Abstract
Since governments all over the world are paying more attention to water quality in water distribution systems (WDS), a method based on mass balance and first-order chlorine decay model was proposed to assess the efficiency of WDS involving water quality (represented by residual chlorine). The concepts of surplus chlorine factor (S) for nodes in individual pipes and comprehensive surplus chlorine factor (CS) for nodes in WDS were put forward to represent the water quality characteristic of nodes in WDS based on the assumption that the structure of the pipe network and quantity of chlorine dose are definite. The proposed method was applied to two examples of WDS and sensitivity analysis regarding chlorine decay coefficient (k0) was discussed. The results indicated that values of CS for nodes in WDS are affected by the inflow of nodes, which is determined by water demand and pipe length from water sources to nodes. In addition, the value of CS increases with k0 when the inflow of the node is larger than the optimized inflow. The results verified that the deduction of S for a single pipe can be generalized to WDS, and can measure the water quality characteristics for nodes in WDS easily.
NOMENCLATURE
- WDS
Water distribution system
- DOM
Dissolved organic matter
- m
Chlorine decay quantity
- q
Flow within the pipe
- Qin
Pipe inflow
- Qout
Pipe outflow
- Cin
Residual chlorine concentration at the inlet
- Cout
Residual chlorine concentration at the outlet
- min
Chlorine entering the pipe at the inlet
- mout
Chlorine leaving the pipe at the outlet
- k0
First-order kinetic decay coefficient
- L
Pipe length
- D
Pipe diameter
- (Cout)max
Maximum chlorine concentration at the outlet
- k
Chlorine transfer efficiency
- (Qin)opt
Optimized pipe inflow
- S
Surplus chlorine factor
- CS
Comprehensive surplus chlorine factor
INTRODUCTION
stands for the chlorine decay coefficient (Hallam et al. 2002; Al-Jasser 2007; Fisher et al. 2011; Kim et al. 2014). In this paper, water quality characteristics of nodes in WDS based on the first-order chlorine decay equation mentioned above were analyzed.The hydraulic characteristic and mechanical reliability of WDS have been much studied in the literature (Vaabel et al. 2006; Tanyimboh & Templeman 2007; Wu et al. 2011; D'Ercole et al. 2018). Mechanical reliability was defined as the probability that a component (new or repaired) experiences no structural failures (Kansal & Kumar 1995). The hydraulic reliability refers to the probability that a water distribution pipe can meet a required water flow level at a required pressure at each nodal demand (Ostfeld 2001). In order to increase the hydraulic capacity of a network and overcome sudden failures in WDS, the concept of hydraulic power capacity with consideration of both flow and pressure was identified (Vaabel et al. 2006). The hydraulic characteristic of nodes represented by the concept of surplus power factor based on hydraulic power was defined and applied to measure network resilience (Wu et al. 2011). However, both flow and water quality are of equal importance for WDS. The capacity concerning water quality for WDS has not been researched sufficiently in past studies (Gupta et al. 2012). Models have been developed of the chlorine concentrations at nodes in WDS with consideration of chlorine decay (Boulos et al. 1995; Mau et al. 1996; Hallam et al. 2002). The fraction of delivered quality (FDQ) was expressed to influence the water quality reliability, which is the ratio of simulation runs of supplied concentration below the threshold concentration to all the simulation runs (Ostfeld et al. 2002). Similarly, the ratio of days that the residual chlorine fulfills the residual chlorine standards to simulated days was proposed to represent water quality reliability (Zhao et al. 2010). The concept of node chlorine availability was proposed to define the water quality reliability of WDS (Li et al. 2013). The optimal operations of booster stations were proposed with the objective of minimizing the chlorine injection quantity (Tryby et al. 2002; Ostfeld & Salomons 2006; Kang & Lansey 2010). However, from the aspect of water quality management, the chlorine injection to WDS often remains fixed with no relationship to water demand. Under such circumstances, the water quality characteristics of nodes are affected by many factors, such as distance from water source to nodes, diameters and flows of pipes connecting with nodes, water demand of nodes, and chlorine decay coefficient, etc. The contribution of this paper is to research the degree of effect of various factors on water quality characteristics of nodes in WDS.
In this paper, first, based on the concepts of surplus chlorine factor (S) for nodes in single pipes deduced from a mass balance and first-order chlorine decay model, the concept of comprehensive surplus chlorine factor (CS) for nodes in WDS is proposed to analyze water quality characteristic of nodes in WDS. Second, CS was applied to measure the water quality characteristics in two examples of WDS based on an EPANET hydraulic and water quality extended simulation, and a sensitivity analysis of k0 to CS is presented. Finally, values of CS of nodes in WDS were measured and compared, and the factors affecting CS are indicated and discussed.
METHODOLOGY
Surplus chlorine factor (S) for individual pipe
is the quantity of chlorine entering the pipe per second at the inlet (mg/s), and 
is the quantity of chlorine leaving the pipe per second at the outlet (mg/s). Suppose residual chlorine decay follows the first-order kinetic reaction equation, the concentration of residual chlorine at the outlet is expressed by Equation (4) as follows: 
is the first-order kinetic decay coefficient (s−1), 
and 
are the length (m) and diameter (m) of the single pipe, respectively.
Flows, concentration and chlorine loss for individual pipe.
, and 
are usually known; however, 
varies depending on the event. Therefore, 
in Equation (4) can be expressed as a function of 
expressed by 
. The aim of this paper is to study the variation of 
, and try to find the variation pattern of 
with 
under the condition that the injection quantity of chlorine is definite. The optimized value of 
is to make the outlet chlorine concentration 
reach the maximum value so as to get the water quality at the best operational condition. Therefore, according to Equation (4), with the assumption that 
, we can deduce that 
gets its maximum value when 
can satisfy the condition expressed by Equation (7), as follows: 
is the optimized value of 
.
reaches the maximum value, which can be expressed by Equation (8) as follows: 
is the available maximum value of 
, corresponding to 
. From Equation (8) we can find out that 
has relationships with the pipe characteristics (length L and diameter D), chlorine decay constant k0, and the input chlorine quantity min. Therefore 
is a definite value for a given pipe. Usually the actual flow in a pipe is different from 
, thus the actual chlorine concentration 
at the outlet is often different from 
. Therefore, to assess the chlorine transfer efficiency, we can obtain Equation (9), expressed as follows: 
The relationship curve between k and 
is shown in Figure 2, where k is presented as a function of 
or 
. The scope of k is between 0 and 1.0. In effect, the water quality characteristics of WDS get better when the chlorine concentration at the outlet is higher. If 
, 
reaches the maximum value 
, which means that the pipe works at the best condition from the viewpoint of water quality corresponding to value of 
being 1.0 (also shown in Figure 2).
Relationship curves of surplus chlorine factor (S) and chlorine transfer coefficient (k) with 
.
reaches 
, which means WDS work with maximum chlorine concentration at the outlet.
decreases or increases from 
, 
always decreases. Combining Equations (13) and (14), we can also conclude that if 
or 
is fixed, k and S remain unchangeable. When 
varies in the range from 0 to 1.0, with the rise of 
, k increases and S decreases. On the contrary, when 
varies in the range from 1.0 to ∞, with the rise of 
, k decreases and S increases. The reason can be explained as follows. From Equation (14), we can deduce Equation (15), expressed as follows: 
From Equation (15), we can find that when 
is bigger than 1.0, i.e. 
is in the scope of (0, 1.0), 
is greater than 0, which means that S decreases with the decrease of 
or the rise of 
. Similarly, when 
is smaller than 1.0, i.e. 
is in the scope of (1.0, ∞), 
is smaller than 0, which means that S increases with the decrease of 
or the rise of 
.
Relationship between surplus chlorine factor (S) and Qin under different chlorine decay constant k0
From Equations (10), (11), and (14), we can find parameters k and S vary with the decay coefficient k0 for the individual pipe. The chlorine decay coefficient k0 varies with flow velocity and temperature (Blokker et al. 2014). From Equation (8) we can draw the conclusion that 
always decreases with the increase of k0 when the other factors remain the same, which means that chlorine decay coefficient k0 affects the reachable maximum chlorine concentration negatively. In addition, with the increase of k0, the value of 
increases according to Equation (7), which means that water flow in the pipe will increase to meet the chlorine concentration of 
at the outlet. With the increased chlorine decay coefficient of k0, the chlorine concentration at the outlet 
decreases. Therefore, the variation of k and S with k0 depends on the relative increased degrees of 
and 
. The relationship curves between S and Qin under different values of k0 are shown in Figure 3. The values of k0 ranged from +20% to −20%.
Relationship curves between surplus chlorine factors (S) and Qin under different k0.
With the increase of 
, S decreases from 1.0 to 0 initially, then increases with the rise of 
. When the value of S is 0, 
reaches the best value of 
. For a specific curve with certain k0, when 
is less than 
, S decreases with the rise of 
. On the contrary, if 
is greater than 
for certain k0, S increases with the rise of 
. The values of 
vary with k0. When k0 increases, 
also increases, which is in accordance with Equation (7). Among all the values of 
, 
is the smallest, corresponding to 0.8k0, and 
is the biggest, corresponding to 1.2k0. From Figure 3 we can also find that when 
is less than 
, the higher the chlorine decay coefficient k0 is, and the greater S is for the same 
. However, when 
is greater than 
, the higher the chlorine decay coefficient k0 is, and the smaller S is for the same 
. The reason can be explained by the fact that when 
is less than 
, 
is always bigger than 1.0. From Equation (15), we can obtain that 
is greater than 0. Therefore, if k0 increases, then 
increases correspondingly according to Equation (11), which leads to the rise of S that coincides with the curve in Figure 3. Moreover, when 
is greater than 
, then 
is always smaller than 1.0, which leads to 
being less than 0. Therefore, if k0 increases, then 
increases correspondingly according to Equation (11), which leads to the decline of S that also coincides with the curve in Figure 3.
Comprehensive surplus chlorine factor (CS) for water distribution network
The consideration for the individual pipe can also be generalized for WDS. For a pipe net consisting of two pipes, shown in Figure 4, node n was connected with two single pipes.
Flows, concentration, and chlorine loss for a pipe net.
For a given WDS that consists of multiple pipes, water sources and nodes, CS for nodes varies with the inflow to nodes, length of pipes from water sources to nodes, diameters of pipes connecting with nodes, water demand of nodes, and chlorine decay coefficient k0, etc. For a certain node in WDS, the length of pipes from water sources to nodes and the diameters of pipes connecting with nodes are determined beforehand. However, the inflow 
of a node is variable depending on water demand by users, including residents and industrial enterprises. The chlorine decay coefficient k0 varies with the water quality, water temperature, and the pipe material. Therefore, values of CS for nodes in a given WDS usually vary with water demand and chlorine decay coefficient k0. For nodes in a WDS, the length of pipes from water sources to nodes and the diameters of pipes connecting with nodes are different. The values of CS for nodes vary with the differences in pipe lengths and pipe diameters.
RESULTS AND DISCUSSION
Example 1
Base run
The proposed methodology was demonstrated for a small illustrative system shown in Figure 5 (Rossman 1994). The initial residual chlorine of the water works was supposed to be 0.5 mg/L. During the hydraulic simulation progress of EPANET2, the Hazen–Williams function was used as the hydraulic simulation model, and the roughness coefficient was assumed to be 100. During the water quality simulation progress, the chlorine decay coefficient k0 was set to be −1.55/day (Rossman 1994).
The system consists of 12 pipes, a water source, a pump, and an elevated storage tank. The system is subject to a 24-h representative demand pattern shown in Table 1.
24-h demand pattern characteristics for example (Rossman 1994)
| Time of day | 1–2 | 3–4 | 5–6 | 7–8 | 9–10 | 11–12 | 13–14 | 15–16 | 17–18 | 19–20 | 21–22 | 23–24 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Multiplier of base demand | 1.0 | 1.2 | 1.4 | 1.6 | 1.4 | 1.2 | 1.0 | 0.8 | 0.6 | 0.4 | 0.6 | 0.8 |
| Time of day | 1–2 | 3–4 | 5–6 | 7–8 | 9–10 | 11–12 | 13–14 | 15–16 | 17–18 | 19–20 | 21–22 | 23–24 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Multiplier of base demand | 1.0 | 1.2 | 1.4 | 1.6 | 1.4 | 1.2 | 1.0 | 0.8 | 0.6 | 0.4 | 0.6 | 0.8 |
The pump is fed by a source that is at a constant water level of 243.8 m. The water is delivered to a storage elevated tank at node 2 (at a ground level of 259.1 m), and to eight consumers located at nodes 11, 12, 13, 21, 22, 23, 31, and 32. The pump has a shutoff head value of 101.3 m and a maximum flow rate of 189.3 L/s. The tank is cylindrical with a diameter of 15.4 m. Its minimum, initial, and maximum levels above ground are 30.5 m, 36.6 m, and 45.7 m, respectively. The results of the base run for the case study are shown in Figure 6 and Table 2.
Average comprehensive surplus chlorine factors (CS) in 24 h for nodes in Figure 5
| Nodes | 11 | 12 | 13 | 21 | 22 | 23 | 31 | 32 |
|---|---|---|---|---|---|---|---|---|
| Average values of CS | 0.55 | 0.71 | 0.49 | 0.68 | 0.55 | 0.25 | 0.59 | 0.47 |
| Nodes | 11 | 12 | 13 | 21 | 22 | 23 | 31 | 32 |
|---|---|---|---|---|---|---|---|---|
| Average values of CS | 0.55 | 0.71 | 0.49 | 0.68 | 0.55 | 0.25 | 0.59 | 0.47 |
Variation curves of time multiplier and comprehensive surplus chlorine factor (CS).
The variation curves of the time multiplier and comprehensive surplus chlorine factors (CS) for all nodes in 24 h calculated by Equation (19) are shown in Figure 6. We can also find that for almost all nodes besides node 13, values of CS were the smallest at the 19th–20th hour when water demand is also the least, which means that values of CS for nodes vary with characteristics of the demand pattern for these nodes. It can be explained that when water demand decreases, the flow 
decreases accordingly, which leads to direct increase of 
under the condition that chlorine injection 
is fixed. The outlet concentration 
increases with 
, which leads to the increase of k and decrease of S. The condition is consistent with the case for a single pipe, shown in Figure 3, when 
is greater than 
.
For comparing water quality characteristics of nodes in WDS, the average values of CS for nodes in 24 h are shown in Table 2.
From Table 2, we can find that the average value of CS for node 23 is significantly less than other nodes. The reason for this is that values of 
for all nodes are greater than 
obtained by Equation (7), and 
is always less than 1.0, but 
for node 23 is closer to 
, since node 23 is the terminal node of the pipe network. Therefore, 
for node 23 is closer to 1.0 than other nodes according to Equation (11), which leads to a smaller value of CS. The conclusion is in accordance with the case shown in Figure 3 for a single pipe; that is, the closer the node inflow is to 
, the smaller CS is for the node.
Sensitivity analysis
Since chlorine decay coefficient k0 is of significant importance in WDS, the sensitivity analysis of k0 to CS is performed. The decay coefficient k0 is set to vary from −20% to +20% compared with k0 in the base run keeping other data the same as that in the base run. The results of average values of CS for all nodes are shown in Table 3. We can observe that values of CS for nodes 11, 13, 22, 23, 31, and 32 decrease with the increase of k0. Although node 12 and node 21 do not comply with the conclusion completely, the variation trends of CS with k0 are the same as other nodes. Based on Figure 3, the relationship between S and k0 depends on Qin for individual pipes. According to the analysis above, that values of 
for all nodes are greater than 
, S decreases with the rise of k0. Although a water distribution network is more complex than an individual pipe, the calculations of CS are on the basis of S of individual pipes. Therefore, the variation trend of average values of CS for all nodes with chlorine decay coefficient k0 can be explained by the conclusions resulting from individual pipes.
Average comprehensive surplus chlorine factor (CS) for all nodes in Example 1 (k0 varies from −20% to +20%)
| Nodes | Chlorine decay coefficient | ||||
|---|---|---|---|---|---|
| 0.8k0 | 0.9k0 | k0 | 1.1k0 | 1.2k0 | |
| 11 | 0.64 | 0.61 | 0.55 | 0.54 | 0.51 |
| 12 | 0.79 | 0.76 | 0.71 | 0.72 | 0.70 |
| 13 | 0.59 | 0.55 | 0.49 | 0.48 | 0.45 |
| 21 | 0.79 | 0.77 | 0.71 | 0.72 | 0.70 |
| 22 | 0.63 | 0.60 | 0.55 | 0.55 | 0.52 |
| 23 | 0.34 | 0.30 | 0.25 | 0.23 | 0.20 |
| 31 | 0.68 | 0.64 | 0.59 | 0.58 | 0.55 |
| 32 | 0.57 | 0.53 | 0.47 | 0.46 | 0.43 |
| Nodes | Chlorine decay coefficient | ||||
|---|---|---|---|---|---|
| 0.8k0 | 0.9k0 | k0 | 1.1k0 | 1.2k0 | |
| 11 | 0.64 | 0.61 | 0.55 | 0.54 | 0.51 |
| 12 | 0.79 | 0.76 | 0.71 | 0.72 | 0.70 |
| 13 | 0.59 | 0.55 | 0.49 | 0.48 | 0.45 |
| 21 | 0.79 | 0.77 | 0.71 | 0.72 | 0.70 |
| 22 | 0.63 | 0.60 | 0.55 | 0.55 | 0.52 |
| 23 | 0.34 | 0.30 | 0.25 | 0.23 | 0.20 |
| 31 | 0.68 | 0.64 | 0.59 | 0.58 | 0.55 |
| 32 | 0.57 | 0.53 | 0.47 | 0.46 | 0.43 |
In addition, the hourly variations of CS for node 21 and node 23 are shown in Figure 7(a) and 7(b).
Variation of comprehensive surplus chlorine factor (CS) for (a) node 21 under various k0 and (b) node 23 under various k0.
We can find from Figure 7(a) that although node 21 does not comply with the conclusion that average values of CS decrease with the increase of k0, the hourly variation of CS corresponding to k0 is the same as other nodes; that is, CS decreases with the increase of k0, which is in accordance with the case shown in Figure 3 when 
is greater than 
.
We can also find from Figure 7(b) that although average values of CS for node 23 follow the conclusion that average values of CS decrease with the increase of k0, the hourly variation of CS does not follow the same conclusion for the valley hour of 18 h–19 h. The reason is that the inflow of node 23 is smaller than other nodes, and when the water demand is lower, the inflow of node 23 becomes lower than 
, which leads to the increase of CS with k0. The conclusion is also is in accordance with the case shown in Figure 3, when 
is less than
; that is, the higher the chlorine decay coefficient k0, the bigger the water quality characteristic CS for the node.
Comparing results in Figure 7(a) and 7(b), hourly values of CS for node 21 are greater than hourly values of CS for node 23. The reason is that 
at node 23 is smaller than 
at node 21, and is closer to 
, which leads to the values of CS for node 23 being less than the values of CS for node 21, which means that the water quality characteristic of node 23 is better than node 21.
The variations of CS for all nodes under various k0 corresponding to peak hour (8th hour) and valley hour (19th hour) were analyzed and are shown in Figure 8(a) and 8(b).
Variation of comprehensive surplus chlorine factor (CS) at (a) peak hour (8th hour) for all nodes under various k0 and (b) valley hour (19th hour) for all nodes under various k0.
From Figure 8(a), we can conclude that at peak hour (8th hour), values of CS for all nodes decrease with the increase of k0, which is in accordance with the results above. Moreover, under various k0, values of CS decrease in the order of node 21 > node 12 > node 31> node 11 > node 13> node 22 > node 32> node 23, which means that at peak hour the water quality characteristic is best at node 23, and worst at node 21. However, from Figure 8(b) at the valley hour (19th hour), we can find that the variation trend of CS for almost all nodes decreases with the rise of k0 except for node 23, which is similar to the peak hour (8th hour). For node 23 at the valley hour (19th hour), the values of CS increased with the rise of k0, which means that the water quality characteristic of node 23 became worse with the increase of k0. The values of CS decreased in the order of node 21 > node 12> node 22 > node 31 > node 11 > node 32> node 13 > node 23.
Comparing results in Figure 8(a) and 8(b), values of CS at the peak hour (8th hour) are greater than at the valley hour (19th hour), which is in accordance with results obtained from Figure 6. When water demands at the valley hour are lower than water demands at the peak hour, the water quality characteristics of nodes are improved.
Example 2
In this case, the concept of CS for the node was applied to a real-life network shown in Figure 9 (Example 3 of EPANET software (Rossman 1994)). The system consists of two sources, one elevated tank, 117 pipes, 97 demand nodes, and two pumps (the complete data used were exactly as that of Example 3, in Rossman (1994), and thus are not repeated here). The coefficient of chlorine decay was set to be −1.55/day.
The results of CS for typical nodes in Example 2 are shown in Table 4.
Average comprehensive surplus chlorine factor (CS) for nodes in Example 2
| Nodes | Water demand of nodes (L/s) | Inflow of nodes (L/s) | Outflow of nodes (L/s) | Diameters of pipes connected with nodes (mm) | Average comprehensive surplus chlorine factor (CS) |
|---|---|---|---|---|---|
| 11 | 11.245 | 11.245 | 0.000 | 400, 400 | 0.848 |
| 15 | 11.983 | 34.687 | 22.704 | 300, 300, 300, 300 | 0.964 |
| 19 | 14.869 | 470.546 | 455.677 | 750, 750, 300, 300 | 0.969 |
| 27 | 0.497 | 8.890 | 8.393 | 200, 200 | 0.873 |
| 30 | 2.332 | 6.953 | 4.621 | 300, 300 | 0.662 |
| 35 | 4.372 | 384.523 | 380.151 | 750, 750 | 0.957 |
| 38 | 0.795 | 359.491 | 358.696 | 750, 750, 350 | 0.990 |
| 40 | 0.219 | 0.219 | 0.000 | 350 | 0.211 |
| 44 | 0.000 | 75.843 | 75.843 | 750, 600 | 0.998 |
| 49 | 0.000 | 5.069 | 5.069 | 200, 300 | 0.998 |
| 64 | 0.000 | 45.475 | 45.475 | 300, 300 | 0.996 |
| 70 | 3.488 | 3.488 | 0.000 | 350 | 0.698 |
| 73 | 1.391 | 1.391 | 0.000 | 300 | 0.520 |
| 86 | 0.000 | 36.040 | 36.040 | 300, 200, 300 | 0.976 |
| 91 | 0.000 | 31.084 | 31.084 | 300, 200, 300 | 0.972 |
| 92 | 0.000 | 30.837 | 30.837 | 300, 200, 300 | 0.996 |
| Nodes | Water demand of nodes (L/s) | Inflow of nodes (L/s) | Outflow of nodes (L/s) | Diameters of pipes connected with nodes (mm) | Average comprehensive surplus chlorine factor (CS) |
|---|---|---|---|---|---|
| 11 | 11.245 | 11.245 | 0.000 | 400, 400 | 0.848 |
| 15 | 11.983 | 34.687 | 22.704 | 300, 300, 300, 300 | 0.964 |
| 19 | 14.869 | 470.546 | 455.677 | 750, 750, 300, 300 | 0.969 |
| 27 | 0.497 | 8.890 | 8.393 | 200, 200 | 0.873 |
| 30 | 2.332 | 6.953 | 4.621 | 300, 300 | 0.662 |
| 35 | 4.372 | 384.523 | 380.151 | 750, 750 | 0.957 |
| 38 | 0.795 | 359.491 | 358.696 | 750, 750, 350 | 0.990 |
| 40 | 0.219 | 0.219 | 0.000 | 350 | 0.211 |
| 44 | 0.000 | 75.843 | 75.843 | 750, 600 | 0.998 |
| 49 | 0.000 | 5.069 | 5.069 | 200, 300 | 0.998 |
| 64 | 0.000 | 45.475 | 45.475 | 300, 300 | 0.996 |
| 70 | 3.488 | 3.488 | 0.000 | 350 | 0.698 |
| 73 | 1.391 | 1.391 | 0.000 | 300 | 0.520 |
| 86 | 0.000 | 36.040 | 36.040 | 300, 200, 300 | 0.976 |
| 91 | 0.000 | 31.084 | 31.084 | 300, 200, 300 | 0.972 |
| 92 | 0.000 | 30.837 | 30.837 | 300, 200, 300 | 0.996 |
For nodes at the extremities of the WDS in Example 2, the average values of CS were lower than other nodes, which are 0.662, 0.221, 0.698, and 0.52 for node 30, 40, 70, and 73, respectively. The reason is that the inflows of nodes 30, 40, 70, and 73 are 6.953, 0.219, 3.488, and 1.391 L/s, respectively, lower than other nodes, which leads to the decrease of CS, and improvement of the water quality characteristic of nodes. Moreover, the smaller inflow of nodes leads to a lower value of CS. For example, the inflows of the four nodes increase in the order of node 40 < node 73 < node 30 < node 70. Accordingly, the values of CS for the four nodes increase in the same order. Although the inflow of node 27 is less than node 11, the pipe diameter connecting to node 27 is smaller than node 11, which leads to the value of CS at node 27 being larger than the value of CS at node 11. The reason for this is that 
is affected by pipe diameter, pipe length from water sources to node, and inflow of nodes together, which is shown in Equation (11). The value of 
determines the water quality characteristic of nodes expressed by CS.
CONCLUSION
In this paper, the concept of surplus chlorine factor (S) for a node in a single pipe based on the mass balance and first-order chlorine decay model was deduced. The relationship between S and 
, and the variation under different k0 for a single pipe were revealed. In addition, comprehensive surplus chlorine factor (CS) for nodes in WDS based on S was put forward, and applied to two examples of WDS during 24 h in a day (shown in Figures 5 and 9). The results indicated that the value of CS decreases with the decrease of inflow at the node, which is caused by lower water demand and longer pipe length from water sources to node. In addition, average values of CS decrease with the increase of chlorine decay coefficient k0. The conclusions are in accordance with the results obtained from a single pipe when 
is greater than
. When the inflow at the node which is further from water sources becomes lower than 
at the valley hour, the value of CS increases with the increase of chlorine decay coefficient k0. In addition, the diameters of pipes connecting with nodes also affect the water quality characteristics of nodes; that is, the larger the pipe diameter, the better the water quality characteristics of nodes. The proposed method was proved to be efficient and easy to use for analyzing the water quality characteristic of nodes in WDS.
ACKNOWLEDGEMENTS
This work was funded by the Special S & T Project on Treatment and Control of Water Pollution from Bureau of Housing and Urban-rural Development of Jiangsu Province numbered as 2014ZX07405002.
