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A marine outfall is a long pipeline that continuously discharges large amounts of effluent streams into the sea. As the number of marine outfalls along the coastal areas is growing, a far field mathematical model with two point sources on a sloping beach is used to assess the coastal water quality following discharges from two outfalls. Asymptotic approximation will be made to the concentration at the beach to measure how well the effluent plumes are mixed and diluted in the coastal waters. The result found agrees with the engineering practice of installing a two-port diffuser at the end of a single outfall to minimize its potential environment impacts.

Most coastal industrial installations and plants, such as municipal sewage treatment plants [

When many marine outfalls are discharging effluents to shallow coastal waters, the adverse long-term impacts are strongly inter-dependent, and compounded from neighbouring outfalls. The interactions of two or more effluent plumes are expected as many outfalls often tend to be closely clustered together along the coastal areas. Newly-constructed coastal plants may need to build two outfalls as a contingency plan for a future increase in the plant’s production capacity [

As coastal industrial plants are built predominantly on the sloping sandy beaches, a mathematical model using a two-dimensional advection diffusion equation with two point sources is presented. The solution is plotted to graphically study the merging of two effluent plumes from two outfalls. While the far field modelling in this paper involves drastic simplifications, key physical mixing and dispersion processes are represented, and thus the analytical solution remains useful in providing a qualitative understanding and in suggesting general behaviour of the marine outfall effluent discharge plumes in coastal environment [

The beach is considered to be straight and the sea wide, and the outfall’s effluent plume is assumed to be vertically well-mixed over the water depth. The coastal (drift) current is assumed to be steady with a speed

As illustrated in

As the water depth is gradually decreasing towards the beach at

with boundary condition

where

the analytical solution of Equation (1) is given by

where

As a higher build-up in concentration is more likely found at the shallow water close to the beach [

It is easy to see for effluent discharges from a single point source at

The concentration at the beach for a point source length

Apart from the effluent discharge rates, the compounded impact of the new outfall is governed by the outfall’s separation distances

We first consider the case where the two outfalls are operated independently by two coastal plants, i.e., when

Since the value

Next, the compounded concentration at the beach for two point sources as given by Equation (3) can be simplified to

Substituting

where

By differentiating,

Note that for a given value of

As shown in

value of the concentration at the beach is

equal rates, i.e.

Modifying the rates of discharge can be achieved through a relatively less expensive method than extending the outfall lengths. For the case where two outfalls is operated by one plant, i.e. when

That is, the maximum value of

To include the case of the modern engineering practice that installs two ports at the end of a marine outfall [

For plotting the contours of the effluent plume, since the point sources are close to each other, and in the limit as

The contours of the solution for a single outfall with two ports are reproduced graphically in

Finally, the maximum value of compounded concentration at the beach for two closely located point sources can be approximated to

Again, the maximum value of

The solutions for an advection diffusion equation with two point sources are applied to study the interaction and merging of effluent discharge plumes from two outfalls on a sloping beach. As a measure for assessing the impact in the coastal environment, the maximum compounded concentration at the beach is formulated. If the two outfalls are independently operated, then the maximum value of the concentration at the beach can be minimized as long as the new outfall length is more than double the old outfall length, and discharging at a rate smaller than the old outfall.

If two outfalls are operated by one plant, then the integrated total effluent load can be shared between them, and it is found that the maximum value of the concentration at the beach is smaller than that of the single outfall. A similar result is also obtained for a single outfall where a two-port diffuser is installed at the end outfall pipe. However, implementation issues related to the control of discharge rates, reliability and cost effectiveness of the marine outfall are not addressed.

The mathematical formulation presented can be extended to the case of multiple outfalls discharging on a coastal area, and in particular, the result for effluent discharges from multiport diffusers has been reported in [

The author is grateful to Sultan Qaboos University for an Internal Grant IG/SCI/DOMS/14/01 which provided financial support for this work.