**6. Design optimization**

### **6.1 Discharge through a single jet creating a submerged plume**

The optimum values of water depth, diameter and discharge velocity needed to dilute a contaminant with excess concentration of Δ*c*<sup>0</sup> to a desired excess concentration of Δ*cth*, with the additional constraint that the plume remains submerged, are given by Eqs. (12)–(15), respectively. *Hd* depends on the mass loading of the contaminant (in excess of ambient concentration, *m*\_ ¼ *Q*0Δ*c*0), buoyancy flux of the effluent (*B*<sup>0</sup> ¼ *Q*0*g*<sup>0</sup> 0 ) and desired concentration (Δ*cth*), but is independent of the flow rate (*Q*0) as shown in Eq. (13). Therefore, the required water depth for salinity as the contaminant of concern and seawater as the pre-diluting stream is independent of *RB* (as *m*\_ and *B*<sup>0</sup> are independent of *RB* in that case).

$$H\_d = \mathbf{1.38} \frac{Q\_0^{2/5}}{\mathbf{g}\_0^{\prime 1/5}} \left(\frac{\Delta c\_0}{\Delta c\_{th}}\right)^{3/5} \tag{12}$$

$$\text{or } H\_d = \frac{1.38}{B\_0^{1/5}} \left(\frac{\dot{m}}{\Delta c\_{th}}\right)^{3/5} \tag{13}$$

$$(D\_0)\_d = 1.18 \frac{Q\_0^{2/5}}{\mathcal{g}\_0^{\prime 1/5}} \left(\frac{\Delta c\_{th}}{\Delta c\_0}\right)^{2/5} \tag{14}$$

$$(u\_0)\_d = 0.91 Q\_0^{1/5} g\_0^{\prime 2/5} \left(\frac{\Delta c\_0}{\Delta c\_{th}}\right)^{4/5} \tag{15}$$

**Figure 3** shows the variation of *Hd*, ð Þ *D*<sup>0</sup> *<sup>d</sup>* and ð Þ *u*<sup>0</sup> *<sup>d</sup>* as functions of *RB* (for different pre-dilution streams) and *RC*. The variables are scaled so that they can be plotted on the same plot. The scaling is the same for all the pre-dilution cases (indicated in the legend for SW blending plot) but is different for the preconcentration case (indicated in the legend for pre-concentration plot) because of the different range of values.

When brine is pre-diluted, the desired physical dilution reduces with an increase in *RB* (except for *RB* >2 for blending with TWE), and thus the discharge velocity also reduces. For the case of brine concentration, the desired physical dilution increases rapidly as *RC* increases and the effluent needs to be discharged with very high velocity to achieve better mixing. For example, the desired physical dilution is equal to 54 for *RC* ¼ 2 and ð Þ *u*<sup>0</sup> *<sup>d</sup>* is equal to 17.7 m/s which is not realistic. ð Þ *u*<sup>0</sup> *<sup>d</sup>* is even higher for higher values of *RC* which suggests that a single jet should not be used to discharge concentrated brine at a location with restriction on plume visibility.

#### **6.2 Discharge through a single jet creating a surfacing plume**

This section explores the optimum design with no restriction on plume visibility, i.e., the design which minimizes total cost without any constraint. For most cases, this design results in a plume which hits the surface. But for some cases, the design

**Figure 3.**

*Variation of H, D*<sup>0</sup> *and u*<sup>0</sup> *with RB and RC for discharge using a single jet with submerged plume for Qb* ¼ <sup>1</sup> *<sup>m</sup>*<sup>3</sup>*=s,* <sup>Γ</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>01</sup> *and a desired excess salinity of* <sup>2</sup> *ppt. (variables are scaled differently for the pre-dilution and pre-concentration cases as indicated in the legend).*

with a submerged plume is also the one which minimizes the total cost and should be adopted. This design optimization results in non-linear equations which are solved using the 'fsolve' function in MATLAB.

**Figure 4** shows the variation of *Hsh*, ð Þ *D*<sup>0</sup> *sh* and ð Þ *u*<sup>0</sup> *sh* as functions of *RB* and *RC*. Unlike the design with a submerged plume where the required water depth is either constant or increases with *RB* (for pre-dilution with SW and CW), the required water depth for the surfacing plume design reduces with *RB* as the desired physical dilution reduces. For pre-dilution with TWE, the required water depth follows the same trend as the desired physical dilution. Thus, it reduces with *RB* for *RB* <2 and increases with *RB* for *RB* > 2. When brine is concentrated, the design with a submerged plume is the optimum design for *RC* >1*:*4 because the smaller flow rate and higher density difference (as compared to brine which is not concentrated) are less likely to lead to shallow conditions. Thus, even when there are no restrictions on plume visibility, the design of a single jet to discharge concentrated brine results in unrealistically high values of *u*0. For all cases (except pre-concentration with *RC* > 1*:*4), the design with a surfacing plume has a higher discharge velocity and lower water depth than the corresponding values for the design with submerged

*Desalination Brine Management: Effect on Outfall Design DOI: http://dx.doi.org/10.5772/intechopen.99180*

**Figure 4.**

*Variation of H, D*<sup>0</sup> *and u*<sup>0</sup> *with RB and RC for discharge using a single jet with surfacing plume for Qb* ¼ <sup>1</sup> *<sup>m</sup>*<sup>3</sup>*=s,* <sup>Γ</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>01</sup> *and a desired excess salinity of* <sup>2</sup> *ppt. (variables are scaled differently for the pre-dilution and pre-concentration cases as indicated in the legend).*

plume. The higher velocity helps in generating the same amount of mixing as the submerged plume case but in smaller water depth.

## **6.3 Discharge through a unidirectional diffuser**

The design optimization for a unidirectional diffuser also results in non-linear equations which are solved using the 'fsolve' function in MATLAB. Optimum design variables are calculated which achieve desired dilution and minimize total cost. However, in some cases the optimized design variables need to be adjusted. For example, to ensure uniform flow through all the ports, the aggregate crosssectional area of the nozzles should be less than two-thirds of the cross-sectional area of the diffuser manifold [31]. Since the manifold diameter is assumed to be related to the discharge flow rate (Eq. (6)), this requires the discharge velocity to be at least equal to 2*Q*<sup>0</sup> <sup>0</sup>*:*28. Thus, if the optimum value of *u*<sup>0</sup> is less than 2*Q*<sup>0</sup> <sup>0</sup>*:*28, *u*<sup>0</sup> is fixed to be equal to 2*Q*<sup>0</sup> <sup>0</sup>*:*<sup>28</sup> and other design variables are re-evaluated to minimize total cost.

For certain cases, the design with a single port is the one which minimizes cost, i.e., any design with multiple ports will have higher total cost than the design with one port. This is observed for cases which require a submerged plume and for which the desired physical dilution is small. The optimum discharge velocity (not adjusted for uniform flow) for such cases is small and adjustment for uniform flow results in a design with total cost higher than the cost of the single jet design. For these cases, the single port design is accepted as the optimum design.

Once the optimum design variables are calculated (which satisfy all constraints), *N* is rounded to the nearest integer and *D*<sup>0</sup> is adjusted such that *Q*<sup>0</sup> ¼ ð Þ *π=*4 *Nu*0*D*<sup>0</sup> 2 .

#### *6.3.1 Discharge through a unidirectional diffuser creating a submerged plume*

**Figure 5** shows the variation of *Hd*, ð Þ *D*<sup>0</sup> *<sup>d</sup>*, ð Þ *u*<sup>0</sup> *<sup>d</sup>* and *Nd* as functions of *RB* and *RC*. The design with a single jet is the optimum design for *RB* >2*:*1 for blending with SW and CW, and for *RB* between 1.4 and 3.8 for blending with TWE. The discharge velocity is fixed to be equal to 2*Q*<sup>0</sup> <sup>0</sup>*:*<sup>28</sup> (to ensure uniform flow through nozzles) for

#### **Figure 5.**

*Variation of H, N, D*<sup>0</sup> *and u*<sup>0</sup> *with RB and RC for discharge using a unidirectional diffuser with submerged plume for Qb* <sup>¼</sup> <sup>1</sup> *<sup>m</sup>*<sup>3</sup>*=s,* <sup>Γ</sup> <sup>¼</sup> <sup>0</sup>*:*<sup>01</sup> *and a desired excess salinity of* <sup>2</sup> *ppt. (variables are scaled differently for the pre-dilution and pre-concentration cases as indicated in the legend).*

### *Desalination Brine Management: Effect on Outfall Design DOI: http://dx.doi.org/10.5772/intechopen.99180*

*RB* between 1.3 and 2.1 (for pre-dilution with SW and CW), and for *RB* between 1.2 and 1.4 and greater than 3.8 (for pre-dilution with TWE).
