طراحی بهینه یک خط فاضلاب با استفاده از برنامه ریزی خطی

Wastewater collection systems greatly contribute to the cost of the overall municipal sewerage system; a cost-effective design of the collection system will provide significant savings towards the cost of wastewater services. It is impossible to evaluate the full impact that each pipe size and slope would have on the overall cost of the collection system with intuitive designs. However, these solutions generally satisfy the design objectives within the given constraints. A survey of the literature indicates that various optimisation techniques are being applied for least-cost solutions. In general these approaches provide continuous pipe sizes, which are converted to closest commercial sizes for adoption, which would heavily dilute the optimal outcome. Search methods are also adopted to obtain cost-effective design solutions using directly commercial pipe sizes, which are computationally expensive. In the design of a sewerage system, a sewer line is a basic unit occurring repeatedly in the design-process and finally the combinations of these basic units formulate the complete sewer system. However, the branch sewer lines, main sewers, trunk sewers, pumping stations, treatment plant and outfall sewers are in general the main components of an urban wastewater collection, treatment and disposal systems. A method has been developed to optimise this basic unit using Linear Programming technique without transforming nonlinear objective function or constraint equations into linear functions and incorporating commercially available pipe sizes directly in the problem formulation. The current research area of optimal sewer system design is focusing equally on economic considerations and hydraulic feasibility and moving away from conventional design guidelines based on only self cleaning velocity concepts for node to node sewer link hydraulic design. This paper is a step forward in developing optimal design approaches of sewer systems.

Wastewater from residential, commercial and industrial areas is collected and transported through the sewerage system to a sewage treatment plant, where it is treated to the specified standards before it is reused or disposed to receiving water. A sewerage system has a tree-like structure and is composed of various sewer lines which terminate at a junction that contains a larger sewer line. This larger sewer line further terminates at the junction of a still larger sewer line – the main sewer line – which eventually terminates at the wastewater treatment plant. The hydraulic design of sewer system has not undergone any major change in the last 100 years; however a lot has been done in the construction and management of these systems. A typical system involves laying out a sewer network along existing and proposed streets which terminates at the wastewater treatment plant − normally at the outskirts of urban boundary. Each sewer link is then designed as a separate element using some relationships governing the hydraulics of flow and a set of limiting constraints [1]. Camp [2] presented a method for the hydraulic design of sewer networks and highlighted the two main functions of the sewer systems: to carry the maximum discharge for which it is designed and to transport suspended solids. Since then many researchers [3], [4], [5], [6], [7], [8] and [9] have contributed to the design of the sewer network and have applied various optimisation techniques. They have described heuristic methodologies for sewer design that could be adapted on microcomputers. Gupta et al. [10] used Powell’s method of conjugate directions for depth–diameter optimisation of wastewater collection systems. Argaman et al. [11], [12], [13] and [14] applied dynamic programming for sewer systems design. Fisher et al. [15], [1], [16], [17] and [18] used piecewise linearization to apply Linear Programming (LP) for estimating the pipe sizes and slopes. Swamee [19] developed a sewer line design method minimising nonlinear cost function and nonlinear constraints by iterative application of the Lagrange multiplier method. Genetic Algorithm (GA) is most popular and widely used search method. There are many examples of its application in sewer system design [20], [21], [22], [23] and [24]. Hanghighi and Bakhshipour [25] highlighted that GAs slowly progress in a random-based framework and thus they are not computationally efficient compared to mathematical methods. As the number of variables and constraints increase the GAs become slow. They developed an adaptive genetic algorithm so that only feasible solutions are developed. The sewer pipe hydraulics estimates continuous pipe diameter which is rounded off to the first larger size in the commercial list for subsequent analysis. To overcome the slow progress of GAs some researchers linked this technique with other optimisation approaches. Cisty [26] hybridized the GA with LP and Haghighi et al. [27] hybridized GA with Integer LP for optimisation of water distribution system for improving efficiency. The integration of GA and LP could be an attractive method; however the transformation of nonlinear functions into linear functions by piecewise linearization destroys the originality of the function. Similarly, Pan and Kao [28] linked GA with quadratic programming (QP) model to improve the solvability. In this approach, the nonlinear functions are transformed into quadric forms and solved using QP instead of their linearization; however implementing such a combination would be very complex. An ant colony optimisation algorithm for the storm sewer deign has also been applied [29] and [30]. In this approach nodal elevations are used as decision variables and the pipe diameters are estimated satisfying hydraulic and other conditions for each pipe link. Such a design cannot be termed as holistic design. Guo et al. [31] and Afshar and Rohni [32] have explored the application of cellular automata based approaches for the optimal design of sewer systems, which is in its early stages of development. Tabu search and simulated annealing techniques are also being applied for the sewer design, which require experience in setting the parameters for their application [33]. In general, these approaches use the Manning equation or Hazen–Williams equation for resistance description; however, ASCE Task Force [34] has disapproved the Manning equation and recommended the use of the Darcy–Weisbach equation for open channel resistance. Liou [35] has strongly discouraged the use of the Hazen–Williams equation and pointed its limitations. Similarly Brown [36] investigated the history of Darcy–Weisbech equation and indicated that due to the general accuracy and complete range of application only the Darcy–Weisbech equation should be applied as the standard and others should be left for historians. Considering above conclusions only the Darcy–Weisbech equation has been adopted in the present formulation of the design problem. Huge amount of public money is invested around the world to provide sewerage services in the existing or upcoming developments. The existing under capacity sewerage services are also upgraded in the growing areas. The design engineers in general make decisions for the sewer systems based on calculations and their experience by analysing limited number of options. Such solutions are seldom optimum. Thus, any small monitory efficiency in the provision of these services on such a large sector would result in the substantial savings in the public funds. Although the Linear Programming (LP) optimisation method has been applied to the optimal design of water distribution networks, however its application in the design of sewer systems without linearization of objective functions is new. In this approach the whole system is designed as single entity and not as individual pipe link. The algorithm terminates in limited number of iterations depending upon the minimum and maximum sizes of commercial pipes used in the optimisation problem formulation. With the commercial pipe sizes used directly in the sewer line design methodology, the conversion of continuous estimated pipe diameters to nearest commercial pipe sizes could be avoided which otherwise would lose the optimality of the whole deign. The overall wastewater collection and treatment systems are comprised of sewer lines, trunk mains, sometimes combined sewer systems, pumping stations, treatment plant and outfall sewer. Now wastewater reuse and recycling components are also becoming integral part of the urban wastewater systems. However, in the design of a sewerage system the sewer line is the basic unit occurring repeatedly in the design-process and finally the combinations of these basic units formulate the complete sewer system. In this paper a method for optimal design of this basic unit is presented applying LP optimisation technique for the estimation of pipe diameters and sewer depths, using the Darcy–Weisbach equation as the resistance equation and commercially available pipe diameters directly in the problem formulation.

An algorithm for the optimal design of a sewerline using the Linear Programming approach has been developed. In this algorithm, the Darcy–Weisbach resistance equation is used in the formulation as the preferred resistance equation. The commercial sewer pipe sizes are directly used in the design of sewer system. This eliminates the problem of rounding off the estimated pipe sizes to the nearest commercial sizes as required in some optimisation techniques, which forfeits the purpose of system optimisation to large extent. At this stage the methodology has been developed for a sewer line having any number of links, which will be extended to a typical sewer network in future.