Analysis of water quality characteristic for water distribution systems

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 ﬁ rst-order chlorine decay model was proposed to assess the ef ﬁ ciency 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 de ﬁ nite. The proposed method was applied to two examples of WDS and sensitivity analysis regarding chlorine decay coef ﬁ cient ( k 0 ) was discussed. The results indicated that values of CS for nodes in WDS are affected by the in ﬂ ow of nodes, which is determined by water demand and pipe length from water sources to nodes. In addition, the value of CS increases with k 0 when the in ﬂ ow of the node is larger than the optimized in ﬂ ow. The results veri ﬁ ed 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.


INTRODUCTION
Water quality in water distribution systems (WDS) is currently of widespread concern, which is influenced by a number of factors including the water age, water storage facility, and disinfectant methods (Mau et al. ). Chlorine is the most widely used disinfectant for preventing finished water from regrowth of microbial pathogens (Kim et al. ).
Most of the chlorine dosed is consumed in reactions with other substances remaining in the water after treatment, particularly dissolved organic matter (DOM) (Fisher et al. ).
Therefore, the concentration of residual chlorine is required to be kept at a certain level, especially in the extremities of WDS (Li et al. ; Blokker et al. ). The most widely used chlorine decay model in WDS is the first-order decay model, expressed by Equation (1) as follows: where C is the concentration of chlorine, t refers to time,  (Kansal & Kumar ). 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  (2) and (3) from the individual pipe shown in Figure 1. The quantity of decayed chlorine is termed m, and the flow within the pipe is termed q.
where Q in is the inflow of the pipe (L/s), Q out is the outflow of the pipe (L/s), C in is the concentration of residual chlorine at the inlet of the pipe (mg/L), C out is the concentration of residual chlorine at the outlet of the pipe (mg/L), m in is the quantity of chlorine entering the pipe per second at the inlet (mg/s), and m out 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: where k 0 is the first-order kinetic decay coefficient (s À1 ), L and D are the length (m) and diameter (m) of the single pipe, respectively.
Obviously, we can obtain Equations (5) and (6) for an individual pipe (shown in Figure 1), expressed as follows: In Equation (4), L, D, k 0 , and m in are usually known; however, Q in varies depending on the event. Therefore, C out in Equation (4) can be expressed as a function of Q in expressed by C out (Q in ). The aim of this paper is to study the variation of C out , and try to find the variation pattern of C out with Q in under the condition that the injection quantity of chlorine is definite. The optimized value of Q in is to make the outlet chlorine concentration C out 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 @C out @Q in ¼ 0, we can deduce that C out gets its maximum value when Q in can satisfy the condition expressed by Equation (7), as follows: where (Q in ) opt is the optimized value of Q in .
Accordingly, C out reaches the maximum value, which can be expressed by Equation (8) as follows: where (C out ) max is the available maximum value of C out , corresponding to (Q in ) opt . From Equation (8) we can find out that (C out ) max has relationships with the pipe characteristics (length L and diameter D), chlorine decay constant k 0 , and the input chlorine quantity m in . Therefore (C out ) max is a definite value for a given pipe. Usually the actual flow in a pipe is different from (Q in ) opt , thus the actual chlorine concentration C out at the outlet is often different from (C out ) max .
Therefore, to assess the chlorine transfer efficiency, we can obtain Equation (9), expressed as follows: where k refers to the chlorine transfer efficiency.
By combining Equations (4) and (7), the coefficient k can also be expressed by Equation (10) as follows: In order to simplify Equation (10), parameter η was defined by Equation (11) as follows: In addition, η can also be expressed by Equation (12) as follows: Therefore, Equation (10) was transformed into Equation (13) as follows: The relationship curve between k and Q in (Q in ) opt i:e: 1 η is shown in Figure 2, where k is presented as a function of Q in (Q in ) opt or η. 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 k ¼ 1, C out reaches the maximum value (C out ) max , which means that the pipe works at the best condition from the viewpoint of water quality corresponding to value of Q in (Q in ) opt being 1.0 (also shown in Figure 2).
Since the coefficient k characterized the potentiality of chlorine concentration of the node, the concept of surplus chlorine factor (S) was proposed to represent the water quality characteristic of the node, expressed by Equation (14) as follows: where S refers to surplus chlorine factor (S), which is applied to assess the characteristic of the node in view of water quality. The scope of S is between 0 and 1, which is similar to k. If S decreases, water quality reliability of WDS will be improved. When S decreases to 0, C out reaches (C out ) max , which means WDS work with maximum chlorine concentration at the outlet.
From Figure 2, we can also find that when Q in decreases or increases from (Q in ) opt , C out always decreases. Combining Equations (13) and (14), we can also conclude that if η or varies in the range from 0 to 1.0, with the rise of , k increases and S decreases. On the contrary, , 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 η is bigger is in the scope of (0, 1.0), dS dη is greater than 0, which means that S decreases with the decrease of η . Similarly, when η is smaller than 1.0, is in the scope of (1.0, ∞), dS dη is smaller than 0, which means that S increases with the decrease of η or the Relationship between surplus chlorine factor (S) and Q in under different chlorine decay constant k 0 From Equations (10), (11), and (14), we can find parameters k and S vary with the decay coefficient k 0 for the individual pipe. The chlorine decay coefficient k 0 varies with flow velocity and temperature (Blokker et al. ). From Equation (8) we can draw the conclusion that (C out ) max always decreases with the increase of k 0 when the other factors remain the same, which means that chlorine decay coefficient k 0 affects the reachable maximum chlorine concentration negatively. In addition, with the increase of k 0 , the value of (Q in ) opt increases according to Equation (7), which means that water flow in the pipe will increase to meet the chlorine concentration of (C out ) max at the outlet.
With the increased chlorine decay coefficient of k 0 , the chlorine concentration at the outlet C out decreases. Therefore, the variation of k and S with k 0 depends on the relative increased degrees of C out and (C out ) max . The relationship curves between S and Q in under different values of k 0 are shown in Figure 3. The values of k 0 ranged from þ20% to À20%.
With the increase of Q in , S decreases from 1.0 to 0 initially, then increases with the rise of Q in . When the value of S is 0, Q in reaches the best value of (Q in ) opt . For a specific curve with certain k 0 , when Q in is less than (Q in ) opt , S decreases with the rise of Q in . On the contrary, if Q in is greater than (Q in ) opt for certain k 0 , S increases with the rise of Q in . The values of (Q in ) opt vary with k 0 . When k 0 increases, (Q in ) opt also increases, which is in accordance with Equation (7). Among all the values of (Q in ) opt , (Q in ) opt1 is the smallest, corresponding to 0.8k 0 , and (Q in ) opt2 is the biggest, corresponding to 1.2k 0 . From Figure 3 we can also find that when Q in is less than (Q in ) opt1 , the higher the chlorine decay coefficient k 0 is, and the greater S is for the same Q in . However, when Q in is greater than (Q in ) opt2 , the higher the chlorine decay coefficient k 0 is, and the smaller S is for the same Q in . The reason can be explained by the fact that when Q in is less than (Q in ) opt1 , η is always bigger than 1.0.
From Equation (15), we can obtain that dS dη is greater than 0. Therefore, if k 0 increases, then η increases correspondingly according to Equation (11), which leads to the rise of S that coincides with the curve in Figure 3. Moreover, when Q in is greater than (Q in ) opt2 , then η is always smaller than 1.0, which leads to dS dη being less than 0. Therefore, if k 0 increases, then η 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.
From Figure 4, we can obtain Equations (16)-(18), expressed as follows: where Q out is the flow of node n, Q in1 is the flow in pipe ① connecting with node n, Q in2 is the flow in pipe ② connecting with node n.
The value of CS for node n is calculated by Equation (19) as follows:  where CS is the comprehensive surplus chlorine factor for node n, S i (i ¼ 1, 2, …, K ) is the surplus chlorine factor for node n through the i th pipe, Q i (i ¼ 1, 2, …, K ) is the flow of the i th pipe connecting with node n, and K is the number of pipes connected with node n. 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 ). 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 k 0 was set to be À1.55/day (Rossman ).
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.    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 Q in decreases accordingly, which leads to direct increase of C in under the condition that chlorine injection m in is fixed. The outlet concentration C out increases with C in , 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 Q in 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 Q in for all nodes are greater than (Q in ) opt obtained by Equation (7), and η is always less than 1.0, but Q in for node 23 is closer to (Q in ) opt , since node 23 is the terminal node of the pipe network. Therefore, η 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 (Q in ) opt , the smaller CS is for the node.

Sensitivity analysis
Since chlorine decay coefficient k 0 is of significant importance in WDS, the sensitivity analysis of k 0 to CS is performed. The decay coefficient k 0 is set to vary from À20% to þ20% compared with k 0 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 k 0 . Although node 12 and node 21 do not comply with the conclusion completely, the variation trends of CS with k 0 are the same as other nodes. Based on Figure 3, the relationship between S and k 0 depends on Q in for individual pipes.
According to the analysis above, that values of Q in for all nodes are greater than (Q in ) opt , S decreases with the rise of 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 k 0 , the hourly variation of CS corresponding to k 0 is the same as other nodes; that is, CS decreases with the increase of k 0 , which is in accordance with the case shown in Figure 3 when Q in 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 k 0 , 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 (Q in ) opt , which leads to the increase of CS with k 0 . The conclusion is also is in accordance with the case shown in Figure 3, when Q in is less than(Q in ) opt ; that is, the higher the chlorine decay coefficient k 0 , 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 Q in at node 23 is smaller than Q in at node 21, and is closer to (Q in ) opt , 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 k 0 corresponding to peak hour (8th hour) and valley hour (19th hour) were analyzed and are shown in Figure 8(a) and 8(b).
From Figure 8(a), we can conclude that at peak hour (8th hour), values of CS for all nodes decrease with the increase of k 0 , which is in accordance with the results above. Moreover, under various k 0 , 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 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 )). 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 (), and thus are not repeated here). The coefficient of chlorine decay was set to be À1.55/day.
The results of CS for typical nodes in Example 2 are shown in Table 4.
For nodes at the extremities of the WDS in Example 2, the average values of CS were lower than other nodes, which are  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 η 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 η 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 Q in (Q in ) opt , and the variation under different k 0 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 k 0 . The conclusions are in accordance with the results obtained from a single pipe when Q in is greater than(Q in ) opt . When the inflow at the node which is further from water sources becomes lower than (Q in ) opt at the valley hour, the value of CS increases with the increase of chlorine decay coefficient k 0 . 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.