Fertigation involves dissolving fertilizer in irrigation water, and requires uniform water distribution and fertilization. A gated conduit can be used to inject liquid fertilizers or chemicals into a pressurized irrigation system. When the conduit gate is partially opened, a vacuum draws the fluid from the suction pipe into the conduit. An experimental study was conducted to investigate the fluid-injection ratio of gated conduits, which was shown to be high. The density and viscosity of the fluid were the most important parameters affecting the fluid-injection ratio. That ratio increased with increasing Froude number. It decreased as the gate opening proportion increased and with increasing conduit length. A design formula related to the Froude number, and the fluid's density and viscosity is presented to enable estimation of the fluid-injection ratio.

  • A study was conducted to investigate the fluid-injection ratio of gated conduits.

  • It was observed from the results that gated conduits had a high fluid-injection ratio.

  • The specific weight and viscosity were the most important parameters affecting the fluid-injection ratio.

  • A design formula was presented for estimating the fluid-injection ratio.

  • The obtained data provide a good basis for the development of numerical models.

     
  • Aw

    water flow cross-sectional area

  •  
  • B

    water surface width

  •  
  • D

    conduit diameter

  •  
  • Fr

    Froude number based on effective depth in conduit

  •  
  • g

    gravitational acceleration

  •  
  • h

    gate opening

  •  
  • ye

    effective depth

  •  
  • L

    conduit length

  •  
  • QF

    fluid flow rate in suction pipe

  •  
  • QF/QW

    fluid-injection ratio (ratio of fluid flow rate to that of water)

  •  
  • QW

    water flow rate in conduit

  •  
  • VW

    water flow velocity at gate location

  •  
  • μ

    dynamic viscosity

  •  
  • γ

    density

  •  
  • φ

    ratio of water cross-sectional flow area to conduit cross-sectional area

Gated conduits ensure that water is transferred downstream from the reservoir, to facilitate lowering of the reservoir level and/or to ensure maintenance of the biological minimum flow required downstream. For small dams and reservoirs, gated conduits often also serve as water intakes for some users, mostly for irrigation (Tanchev 2014).

Gated conduits are hydraulic structures that involve high-velocity air-water flow. If the conduit gate is opened partially, flow velocity increases due to the reduced cross-section. High-velocity flow can cause structural damage, e.g., by cavitation. To reduce or eliminate this damage, a suction pipe is installed after the gate, where flow velocity is high, so that air is drawn into the conduit and the pressure downstream of the gate is maintained at a safe level.

Numerous studies have been reported concerning air-injection ratios in gated conduits, e.g., Aydin et al. (2021), Baylar et al. (2021, 2022); Hohermuth (2019), Hohermuth et al. (2020), Pengchengi et al. (2022). The study results, between them, show that the air-injection ratio is a function of geometry and hydraulics.

Fluid injection created in gated conduits can be used for many purposes such as solving environmental problems such as lack of dissolved oxygen (Baylar et al. 2010). They can also be used to inject liquid fertilizers or chemicals into pressurized irrigation systems, as investigated in this study.

When a conduit cross-section is partially closed by a gate, high-velocity and low-pressure flow occurs downstream of the gate, and pressure below atmospheric can occur within the conduit. Theoretically, these pressures could be as low as the vapor pressure of water. When an atmospheric connection is made by installing a suction pipe after the gate, fluid suction occurs in the suction pipe (Figure 1). The extent of fluid suction inside varies according to the flow's entraining and carrying capacity. The pressure reduction downstream of the gate is a function of the Froude number (Fr), gate opening proportion and conduit length, and the fluid's density and viscosity.

Figure 1

Fluid injection via a gated conduit.

Figure 1

Fluid injection via a gated conduit.

Close modal

The aim of the study was to understand the role of the gated conduit in the fluid-injection ratio. The experimental setup was built at Firat University Hydraulic Laboratory, Elazig, Turkey, and the components used are shown in Figure 2.

Figure 2

Experimental apparatus.

Figure 2

Experimental apparatus.

Close modal

The gated conduit's fluid-injection ratio was investigated using fluids with different densities and viscosities – see Table 1. The oil used in the tests was used engine oil.

Table 1

Fluids injected into gated conduit

FluidDensity γ (kg/m3)Dynamic viscosity μ (kg.s/m2)
1 (air) 1.225 1.85 × 10−5 
2 (oil) 920 8 × 10−3 
3 (water) 1,000 1 × 10−3 
FluidDensity γ (kg/m3)Dynamic viscosity μ (kg.s/m2)
1 (air) 1.225 1.85 × 10−5 
2 (oil) 920 8 × 10−3 
3 (water) 1,000 1 × 10−3 

Gate opening proportions (φ) of 15, 30 and 60% of the conduit's cross-section were used (Figure 3) and water was passed at different flow rates (QW) under the gate. An electromagnetic flow meter was used to measure the flow rate. The conduit diameter (D) was 68 mm, and three conduit lengths (L) were used: 2, 4, and 6 m.

Figure 3

Gate opening proportions.

Figure 3

Gate opening proportions.

Close modal
The values of Fr were between 2.4 and 48.4. In the literature, Fr is often considered in the vena contracta section. Because gated conduits involve high-velocity, two-phase flow, and to avoid the problem of determining flow depths and velocities in the vena contracta section, Fr in this study was based on the conduit's effective depth (Figure 4) and was calculated using Equation (1).
formula
(1)
where VW is water velocity (m/s), g gravitational acceleration (m/s2), and ye the effective depth (m).
formula
(2)
where Aw is the water flow cross-sectional area (m2), and B the water surface width (m).
Figure 4

Water flow cross-sectional area and water surface width.

Figure 4

Water flow cross-sectional area and water surface width.

Close modal

The first injection fluid studied was air, the velocity of which was measured in suction pipes at four locations downstream of the gate with an anemometer (Testo Model 435). The anemometer was placed in the middle of the suction pipe and each measurement lasted for at least 60 seconds. The accuracy of the anemometer used is ±0.2 m/s +1.5% of mv. Precautions were taken to ensure that the anemometer was always perpendicular to the direction of airflow. The fluid-injection ratios for oil and water were measured by delivering them into the system from a graduated container. Thus, after water passed through the conduit, the ends of the suction pipes were immersed in the graduated container for a measured time – about 20 seconds – and the volume change in the graduated container measured, so that the flow rate could be determined. The list of experiments is given in Table 2.

Table 2

List of experiments

φLQWVWAW × 10−4B × 10−2Fr
(%)(m)(l/s)(m/s)(m2)(m)(-)
(a) Air 
15 1.835 5.45 5.515 5.893 
15 3.670 5.45 5.515 11.786 
15 5.505 5.45 5.515 17.679 
15 7.339 5.45 5.515 23.572 
15 9.174 5.45 5.515 29.465 
15 11.009 5.45 5.515 35.359 
15 12.844 5.45 5.515 41.252 
15 14.679 5.45 5.515 47.145 
(b) Oil 
15 1.835 5.45 5.515 5.893 
15 3.670 5.45 5.515 11.786 
15 5.505 5.45 5.515 17.679 
15 7.339 5.45 5.515 23.572 
15 9.174 5.45 5.515 29.465 
15 11.009 5.45 5.515 35.359 
15 12.844 5.45 5.515 41.252 
15 14.679 5.45 5.515 47.145 
(c) Water 
15 2, 4, 6 1.835 5.45 5.515 5.893 
15 2, 4, 6 3.670 5.45 5.515 11.786 
15 2, 4, 6 5.505 5.45 5.515 17.679 
15 2, 4, 6 7.339 5.45 5.515 23.572 
15 2, 4, 6 9.174 5.45 5.515 29.465 
15 2, 4, 6 11.009 5.45 5.515 35.359 
15 2, 4, 6 12.844 5.45 5.515 41.252 
15 2, 4, 6 14.679 5.45 5.515 47.145 
30 2, 4, 6 1.5 1.375 10.91 6.445 3.374 
30 2, 4, 6 2.750 10.91 6.445 6.748 
30 2, 4, 6 4.5 4.125 10.91 6.445 10.122 
30 2, 4, 6 5.500 10.91 6.445 13.496 
30 2, 4, 6 7.5 6.874 10.91 6.445 16.869 
30 2, 4, 6 8.249 10.91 6.445 20.243 
30 2, 4, 6 10.5 9.624 10.91 6.445 23.617 
30 2, 4, 6 12 10.999 10.91 6.445 26.991 
30 2, 4, 6 15 12.374 10.91 6.445 30.365 
60 2, 4, 6 1.377 21.79 6.757 2.448 
60 2, 4, 6 2.754 21.79 6.757 4.896 
60 2, 4, 6 4.130 21.79 6.757 7.343 
60 2, 4, 6 12 5.507 21.79 6.757 9.791 
60 2, 4, 6 15 6.884 21.79 6.757 12.239 
60 2, 4, 6 18 8.261 21.79 6.757 14.687 
60 2, 4, 6 21 9.637 21.79 6.757 17.135 
60 2, 4, 6 25 10.785 21.79 6.757 19.175 
φLQWVWAW × 10−4B × 10−2Fr
(%)(m)(l/s)(m/s)(m2)(m)(-)
(a) Air 
15 1.835 5.45 5.515 5.893 
15 3.670 5.45 5.515 11.786 
15 5.505 5.45 5.515 17.679 
15 7.339 5.45 5.515 23.572 
15 9.174 5.45 5.515 29.465 
15 11.009 5.45 5.515 35.359 
15 12.844 5.45 5.515 41.252 
15 14.679 5.45 5.515 47.145 
(b) Oil 
15 1.835 5.45 5.515 5.893 
15 3.670 5.45 5.515 11.786 
15 5.505 5.45 5.515 17.679 
15 7.339 5.45 5.515 23.572 
15 9.174 5.45 5.515 29.465 
15 11.009 5.45 5.515 35.359 
15 12.844 5.45 5.515 41.252 
15 14.679 5.45 5.515 47.145 
(c) Water 
15 2, 4, 6 1.835 5.45 5.515 5.893 
15 2, 4, 6 3.670 5.45 5.515 11.786 
15 2, 4, 6 5.505 5.45 5.515 17.679 
15 2, 4, 6 7.339 5.45 5.515 23.572 
15 2, 4, 6 9.174 5.45 5.515 29.465 
15 2, 4, 6 11.009 5.45 5.515 35.359 
15 2, 4, 6 12.844 5.45 5.515 41.252 
15 2, 4, 6 14.679 5.45 5.515 47.145 
30 2, 4, 6 1.5 1.375 10.91 6.445 3.374 
30 2, 4, 6 2.750 10.91 6.445 6.748 
30 2, 4, 6 4.5 4.125 10.91 6.445 10.122 
30 2, 4, 6 5.500 10.91 6.445 13.496 
30 2, 4, 6 7.5 6.874 10.91 6.445 16.869 
30 2, 4, 6 8.249 10.91 6.445 20.243 
30 2, 4, 6 10.5 9.624 10.91 6.445 23.617 
30 2, 4, 6 12 10.999 10.91 6.445 26.991 
30 2, 4, 6 15 12.374 10.91 6.445 30.365 
60 2, 4, 6 1.377 21.79 6.757 2.448 
60 2, 4, 6 2.754 21.79 6.757 4.896 
60 2, 4, 6 4.130 21.79 6.757 7.343 
60 2, 4, 6 12 5.507 21.79 6.757 9.791 
60 2, 4, 6 15 6.884 21.79 6.757 12.239 
60 2, 4, 6 18 8.261 21.79 6.757 14.687 
60 2, 4, 6 21 9.637 21.79 6.757 17.135 
60 2, 4, 6 25 10.785 21.79 6.757 19.175 

The variation of the fluid-injection ratio (QF/QW) with Fr for φ and L is shown in Figures 5 and 6. As can be seen, QF/QW increased as Fr increased in all experiments. In other words, there is a direct relationship between Fr and QF/QW.

Figure 5

QF/QW vs Fr for water, for φ = (a) 15%, (b) 30%, (c) 60%.

Figure 5

QF/QW vs Fr for water, for φ = (a) 15%, (b) 30%, (c) 60%.

Close modal
Figure 6

QF/QW vs Fr for water, for L [m] = (a) 2, (b) 4, (c) 6.

Figure 6

QF/QW vs Fr for water, for L [m] = (a) 2, (b) 4, (c) 6.

Close modal

QF/QW decreased with increasing φ because the pressure difference between the upstream and downstream sides of the gate decreases as the gate's open proportion increases. QF/QW also decreased with increasing L, because the pressure difference between the upstream and downstream sides of the gate also decreases as L increases.

Figures 5 and 6 show that the best QF/QW value was achieved in the gated conduit with φ = 15% and L = 2 m. The QF/QW values of all three fluids were, therefore, compared for φ = 15% and L = 2 m. Figure 7 shows plots of QF/QW vs Fr for all three fluids. As can be seen, QF/QW increases with increasing Fr in every case.

Figure 7

QF/QW vs Fr for air, water and oil.

Figure 7

QF/QW vs Fr for air, water and oil.

Close modal

Fluid density (γ) is also an important parameter affecting QF/QW. It was observed that more air was injected than either water or oil. However, water's QF/QW exceeded that of oil, even though its density was higher, because its viscosity (μ) is lower.

Fluid viscosity is also an important influence on QF/QW, which increases with decreasing fluid viscosity, making it easier to inject the fluid into the system.

Many equations have been developed to determine QF/QW in gated conduits, but for a single type of fluid (usually air). In this study, an equation based on Fr, and fluid γ and μ was developed for prediction (Equation (3)). Nonlinear regression was used to determine the equation constants and the correlation coefficient (R2) for the 91 data points used was 0.94. Comparison of the observed values of QF/QW with those predicted by Equation (3) showed good agreement. The quality of the correlation can be seen in Figure 8.
formula
(3)
where Fr is the Froude number, γ the fluid density (kg/m3), and μ its dynamic viscosity (kg.s/m2).
Figure 8

Comparison of observed and predicted values of QF/QW.

Figure 8

Comparison of observed and predicted values of QF/QW.

Close modal

In this study, the fluid-injection ratio of gated conduits was examined. When a gate is partially opened, the water flow velocity under it increases due to the reduced cross-section. As the flow velocity increases, the pressure downstream of the gate decreases, because of which fluid can be drawn into the conduit via a suction pipe opened downstream of the gate. Numerous field measurements and model studies were carried out to determine the air-injection ratio in the gated conduits, but extensive literature searches have not shown any published analytical or physical studies of the fluid-injection ratio in gated conduits. Laboratory experiments were done, therefore, to determine the fluid-injection ratio of gated conduits, from which it was concluded that:

  • Gated conduits have a high fluid-injection ratio.

  • The fluid-injection ratio increases as the Froude number increases as well as when the gate opening proportion decreases, but decreases with increasing conduit length.

  • Fluid density affects the fluid-injection ratio, as does fluid viscosity – the fluid-injection ratio increasing with decreasing fluid viscosity.

We would like to thank the editor and the reviewers for taking the time and effort necessary to review the manuscript. We sincerely appreciate all valuable comments and suggestions, which helped us to improve the quality of the manuscript.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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