Discharge coefficient in the combined weir-gate structure

Highlights

• Discharge coefficient of combined Rectangular compound broad crested weir-rectangular gate structure (C_dt) was investigated.

• Increasing the gate opening and gate width due to increasing of the wetted perimeter of the weir, led to a decline in C_dt.

• Increasing the central weir height (Z) caused an increase in resistance against the flow and thus a decline in C_dt.

• Among the fitted models, the RNN model could have an accurate estimate of the C_dt values.

Abstract

The present study investigated the flow discharge coefficient (C_dt) in the combined rectangular broad crested weir-gate structure. To this end, the effect of the following dimensionless parameters on the C_dt were investigated: the width ratio of the central weir to the width of the total structure (B/B_o), the height ratio of the central weir to the height of the central weir floor (Z/P), the ratio of the gate width to the width of the total structure (b/B_o), the ratio of the gate opening height to the height of the central weir floor (d/P), and the ratio of the head on central weir to the total head behind the structure (h₁/H). The Flow-3D numerical model, artificial intelligence models such as linear multilayer perceptron (MLP), Canfis network (CNN), recurrent network (RNN), modular neural network (MNN), and regression equation, were used to estimate the C_dt. The results indicated that increasing d/P and b/B_o ratios led to a decline in this coefficient. In the case of h₁/H ≤ 0.4, an increase in B/B_o ratio resulted in decreasing the turbulence intensity and C_dt while the impact of enhancing the size of B/B_o was not significant if h₁/H > 0.4. Besides, increasing Z/P ratio caused an increase in resistance against the flow and thus a decline in C_dt. Further, the results of artificial intelligence models and regression equation demonstrated that the MNN model with an RMSE and R² of 0.03 and 0.97, respectively, could have an accurate estimate of the C_dt values.

Introduction

Hydrometry is the knowledge of measuring volumetric or weight flow in under pressure and open channels, which plays an essential role in managing water resources. The intensity of flow or discharge is described as the volume flow per unit time. In open channels, the flow is measured by various methods such as using the pressure difference, velocity-area method, tracer-dilution method, hydraulic structures (flumes and weirs), etc. Abbaspoor and Yasi [1].

Weirs are one of the most straightforward hydraulic structures that are employed for controlling water levels and measuring the flow in the irrigation canals. The ease and accuracy of measuring in different flow conditions led to the design of various types of weirs. The edge of the broad-crested weirs (BCW) is wide enough, and this edge has a suitable size compared to another part of the weir. Furthermore, their crest is flat, horizontal, or has a particular curvature. Although BCW is used to measure the discharge, they are more likely employed as a dam spillway and occasionally as a dam itself (if the water is allowed to pass top of it). In any case, these weirs are used to store large volumes and high level of water Abrishami and Hoseini [2]. The BCW is divided into two types, namely, simple and compound BCW. The compound weirs have several advantages. In low-intensity flows, the central section of the weir acts individually, and the flow is measured with an appropriate degree of accuracy. Moreover, in high-intensity flows, the upper part of the weir prevents from head increasing in the upstream, and discharge is estimated with acceptable accuracy Salmasi et al. [3].

Similarly, gates are applied to measure the flow in the open channels. Various studies have been conducted on the gates and weirs in order to find the relationship between the upstream hydraulic head and passing discharge [[4], [5], [6], [7], [8], [9]], minimizing the sediments and preventing their accumulation in the upstream structure of the flow measurement are the most important advantages of using the combined weir-gate structure, which can have a considerable effect on the structural efficiency. In addition, the water level can appropriately be controlled in the irrigation canals by employing the combined weir-gate structure design. Therefore, it is a self-cleaning system which requires the least operational work and the minimum cost [35].

A large body of research was implemented regarding the compound BCW. Alkhatib and Gogus (2014) examined several models for estimating the discharge coefficient of the compound BCW by changing the height and width of the central section of the weir and finally proposed acceptably accurate regression equations [10]. Salmasi et al. (2013) using artificial intelligence techniques and genetic programming (GEP) investigated the discharge coefficient in the compound rectangular BCW and reported that the results of GEP were more precise regarding discharge coefficient estimation as compared to those of the artificial intelligence methods [11]. In an experimental study on evaluating the discharge coefficient in the combined BCW-culvert structure, Guven et al. (2013) found that the discharge coefficient enhanced by increasing the upstream hydraulic head and reducing the weir crest angle [12]. Furthermore, Negm et al. (2002) studied the discharge coefficient in the combined rectangular sharp-crested weir and rectangular gate and reported that the ratio of the distance between the weir and gate to the gate opening height, among the geometric parameters, had the biggest influence on the discharge coefficient [13]. Gharahgezlou and Gholami (2013) using energy, momentum, and continuity equations proposed a mathematical model for estimating the discharge coefficient-depth equation in the passing flow of the combined BCW-gate structure and obtained considerable results [14].

Based on the above studies, it is clear that the C_dt in the combined structure of rectangular BCW-rectangular gate has less been investigated. Proposing a predictive model for the C_dt is further needed in designing this combined structure. The present study mainly investigated the impact of the five effective dimensionless parameters of the combined structure on C_dt values. Finally, based on the results of the dimensionless parameters evaluation, regression relationship, and artificial intelligence methods were employed for estimating the C_dt and proposing an optimum model for its prediction.

By using of energy equation between the head measurement section and the control section, an equation for the flow rate, $Q_w$, can be obtained. There are few assumptions which are: the streamlines of the flow are parallel, the pressure above the crest is hydrostatic, distribution of velocity over the weir is uniform and thickness of boundary layer is ignored compared to the flow depth over the weir. By using a discharge coefficient ($C_{dw}$ ) for correcting these assumptions, the weirs discharge equation must be adjusted for real fluids:

$$Q_W=C_{dw}W{H}_1^n$$

where $Q_W$ is volumetric flow rate in the weirs; W = weir shape and size coefficient, $H_1$ = total energy head at the weir head measurement section. Also, n is the dimensionless number that depends on geometry properties of control section and for rectangular control sections n = 3/2. [10,15,16]. Measurement of H₁ is difficult, so it is a common practice to relate H₁ and h₁ (head above the weir). Therefore, the velocity term (C_v) is added to Eq. (1):

$$Q_W=C_{dw}{C}_{v}W{h}_1^n$$

where C_v = approach velocity coefficient that is used to correct of velocity head ignoring at the measurement section. There are two cases regarding compound weir. Given the upstream hydraulic head, the weir acts both as a simple (Case 1) or a compound (Case 2) BCW (Fig. c, b), which is described below.

Case 1. ($h_1\le Z$ and $y_c\le Z$)

If the upstream hydraulic head of the compound weir is such that the outlet discharge passes only through the central weir, then the compound weir acts as a simple weir (Fig. 1-c). In this case, the total energy of the weir upstream flow is less than 1.5 Z. In this condition, the discharge of the weir is defined in the form of Eq. (3) [3,[16], [17], [18]].

$$Q_w=\frac{2}{3}{C}_{dw}{C}_{v}B{\left(\frac{2}{3}g\right)}^{1/2}{h_1}^{3/2}$$

where,

$$C_{dw}=\frac{3}{2}\frac{Q}{B{\left(\frac{2}{3}g\right)}^{1/2}{H}_1^{3/2}}$$

$$C_v={\left(\frac{H_1}{h_1}\right)}^{3/2}$$

where B = central weir width, Z = central weir height, g = acceleration of gravity.

Case 2. ($h_1>Z$ and $y_c>Z$)

In another case, the upstream hydraulic head is larger than the central weir height (Z), the outlet discharge occurs from the total compound weir (Fig. 1-b), and the total energy flow is higher than 1.5 Z. Accordingly, the discharge of the compound weir, the discharge and velocity coefficients of the weir, in this case, are obtained by Eqs. (6), (7), (8) [6,19] respectively.

$$Q_w=C_{dw}{C}_v{\left(\frac{g}{B_0}\right)}^{1/2}{\left[BZ+B_0\left(\frac{2}{3}{h}_1-\frac{BZ}{3B_0}-\frac{2Z}{3}\right)\right]}^{3/2}$$

$$C_{dw}=\frac{Q_w}{{\left(\frac{g}{B_0}\right)}^{1/2}{\left[BZ+B_0(\frac{2}{3}{H}_1-\frac{BZ}{3B_0}-\frac{2Z}{3})\right]}^{3/2}}$$

$$C_v={\left[\frac{BZ+B_0(\frac{2}{3}{H}_1-\frac{BZ}{3B_0}-\frac{2Z}{3})}{BZ+B_0(\frac{2}{3}{h}_1-\frac{BZ}{3B_0}-\frac{2Z}{3})}\right]}^{3/2}$$

where B₀ = total structure width and width of head measurement section.

Similarly, by using the energy equation, the discharge equation for gates is written in the form of Eq. (9) [20]:

$$Q_g=c_{dg}db\sqrt{2gH}$$

where, $Q_g$ = the discharge passing through the gate, $C_{dg}$ = the discharge coefficient for the gate, $d$ = height of the gate opening, $b$ = width of gate opening and $H$ = water depth upstream of the structure. For free flow conditions, $C_{dg}$ is given to be only a function of $H/d$.

Discharge related to the combined weir-gate structure ($Q_t$) is obtained by the sum of the outlet discharge from the weir and gate:

$$Q_t=Q_w+Q_g$$

As mentioned above, there are two cases of compound BCW operation (Fig. 1). If the compound weir acts as a simple weir (Fig. 1-c), the combined structure discharge coefficient (C_dt) is as Eq. (11). If the outlet discharge occurs from the total compound weir and weir acts as a compound weir (Fig. 1-b), the combined structure discharge coefficient (C_dt) is as Eq. (12).

$$C_{dt}=\frac{Q_t}{\frac{2}{3}{C}_{v}B{\left(\frac{2}{3}g\right)}^{1/2}{h}_1^{3/2}+db\sqrt{2gH}}$$

$$C_{dt}=\frac{Q_t}{C_v{\left(\frac{g}{B_0}\right)}^{1/2}{\left[BZ+B_0(\frac{2}{3}{h}_1-\frac{BZ}{3B_0}-\frac{2Z}{3})\right]}^{3/2}+db\sqrt{2gH}}$$