^{*}

Multiport diffusers are the effective engineering devices installed at the marine outfall systems for the steady discharge of effluent streams from the modern coastal plants, such as municipal sewage treatment, power generation and seawater desalination. A far field mathematical model using a two-dimensional advection-diffusion equation is presented for continuous discharges of effluent streams from multiple outfalls on a uniformly sloping beach with a current parallel to the shoreline. The analytical solutions are illustrated graphically to replicate and capture the merging process of effluent plumes in shallow coastal waters, and then asymptotic approximation will be made to the maximum shoreline’s concentration to formulate effluent discharge plume dilution from a multiport diffuser.

Along the highly populated coasts of the Arabian Gulf, Gulf of Oman and Red Sea, many large scale municipal sewage treatment and (co-location) power generation and seawater desalination plants are often found to be clustered together [1,2]. Desalination plants generate two products, pure water and brine—a reject concentrate stream. The unwanted brine product is primarily seawater but at a more concentrated level, with a concentration factor of as high as 2.5 more than the typical seawater salinity. Most coastal plants continuously discharge brine streams back into the sea through a submerged outfall, and as a brine stream enters the receiving marine waters, it creates a high salinity plume. Without proper dilution, the brine plume will tend to sink and propagate down the slope for hundreds of meters, harming the ecosystem along the way, and most at risk are the benthic marine organisms living at the sea bottom [2,3]. An engineering solution utilizing the best available technology is required where a multiport diffuser would be installed at the pipe-end to rapidly dilute the concentrate [4-6]. A multiport diffuser is a linear structure consisting of many closely spaced ports designed to discharge a series of effluent streams into the receiving coastal water.

^{3}/h to discharge the cooling water from the power generation plants and mix it with brine reject stream (and other effluents) from seawater desalination plants. The old (currently in use by the existing Barka I and II plants) outfall pipe length is about 650 m, while the new (not yet been used) outfall pipe length is about 1200 m, and the distance between the two discharge points is 1000 m. The old outfall system comprises of four parallel pipes angled at 62 degrees to the coastline, each with a diameter of 2.5 m, buried at 5 m below the seabed (not visible on the surface) and spaced equally at 4.8 m apart. Each pipe has a 62.4 m long multiport diffuser, consisting of nine ports equally spaced at 7.5 m apart, installed at the end of each outfall pipe. The multiport diffusers are arranged in two nested V shapes as illustrated in ^{ }degrees on either side of the outfall pipeline. The two internal pipes of length 653 m have its end at a depth of 9 m below the mean sea level, while the other two shorter external pipes of length 582 m end at a depth of 8.4 m. The ports of each diffuser are oriented in an alternating way each with an angle of 20 degrees to the diffuser pipe. The port diameter is 0.7 m and located at 1 m above the seabed, and the ports are oriented upwards with an angle of 10 degrees against the horizontal.

Owing to the highly variable nature of the sea, we do not yet have a full understanding of the mixing processes of effluent discharge plumes, and the use of mathematic-

cal models has been a key strategy for assessing the potential marine environmental impacts [2,3,6-9]. A clear understanding of these processes is needed so that predictive models can be developed which form the basis of sound engineering design [

Immediately after steady release from the multiport diffusers, vigorous and rapid mixing of the effluent stream is governed by the effluent buoyancy, momentum of the discharge and its interaction with the sea currents [3-5]. At the end of this mixing zone stage, adjacent effluent discharge plumes interact with each other and merge to form a rising curtain, which then continues to drift away with the longshore currents [6-9]. Because of relatively shallow water depth, it is observed that the elongated effluent plumes are spreading towards the shoreline and may cause concentration build-up in the coastal waters [7-9].

As we are only concerned with the effect of seabed depth profile, for simplicity the other complexities such as tidal motions, density and temperature are ignored. The shoreline is assumed to be straight and the sea wide, and we assume that the outfall’s effluent plume is vertically well-mixed over the water depth. The coastal (drift) current is assumed to be steady with a speed U and remains in the x-direction parallel to the beach at all times. The dispersion mechanisms are represented by eddy diffusivities, and diffusion in the x-direction is neglected, as the effluent plumes in steady currents become very elongated in the x-direction. The variations in the y-direction of drift current U and coefficient of dispersivity D are assumed as the power functions only of water depth h, and for application, we take U to be proportional to and D to. These scalings are appropriate for a turbulent shallow-water flow over a smooth bed [9-11].

We also consider the effluent stream to be steadily discharged at a rate from the (original) single outfall at the position, where is an arbitrary reference water depth; at a different rate from the first (new) outfall at the position (); at a rate from the second (new) outfall at; and so on, where is the outfall’s (offshore) and (along the shore) separation distances. For a single outfall, the total effluent load is a function of. As illustrated in

In a uniformly sloping beach, the water depth varies

increasingly linear as, where the beach slope and the beach is at; following [7-9] and by applying a linear superposition, the two-dimensional far field advection-diffusion equation for effluent discharge plume concentration from the multiple outfalls is given by

with the boundary condition at the beach, and is assumed to be ultimately dissolved into the ocean. is the Dirac delta function.

In order to solve Equation (1), the delta function representation of the point source term must be removed as it does not facilitate the solution. However, by doing so, the information about the source strength is also lost. For each long sea outfall at the position, discharging effluent stream continuously at a rate,

is solved separately in the two regions and, and the solutions are then connected by the matching condition

for all.

Since no concentration is lost or produced anywhere, and the longitudinal dispersion has been neglected, the solution must also satisfy

for all;

that is, the flux of concentration by advection across any plane perpendicular to the flow direction must be equal to the rate at which concentration is being released from the point source [

In terms of dimensionless quantities

and by setting

using the Laplace transform

Equation (2) is transformed into a second-order ordinary differential equation

which can be reduced to the modified Bessel’s equation

by writing

where. The general solution in the two regions is given by

and

where and are modified Bessel functions [

Next, to obtain the particular solution, the functions and can be determined from the matching conditions

where. From the table of integrals [

and then using the property of the Bessel’s function

it is found that

Finally, using the inversion of the Laplace transform tabulated in [

After summing for all concentration from the multiple outfalls, the analytical solution of Equation (1) is given by

As the water depth is gradually decreasing towards the beach, the effluent plumes are elongated and turning towards the beach, and the gentler the beach slope, the higher the buildup in concentration in the shallow water close to the beach [7,8]. This is expected since deeper water is a more efficient transport mechanism. The model parameter represents the effluent plume elongation in the -direction; the larger the values of, the more elongated the effluent plumes. To investigate the uncertainty in,

The other parameters related to the multiple outfalls are the (original) single outfall (offshore) distance, outfall’s (offshore) and (along the shore) separation distances. Note that, for a multiport line diffuser with n ports, both values of and are smaller than, and.

For a large volume effluent discharge, the engineering practice is to distribute the effluent stream over a large expanse by installing a multiport diffuser at the end of a marine outfall to substantially improve the mixing and

dilution of effluent plumes in the coastal waters [3-6]. By plotting the results of numerical integrations of Equation (3), the merging processes of effluent discharge plumes from a multiport diffuser with 5 ports are reproduced graphically in

Again following [7-9], the appropriate measure for assessing the impact of effluent discharges from coastal plants would be the shoreline’s concentration values. In the limit as and replacing in Equation (3) by its asymptotic form [

It is easy to see that for a single outfall when, Equation (4) then reduces to

.

By differentiating, this concentration at the beach has a maximum value of

which occurs at downstream of the outfall.

The compounded concentration at the beach for effluxent discharge from a multiport diffuser with five ports is plotted in

falls only changes the value of maximum concentration, but not its position. Note that the position of maximum concentration is proportional to the model parameter [

For the quantitative illustration, we consider a perpendicular line diffuser design, where the line diffuser with n ports is placed in the (offshore) y-direction perpendicular to the current direction, and it consists of a series of ports equally spaced by the offshore separation distance. The maximum compounded concentration at the beach can be approximated by substituting [

Finally, after summing for n ports, the maximum concentration Equation (5) reduces to

where is the (total) length of the line diffuser. As the number of ports increases and the single outfall distance gets longer, the maximum shoreline’s concentration Equation (6) gets smaller than that of the single outfall value.