Optimized orifice geometry to enhance the flow environment and achieve maximum discharge coefficient for effective water resources management

: Flow measurements through pipelines and open channels are very important for the management of water resources effectively. Orifice meter is a very common and widely used flow measuring device in pipes as it is very cheap and simple compared with other devices, however, many parameters affect its performance. Also, the orifice is used as an energy dissipation method in water hammer protection devices and hydroelectric power tunnels. Although traditional circular orifice meters have been extensively studied, many points need to be studied. So, experimental research is carried out to study the effect of orifice geometry on energy loss. The experimental tests are carried out using four different types of orifice plates: circular, square, triangular, and hexagonal, the cross-sectional area is changed four times for each one. The flow rate is changed ten times for each orifice ranging from 13.8 to 49.2 m3/hr. Due to head losses occurring at the orifice. General equations are developed for the coefficient of discharge and head loss through orifices based on dimensional analysis. The experimental results conclude that the triangular shapes are better than the other orifice shapes in terms of performance, with reduced head loss and a larger discharge coefficient. By using computational fluid dynamics techniques, the flow behavior through the orifice is analyzed by ANSYS Fluent software. The numerical results confirmed the experimental ones where the pressure head loss for the triangular orifice is the lowest compared with the other orifice shapes

However, orifice meters are the most widely used flow measurement devices for monitoring pipeline discharge.Its idea is based on measuring the pressure difference created through a pipe constriction, which changes velocity and pressure.The theoretical discharge can be determined by applying Bernoulli's Energy Equation in conjunction with the Continuity Equation.To get on the actual discharge, the theoretical discharge is multiplied by a correction factor which is called the discharge coefficient to take into account the losses that occurred through the orifice meter.Many factors influence the discharge coefficient such as orifice diameter, shape, orifice hole edges, orifice plate thickness, pressure tap position, and fluid type (compressible or incompressible).
Many researchers studied orifice meters from different points of view (El Toukhy and Alsaydalani 2022;Wu Jianhua et al. 2010;Wanzheng and Tianmeng 2019;Wanzheng and Pengfei 2021), they also studied the fluid compressibility impact on several orifice plate types, including the standard concentric orifice plate, quarter circle orifice plate, and square edge orifice plate.Prasanna et al. (2016) investigated the flow characteristics of an orifice flow meter described using commercial CFD, where the flow behavior is studied and the vena contracta location is set by Tukiman et al. (2017) and Danesh and Hassan (2018).Eight different orifice shapes are used to investigate the effect of orifice shapes on the coefficient of discharge, a general correlation equation for computing the discharge is deduced by krishnan et al. (2017).
The orifice coefficients and discharges through circular and rectangular orifices were determined experimentally by Nicholas (2018), with the same cross-sectional area under a constant head of water, the rectangular orifice had a greater orifice coefficient of discharge than the circular orifice.The computational fluid dynamics technique is used to determine the calibration coefficient of an orifice meter numerically by Oliveira et al. (2010).Furthermore, Imada et al. (2013) compared the numerical simulation results with the ISO Standard, it achieved by applying the realizable k -ε and kω turbulence models.It was found that the predicted discharge coefficient values agree well with the ISO Standard, with a maximum error of 4.92%, the agreement is slightly higher for the k -ω model.Shaaban (2014) added a ring downstream of the orifice, which reduced typical energy loss by 33.5 percent.Optimization studies showed that the best improvement in orifice meter pressure loss and energy consumption is obtained at a ring diameter ratio βr = 0.5 and a normal distance Lr/D = 0.38.Sravani and Santhosh (2022) showed the observed differential pressure and discharge coefficient which are affected by the orifice plate thickness, pipe diameter, fluid density, edge shape, inclination angle, number of holes, position of the orifice hole, and upstream-downstream pipe lengths.
A numerical model is used to investigate the laminar and turbulent flow characteristics of an orifice plate by Sahin et al. (2014), the ratio of orifice plate thickness to the orifice diameter (t* = t/d) is varied from 1/12 to 1, while the beta ratio remains constant at 0.6, t*=1 resulted observed decrease in the pressure difference, a reduction in C d readings, and greater flow rate values comparing with t*=1/12.Five orifice plates with beta ratios of 0.30, 0.35, 0.47, 0.59, and 0.71 are investigated by Rahman et al. (2009), with the hole centered in an 8.5 cm pipe diameter that is filled with water.They concluded that the orifice meter with a beta ratio of 0.60 can be used to efficiently measure pipe flow.Abd et al. (2019) conducted experiments for various beta ratios and Reynolds numbers, and the results showed that the C d coefficient increases with an increase in β when R e ≤ 9000, whereas this coefficient decreases with an increase in β when R e > 9000.The performance of the orifice discharge coefficient for laminar and incompressible flow through square-edged concentric orifice was investigated by Ahmed and Ghanem (2020).The flow through a multi-hole orifice meter is studied by Almutairi et al. (2023) with three different β ratios 0.55, 0.6, and 0.7, Reynold's number equaled 10 5 , and it was observed that an increase in the β ratio causes a decrease in pressure drop and an increase in discharge coefficient.Four different flow meter types are tested in laminar and turbulent flow conditions by Hollingshead et al. (2011).The flow through the orifices and the pressure drop are studied experimentally by (Yamaguchi 1976;Yan and Thorpe 1990;Zhang and Cai 1999;Zimmermann 1999).the velocity profile was affected by temperature effects due to viscosity and density changes and installation effects, and the temperature influence of the orifice plate was studied in depth by Büker et al. (2013).
In a study by Peter and Chinedu (2016), a model prediction is made for the constant area, variable pressure drops in orifice plate characteristics in the flow system.Also, these articles (Della et al. 2000;Wan-zheng and Jia-hong 2015;Yoshida et al. 2010;Zeghloul et al. 2017) investigated their results using the numerical method CFD.The behavior of an orifice meter is investigated conditioned by a fractal flow conditioner by using experimental methods and computational simulations by Manshoor and Nicolleau (2011).The performance of perforated plates with optimized hole geometry is studied by Shaaban (2015).
Although much effort has recently been put into the study of the flow through discharge measuring devices in general and orifice meters especially, there are still many questions left.However, most of the previous studies of orifice meters were concentrated on circular orifice.The objectives of the current study are to investigate the effect of different shapes of orifices, i.e., circular, triangular, square, and hexagonal on the discharge coefficient experimentally and numerically.Also, the head loss through the orifice is studied as an energy-dissipating device.

2-Materials and Methods
An orifice plate is a circular metallic plate that is inserted in a pipe to measure flow rate by causing a pressure drop across it.The thin sharp-edged orifice with 45 degrees has been recognized as a standard as shown in the following   Four different sizes of each orifice shape are used of diameters or side lengths as listed in Table 1.The discharge through the orifice is changed ten times in the range from 13.8 to 49.2 cubic meters per hour.A mercury U-tube manometer is used to measure the pressure difference between two taps upstream and downstream of the orifice plate at distances.The upstream tap is one time the diameter of the pipe (1D) from the orifice plate position and the downstream tap is a distance of (0.5D, D, 1.5D) from the orifice plate position as shown in   The higher velocity of fluid traveling through the decreased area of the orifice causes a decrease in pressure as it passes through the opening.As the fluid leaves the vena contracta, the velocity reduces and pressure rises, eventually returning to its original level.With increasing flow rate, the pressure drop across the orifice increases.Bernoulli's theory between two points at the same elevation and equation of continuity, an equation for the volumetric flow rate is derived. (1) Where P is the pressure, V is the velocity, ρ is the water density, and g is the earth's gravity.By applying Bernoulli's equation between the section upstream of the orifice and the section at vena contracta, The following equation can be used for calculating the theoretical discharge through the orifice meter.
Where H is the differential pressure head, A 1 and A 2 are the cross-sectional areas of pipe and jet at vena contracta, respectively, where pressure gauges are placed.Because measuring the diameter of the jet within the pipe was impossible, so A 2 was estimated using the diameter of the orifice.For the same reason, the vena contracta was estimated its position to be 0.5D downstream of the orifice plate.Some irreversible energy is lost when fluid passes through an opening due to blockage, frictional, compression, and vortices.When a constriction is created in a pipe conveying a fluid, the velocity rises, and thus the kinetic energy increases at the location of constriction.According to the energy balance equation provided by Bernoulli's theorem, there must be a corresponding decrease in static pressure.To account for all of these losses, the theoretical discharge is multiplied by a discharge coefficient to calculate the actual discharge.The discharge coefficient is affected by the orifice-to-pipe diameter ratio known as the beta ratio (β), Reynolds number (R e ), and orifice shape.3.2, 4.5, 5.5, 6.4 3, 4, 5, 5.7 4.5, 6, 7.5, 8.5 1.9, 2.5, 3, 3.5 Discharge (m 3 /hr) 14.5 -43.4 13.8 -44.5 14.8 -49.2 14.4 -45 Reynolds number 51000 -152300 48300 -156100 51800 -172750 50600 -158000

2.1.Dimensional Analysis
Dimensional analysis is a methodology for decreasing the quantity and complexity of experimental factors that impact a certain physical phenomenon.The goal of using dimensional analysis is to convert the various parameters to a dimensionless format to avoid dealing with a specific unit system.Saving time and money is one of the many advantages of dimensional analysis.In order to find the factors affecting the discharge coefficient and rate of energy dissipated through the orifice, dimension analysis is performed to simplify and define the problem.
The other is its superiority in solving complex problems that are difficult to deal with using integral theories and differential equations.The challenge is to determine the different factors that affect a certain phenomenon.Here in our case, the different variables that affect the coefficient of discharge (C d ), the pressure drop (ΔP), and different heads at the manometer (ΔH) are pipe diameter (D), orifice hydraulic diameter (d), flow velocity (V), dynamic viscosity of water (μ), water density (ρ) and discharge (Q), which are represented in Eq. ( 3): (3) According to the π-theorem and the principle of dimensional homogeneity, the parameters, D, μ, and ρ are defined as repeated variables and there are six (π) variables.Then the function can be rewritten as dimensionless groups as follows in Eq. ( 4):

2.2.Experimental Setup
The differential pressure head was measured by tapping a differential U-tube mercury manometer 10 cm upstream and 5 cm downstream of the orifice plate.An electric motor pump pumped water, and the needed actual discharge was measured using the volumetric method by using a collecting tank.For each beta ratio, ten random readings of discharge and the related pressure differential were taken.Sixteen orifice plates were installed separately, and the resulting discharges were carefully recorded.The stopwatch readings are taken two times to account for more accuracy, Fig 5 shows the steps of the experiment to calculate the coefficient of discharge.This strategy is made for each orifice ten times with different valve openings.

2.3.CFD Simulation
Computational fluid dynamics simulation is used for flow through orifices with different shapes.CFD simulation is carried out using ANSYS Fluent software.To simulate the steadystate flow through an orifice meter, the governing equations need to be solved.In the present work, the standard k-ε turbulence model is used.Constant flow discharge equal to 40 cubic meters per hour is assumed in a pipe that has a 10 cm diameter, four orifices (circular, triangular, hexagonal, and square) shapes are simulated in the pipe, and a constant area for the four shapes is taken equal to 28 cm 2 .The mesh is generated using ANSYS Workbench with a hexahedral block organized mesh for the whole computational region (5D upstream and 10D downstream) as shown in Fig 6, the number of elements is 1.2x10 6 .

Fig 6.
Hexahedral Mesh for the study region.

3-Results and Discussions
The results of the experimental program and the numerical simulation are presented in this paper.The influence of Reynolds number, orifice shape, and orifice diameter to pipe diameter ratio on discharge coefficient and head loss through the orifice were analyzed.Based on the dimensional analysis, the studied parameters are normalized and presented in dimensionless relations.

Effect of Different Parameters on Discharge Coefficient
As shown in the previous section, many parameters affect the discharge coefficient among them Reynolds number, pressure head difference between upstream original pipe cross-section and orifice cross-section, orifice meter shape, and the ratio of orifice meter diameter to the pipe diameter.The following effect of the different parameters on the discharge coefficient will be presented and discussed.
Figs 7 to 10 show the variation of discharge coefficient with Reynolds number at different orifice opening to pipe diameter ratios for the four shapes of orifice used in the study.For all orifice meter shapes, i.e., circular, triangular, square, and hexagonal, it can be noticed that the flow was turbulent flow having a Reynolds number higher than 50000.Fig. 7 shows the variation of discharge coefficient with Reynolds number for a circular orifice meter with four different orifice diameters, so the orifice diameter to pipe diameter ratio (d/D) was in the range from 0.32 to 0.64.It can be shown from the figure that there is no clear trend between the discharge coefficient and the Reynolds number.For d/D equal to 0.32, the relation is directly proportional, however, for d/D equal to 0.45, the relation is inversely proportional, and for d/D equal to 0.55 and 0.64, nearly there is no variation in discharge coefficient by increasing Reynolds number with a value of discharge coefficient equal to 0.6 for d/D equal to 0.55 and discharge coefficient equal to 0.58 for d/D ratio equal to 0.64.Also, it can be shown from Fig 7 that the discharge coefficient for the circular orifice is in the range of 0.58 to 0.62, and these values are in agreement with ISO standard values (ISO, 2003) presented by Imada et al. (2013).In general, the discharge coefficient for a square orifice is higher than that for circular and hexagonal, however, it has less C d values for triangular.Looking at Fig. 10 for the hexagonal shape, it can be found the same trend for other shapes with a value of discharge coefficient is slightly higher than that for circular orifice.The conclusion is that the discharge coefficient increases by increasing the irregularity in the orifice shape and this may be due to the reduction of the dead zone downstream of the orifice.To explore the variation of theoretical discharge with pressure head difference for different orifice shapes, Figs 11 to 14 depict the variation of theoretical discharge computed using Eq. 1 with pressure head difference through the orifice at different values of d/D for circular, triangular, square, and hexagonal orifice shape respectively.In general, the relation between Q th and pressure head difference is nonlinear, and the theoretical discharge increases by increasing pressure head difference.Also, it can be shown that the three high values of d/D for all orifice shapes have the same trend, however, the smallest d/D value has a different trend and this indicates the scatter in the relations in Figs 7 to 10 for discharge coefficient.

.Discharge Coefficient Equations
As shown in the previous sections and literature review, many parameters affect the discharge coefficient, so in this section, empirical equations will be deduced for computing the discharge coefficient using approximate relations and dimensional analysis concepts.An equation is developed for computing the discharge coefficient as a function of the ratio of the pipe crosssection area to the orifice cross-section area (A 1 /A 2 ) in the form of Eq. 5, where a, b, and c are coefficients depending on the orifice shape, and its values are shown in Table 2, these coefficients are extracted from the relation between relative area and discharge coefficient for the four orifice shapes which observed in Fig 15 .( ) ( ) Where A 1 is pipe cross-sectional area, A 2 is orifice cross-sectional area, C d is coefficient of discharge, and a, b, and c are constants depending on the orifice shape.
Table 2 .Values of the coefficients in Eq. 5. Fig 16 shows a comparison between the actual values of the discharge coefficient using a scatter diagram, and that obtained from Eq. 5.It can be noticed that the maximum error is less than 1% with R square equal to 0.994.A second Eq. 6 for computing the discharge coefficient is obtained as a function of the beta ratio and Reynolds number.This equation is similar to the (Danesh and Hassan 2018) equation but with different values of constant for all the experimental data.The coefficient of determination R 2 for the values computed by this equation is 0.887. (6)

. Equation for Head Loss Estimation
The head loss through the orifice is a very important parameter that has many engineering applications as a tool for energy dissipation.An equation for computing the dimensionless head loss (∆H* = ∆H/d) through the orifice can be deduced from the experimental data in the form of Eq. 7, where a and b are coefficients depending on the orifice shape and beta ratio.Values of coefficients a and b are tabulated in Table 3, Eq.7 shown the following: (7) Where ∆H is head loss, d is hydraulic diameter of the orifices and R e is Reynolds number.

. Numerical Results
Figs 22 to 25 the velocity contours of circular, triangular, square, and hexagonal orifices, respectively.As shown from these figures, the high velocity area for the triangular orifice is larger compared with the other orifice shapes.Also, can be noticed from Fig. 26 that the longitudinal centerline velocity is higher for triangular orifice which confirms the experimental results that the discharge coefficient is the higher for triangular orifice.Fig. 27 shows the pressure variation along the pipe centerline for the four orifice shapes.It is noticeable that the square orifice has the higher-pressure difference and this in agreement with the experimental results.Also, it can be shown from

4-Conclusions
The experimental and numerical study has been carried out for four shapes of orifice meters (circular, square, triangular, and hexagonal) to investigate the effect of orifice shape on discharge coefficient and head loss through the orifice.The findings of this study can be summarized as follows; 1.For all orifice shapes, discharge varies linearly with the Reynolds number and is not affected by the orifice hydraulic diameter to pipe diameter ratio called beta ratio (β). 2.An empirical equation is developed to calculate the head loss through the orifices.3. Triangular orifices have the greatest average discharge coefficient, ranging from 0.62 to 0.8. 4. Circular orifices have the lowest average discharge coefficient, ranging from 0.58 to 0.62. 5.The discharge increases by increasing of pressure head difference through the orifice, however, for small size orifice the rate of increase of discharge by increasing the head is small compared with large orifice size for all orifice shapes.6. Empirical equations are deduced for computing the discharge coefficient based on dimensional theory analysis and approximate relation for the different orifice shapes as a function of beta ratio and Reynold's Number.7. Orifice meter with d/D less than 0.3 is more effective in energy dissipation so it is recommended to be used in surge damping.8.For turbulent flow type of Reynolds number higher than 5000, Reynolds number has a slight effect on discharge coefficient and this effect varies with the shape of the orifice and orifice diameter to pipe ratio.9. Numerical analysis confirmed the experimental results where the triangular orifice has the pressure head difference and the vena contracta far from the orifice distance equal to half the pipe diameter.

Declaration of Conflicting Interests
The Authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Fig 1.The plate's sharp edge is facing the upstream flow.Four shapes of orifice plates made from copper are investigated in this experiment which are triangular, square, circular, and hexagonal orifices, for each one the cross-sectional area is changed four times as shown in the following Fig 2.

Fig 1 .
Fig 1. Cross section at the pipe.

Fig 3 .
Fig 3. Orifice meter components and the pressure taps.

Fig 4
Fig 4 shows the Experimental setup that is used in this study.The pressure upstream of the orifice decreases gradually until it reaches the vena contracta after which it progressively raises until it reaches its original value at a distance of about 5 to 8 orifice diameters downstream (Abd et al. 2019).

Fig 5 .
Fig 5.The flow chart shows the steps of the experiment for sixteen orifices.

Fig 8
Fig 8 shows the variation of discharge coefficient with Reynolds number for triangular orifice meter with four different sizes represented as dimensionless values d/D of 0.26, 0.35, 0.43, and 0.49 respectively.For non-circular orifice meter openings, d values are taken equal to two times the hydraulic radius of the cross-section.In contrast to Fig 8, the discharge coefficient for triangular orifice cross-section increases by increasing Reynolds number, however, there is no clear trend with the d/D value.Also, it can be noticed that discharge coefficient values in the range of 0.62 to 0.8 which considered a high value compared to circular orifice and the highest value for all orifice shapes in Figs 7 to 10. Fig 9 shows the variation of discharge coefficient with Reynolds for square orifice crosssection with four different values of d/D i.e., 0.3, 0.4, 0.5, and 0.57, respectively.It can be shown from that figure, that there is no clear trend for the variation of discharge coefficient with Reynolds number and d/D values.For d/D equal to 0.3 and 0.4, C d is nearly constant with an average value equal to 0.61 and for d/D equal to 0.57, C d is higher than 0.62 and slightly increases by increasing Reynolds number.The highest values are for d/D equal to 0.3.In general, the discharge coefficient for a square orifice is higher than that for circular and hexagonal, however, it has less C d values for triangular.Looking at Fig.10for the hexagonal shape, it can be found the same trend for other shapes with a value of discharge coefficient is slightly higher than that for circular orifice.The conclusion is that the discharge coefficient increases by increasing the irregularity in the orifice shape and this may be due to the reduction of the dead zone downstream of the orifice.

Fig 7 .
Fig 7. Variation of C d with Reynolds number for the circular orifices at different values of d/D.

Fig 8 .
Fig 8. Variation of C d with Reynolds number for the triangular orifices at different values of d/D.

Fig 9 .Fig 10 .
Fig 9. Variation of C d with Reynolds number for the square orifices at different values of d/D.

Fig 11 .Fig 12 .
Fig 11.Variation of theoretical discharge with pressure head difference at different values of d/D for circular orifice shape.

Fig 13 .Fig 14 .
Fig 13.Variation of theoretical discharge with pressure head difference at different values of d/D for square orifice shape.

Fig 15 .
Fig 15.Variation of discharge coefficient with relative area.

Fig 16 .
Fig 16.Average C d from experimental results versus C d computed from Eq. 5.

Fig 17 .
Fig 17.Variation of head loss with Reynolds number at different values of d/D for circular orifice.

Fig 18 .
Fig 18. Variation of head loss with Reynolds number at different values of d/D for triangular orifice.

Fig 19 .
Fig 19.Variation of head loss with Reynolds number at different values of d/D for square orifice.

Fig 20 .
Fig 20.Variation of head loss with Reynolds number at different values of d/D for hexagonal orifice.
Fig 26 and Fig 27, the vena contracta at about 5 cm downstream the orifice plate which is similar to the experimental work suggestion.

Table 1
Range of the experimental variables.

Table 3
Value of coefficients a and b in Eq. (7).
Fig 21.Variation of head loss with Reynolds number for all orifice shapes and sizes.