Numerical simulation of free flow through side orifice in a circular open-channel using response surface method

Highlights

• The flow characteristics of side orifices in a circular channel are numerically investigated.

• The capability of Response Surface Methodology to describe the performance of side orifices is evaluated.

• Discharge coefficient is inversely related to Fr and L/Y_m.

• Discharge coefficient is directly related to L/D_m and W/L.

• A new method is introduced for designing side orifices located in a circular channel.

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Abstract

A side orifice is an important type of hydraulic structure which is used widely in irrigation and waste management systems to divert desired discharges from a main channel or to distribute the flow within the basins. Circular pipes flowing partially full are often used in these systems, but existing predictive relations developed for rectangular channels result in significant error. In the present study, the flow through a side orifice in a circular open-channel is numerically simulated and validated to test the effect of different parameters on the discharge coefficient and propose an appropriate predictive equation. To minimize the number of required simulations and validations, a Response Surface Method-Central Composite Design (RSM-CCD) is employed. Results showed that the discharge coefficient is inversely related to the Froude number (Fr) and the ratio of the side orifice length to the approaching flow water depth. However, any increase in either the ratio of the orifice length to the main channel diameter or the ratio of the lower crest level to the orifice length will increase the discharge coefficient. A new equation is presented to determine the discharge coefficient of side orifice in a circular open-channel using RSM-CCD. The sensitivity analysis showed that all linear terms must considered in the equation but that the interaction terms can be dropped. The maximum error of the equation to predict the training and validation data are 1% and 2% respectively.

Introduction

Side orifices are used in many networks of pressurized pipes and open channels to distribute flow through a network. For example, side orifices are used in sewer treatment systems to distribute wastewater across the width of the basin for flocculation and settlings [1,2]; Hossain et al., 2010, 2016). In many open channel systems, a side weir with an obvert works as a side orifice when the flow touches the ceiling of the side weir [2]. Despite their wide application, discharge coefficients for this type of simple control structure are relatively poorly modeled by available equations.

Side orifices can be classified into large and small orifices. By neglecting the pressure variation across the orifice, Eq. (1) can be applied to flow through a small rectangular side orifice [3,4]:

$$\text{Q}={\text{C}}_{\text{d}}\text{Lb}\sqrt{2\text{g}{\text{H}}_0}$$

where Q is the flow discharge through the side orifice, C_d is the discharge coefficient of side orifice, g is gravitational acceleration, b is the height of rectangular side orifice, L is the orifice length and H₀ is the flow head over the centroid of the vertical section of side orifice. For larger square side orifices, the pressure variation over the orifice should be considered following Hussain et al. [4]:

$$\text{Q}={\text{C}}_{\text{d}}\sqrt{2\text{g}}\frac{2}{3}\text{L}\left({{\text{H}}_1}^{1.5}-{{\text{H}}_2}^{1.5}\right)$$

where, H₁ and H₂ are the flow head above the lower crest and upper crest (or obvert) of the side orifice, respectively.

The approach Froude number has been found by many researchers to have a dominant influence on C_d [5]; Eghbalzadeh et al., 2015, [4,[6], [7], [8], [9]]. Ghodsian [5] and Hussain et al. [6]; Hussain et al. [4] and Hussain et al. [7] confirmed an inverse relation between the approaching flow Froude number (Fr) and C_d. The ratio of the main channel width to the orifice dimension is another effective parameter. Hussain et al. [6]; Hussain et al. [4] and Hussain et al. [7] found that C_d increases by increasing the parameter, while an opposite effect was reported by Ramamurthy et al. [1] for side sluice gates. The effect of the ratio of the sill height to the orifice dimension was investigated by Hussain et al. [6]; Hussain et al. [4] and Hussain et al. [7]; Ebtehaj et al. [8]; Eghbalzadeh et al. (2015) and Vatankhah and Mirnia [9]. Although Hussain et al. [6] and Hussain et al. [4] neglected the effect of the parameter; Hussain et al. [7] stated that C_d increases with increasing the parameter especially in lower values of the parameter. The effect of the ratio of the approaching flow depth to the orifice dimension was considered by Ghodsian [5]; Hussain et al. [6] and Hussain et al. [4]; Eghbalzadeh et al. (2015), Ebtehaj et al. [8] and Vatankhah and Mirnia [9]. Hussain et al. [6] and Hussain et al. [4] neglected the effects of the parameter on C_d of the circular side orifice in the rectangular channel. However, Eghbalzadeh et al. (2015) and Ebtehaj et al. [8] showed that the effect of the parameter should be considered in the analysis. Ghodsian [5] showed that the C_d of the side sluice gate is a monotonic increasing function of the parameter.

Although Ramamurthy et al. [1] derived an equation to estimate the discharge coefficient of a rectangular side orifice in the rectangular channel, Ojha and Subbaiah [3] introduced an elementary discharge coefficient for side orifices. They recommended a large orifice formulation, for which the small orifice formulation is a particular case. Hussain et al. [10] improved Ramamurthy et al. [1] equation to estimate the discharge coefficient of side rectangular orifice. Hussain et al. [6] proposed a new equation based on linear regression to estimate the discharge coefficient of side circular orifices in a rectangular channel. Hussain et al. [7] developed the equations for side circular orifices in submerged conditions. Hussain et al. [6] and Hussain et al. [4] data were used by many researchers such as Ebtehaj et al. [8]; Eghbalzadeh et al. (2015), Azimi et al. [11] and Ezzeldin and Hatata [12] to train and check the appropriate models using soft computing techniques. Their models are more accurate than the original equations proposed by Hussain et al. [6] and Hussain et al. [4].

The above review of literature on side orifices shows that existing equations to estimate the discharge of the orifice are acceptable for use where the main channel is rectangular. The equations are not, however, validated for other channel shapes. Gill [2] derived a semi-analytical equation to estimate the discharge through the side orifices for different type of the main channel section, but the discharge coefficient of the side orifice is an unknown parameter in his approach. Although circular channel is one of the standard profiles in sewers [13], there are no recommendations to estimate the discharge coefficient of the side orifice in circular main channels.

One of the challenges in previous research was the need to carry out large numbers of tests to determine the effect of different factors on Cd. Vatankhah and Mirnia [9]; for example, conducted 570 experiments to derive an equation to estimate Cd as a function of 5 effective parameters. Hussain et al. [4] conducted 42 experiments, while Hussain et al. [7] expanded this to 91 experiments to explore the flow features in partially submerged conditions. For this reason it is worthwhile to consider experimental designs other than the one factor at a time method, including the full factorial method or response surface method (RSM), to achieve the research goals with less effort and high accuracy. Recently, the capability of RSM to design experiments in hydraulic engineering were confirmed by Sangsefidi et al. [14]; Karami et al. [15]; and Pakzad and Azimi [16]. The main advantages of RSM are: i) decrease of costs by reducing the number of experimental runs or numerical simulations; ii) consideration of the interaction between factors; and iii) identification of the optimum point (or range) of the response.

In this study, a series of numerical simulations using Flow-3D software are applied to investigate the flow through the side orifice in the circular main channel for the first time. The effective non-dimensional parameters are derived using dimensional analysis. Using RSM, specific numerical simulations are selected and run to characterize the effect of different non-dimensional parameters on the C_d of a side orifice in a circular main channel. An appropriate equation to estimate the discharge coefficient of side orifice in a circular main channel is derived.