# Sill Role Effect on the Flow Characteristics (Experimental and Regression Model Analytical)

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## Abstract

**:**

_{d}). The physical model of the sills includes rectangular sills of different dimensions. The results show that the gate opening size is inversely related to the C

_{d}for a gate without a sill. In addition, increasing the gate opening size for a given discharge decreases the relative energy dissipation, and increasing the Froude number increases the relative energy dissipation. The results also show that the C

_{d}and relative energy dissipation decrease when the width of the sill is decreased, thus increasing the total area of the flux flowing through the sluice gate and vice versa. According to the experimental results, the relative energy dissipation and the C

_{d}of the sluice gate are larger for all sill widths than without the sill. Finally, non-linear polynomial relationships are presented based on dimensionless parameters for predicting the relative energy dissipation and outflow coefficient.

## 1. Introduction

_{d}). Designers often exercise extreme care when controlling and distributing water in irrigation networks to avoid wasting water.

_{d}value of sluice gates. Rajaratnam [4] studied the free flow in sluice gates and established an equation to estimate the C

_{d}value. Swamee [5] determined the C

_{d}of sluice gates subjected to free flow conditions; the relationship included the dependence of upstream depth and gate opening. Shivapur and Prakash [6] studied the sluice gates at different positions and established a relationship for C

_{d}. Using Mathematica software, Nasehi Oskuyi and Salmasi [7] determined the C

_{d}value of sluice gates. Their results agreed well with the results of Henry [2]. Daneshfaraz et al. [8] studied the edge shapes of sluice gates and their impact on flow characteristics using FLOW-3D. They found a flow contraction coefficient for sluice gates with sharp edges upward and downward, and round edges decrease when the sluice gate opening/specific energy upstream is less than 0.4 and increase for larger ratios. Salmasi et al. [9] evaluated the C

_{d}of inclined gates using multiple machine intelligence models. Their study showed that the C

_{d}value increases with increasing angle. In Salmasi and Abraham study [10], laboratory experiments were performed to quantify the C

_{d}for oblique sluice gates. They showed that the sluice gate slope has an effect on the C

_{d}; using a gate increases the C

_{d}.

_{d}increases when a sill is employed. Salmasi and Norouzi [12] showed that a circular sill increases the C

_{d}value by at least 23–31%. Karami et al. [13] numerically investigated the effect of geometric sill shape on C

_{d}; they found that the semicircular sill has a greater effect. Salmasi and Abraham [14] carried out experiments on sills. They found that a sill plays an effective role in increasing C

_{d}. Ghorbani et al. [15] analyzed sluice gate C

_{d}using the H

_{2}O method and soft computing models. Lauria et al. [16] studied sluice gates for wide-crowned weirs. They found that it is possible to operate a weir so that viscous effects can be neglected.

_{d}, and energy dissipation, which were studied in this research. This research intends to improve the design of hydraulic control structures. Due to the importance of the subject, the C

_{d}and energy dissipation in the conditions without a sill, with a suppressed sill, and with an unsuppressed sill are investigated at various gate openings.

## 2. Materials and Methods

#### 2.1. Experimental Equipment

_{d}of the vertical sluice gate. Figure 2 shows photographs of the physical experiment.

#### 2.2. Dimensional Analysis

_{A}and E

_{B}are the specific energies of water in sections A and B, respectively. y

_{A}is the depth of water in section A or, in other words, the initial depth; y

_{B}is the depth of water in section B or the following depth; V

_{A}and V

_{B}are the velocities averaged over depth in sections A and B, respectively; and g is the gravitational acceleration. In Equation (1), α is the correction factor for kinetic energy and is equal to

_{i}is the velocity in section i, A

_{i}is the cross-sectional area of i, A is the total area of the section (A = ∑A

_{i}), and V is the velocity averaged over depth in the entire section.

_{A}as iterative variables and using the π-Buckingham, Equation (4) emerges as

_{A}and Re

_{A}are the Froude and Reynolds numbers, respectively, in section A. Some of the parameters in the above relationship, such as channel width, thickness, and sill height, have assumed specific values and are not present in the research objectives, so the effects of these parameters have been ignored. In the present study, since the flow is turbulent, the Reynolds number can be ignored [17,18,19]. To make the parameters meaningful, the dimensional analysis of the present study was summarized and calculated in Equation (5) by dividing some of them by each other.

_{0}is the upstream water depth of the sluice gate, G

_{1}is the gate opening with suppressed sill and without sill, WG

_{1}is the flow area under the gate, and C

_{d}is the discharge coefficient.

_{1}, A

_{3}, and A

_{2}are the areas of the flow without the sill and above the unsuppressed sill, respectively (Figure 1).

_{d}in the condition without a sill depends on the upstream depth and the openings. The functional dependence is [5]

_{d}are

_{total}= 2A

_{1}+ A

_{2}. Therefore, the parameters studied in the present study were presented as relation (12):

#### 2.3. Statistical Indicators

_{d}of the gate with and without a sill. For this purpose, the dependent parameters were considered a function of the independent parameters. The statistical indicators of absolute error (AE), percent relative error (RE %), root mean square error (RMSE), and coefficient of determination (R

^{2}) were used to evaluate the equations:

_{regression}and SS

_{total}represent the sum squared regression error and total squared error, respectively. The values of Equations (14)–(16) when close to the number zero and the values of the relation (17) when close to the number one indicate the high accuracy of the presented relations.

## 3. Results and Discussion

#### 3.1. Energy Dissipation of Gate without Sill

_{AB}/E

_{A}and ∆E

_{AB}/E

_{B}). One of the properties of the flow that is very important for understanding the flow behavior is the Froude number. Figure 3a,b shows the rate of change of these parameters at different apertures, where the horizontal axis is the dimensionless parameter Fr

_{A}and the vertical axis is the energy dissipation ratio between sections A and B to the flow-specific energy in sections A and B. According to Figure 3a,b, it can be observed that as the Froude number increases, the ratio of energy dissipation upstream and downstream of the hydraulic jump increases. The depth of the hydraulic jump increases due to the reduction of the gate opening compared to large gate openings at different discharges, resulting in increased energy dissipation. To illustrate the results and provide better agreement with the data, the energy dissipations at different discharges and openings are shown in Figure 3c,d, respectively. The relative energy dissipation is lowest when the aperture is 0.04 m; decreasing the gate opening rate increases the relative energy dissipation. As the opening rate increases, the velocity of the flow that passes under the gate decreases, and, as a result, the initial depth of the flow increases, which decreases the specific energy in section A. The increase in depth results in a decrease in the following depth compared to the smaller openings, resulting in a decrease in specific energy in section B as in section A. As can be seen in Figure 3c,d, the relative energy dissipation at a constant flow is greater for an orifice of 0.01 m than for the orifices of 0.02 and 0.04 m, respectively. Thus, the average ratio of energy dissipation to upstream energy is 15.93% and 56% higher for an orifice of 0.01 m than for orifices of 0.02 and 0.04 m, respectively. For the downstream opening, this value is 41.32% and 83.27%, respectively.

#### 3.2. C_{d} without Sill

_{d}increases as the ratio of upstream water depth to the sluice opening (H

_{0}/G

_{1}) is increased. Moreover, the C

_{d}decreases as the gate opening increases. In other words, the C

_{d}is inversely related to the opening rate. A parameter that affects the C

_{d}is the upstream water depth. If the sluice gate opening increases, the upstream water depth upstream decreases, and this factor will reduce the C

_{d}at larger openings. If you decrease the gate opening, the flow converges, and the area below the gate decreases, increasing the C

_{d}. As shown in Figure 4a, the maximum C

_{d}value for the specific (H

_{0}/G

_{1}) is at an opening of 0.01 m and the lowest value at 0.04 m. Figure 4b shows the diagram of stage discharges for the different sluice openings. For a given discharge, the size of the gate opening is inversely related to the upstream water depth and decreases as the opening increases. Here, the C

_{d}for an opening of 0.01 m is, on average, higher than that for the 0.02 and 0.04 m openings, namely, 7.75% and 16.51%, respectively, and a maximum of 16.62% and 28.9%, respectively.

#### 3.3. Energy Dissipation of the Gate with Sill

_{B}/y

_{A}) is a function of Fr

_{A}. To study the effect of the sill below the gate on the subsequent flow depth, the plot of the relative depth of the hydraulic jump is shown in Figure 7. It can be observed that as the Froude number increases, the relative depth of the hydraulic jump linearly increases. The reason for this is the significant effect of the sill on increasing the sequent depth significantly on the jump depth increase. As the sill width increases, the jump depth is greater for the same discharge than for a sill with a smaller width.

_{A}increases, the ratio between the following depth and the initial depth (y

_{B}/y

_{A}), i.e., the wave height (y

_{B}-y

_{A}), increases. The direct relationship of the energy output to the third power of the expression (y

_{B}-y

_{A}) causes the amount of energy output to be very sensitive to the intensity and strength of the jump. A comparison between the present results and those of [25] shows that applying the sill leads to a decrease in the following depth (Figure 7).

#### 3.4. C_{d} with Sill

_{d}decreases with a decrease in the sill width. Thus, the sill with the smallest width has the smallest C

_{d}. The reason for increasing the C

_{d}is related to the even flow distribution across the gate. When the width of the sill below the gate increases, the sill acts as a barrier; downstream of the sill, the water above the sill is discharged all at once and uniformly. The return flow decreases to a minimum as the width increases, which increases the C

_{d}. In addition, the C

_{d}tends to increase as the ratio between the upstream depth and the width of the sill increases.

_{d}increases with increasing runoff when the sill height is kept constant in all models and the sill width increases. Figure 8c shows the effect of the opening and the flow area under the gate. It is observed that by increasing the ratio of the flow area above the sill to the flow area on both sides of the sill, the C

_{d}increases. Increasing the sill width, the A on both sides of the sill decreases, and, therefore, at the sill with a larger width, the A

_{total}will be less than at the sill with a smaller width, which leads to an increase in C

_{d}. A sill with a lower size below the gate increases the C

_{d}compared to the mode without a sill. This increase is due to the reduction of the total area underneath the gate.

_{d}was plotted against the H

_{0}-Z/G

_{1}to find the best fit and compare the data. A comparison was made between the C

_{d}without a sill and suppressed sills with the same opening (Figure 9). Figure 9 shows that a sill below the sluice gate increases the flow rate and improves the system performance compared to the without sill situation. At constant discharge, the upstream depth with a sill is less than without a sill. The presence of a sill that is the same width as the channel increases the discharge (at an opening of 0.01 m) by an average of 7.75%.

_{d}for the sample. For similar discharges, the C

_{d}for the unsuppressed sill is higher than the no-sill condition, and this trend increases with increasing sill width. Therefore, unsuppressed sills can be considered and used because they increase flow efficiency and prevent sediment accumulation behind the gate.

_{d}of the gate with and without sill conditions. First, the non-linear form of the proposed equations for relative energy dissipation as a function of the dimensionless parameters was determined. The proposed equations’ general forms were considered Equation (18).

_{B}/y

_{A}and the dependent parameters ∆E

_{AB}/E

_{A}and ∆E

_{AB}/E

_{B}to verify the accuracy of Equations (19) and (20). Figure 10b,d shows the graphs of the percent relative error against the effective dimensionless parameter y

_{B}/y

_{A}. In these figures, a large range of data lies within the relative error range of ±5%. This shows that the proposed equations have very good accuracy in predicting the relative energy dissipation.

_{d}of the gate without the sill condition.

_{d}of the present study without the sill state with the results of the previous studies.

_{d}obtained from the proposed Equation (21) into Equation (5), a comparison was made between the discharge rate obtained from the experimental results and Equation (5). In Figure 11b, it can be seen that a large area of the data is within the error range of ±1.5%. This shows that the formula is very accurate, such that more than 82% of the data have an error of less than ±1.5%.

_{total}= A

_{2}. One way to compare the experimental results with Equation (22) to determine its accuracy is to examine the magnitude of the difference between the C

_{d}obtained from the experimental results and the C

_{d}calculated using the predicted equation.

## 4. Conclusions

_{d}with and without sills below the lock gate. The results show that the relative energy dissipation without a sill and at different openings is inversely related to the gate openings. As the opening of the sluice gate increases, the relative energy dissipation decreases due to the increase in initial depth. As a result, a decrease in specific energy in section A and a decrease in exit depth and specific energy in section B can be observed. The flow of the gate without a sill decreases as the opening increases. The C

_{d}of the sluice gate without a sill is most affected by the upstream flow depth and the opening. At constant discharge, as the opening increases, the upstream water depth of the sluice gate decreases, and the C

_{d}tends to decrease in proportion to the lower opening of the sluice gate. The comparison of the sluice gates with different openings shows that the hydrodynamic force on the gate increases with a decrease in the number of openings for the same discharge rate. According to the results, in all cases where the sill is used under the sluice gate and at all discharge rates, the relative energy dissipation is greater than for the free classical hydraulic jump without a sill. The energy dissipation increases as the Fr

_{A}increases. A comparison of the results for the C

_{d}with sill and without sill shows that a sill under the sluice gate increases the C

_{d}. In the present study, the general equation for discharge calculation was developed for an unsuppressed sill, and the calculations were performed based on the new equation of the present study, which can be used for unsuppressed symmetrical sills. The presence of a sill the same width as the channel below the sluice gate increases the C

_{d}compared to the condition without a sill at a fixed opening. From the comparison of the rate of increase of the C

_{d}, it can be concluded that the sill width parameter has the greatest influence on the C

_{d}. Finally, non-linear polynomial regression relationships were determined to calculate the energy dissipation related to the upstream and downstream hydraulic jump. Non-linear regression equations were also established to predict the C

_{d}.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**(

**a**,

**b**) Changes in the relative energy dissipation against Froude number/ (

**c**,

**d**) Changes in the relative energy dissipation in different discharges.

**Figure 4.**(

**a**) Variation of C

_{d}versus H

_{0}/G

_{1}; (

**b**) stage-discharge diagram at various openings.

**Figure 8.**The C

_{d}changes against the (

**a**) upstream depth to sill width; (

**b**) height to sill width; (

**c**) flow area above the sill to flow area at the sill sides.

**Figure 10.**(

**a**,

**c**) Comparison of calculated and experimental values of the relative energy dissipation; (

**b**,

**d**) the percentage relative error dispersion.

**Figure 11.**(

**a**) Comparison diagram of calculated and experimental values of C

_{d}; (

**b**) percentage relative error versus experimental discharge.

**Figure 12.**(

**a**) Comparison diagram of calculated and experimental C

_{d}values with sill in different widths; (

**b**) dispersion of the percentage relative error.

Gate Openings (m) | |||||
---|---|---|---|---|---|

0.01 | 0.02 | 0.04 | |||

q (m ^{3}/s·m) | R (kg/m) | q (m ^{3}/s·m) | R (kg/m) | q (m ^{3}/s·m) | R (kg/m) |

0.0111 | 7.77 | 0.0167 | 1.96 | 0.0278 | 0.21 |

0.0139 | 17.53 | 0.0194 | 4.02 | 0.0306 | 0.49 |

0.0167 | 35.52 | 0.0222 | 7.36 | 0.0319 | 0.69 |

0.0194 | 59.62 | 0.025 | 11.90 | 0.0333 | 0.93 |

0.0222 | 89.92 | 0.0278 | 17.58 | 0.0347 | 1.27 |

- | - | 0.0306 | 24.78 | 0.0361 | 1.61 |

- | - | 0.0333 | 35.61 | 0.0375 | 1.96 |

- | - | - | - | 0.0389 | 2.36 |

- | - | - | - | 0.0417 | 3.58 |

Q (L/min) | Sill widths (m) | |||||||

0.025 | 0.05 | 0.075 | 0.10 | |||||

α_{A} | α_{B} | α_{A} | α_{B} | α_{A} | α_{B} | α_{A} | α_{B} | |

450 | - | - | - | - | 1.024 | 1.002 | 1.045 | 1.003 |

500 | - | - | 1.021 | 1.001 | 1.018 | 1.005 | 1.040 | 1.006 |

550 | 1.022 | 1.001 | 1.013 | 1.002 | 1.015 | 1.008 | 1.030 | 1.009 |

575 | 1.019 | 1.002 | 1.009 | 1.009 | 1.007 | 1.025 | 1.015 | 1.018 |

600 | 1.010 | 1.008 | 1.003 | 1.030 | 1.005 | 1.038 | 1.005 | 1.038 |

625 | 1.008 | 1.020 | 1.002 | 1.065 | 1.002 | 1.065 | 1.001 | 1.062 |

650 | 1.006 | 1.050 | 1.001 | 1.075 | 1.001 | 1.080 | 1.001 | 1.095 |

675 | 1.001 | 1.075 | 1.000 | 1.095 | 1.001 | 1.098 | 1.001 | 1.100 |

700 | 1.001 | 1.090 | 1.000 | 1.100 | 1.000 | 1.105 | 1.001 | 1.108 |

750 | 1.001 | 1.100 | 1.000 | 1.105 | 1.000 | 1.130 | 1.000 | 1.119 |

Q (L/min) | Sill widths (m) | |||||||

0.15 | 0.20 | 0.25 | 0.30 | |||||

α_{A} | α_{B} | α_{A} | α_{B} | α_{A} | α_{B} | α_{A} | α_{B} | |

300 | - | - | - | - | 1.009 | 1.004 | - | - |

325 | - | - | - | - | - | - | 1.001 | 1.050 |

350 | - | - | - | - | 1.007 | 1.009 | 1.001 | 1.084 |

375 | - | - | - | 1.005 | 1.030 | 1.000 | 1.100 | |

400 | - | - | - | - | 1.001 | 1.075 | - | - |

450 | - | - | 1.020 | 1.002 | 1.000 | 1.098 | - | - |

500 | 1.030 | 1.002 | 1.015 | 1.005 | 1.000 | 1.107 | - | - |

550 | 1.028 | 1.004 | 1.012 | 1.009 | - | - | - | - |

575 | 1.028 | 1.006 | 1.007 | 1.020 | - | - | - | - |

600 | 1.015 | 1.008 | 1.003 | 1.055 | - | - | - | - |

625 | 1.010 | 1.010 | 1.001 | 1.085 | - | - | - | - |

650 | 1.008 | 1.020 | 1.000 | 1.140 | - | - | - | - |

675 | 1.001 | 1.055 | - | - | - | - | - | - |

700 | 1.001 | 1.085 | - | - | - | - | - | - |

750 | 1.000 | 1.130 | - | - | - | - | - | - |

Q (L/min) | Sill Widths (m) | |||||||||||

0.025 | 0.05 | |||||||||||

with α E _{A} (m) | without α E _{A} (m) | RE (%) | with α E _{B} (m) | without α E _{B} (m) | RE (%) | with α E _{A} (m) | without α E _{A} (m) | RE (%) | with α E _{B} (m) | without α E _{B} (m) | RE (%) | |

500 | - | - | - | - | - | - | 0.096 | 0.094 | 1.56 | 0.073 | 0.073 | 0.01 |

550 | 0.107 | 0.105 | 1.63 | 0.086 | 0.086 | 0.01 | 0.111 | 0.110 | 1.04 | 0.077 | 0.077 | 0.03 |

575 | 0.115 | 0.113 | 1.43 | 0.079 | 0.079 | 0.03 | 0.121 | 0.120 | 0.71 | 0.081 | 0.081 | 0.11 |

600 | 0.124 | 0.123 | 0.80 | 0.082 | 0.082 | 0.11 | 0.130 | 0.130 | 0.21 | 0.084 | 0.083 | 0.39 |

625 | 0.136 | 0.135 | 0.66 | 0.088 | 0.088 | 0.23 | 0.141 | 0.141 | 0.13 | 0.089 | 0.089 | 0.72 |

650 | 0.146 | 0.145 | 0.50 | 0.090 | 0.090 | 0.59 | 0.151 | 0.151 | 0.07 | 0.092 | 0.092 | 0.81 |

675 | 0.155 | 0.155 | 0.08 | 0.096 | 0.095 | 0.78 | 0.162 | 0.162 | 0.01 | 0.098 | 0.097 | 0.90 |

700 | 0.165 | 0.165 | 0.07 | 0.098 | 0.097 | 0.93 | 0.175 | 0.175 | 0.01 | 0.100 | 0.099 | 0.97 |

750 | 0.186 | 0.186 | 0.04 | 0.106 | 0.105 | 0.93 | 0.204 | 0.204 | 0.01 | 0.107 | 0.106 | 0.93 |

Q (L/min) | Sill widths (m) | |||||||||||

0.075 | 0.10 | |||||||||||

with α E _{A} (m) | without α E _{A} (m) | RE (%) | with α E _{B} (m) | without α E _{B} (m) | RE (%) | with α E _{A} (m) | without α E _{A} (m) | RE (%) | with α E _{B} (m) | without α E _{B} (m) | RE (%) | |

450 | 0.088 | 0.086 | 1.75 | 0.069 | 0.069 | 0.07 | 0.099 | 0.095 | 3.41 | 0.069 | 0.069 | 0.04 |

500 | 0.104 | 0.102 | 1.39 | 0.076 | 0.076 | 0.08 | 0.118 | 0.114 | 3.18 | 0.077 | 0.077 | 0.07 |

550 | 0.123 | 0.122 | 1.22 | 0.083 | 0.083 | 0.08 | 0.134 | 0.131 | 2.46 | 0.083 | 0.083 | 0.09 |

575 | 0.137 | 0.136 | 0.59 | 0.089 | 0.089 | 0.09 | 0.148 | 0.146 | 1.27 | 0.089 | 0.089 | 0.16 |

600 | 0.151 | 0.150 | 0.39 | 0.093 | 0.092 | 0.09 | 0.165 | 0.164 | 0.44 | 0.093 | 0.093 | 0.32 |

625 | 0.164 | 0.164 | 0.13 | 0.095 | 0.094 | 0.09 | 0.182 | 0.182 | 0.09 | 0.097 | 0.096 | 0.50 |

650 | 0.179 | 0.179 | 0.09 | 0.101 | 0.100 | 0.10 | 0.197 | 0.197 | 0.07 | 0.102 | 0.101 | 0.72 |

675 | 0.193 | 0.193 | 0.07 | 0.103 | 0.102 | 0.10 | 0.213 | 0.213 | 0.07 | 0.105 | 0.105 | 0.72 |

700 | 0.207 | 0.207 | 0.01 | 0.107 | 0.107 | 0.11 | 0.230 | 0.230 | 0.05 | 0.109 | 0.109 | 0.75 |

750 | 0.247 | 0.247 | 0.01 | 0.117 | 0.116 | 0.12 | 0.266 | 0.266 | 0.01 | 0.118 | 0.117 | 0.74 |

Q (L/min) | Sill widths (m) | |||||||||||

0.15 | 0.20 | |||||||||||

with α E _{A} (m) | without α E _{A} (m) | RE (%) | with α E _{B} (m) | without α E _{B} (m) | RE (%) | with α E _{A} (m) | without α E _{A} (m) | RE (%) | with α E _{B} (m) | without α E _{B} (m) | RE (%) | |

450 | - | - | - | - | - | - | 0.155 | 0.152 | 1.77 | 0.079 | 0.079 | 0.01 |

500 | 0.135 | 0.131 | 2.51 | 0.084 | 0.084 | 0.02 | 0.189 | 0.186 | 1.36 | 0.088 | 0.088 | 0.03 |

550 | 0.167 | 0.163 | 2.43 | 0.088 | 0.088 | 0.03 | 0.242 | 0.239 | 1.11 | 0.098 | 0.098 | 0.05 |

575 | 0.183 | 0.179 | 2.46 | 0.092 | 0.092 | 0.05 | 0.266 | 0.264 | 0.66 | 0.101 | 0.101 | 0.11 |

600 | 0.205 | 0.202 | 1.35 | 0.098 | 0.098 | 0.07 | 0.302 | 0.302 | 0.29 | 0.105 | 0.105 | 0.30 |

625 | 0.222 | 0.220 | 0.91 | 0.101 | 0.101 | 0.07 | 0.332 | 0.332 | 0.05 | 0.114 | 0.113 | 0.40 |

650 | 0.251 | 0.250 | 0.74 | 0.108 | 0.108 | 0.12 | 0.393 | 0.393 | 0.01 | 0.116 | 0.116 | 0.66 |

675 | 0.280 | 0.280 | 0.09 | 0.111 | 0.111 | 0.32 | - | - | - | - | - | - |

700 | 0.303 | 0.303 | 0.08 | 0.116 | 0.116 | 0.47 | - | - | - | - | - | - |

750 | 0.349 | 0.349 | 0.03 | 0.123 | 0.122 | 0.70 | - | - | - | - | - | - |

Q (L/min) | Sill widths (m) | |||||||||||

0.25 | 0.30 | |||||||||||

with α E _{A} (m) | without α E _{A} (m) | RE (%) | with α E _{B} (m) | without α E _{B} (m) | RE (%) | with α E _{A} (m) | without α E _{A} (m) | RE (%) | with α E _{B} (m) | without α E _{B} (m) | RE (%) | |

300 | 0154 | 0.153 | 0.83 | 0.062 | 0.062 | 0.03 | 0.375 | 0.375 | 0.08 | 0.068 | 0.068 | 0.25 |

325 | - | - | - | - | - | - | 0.425 | 0.425 | 0.06 | 0.073 | 0.072 | 0.40 |

350 | 0.214 | 0.213 | 0.62 | 0.073 | 0.073 | 0.05 | 0.492 | 0.492 | 0.04 | 0.078 | 0.078 | 0.45 |

375 | 0.256 | 0.255 | 0.48 | 0.076 | 0.076 | 0.17 | - | - | - | - | - | - |

400 | 0.315 | 0.315 | 0.08 | 0.085 | 0.084 | 0.34 | - | - | - | - | - | - |

450 | 0.398 | 0.398 | 0.01 | 0.094 | 0.094 | 0.42 | - | - | - | - | - | - |

500 | 0.495 | 0.495 | 0.01 | 0.103 | 0.103 | 0.42 | - | - | - | - | - | - |

Types of Channel | α Values | ||
---|---|---|---|

Min | Mean | Max | |

Regular channels, flumes, Spillways | 1.10 | 1.15 | 1.20 |

Natural channels | 1.15 | 1.30 | 1.50 |

Rivers with ice cover | 1.20 | 1.50 | 2.00 |

River valleys, over flooded | 1.50 | 1.75 | 2.00 |

B (m) | Without Sill G = 0.04 | 0.025 | 0.05 | 0.075 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | ||
---|---|---|---|---|---|---|---|---|---|---|---|

Q (L/min) | 300 | ∆E_{AB}/E_{A} (%) | - | - | - | - | - | - | - | 59.3 | 71.5 |

350 | - | - | - | - | - | - | - | 65.7 | 77.3 | ||

400 | - | - | - | - | - | - | - | 73.2 | - | ||

450 | - | - | - | 19.9 | 27.4 | - | 47.8 | 76.5 | - | ||

500 | 21.5 | - | 22.6 | 25.9 | 32.8 | 36.3 | 52.5 | 79.2 | - | ||

550 | 25.8 | 27.7 | 29.9 | 32 | 36.5 | 46 | 59.2 | - | - | ||

600 | 29.7 | 33.5 | 35.8 | 38.3 | 43.2 | 51.6 | 65.1 | - | - | ||

650 | 34.3 | 38.1 | 39.4 | 44 | 48.9 | 56.9 | 70.6 | - | - | ||

300 | ∆E_{AB}/E_{B} (%) | - | - | - | - | - | - | - | 145.9 | 251.3 | |

350 | - | - | - | - | - | - | - | 191.8 | 340.2 | ||

400 | - | - | - | - | - | - | - | 273.1 | - | ||

450 | - | - | - | 24.8 | 38.8 | - | 91.6 | 326.4 | - | ||

500 | 27.3 | - | 29.2 | 34.9 | 48.8 | 57 | 110.4 | 380.7 | - | ||

550 | 34.7 | 38.3 | 42.7 | 47 | 57.5 | 85.3 | 144.9 | - | - | ||

600 | 42.3 | 50.4 | 55.8 | 62.2 | 76.2 | 106.8 | 186.8 | - | - | ||

650 | 52.2 | 61.5 | 65 | 78.5 | 95.5 | 131.8 | 239.7 | - | - |

B (m) | Without Sill | 0.025 | 0.025 | 0.075 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | ||
---|---|---|---|---|---|---|---|---|---|---|---|

Q (L/min) | 350 | C_{d} (-) | - | - | 0.5554 | 0.5573 | 0.5683 | 0.5969 | 0.6515 | 0.6897 | 0.7797 |

375 | - | - | 0.5592 | 0.5617 | 0.5763 | 0.6083 | 0.6538 | 0.6860 | 0.7760 | ||

400 | 0.5377 | 0.5445 | 0.5604 | 0.5658 | 0.5769 | 0.6143 | 0.6520 | 0.6818 | - | ||

450 | 0.5667 | 0.5675 | 0.5820 | 0.5838 | 0.5938 | 0.6310 | 0.6625 | 0.6868 | - |

**Table 7.**Results of statistical indicators comparing experimental results with Equations (18) and (19).

Mode | Mean AE (-) | Mean RE (%) | Max Relative Error (%) | Min Relative Error (%) | RMSE (-) | R^{2} |
---|---|---|---|---|---|---|

$\frac{{\u2206E}_{AB}}{{E}_{A}}$ | 0.0134 | 3.12 | 14.49 | −15.31 | 0.0194 | 0.985 |

$\frac{{\u2206E}_{AB}}{{E}_{B}}$ | 0.0233 | 1.62 | 5.83 | −5.7 | 0.0410 | 0.998 |

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## Share and Cite

**MDPI and ACS Style**

Abbaszadeh, H.; Norouzi, R.; Sume, V.; Kuriqi, A.; Daneshfaraz, R.; Abraham, J.
Sill Role Effect on the Flow Characteristics (Experimental and Regression Model Analytical). *Fluids* **2023**, *8*, 235.
https://doi.org/10.3390/fluids8080235

**AMA Style**

Abbaszadeh H, Norouzi R, Sume V, Kuriqi A, Daneshfaraz R, Abraham J.
Sill Role Effect on the Flow Characteristics (Experimental and Regression Model Analytical). *Fluids*. 2023; 8(8):235.
https://doi.org/10.3390/fluids8080235

**Chicago/Turabian Style**

Abbaszadeh, Hamidreza, Reza Norouzi, Veli Sume, Alban Kuriqi, Rasoul Daneshfaraz, and John Abraham.
2023. "Sill Role Effect on the Flow Characteristics (Experimental and Regression Model Analytical)" *Fluids* 8, no. 8: 235.
https://doi.org/10.3390/fluids8080235