Rectangular top-hinged plate as portable flow measuring device Open Access

a_i and b_i

empirical coefficients [i = 1, 2 …] (-);

B

rectangular open channel width (m);

b

top-hinged plate width (m);

f

functional symbol;

g

the acceleration due to gravity (m/s²);

h

upstream water depth (m);

L

top-hinged plate length (m);

M

mass of the hinged plate without its axis (kg);

P

plate opening height [=2Lsin²(θ/2)] (m);

Q

flow discharge (m³/s);

Q_cal

calculated flow discharge (m³/s);

Q_m

measured flow discharge (m³/s);

R

Reynolds number (–);

S

channel longitudinal slope (m/m);

t

top-hinged plate thickness (m);

W

Weber number (–);

ρ

water density (kg/m³);

μ

water dynamic viscosity (Ns/m²);

σ

water surface tension (N/m);

∏

dimensionless group;

ψ

functional symbol;

θ

angle of deflection (degree);

ν

kinematic water viscosity [=10⁻⁶] (m²/s).

The use of a measuring device is vital to developing water-saving irrigation methods. The primary basis for implementing precise irrigation is to accurately measure the flow discharge. The use of flow discharge measuring technology is important to better manage the irrigated areas and improve water transfer, water diversion and water allocation. There have been many achievements in research on flow discharge measurement devices such as weirs, flumes, orifices, and gates. These devices are available for flow measurements in water conveyance channels and irrigation and drainage networks. Simple measuring structures are expanded in recent years and different types of them have been developed. Weirs and sluice gates generally used in rectangular and trapezoidal open channels. Application of simple measuring devices is also extended to non-rectangular open channels. Recently, discharge equations of sharp-crested circular weirs are developed for flow discharge estimation in circular open channels under free-flow conditions (Vatankhah 2010; Bijankhan & Ferro 2018; Shabanlou 2018). The hydraulic behavior of the sluice gates located in the circular open channels is also investigated (Hoseini & Vatankhah 2020; Vatankhah & Hoseini 2020). Installing the flow measurement structure may affect the upstream flow conditions in the channel and may cause sediment deposits upstream from the measurement device. The installation of weirs, gates and flumes is difficult and may cause some damages to the channel section.

Generally, simple flow measuring structures are preferred due to low maintenance and construction costs and there is a need to investigate the flow processes leading to the simple flow measuring devices. Flap gates are one of the flow-measuring devices and are designed to automatically control upstream water elevation as well as measuring the flow discharge. They commonly used at downstream end of open channels, especially sewage channels, for controlling and measuring the flow discharge (Belaud et al. 2008). The flap gate behaves as a broad-crested weir for large opening angles, while it behaves as an orifice for small opening angles (Belaud et al. 2008). Burrows (1986) studied a hinged flap gate at the outfall into the downstream tank. Burrows et al. (1997) also studied circular flap gates on river outfalls. The proposed model was based on the principle of the conservation of angular momentum. Raemy & Hager (1998) used flap gates only for controlling upstream water elevation in small open channels and proposed a comprehensive model. The flap gate needs no electric power and no manual adjustment for a range of flow discharge variations (Burt et al. 2001). Litrico et al. (2005) proposed an efficient mathematical model of an automatic upstream water-level control, which was based on the flap gate. This flap gate controls upstream water level close to a reference level by its own counterweight. Litrico et al. (2005) derived a mathematical model then they used experimental data and data from literature to evaluate and validate the proposed mathematical model. Tariq & Masood (2001) used a hinged rod to measure flow velocity. Their device was a rectangular wooden rod. In this study, the mathematical model was developed based on the moment equation, and then calibrated by experimental data. This technique is acceptable for the flow velocity less than 1 m/s and the flow depth less than 45 cm.

More recently, Vatankhah & Ghaderinia (2018) proposed simple hinged semi-circular plates (a plate flowmeter without any counterweight) for discharge measurement along the circular open channels under the free-flow condition. This portable device was established vertically along the channel and deviated due to the flow forces. According to the angle deviation and the upstream water depth, two models were proposed for flow discharge estimation based on Buckingham's theorem of dimensional analysis. They programmed a series of laboratory experiments for two horizontal circular channels to calibrate the proposed models.

Since low-sloping channels are widely used in irrigation and drainage networks, the main objective of this study is to obtain a simple generalized discharge relation for a top-hinged plate located along both horizontal and sloping rectangular, open channel under free-flow conditions. The proposed top-hinged plate can have different width ratio (plate width to channel width varies from 0.2 to 0.8) and thus help to reduce afflux compared to sluice gates. The advantages of the proposed portable device over the sluice gates and weirs are its simple construction, ease of flow discharge measurement, measuring a wider range of flow rates and its low costs.

The following sections present dimensional analysis method for deducing the discharge equation of a top-hinged rectangular plate along with experimental setup used in this research. Results and discussions including a step-by-step procedure to obtain suitable discharge equations are then presented followed by concluding remarks.

in which f₁ is the functional symbol, Q is the flow discharge (m³/s), S is the channel longitudinal slope (m/m), h is the upstream water depth (m), g is the acceleration due to gravity (m/s²), M is the mass of the hinged plate without its axis (kg), θ is the angle of deflection (degree), μ is the water dynamic viscosity (Ns/m²), ρ is the water density (kg/m³) and σ is the water surface tension (N/m).

The increase in flow depth in the rectangular channel led to an increase in the deflection angle of the plate (Figure 1) and thus there is an interaction between upstream flow depth and the deflection angle. The deflection angle, θ, will be fixed when the closing moments are balanced by the opening moments (closing moments are equal to opening moments). The deflection angle of the plate can be considered as either an independent variable or a dependent variable [13]. If θ is obtained from experimental/field measurement data then this variable will be independent (Model I). In this case, the plate mass, M, is irrelevant and should not be considered as an effective variable in functional Equation (1). If θ is considered as a dependent variable (Model II), then this variable is influenced by the plate mass, M, and other variables expressed in Equation (1). In this case, θ should not be considered as an effective variable in Equation (1), and its effects are considered by other variables expressed in Equation (1). It is worth noting that the flow area under the plate [plate opening height is equal to P = 2Lsin²(θ/2)] is a function of L and θ, and is not an independent variable and thus should not be considered as an effective variable in Equation (1).

These original dimensionless variables can be combined to obtain new dimensionless variables.

Roth & Hager (1999) experimentally observed that for small values of the sluice gate opening (1–2 cm), the surface tension dominates the flow. In this study, the gate opening is higher than 2 cm in most cases (82% of the observations) and thus the surface tension can be neglected. In this study, the Reynolds number [B(gh)^0.5/ν in which ν is the kinematic water viscosity; ν = 10⁻⁶ m²/s] was in the range of 0.1 × 10⁶ to 0.38 × 10⁶ which indicates that the effect of viscosity can be neglected.

In Equations (18) and (19), ψ_I and ψ_II are functional symbols. The mathematical shape of the functional relationships can be found using experimental data. In this study to verify the relation of flow discharge with the involved parameters, a comprehensive experimental program was performed.

A series of experiments was conducted to investigate the flow discharge characteristics of top-hinged plates installed at the middle of a rectangular open channel (12 m long, 0.25 m wide, and 0.5 m deep) with two longitudinal slopes of 0 and 0.005. The experiments were performed at the hydraulic laboratory of the Irrigation and Reclamation Engineering Department, University of Tehran, Iran. The laboratory channel was supplied by a pump and was connected to an upstream supply through an inlet stilling/stabilizing part to eliminate water surface fluctuations. From a large constant head reservoir (upper tank), water was supplied to the rectangular channel entrance by a supply pipe equipped with a flow control valve. From the downstream end of the rectangular channel, water passed through a triangular weir and entered an underground reservoir and then it was pumped and recirculated using a pump to the constant head reservoir (upper tank). Within the flow circulation path, a rectangular weir was also installed and used for high discharge measurement. Both triangular and rectangular weirs were calibrated using an electromagnetic flowmeter with an accuracy of 0.5% of the full scale. The tests were carried out in a steady-state flow condition. During the experiments, a constant discharge was adjusted via the flow control valve, and excess water was discharged from the upper reservoir into the underground reservoir through a weir. The upstream gauge station should be far enough from the plate location to eliminate the water surface effects (backwater effects). Upstream flow depths were measured at the centreline of the approach channels using a point gauge with an accuracy of 0.1 mm which installed at a distance of 0.8 m upstream of the plates. The plates were made of Plexiglas sheets with two different thicknesses of t = 5 and 10 mm. The measurements were carried out for two different plate lengths (L = 20 and 30 cm) and four different values of the plate width (b = 5, 10, 15 and 20 cm). The plates were installed vertically in such a way that their lower edges were located in a distance of 5 mm from the channel bottom. All edges of the rectangular plates were cut in a right-angle and thus they acted as a sharp-crested measurement device during the operation. The plates were installed in a distance of 8 m from the channel inlet and flow was fully developed upstream of the plate.

The downstream flow condition of the plate differs from that of traditional sluice gates, which usually produce a regular hydraulic jump for dissipating excess energy of the supercritical flow under the gate. This is due to adjustable plate opening P. There was a clear contracted flow under the plates in all experiments, which indicated free flow conditions. The water surface profile upstream of the plate was still without any fluctuation (Figure 3(b)). Moreover, the plate under operation was completely stable and thus accurate measurement of the deflection angle, θ, was possible for all experiments. For each free flow experiment, the flow discharge and the upstream flow depth were measured. The discharge used in the tests was ranging from 1 to 74 L/s. The deflection angle, θ, ranged from 3.5° to 81°, the flow depth ratio h/B ranged from 0.065 to 0.95 and the width ratio b/B ranged from 0.2 to 0.8. Tables 1 and 2 present the experimental data collected in this research, respectively, for S = 0 and S = 0.005 under free flow conditions.