Numerical simulation of free flow through side orifice in a circular open-channel using response surface method
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]:where Q is the flow discharge through the side orifice, Cd 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]: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 Cd [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 Cd. 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 Cd 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 Cd 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 Cd 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 Cd 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 Cd 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.
Section snippets
Methodology and design of research
The discharge coefficients of the side orifices depend on the approaching flow characteristics and the main channel and the orifice geometries (Fig. 1). Based on the dimensional analysis the effective non-dimensional parameters on discharge coefficients are as follows:where Fr is the approaching flow Froude number, V1 is mean approaching flow velocity, D is the hydraulic depth of the approaching flow, g is the acceleration due to gravity, L is the orifice dimension, Dm
Model verification
Since the FLOW-3D is an appropriate software to model the flow features around side weirs and orifices; in this research the flow features around the square side orifice in a circular main channel is simulated numerically. Fig. 4 shows a representation of the developed numerical model and the boundary conditions. Side orifice was set in the 5 m from the beginning of the main channel. A sluice gate was defined near the end of main channel to regulate the flow depth in the main channel. Mesh
Conclusions
In this research the effect of different parameters on the discharge coefficient of the side square orifice in a circular main channel was investigated based on numerical simulation using FLOW-3D software. The required simulations were determined using design expert RSM-CCD. According to the results, the Froude number of the approaching flow and the ratio of the orifice length to the approaching flow depth decrease the discharge coefficient. However, the increment of the ratio of crest level to
Author statement
Mojtaba Mehraein: Conceptualization, Software, Writing- Original draft preparation, Supervision, Mohammadamin Torabi: Investigation, Writing- Original draft preparation, Yousef Sangsefidi: Software, Validation, Bruce MacVicar: Writing- Reviewing and Editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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