Far Field Model Simulation of the Sea Outfall Plume

The 9 th Asian Symposium on Visualization Hong Kong, 4-9 June, 7 Far Field Model Simulation of the Sea Outfall Plume Anton Purnama 1, H.H. Al-Barwani 1. Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, PO Box 36, Al-Khod PC 13, Muscat, Oman, e-mail: antonp@squ.edu.om. Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, PO Box 36, Al-Khod PC 13, Muscat, Oman, e-mail: hamdi@squ.edu.om Abstract Corresponding author A. Purnama An analytical solution of the far field plume concentration associated with the long sea outfall steady discharge is presented. Due to the oscillatory current, highly undesirable concentration peaks are created in the plumes drifting along and spreading towards the coast. A criterion based on the maximum concentration values at the beach as a measure is proposed for assessing the long term environmental impacts of sea outfall discharge. Keyword: advection-diffusion equation, oscillating flow, sea outfall, steady discharge 1. Introduction Long sea outfall discharge is the practical way to dispose of municipal or industrial effluent wastes as it gives the maximum possible separation of the discharge from people [1]. Such steady discharges are operated from plants or factories situated in coastal areas; e.g., treated effluents from sewage plants [1,], toxic contaminants from industrial installations [1], cooling water discharges from power stations [3,4], and brine wastewater from seawater desalination plants [5,6]. The main objective of sea outfalls is to continuously discharge effluent wastes so that the detrimental impacts to the receiving coastal waters are minimal. However, the elongated effluent plume that results from the sea outfall's steady discharge is observed to be spreading towards the beach []. The coastal areas are developing areas of industry and population, and sandy beaches are popular holiday resorts used by tourists for recreation and swimming. Therefore, reliable predictions of the long term mixing and dispersal of the discharges plume are crucial for effective sea outfall design and operation that ensures a minimal impact on the coastal and marine environments and controls the public health risks from hazards that may be encountered in recreational use of coastal and marine environments. Because of the complexity of mixing processes and highly variable nature of the sea, we do not yet have a full understanding or description of the outfall plume [7,8,9]. In practice, the effluent wastes discharge is made through a long outfall terminating in a diffuser to promote rapid initial dilution. Such a mixing is governed by the effluents buoyancy, momentum of the discharge, and its interaction with the seawaters. At the end of this initial mixing stage, the established far field effluents plume then continues to drift away and spreads with the sea currents [,8]. As the flow in the sea is dominated by the longshore tidal current oscillating with a period of half a lunar day, it is the oscillatory nature of the current which produces physically and unpredictably interesting features [1,11]. As illustrated in Fig. 1, the plume is transported back and forth along the coastline past the outfall, and in the absence of current at the time of a flow reversal, effluent waste is discharged into stationary water close to the outfall. While the far field modelling in this paper involves drastic simplifications, key physical mixing and dispersion processes are represented, and thus the model remains useful in providing qualitative understanding and in suggesting general behaviour of the sea outfall steady discharge plume. Model solutions are derived analytically, but as very little information is available on the model parameters, outfall plume simulations are carried out for some values of the parameter [6]. The long term environmental impact assessments of long sea outfall steady discharge have become increasingly important, both as a result of public concern and scientific awareness and because of the increasing ASV-1 ASV-1

A. Purnama, H.H. Al-Barwani scale of industrial activity. As it has also become a key issue to obtain permits to build a new outfall, often considerably influencing plant commissioning and design [1,13], regulatory requirements and strategies should be properly defined based on the protection of a sustainable marine environment. Fig. 1. Oscillatory motion of the outfall discharge plume over 5 tidal cycles. Outfall steady discharge plume model For simplicity, we assume the shoreline to be straight and that waves at sea generate the net parallelcomponent (longshore) current. This coastal current is assumed to be uniform over water depth and remain in the x-direction parallel to the beach. The effluent waste stream is discharged with a steady rate Q from a long sea outfall located at a distance α > from the beach. As the discharge is made via diffusers and utilizing the best available technology, we also assume that the effluent plume is vertically well-mixed over the water depth. The effect of tidally oscillating flows on the mixing and dispersal of the long sea outfall plume initially discharged at time t i is simulated using a two-dimensional advection-diffusion equation [6]: c + u t c x () t D D = Q δ ( t t ) δ ( x) [ δ ( y α ) + δ ( y + α )] x c y c x y i (1) where c is the plume concentration, u () t the tidally oscillating flow, D x the longitudinal diffusivity, D y the lateral diffusivity and δ the Dirac delta function. In equation (1), the outfall is represented as a point source at ( x =, y = α ), and in order to satisfy the boundary condition at the beach, an imaginary source is added at ( x =, y = α ). A simple model of an oscillating flow consists of a steady (residual) drift and a periodic component [3,4,6,11]: u( t) = v + U sin ω t. Using numerical 4 values of the mean tidal amplitude U =. 4 m/s and the period π ω = 4.5 1 s [9], we define a length scale U ω of order.8 km. The oscillatory displacement of effluent plumes is given by x t () t = dτ u( τ ), and on integration, in its dimensionless form, X ( T ) = VTi T + cos( T T i ) t i cos, where X = xω U, V = v U, T = ω t and Ti = T ω t i. Although the discharged effluent plume is moved back and forth by the oscillatory current over a tidal cycle, as shown by the dashed line in Fig. 1, the net transport depends on the ratio of the drift current to tidal amplitude V. In terms of the dimensionless variables, the solution of equation (1) is given by [6] 9 th Asian Symposium on Visualization, Hong Kong SAR, China, 7. ASV-

Far field model simulation of the sea outfall plume T i ( Y Λ) λη( Y + Λ) dτ λ λη C = exp { X Vτ + cost cos( T τ )} exp + exp τ τ τ τ () where C = 4π c DxDy Q, λ = U 4ωDx, Y = yω U, Λ = αω U and η = D x D y. Note that beside the outfall length Λ, there are 3 model parameters which characterize the outfall steady discharge plume: V the ratio of the drift current to tidal amplitude, λ the distances by which the plume is transported and spread over by advection to that by longitudinal diffusion [1,11], and η the ratio of longitudinal to lateral diffusivities. Due to the unpredictable sea conditions, very little information is available on these parameters [7,8,9]. Because longitudinal diffusion is a very efficient process, the values of η could be at least. The higher its value, the more elongated the shape of the plume. No sea outfall discharge will assure the expected efficiency if the parameters are not known with sufficient precision, or defined based on proven experience. For illustrations, numerical integration of equation () is presented graphically by plotting contours of concentration using the parameter values λ = 15 and η = 5. Fig. shows a characteristic effluent plume drifting along and spreading towards the beach following a steady discharge for over 1 month ( T i =11.5π ) from a long sea outfall located at Λ =. 5. On comparison, larger drift currents efficiently spread the plume over large distances downstream the outfall (Fig. b). Note that the actual plumes are elongated in the x-direction by a factor of 5, and the plume also spreads towards the upstream side of the outfall location. Due to flow oscillations, the concentration peaks are formed on both sides of the outfall [6,11]. Fig.. Simulated outfall steady discharge plume at T i =11.5π when (a) V=.1 and (b) V=.15 3. Assessing the impact of long sea outfall discharge Long term potential environmental impacts of long sea outfall discharge can be addressed and regulated by restricting the effluent waste concentration at the discharge point (emission limit values), which can be achieved by prescribing specific design and using some treatment or recycling technologies, and by imposing the maximum concentration limit within the receiving coastal waters [1,13]. The second regulatory control mechanism requirement (environmental quality standards) puts a direct responsibility on the outfall's operator. As the sensitive areas for the evaluation and assessment of the impact of sea outfall discharge would be at the beach, we therefore use, as an appropriate measure, the concentration values at the beach. On substituting Y = into equation (), we obtain the concentration at the beach 9 th Asian Symposium on Visualization, Hong Kong SAR, China, 7. ASV-3

A. Purnama, H.H. Al-Barwani T i dτ λ ληλ C = exp { X Vτ + cost cos( T τ )} exp τ τ τ (3) If we are only interested in the long time concentration, i.e. in the limit as ( T τ ) T i, then the term cos may be neglected from equation (3) as it has little contribution to the integral [1]. The flow periodicity enables us to restrict our observation time T in a single representative tidal cycle. dx A From the integral formula Bx = K ( AB ), where K is a modified Bessel exp x x function of the second kind [14], the resulting closed form of equation (3) simplifies to ( { }) λv X + cost K λv { X + T} + η C = 4 exp cos Λ (4) Next, using the asymptotic representation K ( x) x exp( x) π, we can approximate further C λv 4π exp λvηλ ( ) X + cost X + cost (5) The concentration at the beach as given by equation (5) is plotted in Fig. 3 at T = π and T = π of a tidal cycle for the parameter values Λ =. 5, λ = 15 and η = 5. For comparison, the concentration at T i is also shown in Fig. 3. The effect of oscillatory flow is clearly showed up to the time when the maximum concentration is reached. As the maximum concentration value remains constant throughout the tidal cycle, it can thus be used as the numerical upper limit for a regulatory measure in the long time impact of long sea outfall discharge [1,13]. the beach as given by equation (3) following steady discharges for over 1 month ( =11.5π ) Fig. 3. Outfall plume concentration along the beach By differentiating equation (5), the maximum concentration at the beach is given by ( 1 λvλ) π eη C =, which occurs at X = λ V ηλ cos T downstream of the outfall. The max max maximum concentration is inversely proportional to the outfall distance Λ. This result agrees with the standard practice of building a longer sea outfall in order to minimize its potential environmental 9 th Asian Symposium on Visualization, Hong Kong SAR, China, 7. ASV-4

Far field model simulation of the sea outfall plume max ± impact [1,1]. It is important to note that over one tidal cycle X = λvηλ 1, and therefore X oscillates over a large distance of U ω along the beach. From a regulatory viewpoint, max X max could also be used as a minimum standard distance to permit the construction of a new outfall from the existing long sea outfall. Fig. 4 shows the values of C max as a function of the outfall length Λ and the drift current V for the parameter values λ = 15 and η = 5. It is evident that the efficient far field mixing of a long sea outfall discharge plume is due to the larger drift current V, and in particular if Λ =. 5, increasing the value V =. 1 to V =. 15 reduces the value of maximum concentration at the beach by a third (Fig. 3). Fig. 4. Maximum concentration at the beach 4. Conclusions As the long term detrimental impacts of a long sea outfall's steady discharge to the receiving coastal waters are believed to be minimal, the number of outfall installations worldwide is rapidly increasing. The practical way to achieve this environmental objective is to build a longer sea outfall, and implement the appropriate technologies to limit the effluent concentration at the discharge point. This controlling discharges strategy however does not consider the long term quality response of the coastal waters, and in particular, several outfalls could accumulatively cause excessive pollutant loadings. Additional measures for assessing the impact of long sea outfall discharges would be the establishment of environmental quality standards such as the maximum numerical values of concentration at the beach that may not be exceeded. Using a far field two-dimensional model based on an advection-diffusion equation for an outfall steady discharges plume concentration in oscillating flows, the long term concentration at the beach is formulated analytically. The result shows that longer sea outfalls minimize its potential environmental impact. Acknowledgement The author (AP) would like to thank Sultan Qaboos University for financial support to attend the symposium through the Conference Central Budget. References [1] Institution of Civil Engineers, Long Sea Outfalls, 1st edition, Thomas Telford Ltd, (1). [] Signell R.P., Jenter H.L. and Blumberg A.F., Predicting the physical effects of relocating Boston's sewage outfall, Journal of Estuarine, Coastal and Shelf Sciences, Vol. 5, (), pp 59-7. 9 th Asian Symposium on Visualization, Hong Kong SAR, China, 7. ASV-5

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