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Optimal Design of Gravity-Fed Sewer Lines Using Linear Programming

  • Deepak SinghEmail author
  • P. S. Mahar
  • R. P. Singh
Original Contribution
  • 59 Downloads

Abstract

The cost of a sewerage system is mainly governed by the size of the sewer pipe, excavation depth and manhole spacing. A linear programming model is developed to minimize the total cost comprising of the pipeline cost, excavation cost and manhole cost of the sewer line. The constraints of the optimization model are related to the distance between two consecutive manholes, and slope of the sewer line to maintain the self-cleansing velocity. The nonlinearity due to the pipe size is eliminated by considering only those available diameters that satisfy the self-cleansing velocity constraint. The model selects the combination of pipe sizes and slope of the sewer line between different manholes maintaining the self-cleansing velocity, which results in the minimum value of the total cost of the entire sewer line. The application of the developed model is illustrated with the help of an existing design problem, and the results are compared with the available solution using forward recursive dynamic programming. It is found that the linear programming model results in lesser value of the total cost of the sewer line.

Keywords

Optimal design Sewer line Linear programming Manholes Self-cleansing velocity 

List of Symbols

A

Area of cross section of sewer pipe

Ce

Excavation cost per unit volume of the sewer system between two manholes

Cm

Manhole cost per unit depth of upstream manhole

Cp

Pipe cost per unit length

Cijk

Cost per meter length of the ith sewer link laid between jth upstream node and kth downstream node

d

Inside diameter of pipe

Ec

Excavation cost

Hijk

Elevation difference between jth upstream node and kth downstream node of ith link

i

Link number

j

Upstream node number for ith link

k

Downstream node number for ith link

L

Length of pipe between two consecutive manholes

Li

Horizontal length between two manholes connected by link

Mc

Manhole cost

Mi

Manhole no. (i = 1, 2, 3,…)

n

Manning’s roughness coefficient

nd(i)

Total number of downstream nodes in the ith link

nl

Total number of links in the sewer line

nu(i)

Total number of upstream nodes of the ith link

P

Wetted perimeter

Pc

Pipe cost

Q

Peak discharge through sewer pipe

R

Hydraulic mean radius of channel

S0

Longitudinal slope of the sewer pipe between two consecutive manholes

T

Total cost of sewer line between two consecutive manholes

V

Velocity of flow in the pipe

Ve

Volume of excavation of the sewer system between two manholes

Xijk

Length of ith sewer link laid between jth upstream node and kth downstream node

Y

Depth of upstream manhole for a link

y

Depth of flow in the sewer pipe

Z

Total cost of the sewer line

αijk

Angle of ith link laid between jth u/s node and kth d/s node

θ

Half of angle subtended at the center of the pipe by water surface

Notes

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Copyright information

© The Institution of Engineers (India) 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringDelhi Technological UniversityNew DelhiIndia
  2. 2.Civil Engineering Department, College of TechnologyGBPUA&T PantnagarDistt. Udham Singh NagarIndia
  3. 3.Irrigation and Drainage Engineering Department, College of TechnologyGBPUA&T PantnagarDistt. Udham Singh NagarIndia

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