MODEL DEVELOPMENT FOR STREAM QUALITY MANAGEMENT

This study is aimed at finding the level of wastewater treatment at two sites that would achieve the desired concentrations at a minimum total cost for a river that received wastewater effluent from some other two point sources located at two sites. Pollutant concentrations and stream flow in the stream at selected reaches were obtained. Linear programming model was developed incorporating waste transfer coefficients. The model was applied at Nworie River in Imo State, Nigeria. The model solution was obtained using graphical method and the results revealed that 80% treatment efficiencies met the stream standards for the design stream flow and waste load condition at a total minimum cost. The study shows that least-cost waste removal efficiencies could be determined without prior knowledge of the cost functions.


INTRODUCTION
Linear programming deals with the problem of allocating limited resources among competing activities in an optimal manner. Linear programming uses a mathematical model to describe the problem of concern. Linear programming has previously been used to study water quality problems. Lynn et al. (I962) demonstrated the use of linear programming to determine the optimal design for a sewage treatment plant. Loucks and Lynn (I967) have used linear programming models to determine the least-cost plan for waste treatment in a river basin. The decision variables were the degrees of BOD removal to be provided by each discharger for individual waste effluents. The constraints were that each discharger must provide partial or complete secondary treatment and that the dissolved oxygen concentration at any point in the stream must not go below a specified minimum value. A linear programming model has been developed and applied to determine the minimum treatment cost to maintain at least a minimum dissolved oxygen concentration at all points in the Delaware estuary . Hence, this study is focused on finding the level of wastewater treatment that would achieve the desired concentrations at a minimum total cost for a river that received wastewater effluent from two point sources located at two sites.

STUDY AREA
Nworie River is a first-order stream that runs about 5km course across Owerri metropolis in Imo State, Nigeria before emptying into another river, the Otamiri River ( Figure 1). Its watershed is subject to intensive human and industrial activities resulting in the discharge of a wide range of pollutants. The river is used for various domestic applications by inhabitants of Owerri. When the public water supply fails, the river further serves as a source of direct drinking water, especially for the poorer segment of the city.

Fig 1: Location of Nworie River in Imo River Basin
Nworie River receives wastewater effluent from two point sources located at Site 1 and 2 ( Figure 2). Sites 2 and 3 constitute source of water supply for some community. Thus, without some wastewater treatment at these sites, the concentration of pollutant, at the sites 2 and 3, would continue to exceed the maximum desired concentration specified by the state sanitation authority. The problem is to find the level of wastewater treatment at sites 1 and 2 that would achieve the desired concentrations at sites 2 and 3 at a minimum total cost. Location of Nworie River in Imo River Basin Nworie River receives wastewater effluent from two point sources located at Site 1 and 2 ( Figure 2). Sites 2 and 3 constitute source of water supply for some wastewater treatment at these sites, the concentration of pollutant, at the sites would continue to exceed the maximum desired concentration specified by the state sanitation authority. The problem is to find the level of wastewater treatment that would achieve the desired at a minimum total cost.
The solution to this problem can be obtained through linear programming.

DATA COLLECTION
At the two sites of point pollution, discharge and velocity measurements were at the river made using current meter. After collecting water samples in plastic bottles per site, the pollutant concentrations at two sites were determined in the laboratory as demand (BOD), using standard procedures (APHA, 1998).

Fig 2: Pollution Sites along Nworie River
The solution to this problem can be obtained through At the two sites of point pollution, discharge and velocity measurements were at the river made using current meter. After collecting water samples in plastic bottles per site, the pollutant concentrations at two sites were determined in the laboratory as biochemical oxygen demand (BOD), using standard procedures (APHA,

ESTIMATION OF TRANSFER COEFFICIENTS
Let P j (mg/l) be the pollutant concentration in the stream at site j having stream flow Q j (m 3 /s). Mass is expressed as kg/day can obtained as.
The fraction a ଵଶ of the mass at site 1 that reaches site 2 is often assumed to be: Where k is a rate constant and t 12 is the time it takes a particle of pollutant to flow from site 1 to site 2. A similar expression, a 23 , applies for the fraction of pollutant mass at site 2 that reaches site 3. The fraction of pollutant mass at site 1 that reaches site 3 is obtained as a ଵଷ = a ଵଶ a ଶଷ (3) P ଶ = P ଵ a ଵଶ (4) The design stream flow condition is the minimum seven-day average flow expected once in ten years and is assumed that the design stream flow just downstream of site1 and just upstream of site 2 are the same and equal to 12m 3 /s (Ibe et al.,1991). Denote the concentration of each pair of sample measurements s in the first reach (just downstream of site 1 and just upstream of site 2) as SP 1s and SP 2s and their combined error as E s , equation (5) becomes In order to obtain the best estimates of the unknown a ଵଶ , the values of a ଵଶ and all E ୱ that minimize the sum of the absolute values of all the error terms E ୱ , are evaluated. This objective combined could be written as minimze |E ୱ | ୱ (7)

MODEL FORMULATION
To find the factor x i of waste removal at site i = 1 and 2 that meet the stream quality standards at the downstream sites 2 and 3 at a minimum total cost, thus from equation 2 P ଶ = ሾP ଵ Q ଵ + W ଵ (1 − x ଵ )ሿa ଵଶ Q ଶ (10) P ୨ ≤ P ୨ ୫ୟ୶ for j = 2 and 3 (12) 0 ≤ x ୧ ≤ 1.0 for i = 1 and 2 (13) Hence, the objective is to minimize the total cost of meeting the stream quality standards P ଶ ୫ୟ୶ and P ଷ ୫ୟ୶ specified in equation 12. Let C ୧ (x ୧ ) represent the wastewater treatment cost function for site i, the objective can be written as: There are four unknown decision variables, x ଵ , x ଶ , P ଶ ,and P ଷ . All variables are assumed to be non-negative. Combining Equations 10 and 12, Combining equations 11 and 12, and using the fraction a ଵଷ (see equation 2) to predict the contribution of the pollutant concentration at site 1 on the pollutant concentration at site 3: Rewriting the water quality management model defined by equations 13 to 17 and substituting the parameter values in place of the parameters, and recalling that kg/day = 86.4 (mg/l)(m 3 /s): Subject to: Water quality constraint at site 2: Water quality constraint at site 3: Restrictions on fractions of waste removal: Parameters values selected for the water quality management problem illustrated in Subject to: x ଵ ≥ 0.78 (22) x ଵ + 1.28x ଶ ≥ 1.79 (23) 0 ≤ x ୧ ≤ 1.0 for i = 1 and 2 (24)

MODEL SOLUTION
The feasible combinations of x 1 and x 2 is shown in Figure 3a. This graph is a plot of each constraint, showing the boundaries of the region of combinations of x 1 and x 2 that satisfy all the constraints. The shaded region is called the feasible region. Since the actual cost functions are not known, their general form was assumed, as shown in Figure 3b. Since the wasteloads produced at Site 1 were substantially greater than those produced at Site 2, and given similar site, transport, labour, and material cost conditions, it seems reasonable to assume that the cost of providing a specified level of treatment at Site 1 would exceed the cost of providing the same specified level of treatment at Site 2. It would also seem the marginal cost at Site 1 would be greater than, or at least not less than, the marginal cost at Site 2 for the same amount of treatment. The relative positions of the cost functions shown in Figure 3b are based on these assumptions.
To find the least-cost solution, it was assumed that c equal c ଵ . Then, let c ଵ x ଵ + c ଶ x ଶ = c and c/c ଵ = 1. Thus, x ଵ + x ଶ = 1.0. The plot of this line is shown in Figure 3c, as line 'a'. Line 'a' represents equal values for c ଵ and c ଶ , and the total cost, c ଵ x ଵ + c ଶ x ଶ , equal to 1. Keeping the slope of this line constant and moving it upward, representing increasing total cost, to line 'b', where it covers the nearest point in the feasible region, will identify the least-cost combination of x ଵ and x ଶ , again assuming marginal costs are equal. In this case the solution is approximately 80% treatment at both sites. If the marginal cost of 80% treatment at site 1 is no less than the marginal cost of 80% treatment at site 2, then c ଵ ≥ c ଶ and indeed the 80% treatment efficiencies will meet the stream standards for the design streamflow and wasteload condition at a total minimum cost. For any other assumption regarding c ଵ and c ଶ , 80% treatment at both sites will result in a least-cost solution to meeting the water quality standards for those design wasteload and streamflow conditions.

Objective Functions and Constraints
Figure 3: Graphical Solution of the Model cost waste load removal efficiencies have been determined without knowing the cost functions. No doubt the actual cost of installing the least-cost treatment efficiencies of 80% will have to be determined for issuing bonds, or making other arrangements for paying the costs. However, knowing the least-cost removal efficiencies means one does not have to spend ୧ ).