1 Introduction

Nowadays, the scarcity of freshwater for drinking, agriculture, and some industries has become a global concern. As the population grows, this crisis will rise further in the future; therefore, using desalination plants is an effective solution that can produce freshwater from seawater [6]. In all kinds of desalination processes, eventually, a large quantity of brine sewage is discharged into the marine environment by different types of outfall systems. This brine sewage contains a high amount of salts and several chemical materials that make the sewage heavier than seawater; so this sewage tends to settle on the seabed and form a stable stratification of brine and other contaminants. This phenomenon may cause harmful impacts on the marine ecosystem [10, 16]. The use of inclined jets is one of the discharge methods that can reduce these hazards by mixing and dilution processes. Because the brine sewage is heavier than seawater, it is preferable to discharge the sewage upward, to increase the trajectory and consequently achieve better dilution at the return point [3].

In recent decades, extensive researches have been conducted on both the geometry and the dilution parameters of inclined dense jets, which investigated the behavior of jets in different conditions of nozzle and ambient. Numerous researches have been done about optimum nozzle orientation in the single port inclined dense jets [8, 14, 15, 17,18,19,20,21,22, 24, 25, 27]. A notable disagreement is visible for the dilution at the return point in the previous studies, but these studies generally showed that the highest dilution at the return point occurs at about 60°. Studies on the effects of the bottom slope showed that, by increasing the slope from 0° to 30°, the optimum angle decrease from 60° to about 40° [12]. Also, some studies have been carried out on the patterns of concentration and velocity distribution in cross sections of the dense effluent trajectories; the results show that the distribution is essentially Gaussian for jets when the momentum flux is dominant. However, it is reported to be half Gaussian for surface discharges or Semi Gaussian for inclined dense jet flows when the buoyancy flux is dominant, actually, the concentration distribution deviates from the Gaussian in the lower part because dense fluid detaches from the jet flow [1, 9, 11, 21, 23, 28]. Several studies have been conducted on multiport outfall systems to determine the optimal distance and arrangement of nozzles [2, 26, 30]. The buoyancy behavior of thermal-saline effluent has also been investigated [4]. Besides the experimental studies, mathematical and numerical modeling have been considered [5, 13, 18, 19, 31] and commercial software such as CORMIX, CORJET and VISJET have been developed, but the experimental results are still the basis of estimating the prototype behavior.

The simplest experimental model for inclined dense jet is a single port jet that discharges the brine (ρ0) through a D0 diameter nozzle with U0 velocity into a stagnant ambient (ρa < ρ0) at θ0 angle toward the horizontal (Fig. 1).

Fig. 1
figure 1

Schematic and time-averaged image of an inclined dense jet modeled in the laboratory for this study

Full size image

In inclined dense jets, when the jet enters the water ambient, the momentum flux is dominant and the jet moves upward; then, by reduction of the flow velocity, the vertical momentum flux fades out and the buoyancy flux dominates [29]. Therefore, the jet turns into a plume and after reaching to the maximum height, it starts to move downward until hitting the bottom. In the flow trajectory, the line with the highest concentration is called centerline and is obtained from time-averaged image. In each of the cross sections of the flow, the point that has maximum concentration (bolder color) is one point of the centerline. The set of these points forms the centerline of the trajectory. The length of the centerline (LC) is used as the length of the flow path.

The maximum height of rise (Zmax) and the distance of centerline return point to the nozzle level (XC) are important geometrical parameters of the flow trajectory [7]. The flow behavior is controlled by a dimensionless number that is defined as the initial densimetric Froude number (F0):

$$F_{0} = \frac{{u_{0} }}{{\sqrt {g^{\prime } .D_{0} } }}$$
(1)
$$g^{\prime } = \frac{{g(\rho _{0} - \rho _{{\text{a}}} )}}{{\rho _{0} }}$$
(2)

The g′ is the modified acceleration of gravity. For densimetric Froude numbers greater than about 20, the normalized trajectory of the flow is a function of the nozzle angle [15]; therefore, this principle also includes the maximum height of upper boundary (Zmax) and the distance of return point (XC), and these can be presented in the following forms:

$${R}_{\mathrm{Z}}=\frac{{Z}_{\mathrm{max}}}{{D}_{0}.{F}_{0}}$$
(3)
$${R}_{\mathrm{X}}=\frac{{X}_{\mathrm{C}}}{{D}_{0}.{F}_{0}}$$
(4)

In this study, an attempt has been made to extract a parameter (Gθ) that can be used to estimate the trajectory length with having the XC and the Zmax. This parameter is defined as follows:

$${G}_{\theta }=\frac{{L}_{\mathrm{C}}}{({X}_{\mathrm{C}}+{Z}_{\mathrm{max}})}$$
(5)

where LC is the length of the centerline that obtained from each experiment.

The dilution (S) at a particular point of the flow is defined as the ratio of the initial salinity of the brine (C0) to the salinity(C) of that point of the flow:

$$S=\frac{({C}_{0}-{C}_{\mathrm{a}})}{(C-{C}_{\mathrm{a}})}$$
(6)

where Ca is the salinity of the ambient.

The dilution can be normalized by F0 and becomes a function of the nozzle angle (θ0) [29]. The minimum dilution at the return point (Si) occurs in the centerline and can be presented in the following form:

$$k=\frac{{S}_{\mathrm{i}}}{{F}_{0}}$$
(7)

Considering the fact that many animals and plants live on the seabed, the angle that causes the greatest dilution at the return point is the optimum angle to minimize the environmental hazards of brine disposal in the marine environment. In the following, the effects of nozzle orientation on dilution and geometrical parameters of the inclined dense jets are investigated for a comprehensive series of nozzle angles by light attenuation method. The results are compared with other available studies. Furthermore a parameter is defined as dimensionless length of the trajectory (RL) and its relationship with the geometry and dilution of the flow has been investigated.

2 Methodology

2.1 Experimental setup

In the current study, an experimental model has been made with the following description. A glass tank, 2 m long, 2 m wide and 1.2 m high, was used for simulating the sea ambient in the experiments (Fig. 2). To prepare the brine, a smaller tank with a capacity of 0.3 m3 is placed approximately 3 m above the ambient tank. There is a mixer in the brine tank to mix salt (NaCl) and water. Also, to observe the effluent path, a color detector (Rhodamine) is mixed with the brine in this tank. An overflow container is located near the brine tank to provide the constant head. There is a rotameter between the constant head and the nozzle to adjust the discharge rate by a regulation valve, therefore the initial densimetric Froude number is adjustable by changing two parameters: the discharge velocity and the brine salinity. The diameter of the nozzle is 4 mm and the discharge angle is adjustable; the nozzle is located 30 cm above the bottom of the ambient so that the boundary conditions don't affect the flow behavior. To measure the salinity of the brine and water ambient before each experiment, a portable multimeter (Hach) with a precision of 0.1 psu is used. The brine and water densities are measured by hydrometers with an accuracy of 0.1 kg/m3. To record the flow path, a full HD camera with the ability to recording 30 frame per second video is used, which is located at a distance of 3 m from the ambient; the manual setting is used on this camera to keep the focus from changing during the experiment. Another camera is used to record the experiment from above in order to measure the lateral spreading of the flow at the return point. The experiments are performed in a dark room and a luminous sheet is used in the backside of the ambient to increase the recording quality.

Fig. 2
figure 2

Schematic of experimental setup

Full size image

2.2 Calibration

In each experiment, a time-average image is provided from 45 s of the brine discharging by MATLAB image processing (Fig. 3). The geometrical parameters are extracted from the processed images. In particular, to find the centerline, a MATLAB image processing code was used, in which a desired number of cross sections perpendicular to the flow path was manually generated. The darkest points in each cross section were defined as points of the centerline, in the form of a broken line. This was converted to a smooth centerline by fitting a SP line in AutoCAD software and its length was obtained by using “list” command.

Fig. 3
figure 3

MATLAB image processing (30° jet): a instantaneous image. b Time-averaged image

Full size image

In this research, the light attenuation method is used to obtain the dilutions. So the dilutions at the return point are obtained from calibration of the light intensity in gray time-averaged images with the salinity. To achieve this purpose, at first the 2D light intensity (uniform cross-sectional profile) obtained from lateral camera should convert to the 3D (Gaussian cross-sectional profile). In order to achieve this purpose it is necessary to know the lateral spreading of the plume in the return point. Therefore, to determine the lateral spreading of the plume at the return point, the flow of the brine is recorded by a camera from above. Due to the simultaneous videos recording in both of the top and the side views, the lateral spreading of the plume can be obtained at the moment of the arrival to the return point. In other words, when in the lateral camera point of view the brine reaches the return point, the lateral spreading of the plume is measurable from the top camera point of view. According to the symmetry of the concentration in cross section of the flow, the distribution of the concentration is Gaussian. Due to the fact that lateral imaging has been done in 2D form, instead of the Gaussian profile, a uniform profile (with the same thickness and area) is used, because the light intensity is the same for both profiles. Thus, if the concentration peak point in the Gaussian profile is b, the concentration in the uniform profile is approximately 0.45b. This method is only for finding the concentration at the return point and is assumed that the lateral spreading of the plume is equal to the width of the Gaussian profile (Fig. 4).

Fig. 4
figure 4

Top view to measure lateral spreading at return point

Full size image

According to the facilities in this study, measuring of the salinity at the return point was not possible, but by using a method that will be explained, the salinity is obtained by the light intensity at the return point. For calibration of the light intensity and the brine salinity, two glass cubical containers with widths of 20 and 40 cm (in the range of lateral spreading of plumes at the return point) are used. Because the lateral spreading of the plumes at the return point are different in the experiments, thus two glass containers with different thicknesses have been used so that the effect of the lateral spreading of the plume on the light intensity can be measurable by interpolation. To find the equivalent salinity of any light intensity, after each experiment, the glass cubical containers are filled with the brine. By adding clear freshwater in the containers, the reduction of the salinity and the light intensity is recorded, by Hach multimeter and video camera, respectively. All images are turned into gray in image processing. In the gray images, a number between 0 and 255 is assigned to each pixel, 0 assigned to absolute black and 255 assigned to absolute white. Eventually, the salinity of uniform profiles is obtained considering the interpolation of the plume thickness. These values are 0.45 of the Gaussian distribution peak point (salinity of centerline in the return point). Then, the dilutions are achieved by considering the salinities of the initial brine, the ambient and the return point (according Eq. 6). An example of typical curve obtained from light intensity (pixel index in gray image) and dilution is shown in Fig. 5.

Fig. 5
figure 5

Calibration curve of dilution and light intensity

Full size image

3 Results and analysis

3.1 Experimental data

In the present study attempts have been made to ensure that the initial parameters and nozzle angles have sufficient diversity and include appropriate range of densimetric Froude numbers. For each nozzle angle, at least 6 experiments have been carried out in various salinities. The specifications of each experiment and the extracted parameters are presented in Table 1. The experiments have been carried out in the range of densimetric Froude number from about 19 to about 62 and the angle of nozzle was set between 15° and 90° by 15° intervals.

Table 1 The main experimental specifications and parameters
Full size table

3.2 Geometrical parameters

The relation of the normalized return point distance (XC/D0) and the densimetric Froude number is presented in Fig. 6. For each nozzle orientation, the normalized return point distance has a significant linear relation with the dens metric Froude number (XC/D0 = RX.F0). Comparisons of the results of this study with the results of other researches are presented quantitatively in Table 2; furthermore, Fig. 7 indicates the RX values for the various nozzle orientations. Two points, 0° and 90°, have zero distance for obvious physical reasons.

Fig. 6
figure 6

Relation of normalized return point distance with densimetric Froude number at different angles

Full size image
Table 2 Results of RX in this study and other researches
Full size table
Fig. 7
figure 7

RX for various nozzle angles and comparison with previous studies

Full size image

The relation of the normalized maximum height of rise (Zmax/D0) and the densimetric Froude number is illustrated in Fig. 8. This indicates a simple linear function between normalized maximum height and dens metric Froude number (Zmax/D0 = RZ.F0). The results of RZ values that obtained in this research and in other studies are presented in Table 3 and Fig. 9.

Fig. 8
figure 8

Relation of normalized maximum height with densimetric Froude number at different angles

Full size image
Table 3 Results of RZ in this study and other researches
Full size table
Fig. 9
figure 9

RZ for various nozzle angles and comparison with previous studies

Full size image

The Gθ coefficient is in range of 0.84–1.11 that are presented in Table 1 for each experiment, According to the self-similarity of the trajectory at each angle, the average of these values is given in Table 4. According to Eq. 5, the parameter of Gθ can relate the length of the centerline (LC) to XC and Zmax.

Table 4 The average of Gθ at each angle
Full size table

3.3 Dilution at the return point

The following diagram (Fig. 10) indicates that there is an approximately linear function between return point dilution and densimetric Froude number (Si = K.F0) with an acceptable correlation coefficient. The values of K are presented in Table 5 for 15°, 30°, 45°, 60° and 75° angles. The dilution in return point is not measurable for 90°. In Fig. 11, the K values of this study are compared with the results of other researches. These results are sufficiently in line with other studies, although, there are slight differences which could be due to the variety of research methods and other initial conditions, such as salinity, temperature, turbulence, nozzle diameter and distance of nozzle from the bottom (boundary effects).

Fig. 10
figure 10

Relation of dilution with densimetric Froude number at different angles

Full size image
Table 5 Results of normalized return point dilution (Si/F0) in this study and other researches
Full size table
Fig. 11
figure 11

Return point dilution for various nozzle angles and comparison with previous studies

Full size image

4 Discussion

In Fig. 9, a decline is visible in the maximum height at about 75° due to the fallout effect and the interaction of the velocity vectors; this result is adequately in accordance with other studies. Figures 7 and 9 show that the maximum return point distance (XC) occurred at about 45˚ and the maximum height of rise (Zmax) happened at about 75°, but in discharge of brine sewage, the length of the trajectory is important. Therefore, the relationship between trajectory length and dilution rate has been investigated. In this research a dimensionless parameter of trajectory length (RL) is defined as an appropriate parameter to estimate the length of the flow. So it is possible to obtain a relatively accurate value of the length of trajectory by using the following equation:

$$R_{{\text{L}}} = G_{\theta } \left( {R_{{\text{X}}} + R_{{\text{Z}}} } \right) = \frac{{L_{{\text{c}}} }}{{Z_{{\max }} + X_{{\text{c}}} }} \cdot \frac{{Z_{{\max }} + X_{{\text{c}}} }}{{D_{0} F_{0} }} = \frac{{L_{{\text{c}}} }}{{D_{0} F_{0} }}$$
(8)

It is worth mentioning that it is not possible to determine the exact length of the centerline (LC) at angle of 90° because despite the maximum height of rise (Zmax) is recognizable, the highest point of the centerline (ZC) is not recognizable in this angle, but considering that at angle of 90˚, the fallout effect is maximum and also the XC is zero, surely the length of trajectory at this angle is less than the length of trajectory at 75°.

As can be seen in Fig. 12, the maximum length of the trajectory occurs between 60˚ and 75˚; but the maximum dilution occurs at about 60˚. These results show that the maximum length of the trajectory has a minor difference with the maximum dilution that is due to the decrease of diffusion at higher angles. It can be concluded that although the maximum length of the path is created at an angle between 60º and 75º, but the nozzle angle at about 60˚ has a better performance in dilution of the brine.

Fig. 12
figure 12

Parameter of trajectory length (RL = Gθ (RX + RZ))

Full size image

5 Conclusion

Several experiments are performed to investigate the nozzle orientation effects on the geometry and dilution behavior of the inclined dense jets, to find the optimum angle that causes the least hazard to marine environment. The results show that the maximum distance of the centerline return point (XC) occurs at about 45˚, the maximum height of rise occurs at about 75˚ and a decline in the height of rise is observed for higher angles (fallout effect). To find the relationship between trajectory length and dilution rate, a parameter is defined as RL to express the dimensionless length of the trajectory. The maximum value of RL occurs between 60° and 75°, but according to the experimental results for maximum dilution that occurs at 60°, it can be said that discharging the brine sewage at about 60˚ and by greater Froude number can cause more dilution, so is preferable for outfall systems.