship obtained by Eq. (9) and field data measured by Samani [1]. According to this literature review, the available stage-discharge formulas were obtained using experimental data with Dc / D ranging from 0.6 to 0.774. In this paper some experimental runs are carried out to extend the range of measurements by considering Dc / D equal to 0.342 and 0.421. Then, the performance of the previously proposed stage-discharge formulas was evaluated using the available literature data and those obtained in this investigation. Buckingham's Theorem of the dimensional analysis and the self-similarity theory are used to deduce a stage-discharge relationship of the circular flume. This analysis highlighted that the parameter Dc / D significantly affects the stagedischarge curves. Finally, using the experimental data collected both by Ontkean and Healy [6] and those obtained in this investigation, the effect of the flume slope on the stage-discharge equation was studied.
2.2. Experimental setup The experimental setup used in this study is located at the hydraulic laboratory of the Water Sciences and Engineering Department of Imam Khomeini International University, IKIU, Qazvin, Iran. The experimental installation consisted of a 12 m smooth bed pipe with the inside diameter of 0.19 m (Fig. 3a). Circular flume was installed at the downstream end of the experimental channel. Two vertical cylinders of the diameters of 0.11 m and 0.125 m were installed vertically inside the main pipe. Flow depth was recorded at a distance of 0.15 m upstream the cylinder (Fig. 3b). A magnetic flow meter with an accuracy of ± 0.5% of the full-scale was installed at the flume entrance pipe to measure the flow rate. Note that, in all experiments the free surface flow condition was established. Both positive and negative longitudinal slopes were tested. The detailed characteristics of the experiments are listed in Table 1. Hydraulic characteristics of the available experimental for horizontal bed flume are listed in Table 2. According to this table, in the previous studies the minimum tested Dc / D value was 0.632. The experimental data obtained in this study cover Dc / D values smaller than 0.632 (0.342 and 0.421).
2. Material and methods 2.1. Laboratory and field data sets The experimental data of Hager [3], Samani et al. [4], and Ontkean and Healy [6] are used in this investigation. In order to extend the applicability range of the proposed formulas, in this work, new experimental program was conducted for the contraction ratio values equal to 0.342 and 0.42. Furthermore, the effect of the flume slope was investigated experimentally for both positive and negative longitudinal slopes. Hager [3] performed experiments using a circular flume of 1 m long, and diameter of 0.488 m. He installed the circular flume in a U-shaped channel with the length of 8.5 m. Three cylinders of diameters 0.1105, 0.14, and 0.1595 m were used. For the smaller cylinder diameter, Hager [3] observed near critical flow condition involving surface waves. In all
2.3. The stage-discharge relationship In order to describe the flow through a horizontal circular flume the following functional relationship can be used [5,7]:
φ (h, Q, D, Dc , μ, g ) = 0
(10)
where, φ is a functional symbol, h is the upstream flow depth, Q is the discharge, D is the pipe diameter, d is the circular column diameter, Dc = D − d, μ is the water viscosity, and g is the acceleration due to gravity.
Fig. 3. Experimental setup a) cross sectional view, b) upstream water measuring location. 78
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Table 1 Characteristics of the experiments performed in this investigation. D (m)
d (m)
Flume slope (%)
h (cm)
Q (l/s)
0.19
0.11
0.19
0.125
0 +0.2 +1.11 −0.5 −0.85 −1.24 0 +0.187 +0.6 +1.375 −0.4625 −1.25 −1.75
7.33–16.9 10.3–16.7 8.5–17.24 8.2–16.75 7.7–16.85 7.4–17.55 8.93–16.5 8.3–17 9.33–16.7 8.5–16.8 9.7–16.64 11.5–17.1 9.9–16.9
1.2–8.8 2.69–8.03 1.63–8.88 1.4–8.56 1.3–8.25 1.12–8.22 1.33–6.16 1.03–6.41 1.64–6.94 1.26–6.45 1.94–6.24 2.83–6.62 1.76–6.46
Table 2 Hydraulic characteristics of the experimental data of different studies for the horizontal flume. Study
Dc/D
h/Dc
Q/[Dc2.5g0.5]
Fig. 5. Comparison between measured Π1 values and calculated by Eq. (15).
Hager [3] Samani et al. [4] Ontkean and Healy [6] Current study
0.673–0.774 0.632–0.7 0.7 0.342–0.421
0.15–1.28 0.25–1.31 0.014–0.494 0.916–2.54
0.005–0.775 0.016–0.735 0.0001–0.118 0.21–1.825
Q h = ϕ1 ⎛ , r ⎞ g1/2 Dc 5/2 D ⎝ c ⎠ ⎜
(11)
in which Π1, Π2, Π3 are dimensionless groups. Considering g, μ, and Dc as independent variables, the following dimensionless groups were obtained [5,7]:
Π1 =
Q g1/2 Dc 5/2
(12a)
Π2 =
D 1 = Dc r
(12b)
Π3 =
h Dc
(13)
where φ1 is a functional symbol. The self-similarity concept can be used to deduce the mathematical shape of Eq. (13) [9,10]. The self-similarity solutions of a physical problem are searched for boundary conditions, where the non-dimensional parameters tend to zero or infinity. For a given Π-group, when the function f tends to a finite limit different from zero, the phenomenon is not influenced by Πn and it can be dropped from the functional relationship Π1 = φ1 (Π2, Π3, ........., Πn − 1) , in which φ1 is a functional symbol. For such a case, the self-similarity is named complete self-similarity (CSS) in a given Πn dimensionless group. When the function φ1 has a limit equal to 0 or ∞, the incomplete self-similarity (ISS) in the parameter Πn occurs In this last case, the phenomenon is expressed by the following functional relationship [9,10]:
Applying the Π-Theorem of the dimensional analysis the following dimensionless functional relationship can be written:
f (Π1, Π2, Π3) = 0
⎟
Π1 = Πnε
φ2 (Π2, Π3, ........., Πn − 1)
(14)
in which φ2 is a functional symbol and ε is a numerical constant, that should be obtained using the experimental data. For a given r value, when h/ Dc → 0 then Q/(g1/2 Dc 5/2) → 0 . Taking into account that when r increases then Π1 increases too and assuming the ISS for h/Dc and r, Eq. (13) can be rewritten as follows:
(12c)
in which Dc / D is the contraction ratio r. Rearranging Eq. (11) the following non-dimensional functional relationship is obtained:
Fig. 4. Experimental stage-discharge curves obtained (a) using the data points of this study and those reported by Hager [3] and Samani et al. [4] and (b) using the data points of Ontkean and Healy [6]. 79
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Fig. 8. Π1 values in terms of h/Dc for different slope installations.
Fig. 6. Cumulative frequency of the relative errors associated with different methods.
Table 3 Mean absolute relative errors and percentages of the data points with the relative errors of less than ± 10% and ± 15% for different stage-discharge equations. Method
Mean absolute relative error
Ferro [5] Eq. (9) Ferro [5] recalibrated Eq. (8) with ao = 0.4187 and a1 = 2.1299 Eq. (15)
Percentages of the data points with the relative errors less than or equal to ± 10%
± 15%
22.2 21.6
40.4 28.4
45.9 41.2
8.8
64.2
84.4
Fig. 9. Comparison between measured Π1 values and those calculated by Eq. (17). Table 4 Mean absolute relative errors and percentages of the data points with the relative errors of less than ± 10% and ± 15% for Eqs. (16) and (17). Method
Eq. (16) Eq. (17)
Mean absolute relative error
11.6 11.1
Percentages of the data points with the relative errors less than or equal to ± 10%
± 15%
69.2 70.2
79.8 79.8
3. Results 3.1. Recalibrating Ferro's equation Eq. (8), originally obtained by Ferro [5] is referred to Ferro's equation in this study. As indicated in Fig. 4, employing the experimental data of Hager [3] and Samani et al. [4] falling in the range 0.63 < r < 0.77, and the data points obtained in this study for r = 0.342 and 0.421, the stage-discharge relationship, Eq. (8), was recalibrated and the coefficients a0 = 0.4187 and a1 = 2.1299 were obtained. The experimental data of Ontkean and Healy [6], for a corrugated circular flume with zero longitudinal slope, were also plotted in Fig. 4b. Using these experimental data, the stage-discharge relationship by
Fig. 7. Mean absolute relative error versus contraction ratio of different methods. b
g1/2
Q h 1 = bo ⎛ ⎞ r b2 5/2 Dc ⎝ Dc ⎠ ⎜
⎟
(15)
in which bo, b1, and b2 are coefficients that should be obtained experimentally.
80
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Fig. 10. Observed and calculated values of Π1 versus h/Dc along with the associated relative errors for the case of r = 0.421 with positive and negative bed slopes.
current study were estimated by Eq. (15) with the relative error range of ± 10%. The same comparison shows that 40% and 28.4% of the data points which is calculated by the original and recalibrated Ferro's [5] methods respectively are in the relative error range of ± 10%. The mean absolute relative error associated both to recalibrated Ferro's [5] method, and current study were 21.59%, and 8.8% respectively. In summary, the developed analysis based on the experimental data of Hager [3], Samani et al. [4] and the measurements carried out in this study indicated that the contraction ratio would affect significantly the stage-discharge formula and it should not be neglected from the functional relationship. To better identify the effect of the contraction ratio on the stagedischarge formula, the mean absolute relative error associated with each value of the contraction ratio was calculated and the results were plotted in Fig. 7. The figure shows clearly that both Ferro's method, Eq. (9), and its recalibrated formula, Eq. (4) with a0 = 0.4187 and a1 = 2.1299, are not sufficiently accurate for the smallest values of the contraction ratios. However, as indicated in Fig. 7, the stage-discharge formula developed in this study, i.e. Eq. (15), for which the effect of the contraction ratio is considered, can be used accurately in the range of 0.342 ≤ Dc / D ≤ 0.774.
Ferro [5] was recalibrated and the coefficients a0 = 0.4442 and a1 = 1.9671 were obtained. According to this figure the overall trend of the corrugated circular flume, i.e. the experimental data of Ontkean and Healy [6], is different from the data points of the smooth bed circular flumes. 3.2. Effect of Dc/D on the stage-discharge formula In the previous studies [1,3–5], the experimental data of Hager [3] and Samani et al. [4], characterized by 0.632 ≤ Dc / D ≤ 0.774 were considered to find the stage-discharge formulas. Consequently, due to a narrow range of the contraction ratio, the effect of Dc / D could not be distinguished, and therefore the stage-discharge relationship of the circular flume was considered to be independent of the contraction ratio. Eq. (15) proposed in this study, can be used to consider the effect of the parameter Dc / D. Using the data by Hager [3], Samani et al. [4], and those obtained in this study, bo = 0.8195, b1 = 2.3908 and b2 = 1.3349 were obtained. Note that Eq. (15) is applicable for smooth bed circular flumes and in the wide range 0.342 ≤ Dc / D ≤ 0.774. Fig. 5 shows the comparison between measured Π1 values and those calculated by Eq. (15). The cumulative frequency distributions of the relative errors associated with different design methods were calculated and plotted in Fig. 6. For a given relative error value, the cumulative frequency indicates the percentage of the data points with a relative error less than or equal to the selected value. According to this figure, detailed error analyses of different stage-discharge formulas were extracted and listed in Table 3. Fig. 6 and Table 3 show that, 64.2% of the data points of the
3.3. Effect of the slope on the stage-discharge equation The effects of bed roughness and bed slope are not included in Eq. (10). Consequently, the constant parameters of b0, b1, and b2 appeared in Eq. (15) may differ from that of being smooth and level. In the following sections the effects of corrugated pipes and non-level conditions are investigated. 81
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Fig. 11. Observed and calculated values of Π1 versus h/Dc along with the associated relative errors for the case of r = 0.342 with positive and negative bed slopes.
is obtained:
Table 5 Mean absolute relative errors associated with Eq. (15) for different flume bed slopes. Dc/D
Slope (%)
Mean absolute relative error (%)
0.421
0.2 1.11 −0.5 −0.85 −1.24 0.187 0.6 1.375 −0.46 −1.25 −1.75
2.83 2.78 4.87 2.87 8.62 8.27 6.43 8.6 4.3 5.25 2.07
0.342
2.1229
g1/2
⎟
s−0.03287
(17)
3.3.2. Smooth bed circular flume The experimental data obtained in this study were employed to evaluate the effect of both positive and negative flume bed slopes on the stage-discharge relationship of the circular flume. In this regard the applicability of Eq. (15) was examined to determine the discharge through the non-level circular flume. Observed and calculated Π1-values were plotted versus h / Dc in Figs. 10 and 11 for r = 0.421 and r = 0.342 respectively. Although for r = 0.342 with positive bed slopes Eq. (15) underestimates the discharge values, this equation can be used for the non-level circular flumes with acceptable accuracy. To better identify the effect of the longitudinal slopes on the stage-
2.1226
⎜
⎜
in which s = longitudinal slope expressed in percentage (%). Comparison between measured Π1 values and those calculated by Eq. (17) is plotted in Fig. 9. Also, Mean absolute relative errors and percentages of the data points with the relative errors of less than ± 10% and ± 15% for Eqs. (16) and (17) are listed in Table 4. According to Table 4 it was revealed that the mean absolute relative error of Eq. (17) is slightly less than Eq. (16). Consequently, no significant improvement was observed by introducing the flume slope in the stage-discharge formula. Note that the experimental data of Ontkean and Healy [6] are characterized by very limited range of Π1-values, i.e. Π1 < 0.12., and, consequently, the effect of the flume slope for wider ranges of Π1 should be investigated.
3.3.1. Corrugated circular flume The experimental data collected by Ontkean and Healy [6] were used to study the effect of the slope installation on the corrugated circular flume. Using the data points with the flume slopes of +1%, +3%, +5%, and +7%, for Dc / D = 0.7, the pairs (Π1, h / Dc) were plotted in Fig. 8 and the following stage-discharge formula was obtained:
Q h = 0.5062 ⎛ ⎞ g1/2 Dc 5/2 D ⎝ c⎠
Q h = 0.4523 ⎛ ⎞ Dc 5/2 ⎝ Dc ⎠
⎟
(16)
Using the experimental data of Ontkean and Healy [6], and taking into account the slope effect on the stage discharge relationship, the following equation (by employing LINEST function in Microsoft Excel) 82
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literature, is more accurate than the equation proposed by Ferro [5] and that recently applied by Samani [1] for the circular flumes. Using of the experimental data collected by Ontkean and Healy [6] allowed to demonstrate the effect of the flume slope on the stage-discharge equation for the corrugated pipes. No significant improvement was observed by introducing the bed slope in the stage-discharge formula. Experimental data obtained in this study were employed to study the effect of both positive and negative longitudinal slopes on the stagedischarge formula of the smooth bed circular flume. Accordingly, it was suggested to use the proposed stage-discharge formula only within the bed slope range of ± 1%. However, more experimental studies should be carried out to illustrate the effects of the flume slope and bed roughness on the proposed stage-discharge equation.
discharge curve, mean absolute relative errors associated with Eq. (15) were calculated for different longitudinal slopes and the results were listed in Table 5. The maximum value of the mean absolute relative error associated with the positive longitudinal slopes was observed for Dc / D= 0.342 and s = 1.375%. For the negative longitudinal slope, however, the highest mean absolute relative error was observed for Dc / D = 0.421 and s = −1.24%. Accordingly, Eq. (15) is suggested to be used only for the longitudinal bed slopes, within the range of ± 1%. According to the results of this study, in order to measure the flow rates in the field, the following points should be considered: 1. The flume must work in a free flow condition. It is advisable to install the flume at the end of the canal/pipe where a free overfall occurs downstream the measuring flume; 2. It is strongly suggested to construct the circular flume according to the following advises: a) The vertical pipe should be installed at a distance of 0.5 m from the downstream end. b) A distant of at least 1.5 m of the pipe having no flinches, bends, and any other obstacles should be considered at the flume inlet.
Acknowledgements This work was supported by the Center for International Scientific Studies & Collaboration, CISSC under the Contract number of 1663. The authors appreciate also the assists of Mr. Qolamreza Babaei, the hydraulic lab supervisor, for his efforts to construct the experimental setup.
Note that the circular flumes tested in this study are constructed based on the above dimensions.
References 3. Non-level conditions with the bed slopes out of the range of ± 1% should be avoided; 4. For the smooth beds Eq. (15) can be used in the range of 0.342 ≤ Dc / D ≤ 0.774 with the mean absolute relative error of less than 8.8%; 5. For the corrugated pipes, Eq. (17) can be used only for Dc / D = 0.7 with the mean absolute relative error of 11.06%. Experiments should be carried out to extend the applicability range of the circular flumes for the corrugated pipes.
[1] Z. Samani, Three simple flumes for flow measurement in open channels, J. Irrig. Drain. Eng. 2 (2017) 2–5. [2] W.H. Hager, Modified venturi channel, J. Irrig. Drain. Eng. 111 (1) (1985) 19–35. [3] W.H. Hager, Mobile flume for circular channel, J. Irrig. Drain. Eng. 114 (3) (1988) 520–534. [4] Z. Samani, S. Jorat, M. Yousef, Hydraulic characteristics of circular flume, J. Irrig. Drain. Eng. ASCE 117 (4) (1991) 558–556. [5] V. Ferro, Discussion of “simple flume for flow measurement in open channel” by Zohrab Samani and Henry Magallanez, J. Irrig. Drain. Eng. ASCE 128 (2) (2002) 129–132. [6] G.R. Ontkean, L.H. Healy, Impact of non – level operation of a circular flume on discharge measurements, in: Proceedings of the CSBE/SCGAB 2015 Annual Conference, Edmonton, Alberta, 2015. [7] I.M.H. Rashwan, M.I. Idress, Evaluation efficiency for mobile as discharge measurement device for partially filled circular channel, Ain Shams Eng. J. 4 (2013) 199–206. [8] V. Ferro, Discussion of “three simple flumes for flow measurement in open channels” by Zohrab Samani, J. Irrig. Drain. Eng. ASCE (2018) (In press). [9] G.I. Barenblatt, Similarity, Self-Similarity and Intermediate Asymptotics, Consultans Bureau, New York, 1979. [10] G.I. Barenblatt, Dimensional Analysis, Gordon and Breach Science, Amsterdam, The Netherlands, 1987.
4. Conclusion The developed analysis carried out by the Buckingham's Theorem of dimensional analysis and self-similarity theory showed that the contraction ratio would significantly affect the stage-discharge formula of a circular flume. New experimental program was conducted in this study to investigate the effect of both contraction ratio and longitudinal bed slope on the performance of the circular flume. The theoretically deduced stage-discharge relationship calibrated by the experimental data obtained in this study and those available in the
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