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Optimal design of a sewer line using Linear Programming
Applied Mathematical Modelling 37 (2013) 4430–4439
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Optimal design of a sewer line using Linear Programming Prabhata K. Swamee a,1, Ashok K. Sharma b,⇑ a b
Dept. of Civil Eng., Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India CSIRO Land and Water, P.O. Box 56, Highett 3190, Victoria, Australia
a r t i c l e
i n f o
Article history: Received 14 March 2012 Received in revised form 5 August 2012 Accepted 11 September 2012 Available online 20 September 2012 Keywords: Design Optimisation Roughness height Sewer Sewerage and wastewater collection
a b s t r a c t 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. Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved.
1. Introduction 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 ⇑ Corresponding author. 1
E-mail addresses: [email protected], [email protected] (P.K. Swamee), [email protected] (A.K. Sharma). Visiting Faculty.
0307-904X/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.041
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Notations A ce cr cs D d d0 df dmax dn g hmin kD ke kh km L m n P Q Qmax R So V Vmax Vs Vsc w
e c m g
flow area unit excavation cost at ground level increase in unit excavation cost per unit depth sheeting and shoring cost coefficient link diameter nodal depth initial nodal depth outfall depth maximum nodal depth terminal nodal depth gravitational acceleration minimum cover depth diameter coefficient earthwork cost coefficient man hole cost coefficient pipe cost parameter link length pipe cost parameter number of links flow perimeter link discharge carrying capacity of a link hydraulic radius invert slope of a link average fluid velocity maximum fluid velocity scouring velocity self-cleaning velocity trench width average roughness height Lagrange multiplier kinematic viscosity relative depth
Subscripts i link or node; and j index
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 [3–9] 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. [11–14] applied dynamic programming for sewer systems design. Fisher et al. [15,1,16–18] 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 [20–24]. 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
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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 [29,30]. 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. 2. Resistance relationships The ASCE Task Force [34] on Friction in Open Channels has recommended the use of Darcy–Weisbach equation. Thus, the invert slope So is given by
So ¼
fQ
2
8gRA2
ð1Þ
;
where f = friction factor; Q = discharge; g = gravitational acceleration; and R = hydraulic radius defined as the ratio of the flow area A to the flow perimeter P. For a circular section A and P are given by
h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii A ¼ 0:25 D2 cos1 ð1 2gÞ 2ð1 2gÞ gð1 gÞ ;
ð2Þ
P ¼ D cos1 ð1 2gÞ;
ð3Þ
where D = sewer diameter; g = relative flow depth defined as the ratio of the normal depth y (depth of uniform flow as depicted in Fig. 1) to sewer diameter. For obtaining f ASCE Task Force [34] recommended the following variant of the Colebrook equation:
" f ¼ 1:325 ln
e 12R
þ
!#2 0:625m pffiffiffi ; VR f
ð4Þ
where e = average roughness height of the surface; and m = kinematic viscosity of water. Swamee [37] converted Eq. (4) to the following explicit form
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Fig. 1. Definition sketch: Circular sewer section.
f ¼
( " 0:9 #)2 4 e vP ln : þ 1:63 3 Q 12R
ð5Þ
Using Eqs. (1) and (5) the bed slope So for open channel flow is obtained as
(
0:9 #)2 vP So ¼ : ln þ 1:63 2 Q 12R 6gA R Q2
"
e
ð6Þ
Swamee [37] gave the following equation for the maximum average velocity Vmax and the maximum discharge Qmax occurring at the flow depths 0.813D and 0.943D respectively as
pffiffiffiffiffiffiffiffiffiffiffi V max ¼ 1:355 gDSo ln
Q max
pffiffiffiffiffiffiffiffiffiffiffi ¼ D gDSo ln 2
! 1:32m þ pffiffiffiffiffiffiffiffiffiffiffi ; 3:65D D gDSo
e
! 1:43m þ pffiffiffiffiffiffiffiffiffiffiffi : 3:46D D gDSo
e
ð7Þ
ð8Þ
Further for circular sewer section, Swamee and Swamee [38] gave the following equation for the normal depth yn:
"
#1 0:665 Q max yn ¼ 0:943D 0:823 þ1 ; 1 Q
ð9Þ
where Qmax is obtained from Eq. (8). Barring small depths g 6 0.3, the maximum error involved in Eq. (9) is 1%. 3. Cost function The cost of a sewer line primarily consists of the link costs and nodal costs. Sewer pipes cost is a link cost whereas the cost of manholes is a nodal cost. Although the cost of excavation is a link cost, for mathematical convenience, it would be lumped at nodes. A sewer line and its longitudinal profile are shown in Figs. 2 and 3. Considering Fig. 2, the cost of ith link of sewer pipe Cpi is written as
C pi ¼ ci Li ;
ð10Þ
where ci = cost of pipe per unit length; and Li = length of ith link The coefficient ci depends on sewer diameter Di. As per Fig. 3, the cost of ith manholes Cmi is
C mi ¼ khi di ;
ð11Þ
where khi = cosy of manhole per unit depth; and di = ith nodal depth. The coefficient khi depends on di and the maximum sewer pipe diameter D connecting the manhole as there will be two or more pipes connected at a manhole. Local design guidelines should be consulted for manhole size based on manhole depths for estimating cost of manholes and corresponding cost function developed. The following internal sizes of circular manholes for varying depths have been adopted for costing [39]:
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Fig. 2. Definition sketch: Plan of a sewer line.
Fig. 3. Definition sketch: Longitudinal profile of sewer line.
Fig. 4. Definition sketch: Sewer trench.
For For For For
depths depths depths depths
above above above above
0.90 m 1.65 m 2.30 m 9.00 m
and and and and
up up up up
to to to to
1.65 m 2.30 m 9.00 m 14.0 m
– – – –
manhole manhole manhole manhole
0.90 m 1.20 m 1.50 m 1.80 m
diameter diameter diameter diameter
The ith link of sewer trench has a trapezoidal cross-section having a side slope 0.1 horizontal to 1 vertical (Fig. 4). As an approximation, assuming vertical sides, the earthwork cost of the ith link Cei is
C ei ¼
1 1 2 2 Li wi ce ðdi1 þ di Þ þ cr di1 þ di1 di þ di þ cs Li ðdi1 þ di Þ; 2 3
ð12Þ
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where ce = unit excavation cost at ground level (monetary unit per unit volume of earthwork, e.g., $m3); cr = increase in unit excavation cost per unit depth (monetary unit per unit volume per unit depth, e.g., $m4) and cs = cost of sheeting and shoring per unit area (monetary unit per unit area, e.g., $m2). Generally timbering or steel trench shields are used to protect pipeline workers from trench collapse, including maintaining the ground slope and working space. Swamee et al. [40] have studied excavation cost of various types of strata and found that ce/cr varied between 6.96 and 13.86 m. According to CPHEEO [39] the bottom sewer trench width wi is given by the following criteria:
wi ¼ 1m; Di 6 0:6m;
ð13Þ
wi ¼ Di þ 0:4m; Di > 0:6m:
ð14Þ
Lumping half of ith link cost coefficient and half of the (i + 1)st link cost coefficient and adding the manhole cost coefficient as defined by Eq. (11), the ith nodal cost coefficient kni is written as
kni ¼ khi þ 14 ce ½Li wi ðmi1 þ 1Þ þ Liþ1 wiþ1 ð1 þ miþ1 Þ þ 12 cs ½Li ðmi1 þ 1Þ þ Liþ1 ð1 þ miþ1 Þ
1 þ 12 cr di Li wi ðm2i1 þ mi1 þ 1Þ þ Liþ1 wiþ1 ð1 þ miþ1 þ m2iþ1 Þ ; i ¼ 1; 2; 3; . . . ; ðiL 1Þ;
ð15Þ
where mi1 = di1/di;mi+1 = di+1/di; Li+1 and wi+1 = length and width respectively of (i + 1)st link; and iL = total number of links. Lumping half of the 1st link cost to 0th node and adding the 0th manhole cost coefficient given by Eq. (11), the nodal cost coefficient is written as:
1 1 1 kn0 ¼ kh0 þ L1 w1 ce ð1 þ m1 Þ þ cr d0 ð1 þ m1 þ m21 Þ þ cs L1 ð1 þ miþ1 Þ: 4 3 2
ð16Þ
On the other hand, lumping half of the iLth link cost to iLth node and adding the iLth manhole cost coefficient given by Eq. (11), the nodal cost coefficient kniL is written as
1 1 1 kniL ¼ kLhiL þ LiL wiL ce ðmiL1 þ 1Þ þ cr diL ðm2iL1 þ miL1 þ 1Þ þ cs LiL ðmiL1 þ 1Þ; 4 3 2
ð17Þ
Thus, the cost function of a sewerline Csewer is written as
XiL
C sewer ¼ kn0 d0 þ
i¼1
ðkni di þ ci Li Þ;
ð18Þ
where d0 = initial nodal depth. Swamee [37] recommended the following equation for d0 which provides a constant value of the manhole cost.
d0 ¼ hmin þ 0:5m; where hmin = minimum cover. As the sewer depth d0 is fixed, first term representing the manhole cost is also fixed. 4. Constraints description The invert slope of ith link Soi is given by
Soi ¼
di di1 þ zi1 zi ; Li
ð19Þ
where zi1 and zi = elevations at (i 1)st and ith nodes. See Fig. 3, Eq. (6) for ith link is
Soi ¼
(
Q 2i
"
ln
6gA2i Ri
ei
12Ri
þ 1:63
mPi
0:9 #)2 :
Qi
ð20Þ
Using Eqs. (2) and (3) for circular sewer sections, Eq. (20) modifies to
Soi ¼
k1 Q 2i gD5i
( ln
"
ei
k2 Di
þ k3
mDi Qi
0:9 #)2 ;
ð21Þ
where
32 cos1 ð1 2gÞ k1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ; 3 cos1 ð1 2gÞ 2ð1 2gÞ gð1 gÞ k2 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 cos1 ð1 2gÞ 2ð1 2gÞ gð1 gÞ ; cos1 ð1 2gÞ
ð22Þ
ð23Þ
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0:9 k3 ¼ 1:63 cos1 ð1 2gÞ :
ð24Þ
The relative depth g depends on the sewer diameter. Table 1 provides g for various diameter ranges and the corresponding values of slope coefficients k1, k2 and k3. The local design guidelines should be referred for relative depth g and corresponding slope coefficients k1, k2 and k3 estimated to develop Table 1. Eliminating Soi between Eqs. (19) and (21)
di di1 þ zi1 zi ¼
k1 Li Q 2i gD5i
2 ei ln ; k2 Di
ð25Þ
where, the term containing m being small has been dropped. 5. Optimisation Dropping the initial manhole cost as constant term, the cost function F of the sewer line having iL links is
F¼
XiL i¼1
ðkni di þ ci Li Þ;
ð26Þ
where d1, d2, d3,. . ., diL are decision variables. Splitting the sewer length Li into two parts x1i and x2i having diameters Di1 and Di2 for LP application, Eq. (26) is rewritten as
F¼
iL X ðkni di þ ci1 xi1 þ ci2 xi2 Þ;
ð27Þ
i¼1
where ci1 and ci2 = unit costs of sewer pipes of diameter Di1 and Di2 respectively. The decision variables xi1 and xi2 should satisfy the following constrains
xi1 þ xi2 ¼ Li ;
i ¼ 1; 2; 3 . . . iL :
ð28Þ
Further, applying Eq. (25) to the segments of lengths xi1 and xi2, the following constraint is obtained:
di1 þ di
k1 xi1 Q 2i gD5i1
ln
ei
k2 Di1
2
k1 xi2 Q 2i gD5i2
2 ei ln ¼ zi1 þ zi ; k2 Di2
i ¼ 1; 2; 3 . . . iL :
ð29Þ
Normally the application of LP technique requires the linearization of objective function and constraints. Here it has been applied in such a way that the originality of the objective function and constraints are maintained by selecting the pipe lengths x1i or x2i of commercial pipe diameters Di1 and Di2 in the optimisation process (Eq. (28)). The formulation of constraint equations (Eq. (29)) for each pipe link at its downstream node J2(i) will ensure that the entire sewer line containing iL links is designed as single entity. Eqs. (27)–(29) constitute a LP problem which involves 3iL decision variables, consisting of 2iL equality constraints. By including the lower and upper ranges of commercially available pipe sizes, such as Di1 and Di2, the problem is solved by using simplex algorithm. Thus, the LP solution provides the minimum system cost according to the corresponding pipe diameters. In order to begin the LP algorithm, the uniform sewer material is selected for all the links; and for this material the roughness height is obtained. This roughness height is used for all the sewer links. For the assumed pipe diameters, Di1 and Di2, relative flow depths gi1 and gi2 are obtained by using Table 1. Once gi1 and gi2 are known, the coefficients k1 and k2 are then computed using Eqs. (22) and (23). Furthermore, for known diameters Di1 and Di2 the sewer cost coefficients ci1 and ci2 are obtained. Similarly, as the sewer diameters are known, the bottom sewer trench widths wi and wi+1 are obtained by Eqs. (13) and (14). The nodal cost coefficients kni are obtained by using Eqs. (15)–(17). For this purpose a tentative sewer invert profile is assumed for obtaining the depth ratios mi and mi+1. To initiate the LP algorithm, the pipe invert levels can be estimated based on the pipe gradients recommended in local manuals/design codes. Thus, the invert levels of all the pipes of the sewerline can be estimated. Using continuity conditions the sewer discharges Qi are computed, which is based on the upstream pipe flow and the population load on the pipe i. Thus, the LP Eqs. (27)–(29) can be written. The LP solution indicates preference for one diameter (lower or higher) in each pipe link. By knowing such preferences, the sewer pipe diameter not preferred by LP is rejected and another diameter replacing it is introduced as Di1 or Di2. Once the new diameters and the nodal depths are known, Eqs. (27) and (29) are then revised. After completing the replacement process for i = 1, 2,
Table 1 Relative Depth and Slope Coefficients. Sewer diameter, D (m)
Relative depth, g
k1
k2
k3
0.15–0.25 0.30–0.50 0.55–1.20 Above 1.20
0.50 0.60 0.70 0.75
4.323 2.471 1.632 1.384
3.00 3.332 3.555 3.620
2.447 2.728 3.018 3.171
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Fig. 5. Plan of the sewer line.
Table 2 Sewer line data. Pipe link/node i/j
1st node of i J1(i)
2nd node of i J2(i)
Length of link i L(i) (m)
Elevation of node j z(j) (m)
Discharge at node j q(j) (m3/s)
1 2 3 4 5 6
1 2 3 4 5
2 3 4 5 6
300 350 300 350 400
120.00 120.50 121.00 120.00 119.50 120.00
0.01 0.02 0.02 0.03 0.05 0
Table 3 Cost of various commercial pipe sizes including the cost of laying. S. No.
Pipe diameter (mm)
Cost per meter (IRs)
S. No.
Pipe diameter (mm)
Cost per meter (IRs)
1 2 3 4 5 6 7 8 9
150 200 250 300 350 400 450 500 600
345 452 542 710 1181 1289 1450 1640 2126
10 11 12 13 14 15 16 17 18
700 800 900 1000 1100 1200 1400 1600 1800
2580 3485 3981 4450 5103 5774 6992 8906 11,272
3,. . .iL, another LP solution is carried out to obtain the new preferred diameters and nodal depths. The process of LP and pipe size replacement is continued until Di1 and Di2 are two consecutive commercial sewer pipe sizes. One more LP cycle now obtains the diameters and depths to be adopted. The algorithm is further explained with the following example.
Example It is proposed to design a sewerline containing five sewer links and six nodes as depicted in Fig. 5. The nodal elevations and nodal discharges are listed in Table 2. The commercial pipe sizes and costs are listed in Table 3. Soil excavation cost ce = 100 IRs/m3 and additional cost of excavation cr = 30 IRs/m3/m depth were also considered. The cost of timbering the trenches has been considered as 300 IRs/m. The algorithm terminated in LP 17 iterations. The variation of system cost with LP iterations is shown in Fig. 6. Thus the system cost is minimum at the terminal iteration, which is the optimal solution. The final pipe sizes and invert depths are depicted in Fig. 7.
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Fig. 6. Variation of sewer line cost with LP cycles.
Fig. 7. Estimated pipe sizes and invert depths after 17 LP cycles.
6. Conclusions 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. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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[23] Y. Guo, G.A. Walters, S.-T. Khu, E. Keedwell, Efficient multiobjective storm sewer design using cellular automata and genetic algorithm hybrid, J. Water Resour. Plng. Mangt. 134 (6) (2008) 511–515. [24] M.H. Afshar, Rebirthing genetic algorithm for storm sewer network design, Scientia Iranica 19 (1) (2012) 11–19. [25] A. Haghighi, A.E. Bakhshipour, Optimization of sewer networks using an adaptive genetic algorithm, Water Resour. Manag. (2012), http://dx.doi.org/ 10.1007/s11269-012-0084-3. [26] M. Cisty, Hybrid genetic algorithm and linear programming method for least-cost design of water distribution systems, Water Resour. Manag. 24 (1) (2010) 1–24. [27] H. Haghighi, M.V. Samani, Z. Samani, GA-ILP method for optimization of water distribution networks, Water Resour. Manage 25 (7) (2011) 1791–1808. [28] T. Pan, J. Kao, GA-QP model to optimize sewer system design, J. Environ. Eng. 135 (1) (2009) 17–24. [29] M.H. Afshar, A parameter free continuous ant colony optimization algorithm for the optimal design of storm sewer networks: constrained and unconstrained approach, Adv. Eng. Softw. 41 (2) (2010) 188–195. [30] R. Moeini, M.H. Afshar, Layout and size optimization of sanitary sewer network using intelligent ants, Adv. Eng. Softw. 51 (2012) 49–62. [31] Y. Guo, G.A. Walters, S.T. Khu, E. Keedwell, A novel cellular automata based approach to storm sewer design, Eng. Optim. 39 (3) (2007) 345–364. [32] M.H. Afshar, M. Rohani, Optimal design of sewer networks using cellular automata-based hybrid methods: discrete and continuous approaches, Eng. Optim. 44 (1) (2012) 1–22. [33] S.F. Yeh, C.W. Chu, et al, Applying tabu search and simulated annealing to the optimal design of sewer networks, Eng. Optim. 43 (2) (2011) 159–174. [34] ASCE Task Force, Friction factors in open channels: Progress report of task force on friction factors in open channels, J. Hydraul. Eng. Div., ASCE 89 (2) (1963) 97–143. [35] C.P. Liou, Limitation and proper use of the Hazen–Williams equation, J. Hydraul. Eng. ASCE 124 (9) (1998) 951–954. [36] G.O. Brown, The history of the Darcy–Weisbach equation for pipe flow resistance, in: J.R. Rogers, A.J. Fredrich (Eds.), Environmental and Water Resources History, Proceedings and Invited Papers for the ASCE 150th Anniversary presented at the ASCE Civil Engineering Conference and Exposition, Washington, 2002, pp. 34–43. [37] P.K. Swamee, Critical slope equations for open channels, ISH J. Hydraul. Eng. 8 (2) (2002) 44–49. [38] P.K. Swamee, N. Swamee, Design of noncircular sewers sections, J. Hydraul. Res. IAHR 46 (2) (2008) 277–281. [39] CPHEEO, Manual on Sewerage and Sewage Treatment, second ed., Central Public Health and Environmental Engineering Organisation, Ministry of Urban Development, New Delhi, India, 1993. [40] P.K. Swamee, G.C. Mishra, B.R. Chahar, Minimum cost design of lined canal sections, J. Wat. Resour. Manag. 14 (1) (2000) 1–12.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Optimal design of a sewer line using Linear Programming Prabhata K. Swamee a,1, Ashok K. Sharma b,⇑ a b
Dept. of Civil Eng., Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India CSIRO Land and Water, P.O. Box 56, Highett 3190, Victoria, Australia
a r t i c l e
i n f o
Article history: Received 14 March 2012 Received in revised form 5 August 2012 Accepted 11 September 2012 Available online 20 September 2012 Keywords: Design Optimisation Roughness height Sewer Sewerage and wastewater collection
a b s t r a c t 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. Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved.
1. Introduction 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 ⇑ Corresponding author. 1
E-mail addresses: [email protected], [email protected] (P.K. Swamee), [email protected] (A.K. Sharma). Visiting Faculty.
0307-904X/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.041
P.K. Swamee, A.K. Sharma / Applied Mathematical Modelling 37 (2013) 4430–4439
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Notations A ce cr cs D d d0 df dmax dn g hmin kD ke kh km L m n P Q Qmax R So V Vmax Vs Vsc w
e c m g
flow area unit excavation cost at ground level increase in unit excavation cost per unit depth sheeting and shoring cost coefficient link diameter nodal depth initial nodal depth outfall depth maximum nodal depth terminal nodal depth gravitational acceleration minimum cover depth diameter coefficient earthwork cost coefficient man hole cost coefficient pipe cost parameter link length pipe cost parameter number of links flow perimeter link discharge carrying capacity of a link hydraulic radius invert slope of a link average fluid velocity maximum fluid velocity scouring velocity self-cleaning velocity trench width average roughness height Lagrange multiplier kinematic viscosity relative depth
Subscripts i link or node; and j index
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 [3–9] 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. [11–14] applied dynamic programming for sewer systems design. Fisher et al. [15,1,16–18] 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 [20–24]. 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
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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 [29,30]. 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. 2. Resistance relationships The ASCE Task Force [34] on Friction in Open Channels has recommended the use of Darcy–Weisbach equation. Thus, the invert slope So is given by
So ¼
fQ
2
8gRA2
ð1Þ
;
where f = friction factor; Q = discharge; g = gravitational acceleration; and R = hydraulic radius defined as the ratio of the flow area A to the flow perimeter P. For a circular section A and P are given by
h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii A ¼ 0:25 D2 cos1 ð1 2gÞ 2ð1 2gÞ gð1 gÞ ;
ð2Þ
P ¼ D cos1 ð1 2gÞ;
ð3Þ
where D = sewer diameter; g = relative flow depth defined as the ratio of the normal depth y (depth of uniform flow as depicted in Fig. 1) to sewer diameter. For obtaining f ASCE Task Force [34] recommended the following variant of the Colebrook equation:
" f ¼ 1:325 ln
e 12R
þ
!#2 0:625m pffiffiffi ; VR f
ð4Þ
where e = average roughness height of the surface; and m = kinematic viscosity of water. Swamee [37] converted Eq. (4) to the following explicit form
P.K. Swamee, A.K. Sharma / Applied Mathematical Modelling 37 (2013) 4430–4439
4433
Fig. 1. Definition sketch: Circular sewer section.
f ¼
( " 0:9 #)2 4 e vP ln : þ 1:63 3 Q 12R
ð5Þ
Using Eqs. (1) and (5) the bed slope So for open channel flow is obtained as
(
0:9 #)2 vP So ¼ : ln þ 1:63 2 Q 12R 6gA R Q2
"
e
ð6Þ
Swamee [37] gave the following equation for the maximum average velocity Vmax and the maximum discharge Qmax occurring at the flow depths 0.813D and 0.943D respectively as
pffiffiffiffiffiffiffiffiffiffiffi V max ¼ 1:355 gDSo ln
Q max
pffiffiffiffiffiffiffiffiffiffiffi ¼ D gDSo ln 2
! 1:32m þ pffiffiffiffiffiffiffiffiffiffiffi ; 3:65D D gDSo
e
! 1:43m þ pffiffiffiffiffiffiffiffiffiffiffi : 3:46D D gDSo
e
ð7Þ
ð8Þ
Further for circular sewer section, Swamee and Swamee [38] gave the following equation for the normal depth yn:
"
#1 0:665 Q max yn ¼ 0:943D 0:823 þ1 ; 1 Q
ð9Þ
where Qmax is obtained from Eq. (8). Barring small depths g 6 0.3, the maximum error involved in Eq. (9) is 1%. 3. Cost function The cost of a sewer line primarily consists of the link costs and nodal costs. Sewer pipes cost is a link cost whereas the cost of manholes is a nodal cost. Although the cost of excavation is a link cost, for mathematical convenience, it would be lumped at nodes. A sewer line and its longitudinal profile are shown in Figs. 2 and 3. Considering Fig. 2, the cost of ith link of sewer pipe Cpi is written as
C pi ¼ ci Li ;
ð10Þ
where ci = cost of pipe per unit length; and Li = length of ith link The coefficient ci depends on sewer diameter Di. As per Fig. 3, the cost of ith manholes Cmi is
C mi ¼ khi di ;
ð11Þ
where khi = cosy of manhole per unit depth; and di = ith nodal depth. The coefficient khi depends on di and the maximum sewer pipe diameter D connecting the manhole as there will be two or more pipes connected at a manhole. Local design guidelines should be consulted for manhole size based on manhole depths for estimating cost of manholes and corresponding cost function developed. The following internal sizes of circular manholes for varying depths have been adopted for costing [39]:
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Fig. 2. Definition sketch: Plan of a sewer line.
Fig. 3. Definition sketch: Longitudinal profile of sewer line.
Fig. 4. Definition sketch: Sewer trench.
For For For For
depths depths depths depths
above above above above
0.90 m 1.65 m 2.30 m 9.00 m
and and and and
up up up up
to to to to
1.65 m 2.30 m 9.00 m 14.0 m
– – – –
manhole manhole manhole manhole
0.90 m 1.20 m 1.50 m 1.80 m
diameter diameter diameter diameter
The ith link of sewer trench has a trapezoidal cross-section having a side slope 0.1 horizontal to 1 vertical (Fig. 4). As an approximation, assuming vertical sides, the earthwork cost of the ith link Cei is
C ei ¼
1 1 2 2 Li wi ce ðdi1 þ di Þ þ cr di1 þ di1 di þ di þ cs Li ðdi1 þ di Þ; 2 3
ð12Þ
P.K. Swamee, A.K. Sharma / Applied Mathematical Modelling 37 (2013) 4430–4439
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where ce = unit excavation cost at ground level (monetary unit per unit volume of earthwork, e.g., $m3); cr = increase in unit excavation cost per unit depth (monetary unit per unit volume per unit depth, e.g., $m4) and cs = cost of sheeting and shoring per unit area (monetary unit per unit area, e.g., $m2). Generally timbering or steel trench shields are used to protect pipeline workers from trench collapse, including maintaining the ground slope and working space. Swamee et al. [40] have studied excavation cost of various types of strata and found that ce/cr varied between 6.96 and 13.86 m. According to CPHEEO [39] the bottom sewer trench width wi is given by the following criteria:
wi ¼ 1m; Di 6 0:6m;
ð13Þ
wi ¼ Di þ 0:4m; Di > 0:6m:
ð14Þ
Lumping half of ith link cost coefficient and half of the (i + 1)st link cost coefficient and adding the manhole cost coefficient as defined by Eq. (11), the ith nodal cost coefficient kni is written as
kni ¼ khi þ 14 ce ½Li wi ðmi1 þ 1Þ þ Liþ1 wiþ1 ð1 þ miþ1 Þ þ 12 cs ½Li ðmi1 þ 1Þ þ Liþ1 ð1 þ miþ1 Þ
1 þ 12 cr di Li wi ðm2i1 þ mi1 þ 1Þ þ Liþ1 wiþ1 ð1 þ miþ1 þ m2iþ1 Þ ; i ¼ 1; 2; 3; . . . ; ðiL 1Þ;
ð15Þ
where mi1 = di1/di;mi+1 = di+1/di; Li+1 and wi+1 = length and width respectively of (i + 1)st link; and iL = total number of links. Lumping half of the 1st link cost to 0th node and adding the 0th manhole cost coefficient given by Eq. (11), the nodal cost coefficient is written as:
1 1 1 kn0 ¼ kh0 þ L1 w1 ce ð1 þ m1 Þ þ cr d0 ð1 þ m1 þ m21 Þ þ cs L1 ð1 þ miþ1 Þ: 4 3 2
ð16Þ
On the other hand, lumping half of the iLth link cost to iLth node and adding the iLth manhole cost coefficient given by Eq. (11), the nodal cost coefficient kniL is written as
1 1 1 kniL ¼ kLhiL þ LiL wiL ce ðmiL1 þ 1Þ þ cr diL ðm2iL1 þ miL1 þ 1Þ þ cs LiL ðmiL1 þ 1Þ; 4 3 2
ð17Þ
Thus, the cost function of a sewerline Csewer is written as
XiL
C sewer ¼ kn0 d0 þ
i¼1
ðkni di þ ci Li Þ;
ð18Þ
where d0 = initial nodal depth. Swamee [37] recommended the following equation for d0 which provides a constant value of the manhole cost.
d0 ¼ hmin þ 0:5m; where hmin = minimum cover. As the sewer depth d0 is fixed, first term representing the manhole cost is also fixed. 4. Constraints description The invert slope of ith link Soi is given by
Soi ¼
di di1 þ zi1 zi ; Li
ð19Þ
where zi1 and zi = elevations at (i 1)st and ith nodes. See Fig. 3, Eq. (6) for ith link is
Soi ¼
(
Q 2i
"
ln
6gA2i Ri
ei
12Ri
þ 1:63
mPi
0:9 #)2 :
Qi
ð20Þ
Using Eqs. (2) and (3) for circular sewer sections, Eq. (20) modifies to
Soi ¼
k1 Q 2i gD5i
( ln
"
ei
k2 Di
þ k3
mDi Qi
0:9 #)2 ;
ð21Þ
where
32 cos1 ð1 2gÞ k1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ; 3 cos1 ð1 2gÞ 2ð1 2gÞ gð1 gÞ k2 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 cos1 ð1 2gÞ 2ð1 2gÞ gð1 gÞ ; cos1 ð1 2gÞ
ð22Þ
ð23Þ
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0:9 k3 ¼ 1:63 cos1 ð1 2gÞ :
ð24Þ
The relative depth g depends on the sewer diameter. Table 1 provides g for various diameter ranges and the corresponding values of slope coefficients k1, k2 and k3. The local design guidelines should be referred for relative depth g and corresponding slope coefficients k1, k2 and k3 estimated to develop Table 1. Eliminating Soi between Eqs. (19) and (21)
di di1 þ zi1 zi ¼
k1 Li Q 2i gD5i
2 ei ln ; k2 Di
ð25Þ
where, the term containing m being small has been dropped. 5. Optimisation Dropping the initial manhole cost as constant term, the cost function F of the sewer line having iL links is
F¼
XiL i¼1
ðkni di þ ci Li Þ;
ð26Þ
where d1, d2, d3,. . ., diL are decision variables. Splitting the sewer length Li into two parts x1i and x2i having diameters Di1 and Di2 for LP application, Eq. (26) is rewritten as
F¼
iL X ðkni di þ ci1 xi1 þ ci2 xi2 Þ;
ð27Þ
i¼1
where ci1 and ci2 = unit costs of sewer pipes of diameter Di1 and Di2 respectively. The decision variables xi1 and xi2 should satisfy the following constrains
xi1 þ xi2 ¼ Li ;
i ¼ 1; 2; 3 . . . iL :
ð28Þ
Further, applying Eq. (25) to the segments of lengths xi1 and xi2, the following constraint is obtained:
di1 þ di
k1 xi1 Q 2i gD5i1
ln
ei
k2 Di1
2
k1 xi2 Q 2i gD5i2
2 ei ln ¼ zi1 þ zi ; k2 Di2
i ¼ 1; 2; 3 . . . iL :
ð29Þ
Normally the application of LP technique requires the linearization of objective function and constraints. Here it has been applied in such a way that the originality of the objective function and constraints are maintained by selecting the pipe lengths x1i or x2i of commercial pipe diameters Di1 and Di2 in the optimisation process (Eq. (28)). The formulation of constraint equations (Eq. (29)) for each pipe link at its downstream node J2(i) will ensure that the entire sewer line containing iL links is designed as single entity. Eqs. (27)–(29) constitute a LP problem which involves 3iL decision variables, consisting of 2iL equality constraints. By including the lower and upper ranges of commercially available pipe sizes, such as Di1 and Di2, the problem is solved by using simplex algorithm. Thus, the LP solution provides the minimum system cost according to the corresponding pipe diameters. In order to begin the LP algorithm, the uniform sewer material is selected for all the links; and for this material the roughness height is obtained. This roughness height is used for all the sewer links. For the assumed pipe diameters, Di1 and Di2, relative flow depths gi1 and gi2 are obtained by using Table 1. Once gi1 and gi2 are known, the coefficients k1 and k2 are then computed using Eqs. (22) and (23). Furthermore, for known diameters Di1 and Di2 the sewer cost coefficients ci1 and ci2 are obtained. Similarly, as the sewer diameters are known, the bottom sewer trench widths wi and wi+1 are obtained by Eqs. (13) and (14). The nodal cost coefficients kni are obtained by using Eqs. (15)–(17). For this purpose a tentative sewer invert profile is assumed for obtaining the depth ratios mi and mi+1. To initiate the LP algorithm, the pipe invert levels can be estimated based on the pipe gradients recommended in local manuals/design codes. Thus, the invert levels of all the pipes of the sewerline can be estimated. Using continuity conditions the sewer discharges Qi are computed, which is based on the upstream pipe flow and the population load on the pipe i. Thus, the LP Eqs. (27)–(29) can be written. The LP solution indicates preference for one diameter (lower or higher) in each pipe link. By knowing such preferences, the sewer pipe diameter not preferred by LP is rejected and another diameter replacing it is introduced as Di1 or Di2. Once the new diameters and the nodal depths are known, Eqs. (27) and (29) are then revised. After completing the replacement process for i = 1, 2,
Table 1 Relative Depth and Slope Coefficients. Sewer diameter, D (m)
Relative depth, g
k1
k2
k3
0.15–0.25 0.30–0.50 0.55–1.20 Above 1.20
0.50 0.60 0.70 0.75
4.323 2.471 1.632 1.384
3.00 3.332 3.555 3.620
2.447 2.728 3.018 3.171
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Fig. 5. Plan of the sewer line.
Table 2 Sewer line data. Pipe link/node i/j
1st node of i J1(i)
2nd node of i J2(i)
Length of link i L(i) (m)
Elevation of node j z(j) (m)
Discharge at node j q(j) (m3/s)
1 2 3 4 5 6
1 2 3 4 5
2 3 4 5 6
300 350 300 350 400
120.00 120.50 121.00 120.00 119.50 120.00
0.01 0.02 0.02 0.03 0.05 0
Table 3 Cost of various commercial pipe sizes including the cost of laying. S. No.
Pipe diameter (mm)
Cost per meter (IRs)
S. No.
Pipe diameter (mm)
Cost per meter (IRs)
1 2 3 4 5 6 7 8 9
150 200 250 300 350 400 450 500 600
345 452 542 710 1181 1289 1450 1640 2126
10 11 12 13 14 15 16 17 18
700 800 900 1000 1100 1200 1400 1600 1800
2580 3485 3981 4450 5103 5774 6992 8906 11,272
3,. . .iL, another LP solution is carried out to obtain the new preferred diameters and nodal depths. The process of LP and pipe size replacement is continued until Di1 and Di2 are two consecutive commercial sewer pipe sizes. One more LP cycle now obtains the diameters and depths to be adopted. The algorithm is further explained with the following example.
Example It is proposed to design a sewerline containing five sewer links and six nodes as depicted in Fig. 5. The nodal elevations and nodal discharges are listed in Table 2. The commercial pipe sizes and costs are listed in Table 3. Soil excavation cost ce = 100 IRs/m3 and additional cost of excavation cr = 30 IRs/m3/m depth were also considered. The cost of timbering the trenches has been considered as 300 IRs/m. The algorithm terminated in LP 17 iterations. The variation of system cost with LP iterations is shown in Fig. 6. Thus the system cost is minimum at the terminal iteration, which is the optimal solution. The final pipe sizes and invert depths are depicted in Fig. 7.
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Fig. 6. Variation of sewer line cost with LP cycles.
Fig. 7. Estimated pipe sizes and invert depths after 17 LP cycles.
6. Conclusions 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. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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