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Integrated modeling of single port brine discharges into unstratified stagnant ambient

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 co-financed 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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Correspondence to I. K. Nikiforakis.

Appendices

Appendix 1: Local parameters in ZEF

$$ u_{c} = \frac{2M}{Q} $$
(23)
$$ b = \frac{Q}{{\sqrt {2\pi M} }} \, $$
(24)
$$ g_{c}^{{\prime }} = \frac{J}{{Q_{scalar} }} \, $$
(25)
$$ C_{c}^{{}} = \frac{{Q_{c} }}{{Q_{scalar} }} \, $$
(26)
$$ Q_{scalar} = \pi b^{2} u_{c} \left( {\frac{{\lambda^{2} }}{{\lambda^{2} + 1}}} \right) $$
(27)

Appendix 2: Jet parameters at the end of ZOFE

$$ L_{e} = 5D_{jeto} \left( {1 - e^{{{{ - 2F_{o} } \mathord{\left/ {\vphantom {{ - 2F_{o} } {F_{lp} }}} \right. \kern-0pt} {F_{lp} }}}} } \right) $$
(28)
$$ X_{e} = L_{e} \cos \theta ,\,Z_{e} = h_{o} + L_{e} \sin \theta $$
(29)
$$ Q_{e} = \sqrt 2 {\text{ Q}}_{\text{o}} $$
(30)
$$ M_{e} = M_{o} $$
(31)
$$ J_{e} = J_{o} $$
(32)
$$ Q_{ce} = Q_{co} $$
(33)

In Eqs. 2833, the subscripts “o” and “e” refer to the values of the variables at the source and at the end of ZOFE, respectively. Le is the modified ZOFE length and Flp 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, S1 and S2. The coordinates of the jet axis (Xjet, Yjet, Zjet), and therefore, the coordinates of centerline points S1 (Xjet1, Yjet1, Zjet1) and S2 (Xjet2, Yjet2, Zjet2), can be derived by solving the equations of the MCM. However, the coordinates of the (a) upper and lower jet boundaries, Sobup (Xobup, Yjet, Zobup) and Soblow (Xoblow, Yjet, Zoblow), and (b) lateral jet boundaries, Sobl (Xjet, Yobl, Zjet) and Sobr (Xjet, Yobr, Zjet), can be determined by employing the equations of (a) the Euclidean distance between the points S2 and Si2, 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 S1, S2 and Si2, see Eq. 35. In the following, the determination of the coordinates of Sobup and Soblow is only described; however, the determination of the coordinates for Sobl and Sobr can be performed similarly.

$$ \left( {X_{jet2} - X_{i2} } \right)^{2} + \left( {Y_{jet2} - Y_{i2} } \right)^{2} + \left( {Z_{jet2} - Z_{i2} } \right)^{2} = b^{2} $$
(34)
$$ \left( {X_{jet2} - X_{jet1} } \right)\left( {X_{i2} - X_{jet2} } \right) + \left( {Y_{jet2} - Y_{jet1} } \right)\left( {Y_{i2} - Y_{jet2} } \right) + \left( {Z_{jet2} - Z_{jet1} } \right)\left( {Z_{i2} - Z_{jet2} } \right) = 0 $$
(35)

After some algebraic manipulations, Eq. 35 gives

$$ X_{i2} = \alpha_{1} Z_{i2} + \beta_{1} , $$
(36)

where

$$ \alpha_{1} = - \frac{{{\rm Z}_{jet2} - {\rm Z}_{jet1} }}{{X_{jet2} - X_{jet1} }},\beta_{1} = \frac{{X_{jet2}^{2} + {\rm Z}_{jet2}^{2} - X_{jet1} X_{jet2} - Z_{jet1} Z_{jet2} }}{{X_{jet2} - X_{jet1} }}. $$
(37)

By substituting Eq. 36 to Eq. 34, Eq. 38 is obtained,

$$ kZ_{i2}^{2} + \mu Z_{i2} + \nu = 0, $$
(38)

where

$$ k = \alpha_{1}^{2} + 1,\mu = 2\alpha_{1} \beta_{1} - 2\alpha_{1} X_{jet2} - 2Z_{jet2} ,\nu = - b^{2} + X_{jet2}^{2} + Z_{jet2}^{2} - 2\beta_{1} X_{jet2} + \beta_{1}^{2} . $$
(39)

Finally, the coordinates of Si2 can be derived, see Eqs. 40 and 41, by solving the quadratic Eq. 38 and inserting the solutions to Eq. 36.

$$ S_{obup2} = \left( {X_{obup2} ,Y_{jet2} ,Z_{obup2} } \right) = \left( {\alpha_{1} \frac{{ - \mu + \sqrt {\mu^{2} - 4k\nu } }}{2k} + \beta_{1} ,Y_{jet2} ,\frac{{ - \mu + \sqrt {\mu^{2} - 4k\nu } }}{2k}} \right)$$
(40)
$$ S_{oblow2} = \left( {X_{oblow2} ,Y_{jet2} ,Z_{oblow2} } \right) = \left( {\alpha_{1} \frac{{ - \mu - \sqrt {\mu^{2} - 4k\nu } }}{2k} + \beta_{1} ,Y_{jet2} ,\frac{{ - \mu - \sqrt {\mu^{2} - 4k\nu } }}{2k}} \right)$$
(41)

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 (xm, ym, Zm) 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 (ux), the calculations of the vertical (uz) and horizontal in y direction (uy) velocities can be performed similarly. The steps of the developed methodology are:

  1. 1.

    The equation of the Euclidean distance between the center (m) of a trapezoid with coordinates (xm, ym, Zm) and a jet axis point (S) with coordinates (xjet,yjet, Zjet) 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. 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 (uc) 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.

    Fig. 14
    figure 14

    Sketch of the Gaussian distribution of local jet velocities at the location of the centerline point S; the jet velocity (um) at the point “m” is also depicted

$$ u_{m} = u_{c} e^{{ - {{d^{2} } \mathord{\left/ {\vphantom {{d^{2} } {b^{2} }}} \right. \kern-0pt} {b^{2} }}}} $$
(43)
$$ u_{x} = u_{m} \cos \theta $$
(44)

The minimization of d can be achieved by inserting the coordinates (xjet, yjet, Zjet) 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 Qimp is the flowrate of the NBJ at its impingement location and that the shape of the impingement region is rectangular, see Fig. 15b. Qimp is divided into four flowrates, one in the upslope (Qups), one in the downslope (Qds) and two in the lateral direction (Qlt). 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 Qimp; i.e. 2Qlt = αQimp; where 0 ≤ α ≤ 1.

Fig. 15
figure 15

a, b Definition sketch of the BIR of a NBJ denoted by the hatched FFCV and its velocity field as considered in the calculations

The continuity equation gives:

$$ (1 - \alpha )Q_{imp} = Q_{ds} + Q_{ups} \Rightarrow $$
(45a)
$$ (1 - \alpha )\frac{{U_{imp} }}{1.7}A_{imp} = \left( {L_{y} h_{ds} } \right)Ud_{ds} + \left( {L_{y} h_{ups} } \right)Ud_{ups} . $$
(45b)

The conservation of momentum along the upslope–downslope direction of the bottom gives:

$$ (1 - \alpha )Q_{imp} \frac{{U_{imp} }}{1.7}\cos (\theta_{imp} ) = Q_{ds} Ud_{ds} - Q_{ups} Ud_{ups} = \left( {L_{y} h_{ds} } \right)Ud_{ds}^{2} - \left( {L_{y} h_{ups} } \right)Ud_{ups}^{2} . $$
(46)

The conservation of energy gives:

$$ \frac{{Ud_{ds}^{2} }}{2g} = \frac{{\left( {1 - K_{L} } \right)}}{2g}\left( {\frac{{U_{imp}^{{}} }}{1.7}} \right)^{2} - \frac{\Delta \rho }{\rho }(Z_{ds} - Z_{imp} ) - \frac{\Delta \rho }{\rho }(h_{ds} - h_{imp} ) $$
(47a)
$$ \frac{{Ud_{ups}^{2} }}{2g} = \frac{{(1 - K_{L} )}}{2g}\left( {\frac{{U_{imp}^{{}} }}{1.7}} \right)^{2} - \frac{\Delta \rho }{\rho }(Z_{ups} - Z_{imp} ) - \frac{\Delta \rho }{\rho }(h_{ups} - h_{imp} ). $$
(47b)

Since himp = hlt = (hds + hups)/2, Eqs. 47a and 47b give Eqs. 48a and 48b, respectively.

$$ Ud_{ds}^{2} = \left( {1 - K_{L} } \right)\left( {\frac{{U_{imp}^{{}} }}{1.7}} \right)^{2} - 2g\frac{\Delta \rho }{\rho }(Z_{ds} - Z_{imp} ) - g\frac{\Delta \rho }{\rho }(h_{ds} - h_{ups} ) $$
(48a)
$$ Ud_{ups}^{2} = \left( {1 - K_{L} } \right)\left( {\frac{{U_{imp}^{{}} }}{1.7}} \right)^{2} - 2g\frac{\Delta \rho }{\rho }(Z_{ups} - Z_{imp} ) - g\frac{\Delta \rho }{\rho }(h_{ups} - h_{ds} ) $$
(48b)
$$ Ud_{lt} = \frac{{\alpha Q_{imp} }}{{2h_{imp} L_{x} }}. $$
(49)

In the above equations, Uimp and Aimp 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]; Lx and Ly are the dimensions of the impingement region parallel to x and y axes, respectively; θimp is the jet impingement angle, see Fig. 15a; KL 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 − ρα, where ρα is the ambient density; Zds and Zups are the vertical coordinates of the downslope and upslope bottom boundaries of BIR and Zlt 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 Udds, Udups, Udlt and the heights hds, hups and hlt. The steps of the determination process are described in the following:

  1. 1.

    From the application of the MCM, the values of Uimp, Aimp, ρ, Δρ and θimp at the level of the bottom are determined.

  2. 2.

    We select a pair of values for the heights hds and hups, and from Eqs. 47a and 47b, the velocities Udds and Udups are calculated.

  3. 3.

    We insert the values of hds, hups, Udds and Udups 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 hds and hups, and therefore Udds and Udups; the selection of these values continues until we find the proper ones.

  4. 4.

    From the determined hds and hups the value of himp is obtained. The MCM is applied again and the values of Uimp, Aimp, ρ and Δρ are determined at the height of himp. The trial and error process of hups, hds, Udds and Udups begins once again, until the continuity and momentum equations are satisfied.

  5. 5.

    At the end of this process, the final values of hds, hups, himp (or hlt), Udds, Udups and Udlt, 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 Lx and Ly 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/s10652-016-9473-0

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Keywords

  • Desalination
  • Brine discharges
  • Near-field models
  • Far-field models
  • Integrated models