PIPE NETWORK ANALYSIS
3.8. MULTI-INPUT SOURCE WATER NETWORK ANALYSIS
Generally, urban water distribution systems have looped configurations and receive water from multi-input points (sources). The looped configuration of pipelines is preferred over
3.8. MULTI-INPUT SOURCE WATER NETWORK ANALYSIS 67
branched configurations due to high reliability (Sarbu and Kalmar, 2002) and low risk from the loss of services. The analysis of a single-input water system is simple. On the other hand in a multi-input point water system, it is difficult to evaluate the input point discharges, based on input head, topography, and pipe layout. Such an analysis requires either search methods or formulation of additional nonlinear equations between input points.
A simple alternative method for the analysis of a multi-input water network is described in this section. In order to describe the algorithm properly, a typical water distribution network as shown in Fig. 3.14 is considered. The geometry of the network is described by the following data structure.
3.8.1. Pipe Link Data
The pipe link i has two end points with the nodes J1(i) and J2(i) and has a length Lifor i ¼ 1, 2, 3, . . . , iL; iLbeing the total number of pipe links in the network. The pipe nodes are defined such that J1(i) is a lower-magnitude node and J2(i) is a higher-magnitude node of pipe i. The total number of nodes in the network is JL. The elevations of the end points are z(J1i) and z(J2i). The pipe link population load is P(i), diameter of pipe i is D(i), and total form-loss coefficient due to valves and fittings is kf(i). The pipe data structure is shown in Table 3.1.
3.8.2. Input Point Data
The nodal number of the input point is designated as S(n) for n ¼ 1 to nL(total number of input points). The two input points at nodes 1 and 13 are shown in Fig. 3.14. The
Figure 3.14. A looped water supply network.
TABLE3.1.NetworkPipeLinkData Pipe(i)/ Node(j)Node1 J1(i)Node2 J2(i)Loop1 K1(i)Loop2 K2(i)LengthL(i) (m)Form-Loss Coefficientkf(i)Population P(i)PipeSize D(i)(m)Elevationz(j) (m) 112108000.154000.40101.5 223208000.154000.30100.5 3343080004000.20101.0 4453060003000.20100.5 5362360003000.20100.5 6271260003000.20100.5 7181060003000.40100.5 8781680004000.20100.5 9672580004000.20100.0 1056348000.24000.20100.0 115124060003000.20101.0 126114560003000.20101.0 137105660003000.20100.0 14896060003000.20100.5 159106780004000.20101.0 1610115880004000.20100.0 1711124980004000.20— 181213906000.153000.20— 1911148960003000.20— 2010157860003000.20— 219167060003000.20— 2215167080004000.20— 2314158080004000.30— 241313908000.154000.40—
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corresponding input point pressure heads h0(S(n)) for n ¼ 1 to nLfor analysis purposes are given in Table 3.2.
3.8.3. Loop Data
The pipe link i can be the part of two loops K1(i) and K2(i). In case of a branched pipe configuration, K1(i) and K2(i) are zero. However, the description of loops is not indepen-dent information and can be generated from pipe – node connectivity data.
3.8.4. Node – Pipe Connectivity
There are Np( j ) number of pipe links meeting at the node j. These pipe links are num-bered as Ip( j, ‘) with ‘ varying from 1 to Np( j ). Scanning Table 3.1, the node pipe con-nectivity data can be formed. For example, pipes 6, 8, 9, and 13 are connected to node 7.
Thus, Np( j ¼ 7) ¼ 4 and pipe links are Ip(7,1) ¼ 6, Ip(7,2) ¼ 8, Ip(7,3) ¼ 9, and Ip(7,4) ¼ 13. The generated node – pipe connectivity data are given in Table 3.3.
TA B L E 3.2. Input Point Nodes and Input Heads Input Point Number
TA B L E 3.3. Node – Pipe Connectivity
Ip( j, ‘) ‘ ¼1 to Np( j)
3.8.5. Analysis
Analysis of a pipe network is essential to understand or evaluate a physical system. In case of a single-input system, the input discharge is equal to the sum of withdrawals.
The known parameters in a system are the input pressure heads and the nodal with-drawals. In the case of a multi-input network system, the system has to be analyzed to obtain input point discharges, pipe discharges, and nodal pressure heads. Walski (1995) indicated the numerous pipe sizing problems that are faced by practicing engin-eers. Similarly, there are many pipe network analysis problems faced by water engineers, and the analysis of a multi-input points water system is one of them. Rossman (2000) described the analysis method used in EPANet to estimate pipe flows for the given input point heads.
To analyze the network, the population served by pipe link i was distributed equally to both nodes at the ends of pipe i, J1(i), and J2(i). For pipes having one of their nodes as an input node, the complete population load of the pipe is transferred to another node.
Summing up the population served by the various half-pipes connected at a particular node, the total nodal population Pjis obtained. Multiplying Pjby the per-capita water demand w and peak discharge factor uP, the nodal withdrawals qjare obtained. If v is in liters per person per day and qj is in cubic meters per second, the results can be written as
qj¼ upvPj
86,400,000: (3:29)
The nodal water demand due to industrial and firefighting usage if any can be added to nodal demand. The nodal withdrawals are assumed to be positive and input discharges as negative. The total water demand of the system QTis
QT ¼jXLnL
j¼1
q( j): (3:30)
The most important aspect of multiple-input-points water distribution system analysis is to distribute QTamong all the input nodes S(n) such that the computed head h(S(n)) at input node is almost equal to given head h0(S(n)).
For starting the algorithm, initially total water demand is divided equally on all the input nodes as
QTn¼QT
nL
: (3:31)
In a looped network, the pipe discharges are derived using loop head-loss relationships for known pipe sizes and nodal linear continuity equations for known nodal withdrawals.
A number of methods are available to analyze such systems as described in this chapter.
Assuming an arbitrary pipe discharge in one of the pipes of all the loops and using
3.8. MULTI-INPUT SOURCE WATER NETWORK ANALYSIS 71
continuity equation, the pipe discharges are calculated. The discharges in loop pipes are corrected using the Hardy Cross method, however, any other analysis method can also be used. To apply nodal continuity equation, a sign convention for pipe flows is assumed that a positive discharge in a pipe flows from a lower-magnitude node to a higher-magnitude node.
The head loss in the pipes is calculated using Eqs. (2.3b) and (2.7b):
hfi¼8fiLiQ2i p2gD5i þ kfi
8Q2i
p2gD4i , (3:32)
where hfiis the head loss in the ith link in which discharge Qiflows, g is gravitational acceleration, kfiis form-loss coefficient for valves and fittings, and fi is a coefficient of surface resistance. The friction factor fican be calculated using Eq. (2.6a).
Thus, the computed pressure heads of all the nodes can be calculated with reference to an input node of maximum piezometric head (input point at node 13 in this case). The calculated pressure head at other input point nodes will depend upon the correct division of input point discharges. The input point discharges are modified until the computed pressure heads at input points other than the reference input point are equal to the given input point heads h0(n).
A discharge correction DQ, which is initially taken equal to 0.05QT/nL, is applied at all the point nodes discharges, other than that of highest piezometric head input node.
The correction is subtractive if h(S(n)) . h0(n) and it is additive otherwise. The input discharge of highest piezometric head input node is obtained by continuity consider-ations. The process of discharge correction and network analysis is repeated until the
error¼jh0(n) h(S(n))j
h0(n) 0:01 for all values of n (input points): (3:33) The designer can select any other suitable value of minimum error for input head correc-tion. The next DQ is modified as half of the previous iteration to safeguard against any repetition of input point discharge values in alternative iterations. If such a repetition is not prevented, Eq. (3.33) will never be satisfied and the algorithm will never terminate.
The water distribution network as shown in Fig. 3.14 was analyzed using the described algorithm. The rate of water supply 300 liters per person per day and a peak factor of 2.0 were adopted for the analysis. The final input point discharges obtained are given in Table 3.4.
TA B L E 3.4. Input Point Discharges Input Point Input Point Node
Input Point Head
The variation of computed input point head with analysis iterations is shown in Fig. 3.15. The constant head line for input point 2 indicates the reference point head.
Similarly, the variation of input point discharges with analysis iterations is shown in Fig. 3.16. It can be seen that input discharge at input point 2 (node 13) is higher due to higher piezometric head meaning thereby that it will supply flows to a larger popu-lation than the input node of lower piezometric head (input point 1).
The computed pipe discharges are given in Table 3.5. The sum of discharges in pipes 1 and 7 is equal to discharge of source node 1, and similarly the sum of discharges in pipes 18 and 24 is equal to the discharge of source node 2. The negative discharge in pipes indicates that the flow is from a higher-magnitude node to a lower-magnitude node of the pipe. For example, discharge in pipe 4 is 20.003 meaning thereby that the flow in the pipe is from pipe node number 5 to node number 4.
Figure 3.15.Variation of computed input heads with analysis iterations.
Figure 3.16. Variation of computed input discharges with analysis iterations.
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