SINGLE-INPUT SOURCE, LOOPED SYSTEMS
9.2. PUMPING SYSTEM
The town water supply systems are generally single-input source, pumping, looped pipe networks. Pumping systems are provided where topography is generally flat or demand nodes are at higher elevation than the input node (source). In these circumstances, exter-nal energy is required to deliver water at required quantity and prescribed pressure.
Figure 9.3. Looped water distribution network.
Figure 9.4. Number of LP iterations in system design.
The pumping systems include pipes, pumps, reservoirs, and treatment units based on the raw water quality. In case of bore water sources, generally disinfection may be sufficient. If the raw water is extracted from surface water, a water treatment plant will be required.
The system cost includes the cost of pipes, pumps, treatment, pumping (energy), and operation and maintenance. As described in Chapter 8, the optimization of such systems is therefore important due to the high recurring energy cost involved in it. This makes the pipe sizes in the system an important factor, as there is an economic trade-off between the pumping head and pipe diameters. Thus, there exists an optimum size of pipes and pump for every system, meaning that the pipe diameters are selected in such a way that the capitalized cost of the entire system is minimum. The cost of the treatment plant is not included in the cost function as it is constant for the desired degree of treatment based on raw water quality.
The design method is described using an example of a town water supply system shown in Fig. 9.5, which contains 18 pipes, 13 nodes, 6 loops, a single pumping source, and reservoir at node 0. The network data is listed in Table 9.9.
As the pipe discharges in looped water distribution networks are not unique, they require a looped network analysis technique. The Hardy Cross analysis method was applied similar as described in Section 9.1. Based on the population load on pipes, the nodal discharges were estimated using the method described in Chapter 3, Eq. (3.29). The rate of water supply 400 L/c/d and peak factor of 2.5 was considered for pipe flow estimation. The pipe discharges estimated for initially assumed pipes size 0.20 m of CI pipe material are listed in Table 9.10.
Figure 9.5. Looped, pumping water distribution system.
9.2. PUMPING SYSTEM 173
9.2.1. Continuous Diameter Approach
The approach is similar to the gravity-sustained, looped network design described in Section 9.1.1. The entire looped water distribution system is converted into a number of distribution mains. Each distribution main is then designed separately using the methodology described in Chapter 7. The total number of such distribution mains is equal to the number of pipes in the distribution main as each pipe would generate a flow path forming a distribution main. Such conversion/decomposition is essential to calculate pumping head for the network.
TA B L E 9.9. Pumping, Looped Water Distribution Network Data
Pipe/
TA B L E 9.10. Looped Network Pipe Discharges Pipe i
1 0.06134 7 0.00183 13 0.00388
2 0.01681 8 0.01967 14 0.00187
3 0.01391 9 0.00377 15 20.00247
4 0.00658 10 0.00782 16 0.00415
5 0.00252 11 0.02052 17 0.00786
6 20.00674 12 0.00774 18 20.00439
The flow paths for all the pipes of the looped water distribution network were generated using the network geometry data (Table 9.9) and pipe discharges (Table 9.10). Applying the flow path methodology described in Section 3.9, the pipe flow paths along with their originating nodes Jt(i) are listed in Table 9.11.
The continuous pipe diameters can be obtained using Eq. (7.11b), which is modified and rewritten as:
Di ¼ 40kTfiQTQ2i p2mkm
mþ51
, (9:5)
where QTis the total pumping discharge.
The optimal diameters were obtained applying Eq. (9.5) and pipe discharges from Table 9.10. Pipe and pumping cost parameters were similar to these adopted in Chapters 7 and 8. To apply Eq. (9.5), the pipe friction f was considered as 0.01 in all the pipes initially, which was improved iteratively until the two consecutive solutions were close. The calculated pipe diameters and adopted commercial sizes are listed in Table 9.12. The pumping head required for the system can be obtained using Eq. (7.12), which is rewritten as:
h0¼ znþ H z0þ 8
TA B L E 9.11. Pipe Flow Paths Treated as Water Distribution Main
Pipe i
Flow Paths Pipes Connecting to Input Point Node 0 and Generating Water Distribution Pumping Mains It(i, ‘)
‘ ¼ 1 ‘ ¼ 2 ‘ ¼ 3 ‘ ¼ 4 ‘ ¼ 5 Nt(i) Jt(i)
Equation (9.6) is applied for all the pumping water distribution mains (flow paths), which are equal to the total number of pipes in the distribution system. Thus, the variable n in Eq. (9.6) is equal to Nt(i) and pipes p in the distribution main It(i, ‘), ‘ ¼ 1, Nt(i).
The elevation znis equal to the elevation of originating node Jt(i) of flow path for pipe i generating pumping distribution main. Q1is similar to that defined for Eq. (8.21). Thus, applying Eq. (9.6), the pumping heads for all the pumping mains listed in Table 9.11 were calculated. The minimum terminal pressure prescribed for Fig. 9.4 is 10 m. It can be seen from Table 9.12 that the pumping head for the network is 15.30 m if con-tinuous pipe sizes are adopted. The flow path for pipe 2 provides the critical pumping head. The pumping head reduced to 13.1 m for adopted commercial sizes.
The adopted commercial sizes in Table 9.12 are based on the estimated pipe dis-charges, which are based on the initially assumed pipe diameters. Using the adopted commercial pipe sizes, the pipe network should be reanalyzed for new pipe discharges.
This will again generate new pipe flow paths. The process of network analysis and pipe sizing should be repeated until the two solutions are close. The pumping head is esti-mated for the final pipe discharges and pipe sizes. Once the network design with initially assumed pipe material is obtained, the economic pipe material for each pipe link can be selected using the methodology described in Section 8.3. The process of network analy-sis and pipe sizing should be repeated for economic pipe material and pumping head TA B L E 9.12. Pumping, Looped Network Design: Continuous Diameter Approach
Pipe
1 900 0.06134 0.269 0.300 129 13.0 11.4
2 420 0.01681 0.178 0.200 130 15.3 13.1
3 640 0.01391 0.167 0.200 125 12.2 8.9
4 900 0.00658 0.132 0.150 120 7.2 3.9
5 580 0.00252 0.097 0.100 120 7.2 3.9
6 900 0.00674 0.133 0.150 120 8.2 4.8
7 420 0.00183 0.088 0.100 125 12.9 9.1
8 640 0.01967 0.187 0.200 125 9.7 7.6
9 580 0.00377 0.110 0.125 121 9.3 5.9
10 580 0.00782 0.139 0.150 125 12.5 9.6
11 580 0.02052 0.189 0.200 127 12.9 10.8
12 640 0.00774 0.139 0.150 125 12.5 9.7
13 900 0.00388 0.111 0.125 121 10.3 6.4
14 580 0.00187 0.089 0.100 121 11.3 6.8
15 900 0.00247 0.097 0.100 121 6.9 4.8
16 580 0.00415 0.114 0.125 126 13.5 10.6
17 580 0.00786 0.139 0.150 128 13.9 11.8
18 640 0.00439 0.116 0.125 126 11.9 9.8
estimated based on finally adopted commercial sizes. The pipe sizes and pumping head listed in Table 9.12 are shown in Fig. 9.6 for CI pipe material.
9.2.2. Discrete Diameter Approach
Similar to a branch system (Section 8.2.2.2), the cost function for the design of a looped system is formulated as
min F ¼ XiL
i¼1
(ci1xi1þ ci2xi2)þ rg kTQTh0, (9:7) subject to
xi1þ xi2¼ Li; i¼ 1, 2, 3 . . . iL, (9:8) X
p¼It(i,‘)
8fp1Q2p
p2gD5p1xp1þ 8fp2Q2p p2gD5p2xp2
" #
z0þ h0 zJtð Þi H
X
p¼It(i,‘)
8 kfpQ2p p2gD4p2;
‘¼ 1, 2, 3 Nt(i) For i¼ 1, 2, 3 . . . iL (9:9) As described in Section 9.1.2, the head-loss constraints Inequations (9.9) are devel-oped for all the originating nodes of pipe flow paths. Thus, it will bring all the pipes of the network in the optimization process.
Figure 9.6. Pumping, looped water supply system: continuous diameter approach.
9.2. PUMPING SYSTEM 177
Figure 9.7. Pumping, looped water network design.
TA B L E 9.13. Pumping, Looped Network Design Pipe
Equations (9.7) and (9.8) and Inequation (9.9) constituting a LP problem involve 2iL
decision variables, iLequality constraints, and iLinequality constraints. As described in Section 9.1.2, the LP iterations can be reduced if the starting diameters are taken close to the final solution. Using Eq. (9.5), the continuous optimal pipe diameters Di can be cal-culated. Selecting the two consecutive commercially available sizes such that Di1 Di Di2, significant computer time can be saved. Once the design for an initially assumed pipe material (CI) is obtained, economic pipe material is then selected applying the method described in Section 8.3. The network is reanalyzed and designed for new pipe material. The looped water distribution system shown in Fig. 9.6 was redesigned using the discrete diameter approach. The solution thus obtained is shown in Fig. 9.7 and listed in Table 9.13. The optimal pumping head was 12.90 m for 10-m terminal pressure head. The variation of pumping head with LP iterations is plotted in Fig. 9.8. From a perusal of Fig. 9.8, it can be seen that four LP iterations were sufficient using starting pipe sizes close to continuous diameter solution.
It can be concluded that the discrete pipe diameter approach provides an economic solution as it formulates the problem for the system as a whole, whereas piecemeal design is carried out in the continuous diameter approach and also conversion of continuous sizes to commercial sizes misses the optimality of the solution.
EXERCISES
9.1. Describe the advantages and disadvantages of looped water distribution systems.
Provide examples for your description.
9.2. Design a gravity water distribution network by modifying the data given in Table 9.2. The length and population can be doubled for the new data set. Use continuous and discrete diameter approaches.
9.3. Create a single-loop, four-piped system with pumping input point at one of its nodes. Assume arbitrary data for this network, and design manually using discrete diameter approach.
Figure 9.8. Variation of pumping head with LP iterations.
EXERCISES 179
9.4. Describe the drawbacks if the constraint inequations in LP formulation are devel-oped only node-wise for the design of a lodevel-oped pipe network.
9.5. Design a pumping, looped water distribution system using the data given in Table 9.9 considering the flat topography of the entire service area. Apply continu-ous and discrete diameter approaches.
REFERENCE
Swamee, P.K., and Sharma, A.K. (2000). Gravity flow water distribution system design. Journal of Water Supply Research and Technology-AQUA 49(4), 169 – 179.