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Flow through lateral circular orifice under free and submerged flow conditions
Author’s Accepted Manuscript Flow through lateral circular orifice under free and submerged flow conditions A. Hussain, Z. Ahmad, C.S.P. Ojha
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S0955-5986(16)30169-8 http://dx.doi.org/10.1016/j.flowmeasinst.2016.09.007 JFMI1256
To appear in: Flow Measurement and Instrumentation Received date: 17 November 2015 Revised date: 23 August 2016 Accepted date: 30 September 2016 Cite this article as: A. Hussain, Z. Ahmad and C.S.P. Ojha, Flow through lateral circular orifice under free and submerged flow conditions, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2016.09.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Flow through lateral circular orifice under free and submerged flow conditions A. Hussaina*, Z. Ahmadb, C.S.P. Ojhac a
Assistant Professor, Department of Civil Engineering, Aligarh Muslim University, Aligarh-202002, India. b,c Professor, Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India. [email protected] (A. Hussain) [email protected] (Z. Ahmad) [email protected] (C.S.P. Ojha) * Corresponding author. Abstract Open channels, with flow diversion structures such as orifices, weirs and sluice gates; are prevalent in irrigation systems, both for conveying water from the source to the irrigated areas, and for distributing the water within the irrigated area. The present study was broadly aimed at to investigate the flow characteristics of sharp-crested side circular orifices under free and submerged flow conditions through analytical and experimental considerations. It was also intended to develop relationships for coefficient of discharge for orifices under free and submerged flow conditions. The computed discharges using developed relationships were within s5% and s10% of the observed ones for free and submerged orifices, respectively. Sensitivity analysis reveals that the discharge through side orifice is more sensitive to the low head above the center of the orifice. Various parameters affecting the jet angles have been identified and relevant parameters are used for proposing relationships for the jet angle under different flow conditions. Notations B
Width of the main channel, m
Cd
Coefficient of discharge
D
Diameter of the orifice, m
Fr
Froude number
g
Acceleration due to gravity, m/s2
H
Head of water above the centerline of the orifice, m
Q
Discharge through the orifice, m3/s
Qm
Discharge in the main channel, m3/s
T
Length of the elemental strip, m
V1
Velocity in the main channel, m/s
W
Sill height, m
Ym
Depth of flow in the main channel, m 1
z
Head over the elemental strip, m
z
Height of the elemental strip from the base of the orifice, m
α
Inclination of left nappe of water jet, degree
β
Inclination of right nappe of water jet, degree
φ
jet angle, degree
ρ
Mass density, kg/m3
μ
Viscosity, N.s/m2
A1 & A2 Cross sectional area of the free and the submerged portions of the orifice, m2 h
Difference of water levels of on the either sides of the orifice, m
SA
Sensitivity analysis
RSA
Relative sensitivity
Keywords: Flow diversion, Side circular orifice, Coefficient of Discharge, Froude number, Jet angle 1.
Introduction Orifices built in channels are intended to divert flow from the channel for various purposes of
use, including irrigation, potable water supply, hydroelectric power, industrial water supply, water and waste water treatment plant etc. (Ramamurthy et al. 1986, Ojha and Subbaiah 1997, Hussain et al. 2010, 2011, 2014a, Gill 1987). Orifices and sluice gates are commonly used flow control and metering devices (Shammaa et al. 2005). A well defined opening in the wall, the top of which is placed well below the upstream water level, is classified as an orifice (Athar and Ansari 1996). For true orifice flow to occur, the upstream water level must always be well above the top of the opening, such that vortex-flow with air entrainment is not occurring. If the upstream water level drops below the top of the opening, it no longer performs as an orifice but as a weir. This study is in continuation of the work carried out by Hussain et al. (2010) on the flow characteristics for side circular orifice under free flow. Godson (2003) found that the coefficient of discharge of sluice gate depends on the approach flow Froude number. However; Swamee et al. (1993) related the discharge coefficient of the side sluice gate with depth of flow in the main channel and gate opening for free flow conditions. Gregg et al. (2003) and Werth et al. (2005) reported studies on orifices cut into a thin-walled pipe, where the flow exits as a pressurized pipe through multiple orifices. Gill (1987), Ramamurthy et al. (1986), Ojha and Subbaiah (1997) and Ramamurthy et al. (1987) carried out an experimental study on side rectangular orifice and derived a discharge equation for the side rectangular orifice. Zahiri et al. (2013) carried out an experimental study on coefficient of discharge for compound sharp crested side weirs in subcritical flow conditions. They reported that the coefficient of discharge of a compound side weir is a function of the upstream Froude number, ratio of the weighted crest height of the weir to the upstream depth and ratio of weir length to upstream water depth. 2
Since under the free flow conditions, the jet emerging out from the orifice into the air therefore, atmospheric pressure prevails over the cross-section of the jet (Fig. 1a). For fully submerged orifice (Fig. 1b), water level in the diverted channel is above the upper crest of the orifice. The jet coming out of the orifice is within water and experiences higher pressure than free jet. In case of the partially submerged orifice, water level in the diverted channel is above lower edge of the orifice and below the upper edge of the orifice (Fig. 1c). Analytical equations for free flow, fully submerged flow and partially submerged flow have been derived. The data collected from the experimental program for orifices under free and submerged conditions are used to evolve equations for coefficient of discharge.
Fig. 1 Schematic view of side circular orifice under different flow conditions: (a) Free flow, (b) Fully submerged flow and (c) Partially submerged flow
Consider a large orifice of diameter D fitted in the side of the wall of an open channel at crest height W as shown in Fig. 2.
Fig. 2 Circular side orifice in an open channel Considering varying pressure head over the flow area of the orifice, the discharge relation developed by Hussain et al. (2010) is 1/2
1/2
1/2
æH 1 z ö æz ö æ z ö Q = 2 2 g D3/2Cd ò ç + - ÷ ç ÷ ç1 - ÷ dz D 2 Dø èDø è Dø 0è D
From geometry
z =H +
D -z 2
1/2
(1)
1/2
æz ö æ z ö ÷ ç1- ÷ èDø è Dø
and T =2D ç
For known value of Cd, D and H, discharge Q flowing through the orifice can be computed by integrating Eq. (1) numerically. For a small orifice with constant pressure distribution over the flow area, the discharge equation for free flow condition can be written as (Rouse 1970, Yuan 1988)
p Q = Cd 2 gH . D 2 4
(2)
The discharge through a partially submerged orifice may be determined by computing, separately the discharge through the free and the submerged portions and then adding the two discharges. Let H1 & H2 be the height of the liquid on the upstream side above the bottom edge and top edge of the orifice as shown in Fig. 3. Difference of water levels on the either sides of the orifice is h. Cross sectional area of the free and the submerged portions of the orifice are A1 & A2, respectively.
3
Fig. 3 Lay-out of the partially submerged circular orifice Let discharges through free and submerged portions of the orifice are Q1 & Q2, respectively, discharge through orifice Q is (Ojha et. al. 2010)
Q = Q1 + Q2
(3)
2
From geometry, A2 = p
D 1 D2 - (q - sin q ) 4 2 4
(4)
æ H3 - D / 2 ö ÷ è D/2 ø
(5)
q = 2 cos -1 ç
and
Here θ is in radian. Discharge through the submerged portion of the orifice 2 é æ D ö2 æ æ H - D / 2 ö ö æ æ -1 æ H 3 - D / 2 ö ö ö æ D ö ù sin cos Q2 = Cd êp ç ÷ - 0.5 ç 2cos -1 ç 3 ç ÷ ç ÷÷ ç D / 2 ÷ ÷ ç 2 ÷ ú 2 gh è D / 2 øø è è è ø ø ø è ø úû è ëê è 2 ø
(6)
From geometry, discharge through the free flow portion of the orifice D - H3
Q1 = Cd
ò 0
1/2
1/2
æ H - y ö æ H1 - y ö 2D ç 1 ÷ ç1 ÷ D ø è D ø è
2 gydy
(7)
For known value of Cd, D and H, discharge Q1 flowing through the free flow portion of the orifice can be computed by integrating Eq. (7) numerically while discharge through the submerged part of orifice can be computed using Eq. (5). Consider a fully submerged orifice of diameter D fitted in the side of the wall of an open channel at crest height W as shown in Fig. 1b. The discharge can be obtained from (Yuan 1988, Ojha et. al. 2010)
Q = Cd (p D2 ) 2 gh 4
(8)
Where h is the difference between the water levels on either side of the orifice This paper deals with hydraulics of the side circular orifice under free flow, fully and partially submerged conditions. 2.
Experimental program Circular orifices of diameter 5 cm, 10 cm and 15 cm were used in the experiment. The flume
used in the experiments is the same as that given in Hussain et al. 2010 (Fig. 4). Experiments were performed for crest height of orifice W = 0, 0.05, 0.10, 0.15, 0.20 and 0.25 m and for each set of D and W three to four discharge Qm in the main channel. For each Qm, different depths of flow were maintained in the main channel by regulating the downstream sluice gate. For each run, flow depths in the main channel in the vicinity of the orifice and head over the crests of weir-A and weir-B were measured by digital pointer gauge of accuracy ±0.1 mm. Ultrasonic flow meter has been used for the calibration of weirs for measuring discharge through orifice and the main channel. In all the 4
conducted experiments, the nappes were fully ventilated under free flow condition. Splitter plates and wave suppressor were provided in the upstream of each channel to break large size eddies and to dissipate the surface disturbances, respectively. Experiments were performed under no-formation-ofvortex condition in the main channel in the vicinity of the orifice. Fig. 4 Layout of the experimental set-up Plan and Section Y-Y In case of submerged orifice a sluice gate was provided downstream of the diversion channel to regulate the flow in the diversion channel to submerge the orifice. Procedure of measurements under submerged flow condition was same as that of free flow condition. For each Qm in the main channel, orifice submergence depth in the downstream direction was varied with the help of the sluice gate. For each run, flow depth in the main channel and head over the crests of weir-A and weir-B were measured. The ranges of data collected in the present study for circular orifices are given in Tables 1. Experiments were conducted for subcritical flow condition in the main channel.
Table 1 Ranges of data collected for circular orifices in the present study Range of data Parameters
Unit
Qm Q D Ym V W Fr
m3/s m3/s m m m/s m Dimensionless
Min. 0.01654 0.00103 0.05 0.113 0.07 0.0 0.036
3.
Results and analysis for free flow orifice
3.1
Effect of various parameters on coefficient of discharge, Cd
Max. 0.22222 0.02942 0.15 0.593 1.50 0.25 0.931
Hussain et al. (2010) reported that the final functional relationship for Cd may be expressed as
æB W H ö Cd = f 2 ç , , ,Fr ÷ èD D D ø
(9)
Here B is the width of the main channel and Fr is the Froude number of approach flow expressed
as Fr = V / gYm The data collected in the present study have been used to analyse the effect of non dimensional parameters of Eq. (9) on coefficient of discharge, Cd. Figure 5a shows the variation of the observed coefficient of discharge Cd with the Froude number Fr. It reveals that the value of Cd of the orifice decreases with the increase of the Froude number Fr when other parameters such as head of 5
water above the centerline of the orifice, the size of the orifice, sill height etc. are constant. Figure 5b shows the effect of dimensionless upstream depth of flow in the main channel on the coefficient of discharge. It is apparent from the Fig. 5b that the coefficient of discharge increases with increase of H/D for other parameters kept constant. Such variation was also noticed during the experimentation for each size of the orifice and different sill height. Figure 5c shows that the coefficient of discharge increases with the increase in B/D value when other parameters such as Froude number, sill height and head above the centerline of the orifice are kept constant. This could be attributed to opposite side solid boundary. Flow near the opposite side does not participate in the flow coming out from the small orifice; however, it does for larger orifice size. The influence of the crest height (W) above the floor on the Cd is shown in Fig. 5d, which reveals that the values of Cd increases with the increase of crest height of orifice but for large crest height location effect are not prominent. This is due to the fact that an orifice at larger crest height will receive more stream lines as compared to low crest height, i.e. orifice having a larger crest height will be exposed to the large flow area in its vicinity and as a result more discharge will enter into the orifice. With further increase in the crest height boundary may not have any affect the streamlines entering into the orifice and thus Cd will become constant for any crest height. Fig. 5 Variation of Cd with (a) Froude number, (b) H/D, (c) B/D and (d) W/D 3.2
Formulation of equations for free flow orifice It is common practice to use 70% to 75% of the total available data randomly selected for
proposing/training a relationship and remaining data for its validation (Hussain et al., 2010, 2014b; Hammerstrom, 1993; Kumar et al., 2010; Bowden et al., 2005 and Khan et al., 2013). Such approach is also adopted in this study for proposing an equation for Cd using the collected data. Data obtained in the present experimental program and Hussain et al. (2010) are used for developing the relations for free flow side circular orifice. Trapezoidal method is used to solve the Eq. (1) numerically for obtaining Cd. Out of 458 data sets collected in the present study, 304 data sets selected randomly have been used to evolve relationship for Cd and remaining for validating the proposed the relationship. Using the least squares technique, Eqs. (10) & (11) are proposed for Cd, considering orifice to be large and small, respectively.
æDö Cd = 0.669 - 0.156Fr - 0.715 ç ÷ èBø
3.5
æDö Cd = 0.662 - 0.153Fr - 0.758 ç ÷ èBø
(10) 3.8
(11)
The remaining 154 unused data sets have been used to validate the proposed relationships for Cd for the computation of discharge through the orifice. The observed and computed values of discharge through orifice for the calibration data are compared graphically in Figs. 6 and 7, 6
considering orifice to be large and small, respectively. Figures 6 & 7 reveal that the computed discharge is within ±5% of the observed ones, which is a satisfactory prediction of discharge through the orifice. It is to be noted that computation of discharge using Eq. (1) requires integration of this equation for given values of D and H.
Fig. 6 Comparison of computed discharge through orifice using Eqs. (1) and (10) with observed ones considering orifice as large for validation Fig. 7 Comparison of computed discharge through orifice using Eqs. (2) and (11) with observed ones considering orifice as small for validation The average percentage error in computation of discharge through the orifice considering it as large and small are 3.99% and 4.04%, respectively. As these two values are practically the same, it is concluded that computation of discharge through the orifice can be performed considering it as small orifice within range of H/D i.e., 0.75 to 8.17 collected in the present study. Equations (10) and (11) are valid for Fr < 1 (sub critical) and 0.109 < D/B > 0.303 m. 3.3
Sensitivity analysis for free flow orifice Sensitivity analysis (SA) is the study of how the variation (uncertainty) in the output of a
mathematical model can be apportioned qualitatively or quantitatively to different sources of variation in the input of a model (Saltelli et al., 2008). In broad, sensitivity analysis is the investigation of the potential changes and errors in the parameters of any model and their impacts on the model (McCuen and Synder, 1986). It can help build confidence in the model by studying the uncertainties that are often associated with parameters in the model. Thus, sensitivity analysis is conducted to develop a comprehensive understanding of the response of the model due to change in the calibration parameter values. Sensitivity analysis can also be used to check the range of parameter values and assist in the selection of parameters for model calibration, operations and improvement of model capabilities. Keeping this in view, in the subsequent sections, an effort has been made to investigate the sensitivity of the parameters of the proposed models. The analysis presented in this study is confined to univariate case, which deals with single variable at a time. Sensitivity values computed using Eq. (12) is in absolute form and quantifies the changes in factors that results from changes in factor Fi. Relative sensitivity (RSA) can be defined as the relative change in Fo with respect to a relative change in Fi and represented by Eq. (13).
SA =
¶F0 ( f ( Fi + DFi , Fj / j ¹ i ) - f ( F1 , F2 ,......Fn ) = ¶Fi ¶Fi
RSA =
¶F0 F0 æ ¶F0 ö Fi =ç ÷´ ¶Fi Fi è ¶Fi ø F0
(12)
(13)
7
Sensitivity analysis, i.e., change in discharge due to a unit change in the head of water is carried out for the proposed discharge equation of the side circular orifice. Using Eq. (11) and (2), discharge through a side orifice in terms of variable head can be written as
æ ö V1 D ÷ A 2 gH Q = ç 0.69 - 0.035 - 0.154 ç ÷ B 0.5 g H W D + + ( ) è ø
(14)
Replacing V1 by Q/(B(H+W+0.5D)) and rearranging Eq. (14), it can be written as
é ù ( 0.69 - 0.035 D B ) A 2 g Q = ê -0.5 ú -1.5 êë H + 0.2177 AB -1 ( H + W + 0.5D ) úû
(15)
Differentiating Q with respect to H and arranging the terms, one can get
(
Dö -2.5 æ 0.69 - 0.035 ÷ A 2 g 0.5H -1.5 + 0.326 AB -1 ( H + W + D / 2 ) ç dQ è Bø = -1.5 2 dH H -0.5 + 0.217 A / B H + W + D / 2
(
(
)
)
( (
-1.5 -1 dQ a ¢ 0.5H + 0.326 AB ( H + W + D / 2 ) = -1.5 2 dH H -0.5 + 0.217 A / B ( H + W + D / 2 )
æ è
Where a ¢ = ç 0.69 - 0.035
-2.5
)
)
)
(16)
(17)
Dö ÷ A 2g Bø
The larger value of dQ/dH implies higher sensitivity. Data collected in the present study was used to compute dQ/dH for different values of diameter of the orifice. The variation of dQ/dH with H/D is depicted in the Fig. 8. A perusal of Fig. 8 reveals that the discharge through a side circular orifice is more sensitive to the low head. As the head increases, the sensitivity decreases due to the decrease in pressure variation across the cross sectional area of the orifice. Further, sensitivity is higher for the higher size of the orifice. Fig. 8 Sensitivity of the side circular orifice as a function of H/D Sensitivity values computed using Eq. (16) is in absolute form. These values cannot be used for comparision of parametric sensitivities because the computed values are not invariant to the dimensions of either factor Q or H. Dividing the numerator by Q and the denominator by H provides an estimate of the relative change in Q with respect to a relative change in H. From Eqs. (15) and (16), one can get
(
-2.5
0.5H -1.5 + 0.326 AB -1 ( H + W + D / 2 ) dQ = -1.5 Q H -0.5 + 0.217 AB -1 ( H + W + D / 2 ) Multiplying by H/H and rearranging
8
) dH
(18)
æ dQ / Q ç è dH / H
-2.5
-0.5 -1 ö 0.5H + 0.326 AB ( H + W + D / 2 ) H = ÷ -1.5 ø H -0.5 + 0.217 AB -1 ( H + W + D / 2 )
(19)
Relative sensitivity with respect to discharge and head for circular orifice of D = 15 cm is shown in Fig. 9. It is found that with the increase in head, relative sensitivity decreases. Relative sensitivity is highly pronounced for lower head and becomes constant for higher head in the main channel. Fig. 9 Relative sensitivity of the side circular orifice as a function of H/D 3.4
Variation of jet angles for free flow orifice Several aspects of flow in the main channel strongly affect the geometry of the spilling jet. It
is likely that variables affecting the jet angle of the side weir would also affect the jet angle of a side circular orifice. Thus, from the review of literature, it is found that probable variables affecting the jet angle φ for square side orifice would be L, W, V, H, mass density ρ, viscosity μ, and g. The jet angle φ is considered as the average of the inclination of the left nappe of water jet a (degree) and the inclination of the right nappe b (degree). The functional relationship for φ may, thus, be written as
j = f1 ( D,W ,V1 , Ym , H , r , m , g )
(20)
Where φ = (a+b)/2 Taking ρ, V and D as the repeating variables, the functional relationship for Cd in terms of nondimensional parameters may, thus be written as
æ H W V1 , , ç D D gY m è
j = f2 ç
ö ÷ ÷ ø
(21)
Experimental study carried out in this study is persuaded to investigate the effect of the identified non-dimensional parameters on the φ. 3.4.1
Observations during experimentation & equation development for jet angle Water jet, issuing out from the circular orifice, was not truly circular downstream of the
orifice due to gravitational forces. Further, the centerline of water jet was inclined to the direction that is normal to the flow in the main channel as shown in Fig. 10. As the Froude number of the main channel decreases the inclination angle decreases and the jet turns normal to the flow in the main channel. Fig. 11(a & b) shows the jet and its inclination for high and low Froude numbers in the main channel, respectively. The jet coming out of the orifice takes the shape of the circular to venacontracta and then distorted downstream. Further, inclination of the left nappe of water jet a was higher than the inclination of the right nappe b (Fig. 12). Fig. 10 Jet issuing out of the orifice 9
To seek the importance of various independent variables in predicting φ, feature selection and variable screening, i.e. F-test has been carried out. When F-value of any parameter compared to the Fvalue of another parameter having very low F-value, then the least F value parameter is dropped because this parameter is considered as not affecting the whole value of the equation (Lomax, 2007; Hussain et al., 2014b; Khan et al., 2013). All the three variables of Eq. (21) were taken for feature selection and variable screening process with φ as the dependent variable. As shown in Fig. 13, Fr possesses a high F-value, followed by H/D. The parameter W/D shows least importance and is, therefore, dropped while deriving the relationship for φ. The effect of the dimensionless parameters Fr, H/D and W/D on the jet angle was examined. A thorough data analysis reveals that Fr and H/D are, indeed, the predominant parameters which affect the φ. A further assessment revealed that the Froude number dominantly effect on the jet angle: as the correlation coefficient between φ and Fr is 0.79 (Table 2). The jet angle φ increases with increase of Fr (Fig. 14a). For the range of data used in the present study, φ is unaffected by the parameters W/D -that is also clear from the Table 2 i.e., in comparison to other parameters W/D has a least correlation coefficient. A perusal of Fig. 14b indicates that with the increase of H/D, there is decrease in φ. Fig. 11 Angular variation of the jet for circular orifice for (a) high Fr = 0.35 and (b) low Fr = 0.15 Fig. 12 Variation of the jet angle a and b with H/D
Table 2 Correlation matrix among variables of Eq. (21)
φ(obs) Fr W/D H/D
φ(obs) 1 0.79 0.15 0.59
Fr 1 -0.21 0.31
W/D
1 0.14
H/D
1
Fig. 13 Importance of various independent inputs in predicting output (φ) Fig. 14 Variation of φ with (a) Fr and (b) H/D The following equation is proposed for φ using 206 data sets selected randomly out of 276 data sets collected in the present study and invoking the least squares technique,
j = 16 + 44Fr - 2.5
H D
(22)
10
The R2 value of proposed Eq. (22) is 0.8. Eq. (22) is validated using the remaining 70 data sets for the computation of the jet angle of circular orifice. A graphical comparison between the observed and computed φ for the validation data is shown in Fig. 15. It is apparent from the Fig. 15 that the computed φ is within ±20% of the observed ones. It can be concluded that Froude number has a significant effect on the jet angle. The spilling jet angle (φ) is positively correlated with the upstream Froude number and negatively correlated with H/D. Ultimately; an expression for estimating of φ is developed. Fig. 15 Comparison between the computed and observed φ values for the validation 4.
Results and analysis for submerged orifice
4.1
Discharge calculation for fully submerged orifice Out of the 170 data sets collected in the present study, selected 127 data sets have randomly
been used to evolve the relationship for Cd and remaining for validating the proposed relationship. Using the least square technique, the following relationship for Cd has been evolved.
Cd = 0.7 - 0.294Fr - 0.023
D B
(23)
The observed and computed values of discharge through orifice using Eqs. (9) and (23) for the 43 test data are compared graphically in Fig. 16, which reveals that the computed discharge is within ±10% of the observed ones, which can be considered satisfactory prediction of discharge through the submerged orifice. Fig. 16 Comparison between the observed and computed discharge values using the proposed equation for the validation data 4.2
Discharge calculation for partially submerged orifices The above technique of least square for finding the relationship for Cd of submerged flow is
also used for partially submerged orifice. A total 91 data sets were collected in the study for the partially submerged flow orifice. Out of the 91 data sets, selected 64 data sets have randomly been used to evolve relationship for Cd and remaining for validating the proposed the relationship. Using the least square technique, the following relationship for Cd is
Cd = 0.59 - 0.074F - 0.165
D B
(24)
The remaining 25 data sets, not used in the derivation of Eq. (24), were used to validate the proposed relationship for Cd i.e., Eq. (24), for the computation of discharge through the orifice. The observed and computed values of discharge through orifice using Eqs. (4) and (24) for the test data are compared graphically in Fig. 17, which reveals that the computed discharge is within ±20% of the 11
observed ones, which can be considered satisfactory prediction of discharge under the submerged condition. Fig. 17 Comparison between the observed and computed discharge values using the proposed equations for the validation 4.3
Cd of submerged orifice in terms of Cd of free flow orifice Under submerged flow condition, the jet coming out of the orifice is fully or partly
submerged. The pressure of the issued jet is high compared to the free flow condition and it further increases with the increase of submergence resulting in a reduced value of the diverted discharge. The Cd of the orifice under submerged flow condition may be related to H3/ H1 and Cd under free flow conditions as
é æ H ö a1 ù Cd ( sub) = Cd ( free) ê1 - çç 3 ÷÷ ú êë è H1 ø úû
b1
(25)
Where a1 and b1 are constants and can be obtained by fitting the data of the coefficient of discharge and H3/H1. Eq. (25) is non-linear; therefore, a computer code is written for a grid search method to obtain values of constants a1 and b1 of Eq. (25) for best matching of computed Cd with the observed values. Out of 261 data sets collected in the present study for the submerged orifice, 190 data sets selected randomly were used to evolve the relationship for the Cd (sub). Here Cd (free) shall be calculated from the Eq. (10). For submerged circular orifice, the obtained value of a1 = 1.5 and b1 = 0.6, thus
é æ H ö1.5 ù Cd ( sub ) = Cd ( free ) ê1 - ç 3 ÷ ú êë è H1 ø úû
0.6
(26)
The remaining 71 data sets, not used in the derivation of Eq. (26), were used next to validate the proposed relationship for Cd. The observed and computed values of discharge through orifice using Eqs. (26) and (8) for the validation data are compared graphically in Fig. 18, which reveals that the computed discharge is within ±12% of the observed values, which is a satisfactory prediction of discharge through the submerged orifice.
Fig. 18 Comparison between the observed and computed discharge values using the proposed equation for the (a) calibration and (b) validation It may be noted that the effect of other variables viz: Fr, B and D is already accounted for through the use of Eq. (10) for the determination of the Cd (submerged). Eq. (26) is valid for both fully submerged and partially submerged flow. Thus, for calculating Q under submerged conditions 12
first the Cd is calculated for free flow conditions using Eq. (10) and then it will be used in Eq. (26). The obtained Cd will be used in Eq. (8) to compute Q. It is advocated to use Eq. (26) for submerged orifice as it gives more accurate results compared to Eq. (23). 5.
Conclusions It is concluded from the present study that the coefficient of discharge depends mainly on the
approach channel Froude number and the ratio of width of the orifice and bed width of the channel under free flow conditions. The computed discharges using the proposed relationships for circular orifice, by considering the orifice as large and small, were within ±5% of the observed values. The average percentage error in computation of discharge through the circular orifice considering it as large and small are 3.99% and 4.04%, respectively, which are practically the same. Therefore, discharge through the side orifices in open channel can be computed treating it as small orifice within range of H/D = 0.75 to 8.17 for circular orifices. Discharge through side orifice is more sensitive to the low head above the center of the orifice. As the head increases, the sensitivity decreases due to the decrease in pressure variation across the cross sectional area of the orifice. Further, the sensitivity is higher for the higher size of the orifice. Relative sensitivity is pronounced for lower head above the center of the orifice and becomes constant for higher head in the main channel. The Froude number of the flow in the main channel is a dominant factor that affects the jet angle. The computed discharges using the proposed relationships of the coefficient of discharge for circular orifices under full and partial submerged conditions were within ±10% and ±20% of the observed values, respectively. The computed values of discharge through side circular orifice using Cd in terms of Cd of free flow are within ±10% of the observed values. It is advisable to use Eq. (26) for finding the Cd (sub) for partial or full submerged circular orifice, respectively. References [1]
Athar, M., Ansari, M.A. and Haque, M., 1996. Effect of Relative Roughness on Critical Reynolds Number in Pipes. II National Conference of Fluid Machinery at P.S.G. College of Tech, Coimbatore, Proc. 28-29, 269-277.
[2]
Bowden, G.J., Maier, H.R., Dandy, G.C., 2005. Input determination for neural network models in water resources applications. Part 2. Case study: forecasting salinity in a river. Journal of hydrology, Elsevier 301, 93-107.
[3]
Ghodsian, M., 2003. Flow through side sluice gate. Journal of Irrigation and Drainage Engineering, ASCE 129(6), 458-62.
[4]
Gill, M.A., 1987. Flow through side slots. Journal of Environmental Engineering, ASCE 113(5), 1047-1057.
13
[5]
Gregg, W.B., Werth, D.E., Frizzell, C., 2003. Determination of discharge coefficients for hydraulic sparger design. Journal of Pressure Vessel Technology, ASME 126(3), 354 – 359.
[6]
Hammerstrom, D., 1993. Working with neural networks. IEEE Spectrum, 30(7), 46-53.
[7]
Hussain, A., Ahmad, Z., Asawa, G.L., 2010. Discharge characteristics of sharp-crested circular side orifices in open channels. Journal of Flow Measurement and Instrumentation, Elsevier 21(3), 418-24.
[8]
Hussain, A., Ahmad, Z., Asawa, G.L., 2011. Flow through sharp-crested rectangular side orifices under free flow condition in open channels. Agricultural Water Management, Elsevier 98(10), 1536-1544.
[9]
Hussain, A., Ahmad, Z., Ojha, C.S.P., 2014a. Analysis of flow through lateral rectangular orifices in open channels. Journal of Flow Measurement and Instrumentation. Elsevier 36(10), 32-35.
[10] Hussain, S., Hussain A., Ahmad, Z., 2014b. Discharge characteristics of orifice spillway under oblique approach flow. Journal of Flow Measurement and Instrumentation, Elsevier 39(3), 0918. [11] Khan, D., Hussain, A., Ahmad, Z., 2013. Energy dissipation of skimming flow over stepped spillway. International water power and dam construction, Dam Engineering XXIII (4), 187206. [12] Kumar, S., Ahmad, Z., Kothyari, U.C., Mittal, M.K., 2010. Discharge characteristics of a trench weir. Flow Measurement and Instrumentation. Elsevier 21(2), 80-87. [13] Lomax, R.G., 2007. Statistical Concepts: A Second Course. Lawrence Erlbaum Associates, Mahwah, NJ. [14] McCuen, R.H., Synder, W.M., 1986. Hydrologic Modeling: Statistical Methods and Applications, Prentice-Hall, Englewood Cliffs, NJ. [15] Ojha, C.S.P., Berndtsson, R., Chandramoulli, P.N., 2010. Fluid Mechanics and Fluid Machinery. Oxford University Press, New Delhi. [16] Ojha, C.S.P., Subbaiah, D., 1997. Analysis of flow through lateral slot. Journal of Irrigation and Drainage Engineering, ASCE 123(5), 402-405. [17] Ramamurthy, A.S., Udoyara S.T., Serraf S., 1986. Rectangular lateral orifices in open channel. ASCE Journal of Environmental Engineering 135(5): 292-98. [18] Ramamurthy, A.S., Udoyara ST, Rao, M.V.J., 1987. Weir orifice units for uniform flow distribution. ASCE Journal of Environmental Engineering 113(1): 155-166. [19] Rouse, H., 1970. Elementry Mechanics of Fluids. Wiley Eastern, New Delhi. [20] Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S., 2008. Global Sensitivity Analysis: The Primer John Wiley & Sons, Chichester. [21] Shammaa, Y., Zhu, D.Z., Rajaratnam, N., 2005. Flow Upstream of Orifices and Sluice Gates. Journal of Hydraulic Engineering, ASCE 131(2), 127-133. 14
[22] Swamee, P.K., Pathak, S.K., Ali, M.S., 1993. Weir orifice units for uniform flow distribution. Journal of Irrigation and Drainage Engineering, ASCE 119(6), 1026-1035. [23] Werth, D.E., Khan, A.A., Gregg, W.B., 2005. Experimental study of wall curvature and bypass flow effects on orifice discharge coefficients. Experiments in Fluids, 39, 485–491. [24] Yuan, S.W., 1988. Fundamentals of Fluid Mechanics, Prentice-Hall of India, New Delhi. [25] Zahiri, A., Azamathulla, H.Md., Bagheri, S., (2013). Discharge coefficient for compound sharp crested side weirs in subcritical flow conditions. Journal of Hydrology, Elseveir 480, 162-166.
Brief Bio-Data of Authors A. Hussain is currently Assistant Professor at Civil Engineering Department of Aligarh Muslim University (AMU), India. He obtained his B.Tech. (Civil) from AMU, Aligarh and M.Tech. (Hyd) & Ph.D. degree from IIT Roorkee, India. He has published 08 papers in referred international journal 10 national and international conference. His areas of interests are open channel flow, hydro power, computational hydraulics and hydraulic structures. Z. Ahmad is currently Professor of Civil Engineering at IIT Roorkee, India. He obtained his B.Tech. (Civil) degree from AMU Aligarh, M.Tech. (Hyd) degree from Univ. of Roorkee and Ph.D. degree from T.I.E.T., Patiala. He has published about 75 papers in referred national and international journals and conference proceedings. His area of research is surface water quality management, computational hydraulics and hydraulic structures. He has written two Monographs on Transport of Pollutants in Open Channels and Control Section in Open Channels for AICTE, New Delhi. He has been recipient of G.N. Nawathe, Jal Vigyan Puraskar (twice); Department of Irrigation Award. CSP Ojha joined the Department of Civil Engineering at the Indian Institute of Technology Roorkee (formerly University of Roorkee) in 1971 and is presently Professor. He has supervised more than 39 PhD. He has been teaching hydraulic engineering courses at both undergraduate and postgraduate levels, and also associated with research and consultancy activities in different aspects of water resources engineering. He has published 04 books, more than 100 research papers, and written many reports on consultancy projects. He has been recipient of several awards and has travelled abroad for various academic programmes.
Highlights Ø Relationships for flow through side circular orifice under free and submerged conditions have been developed. Ø Relationship for estimating jet angle (φ) for circular orifice has been developed. Ø Relative sensitivity is highly pronounced for low head above the center of the orifice.
15
Figure 1a
Figure 1b
Figure 1c
Figure 2
Figure 3
Figure 4
Figure 5a
Figure 5b
Figure 5c
Figure 5d
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11a
Figure 11b
Figure 12
Figure 13
Figure 14a
Figure 14b
Figure 15
Figure 16
Figure 17
Figure 18
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PII: DOI: Reference:
S0955-5986(16)30169-8 http://dx.doi.org/10.1016/j.flowmeasinst.2016.09.007 JFMI1256
To appear in: Flow Measurement and Instrumentation Received date: 17 November 2015 Revised date: 23 August 2016 Accepted date: 30 September 2016 Cite this article as: A. Hussain, Z. Ahmad and C.S.P. Ojha, Flow through lateral circular orifice under free and submerged flow conditions, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2016.09.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Flow through lateral circular orifice under free and submerged flow conditions A. Hussaina*, Z. Ahmadb, C.S.P. Ojhac a
Assistant Professor, Department of Civil Engineering, Aligarh Muslim University, Aligarh-202002, India. b,c Professor, Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India. [email protected] (A. Hussain) [email protected] (Z. Ahmad) [email protected] (C.S.P. Ojha) * Corresponding author. Abstract Open channels, with flow diversion structures such as orifices, weirs and sluice gates; are prevalent in irrigation systems, both for conveying water from the source to the irrigated areas, and for distributing the water within the irrigated area. The present study was broadly aimed at to investigate the flow characteristics of sharp-crested side circular orifices under free and submerged flow conditions through analytical and experimental considerations. It was also intended to develop relationships for coefficient of discharge for orifices under free and submerged flow conditions. The computed discharges using developed relationships were within s5% and s10% of the observed ones for free and submerged orifices, respectively. Sensitivity analysis reveals that the discharge through side orifice is more sensitive to the low head above the center of the orifice. Various parameters affecting the jet angles have been identified and relevant parameters are used for proposing relationships for the jet angle under different flow conditions. Notations B
Width of the main channel, m
Cd
Coefficient of discharge
D
Diameter of the orifice, m
Fr
Froude number
g
Acceleration due to gravity, m/s2
H
Head of water above the centerline of the orifice, m
Q
Discharge through the orifice, m3/s
Qm
Discharge in the main channel, m3/s
T
Length of the elemental strip, m
V1
Velocity in the main channel, m/s
W
Sill height, m
Ym
Depth of flow in the main channel, m 1
z
Head over the elemental strip, m
z
Height of the elemental strip from the base of the orifice, m
α
Inclination of left nappe of water jet, degree
β
Inclination of right nappe of water jet, degree
φ
jet angle, degree
ρ
Mass density, kg/m3
μ
Viscosity, N.s/m2
A1 & A2 Cross sectional area of the free and the submerged portions of the orifice, m2 h
Difference of water levels of on the either sides of the orifice, m
SA
Sensitivity analysis
RSA
Relative sensitivity
Keywords: Flow diversion, Side circular orifice, Coefficient of Discharge, Froude number, Jet angle 1.
Introduction Orifices built in channels are intended to divert flow from the channel for various purposes of
use, including irrigation, potable water supply, hydroelectric power, industrial water supply, water and waste water treatment plant etc. (Ramamurthy et al. 1986, Ojha and Subbaiah 1997, Hussain et al. 2010, 2011, 2014a, Gill 1987). Orifices and sluice gates are commonly used flow control and metering devices (Shammaa et al. 2005). A well defined opening in the wall, the top of which is placed well below the upstream water level, is classified as an orifice (Athar and Ansari 1996). For true orifice flow to occur, the upstream water level must always be well above the top of the opening, such that vortex-flow with air entrainment is not occurring. If the upstream water level drops below the top of the opening, it no longer performs as an orifice but as a weir. This study is in continuation of the work carried out by Hussain et al. (2010) on the flow characteristics for side circular orifice under free flow. Godson (2003) found that the coefficient of discharge of sluice gate depends on the approach flow Froude number. However; Swamee et al. (1993) related the discharge coefficient of the side sluice gate with depth of flow in the main channel and gate opening for free flow conditions. Gregg et al. (2003) and Werth et al. (2005) reported studies on orifices cut into a thin-walled pipe, where the flow exits as a pressurized pipe through multiple orifices. Gill (1987), Ramamurthy et al. (1986), Ojha and Subbaiah (1997) and Ramamurthy et al. (1987) carried out an experimental study on side rectangular orifice and derived a discharge equation for the side rectangular orifice. Zahiri et al. (2013) carried out an experimental study on coefficient of discharge for compound sharp crested side weirs in subcritical flow conditions. They reported that the coefficient of discharge of a compound side weir is a function of the upstream Froude number, ratio of the weighted crest height of the weir to the upstream depth and ratio of weir length to upstream water depth. 2
Since under the free flow conditions, the jet emerging out from the orifice into the air therefore, atmospheric pressure prevails over the cross-section of the jet (Fig. 1a). For fully submerged orifice (Fig. 1b), water level in the diverted channel is above the upper crest of the orifice. The jet coming out of the orifice is within water and experiences higher pressure than free jet. In case of the partially submerged orifice, water level in the diverted channel is above lower edge of the orifice and below the upper edge of the orifice (Fig. 1c). Analytical equations for free flow, fully submerged flow and partially submerged flow have been derived. The data collected from the experimental program for orifices under free and submerged conditions are used to evolve equations for coefficient of discharge.
Fig. 1 Schematic view of side circular orifice under different flow conditions: (a) Free flow, (b) Fully submerged flow and (c) Partially submerged flow
Consider a large orifice of diameter D fitted in the side of the wall of an open channel at crest height W as shown in Fig. 2.
Fig. 2 Circular side orifice in an open channel Considering varying pressure head over the flow area of the orifice, the discharge relation developed by Hussain et al. (2010) is 1/2
1/2
1/2
æH 1 z ö æz ö æ z ö Q = 2 2 g D3/2Cd ò ç + - ÷ ç ÷ ç1 - ÷ dz D 2 Dø èDø è Dø 0è D
From geometry
z =H +
D -z 2
1/2
(1)
1/2
æz ö æ z ö ÷ ç1- ÷ èDø è Dø
and T =2D ç
For known value of Cd, D and H, discharge Q flowing through the orifice can be computed by integrating Eq. (1) numerically. For a small orifice with constant pressure distribution over the flow area, the discharge equation for free flow condition can be written as (Rouse 1970, Yuan 1988)
p Q = Cd 2 gH . D 2 4
(2)
The discharge through a partially submerged orifice may be determined by computing, separately the discharge through the free and the submerged portions and then adding the two discharges. Let H1 & H2 be the height of the liquid on the upstream side above the bottom edge and top edge of the orifice as shown in Fig. 3. Difference of water levels on the either sides of the orifice is h. Cross sectional area of the free and the submerged portions of the orifice are A1 & A2, respectively.
3
Fig. 3 Lay-out of the partially submerged circular orifice Let discharges through free and submerged portions of the orifice are Q1 & Q2, respectively, discharge through orifice Q is (Ojha et. al. 2010)
Q = Q1 + Q2
(3)
2
From geometry, A2 = p
D 1 D2 - (q - sin q ) 4 2 4
(4)
æ H3 - D / 2 ö ÷ è D/2 ø
(5)
q = 2 cos -1 ç
and
Here θ is in radian. Discharge through the submerged portion of the orifice 2 é æ D ö2 æ æ H - D / 2 ö ö æ æ -1 æ H 3 - D / 2 ö ö ö æ D ö ù sin cos Q2 = Cd êp ç ÷ - 0.5 ç 2cos -1 ç 3 ç ÷ ç ÷÷ ç D / 2 ÷ ÷ ç 2 ÷ ú 2 gh è D / 2 øø è è è ø ø ø è ø úû è ëê è 2 ø
(6)
From geometry, discharge through the free flow portion of the orifice D - H3
Q1 = Cd
ò 0
1/2
1/2
æ H - y ö æ H1 - y ö 2D ç 1 ÷ ç1 ÷ D ø è D ø è
2 gydy
(7)
For known value of Cd, D and H, discharge Q1 flowing through the free flow portion of the orifice can be computed by integrating Eq. (7) numerically while discharge through the submerged part of orifice can be computed using Eq. (5). Consider a fully submerged orifice of diameter D fitted in the side of the wall of an open channel at crest height W as shown in Fig. 1b. The discharge can be obtained from (Yuan 1988, Ojha et. al. 2010)
Q = Cd (p D2 ) 2 gh 4
(8)
Where h is the difference between the water levels on either side of the orifice This paper deals with hydraulics of the side circular orifice under free flow, fully and partially submerged conditions. 2.
Experimental program Circular orifices of diameter 5 cm, 10 cm and 15 cm were used in the experiment. The flume
used in the experiments is the same as that given in Hussain et al. 2010 (Fig. 4). Experiments were performed for crest height of orifice W = 0, 0.05, 0.10, 0.15, 0.20 and 0.25 m and for each set of D and W three to four discharge Qm in the main channel. For each Qm, different depths of flow were maintained in the main channel by regulating the downstream sluice gate. For each run, flow depths in the main channel in the vicinity of the orifice and head over the crests of weir-A and weir-B were measured by digital pointer gauge of accuracy ±0.1 mm. Ultrasonic flow meter has been used for the calibration of weirs for measuring discharge through orifice and the main channel. In all the 4
conducted experiments, the nappes were fully ventilated under free flow condition. Splitter plates and wave suppressor were provided in the upstream of each channel to break large size eddies and to dissipate the surface disturbances, respectively. Experiments were performed under no-formation-ofvortex condition in the main channel in the vicinity of the orifice. Fig. 4 Layout of the experimental set-up Plan and Section Y-Y In case of submerged orifice a sluice gate was provided downstream of the diversion channel to regulate the flow in the diversion channel to submerge the orifice. Procedure of measurements under submerged flow condition was same as that of free flow condition. For each Qm in the main channel, orifice submergence depth in the downstream direction was varied with the help of the sluice gate. For each run, flow depth in the main channel and head over the crests of weir-A and weir-B were measured. The ranges of data collected in the present study for circular orifices are given in Tables 1. Experiments were conducted for subcritical flow condition in the main channel.
Table 1 Ranges of data collected for circular orifices in the present study Range of data Parameters
Unit
Qm Q D Ym V W Fr
m3/s m3/s m m m/s m Dimensionless
Min. 0.01654 0.00103 0.05 0.113 0.07 0.0 0.036
3.
Results and analysis for free flow orifice
3.1
Effect of various parameters on coefficient of discharge, Cd
Max. 0.22222 0.02942 0.15 0.593 1.50 0.25 0.931
Hussain et al. (2010) reported that the final functional relationship for Cd may be expressed as
æB W H ö Cd = f 2 ç , , ,Fr ÷ èD D D ø
(9)
Here B is the width of the main channel and Fr is the Froude number of approach flow expressed
as Fr = V / gYm The data collected in the present study have been used to analyse the effect of non dimensional parameters of Eq. (9) on coefficient of discharge, Cd. Figure 5a shows the variation of the observed coefficient of discharge Cd with the Froude number Fr. It reveals that the value of Cd of the orifice decreases with the increase of the Froude number Fr when other parameters such as head of 5
water above the centerline of the orifice, the size of the orifice, sill height etc. are constant. Figure 5b shows the effect of dimensionless upstream depth of flow in the main channel on the coefficient of discharge. It is apparent from the Fig. 5b that the coefficient of discharge increases with increase of H/D for other parameters kept constant. Such variation was also noticed during the experimentation for each size of the orifice and different sill height. Figure 5c shows that the coefficient of discharge increases with the increase in B/D value when other parameters such as Froude number, sill height and head above the centerline of the orifice are kept constant. This could be attributed to opposite side solid boundary. Flow near the opposite side does not participate in the flow coming out from the small orifice; however, it does for larger orifice size. The influence of the crest height (W) above the floor on the Cd is shown in Fig. 5d, which reveals that the values of Cd increases with the increase of crest height of orifice but for large crest height location effect are not prominent. This is due to the fact that an orifice at larger crest height will receive more stream lines as compared to low crest height, i.e. orifice having a larger crest height will be exposed to the large flow area in its vicinity and as a result more discharge will enter into the orifice. With further increase in the crest height boundary may not have any affect the streamlines entering into the orifice and thus Cd will become constant for any crest height. Fig. 5 Variation of Cd with (a) Froude number, (b) H/D, (c) B/D and (d) W/D 3.2
Formulation of equations for free flow orifice It is common practice to use 70% to 75% of the total available data randomly selected for
proposing/training a relationship and remaining data for its validation (Hussain et al., 2010, 2014b; Hammerstrom, 1993; Kumar et al., 2010; Bowden et al., 2005 and Khan et al., 2013). Such approach is also adopted in this study for proposing an equation for Cd using the collected data. Data obtained in the present experimental program and Hussain et al. (2010) are used for developing the relations for free flow side circular orifice. Trapezoidal method is used to solve the Eq. (1) numerically for obtaining Cd. Out of 458 data sets collected in the present study, 304 data sets selected randomly have been used to evolve relationship for Cd and remaining for validating the proposed the relationship. Using the least squares technique, Eqs. (10) & (11) are proposed for Cd, considering orifice to be large and small, respectively.
æDö Cd = 0.669 - 0.156Fr - 0.715 ç ÷ èBø
3.5
æDö Cd = 0.662 - 0.153Fr - 0.758 ç ÷ èBø
(10) 3.8
(11)
The remaining 154 unused data sets have been used to validate the proposed relationships for Cd for the computation of discharge through the orifice. The observed and computed values of discharge through orifice for the calibration data are compared graphically in Figs. 6 and 7, 6
considering orifice to be large and small, respectively. Figures 6 & 7 reveal that the computed discharge is within ±5% of the observed ones, which is a satisfactory prediction of discharge through the orifice. It is to be noted that computation of discharge using Eq. (1) requires integration of this equation for given values of D and H.
Fig. 6 Comparison of computed discharge through orifice using Eqs. (1) and (10) with observed ones considering orifice as large for validation Fig. 7 Comparison of computed discharge through orifice using Eqs. (2) and (11) with observed ones considering orifice as small for validation The average percentage error in computation of discharge through the orifice considering it as large and small are 3.99% and 4.04%, respectively. As these two values are practically the same, it is concluded that computation of discharge through the orifice can be performed considering it as small orifice within range of H/D i.e., 0.75 to 8.17 collected in the present study. Equations (10) and (11) are valid for Fr < 1 (sub critical) and 0.109 < D/B > 0.303 m. 3.3
Sensitivity analysis for free flow orifice Sensitivity analysis (SA) is the study of how the variation (uncertainty) in the output of a
mathematical model can be apportioned qualitatively or quantitatively to different sources of variation in the input of a model (Saltelli et al., 2008). In broad, sensitivity analysis is the investigation of the potential changes and errors in the parameters of any model and their impacts on the model (McCuen and Synder, 1986). It can help build confidence in the model by studying the uncertainties that are often associated with parameters in the model. Thus, sensitivity analysis is conducted to develop a comprehensive understanding of the response of the model due to change in the calibration parameter values. Sensitivity analysis can also be used to check the range of parameter values and assist in the selection of parameters for model calibration, operations and improvement of model capabilities. Keeping this in view, in the subsequent sections, an effort has been made to investigate the sensitivity of the parameters of the proposed models. The analysis presented in this study is confined to univariate case, which deals with single variable at a time. Sensitivity values computed using Eq. (12) is in absolute form and quantifies the changes in factors that results from changes in factor Fi. Relative sensitivity (RSA) can be defined as the relative change in Fo with respect to a relative change in Fi and represented by Eq. (13).
SA =
¶F0 ( f ( Fi + DFi , Fj / j ¹ i ) - f ( F1 , F2 ,......Fn ) = ¶Fi ¶Fi
RSA =
¶F0 F0 æ ¶F0 ö Fi =ç ÷´ ¶Fi Fi è ¶Fi ø F0
(12)
(13)
7
Sensitivity analysis, i.e., change in discharge due to a unit change in the head of water is carried out for the proposed discharge equation of the side circular orifice. Using Eq. (11) and (2), discharge through a side orifice in terms of variable head can be written as
æ ö V1 D ÷ A 2 gH Q = ç 0.69 - 0.035 - 0.154 ç ÷ B 0.5 g H W D + + ( ) è ø
(14)
Replacing V1 by Q/(B(H+W+0.5D)) and rearranging Eq. (14), it can be written as
é ù ( 0.69 - 0.035 D B ) A 2 g Q = ê -0.5 ú -1.5 êë H + 0.2177 AB -1 ( H + W + 0.5D ) úû
(15)
Differentiating Q with respect to H and arranging the terms, one can get
(
Dö -2.5 æ 0.69 - 0.035 ÷ A 2 g 0.5H -1.5 + 0.326 AB -1 ( H + W + D / 2 ) ç dQ è Bø = -1.5 2 dH H -0.5 + 0.217 A / B H + W + D / 2
(
(
)
)
( (
-1.5 -1 dQ a ¢ 0.5H + 0.326 AB ( H + W + D / 2 ) = -1.5 2 dH H -0.5 + 0.217 A / B ( H + W + D / 2 )
æ è
Where a ¢ = ç 0.69 - 0.035
-2.5
)
)
)
(16)
(17)
Dö ÷ A 2g Bø
The larger value of dQ/dH implies higher sensitivity. Data collected in the present study was used to compute dQ/dH for different values of diameter of the orifice. The variation of dQ/dH with H/D is depicted in the Fig. 8. A perusal of Fig. 8 reveals that the discharge through a side circular orifice is more sensitive to the low head. As the head increases, the sensitivity decreases due to the decrease in pressure variation across the cross sectional area of the orifice. Further, sensitivity is higher for the higher size of the orifice. Fig. 8 Sensitivity of the side circular orifice as a function of H/D Sensitivity values computed using Eq. (16) is in absolute form. These values cannot be used for comparision of parametric sensitivities because the computed values are not invariant to the dimensions of either factor Q or H. Dividing the numerator by Q and the denominator by H provides an estimate of the relative change in Q with respect to a relative change in H. From Eqs. (15) and (16), one can get
(
-2.5
0.5H -1.5 + 0.326 AB -1 ( H + W + D / 2 ) dQ = -1.5 Q H -0.5 + 0.217 AB -1 ( H + W + D / 2 ) Multiplying by H/H and rearranging
8
) dH
(18)
æ dQ / Q ç è dH / H
-2.5
-0.5 -1 ö 0.5H + 0.326 AB ( H + W + D / 2 ) H = ÷ -1.5 ø H -0.5 + 0.217 AB -1 ( H + W + D / 2 )
(19)
Relative sensitivity with respect to discharge and head for circular orifice of D = 15 cm is shown in Fig. 9. It is found that with the increase in head, relative sensitivity decreases. Relative sensitivity is highly pronounced for lower head and becomes constant for higher head in the main channel. Fig. 9 Relative sensitivity of the side circular orifice as a function of H/D 3.4
Variation of jet angles for free flow orifice Several aspects of flow in the main channel strongly affect the geometry of the spilling jet. It
is likely that variables affecting the jet angle of the side weir would also affect the jet angle of a side circular orifice. Thus, from the review of literature, it is found that probable variables affecting the jet angle φ for square side orifice would be L, W, V, H, mass density ρ, viscosity μ, and g. The jet angle φ is considered as the average of the inclination of the left nappe of water jet a (degree) and the inclination of the right nappe b (degree). The functional relationship for φ may, thus, be written as
j = f1 ( D,W ,V1 , Ym , H , r , m , g )
(20)
Where φ = (a+b)/2 Taking ρ, V and D as the repeating variables, the functional relationship for Cd in terms of nondimensional parameters may, thus be written as
æ H W V1 , , ç D D gY m è
j = f2 ç
ö ÷ ÷ ø
(21)
Experimental study carried out in this study is persuaded to investigate the effect of the identified non-dimensional parameters on the φ. 3.4.1
Observations during experimentation & equation development for jet angle Water jet, issuing out from the circular orifice, was not truly circular downstream of the
orifice due to gravitational forces. Further, the centerline of water jet was inclined to the direction that is normal to the flow in the main channel as shown in Fig. 10. As the Froude number of the main channel decreases the inclination angle decreases and the jet turns normal to the flow in the main channel. Fig. 11(a & b) shows the jet and its inclination for high and low Froude numbers in the main channel, respectively. The jet coming out of the orifice takes the shape of the circular to venacontracta and then distorted downstream. Further, inclination of the left nappe of water jet a was higher than the inclination of the right nappe b (Fig. 12). Fig. 10 Jet issuing out of the orifice 9
To seek the importance of various independent variables in predicting φ, feature selection and variable screening, i.e. F-test has been carried out. When F-value of any parameter compared to the Fvalue of another parameter having very low F-value, then the least F value parameter is dropped because this parameter is considered as not affecting the whole value of the equation (Lomax, 2007; Hussain et al., 2014b; Khan et al., 2013). All the three variables of Eq. (21) were taken for feature selection and variable screening process with φ as the dependent variable. As shown in Fig. 13, Fr possesses a high F-value, followed by H/D. The parameter W/D shows least importance and is, therefore, dropped while deriving the relationship for φ. The effect of the dimensionless parameters Fr, H/D and W/D on the jet angle was examined. A thorough data analysis reveals that Fr and H/D are, indeed, the predominant parameters which affect the φ. A further assessment revealed that the Froude number dominantly effect on the jet angle: as the correlation coefficient between φ and Fr is 0.79 (Table 2). The jet angle φ increases with increase of Fr (Fig. 14a). For the range of data used in the present study, φ is unaffected by the parameters W/D -that is also clear from the Table 2 i.e., in comparison to other parameters W/D has a least correlation coefficient. A perusal of Fig. 14b indicates that with the increase of H/D, there is decrease in φ. Fig. 11 Angular variation of the jet for circular orifice for (a) high Fr = 0.35 and (b) low Fr = 0.15 Fig. 12 Variation of the jet angle a and b with H/D
Table 2 Correlation matrix among variables of Eq. (21)
φ(obs) Fr W/D H/D
φ(obs) 1 0.79 0.15 0.59
Fr 1 -0.21 0.31
W/D
1 0.14
H/D
1
Fig. 13 Importance of various independent inputs in predicting output (φ) Fig. 14 Variation of φ with (a) Fr and (b) H/D The following equation is proposed for φ using 206 data sets selected randomly out of 276 data sets collected in the present study and invoking the least squares technique,
j = 16 + 44Fr - 2.5
H D
(22)
10
The R2 value of proposed Eq. (22) is 0.8. Eq. (22) is validated using the remaining 70 data sets for the computation of the jet angle of circular orifice. A graphical comparison between the observed and computed φ for the validation data is shown in Fig. 15. It is apparent from the Fig. 15 that the computed φ is within ±20% of the observed ones. It can be concluded that Froude number has a significant effect on the jet angle. The spilling jet angle (φ) is positively correlated with the upstream Froude number and negatively correlated with H/D. Ultimately; an expression for estimating of φ is developed. Fig. 15 Comparison between the computed and observed φ values for the validation 4.
Results and analysis for submerged orifice
4.1
Discharge calculation for fully submerged orifice Out of the 170 data sets collected in the present study, selected 127 data sets have randomly
been used to evolve the relationship for Cd and remaining for validating the proposed relationship. Using the least square technique, the following relationship for Cd has been evolved.
Cd = 0.7 - 0.294Fr - 0.023
D B
(23)
The observed and computed values of discharge through orifice using Eqs. (9) and (23) for the 43 test data are compared graphically in Fig. 16, which reveals that the computed discharge is within ±10% of the observed ones, which can be considered satisfactory prediction of discharge through the submerged orifice. Fig. 16 Comparison between the observed and computed discharge values using the proposed equation for the validation data 4.2
Discharge calculation for partially submerged orifices The above technique of least square for finding the relationship for Cd of submerged flow is
also used for partially submerged orifice. A total 91 data sets were collected in the study for the partially submerged flow orifice. Out of the 91 data sets, selected 64 data sets have randomly been used to evolve relationship for Cd and remaining for validating the proposed the relationship. Using the least square technique, the following relationship for Cd is
Cd = 0.59 - 0.074F - 0.165
D B
(24)
The remaining 25 data sets, not used in the derivation of Eq. (24), were used to validate the proposed relationship for Cd i.e., Eq. (24), for the computation of discharge through the orifice. The observed and computed values of discharge through orifice using Eqs. (4) and (24) for the test data are compared graphically in Fig. 17, which reveals that the computed discharge is within ±20% of the 11
observed ones, which can be considered satisfactory prediction of discharge under the submerged condition. Fig. 17 Comparison between the observed and computed discharge values using the proposed equations for the validation 4.3
Cd of submerged orifice in terms of Cd of free flow orifice Under submerged flow condition, the jet coming out of the orifice is fully or partly
submerged. The pressure of the issued jet is high compared to the free flow condition and it further increases with the increase of submergence resulting in a reduced value of the diverted discharge. The Cd of the orifice under submerged flow condition may be related to H3/ H1 and Cd under free flow conditions as
é æ H ö a1 ù Cd ( sub) = Cd ( free) ê1 - çç 3 ÷÷ ú êë è H1 ø úû
b1
(25)
Where a1 and b1 are constants and can be obtained by fitting the data of the coefficient of discharge and H3/H1. Eq. (25) is non-linear; therefore, a computer code is written for a grid search method to obtain values of constants a1 and b1 of Eq. (25) for best matching of computed Cd with the observed values. Out of 261 data sets collected in the present study for the submerged orifice, 190 data sets selected randomly were used to evolve the relationship for the Cd (sub). Here Cd (free) shall be calculated from the Eq. (10). For submerged circular orifice, the obtained value of a1 = 1.5 and b1 = 0.6, thus
é æ H ö1.5 ù Cd ( sub ) = Cd ( free ) ê1 - ç 3 ÷ ú êë è H1 ø úû
0.6
(26)
The remaining 71 data sets, not used in the derivation of Eq. (26), were used next to validate the proposed relationship for Cd. The observed and computed values of discharge through orifice using Eqs. (26) and (8) for the validation data are compared graphically in Fig. 18, which reveals that the computed discharge is within ±12% of the observed values, which is a satisfactory prediction of discharge through the submerged orifice.
Fig. 18 Comparison between the observed and computed discharge values using the proposed equation for the (a) calibration and (b) validation It may be noted that the effect of other variables viz: Fr, B and D is already accounted for through the use of Eq. (10) for the determination of the Cd (submerged). Eq. (26) is valid for both fully submerged and partially submerged flow. Thus, for calculating Q under submerged conditions 12
first the Cd is calculated for free flow conditions using Eq. (10) and then it will be used in Eq. (26). The obtained Cd will be used in Eq. (8) to compute Q. It is advocated to use Eq. (26) for submerged orifice as it gives more accurate results compared to Eq. (23). 5.
Conclusions It is concluded from the present study that the coefficient of discharge depends mainly on the
approach channel Froude number and the ratio of width of the orifice and bed width of the channel under free flow conditions. The computed discharges using the proposed relationships for circular orifice, by considering the orifice as large and small, were within ±5% of the observed values. The average percentage error in computation of discharge through the circular orifice considering it as large and small are 3.99% and 4.04%, respectively, which are practically the same. Therefore, discharge through the side orifices in open channel can be computed treating it as small orifice within range of H/D = 0.75 to 8.17 for circular orifices. Discharge through side orifice is more sensitive to the low head above the center of the orifice. As the head increases, the sensitivity decreases due to the decrease in pressure variation across the cross sectional area of the orifice. Further, the sensitivity is higher for the higher size of the orifice. Relative sensitivity is pronounced for lower head above the center of the orifice and becomes constant for higher head in the main channel. The Froude number of the flow in the main channel is a dominant factor that affects the jet angle. The computed discharges using the proposed relationships of the coefficient of discharge for circular orifices under full and partial submerged conditions were within ±10% and ±20% of the observed values, respectively. The computed values of discharge through side circular orifice using Cd in terms of Cd of free flow are within ±10% of the observed values. It is advisable to use Eq. (26) for finding the Cd (sub) for partial or full submerged circular orifice, respectively. References [1]
Athar, M., Ansari, M.A. and Haque, M., 1996. Effect of Relative Roughness on Critical Reynolds Number in Pipes. II National Conference of Fluid Machinery at P.S.G. College of Tech, Coimbatore, Proc. 28-29, 269-277.
[2]
Bowden, G.J., Maier, H.R., Dandy, G.C., 2005. Input determination for neural network models in water resources applications. Part 2. Case study: forecasting salinity in a river. Journal of hydrology, Elsevier 301, 93-107.
[3]
Ghodsian, M., 2003. Flow through side sluice gate. Journal of Irrigation and Drainage Engineering, ASCE 129(6), 458-62.
[4]
Gill, M.A., 1987. Flow through side slots. Journal of Environmental Engineering, ASCE 113(5), 1047-1057.
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[5]
Gregg, W.B., Werth, D.E., Frizzell, C., 2003. Determination of discharge coefficients for hydraulic sparger design. Journal of Pressure Vessel Technology, ASME 126(3), 354 – 359.
[6]
Hammerstrom, D., 1993. Working with neural networks. IEEE Spectrum, 30(7), 46-53.
[7]
Hussain, A., Ahmad, Z., Asawa, G.L., 2010. Discharge characteristics of sharp-crested circular side orifices in open channels. Journal of Flow Measurement and Instrumentation, Elsevier 21(3), 418-24.
[8]
Hussain, A., Ahmad, Z., Asawa, G.L., 2011. Flow through sharp-crested rectangular side orifices under free flow condition in open channels. Agricultural Water Management, Elsevier 98(10), 1536-1544.
[9]
Hussain, A., Ahmad, Z., Ojha, C.S.P., 2014a. Analysis of flow through lateral rectangular orifices in open channels. Journal of Flow Measurement and Instrumentation. Elsevier 36(10), 32-35.
[10] Hussain, S., Hussain A., Ahmad, Z., 2014b. Discharge characteristics of orifice spillway under oblique approach flow. Journal of Flow Measurement and Instrumentation, Elsevier 39(3), 0918. [11] Khan, D., Hussain, A., Ahmad, Z., 2013. Energy dissipation of skimming flow over stepped spillway. International water power and dam construction, Dam Engineering XXIII (4), 187206. [12] Kumar, S., Ahmad, Z., Kothyari, U.C., Mittal, M.K., 2010. Discharge characteristics of a trench weir. Flow Measurement and Instrumentation. Elsevier 21(2), 80-87. [13] Lomax, R.G., 2007. Statistical Concepts: A Second Course. Lawrence Erlbaum Associates, Mahwah, NJ. [14] McCuen, R.H., Synder, W.M., 1986. Hydrologic Modeling: Statistical Methods and Applications, Prentice-Hall, Englewood Cliffs, NJ. [15] Ojha, C.S.P., Berndtsson, R., Chandramoulli, P.N., 2010. Fluid Mechanics and Fluid Machinery. Oxford University Press, New Delhi. [16] Ojha, C.S.P., Subbaiah, D., 1997. Analysis of flow through lateral slot. Journal of Irrigation and Drainage Engineering, ASCE 123(5), 402-405. [17] Ramamurthy, A.S., Udoyara S.T., Serraf S., 1986. Rectangular lateral orifices in open channel. ASCE Journal of Environmental Engineering 135(5): 292-98. [18] Ramamurthy, A.S., Udoyara ST, Rao, M.V.J., 1987. Weir orifice units for uniform flow distribution. ASCE Journal of Environmental Engineering 113(1): 155-166. [19] Rouse, H., 1970. Elementry Mechanics of Fluids. Wiley Eastern, New Delhi. [20] Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S., 2008. Global Sensitivity Analysis: The Primer John Wiley & Sons, Chichester. [21] Shammaa, Y., Zhu, D.Z., Rajaratnam, N., 2005. Flow Upstream of Orifices and Sluice Gates. Journal of Hydraulic Engineering, ASCE 131(2), 127-133. 14
[22] Swamee, P.K., Pathak, S.K., Ali, M.S., 1993. Weir orifice units for uniform flow distribution. Journal of Irrigation and Drainage Engineering, ASCE 119(6), 1026-1035. [23] Werth, D.E., Khan, A.A., Gregg, W.B., 2005. Experimental study of wall curvature and bypass flow effects on orifice discharge coefficients. Experiments in Fluids, 39, 485–491. [24] Yuan, S.W., 1988. Fundamentals of Fluid Mechanics, Prentice-Hall of India, New Delhi. [25] Zahiri, A., Azamathulla, H.Md., Bagheri, S., (2013). Discharge coefficient for compound sharp crested side weirs in subcritical flow conditions. Journal of Hydrology, Elseveir 480, 162-166.
Brief Bio-Data of Authors A. Hussain is currently Assistant Professor at Civil Engineering Department of Aligarh Muslim University (AMU), India. He obtained his B.Tech. (Civil) from AMU, Aligarh and M.Tech. (Hyd) & Ph.D. degree from IIT Roorkee, India. He has published 08 papers in referred international journal 10 national and international conference. His areas of interests are open channel flow, hydro power, computational hydraulics and hydraulic structures. Z. Ahmad is currently Professor of Civil Engineering at IIT Roorkee, India. He obtained his B.Tech. (Civil) degree from AMU Aligarh, M.Tech. (Hyd) degree from Univ. of Roorkee and Ph.D. degree from T.I.E.T., Patiala. He has published about 75 papers in referred national and international journals and conference proceedings. His area of research is surface water quality management, computational hydraulics and hydraulic structures. He has written two Monographs on Transport of Pollutants in Open Channels and Control Section in Open Channels for AICTE, New Delhi. He has been recipient of G.N. Nawathe, Jal Vigyan Puraskar (twice); Department of Irrigation Award. CSP Ojha joined the Department of Civil Engineering at the Indian Institute of Technology Roorkee (formerly University of Roorkee) in 1971 and is presently Professor. He has supervised more than 39 PhD. He has been teaching hydraulic engineering courses at both undergraduate and postgraduate levels, and also associated with research and consultancy activities in different aspects of water resources engineering. He has published 04 books, more than 100 research papers, and written many reports on consultancy projects. He has been recipient of several awards and has travelled abroad for various academic programmes.
Highlights Ø Relationships for flow through side circular orifice under free and submerged conditions have been developed. Ø Relationship for estimating jet angle (φ) for circular orifice has been developed. Ø Relative sensitivity is highly pronounced for low head above the center of the orifice.
15
Figure 1a
Figure 1b
Figure 1c
Figure 2
Figure 3
Figure 4
Figure 5a
Figure 5b
Figure 5c
Figure 5d
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11a
Figure 11b
Figure 12
Figure 13
Figure 14a
Figure 14b
Figure 15
Figure 16
Figure 17
Figure 18