Introduction

A sewerage system involves the collection, treatment and disposal of the sewage. The sewage is collected from individual homes and carried to the treatment plant through a network of sewer pipes. The network of sewer pipes may consist of house sewers, main/trunk sewers and out fall sewers. Main sewer lines are generally provided with manholes at suitable interval for facilitating their cleaning and inspection. The sewer pipes are laid at downward gradient to carry sewage under gravity up to the outfall point. The design, construction, modification, operation and maintenance of sewerage systems involve a large capital. The sewer lines being the basic unit of a sewerage system, any saving in its cost due to proper design may affect the overall cost of the sewerage system. In spite of the growing concern for the urban environment and the future expenses and effort involved, conventional design methods for storm sewers fail to explicitly account for the cost interactions of the various components of storm sewer systems [18].

The earliest effort to consider hydraulic design of sewer lines was made by Camp [3]. Since then, a large number of research workers contributed to this subject. Some of the researchers employed heuristic methodologies [4, 5, 9, 10, 15, 17, 21]. Many researchers used optimization methods for design of sewer lines. Dynamic programming approach was used by Argaman et al. [2], Walsh and Brown [31], Mays and Yen [18], Mays and Wenzel [19], Gupta et al. [12] and Kulkarni and Khanna [14], to obtain optimal design of sewer networks. Gupta et al. [13] used Powell’s method of conjugate directions to optimize the cost function. Swamee [29] applied the Lagrange multiplier method for minimization of the cost function. Afshar and Rohani [1] proposed a hybrid method by considering two sub-optimization problems.

Dajani et al. [6] addressed the problem using a separable convex programming model. Dajani and Hasit [7] compared three models based on separable-convex and mixed integer programming for the optimization of drainage networks. Dajani et al. [8] minimized the cost of a wastewater collection network using separable convex, dynamic and geometric programming. Swamee and Sharma [30] addressed the optimal design of the sewer system by incorporating commercially available pipe sizes directly in the linear programming model. In the recent past, different multi-objective optimization approaches have been developed mainly to minimize the pollution load in combined sewer water and treatment costs by Rathnayake and Tanyimboh [24,25,26] and Rathnayhke [22, 23].

Theoretical Considerations

Sewer pipes are designed to carry sewage such that the solid particles remain in suspension to avoid occurrence of clogging due to settlement of particles. The particles are kept in suspension by laying the sewer pipes at suitable gradients to achieve self-cleansing velocity at different possible discharges. A trunk sewer line collects sewage from branched sewer pipes at manholes causing variation of flow in its different sections. This requires use of different pipe sizes and gradients to maintain self-cleansing velocity from section to section. The cost of a trunk sewer line includes the costs of sewer pipes, excavation and manholes. The cost of sewer pipes depends on the size and material of the pipe. The cost of the excavation depends upon the gradient adopted, and the cost of the manhole depends upon the layout of sewer system, size and materials used. The aspects related to hydraulics of flow through sewer pipes and cost estimation are described in the following subsections.

Hydraulics of Sewer Pipe

The flow in a sewer pipe under gravity is considered as open-channel flow for designing the pipe size. The velocity of flow through a sewer pipe can be calculated using the Manning’s formula as:

$$V = \frac{1}{n}R^{2/3} S_{0}^{1/2}$$
(1)

where V is the velocity of flow in the channel (m/s), n is the Manning’s roughness coefficient, R is the hydraulic mean radius of channel (m) and S0 is the longitudinal slope of the sewer pipe between two consecutive manholes. The hydraulic mean radius is calculated as ratio of area of cross section A (m2) to wetted perimeter, P (m). The value of A can be calculated as [28]:

$$A = \frac{{d^{2} }}{8}(2\theta - \text{Sin} 2\theta )$$
(2)

where d is inside diameter of pipe (m) and \(2\theta\) is the angle (radian) subtended at the center of the pipe by water surface as shown in Fig. 1. The value of P can be calculated for any depth of flow y (m) as:

$$P = d\theta .$$
(3)
Fig. 1
figure 1

Circular channel running partially full

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The discharge through a sewer pipe can be calculated as:

$$Q = VA$$
(4)

where Q is the peak discharge through the sewer pipe (m3/s).

Using the value of θ as 2.094395 and for y as 0.75d, and arranging Eqs. (1), (2), (3) and (4), the value of d can be obtained as:

$$d = \left( {3.5184\frac{n}{{\sqrt {S_{0} } }}Q} \right)^{3/8}$$
(5)

The value of V should be such that neither the suspended particles in sewage get silted up nor the pipe material gets scoured out. Therefore, the velocity in the sewer pipe is maintained within a permissible range from 0.6 to 3 m/s [11].

Pipe Cost

The cost of a sewer pipe line can be calculated according to diameter and length as:

$$P_{\text{c}} = C_{\text{p}} L$$
(6)

where Pc is the pipe cost ($), Cp is the pipe cost per unit length ($/m) that depends upon the diameter and L is the length of pipe between two consecutive manholes (m).

Excavation Cost

The excavation cost based on the average depth (sum of pipe diameter and minimum safe excavation cover) and width of excavation can be calculated as:

$$E_{\text{c}} = C_{\text{e}} V_{\text{e}}$$
(7)

where Ec is the excavation cost ($), Ce is the excavation cost per unit volume of the sewer system between two manholes ($/m3) and Ve is the volume of excavation of the sewer system between two manholes (m3).

Manhole Cost

Manholes are generally provided at every bend, junction, change of gradient and change of sewer pipe size. The discharge between different pairs of two consecutive manholes may vary according to the addition of flow from lateral lines at the upstream manhole. Manhole cost of a sewer line between two consecutive manholes can be expressed as:

$$M_{c} \, = \,\,C_{m} \,\,Y$$
(8)

where Mc is the cost of manhole ($), Cm is the cost per unit depth of upstream manhole ($/m) and Y is the depth of upstream manhole (m).

Total Cost

The total cost of a sewer line between two consecutive manholes, T ($), can be expressed by combining Eqs. (6), (7) and (8) as:

$$T = P_{\text{c}} + E_{\text{c}} + M_{\text{c}}$$
(9)

Linear Programming Model

A linear programming model is formulated for minimizing the total cost of the sewer line. The sewer line is designed to carry the design discharge satisfying the requirements of the length, velocity and slope in different sections. The sewer line between two consecutive manholes is termed as a link and the manholes as nodes. The lengths of the available discrete pipe sizes satisfying the permissible values of the minimum and maximum velocities in each link are treated as decision variables. For each link, the sewer pipe can be laid at different slopes and accordingly, there will be as many possible diameters as the number of possible slopes to carry the design discharge for the selected pipe material. The preliminary value of diameter corresponding to each possible slope for each link is first calculated using Eq. (5) and then rounded to the next higher available pipe size. If the velocity for the rounded pipe size is within the permissible range, the diameter is selected as candidate diameter for the link. This process eliminates direct use of nonlinear cost functions in the optimization model.

Objective Function

The objective function of the linear programming model to minimize the total cost of the sewer line can be written as:

$${\text{Minimize}}\;Z = \sum\limits_{i = 1}^{\text{nl}} {\sum\limits_{j = 1}^{{{\text{nu}}\,\left( i \right)}} {\sum\limits_{k = 1}^{{{\text{nd}}\,\left( i \right)}} {C_{ijk} X_{ijk} } } }$$
(10)

where Z is the optimal total cost of the sewer line ($), i is link number, nl is the total number of links in the sewer line, j is upstream node number of the ith link, nu(i) is the total number of upstream nodes of the ith link, k is the downstream node number of the ith link, nd(i) is the total number of downstream nodes of the ith link, Cijk is the total cost per meter length of the ith sewer link laid between the jth and the kth nodes ($/m) and Xijk is the length of the ith sewer link laid between the jth and the kth nodes (m). For each link, the combination of upstream and downstream nodes is chosen in such a way that downstream node is at a lower elevation. For each set of i, j, k, there will be a single diameter for which Cijk is calculated from Eq. (9).

Constraints

The length constraint for each link is imposed by making the horizontal length between the two consecutive manholes equal to the sum of the horizontal equivalent of lengths of different candidate pipe sizes as:

$$\sum\limits_{j = 1}^{{{\text{nu}}\,\left( i \right)}} {\sum\limits_{k = 1}^{{{\text{nd}}\left( i \right)}} {X_{ijk} \text{Cos} \alpha_{ijk} = L_{i} \quad \forall ,\;i = 1,nl} }$$
(11)

where Li is the horizontal length between two manholes connected by link i and αijk is the angle in degree for Xijk such that:

$$\tan \alpha_{ijk} = \frac{{\Delta H_{ijk} }}{{L_{i} }}$$
(12)

where ∆Hijk is the elevation difference between jth upstream node and kth downstream node of the ith link. The non-negativity constraints of the decision variables are included as:

$$X_{ijk} \ge 0\quad \forall \;i,j,k$$
(13)

Design Example

To illustrate the applicability of the proposed linear programming model, a design problem is taken for Goodwin Avenue sewer from Mays and Tung [20]. The data related to the length and peak design inflows for the sewer links named 1, 2 and 3 are given in Table 1. The sewer line along with the ground elevation of manholes and possible elevations of the u/s and d/s ends of different links considered between two consecutive manholes are shown in Fig. 2. The manholes are named M1, M2, M3 and M4, respectively, from upstream (u/s) to downstream (d/s). The minimum cover depth over the sewer pipe is 1.0668 m (3.5 ft). At each manhole, the crown elevation of the downstream pipe draining out the manhole is equal to or less than the crown elevation of the upstream pipe draining into the manhole. Three possible elevations are considered for both upstream and downstream ends of each sewer link between two consecutive manholes. Thus, total number of possible paths for each sewer link is nine. The difference between two consecutive possible elevations of the sewer link at upstream (or downstream) end is taken as 0.3048 m. The unit costs of excavation and manholes are $ 6.00/yd3 and $ 100.00/ft depth, respectively. The average excavation width for all pipes is taken as 1.524 m (5 ft). The unit cost of available commercial pipe sizes is given in Table 2. The value of Manning’s coefficient is assumed as 0.014 for all pipe sizes.

Table 1 Data related to Goodwin Avenue sewer line [20]
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Fig. 2
figure 2

Schematic diagram showing elevations and possible paths of sewer links

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Table 2 Cost of different pipe sizes of sewer line
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Solution

The cost coefficients of Eq. (10) for different possible paths of each link of the sewer line are computed in Table 3. The slope for all possible paths of each link is taken using the elevation of the u/s and d/s ends. The values of slope and peak flow rate are used in Eq. (5) to calculate the value of diameter for all possible paths of each link. The pipe diameters are then rounded to just larger available pipe size, and their corresponding costs are taken from Table 2 to obtain the pipe cost from Eq. (6). To compute the excavation cost using Eq. (7), Ve is obtained by multiplying the average width of excavation with the depth of excavation (equal to sum of pipe diameter and minimum safe excavation cover). The computation of total cost of the sewer line using Eq. (9) is given in Table 3. The diameters given in Table 3 for each link are satisfying the permissible velocity limits. Further, the cases of adverse slope (i.e., d/s crown elevation higher than u/s) are considered infeasible as represented by dashed lines in Table 3. Therefore, the cost per unit length of the links corresponding to adverse slope is taken as a large number B in Table 3 using the concept of big-M method [27]. The values of total cost per unit length and slope from Table 3 with B as 2 × 1020 and the lengths from Table 1 are substituted in the linear programming model (Eqs. 10–13) for all the three links resulting in Eqs. (14–20).

Table 3 Computation of cost of sewer system
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$$\begin{aligned} {\text{Minimize}}\;Z & = 77.0227X_{111} + 67.2390X_{112} + 69.017X_{113} + 98.77205X_{121} + 82.0906X_{122} \\ & \quad + 72.31086X_{123} + 101.9619X_{131} + 103.7878X_{132} + 87.1625X_{133} + 80.6871X_{211} \\ & \quad + 70.7543X_{212} + 72.5769X_{213} + 103.3010X_{221} + 86.3701X_{222} + 76.4373X_{223} \\ & \quad + 129.5241X_{231} + 108.9844X_{232} + 92.05303X_{233} + 169.0491X_{311} + 118.5110X_{312} \\ & \quad + 98.3365X_{313} + 2 \times 10^{20} X_{321} + 174.0015\,X_{322} + 123.4636X_{323} + 2 \times 10^{20} X_{331} \\ & \quad + 2 \times 10^{20} X_{332} + 178.9475X_{333} \\ \end{aligned}$$
(14)

subject to:

$$0.9999X_{111} + 0.9998X_{112} + 0.99989X_{113} + 0.9999X_{121} + 0.9998X_{122} + 0.9997X_{123} + 0.9999X_{131} + 0.9998X_{132} + 0.9998X_{133} = 70.104$$
(15)
$$0.9998X_{211} + 0.9997X_{212} + 0.9996X_{213} + 0.9999X_{221} + 0.9998X_{222} + 0.9997X_{223} + 0.9999X_{231} + 0.9999X_{232} + 0.9998X_{233} = 49.0728$$
(16)
$$0.9999X_{311} + 0.9999X_{312} + 0.9999X_{313} + 0.0X_{321} + 0.9999X_{322} + 0.9999X_{323} + 0.0X_{331} + 0.0X_{332} + 0.9999X_{333} = 76.5048$$
(17)
$$X_{111} ,X_{112} ,X_{113} ,X_{121} ,X_{122} ,X_{123} ,X_{131} ,X_{132} ,X_{133} \ge 0$$
(18)
$$X_{211} ,X_{212} ,X_{213} ,X_{221} ,X_{222} ,X_{223} ,X_{231} ,X_{232} ,X_{233} \ge 0$$
(19)
$$X_{311} ,X_{312} ,X_{313} ,X_{321} ,X_{322} ,X_{323} ,X_{331} ,X_{332} ,X_{333} \ge 0$$
(20)

Optimal Design

The optimization model formulated as per Eqs. 14–20 is solved using LINGO 8.0 software [16] in demo mode. Table 4 shows the optimal lengths for the first, second and third links as 70.115 m, 49.086 m and 76.508 m having diameters as 68.52 cm (27 inch), 68.58 cm (27 inch) and 91.44 cm (36 inch) with their corresponding paths as 1–2, 1–2 and 1–3, respectively. The optimal cost of sewer line is obtained as $ 15,712.25. The optimal solution of the same problem obtained by Mays and Tung [20] using the forward recursive dynamic programming optimization technique is also given in Table 4. The comparison of the optimal results shows that the linear programming model resulted in lesser cost of the sewer line with the same values of optimal diameters due to different optimal path obtained for link II.

Table 4 Comparison of results by linear programming and dynamic programming models
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Effect of Slope on Optimal Results

The developed linear programming model is solved for two additional sets of slopes, set I and set II, as given in Table 5. The set I and set II of slopes are taken by lowering the downstream elevation of each link by 0.1524 m (half a foot) and 0.3048 m (one foot), respectively, from the values for the case shown in Fig. 2. The computations of total cost for set I and set II of slopes of the sewer line are given in Tables 6 and 7, respectively. The optimal solutions for these two sets of slopes using the developed linear programming model are given in Table 8. On comparing the results of Tables 4 and 8, it is observed that the increase in optimal costs for set I and set II of slopes is $ 253.53 and $ 491.64, respectively, showing nonlinear nature of increase. Further, the results show there is no change in optimal diameters and optimal lengths for both the sets of slope whereas there is change in optimal path of link III for the slopes of set II. However, the optimal cost of the sewer line is increased due to higher excavation cost at larger slope. This indicates that the optimal solution in terms of length and diameter of the sewer line may or may not be affected due to change in slope.

Table 5 Set of slopes for sensitivity analysis of sewer line design
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Table 6 Computation of cost of sewer system for set I of slopes
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Table 7 Computation of cost of sewer line for set II of slopes
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Table 8 Optimal solutions for two sets of slopes
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Conclusions

The developed linear programming model can be used to minimize the total cost comprising of pipe cost, excavation cost and manhole cost of a sewer line connecting a series of manholes. The developed model results in optimal values of pipe diameter and slope from different alternatives for carrying the desired discharge and satisfying the constraints of self-cleansing velocity and lengths of sewer line between each pair of successive manholes. The optimal solution of the developed linear programming model is found economical over the optimal solution obtained by forward recursive dynamic programming model. The optimal cost of the sewer line increases nonlinearly with an increase in slope. The developed model can also be used to obtain the optimal range of discharge for an existing sewer line. The developed linear programming model will be useful for wastewater management.