Abstract
An integrated model is presented for the calculation of the characteristics in the intermediate field region of brine discharges from reverse osmosis desalination plants into unstratified stagnant coastal waters. The model consists of the near field model Modified CorJet Model and the far field model, which are interconnected via a coupling algorithm. This algorithm has been developed to simulate the flow and concentration characteristics of negatively buoyant jets (NBJ) after their impingement on the bottom. The coupling method was developed to be active according to literature, however further work and investigation is needed to be applicable for NBJ discharged into other ambient environments and especially in cases where the background values of ambient flow and concentrations affect the NF values and vice versa. The integrated model was validated with data from the literature as well as with data from experiments conducted in this study showing a good agreement. The coupling algorithm was also compared to other coupling techniques used in the literature for NBJ discharges showing better estimations of the experimental data.
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Acknowledgments
This research has been cofinanced by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund.
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Appendices
Appendix 1: Local parameters in ZEF
Appendix 2: Jet parameters at the end of ZOFE
In Eqs. 28–33, the subscripts “o” and “e” refer to the values of the variables at the source and at the end of ZOFE, respectively. L_{e} is the modified ZOFE length and F_{lp} is the asymptotic value of the local densimetric Froude number of a pure plume.
Appendix 3: Determination of the outer jet boundaries
In Fig. 2a, b, sketches of a NBJ are presented showing the trajectories of the jet axis, the jet outer boundaries and the jet CV defined by two axial points, S_{1} and S_{2}. The coordinates of the jet axis (X_{jet}, Y_{jet}, Z_{jet}), and therefore, the coordinates of centerline points S_{1} (X_{jet1}, Y_{jet1}, Z_{jet1}) and S_{2} (X_{jet2}, Y_{jet2}, Z_{jet2}), can be derived by solving the equations of the MCM. However, the coordinates of the (a) upper and lower jet boundaries, S_{obup} (X_{obup}, Y_{jet}, Z_{obup}) and S_{oblow} (X_{oblow}, Y_{jet}, Z_{oblow}), and (b) lateral jet boundaries, S_{obl} (X_{jet}, Y_{obl}, Z_{jet}) and S_{obr} (X_{jet}, Y_{obr}, Z_{jet}), can be determined by employing the equations of (a) the Euclidean distance between the points S_{2} and S_{i2}, see Eq. 34, where the subscript “i” may stand for “obup”, “oblow”, “obr” and “obl”, and (b) the perpendicularity between two vectors defined by the points S_{1}, S_{2} and S_{i2}, see Eq. 35. In the following, the determination of the coordinates of S_{obup} and S_{oblow} is only described; however, the determination of the coordinates for S_{obl} and S_{obr} can be performed similarly.
After some algebraic manipulations, Eq. 35 gives
where
By substituting Eq. 36 to Eq. 34, Eq. 38 is obtained,
where
Finally, the coordinates of S_{i2} can be derived, see Eqs. 40 and 41, by solving the quadratic Eq. 38 and inserting the solutions to Eq. 36.
The coordinates of the complete outer vertical boundaries can be derived by following a marching in space procedure, i.e. to the next centerline and boundary points, until the end of the NF region.
Appendix 4: Determination of local jet velocities
Since the coordinates (x_{m}, y_{m}, Z_{m}) of the center of a rectangle or trapezoid have been determined, the jet velocity at this center can be calculated by using the following methodology. Although this methodology describes the calculation of the horizontal jet velocities in x direction (u_{x}), the calculations of the vertical (u_{z}) and horizontal in y direction (u_{y}) velocities can be performed similarly. The steps of the developed methodology are:

1.
The equation of the Euclidean distance between the center (m) of a trapezoid with coordinates (x_{m}, y_{m}, Z_{m}) and a jet axis point (S) with coordinates (x_{jet},y_{jet}, Z_{jet}) is given by Eq. 42.
$$ \left( {x_{jet}  x_{m} } \right)^{2} + \left( {y_{jet}  y_{m} } \right)^{2} + \left( {Z_{jet}  Z_{m} } \right)^{2} = d^{2} $$(42) 
2.
Since the Gaussian distribution of velocities, described from Eq. 7, gives the local jet velocity field at distances perpendicular to the jet axis at a centerline point, see Fig. 14, the requirement for d is to be minimized (d → min). This minimization will provide that m is perpendicular to the jet axis at the axis point S, and therefore, by calculating (a) the jet centerline velocity (u_{c}) from Eq. 23 and (b) taking into account the local jet angle (θ), the horizontal jet velocity at m can be calculated by employing Eq. 42 and thereafter Eq. 44.
The minimization of d can be achieved by inserting the coordinates (x_{jet}, y_{jet}, Z_{jet}) of each jet axis point in Eq. 42, as they have been calculated by MCM, and then see which point gives the minimum distance d. It is noted that d should be in the range 0–b.
Appendix 5: Determination of the initial velocity field of the density current
The determination of the initial velocity field of density currents after the impingement of round NBJ is performed by employing the equations of continuity, momentum and energy, similar to Jirka and Harleman [11] and Lee and Jirka [20].
Firstly, we assume that Q_{imp} is the flowrate of the NBJ at its impingement location and that the shape of the impingement region is rectangular, see Fig. 15b. Q_{imp} is divided into four flowrates, one in the upslope (Q_{ups}), one in the downslope (Q_{ds}) and two in the lateral direction (Q_{lt}). The two lateral flowrates are assumed to be equal to each other, due to the symmetrical flow field in y direction, and that they are a percentage of Q_{imp}; i.e. 2Q_{lt} = αQ_{imp}; where 0 ≤ α ≤ 1.
The continuity equation gives:
The conservation of momentum along the upslope–downslope direction of the bottom gives:
The conservation of energy gives:
Since h_{imp} = h_{lt} = (h_{ds} + h_{ups})/2, Eqs. 47a and 47b give Eqs. 48a and 48b, respectively.
In the above equations, U_{imp} and A_{imp} is the jet centerline velocity and cross section area at a specific height from the bottom, respectively; 1.7 is the factor that converts the jet centerline velocity values to average jet values along a jet cross section [9]; L_{x} and L_{y} are the dimensions of the impingement region parallel to x and y axes, respectively; θ_{imp} is the jet impingement angle, see Fig. 15a; K_{L} is a head loss coefficient [20] that in the present study is assumed to be equal to zero; ρ is the average density of the jet at the specific height from the bottom; Δρ = ρ − ρ_{α}, where ρ_{α} is the ambient density; Z_{ds} and Z_{ups} are the vertical coordinates of the downslope and upslope bottom boundaries of BIR and Z_{lt} is the vertical coordinate of the bottom at the point where the jet axis is inserting into the BIR, see Fig. 15a; in the present study, α is assumed to be equal to 0.5.
The unknown values that should be determined through the trial and error process are the velocities Ud_{ds}, Ud_{ups}, Ud_{lt} and the heights h_{ds}, h_{ups} and h_{lt}. The steps of the determination process are described in the following:

1.
From the application of the MCM, the values of U_{imp}, A_{imp}, ρ, Δρ and θ_{imp} at the level of the bottom are determined.

2.
We select a pair of values for the heights h_{ds} and h_{ups}, and from Eqs. 47a and 47b, the velocities Ud_{ds} and Ud_{ups} are calculated.

3.
We insert the values of h_{ds}, h_{ups}, Ud_{ds} and Ud_{ups} in Eqs. 45b and 46 and, then, we see if they are satisfied. If they are satisfied the process continues to step 4. If they are not satisfied, we select another pair of h_{ds} and h_{ups}, and therefore Ud_{ds} and Ud_{ups}; the selection of these values continues until we find the proper ones.

4.
From the determined h_{ds} and h_{ups} the value of h_{imp} is obtained. The MCM is applied again and the values of U_{imp}, A_{imp}, ρ and Δρ are determined at the height of h_{imp}. The trial and error process of h_{ups}, h_{ds}, Ud_{ds} and Ud_{ups} begins once again, until the continuity and momentum equations are satisfied.

5.
At the end of this process, the final values of h_{ds}, h_{ups}, h_{imp} (or h_{lt}), Ud_{ds}, Ud_{ups} and Ud_{lt}, calculated for zero and small bottom slopes from Eq. 49, are determined and these values are used in the calculations of the integrated model.
It is noted that in the calculations, the lengths L_{x} and L_{y} were determined by multiplying the number of the small rectangles, see Fig. 15a, with their dx and dy dimensions, respectively.
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Nikiforakis, I.K., Stamou, A.I. & Christodoulou, G.C. Integrated modeling of single port brine discharges into unstratified stagnant ambient. Environ Fluid Mech 17, 247–275 (2017). https://doi.org/10.1007/s1065201694730
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DOI: https://doi.org/10.1007/s1065201694730
Keywords
 Desalination
 Brine discharges
 Nearfield models
 Farfield models
 Integrated models