# Theoretical and Experimental Analysis of Operating Conditions of a Circular Flap Gate for an Automatic Upstream Water Level Control

^{1}

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## Abstract

**:**

_{min}< Q < Q

_{max}and the water depth H

_{0}, exceeding which causes the gate to open. Q

_{min}flow denotes the minimum flow rate that allows water to accumulate upstream of the controller. Above the maximum flow rate Q

_{max}, the gate remains in the open position. In the present study, the position of the regulator’s gate axis was related to the water depth H

_{0}in front of the device. Derived dependencies were verified in hydraulic experiments. The results confirmed the regulator’s usefulness for controlling the water level.

## 1. Introduction

## 2. The Principle of Operation of the Circular Flap Gate Design

_{min}leaks at its contact with half-rings and through its fixation of an axis of rotation. At flow rates greater than Q

_{min}, water is retained and the upstream water level increases to the designed level of H

_{0}. When the upstream water level exceeds H

_{0}, the flap gate opens automatically and water flows out through the pipe. Another parameter characterizing the device is the maximum flow rate of Q

_{max}, at which the flap gate remains opened. When the water flow rate exceeds Q

_{max}, the flap gate will not be able to close because the inflow to the device is greater than the outflow. The device operating range determines the flow variability of Q

_{min}< Q < Q

_{max}and the upstream water level of H

_{0}, beyond which the flap gate opens. After lowering of the upstream water level, the flap gate automatically closes due to its weight and the weight of the additional plates and the process of water retaining begins again. The aim of this work was to analyze the operating conditions of the designed device, experimentally determine the values Q

_{min}and Q

_{max}, and verify the theoretically derived formulas.

## 3. Analysis of Circular Flap Gate Working Conditions

_{1}is higher than the gate-closing moment caused by the hydrostatic force P

_{2}.

_{1}is the hydrostatic force acting on a surface of the circular flap gate above its axis of rotation, r

_{1}is the moment arm of force P

_{1}, P

_{2}is the hydrostatic force acting on a surface of the circular flap gate above its axis of rotation, and r

_{2}is a moment arm of force P

_{2}.

_{ax}. Due to the complex form of the integrals in Equations (8) and (9), Simpson’s rule was used to calculate them. Firstly, after assuming the value of the upstream water level, the position of the circular flap gate axis of rotation was calculated. Exceeding the assumed upstream water level should cause a circular flap gate to open. The results of the calculations allowed us to establish a dimensionless relation between the elevation of the axis of rotation of the flap gate above its bottom h and the upstream water depth at which the flap gate opens H

_{0}(Figure 5). This relation can be used to determine, at the given upstream water depth H

_{0}, the elevation of the axis of rotation of the flap gate above its bottom which causes it to open.

_{1}acts. Since the radius of the upper half-ring affects the opening moment, the relation between the elevation of the flap gate axis of rotation above its bottom h and the upstream water depth at which a flap gate opens H

_{0}depends on the radius of the upper half-ring R

_{r}relative to the radius of the flap gate R. The use of a half-ring with a radius of (R − R

_{r})/R = 0.1082 for a 2R diameter circular flap gate described later resulted in a decrease in elevation of the flap gate axis of rotation above its bottom h and the upstream water depth at which a flap gate automatically opens H

_{0}(Figure 5). The influence of the half-ring with the radius of R−R

_{r}= 0.1082 R on the elevation of the flap gate axis of rotation above its bottom is shown in Figure 5. For example, if the circular flap gate has no upper half-ring and were designed assuming that it opens at H

_{0}/(2R) = 1.0, then the elevation flap gate axis of rotation above its bottom should be equal to h/(2R) = 0.3750, so h = 0.3750 (2R). For comparison, assuming that the circular flap gate with the upper half-ring with the difference radius of R−R

_{r}= 0.1082 R has to maintain the same upstream water depth of H

_{0}/2R = 1.0, the elevation flap gate axis of rotation above its bottom should be equal to h/(2R) = 0.3600, so h = 0.3600 (2R).

_{1}and G

_{2}are the weights of the flap gate parts above and below the axis of rotation, g is the gravitational acceleration, m

_{1}and m

_{2}are the masses of the flap gate parts above and below the axis of rotation, r

_{C}

_{1}and r

_{C}

_{2}are the distances between the center of mass and the flap gate axis of rotation for flap gate parts above and below the axis of rotation, and φ is the angle between the gate and vertical axis.

_{C}

_{3}must be attached to the surface below the axis of rotation (Figure 6).

_{3}in relation to the mass of the lower part of the flap gate m

_{2}are presented in Figure 7 as a function of h/(2R) and H

_{0}/(2R). Due to the fact that in theoretical considerations friction was neglected, the required mass of the additional plate must be slightly greater than calculated. An increase of the additional m

_{3}mass increases the water level necessary to close the gate and, in this way, reduces the variability of the water levels upstream of the gate.

_{0}, at which the flap gate opens. For h/(2R) = 0.4, the mass of the additional plates is equal to the mass of part of the flap gate below its axis of rotation (i.e., m

_{2}= m

_{3}). Lowering the position of the flap gate axis of rotation leads to a decrease in the upstream water level at which the flap gate opens H

_{0}, but on the other hand, makes it necessary to increase the mass of the attached plates m

_{3}in order for the flap gate to close automatically. Maintaining the upstream water level by lowering and raising the position of the flap gate axis of rotation is not convenient. The increase of the upstream water level at a given position of the flap gate axis of rotation can be achieved by applying force on the upper edge of the flap gate that counteracts its opening. For this reason, a magnet with adjustable position was installed in the upper half-ring. The shorter the distance between the magnet and the flap gate, the larger the closing moment. Introduction of the magnet increases the gate-closing moment:

## 4. Experimental Verification of the Device’s Operating Conditions

_{p}= 0.080 m. The circular flap gate had a diameter of 2R = 0.0739 m and a mass of m = m

_{1}+ m

_{2}= 31.9 g (Figure 8). The axis of rotation of the flap gate was located at a height of h = 0.0258 m above its bottom. Based on the dependence shown in Figure 5, for R−R

_{r}= 0.1082R, the values of ratios h/2R = 0.3490 and H

_{0}/(2R) = 0.9509 were calculated. This allowed us to calculate the value of the upstream water depth at which the flap gate opens H

_{0}= 0.9509 (2R) = 0.070 m. The calculated mass necessary to close the flap gate was equal to m

_{3}= 22.0 g. When designing the device, it was assumed that the additional mass would be attached in the form of metal plates with a height of l = 0.020 m cut from a circular plate with a diameter equal to the diameter of the flap gate 2R. Therefore, four plates were attached to the flap gate with two screws of a total mass of 29.8 g. Since 29.8 g > m

_{3}= 22.0 g, the circular flap gate should close automatically when the upstream water level drops. The upstream water level at which the flap gate opens was changed by the position of the magnet screw from the “0” to the “3.5” position. The “0” position means that the magnet does not affect the flap gate. Changing the position of the magnet to “1” meant turning the screw with a magnet 360° and bringing the magnet closer to the flap gate by 0.0015 m (Figure 2).

## 5. Results of the Experimental Tests

_{0}= 0.074 m at the flow rate Q > Q

_{min}. This value was 0.004 m higher than that which was theoretically calculated. The difference in depth in relation to the calculated value can be explained by the fact that the resistance of the rotating flap gate was not included in the calculations. Then, the minimum and maximum flow rates at which the circular flap gate opened and closed were determined. Results of the tests for different positions of the screw magnet are presented in Figure 9.

_{0}and the minimum flow rate Q

_{min}. On the other hand, exceeding the maximum flow values Q

_{max}made that the circular flap gate unable to close. This was due to the fact that the inflow to the device was greater than the outflow. Values of the upstream water depth H

_{0}causing the circular flap gate to open for different positions of the magnet are presented in Figure 10.

^{3}/s) significantly influenced the length of the controller’s work cycle, as it depends on the volume of water retained upstream of the controller.

## 6. Conclusions

_{0}, the radius of the flap gate R, and the radius of the upper half-ring R

_{r}. When the circular flap gate is closed, the volumetric flow rate of Q

_{min}leaks at its contact with the half-rings and through its fixation of an axis of rotation. At flow rates greater than Q

_{min}, water is retained and the upstream water depth increases to the designed depth of H

_{0}. When the upstream water depth exceeds H

_{0}, the circular flap gate automatically opens.

_{max}at which the flap gate remains closed. When the water flow rate exceeds Q

_{max}, the flap gate is unable to close because the inflow to the device is greater than the outflow.

_{min}< Q < Q

_{max}and the upstream water depth of H

_{0}, beyond which the flap gate opens. When the upstream water depth is lower than H

_{0}, the flap gate automatically closes due to its weight and the weight of the additional steel plates. By using a magnet-ended screw which “holds” the flap gate, smooth adjustment of the upstream water depth H

_{0}can be achieved. The upstream water depth at which a circular flap gate opens was the same at different flow rates, which indicates that the flow rate has no effect on the H

_{0}. Changing the position of a magnet by bringing it closer to the flap causes an increase of the gate-closing moment, which results in an increase of the upstream water level at which the flap gate opens H

_{0}and the minimum flow rate Q

_{min}.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The water level control mounted in the pipe: (

**a**) view of upstream-flap gate closed, (

**b**) view of upstream-flap gate open, (

**c**) view of downstream-flap gate closed, (

**d**) view of downstream-flap gate open, where 1—flap gate, 2—upper half-ring, 3—screw ended with a magnet, 4—additional metal plate, 5—flap stop, 6—horizontal axis of rotation, 7—lower half-ring.

**Figure 2.**Scheme of the circular flap gate mounted in the pipe, where R

_{r}—internal radius of the half-rings, R

_{p}—internal radius of the pipe, h—elevation of the axis of rotation above the bottom of the flap gate, l—height of the additional metal plate, magnet–magnet screw.

**Figure 3.**Hydrostatic force acting on the flap gate in the case of (

**a**), (

**b**) free surface water flow in a pipe, and (

**c**) pressure water flow at overpressure of p

_{n}.

**Figure 4.**The scheme for calculating the position of the axis of rotation of the circular flap gate in the case of pressure flow.

**Figure 5.**Relation between position of the circular flap gate’s axis of rotation h/(2R) and upstream water depth at which the flap gate opens for a device with and without an upper half-ring.

**Figure 6.**The distribution of forces used to calculate the mass of the additional metal plate attached to the bottom part of the flap gate (S

_{C1}–S

_{C3}are the centers of mass m

_{1}–m

_{3}).

**Figure 7.**The values of the required mass of the additional plates m

_{3}in relation to the mass of the lower part of the flap gate m

_{2}as a function of h/(2R) and H

_{0}/(2R) ratios for the upper half-ring with the radius of (R − R

_{r})/R = 0.1082.

**Figure 9.**Minimum and maximum flow rates at which the circular flap gate opens or closes as a function of the position of the magnet screw.

**Figure 10.**Values of the upstream water depth H

_{0}at which the circular flap gate opens for different positions of the magnet screw.

**Figure 11.**Hydrograph of upstream water level at different flow rates at the inflow for the “1” position of the magnet screw.

**Figure 12.**Hydrograph of upstream water level at different flow rates at the inflow for the “2” position of the magnet screw.

**Figure 13.**Hydrograph of upstream water level at different flow rates at the inflow for the “3” position of the magnet screw.

**Figure 14.**Hydrograph of upstream water level at different flow rates at the inflow for the “3.5” position of the magnet screw.

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**MDPI and ACS Style**

Kubrak, J.; Kubrak, E.; Kaca, E.; Kiczko, A.; Kubrak, M. Theoretical and Experimental Analysis of Operating Conditions of a Circular Flap Gate for an Automatic Upstream Water Level Control. *Water* **2019**, *11*, 2576.
https://doi.org/10.3390/w11122576

**AMA Style**

Kubrak J, Kubrak E, Kaca E, Kiczko A, Kubrak M. Theoretical and Experimental Analysis of Operating Conditions of a Circular Flap Gate for an Automatic Upstream Water Level Control. *Water*. 2019; 11(12):2576.
https://doi.org/10.3390/w11122576

**Chicago/Turabian Style**

Kubrak, Janusz, Elżbieta Kubrak, Edmund Kaca, Adam Kiczko, and Michał Kubrak. 2019. "Theoretical and Experimental Analysis of Operating Conditions of a Circular Flap Gate for an Automatic Upstream Water Level Control" *Water* 11, no. 12: 2576.
https://doi.org/10.3390/w11122576