# Application of Simple Crested Weirs to Control Outflows from Tiles Drainage

## Abstract

**:**

^{2}) values were 0.9955, 0.9981, and 0.9980, respectively. However, the best-fitted equation for a 22.5° V-notch weir was for a slope angle of 45°. The flow equation was Q = 0.2235H

^{2.4182}, with Q in liters per minute and H in centimeters. This equation can be used for measuring flow through tubular-controlled drainage structures.

## 1. Introduction

_{3−}) to the environment and contributes to the eutrophication of aquatic ecosystems worldwide [1,2,3]. Reducing their use is the first essential stage in limiting the quantities of pollutants reaching aquatic environments and responding to the requirements of the European Water Framework Directive (2000/60/CE) [4]. To reduce the export of nitrate and other soluble nutrients from agricultural fields to surface waters, a number of mitigation measures have been developed to reduce the load of these ingredients in drainage water from tile drains or ditches [5,6]. Drainage water management (DWM), also known as controlled drainage (CD), is one of the edge-of-field strategies mainly designed to reduce nitrate load from subsurface drainage systems [7]. In North America, controlled drainage is currently employed as an agricultural beneficial management practice (BMP).

#### 1.1. Instrumental Methods to Measure Flow from Controlled Tile Drainages

^{−1}.

#### 1.2. Commonly Used V-Notch Weir Equations

_{ideal}) is described by the Formula (1) results from the application of Bernoulli’s equation.

- Q
_{ideal}is the discharge (m^{3∙}s^{−1}); - $\theta $ is the angle of the notch (°);
- $g$ is the gravitational acceleration (m∙s
^{−2}); - $H$ is the height of water above the crest (m).

- Q
_{Vnotch}is the flow discharge (m^{3∙}s^{−1}); - C
_{d}is the discharge coefficient (-).

_{d}) accounts for the geometric, viscosity, and surface tension effects. C

_{d}has been the subject of many studies as evidenced by a detailed literature review [44]. The actual measured flow is about 40% less than this, due to contraction, similar to a thin-plate orifice. In terms of the experimental discharge coefficient (C

_{d}), the recommended formula is:

- Q
_{Vnotch}is the flow discharge (m^{3∙}s^{−1}); - C is the discharge coefficient (-);
- h is the depth of water (head) behind the weir (m);
- k is the head correction factor (m).

_{d}and k) were developed for large volumes and are highly influenced by the weir geometry. Modification of this formula was used for fully contracted notches of any angle between 25° and 100° [47], and for three types of tested and calibrated water flow measurement devices [46]. This equation is written as:

- Q
_{Vnotch}is the flow discharge (m^{3∙}s^{−1}); - h
_{1e}= h_{1}+ k_{h}(m); - h
_{1}is the high of water above the crest (m); - k
_{h}is the head correction factor (0.001 m).

- Q
_{Vnotch}is the flow rate (m^{3}∙s^{−1}); - $H$ is the height of water above the crest (m).

- Q
_{Vnotch}is the flow rate (m^{3}∙s^{−1}); - $H$ is the height of water above the crest (m).

- H is the head over the weir measured [m];
- a and b are coefficients determined experimentally.

- (a)
- with a 90° notch (also known as Thomson weir) and flow formula:$${Q}_{Vnotch}=1.365{H}^{2.5}$$
- (b)
- with a half 90° notch (53°8’) and flow formula:$${Q}_{Vnotch}=0.682{H}^{2.5}$$
- (c)
- with a quarter 90° notch (9 = 28°4’) and flow formula:$${Q}_{Vnotch}=0.347{H}^{2.5}$$

^{3}∙s

^{−1}, and height (H) in m.

## 2. Materials and Methods

#### 2.1. Experimental Setup and Flow Measurement

^{3}min

^{−1}, with a ±0.5% accuracy. For the lower flow rates (3–9 dm

^{3}min

^{−1}), the same flow meter was used, employing the manufacturer’s calibration with 2.5% accuracy. For precise flow control, a measurement valve with calibration was additionally used (Boa Control Sar, KSB, Frankenthal, Germany). The air temperature during the research was practically constant and fluctuated between 20 and 21.5 °C. The water temperature varied from 19.0 to 19.5 °C. For research purposes, the measuring flume was partially modified by placing it inside the circular tube with stoplogs imitating a controlled drainage structure installed at the outlet of the main drain to the channel (Figure 1a,b).

- H is the water table level (m
^{3}∙s^{−1}); - p
_{h}is the hydrostatic pressure (kPa); - p
_{atm}is the atmospheric pressure (kPa); - $g$ is the gravitational acceleration (m∙s
^{−2}); - $H$ is the height of water above the crest (m);
- ρ is the liquid density (kg∙m
^{−3}); - ×100 is the conversion factor resulting from the conversion of units;
- +2 is the correction for the logger, determining its position relative to the bottom of the well (cm).

#### 2.2. Developing the Empirical Flow Equation

_{EXP}). Measured flow rates were plotted against the corresponding head measurements, and then a regression equation was calculated. The equation for flow over the V-notch weir took the form of power Equation (8). This regression model was obtained by linearization of a non-linear equation. To check the accuracy of this formula, the coefficient of determination (R-squared) and the root mean square error (RMSE) values were computed by using Equations (13) and (14), respectively.

^{2}indicates that more variability is explained by the model, while a lower RMSE means that a given model is better able to “fit” a dataset. This is a useful value to know because it gives us an idea of the average distance between the observed data values and the predicted data values. This is in contrast to the R-squared value of the model, which tells us the proportion of the variance in the response variable that can be explained by the predictor variable(s) in the model. Laboratory data were compared using analysis of variance testing (α = 0.05) in the software Statistica 13.3 (TIBCO Software Inc. StatSoft GmbH, Hamburg, Germany). All assumptions of data normality and equal variance (evaluated using the Shapiro–Wilk test) were met.

#### 2.3. Comparison between Actual and Theoretical Discharges

_{EXP}) was compared with seven different equations from similar research. The first analysis equation (Q

_{CAL1}) was a power model resulting from the least squares regression analysis. The second formula (Q

_{CAL2}) was a simplified equation, primarily used in practice [49].

- Q
_{Vnotch}is the flow rate (L∙s^{−1}); - $H$ is the height of water above the crest (cm).

_{CAL3}). The fourth formula (Q

_{CAL4}) was used from reference [10]. This equation has the following form:

- Q
_{Vnotch}is the flow rate (dm^{3}∙min^{−1}); - $H$ is the high of water above the crest (cm).

_{CAL5}and Q

_{CAL6}equations. The last equation for analysis (Q

_{CAL7}) was taken from reference [50]. This equation has the following form:

- Q
_{Vnotch}is the flow rate (dm^{3}∙min^{−1}); - $H$ is the height of water above the crest (cm).

## 3. Results and Discussion

#### 3.1. Empirical Flow Equations with Regression Analysis

^{2}= 0.9955 and R

^{2}= 0.9980, respectively (for the 30° and 60°). The smallest RMSE was 0.255 dm

^{3}∙min

^{−1}for 44 flow rate calibration measurements (slope of crest angle 45°). In the other cases, the RMSE was higher by 36% and 30%, respectively (0.346 (30°) and 0.332 (60°) dm

^{3}∙min

^{−1}). Regression equations showing the relationship between measured (Q

_{REF}) and calculated discharge (Q

_{CAL}) were developed based on checked readings. These equations were shown in plots of dispersion (Figure 3, Figure 4 and Figure 5). The red color indicates a regression line with the 95% confidence interval marked with a red dotted line. Measured values were marked with blue circles. The literature on V-notch weirs does not clearly specify which angle is the most appropriate in this case. According to Shen [51], if the tip of the weir plate is thicker than 1/16 inch, the downstream edges of the notch should be chamfered to make an angle of not less than 45° with the surface of the crest. Prakash [52] tested weirs with similar chamfered crests, but they were additionally inclined. Herschy [53], in his chapter titled “Flow through weirs, flumes, orifices, sluices and pipes”, gave instructions on how to chamfer the downstream edge of the notch. According to his recommendation, this angle should not be less then 60°. The same suggestions were also made in the books by Bos [54] and Brassington [55]: “The V-notch weir should have a lip of between 1 mm and 2 mm and the downstream face should slope away from the lip at an angle of at least 60°”. Eli [56] also stated that this angle should be 60° ± 1° at ¼” (6.35 cm) thickness.

_{EXP}and calculated discharges Q

_{CAL}from the equations (Figure 3, Figure 4 and Figure 5), significant agreement is observed between calculated and measured discharges. Residuals of observed values versus calculated discharges were generally within the ±4% error limit and are acceptable (Figure 6). Taking into account the ranges of analyzed data, it seems that all the proposed equations predict discharges with high accuracy.

#### 3.2. Comparison with Previously Reported V-Notch Weirs

_{CAL6}, gave the best results of fitting to the empirical data for all the examined chamfered angles. However, the smallest measurement deviations were observed at the angle of 30° and 45°, and the largest at the angle of 60° (Figure 7, Figure 8 and Figure 9). This situation results from the fact that for the last analyzed variant, the height of operation of the V-notch when the nappe was stabilized, was the smallest. This is the moment when the measurement error is usually the greatest for all V-notch weirs. It should be noted that, according to the authors’ recommendations [46,47], the use of Formulas (4) and (5) requires the appropriate Cd to be maintained so that the overflows operate with full contraction. In the case of drainage wells, due to their small size, the installed V-notch weirs always operate with partial contraction. This means that the developed constant Cd indicators cannot be used in practice, and it is necessary to calibrate such weirs individually. Similarly, Shokrana and Ghane [39] claim, when using a cylindrical control structure, developing a step-discharge equation before monitoring drainage is recommended. According to the United States Bureau of Reclamation [48], the practical criterion for a partially contracted V-notch weir is H/B ≤ 0.4. The next equations describing the examined relationship reasonably well were Formulas (15) (Q

_{CAL2}) and (3) (Q

_{CAL3}). The Q

_{CAL2}equation is a formula that typically relates to an angle of 22.5°. It was used in studies including references [30,31,32,33,34]. In the case of the Q

_{CAL3}equation, a simplification of Cd = 0.44 was adopted for the entire measuring range. Interestingly, using a Cd of 0.578 [57] in the second Equation (2) would give virtually the same flow outcome. According to Shen [51], for h/p ratios higher than 0.16, the discharge coefficient is mainly a function of angle θ, and it is very stable with Re values. The performed analyses clearly show that the results of equations Q

_{CAL2}and Q

_{CAL3}for all analyzed angles are higher than the conducted measurements (Figure 7, Figure 8 and Figure 9). Also, in the case of Equations (16)—Q

_{CAL4}and (17)—Q

_{CAL5}, the flow values are overestimated in relation to the measured data. It is important that for these equations, especially for angles of 45° and 60° (Figure 8 and Figure 9), the translation plot is proportional over the entire range of measurements. However, it should be borne in mind that Formula (16) from reference [10] uses a 15° V-notch weir. The last equation to compare the analysis (Q

_{CAL7}) was taken from Formula (17). Although this formula applies to triangular weirs with a notch of 45°, the results obtained for the chamfer angle of 60° are very similar to formulas Q

_{CAL2}, QC

_{AL3}, Q

_{CAL4}, and Q

_{CAL5}

_{.}These comparisons show how significant discrepancies appear in similar applications. They are also drawing attention to how important the slope of angle of the sharp-crested weir can be here.

## 4. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Analyzed triangular weir: (

**a**) front view of V-notch weir with angle of 22.5° and characteristic parameters, (

**b**) different angles of downstream of the sharp edge: 30°, 45°, and 60°.

**Figure 7.**Comparison of the V-notch equation discharge (Q

_{30}) determined in this study to the values in other studies.

**Figure 8.**Comparison of the V-notch equation discharge (Q

_{45}) determined in this study to the values in other studies.

**Figure 9.**Comparison of the V-notch equation discharge (Q

_{60}) determined in this study to the values in other studies.

Angle of slope | Calculated Equation (Q_{CAL1}) | R-Squared (R^{2}) | Root Mean Square Error (RMSE) | Height of Water above the Crest H (cm) (Range) | Range of Flow Measurement Q (L∙min^{−1}) |
---|---|---|---|---|---|

30° | Q = 0.298H^{2.266} | 0.9955 | 0.346 | 5.4–7.8 (2.45) | 13.60–31.38 |

45° | Q = 0.224H^{2.418} | 0.9981 | 0.255 | 4.2–7.3 (3.10) | 7.31–27.29 |

60° | Q = 0.256H^{2.325} | 0.9980 | 0.332 | 3.8–7.7 (3.90) | 6.00–29.45 |

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**MDPI and ACS Style**

Napierała, M.
Application of Simple Crested Weirs to Control Outflows from Tiles Drainage. *Water* **2023**, *15*, 3248.
https://doi.org/10.3390/w15183248

**AMA Style**

Napierała M.
Application of Simple Crested Weirs to Control Outflows from Tiles Drainage. *Water*. 2023; 15(18):3248.
https://doi.org/10.3390/w15183248

**Chicago/Turabian Style**

Napierała, Michał.
2023. "Application of Simple Crested Weirs to Control Outflows from Tiles Drainage" *Water* 15, no. 18: 3248.
https://doi.org/10.3390/w15183248