# Determination of Tile Drain Discharge under Variable Hydraulic Conditions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{g}), according to Scheme 1 and Scheme 2a or high water level on the outflow simulated in laboratory conditions with an adjustable weir located at the end of the experimental stand (Scheme 2b,c).

_{AV}, V

_{REF}). The measurements of water level, average flow velocity, and average flow rate were conducted in steady, identical 15 s intervals for both flow meters. The determination of the pipe submergence’s influence on its discharge required checking AV behavior in the possible modelling and variable measuring conditions occurring at the end of the drainage pipe in the field. The investigations reproduced variable flow conditions in a drainage pipe typical of drainage pipe systems during spring melting and sudden, intensive summer rains. Our research aimed to model the influence of possible high water level outflow on pipe cross-section flow capacity. The discharge simulation was performed due to the groundwater raising Q = f (T) for a constant level of lower water H

_{d}. Depending on the submergence conditions of the outflow weir P

_{g}and the fulfilment of the control section H

_{g}, three flow phases with different hydraulic calculation roles were defined:

## 3. Results and Discussion

_{REF}related to dimensionless pipe filling (H

_{g}/P) were elaborated (Figure 1). The calculated average values of V

_{REF}flow related to dimensionless pipe filling (H

_{g}/P) formed three lines arranged in a circular fashion. The P value equalled the P

_{g}value for the free flow phase, otherwise it was equal to the H

_{d}value. The presented graph shows the three flow phases defined earlier. The upper limit line reflected Phase I flow conditions (Scheme 2a). A flow transition zone is placed between the limit lines. Points situated in this area represent the change of the free flow form with the outflow weir submergence to pressure flow (Scheme 2b). Points representing transient flow (Phase II) gather along lines within the considered zone. The position of these lines depends on the submergence of the weir. Points marked as Phase 2 (Figure 1) were obtained for H

_{d}= 0.5D. The bottom limit line was determined using pipe pressured flow values (Scheme 2c).

_{d}has no influence on the pipeline flow velocity. The assumption of flow velocity in a pipe as an analyzing measurement parameter is reasonable because the “Area-Velocity” flow meter option was applied in the software and the discharge was calculated according to average flow velocity (the AV software also has different options of discharge calculation, for example, Manning formula or two-term polynomial equations) [26]. Thus, the aim of the further analysis was the determination of the actual flow velocity in the control section. It was assumed that the actual velocity was equal to the velocity measured by the laboratory flow meter (V

_{REF}) and the independent variable was the velocity measured by area velocity (V

_{AV}). The relationship between the actual and the measured velocity values is presented in Figure 2.

_{AV}= 0.5 m s

^{−1}(Figure 2a,b). The most accurate fitting was obtained for the pressured flow (R

^{2}= 0.987), where the actual velocity V

_{REF}was equal to 0.428 of the measured velocity V

_{AV}(Figure 2c). The transient flow phase was relatively short because the rapid increase of lower water level H

_{d}limited the possibility of outflow, leading to pressured flow (Phase III). Thus, a limited number of Phase II measured values gave rise to an unsatisfying fitting (R

^{2}= 0.48). The flow Phase I (Figure 2a), as well as all the phases of flow (Figure 2d) fittings, gave similar results. The actual velocity V

_{REF}amounted to 0.508 and 0.849 of the measured velocity V

_{AV}, respectively (R

^{2}= 0.85). From the practical point of view, it is difficult to watch for different outflow phases (especially Phase II) in a drainage pipeline. However, the most frequent flow case in a drainage pipeline is free flow. As proven above, the measured velocity V

_{AV}over 0.5 m s

^{−1}was characterized by a significant instability. Thus, the calculated discharge values Q

_{AV}would be inaccurate. Further research was focused on the correlation between the discharge calculated by the chosen formula and the reference discharge Q

_{REF}. The discharge along the control section (Scheme 1), working in the free outflow conditions (Scheme 2a, Phase I) can be determined according to the well-known Cipolletti formula [27,28]:

^{−2}), H is the water level over the weir head (cm), and μ is the flow rate coefficient calculated by the following equation:

_{REF}values related to water level over weir head H using the non-linear regression method. As a result, the following form of the California pipe method was devised (called the California pipe outlet weir method—Cpw for contradistinction):

_{REF}) and the discharge values calculated using Equations (1) and (4) is presented in Figure 3. For an accurate evaluation of the considered relationship, the following statistical criteria were applied: The maximum error value (ME), the root-mean-square error value (RMSE), the coefficient of determination value (CD), and the coefficient of residual mass value (CRM) [29]. The ME value is the maximum difference between the observed and the calculated value and indicates the worst case calculated by equation. The RMSE value indicates to what extent the calculations are over- or underestimating the measurements, expressed as a percentage of the averaged value of the measurements. The CD describes the ratio between the scattering of the calculated values and the scattering of the measurements. The CD value proves the dynamics in the measured and calculated values’ agreement.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Scheme 1.**Control section with an area velocity ultrasonic flow meter (AV) sensor inside, where D is the pipe diameter, P

_{g}is the outlet weir height, and H

_{g}is the control section water level [24].

**Scheme 2.**Unsubmerged weir and unpressured pipe (

**a**); submerged weir and unpressured pipe (

**b**); submerged weir and pressured pipe (

**c**); Q is the discharge (dm

^{3}s

^{−1}), T is time (s), H is the water level above the weir (mm), and H

_{d}is the lower water level (mm).

**Figure 1.**Velocity average values V

_{REF}related to dimensionless pipe filling H

_{g}/P: Phase 1 is free flow, Phase 2 is transient flow, and Phase 3 is pressured flow.

**Figure 2.**Relationship between the real velocity V

_{REF}and the measured V

_{AV}for free flow (

**a**), transient flow (

**b**), pressured flow (

**c**), and all phases of flow together (

**d**).

**Figure 3.**Relationship between the measured discharge Q

_{REF}and the calculated Q by Equation 1 (

**a**) and Equation 4 (

**b**).

**Table 1.**Description of statistical criteria values for measured and calculated discharge. ME: maximum error, RMSE: root-mean-square error, CD: coefficient of determination, CRM: residual mass

Statistical Criteria | Equation (1) | Equation (4) | |
---|---|---|---|

ME (dm^{3} s^{−1}) | range: (0; ∞) best value: 0 | 0.406 | 0.101 |

RMSE (%) | range: (−∞; ∞) best value: 0 | 21.12 | 8.24 |

CD (-) | range: (0; ∞) best value: 1 | 1.600 | 1.028 |

CRM (-) | range: (−∞; ∞) best value: 0 | −0.105 | 0 |

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**MDPI and ACS Style**

Szejba, D.; Bajkowski, S.
Determination of Tile Drain Discharge under Variable Hydraulic Conditions. *Water* **2019**, *11*, 120.
https://doi.org/10.3390/w11010120

**AMA Style**

Szejba D, Bajkowski S.
Determination of Tile Drain Discharge under Variable Hydraulic Conditions. *Water*. 2019; 11(1):120.
https://doi.org/10.3390/w11010120

**Chicago/Turabian Style**

Szejba, Daniel, and Sławomir Bajkowski.
2019. "Determination of Tile Drain Discharge under Variable Hydraulic Conditions" *Water* 11, no. 1: 120.
https://doi.org/10.3390/w11010120