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Discharge coefficient of a semi-elliptical side weir in subcritical flow
Flow Measurement and Instrumentation 22 (2011) 25–32
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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Discharge coefficient of a semi-elliptical side weir in subcritical flow Nihat Kaya a , M. Emin Emiroglu a,∗ , Hayrullah Agaccioglu b a
Department of Civil Engineering, Firat University, 23119, Elazig, Turkey
b
Department of Civil Engineering, Yildiz Technical University, 34210, Esenler, Istanbul, Turkey
article
info
Article history: Received 28 October 2009 Received in revised form 30 October 2010 Accepted 4 November 2010 Keywords: Side weir Discharge coefficient Intake Labyrinth Semi-elliptical Channel flow
abstract A labyrinth weir is an overflow weir, folded in plan view to provide a longer total effective length for a given overall weir width. The total length of the labyrinth weir is typically three to five times the weir width. In this study, a semi-elliptical labyrinth weir was used as a side weir structure. Rectangular side weirs have attracted considerable research interest. The same, however, is not true for labyrinth side weirs. The present study investigated the hydraulic effects of semi-elliptical side weirs in order to increase their discharge capacity. To estimate the outflow over a semi-elliptical side weir, the discharge coefficient in the side weir equation needs to be determined. A comprehensive laboratory study including 677 tests was conducted to determine the discharge coefficient of the semi-elliptical side weir. The results were analyzed to find the influence of the dimensionless weir length L/B, the dimensionless effective length L/ℓ, the dimensionless weir height p/h1 , the dimensionless ellipse radius b/a, and upstream Froude number F1 on the discharge coefficient. It was found that the discharge coefficient of semi-elliptical side weirs is higher than that of classical side weirs. Additionally, a reliable equation for calculating the discharge coefficient of semi-elliptical side weirs is presented. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Side weirs have been extensively used in hydraulic and environmental engineering applications. They are substantial parts of the distribution channel in irrigation systems and treatment units. In combined sewer systems suitable locations are selected where, during storms, the approach flow can be partitioned so that only a small fraction of water remains in the system and continues towards the sewage treatment station. The remainder is either discharged into a rainwater storage basin or fed directly into a receiving water body. To achieve this, two structures have been favored in recent decades: bottom openings and side weirs. The bottom opening is particularly suitable when the upstream flow is supercritical, whereas the side weir is usually adopted with subcritical upstream flow. The required water for irrigation can also be obtained from any irrigation channel by using a side weir. A side weir is an overflow weir formed at the side of a channel, which allows lateral flow of the water when the surface of the water in the channel rises above the weir crest. The ability to predict the flow that is diverted in this way is useful in the design of diversion structures and in flood alleviation works. A review of previous studies indicated that rectangular sharpcrested side weirs have been investigated extensively, including
∗
Corresponding author. Tel.: +90 4242370000x5441; fax: +90 4242415526. E-mail addresses: [email protected] (N. Kaya), [email protected], [email protected] (M.E. Emiroglu), [email protected] (H. Agaccioglu). 0955-5986/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.flowmeasinst.2010.11.002
work by Ackers [1], Collings [2], Frazer [3], Subramanya and Awasthy [4], El-Khashab and Smith [5], Uyumaz and Muslu [6], Helweg [7] and, Agaccioglu and Yüksel [8]. Borghei et al. [9] studied the discharge coefficient for sharp-crested side weirs in subcritical flow, and developed an equation for the discharge coefficient of sharp-crested rectangular side weirs. Also, in order to study the variation of the discharge coefficient along side weirs, Swamee et al. [10] used an elementary analysis approach to estimate the discharge in smooth side weirs through an elementary strip along the side weir. The hydraulic behavior and the discharge coefficient of different types of weirs have been studied by many researchers, including: Nandesomoorthy and Thomson [11], Singh et al. [12], Yu-tech [13], Cheong [14], and others. Ranga Raju et al. [15] investigated the discharge coefficient of a broad-crested rectangular side weir, based on the width of the main channel, Froude number and head/weir width ratio. Kumar and Pathak [16] investigated the discharge coefficient of sharp and broad-crested triangular side weirs. Ghodsian [17] studied supercritical flow in rectangular side weirs. Khorchani and Blanpain [18] investigated flow over side weirs using video monitoring techniques. Coşar and Agaccioglu [19] studied the discharge coefficient of a triangular side weir both on straight and curved channels. Yüksel [20] modeled the effect on flow of changes in specific energy height along a side weir. Aghayari et al. [21] investigated experimentally the effect of height, width and side weir crest slope on the spatial discharge coefficient over broad-crested inclined side weirs under subcritical flow conditions in a rectangular channel.
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N. Kaya et al. / Flow Measurement and Instrumentation 22 (2011) 25–32
Nomenclature a b B Cd e E F1 g h0 h1 h2 L
ℓ p P Q0 Q1 Q2 Qw q dQ /ds R s V V1 V2 Vs
θ ψ
Major radius of ellipse, m Minor radius of ellipse, m Width of channel, m Side weir discharge coefficient (De Marchi coefficient) Eccentricity of ellipse, m Specific energy in the main channel, m Froude number at upstream end of side weir, Acceleration due to gravity, m/s2 Main channel flow depth, m Flow depth at upstream end of side weir at channel center, m Flow depth at downstream end of side weir at channel center, m Width of side weir, m Weir crest length (overflow length), m Height of weir crest, m Perimeter of ellipse, m Discharge at the upstream of the side weir in the main channel, m3 /s Discharge at the upstream end of the side weir in the main channel, m3 /s Discharge at the downstream end of the side weir in the main channel, m3 /s Total discharge overflow of the side weir, m3 /s Discharge per unit length over side weir, m2 /s Discharge per unit length of side weir, m2 /s correlation coefficient Distance along side weir measured from upstream end of side weir, m mean velocity in any section of channel, m/s Mean velocity of flow at the upstream end of side weir, m/s Mean velocity of flow at the downstream of sideweir, m/s Velocity of flow dQs over the brink, m/s Labyrinth side weir included angle, ° Deviation angle of flow, °
The flow over a side weir falls within the category of spatially varied flow. The existing studies deal mainly with the application of the energy principle in the analysis of side weir flow. The concept of constant specific energy [22] is often adopted for studying the flow characteristics of these weirs [23,12,14,16,15,4]. There is considerable interest, particularly in rectangular side weirs. De Marchi [22] was one of the first researchers to provide equations for flow over side weirs. Considering the discharge dQ through an elementary strip of length ds along the side weir in a rectangular main channel as a De Marchi equation, one gets q=−
dQ ds
=
2 3
Cd
2g [h − p]3/2
(1)
where Q is the discharge in the main channel, s is the distance from the beginning of the side weir, dQ /ds (or q) is the spill discharge per unit length of the side opening, g is acceleration due to gravity, p is the crest height of the side weir, h is the depth of flow measured from the channel bottom along the channel centerline, and Cd is the discharge coefficient (De Marchi coefficient) of the side weir. Thus, the side weir discharge equation can be written as: Qw =
2 3
Cd L 2g [h − p]3/2
(2)
in which total flow over side weir Qw is in m3 /s, the discharge coefficient Cd is dimensionless, the width of side weir L is in meters, and h and p are in meters (see Fig. 1). This equation is usually used for flow over flat, broad-crested, quarter-round, half-round, and nappe (ogee) profile weirs. Some of the proposed formulas for the discharge coefficient of the rectangular side weirs (see Fig. 1(a), (b)) are as follows: Cd = 0.864
1 − F21
0.5
Cd = 0.81 − 0.6F1 Cd = 0.485
Subramanya and Awasthy [4]
(3)
Ranga Raju et al. [15]
(4)
for p = 0 Hager [24]
(5)
2 + F21
2+ 2+
F21 F21
0.5
Cd = 0.7 − 0.48F1 − 0.3
p h1
+ 0.06
L B
Borghei et al. [9].
(6)
Most of presented equations for Cd depend on the Froude number. Most researchers have concentrated on investigating rectangular and triangular side weirs in straight channels. Kumar and Pathak [16] investigated the variation of discharge coefficient for a sharp-crested triangular side weir having 60°, 90°, and 120° apex angles, and presented equations depending only on the Froude number and the apex angle. Hager [25] studied supercritical flow in circular-shaped side weirs. Oliveto et al. [26] studied the hydraulic characteristics of side weirs in circular channels when flow along the side weir is supercritical. As mentioned above, the most common type of side weir is rectangular. Moreover, triangular and circular types are also used in hydraulic and environmental engineering applications. Emiroglu et al. [23] studied the discharge coefficient of sharpcrested triangular labyrinth side weirs on a straight channel. Dimensionless parameters for triangular labyrinth side weir discharge coefficient on a straight channel given by them are Cd = f (F1 , L/B, L/ℓ, p/h1 , θ , ψ)
(7)
in which, F1 is the upstream Froude number at the beginning of the side weir in the main channel, Cd is the discharge coefficient (De Marchi coefficient), p is the crest height of the side weir, L is the width (length) of the side weir; B is the width of the main channel; ℓ is the overflow length of the side weir, h1 is the depth of flow at the upstream end of the side weir in the main channel centerline. The dimensionless parameters were explained as follows: L/B is the dimensionless weir length (width), L/ℓ is the dimensionless effective side weir length, p/h1 is the dimensionless weir crest height, θ is the included angle of the triangular labyrinth side weir. The water nape deviation or deflection angle ψ is defined as the deflection of the side weir nape from the water surface toward the weir side, and is given as follows [4]:
sin ψ =
1−
V1 Vs
2 (8)
in which, Vs is velocity of flow dQs over the brink. According to Eq. (7), ψ takes different values for each fluid particle and varies with the Froude number, which changes along the side weir due to spilling over the side weir. The deviation angle increases towards the weir side when the Froude number in the main channel decreases towards the downstream direction. El Khashab [5] also mentioned that the dimensionless length of the side weir (L/B) includes the effect of the deviation angle on the discharge coefficient. Therefore, the deviation angle ψ is not included in the side weir discharge coefficient equations in the literature. Emiroglu et al. [23] obtained the following results regarding the labyrinth side weir discharge coefficient:
N. Kaya et al. / Flow Measurement and Instrumentation 22 (2011) 25–32
a
c
b
d
27
Fig. 1. Definition sketch of subcritical flow over a rectangular and a semi-elliptical side weir. (a) Longitudinal cross-section for the rectangular side weir, (b) Plan for the rectangular side weir, (c) Longitudinal cross-section for the semi-elliptical side weir, (d) Plan for the semi-elliptical side weir.
1. Discharge coefficient of the labyrinth side weir is 1.5–4.5 times higher than that of the rectangular side weir. 2. The discharge coefficient Cd increases when the L/B ratio increases. A decrease in the labyrinth weir included angle θ causes a considerable increase in Cd , due to increasing the overflow length. The labyrinth side weir with θ = 45° has the greatest Cd values among the weir included angles that were tested. 3. The following proposed equation for Cd , the De Marchi coefficient, for subcritical flow can be used reliably:
18.6 − 23.535
Cd =
− 0.502
4.024
p h1
0.012 L
B
+ 6.769
0.112 L
Note that when a = b, the ellipse is a circle. If h is equal to zero, then the perimeter becomes 2π a. The Ramanujan formula is
ℓ
P ∼ = π 3(a + b) −
−1.431 + 0.094. sin θ − 0.393F21.155
. (9)
An ellipse is the finite or bounded case of a conic section, the geometric shape that results from cutting a circular conical or cylindrical surface with an oblique plane. It is also the locus of all points of the plane whose distances to two fixed points add to the same constant. By using an appropriate coordinate system, the ellipse can be described by the canonical implicit equation x2 a2
+
y2 b2
=1
(10)
where, (x, y) are the point coordinates in the canonical system. In this system, the center is the origin (0, 0) and the foci are (−ea, 0) and (+ea, 0). a is called the major radius, and b is the minor radius. The quantity e = (1 − b2 /a2 ) is the eccentricity of the ellipse (Fig. 2). The perimeter of an ellipse can be calculated by using Eq. (13) or Eq. (14). Eq. (13) is an ‘‘infinite sum’’ formula. Eq. (14) was developed by the Indian mathematician Ramanujan.
( a − b) ( a + b) 2 2
h=
(11)
P = π (a + b)
2 ∞ − 0.5 n =0
n
hn
(12)
where, P is the perimeter of the ellipse. Eq. (11) expands to this series of calculations
P = π (a + b) 1 +
+
1 256
3
h +
1 4
h+
1 16 384
1 64
h2
h + ··· . 4
Fig. 2. The ellipse and some of its mathematical properties.
(13)
(3a + b)(a + 3b) √ = π (a + b) 3 − 4 − h .
(14)
A semi-elliptical side weir is defined as a weir crest that is not straight in planform. The increased sill length provided by the semi-elliptical side weirs effectively reduces upstream head to the particular discharge. They can therefore be used to particular advantage where the width of a channel is restricted and a weir is required to pass a range of discharges with a limited variation in upstream water level. To the best of our knowledge, no previous work has reported on semi-elliptical side weirs. This paper investigates the discharge coefficient of semi-elliptical side weirs for a subcritical flow regime and, in particular, the effect of the Froude number F1 , the dimensionless weir crest height p/h1 , the dimensionless weir width L/B, the dimensionless effective side weir length L/ℓ and, the dimensionless ellipse radius b/a on the discharge coefficient. 2. Experimental set-up and experiments Semi-elliptical side weir experiments were conducted at the Hydraulic Laboratory of Firat University, Elazig, Turkey. A schematic representation of the experimental set-up is shown in Fig. 3. The experimental set-up consisted of a main channel and a discharge collection channel. The main channel was 12 m long and the bed had a rectangular cross-section. The main channel was 0.50 m wide, 0.50 m deep, with a 0.001 bed slope. The channel was constructed from a smooth, horizontal, well-painted steel bed with vertical glass sidewalls. A sluice gate was fitted at the end of the main channel in order to control the depth of flow. The collection channel was 0.50 m wide and 0.70 m deep, and was situated parallel to the main channel. The width of the collection channel across the side weir was 1.3 m and constructed in a circular shape, to provide free overflow conditions. A rectangular weir was placed at the end of the collection channel, in order to measure
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N. Kaya et al. / Flow Measurement and Instrumentation 22 (2011) 25–32
Fig. 3. Experimental arrangement.
the discharge of the side weir. A Mitutoyo digital point gauge with ±0.01 mm sensitivity was fixed at a location 0.40 m from the weir. Semi-elliptical side weirs were produced from steel plates, which had sharp edges and were fully aerated. These were installed flush with the main channel wall. Water was supplied to the main channel, through a supply pipe, from a sump and the flow was controlled by a gate valve. The discharge was measured to an accuracy of ±0.01 L/s, by means of a Siemens electromagnetic flow-meter installed in the supply line. The results were compared by a calibrated 90° V-notched weir. The overflow rate was measured by a calibrated standard rectangular weir, located at the downstream end of the collection channel. Water depth measurements were conducted using the digital point gauges at the side weir region, along the channel centerline and the weir-side of the main. Water surface measurements were made by using a special type of measurement car, which can move in both directions (i.e., x and y) on a rail (see Fig. 3). Experiments were conducted at subcritical flow, stable flow conditions and free overflow conditions. Coleman and Smith [27] stated that minimum nape height over side weirs should not be less than 19 mm because of the surface tension over the weir crest. Therefore, the minimum nape height is taken into account as 20 mm. The experiments were conducted for the widths of the semi-elliptical side weir (L = 0.25, 0.50 and 0.75 m), heights of the weir (p = 0.12, 0.16 and 0.20 m), the major radius of the ellipse (a = 0.125, 0.250 m and 0.375 m) and the ratios of b/a (0.5, 1.0, and 1.5). L is equal to 2a (see Fig. 4). ℓ is the developed crest length of the semi-elliptical side weir which is computed with Eqs. (13) and (14) or. In other words, ℓ is equal to P /2. The notations and location of the semi-elliptical side weir and are given in Figs. 3 and 4, and the range of test variables is shown in Table 1. It should be noted that the experiments were conducted for a semi-elliptical side weir with one cycle, as shown in Fig. 4. Upon completion of a sufficient physical description of the semielliptical side weir flow at the straight rectangular channel, the flow rates were tested for different Froude numbers, different p/h1 ratios, different L/B ratios, different L/ℓ ratios and different b/a
Fig. 4. Plan view of the semi-elliptical side weir located on the main channel.
ratios, in order to obtain the variation of the discharge coefficient. Hence, a total of 677 test runs were performed in the current study to determine the discharge coefficient. 3. Experimental results and analysis Experiments in this study were conducted to determine the discharge coefficient of semi-elliptical side weirs. The discharge coefficient was computed using the De Marchi equation (Eq. (2)). Dimensional analysis yields the following dimensionless parameters for the semi-elliptical side weir discharge coefficient as:
Cd = f
F1 ,
p b , , , ,ψ . B ℓ h1 a L L
(15)
As previously pointed out, deflection of the water nape or deviation angle ψ varies along the side weir and takes different values for each fluid particle, depending on the Froude number, which changes along the side weirs due to lateral flow. Subramanya and Awasty [4], El-Khashab [28], El Khashab and Smith [5], Agaccioglu and Yüksel [8] and Borghei et al. [9] also indicated that L/B includes the effects of ψ on Cd . Moreover, the deviation angle ψ
Table 1 The range of test variables. L (m)
a (m)
b (m)
ℓ = P /2 (m)
0.25 0.25 0.25
0.125 0.125 0.125
0.0625 0.1250 0.1875
0.302764 0.392699 0.495795
L/B (–)
b/a (–)
L/ℓ (–)
p/h1 (–)
Q0 (L/s)
F1 (–)
0.12, 0.16 and 0.20
0.50
0.5 1.0 1.5
0.8257 0.6366 0.5042
0.4–0.91
10–145
0.07–0.80
0.50 0.50 0.50
0.250 0.250 0.250
0.1250 0.2500 0.3750
0.605528 0.785398 0.991590
0.12, 0.16 and 0.20
1.00
0.5 1.0 1.5
0.8257 0.6366 0.5042
0.4–0.91
10–145
0.07–0.80
0.75 0.75 0.75
0.375 0.375 0.375
0.1875 0.3750 0.5625
0.908292 1.178097 1.487385
0.12, 0.16 and 0.20
1.50
0.5 1.0 1.5
0.8257 0.6366 0.5042
0.4–0.91
10–145
0.07–0.80
p (m)
N. Kaya et al. / Flow Measurement and Instrumentation 22 (2011) 25–32
29
a
b
Fig. 5. Cd versus different p/h1 values for L/B = 0.50 and L/B = 1.50.
c Fig. 6. Side weir coefficient (Cd ) versus F1 together with different dimensionless weir heights (p/h1 ) for b/a = 0.50 and L/B = 0.50.
was not included in Eqs. (3)–(6). Therefore, the present study also excludes the effect of ψ on Cd . Thus, the dimensionless parameters for the semi-elliptical side weir discharge coefficient can be taken as follows:
Cd = f
F1 ,
p b , , , . B ℓ h1 a L L
(16)
To study the effect of parameter p/h1 on the discharge coefficient, the values of Cd are plotted versus p/h1 in Fig. 5. This reveals that Cd increases slightly with an increase in p/h1 for b/a = 0.5 and Cd values corresponding to the same p/h1 values are very different from each other. The scatter of the data is attributed to the effect of the Froude number, together with the other parameters. Therefore, the effect of p/h1 on Cd is investigated in more detail for all ranges of test variables. Cd is plotted against F1 , together with different dimensionless weir heights (p/h1 ) for the dimensionless ellipse radius b/a = 0.50 and the dimensionless weir length L/B = 0.50 in Fig. 6. The effect of p/h1 on the discharge coefficient is very significant for the same Froude number in all the side weir dimensions, and the Cd value increases with an increase in p/h1 values. The effect of p/h1 on Cd can be explained by the existence of a discontinuity region. This discontinuity region has a strong secondary motion next to the boundary of the weir-side. The intensity of this secondary motion
Fig. 7. Cd for different F1 values together with b/a and L/B ratios (a) L/B = 0.5; (b) L/B = 1.0; and L/B = 1.5.
next to the boundary depends on the crest height of the side weir and decreases with the crest height increase of side weir, due to the friction of the weir surface. Fig. 7(a)–(c) shows the values of Cd , plotted against F1 , together with the different b/a and L/B values. Fig. 7 was plotted for p = 20 cm. Moreover, Fig. 7(a)–(c) shows also the Cd values of the
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N. Kaya et al. / Flow Measurement and Instrumentation 22 (2011) 25–32
rectangular side weir. The discharge coefficients of semi-elliptical side weirs have much higher values than those of rectangular side weirs. In other words, the discharge over semi-elliptical labyrinth side weirs is greater than that over an equivalent straight weir in the ratio of the weir sill lengths. In particular, the semi-elliptical side weir with b/a = 1.50 has higher Cd values. The overflow length of the semi-elliptical side weir is always longer than that of a classical rectangular side weir. The main reason is the higher length of the crest and the more severe secondary flow in semielliptical side weirs. In other words, the intensity of secondary motion created by lateral flow increases with increasing overflow length. An increase in the secondary flow results in the growth of the deviation angle and the kinetic energy towards the side weir as the relative side weir length increases. The variation of the discharge coefficient of the side weir located on the straight channel was investigated for different L/B ratios in Fig. 8(a)–(c). It can be seen that, as the L/B ratio increases, the Cd values also increase. In other words, higher Cd values are obtained at high L/B ratios due to an increase at the intensity of secondary flow created by lateral flow. El-Khashab and Smith [5] pointed out that the secondary flow condition due to lateral flow is dominant when a side weir is relatively long (i.e., L/B > 1). Empirical correlations to predict discharge coefficient Cd were developed for semi-elliptical side weirs, according to the result of dimensionless analysis. The resulting correlation is given in Eq. (17).
−6
Cd = 5 × 10
+ 0.003 + 3.345
854 F01.003
1.086 − 0.
−0.195 L
ℓ 0.346 b a
+ 0.037
+ 4.586 p
L
B
1.482
h1
6.662 + 0.607
−9.058 b a
N 1 − RMSE = (Yi observed − Yi estimated )2 N i=1
N 1 −
N i =1
b
0.007
(17)
where the Froude number F1 is dimensionless, the discharge coefficient Cd is dimensionless, the crest height of the side weir p, the width (length) of the side weir L, the width of the main channel B, the overflow length of side weir ℓ, and the depth of flow at the upstream end of the side weir in the main channel centerline h1 are in meters. Eq. (17) is valid for 0.50 ≤ b/a ≤ 1.50. It should be noted that the discharge coefficient of the semi-elliptical side weir is calculated for a single cycle. The Cd that is calculated is substituted into Eq. (2). The depth of flow (h) in Eq. (2) is taken as the flow depth at the upstream end of side weir at channel center (h1 ). Hence, the total flow over the side weir (Qw ) can easily be obtained from Eqs. (2) and (17). To evaluate the accuracy of the estimates produced using nonlinear equations, the root mean square errors (RMSE), mean absolute errors (MAE) and correlation coefficient (R) criteria were used. The R shows the degree to which two variables are linearly related. Different types of information about the predictive capabilities of the estimated nonlinear equation are measured through RMSE and MAE. The RMSE indicates the goodness of the fit related to high discharge coefficient values, whereas the MAE measures a more balanced perspective of the goodness of the fit at moderate discharge coefficients [29]. The RMSE and MAE are defined as:
MAE =
a
|Yi observed − Yi estimated |
(18)
(19)
in which N is the number of data set, Yi is the discharge coefficient.
c
Fig. 8. Cd versus different F1 values for L/B = 0.50, 1.00, and 1.50.
The root mean square errors (RMSE), the mean absolute errors (MAE) and correlation coefficient (R) values for Eq. (17) are 0.07638, 0.0586 and 0.916, respectively. Good agreements between the measured values and the values computed from the predictive equation are obtained. Thus, an accurate equation for the coefficient of discharge of the semi-elliptical side weirs in subcritical flow conditions is introduced.
N. Kaya et al. / Flow Measurement and Instrumentation 22 (2011) 25–32
31
a
Fig. 9. Comparison of the Cd values of the semi elliptical side weir with the values obtained by the equations of Subramanya and Awasty [4], Ranga Raju et al. [15], Hager [24] and Borghei et al. [9].
The discharge coefficient values of the semi-elliptical side weir with b/a = 1.50 are compared with those of Subramanya and Awasty [4], Ranga Raju et al. [15], Hager [24], and Borghei et al. [9], as shown in Fig. 9. The equations presented by Subramanya and Awasty [4], Ranga Raju et al. [15], Hager [24] and Borghei et al. [9] are for rectangular side weirs. It can be seen in Fig. 9 that the values of Cd for the semi-elliptical side weir with b/a = 1.50 are significantly higher than those of the other studies. The primary reason for the increase in the discharge capacity of the semielliptical side weir can be attributed to the increase in the overflow length of the side weir. A comparison between the results of the present study and those of Emiroglu et al. [23] at L/B = 1.50, L/ℓ ∼ = 0.50 cm, for p = 12, 16 and 20 cm for a straight channel, is presented in Fig. 10(a)–(c). The overflow length ℓ was 150 cm in the study by Emiroglu et al. [23] and 148.74 cm in the present study. As shown in Fig. 10, the values of Cd for the triangular labyrinth side weir are higher than those of the semi elliptical side weir. The differences between the two studies are due to the differing types of side weirs and their different hydraulic behaviors. The intensity of the secondary flow due to lateral flow is more dominant when the weir is a labyrinth type.
b
c
4. Conclusions Laboratory experiments using a semi-elliptical side weir located on a straight channel were carried out, in order to investigate the effect of the dimensionless parameters F1 , p/h1 , L/B, L/ℓ and b/a on the discharge coefficient. An empirical correlation predicting the discharge coefficient of semi-elliptical side weirs was developed. The following conclusions can be drawn based on these findings:
• As a result of dimensional analysis, the following dimensionless parameters are found; Cd = f (F1 , L/B, L/ℓ, p/h1 , b/a). • The discharge coefficient of the semi-elliptical side weir is higher than that of the rectangular side weir, but lower than that of the triangular labyrinth side weir. • The discharge coefficient Cd increases with increasing L/B ratio. • The proposed equation (Eq. (17)) for Cd , the De Marchi coefficient, was shown to be reliable for the subcritical flow conditions in this study. Acknowledgement The authors are grateful to the Scientific and Technological Research Council of Turkey (TUBITAK) for their financial support.
Fig. 10. Comparison of the Cd values of present study with the results from Emiroglu et al. [23].
References [1] Ackers P. A theoretical consideration of side weirs as storm water overflows. Proceedings of the Institution of Civil Engineers (London) 1957;6:250–69. [2] Collings VK. Discharge capacity of side weirs. Proceedings of the Institution of Civil Engineers (London) 1957;6:288–304. [3] Frazer W. The behavior of side weirs in prismatic rectangular channels. Proceedings of the Institution of Civil Engineers (London) 1957;6:305–27.
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[4] Subramanya K, Awasthy SC. Spatially varied flow over side weirs. Journal of the Hydraulics Division-ASCE 1972;98(1):1–10. [5] El-Khashab AMM, Smith KVH. Experimental investigation of flow over side weirs. Journal of the Hydraulics Division-ASCE 1976;102(9):1255–68. [6] Uyumaz A, Muslu Y. Flow over side weir in circular channels. ASCE Journal of Hydraulic Engineering 1985;111(1):144–60. [7] Helweg OJ. Microcomputer applications in water resources. Englewood Cliffs: Prentice-Hall; 1991. [8] Agaccioglu H, Yüksel Y. Side-weir flow in curved channels. ASCE Journal of Irrigation and Drainage Engineering 1998;124(3):163–75. [9] Borghei M, Jalili MR, Ghodsian M. Discharge coefficient for sharp-crested side weir in subcritical flow. ASCE Journal of Hydraulic Engineering 1999;125(10): 1051–6. [10] Swamee PK, Santosh KP, Masoud SA. Side weir analysis using elementary discharge coefficient. ASCE Journal of Irrigation and Drainage Engineering 1994;120(4):742–55. [11] Nandesamoorthy T, Thomson A. Discussion of spatially varied flow over side weir. Journal of the Hydraulics Division-ASCE 1972;98(12):2234–5. [12] Singh R, Manivannan D, Satyanarayana T. Discharge coefficient of rectangular side weirs. ASCE Journal of Irrigation and Drainage Engineering 1994;120(4): 814–9. [13] Yu-Tech L. Discussion of spatially varied flow over side weir. Journal of the Hydraulics Division-ASCE 1972;98(11):2046–8. [14] Cheong HF. Discharge coefficient of lateral diversion from trapezoidal channel. ASCE Journal of Irrigation and Drainage Engineering 1991;117(4):321–33. [15] Ranga Raju KG, Prasad B, Grupta SK. Side weir in rectangular channel. Journal of the Hydraulics Division-ASCE 1979;105(5):547–54. [16] Kumar CP, Pathak SK. Triangular side weirs. ASCE Journal of Irrigation and Drainage Engineering 1987;113(1):98–105.
[17] Ghodsian M. Supercritical flow over rectangular side weir. Canadian Journal of Civil Engineering 2003;30(3):596–600. [18] Khorchani M, Blanpain O. Free surface measurement of flow over side weirs using the video monitoring concept. Flow Measurement and Instrumentation 2004;15:111–7. [19] Coşar A, Agaccioglu H. Discharge coefficient of a triangular side weir located on a curved channel. ASCE Journal of Irrigation and Drainage Engineering 2004; 130(5):321–33. [20] Yüksel E. Effect of specific energy variation on lateral overflows. Flow Measurement and Instrumentation 2004;15:259–69. [21] Aghayari F, Honar T, Keshavarzi A. A Study of spatial variation of discharge efficient in broad-crested inclined side weirs. Irrigation and Drainage 2009; 58:246–54. [22] De Marchi G. Essay on the performance of lateral weirs. L’ Energia Electtrica Milan 1934;11(11):849–60 [in Italian]. [23] Emiroglu ME, Kaya N, Agaccioglu H. Discharge capacity of labyrinth sideweir located on a straight channel. ASCE Journal of Irrigation and Drainage Engineering 2010;136(1):37–46. [24] Hager WH. Lateral outflow over side weirs. ASCE Journal of Hydraulic Engineering 1987;113(4):491–504. [25] Hager WH. Supercritical flow in circular-shaped side weirs. ASCE Journal of Irrigation and Drainage Engineering 1994;120(1):1–12. [26] Oliveto G, Biggiero V, Fiorentino M. Hydraulic features of supercritical flow along prismatic side weirs. Journal of Hydraulic Research 2001;39(1):73–82. [27] Coleman GS, Smith D. The discharging capacity of side weirs. Proceedings of the Institution of Civil Engineers (London) 1923;6:288–304. [28] El-Khashab AMM. Hydraulics of flow over side weirs. Ph.D. thesis. England: University of Southampton; 1975. [29] Karunanithi N, Grenney WJ, Whitley D, Bovee K. Neural networks for river flow prediction. ASCE Journal of Computing in Civil Engineering 1994;8(2):201–20.
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Discharge coefficient of a semi-elliptical side weir in subcritical flow Nihat Kaya a , M. Emin Emiroglu a,∗ , Hayrullah Agaccioglu b a
Department of Civil Engineering, Firat University, 23119, Elazig, Turkey
b
Department of Civil Engineering, Yildiz Technical University, 34210, Esenler, Istanbul, Turkey
article
info
Article history: Received 28 October 2009 Received in revised form 30 October 2010 Accepted 4 November 2010 Keywords: Side weir Discharge coefficient Intake Labyrinth Semi-elliptical Channel flow
abstract A labyrinth weir is an overflow weir, folded in plan view to provide a longer total effective length for a given overall weir width. The total length of the labyrinth weir is typically three to five times the weir width. In this study, a semi-elliptical labyrinth weir was used as a side weir structure. Rectangular side weirs have attracted considerable research interest. The same, however, is not true for labyrinth side weirs. The present study investigated the hydraulic effects of semi-elliptical side weirs in order to increase their discharge capacity. To estimate the outflow over a semi-elliptical side weir, the discharge coefficient in the side weir equation needs to be determined. A comprehensive laboratory study including 677 tests was conducted to determine the discharge coefficient of the semi-elliptical side weir. The results were analyzed to find the influence of the dimensionless weir length L/B, the dimensionless effective length L/ℓ, the dimensionless weir height p/h1 , the dimensionless ellipse radius b/a, and upstream Froude number F1 on the discharge coefficient. It was found that the discharge coefficient of semi-elliptical side weirs is higher than that of classical side weirs. Additionally, a reliable equation for calculating the discharge coefficient of semi-elliptical side weirs is presented. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Side weirs have been extensively used in hydraulic and environmental engineering applications. They are substantial parts of the distribution channel in irrigation systems and treatment units. In combined sewer systems suitable locations are selected where, during storms, the approach flow can be partitioned so that only a small fraction of water remains in the system and continues towards the sewage treatment station. The remainder is either discharged into a rainwater storage basin or fed directly into a receiving water body. To achieve this, two structures have been favored in recent decades: bottom openings and side weirs. The bottom opening is particularly suitable when the upstream flow is supercritical, whereas the side weir is usually adopted with subcritical upstream flow. The required water for irrigation can also be obtained from any irrigation channel by using a side weir. A side weir is an overflow weir formed at the side of a channel, which allows lateral flow of the water when the surface of the water in the channel rises above the weir crest. The ability to predict the flow that is diverted in this way is useful in the design of diversion structures and in flood alleviation works. A review of previous studies indicated that rectangular sharpcrested side weirs have been investigated extensively, including
∗
Corresponding author. Tel.: +90 4242370000x5441; fax: +90 4242415526. E-mail addresses: [email protected] (N. Kaya), [email protected], [email protected] (M.E. Emiroglu), [email protected] (H. Agaccioglu). 0955-5986/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.flowmeasinst.2010.11.002
work by Ackers [1], Collings [2], Frazer [3], Subramanya and Awasthy [4], El-Khashab and Smith [5], Uyumaz and Muslu [6], Helweg [7] and, Agaccioglu and Yüksel [8]. Borghei et al. [9] studied the discharge coefficient for sharp-crested side weirs in subcritical flow, and developed an equation for the discharge coefficient of sharp-crested rectangular side weirs. Also, in order to study the variation of the discharge coefficient along side weirs, Swamee et al. [10] used an elementary analysis approach to estimate the discharge in smooth side weirs through an elementary strip along the side weir. The hydraulic behavior and the discharge coefficient of different types of weirs have been studied by many researchers, including: Nandesomoorthy and Thomson [11], Singh et al. [12], Yu-tech [13], Cheong [14], and others. Ranga Raju et al. [15] investigated the discharge coefficient of a broad-crested rectangular side weir, based on the width of the main channel, Froude number and head/weir width ratio. Kumar and Pathak [16] investigated the discharge coefficient of sharp and broad-crested triangular side weirs. Ghodsian [17] studied supercritical flow in rectangular side weirs. Khorchani and Blanpain [18] investigated flow over side weirs using video monitoring techniques. Coşar and Agaccioglu [19] studied the discharge coefficient of a triangular side weir both on straight and curved channels. Yüksel [20] modeled the effect on flow of changes in specific energy height along a side weir. Aghayari et al. [21] investigated experimentally the effect of height, width and side weir crest slope on the spatial discharge coefficient over broad-crested inclined side weirs under subcritical flow conditions in a rectangular channel.
26
N. Kaya et al. / Flow Measurement and Instrumentation 22 (2011) 25–32
Nomenclature a b B Cd e E F1 g h0 h1 h2 L
ℓ p P Q0 Q1 Q2 Qw q dQ /ds R s V V1 V2 Vs
θ ψ
Major radius of ellipse, m Minor radius of ellipse, m Width of channel, m Side weir discharge coefficient (De Marchi coefficient) Eccentricity of ellipse, m Specific energy in the main channel, m Froude number at upstream end of side weir, Acceleration due to gravity, m/s2 Main channel flow depth, m Flow depth at upstream end of side weir at channel center, m Flow depth at downstream end of side weir at channel center, m Width of side weir, m Weir crest length (overflow length), m Height of weir crest, m Perimeter of ellipse, m Discharge at the upstream of the side weir in the main channel, m3 /s Discharge at the upstream end of the side weir in the main channel, m3 /s Discharge at the downstream end of the side weir in the main channel, m3 /s Total discharge overflow of the side weir, m3 /s Discharge per unit length over side weir, m2 /s Discharge per unit length of side weir, m2 /s correlation coefficient Distance along side weir measured from upstream end of side weir, m mean velocity in any section of channel, m/s Mean velocity of flow at the upstream end of side weir, m/s Mean velocity of flow at the downstream of sideweir, m/s Velocity of flow dQs over the brink, m/s Labyrinth side weir included angle, ° Deviation angle of flow, °
The flow over a side weir falls within the category of spatially varied flow. The existing studies deal mainly with the application of the energy principle in the analysis of side weir flow. The concept of constant specific energy [22] is often adopted for studying the flow characteristics of these weirs [23,12,14,16,15,4]. There is considerable interest, particularly in rectangular side weirs. De Marchi [22] was one of the first researchers to provide equations for flow over side weirs. Considering the discharge dQ through an elementary strip of length ds along the side weir in a rectangular main channel as a De Marchi equation, one gets q=−
dQ ds
=
2 3
Cd
2g [h − p]3/2
(1)
where Q is the discharge in the main channel, s is the distance from the beginning of the side weir, dQ /ds (or q) is the spill discharge per unit length of the side opening, g is acceleration due to gravity, p is the crest height of the side weir, h is the depth of flow measured from the channel bottom along the channel centerline, and Cd is the discharge coefficient (De Marchi coefficient) of the side weir. Thus, the side weir discharge equation can be written as: Qw =
2 3
Cd L 2g [h − p]3/2
(2)
in which total flow over side weir Qw is in m3 /s, the discharge coefficient Cd is dimensionless, the width of side weir L is in meters, and h and p are in meters (see Fig. 1). This equation is usually used for flow over flat, broad-crested, quarter-round, half-round, and nappe (ogee) profile weirs. Some of the proposed formulas for the discharge coefficient of the rectangular side weirs (see Fig. 1(a), (b)) are as follows: Cd = 0.864
1 − F21
0.5
Cd = 0.81 − 0.6F1 Cd = 0.485
Subramanya and Awasthy [4]
(3)
Ranga Raju et al. [15]
(4)
for p = 0 Hager [24]
(5)
2 + F21
2+ 2+
F21 F21
0.5
Cd = 0.7 − 0.48F1 − 0.3
p h1
+ 0.06
L B
Borghei et al. [9].
(6)
Most of presented equations for Cd depend on the Froude number. Most researchers have concentrated on investigating rectangular and triangular side weirs in straight channels. Kumar and Pathak [16] investigated the variation of discharge coefficient for a sharp-crested triangular side weir having 60°, 90°, and 120° apex angles, and presented equations depending only on the Froude number and the apex angle. Hager [25] studied supercritical flow in circular-shaped side weirs. Oliveto et al. [26] studied the hydraulic characteristics of side weirs in circular channels when flow along the side weir is supercritical. As mentioned above, the most common type of side weir is rectangular. Moreover, triangular and circular types are also used in hydraulic and environmental engineering applications. Emiroglu et al. [23] studied the discharge coefficient of sharpcrested triangular labyrinth side weirs on a straight channel. Dimensionless parameters for triangular labyrinth side weir discharge coefficient on a straight channel given by them are Cd = f (F1 , L/B, L/ℓ, p/h1 , θ , ψ)
(7)
in which, F1 is the upstream Froude number at the beginning of the side weir in the main channel, Cd is the discharge coefficient (De Marchi coefficient), p is the crest height of the side weir, L is the width (length) of the side weir; B is the width of the main channel; ℓ is the overflow length of the side weir, h1 is the depth of flow at the upstream end of the side weir in the main channel centerline. The dimensionless parameters were explained as follows: L/B is the dimensionless weir length (width), L/ℓ is the dimensionless effective side weir length, p/h1 is the dimensionless weir crest height, θ is the included angle of the triangular labyrinth side weir. The water nape deviation or deflection angle ψ is defined as the deflection of the side weir nape from the water surface toward the weir side, and is given as follows [4]:
sin ψ =
1−
V1 Vs
2 (8)
in which, Vs is velocity of flow dQs over the brink. According to Eq. (7), ψ takes different values for each fluid particle and varies with the Froude number, which changes along the side weir due to spilling over the side weir. The deviation angle increases towards the weir side when the Froude number in the main channel decreases towards the downstream direction. El Khashab [5] also mentioned that the dimensionless length of the side weir (L/B) includes the effect of the deviation angle on the discharge coefficient. Therefore, the deviation angle ψ is not included in the side weir discharge coefficient equations in the literature. Emiroglu et al. [23] obtained the following results regarding the labyrinth side weir discharge coefficient:
N. Kaya et al. / Flow Measurement and Instrumentation 22 (2011) 25–32
a
c
b
d
27
Fig. 1. Definition sketch of subcritical flow over a rectangular and a semi-elliptical side weir. (a) Longitudinal cross-section for the rectangular side weir, (b) Plan for the rectangular side weir, (c) Longitudinal cross-section for the semi-elliptical side weir, (d) Plan for the semi-elliptical side weir.
1. Discharge coefficient of the labyrinth side weir is 1.5–4.5 times higher than that of the rectangular side weir. 2. The discharge coefficient Cd increases when the L/B ratio increases. A decrease in the labyrinth weir included angle θ causes a considerable increase in Cd , due to increasing the overflow length. The labyrinth side weir with θ = 45° has the greatest Cd values among the weir included angles that were tested. 3. The following proposed equation for Cd , the De Marchi coefficient, for subcritical flow can be used reliably:
18.6 − 23.535
Cd =
− 0.502
4.024
p h1
0.012 L
B
+ 6.769
0.112 L
Note that when a = b, the ellipse is a circle. If h is equal to zero, then the perimeter becomes 2π a. The Ramanujan formula is
ℓ
P ∼ = π 3(a + b) −
−1.431 + 0.094. sin θ − 0.393F21.155
. (9)
An ellipse is the finite or bounded case of a conic section, the geometric shape that results from cutting a circular conical or cylindrical surface with an oblique plane. It is also the locus of all points of the plane whose distances to two fixed points add to the same constant. By using an appropriate coordinate system, the ellipse can be described by the canonical implicit equation x2 a2
+
y2 b2
=1
(10)
where, (x, y) are the point coordinates in the canonical system. In this system, the center is the origin (0, 0) and the foci are (−ea, 0) and (+ea, 0). a is called the major radius, and b is the minor radius. The quantity e = (1 − b2 /a2 ) is the eccentricity of the ellipse (Fig. 2). The perimeter of an ellipse can be calculated by using Eq. (13) or Eq. (14). Eq. (13) is an ‘‘infinite sum’’ formula. Eq. (14) was developed by the Indian mathematician Ramanujan.
( a − b) ( a + b) 2 2
h=
(11)
P = π (a + b)
2 ∞ − 0.5 n =0
n
hn
(12)
where, P is the perimeter of the ellipse. Eq. (11) expands to this series of calculations
P = π (a + b) 1 +
+
1 256
3
h +
1 4
h+
1 16 384
1 64
h2
h + ··· . 4
Fig. 2. The ellipse and some of its mathematical properties.
(13)
(3a + b)(a + 3b) √ = π (a + b) 3 − 4 − h .
(14)
A semi-elliptical side weir is defined as a weir crest that is not straight in planform. The increased sill length provided by the semi-elliptical side weirs effectively reduces upstream head to the particular discharge. They can therefore be used to particular advantage where the width of a channel is restricted and a weir is required to pass a range of discharges with a limited variation in upstream water level. To the best of our knowledge, no previous work has reported on semi-elliptical side weirs. This paper investigates the discharge coefficient of semi-elliptical side weirs for a subcritical flow regime and, in particular, the effect of the Froude number F1 , the dimensionless weir crest height p/h1 , the dimensionless weir width L/B, the dimensionless effective side weir length L/ℓ and, the dimensionless ellipse radius b/a on the discharge coefficient. 2. Experimental set-up and experiments Semi-elliptical side weir experiments were conducted at the Hydraulic Laboratory of Firat University, Elazig, Turkey. A schematic representation of the experimental set-up is shown in Fig. 3. The experimental set-up consisted of a main channel and a discharge collection channel. The main channel was 12 m long and the bed had a rectangular cross-section. The main channel was 0.50 m wide, 0.50 m deep, with a 0.001 bed slope. The channel was constructed from a smooth, horizontal, well-painted steel bed with vertical glass sidewalls. A sluice gate was fitted at the end of the main channel in order to control the depth of flow. The collection channel was 0.50 m wide and 0.70 m deep, and was situated parallel to the main channel. The width of the collection channel across the side weir was 1.3 m and constructed in a circular shape, to provide free overflow conditions. A rectangular weir was placed at the end of the collection channel, in order to measure
28
N. Kaya et al. / Flow Measurement and Instrumentation 22 (2011) 25–32
Fig. 3. Experimental arrangement.
the discharge of the side weir. A Mitutoyo digital point gauge with ±0.01 mm sensitivity was fixed at a location 0.40 m from the weir. Semi-elliptical side weirs were produced from steel plates, which had sharp edges and were fully aerated. These were installed flush with the main channel wall. Water was supplied to the main channel, through a supply pipe, from a sump and the flow was controlled by a gate valve. The discharge was measured to an accuracy of ±0.01 L/s, by means of a Siemens electromagnetic flow-meter installed in the supply line. The results were compared by a calibrated 90° V-notched weir. The overflow rate was measured by a calibrated standard rectangular weir, located at the downstream end of the collection channel. Water depth measurements were conducted using the digital point gauges at the side weir region, along the channel centerline and the weir-side of the main. Water surface measurements were made by using a special type of measurement car, which can move in both directions (i.e., x and y) on a rail (see Fig. 3). Experiments were conducted at subcritical flow, stable flow conditions and free overflow conditions. Coleman and Smith [27] stated that minimum nape height over side weirs should not be less than 19 mm because of the surface tension over the weir crest. Therefore, the minimum nape height is taken into account as 20 mm. The experiments were conducted for the widths of the semi-elliptical side weir (L = 0.25, 0.50 and 0.75 m), heights of the weir (p = 0.12, 0.16 and 0.20 m), the major radius of the ellipse (a = 0.125, 0.250 m and 0.375 m) and the ratios of b/a (0.5, 1.0, and 1.5). L is equal to 2a (see Fig. 4). ℓ is the developed crest length of the semi-elliptical side weir which is computed with Eqs. (13) and (14) or. In other words, ℓ is equal to P /2. The notations and location of the semi-elliptical side weir and are given in Figs. 3 and 4, and the range of test variables is shown in Table 1. It should be noted that the experiments were conducted for a semi-elliptical side weir with one cycle, as shown in Fig. 4. Upon completion of a sufficient physical description of the semielliptical side weir flow at the straight rectangular channel, the flow rates were tested for different Froude numbers, different p/h1 ratios, different L/B ratios, different L/ℓ ratios and different b/a
Fig. 4. Plan view of the semi-elliptical side weir located on the main channel.
ratios, in order to obtain the variation of the discharge coefficient. Hence, a total of 677 test runs were performed in the current study to determine the discharge coefficient. 3. Experimental results and analysis Experiments in this study were conducted to determine the discharge coefficient of semi-elliptical side weirs. The discharge coefficient was computed using the De Marchi equation (Eq. (2)). Dimensional analysis yields the following dimensionless parameters for the semi-elliptical side weir discharge coefficient as:
Cd = f
F1 ,
p b , , , ,ψ . B ℓ h1 a L L
(15)
As previously pointed out, deflection of the water nape or deviation angle ψ varies along the side weir and takes different values for each fluid particle, depending on the Froude number, which changes along the side weirs due to lateral flow. Subramanya and Awasty [4], El-Khashab [28], El Khashab and Smith [5], Agaccioglu and Yüksel [8] and Borghei et al. [9] also indicated that L/B includes the effects of ψ on Cd . Moreover, the deviation angle ψ
Table 1 The range of test variables. L (m)
a (m)
b (m)
ℓ = P /2 (m)
0.25 0.25 0.25
0.125 0.125 0.125
0.0625 0.1250 0.1875
0.302764 0.392699 0.495795
L/B (–)
b/a (–)
L/ℓ (–)
p/h1 (–)
Q0 (L/s)
F1 (–)
0.12, 0.16 and 0.20
0.50
0.5 1.0 1.5
0.8257 0.6366 0.5042
0.4–0.91
10–145
0.07–0.80
0.50 0.50 0.50
0.250 0.250 0.250
0.1250 0.2500 0.3750
0.605528 0.785398 0.991590
0.12, 0.16 and 0.20
1.00
0.5 1.0 1.5
0.8257 0.6366 0.5042
0.4–0.91
10–145
0.07–0.80
0.75 0.75 0.75
0.375 0.375 0.375
0.1875 0.3750 0.5625
0.908292 1.178097 1.487385
0.12, 0.16 and 0.20
1.50
0.5 1.0 1.5
0.8257 0.6366 0.5042
0.4–0.91
10–145
0.07–0.80
p (m)
N. Kaya et al. / Flow Measurement and Instrumentation 22 (2011) 25–32
29
a
b
Fig. 5. Cd versus different p/h1 values for L/B = 0.50 and L/B = 1.50.
c Fig. 6. Side weir coefficient (Cd ) versus F1 together with different dimensionless weir heights (p/h1 ) for b/a = 0.50 and L/B = 0.50.
was not included in Eqs. (3)–(6). Therefore, the present study also excludes the effect of ψ on Cd . Thus, the dimensionless parameters for the semi-elliptical side weir discharge coefficient can be taken as follows:
Cd = f
F1 ,
p b , , , . B ℓ h1 a L L
(16)
To study the effect of parameter p/h1 on the discharge coefficient, the values of Cd are plotted versus p/h1 in Fig. 5. This reveals that Cd increases slightly with an increase in p/h1 for b/a = 0.5 and Cd values corresponding to the same p/h1 values are very different from each other. The scatter of the data is attributed to the effect of the Froude number, together with the other parameters. Therefore, the effect of p/h1 on Cd is investigated in more detail for all ranges of test variables. Cd is plotted against F1 , together with different dimensionless weir heights (p/h1 ) for the dimensionless ellipse radius b/a = 0.50 and the dimensionless weir length L/B = 0.50 in Fig. 6. The effect of p/h1 on the discharge coefficient is very significant for the same Froude number in all the side weir dimensions, and the Cd value increases with an increase in p/h1 values. The effect of p/h1 on Cd can be explained by the existence of a discontinuity region. This discontinuity region has a strong secondary motion next to the boundary of the weir-side. The intensity of this secondary motion
Fig. 7. Cd for different F1 values together with b/a and L/B ratios (a) L/B = 0.5; (b) L/B = 1.0; and L/B = 1.5.
next to the boundary depends on the crest height of the side weir and decreases with the crest height increase of side weir, due to the friction of the weir surface. Fig. 7(a)–(c) shows the values of Cd , plotted against F1 , together with the different b/a and L/B values. Fig. 7 was plotted for p = 20 cm. Moreover, Fig. 7(a)–(c) shows also the Cd values of the
30
N. Kaya et al. / Flow Measurement and Instrumentation 22 (2011) 25–32
rectangular side weir. The discharge coefficients of semi-elliptical side weirs have much higher values than those of rectangular side weirs. In other words, the discharge over semi-elliptical labyrinth side weirs is greater than that over an equivalent straight weir in the ratio of the weir sill lengths. In particular, the semi-elliptical side weir with b/a = 1.50 has higher Cd values. The overflow length of the semi-elliptical side weir is always longer than that of a classical rectangular side weir. The main reason is the higher length of the crest and the more severe secondary flow in semielliptical side weirs. In other words, the intensity of secondary motion created by lateral flow increases with increasing overflow length. An increase in the secondary flow results in the growth of the deviation angle and the kinetic energy towards the side weir as the relative side weir length increases. The variation of the discharge coefficient of the side weir located on the straight channel was investigated for different L/B ratios in Fig. 8(a)–(c). It can be seen that, as the L/B ratio increases, the Cd values also increase. In other words, higher Cd values are obtained at high L/B ratios due to an increase at the intensity of secondary flow created by lateral flow. El-Khashab and Smith [5] pointed out that the secondary flow condition due to lateral flow is dominant when a side weir is relatively long (i.e., L/B > 1). Empirical correlations to predict discharge coefficient Cd were developed for semi-elliptical side weirs, according to the result of dimensionless analysis. The resulting correlation is given in Eq. (17).
−6
Cd = 5 × 10
+ 0.003 + 3.345
854 F01.003
1.086 − 0.
−0.195 L
ℓ 0.346 b a
+ 0.037
+ 4.586 p
L
B
1.482
h1
6.662 + 0.607
−9.058 b a
N 1 − RMSE = (Yi observed − Yi estimated )2 N i=1
N 1 −
N i =1
b
0.007
(17)
where the Froude number F1 is dimensionless, the discharge coefficient Cd is dimensionless, the crest height of the side weir p, the width (length) of the side weir L, the width of the main channel B, the overflow length of side weir ℓ, and the depth of flow at the upstream end of the side weir in the main channel centerline h1 are in meters. Eq. (17) is valid for 0.50 ≤ b/a ≤ 1.50. It should be noted that the discharge coefficient of the semi-elliptical side weir is calculated for a single cycle. The Cd that is calculated is substituted into Eq. (2). The depth of flow (h) in Eq. (2) is taken as the flow depth at the upstream end of side weir at channel center (h1 ). Hence, the total flow over the side weir (Qw ) can easily be obtained from Eqs. (2) and (17). To evaluate the accuracy of the estimates produced using nonlinear equations, the root mean square errors (RMSE), mean absolute errors (MAE) and correlation coefficient (R) criteria were used. The R shows the degree to which two variables are linearly related. Different types of information about the predictive capabilities of the estimated nonlinear equation are measured through RMSE and MAE. The RMSE indicates the goodness of the fit related to high discharge coefficient values, whereas the MAE measures a more balanced perspective of the goodness of the fit at moderate discharge coefficients [29]. The RMSE and MAE are defined as:
MAE =
a
|Yi observed − Yi estimated |
(18)
(19)
in which N is the number of data set, Yi is the discharge coefficient.
c
Fig. 8. Cd versus different F1 values for L/B = 0.50, 1.00, and 1.50.
The root mean square errors (RMSE), the mean absolute errors (MAE) and correlation coefficient (R) values for Eq. (17) are 0.07638, 0.0586 and 0.916, respectively. Good agreements between the measured values and the values computed from the predictive equation are obtained. Thus, an accurate equation for the coefficient of discharge of the semi-elliptical side weirs in subcritical flow conditions is introduced.
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a
Fig. 9. Comparison of the Cd values of the semi elliptical side weir with the values obtained by the equations of Subramanya and Awasty [4], Ranga Raju et al. [15], Hager [24] and Borghei et al. [9].
The discharge coefficient values of the semi-elliptical side weir with b/a = 1.50 are compared with those of Subramanya and Awasty [4], Ranga Raju et al. [15], Hager [24], and Borghei et al. [9], as shown in Fig. 9. The equations presented by Subramanya and Awasty [4], Ranga Raju et al. [15], Hager [24] and Borghei et al. [9] are for rectangular side weirs. It can be seen in Fig. 9 that the values of Cd for the semi-elliptical side weir with b/a = 1.50 are significantly higher than those of the other studies. The primary reason for the increase in the discharge capacity of the semielliptical side weir can be attributed to the increase in the overflow length of the side weir. A comparison between the results of the present study and those of Emiroglu et al. [23] at L/B = 1.50, L/ℓ ∼ = 0.50 cm, for p = 12, 16 and 20 cm for a straight channel, is presented in Fig. 10(a)–(c). The overflow length ℓ was 150 cm in the study by Emiroglu et al. [23] and 148.74 cm in the present study. As shown in Fig. 10, the values of Cd for the triangular labyrinth side weir are higher than those of the semi elliptical side weir. The differences between the two studies are due to the differing types of side weirs and their different hydraulic behaviors. The intensity of the secondary flow due to lateral flow is more dominant when the weir is a labyrinth type.
b
c
4. Conclusions Laboratory experiments using a semi-elliptical side weir located on a straight channel were carried out, in order to investigate the effect of the dimensionless parameters F1 , p/h1 , L/B, L/ℓ and b/a on the discharge coefficient. An empirical correlation predicting the discharge coefficient of semi-elliptical side weirs was developed. The following conclusions can be drawn based on these findings:
• As a result of dimensional analysis, the following dimensionless parameters are found; Cd = f (F1 , L/B, L/ℓ, p/h1 , b/a). • The discharge coefficient of the semi-elliptical side weir is higher than that of the rectangular side weir, but lower than that of the triangular labyrinth side weir. • The discharge coefficient Cd increases with increasing L/B ratio. • The proposed equation (Eq. (17)) for Cd , the De Marchi coefficient, was shown to be reliable for the subcritical flow conditions in this study. Acknowledgement The authors are grateful to the Scientific and Technological Research Council of Turkey (TUBITAK) for their financial support.
Fig. 10. Comparison of the Cd values of present study with the results from Emiroglu et al. [23].
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