JKAU: Mar. Sci., Vol. 18, pp: 149-167 (2007 A.D. /1428 A.H.) Application of a Mathematical Model for Estimating the Pattern of Pollution Dispersion at the Coast of Alexandria Ebtessam E.E. Mohamed, El-Sayed M. El-sayed*, Nabil N. Saad, and Hassan A. Abou Tahoun Marine Physics Lab., National Institute of Oceanography and Fisheries, Alexandria *Department of physics, Ain Shams University, Cairo, Egypt Abstract. During December 1996, 27 sampling stations were explored along the coastal zone in front of Alexandria situated between El- Agamy beach in the West and Abu-Qir headland in the East. Also, current measurements have been obtained by using current meter and Drougs. The dispersion coefficient Kx and Ky of water during the time of measurements in front of Alexandria were calculated. A predictive model based on the equations described by Bonazountas (1987) was applied to estimate initial dilution and wastewater field in front of Alexandria coastal zone. Introduction The state of pollutants in inshore water of the Mediterranean has reached critical levels. This is mainly due to high quantities of domestic sewage discharged either partially treated or even untreated into the sea through rivers, outlets and pipelines. Alexandria is one of the largest coastal cities on the Mediterranean; the waste disposal of the city is mainly discharged into the Mediterranean Sea through many short outlets. Some of these disposal sites, e.g., the pumping station at Abu Qir and El Max bay, discharge municipal wastewater mixed with drainage water. El Max bay is mainly polluted by El Umum drainage water, which 6 pours about 6x10 (m3/day) of untreated industrial, agricultural and domestic’s effluents. On the other hand, a major outfall at Kayet Bay and minor at different coastal sites extending from the Eastern Harbour to 149 150 Ebtessam E.E. Mohamed et al., El Maamora dispose sewage wastes into the beaches. The bulk volume of domestic wastewater, which exceeds 400,000 m3/day, is discharged into the sea through the principal pipe at Kayet Bay outfall. Much intensive attention was paid to study the environmental effect on El-max and Eastern Harbour. The major hydrographic, current measurements, the physico-chemical characteristics and sediment of the area under investigation had been studied by; El-Wakeel and El-Sayed (1978); Abdallah (1979); Farag (1982); Rifaat (1982); Abul-Dahab (1986), Halim (1988, 2004); El-Nagar (1994); Said et al (1994); Farag. et al (2000). This study is carried out to evaluate the effect of sewage and drainage wastewater on the coast of Alexandria particularly in front of Kayet Bay outfall and El Umum drain. Also, to illustrate the final patterns resulting from mixing between the ambient seawater and the discharge pollutant. Data and Methods of Analysis During Dec. 1996, the coastal zone in front of Alexandria between El Agami and Abu Qir headline was surveyed. Through this period 27 sampling stations were explored along five sections almost normal to the coast. Also, some stations were taken scattered around Kayet Bay outfall and inside the eastern harbour. Current measurements have been obtained by two different ways: 1. Current meter model Anderaa RCM-7 had been used to gather current data at constant depth 3 m under surface at station 1-22 after anchoring the boat in every station. The current meter was adjusted to average and record one reading every five minutes. 2. Detected the paths of wind-induced currents generated in the upper surface layer by using two drogues submerged to about 40 cm under sea surface. A motorboat followed up the movement of the two drogues and their locations were fixed approximately every thirty minutes. The locations of sampling stations taken during December 1996 are shown in Fig. 1. Application of a Mathematical Model for … 151 Kayet Bay Latitude El Max Bay Longitude Fig. 1. The locations of sampling stations taken during December 1996. Current Regime and its Dispersing Ability It is clear from the current regime in the area of investigation (Fig. 2) the predominant direction of flow vectors was to the southwest, except El Max sector, with their speed fluctuating between 1.007 and 7.5 cm/sec. The current was coastward in front of the area between Kayet Bay and Ras El teen and inside the Eastern Harbour. Longitude Fig. 2. Current regime in the area of study during December 1996, (current meter data). 152 Ebtessam E.E. Mohamed et al., In El Max sector there was no predominant direction. The near shore current was flowing seawards, while the offshore one moved to the west and it was almost parallel to the coastline. Generally, the tide is ignored in this study because its components was very small and contribute nothing in the current intensity. The tracks of the drogues are illustrated in Fig. 3. It is clear that the predominant direction of the surface currents was the southwest. The current value calculated from drogues experiments are used in the calculating of diffusion coefficient (Kx and Ky) using the method described by List et al (1990) and can be summarized as follow: Consider the location coordinates (x,y) of a drogue J at a time I are defined by (Xij, Yij).From these coordinates it is possible to develop estimates of the Lagrangian velocity of the drogue and the position of centroid of the drogue set may be computed. The centroid of set N drogues at a time I is given by: B B Longitude Fig. 3. Tracks of Drogues experiment in front of Alexandria coast during December 1996. Application of a Mathematical Model for … 153 ∑ X ij (1) j X i = N And similar for Y coordinate ∑Yij (2) j Y i = N The mean velocity of the drogue set is simply the time derivative of the position coordinates. The variance in the drogue position defined by: 2 (3) ∑()XXij− i j σ 2 = xi N −1 2 (4) ∑()YYij− i j σ 2 = yi N −1 Following Okubo (1974), the dispersion of the drogue distribution can be estimated as: 2 2 2 σi =( σxi + σ yi )/ 2 (5) And the relative dispersion coefficient K is given by: 1 ∂σ2 1 Δ σ 2 (6) K() ti = i ≅ i 2 ∂t2 Δ t The spatially dependent relative dispersion coefficient is given by: 1 ∂σ2 1 Δ σ 2 (7) K() ti = xi ≅ xi x 2 ∂t2 Δ t 2 2 1 ∂σyi 1 Δ σ yi (8) K() ti = ≅ y 2 ∂t2 Δ t 154 Ebtessam E.E. Mohamed et al., Then KKK=( x + y )/ 2 (9) The dispersion coefficients of the coastal water in front of Alexandria were found to be: Kx =45049 cm2/sec Ky = 870609cm2/sec The obtained value of Kx and Ky means that the dispersion parallel to the coastline is much higher than that normal to the coast. Theoretical Approach for the Model There is a great diversity in the flow patterns that may develop as the discharged wastes stream mixes with the ambient water. These flow patterns will determine the configuration, size and intensity of the mixing process as well as any impact of the discharged pollutants on water body surface, bottom and shoreline of the critical area. In the early decades of th the 20 century, many scientists developed theories and equations, which deal with the physical and hydrodynamic features of mixing process between discharge effluents and their receiving ambient water bodies. The behavior of an outfall discharge wastewater at a certain depth beneath sea surface can be divided onto three successive steps: 1. Near the orifice of the outfall, the discharges are influenced by jet momentum. 2. Near the sea surface, the discharges are governed by buoyancy. 3. The dispersion and disposition of discharges by current action from their final level away to the surrounding areas. The construction of the used constructive model was based on the equations described by Bonazountas (1987) to estimate initial dilution and waste water field. Bonazountas (1987) calculated the thickness of the wastewater field by assuming that, the average dilution at elevation Y is proportional to Y, this is nearly true in all buoyant line plume and jet cases for Y sufficient Application of a Mathematical Model for … 155 large, thus assuming Saw is the same as the average dilution in the plume and Y=Yb then Saw/Sa = Yb/Ymax , where: Saw : the average dilution in the waste field Sa : the calculated value at the top of the plume. Yb : the Y coordinates at the bottom of the waste field Ymax: the Y coordinates at the top of the waste field. Qo : the source discharge (volume flux). U : the current speed. b : the width of the waste field (normal to the current). g : gravitational acceleration. q : outfall discharge. Δρ :residual density difference at top of plume. ρ :density of seawater. The thickness of the waste field is H =Ymax – Yb =Ymax (1-Yb /Ymax) or H = Ymax((QoSa / ubYmax)/(1 +QoSa /ubYmax)) To apply this equation, it is necessary to determine u, b and Sa, also, it should be noted that: If the product (ub) is very small, then the analysis fails because the assumption Saw /Sa =Yb/Ymax break down for a current which is perpendicular to the diffuser, (b) may simply taken as the length of the diffuser. For a current parallel to the diffuser, the width (b) must be estimated differently. For uniform ambient density, Bonazountas (1987) demonstrated that: The actual average dilution including the effect of blocking is given by: 1/3 Saw = 0.38 (gq) Ymax / q G = g (Δρ / ρ) And the average dilution Sa in the absence of blocking is given by: 1/3 Sa =0.54 (Gq) Ymax/q 156 Ebtessam E.E.