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

Abstract

This article introduces a flow controller for an upstream water head designed for pipe culverts used in drainage ditches or wells. The regulator is applicable to water flow rates in the range of Q_min < Q < Q_max and the water depth H₀, 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 H₀ 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

For the automatic control of the upstream water level in irrigation and drainage channels, various devices are used in different countries. The most convenient and reliable types of automatic gates are flap gates controlled directly by a differential hydrostatic force between the face side and the back of the gate. In the literature, one can find descriptions of models of flap gates pivoting around a horizontal axis [1,2,3,4,5,6]. Most of the concepts for the flap-gate-type design use a counterweight which counteracts the hydrostatic force exerted on the gate. When the water level exceeds a design value, the water pressure force opens the gate. With the lowering water level, when pressure decreases, the counterweight installed on the gate closes it. The operation of flap gates, described in the literature, requires the gate-closing and opening couples around the pivot axis to be balanced (i.e., gate opens and closes at the same water level). If these two couples are properly balanced, the flap gate is able to automatically maintain the upstream water level within a few centimeters at all flow rates.

This paper describes the operating conditions of a control device for the upstream water level, consisting of a circular flap gate rotating around a horizontal axis located below the geometric center of mass [7]. The design concept of the proposed flap gate is not the same as previously presented. It was decided that opening and closing water heads would be different, allowing for cyclical emptying of the upstream reservoir and flushing stored sediments. Presently used gate controllers maintain a constant water level, while the proposed design allows for cyclical fluctuations of water levels. The device has been adapted to operate in culverts in drainage ditches and was designed to maintain a certain upstream water level. It was also planned to use the device in the drainage wells of irrigation systems. Compared with other flow flap gate controllers, the proposed design is very simple, easy to install, maintenance-free, does not block the transport of sediments, and is relatively inexpensive.

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

The upstream water level control device consists of a rotating, circular metal flap gate placed in a pipe and fixed on a horizontal axis of rotation (Figure 1).

In the absence of water in the downstream side of the regulator, the circular flap gate opens when the opening moment caused by the hydrostatic force exerted on the flap gate surface above its axis of rotation exceeds the value of the closing moment caused by the hydrostatic force exerted on the flap gate surface below its axis of rotation. By adjusting the position of the circular flap gate’s axis of rotation h, it can open during both a free surface flow, when the water level is lower than the pipe diameter, and pressurized flow conditions. The angle of gate opening can be regulated with a flap stop mounted in a pipe (5 in Figure 1). Since the axis of rotation of the circular flap gate lies below its center of mass, in order to allow the flap gate to automatically close, it is necessary to increase the weight of the bottom part of the flap gate. This was achieved by attaching an additional mass in the form of metal plates with a height of l cut from a circular plate with a diameter equal to the diameter of the flap gate 2R (Figure 2). To reduce the water flow between wall of the pipe and the flap gate, two half-rings were placed in the pipe on both sides of the flap gate.

In order to smoothly adjust the closing moment, in the upper half-ring, a screw ended with a magnet is installed (Figure 2). By changing the position of the magnet “0” to “3.5”, an increase in the upstream water level at which a gate opens can be achieved. The “0” position means that the magnet does not affect the closing moment.

It should be noted that closing the flap gate does not stop the entire water flow. When the flap gate is closed, the volumetric flow rate of Q_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 H₀. When the upstream water level exceeds H₀, 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 H₀, 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

The conditions for opening the circular flap gate were analyzed for a case when the downstream water level is equal to 0. Due to the presence of the lower half-ring, the circular flap gate will not be able to open when the upstream water level H is lower than the elevation of the axis of rotation above the lower edge of the flap h (H < h in Figure 3). The flap gate opening is possible when the upstream water level is higher than the elevation of the flap gate axis of rotation above its bottom (H > h in Figure 3) and when the gate-opening moment caused by the hydrostatic force P₁ is higher than the gate-closing moment caused by the hydrostatic force P₂.

The necessary condition for the opening of the flap gate may be written as follows:

$$P_1{r}_1>P_2{r}_2$$

(1)

where P₁ is the hydrostatic force acting on a surface of the circular flap gate above its axis of rotation, r₁ is the moment arm of force P₁, P₂ is the hydrostatic force acting on a surface of the circular flap gate above its axis of rotation, and r₂ is a moment arm of force P₂.

The differential hydrostatic force dP acting on a differential surface of the flap dA can be calculated on the basis of the fact that the pressure of a liquid with a specific weight of γ at the depth of z is equal to p = γz. The scheme of the circular flap gate in a pipe is shown in Figure 4. It illustrates the notation used in the analysis.

The differential hydrostatic force dP acting on a differential surface of the flap dA is equal to

$$dP=pdA=\gamma zdA$$

(2)

The differential surface area can be written as follows:

$$dA=xdz$$

(3)

The length x can be determined from the Pythagorean theorem:

$${\left(\frac{x}{2}\right)}^2=R^2-{(z_0-z)}^2$$

(4)

Hence,

$$x=2\sqrt{R^2-{(z_0-z)}^2}$$

(5)

The differential moment acting on a surface of dA is equal to

$$d{M}_1=\gamma z(z_{ax}-z)xdz$$

(6)

$$d{M}_1=\gamma z(z_{ax}-z)2\sqrt{R^2-{(z_0-z)}^2}dz.$$

(7)

On the surface of the circular flap gate above its axis of rotation, pressure exerts a gate-opening moment equal to

$$M_1=2\gamma {\displaystyle \underset{z_g}{\overset{z_{ax}}{\int }}z(z_{ax}-z)\sqrt{R^2-{(z_0-z)}^2}}dz.$$

(8)

The closing moment acting on a surface of the circular flap gate below its axis of rotation can be written as follows:

$$M_2=2\gamma {\displaystyle \underset{z_{ax}}{\overset{z_d}{\int }}z(z-z_{ax})\sqrt{R^2-{(z_0-z)}^2}}dz.$$

(9)

From the boundary condition,

$$M_1=M_2$$

(10)

One can calculate the depth of the axis of rotation of the circular flap gate z_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 H₀ (Figure 5). This relation can be used to determine, at the given upstream water depth H₀, the elevation of the axis of rotation of the flap gate above its bottom which causes it to open.

It should be noted that the upper half-ring affects the opening moment, as it reduces the surface area of the circular flap gate. The larger the radius of the upper half-ring, the lower the surface area on which hydrostatic force P₁ 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 H₀ 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 H₀ (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 H₀/(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 H₀/2R = 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).

Since the axis of rotation of a circular flap gate is located below its center of mass, the gate will automatically close when (Figure 6)

$$G_2{r}_{C2}\sin \mathsf{\phi }>G_1{r}_{C1}\sin \mathsf{\phi }$$

(11)

that is,

$$m_{2}g{r}_{C2}\sin \mathsf{\phi }>m_{1}g{r}_{C1}\sin \mathsf{\phi }$$

(12)

where G₁ and G₂ are the weights of the flap gate parts above and below the axis of rotation, g is the gravitational acceleration, m₁ and m₂ are the masses of the flap gate parts above and below the axis of rotation, r_C1 and r_C2 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.

The total mass of the circular flap gate is the sum of both parts:

$$m=m_1+m_2$$

(13)

In order for the flap gate to return to its original position and meet the condition (11), an additional weight $G_3=m_{3}g$ with a center of mass distant from the axis of rotation r_C3 must be attached to the surface below the axis of rotation (Figure 6).

When designing the flow regulator, it was assumed that the additional mass would be attached in the form of metal plates with a height of l cut from a circular plate with a diameter equal to the diameter of the flap gate 2R. The required mass of the plates was calculated using the static equilibrium equation for the horizontal position of the gate (φ = 90°, Figure 6):

$$m_{2}g{r}_{C2}+m_{3}g{r}_{C3}=m_{1}g{r}_{C1}.$$

(14)

The values of the required mass of the additional plates m₃ in relation to the mass of the lower part of the flap gate m₂ are presented in Figure 7 as a function of h/(2R) and H₀/(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 m₃ 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.

The relation presented in Figure 7 shows that the required mass of additional plates for the automatic closing of the flap gate increases with the increase of the axis of rotation h and the decrease of the upstream water level H₀, 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., m₂ = m₃). 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 H₀, but on the other hand, makes it necessary to increase the mass of the attached plates m₃ 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:

$$M_2=2\gamma \underset{z_{ax}}{\overset{z_d}{{\displaystyle \int }}}z\left(z-z_{ax}\right)\sqrt{R^2-{\left(z_0-z\right)}^2}dz+F{r}_F$$

(15)

where F is the magnet force, which maintains the gate in the closed position, and $r_F$ is the distance from the flap gate axis of rotation to the force F in the center of the magnet screw.

4. Experimental Verification of the Device’s Operating Conditions

Experimental tests of the effectiveness of the circular flap gate for automatic upstream water level control were carried out at the Hydraulic Laboratory of the Faculty of Civil and Environmental Engineering at the Warsaw University of Life Sciences. In the presented tests, the single model of the controller was analyzed. The circular flap gate was made of a 1 mm thick metal plate. The device was installed in a drainage pipe with an internal diameter of 2R_p = 0.080 m. The circular flap gate had a diameter of 2R = 0.0739 m and a mass of m = m₁ + m₂ = 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 H₀/(2R) = 0.9509 were calculated. This allowed us to calculate the value of the upstream water depth at which the flap gate opens H₀ = 0.9509 (2R) = 0.070 m. The calculated mass necessary to close the flap gate was equal to m₃ = 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 > m₃ = 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).

The device was placed in a drainage pipe installed in a wall separating a rectangular channel of 200 mm width and 400 mm height. The volumetric flow rate was measured by an inductive flow meter. The water level was measured using a gauge with an accuracy of 0.0002 m. The scheme of the experimental setup is shown in Figure 8.

5. Results of the Experimental Tests

Firstly, the device operation was tested with the magnet screw in the “0” position. It was found that the circular flap gate opened at the upstream water depth of H₀ = 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.

Changing the position of the magnet by bringing it closer to the flap caused an increase of the gate-closing moment, which resulted in an increase of the upstream water depth at which the flap gate opens H₀ 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 H₀ causing the circular flap gate to open for different positions of the magnet are presented in Figure 10.

Operating conditions of the device at various flow rates and positions of the magnet were also analyzed. Figure 11, Figure 12, Figure 13 and Figure 14 present hydrographs of upstream water levels in the channel for different configurations of flow rates at the inflow and magnet positions. Downstream water levels had no effect on the flow through the device. As it can be seen, even a small change of the flow rate at the inflow to the controller (Q = 0.251, 0.397 dm³/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

The operating conditions of the upstream water level circular flap control gate were analyzed. The dimensionless relations were given for determining the elevation of the circular flap gate axis of rotation as a function of upstream water depth at which the flap gate opens H₀, 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 H₀. When the upstream water depth exceeds H₀, the circular flap gate automatically opens.

Another parameter characterizing the device is the maximum flow rate of Q_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.

The device operating range determines the flow variability of Q_min < Q < Q_max and the upstream water depth of H₀, beyond which the flap gate opens. When the upstream water depth is lower than H₀, 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 H₀ 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 H₀. 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 H₀ and the minimum flow rate Q_min.