The water retention time in the water distribution network is an important indicator for water quality. The water age fluctuates with the system demand. The residual chlorine concentration varies with the water age. In general, the concentration of residual chlorine is linearly dependent on the water demand. A novel statistical model using monitoring data of residual chlorine to estimate the nodal water age in water distribution networks is put forward in the present paper. A simplified two-step procedure is proposed to solve this statistical model. It is verified by two virtual systems and a practical application to analyze the water distribution system of Hangzhou city, China. The results agree well with that from EPANET. The model provides a low-cost and reliable solution to evaluate the water retention time.

Water quality will deteriorate with the increment of retention time in the water distribution system, leading to malfunctions such as disinfection by-product formation, disinfectant decay, corrosion, taste, and odor. Water age is very important for the water quality of water distribution system. The water age primarily depends on the water distribution system design and its demands. Although Brandt et al. [

There are two types of tools to estimate the water age: tracer studies and numerical models. Tracer studies involve injecting chemical into the water distribution system for a fixed period, and sensors are set up at downstream nodes to determine the duration before the water containing the chemicals passes the monitoring stations. This method has been applied to calculate the water age throughout the water distribution system and calibrate the water quality and hydraulic models [

You et al. [

A water distribution system consists of pipes, pumps, valves, fittings, and storage facilities that are used to convey water from the source to consumers. The dissolved substance travels along the pipe with the same average velocity as the carrier fluid while reacting (either growing or decaying) at certain rates. The equations governing the water quality are based on the principle of conservation of mass coupled with reaction kinetics [

When junctions receive inflow from two or more pipes, it is assumed that the complete mixing of fluid is accomplished simultaneously. Thus, the concentration of a substance in water when water leaves the junction is simply the flow-weighted sum of the concentrations from the inflowing pipes. For a specific node ^{3}/s) in pipe

Although more complicated models are available for modeling the decay of chlorine (e.g., [

Traveling along with the water trace line, (

The solution of (

Water travels from the water station to the consumer through many pipes. In most skeletal pipes, the influence of mixing is neglected. The solution of (

The water age at node

The residual chlorine concentration varies with the water age. The variation of residual chlorine can show the fluctuations of water age. The first-order expansion of (

Assume that the average velocity is linearly dependent on the water demand in water distribution systems. Thus,

The above derivation shows that the concentration of residual chlorine at node

Assuming that the standard deviations of

Equation (

Assuming

Equation (

Because the distance from the source to the monitor node

Since the monitored data from SCADA is discrete, the above model is transformed to a discrete model. The sampling period is

According to (

In order to verify the proposed model, two virtual water distribution systems, namely, the simplest system consisting of one pipeline and a complicated multisource water system, are modeled. They are also modeled using

The simplest system consisting of one pipe is shown in Figure

The correlation coefficient between the residual chlorine concentration and the mean demand is shown in Figures

Scenario

Water demand pattern factors for Scenario

Objective function of Scenario

Objective function of Scenario

Water age of Scenario

Water age of Scenario

A multisource system is shown in Figure

It is assumed that the residual chlorine concentration at two sources is 2.0 mg/L and the decay coefficient is 2.0 mg/(L

Water distribution network for case 2.

Water age at node 211.

Water age at node 193.

Hangzhou city is the capital city of Zhejiang province, located at the east of China. Its water distribution system consists of 2,639 km pipelines and 5 sources as shown in Figure ^{6} m^{3} of water per day. Figure

Water distribution network in Hangzhou.

5 monitors (S1, S2, S3, S4, and S5) are set up at water stations. 2 monitors (M7 and M8) are around the division line of different sources as node 193 in Scenario

Figure

Residual chlorine concentration at water stations and monitoring points, the total water demand. Red solid lines are filtered data.

Objective function for different water age.

The statistical model always needs more monitoring samples. The decay coefficient of the residual chlorine varies with the water temperature. The decay coefficient is different with seasons. Another serious problem is the drift error of chlorine sensors. The drift error appears in the long-term working chlorine sensor, and it becomes serious gradually. Calibration will make the monitoring serial discontinue. The more the monitoring samples are, the more the unpredicted factors affect the model. We should select the sample when the water temperature and sensors are stable. It is believed that 15 days’ to 30 days’ samples are enough for this statistical model. In the present paper, 15 days’ samples are used to estimate the water age. The sample period

Water age at monitors.

Monitoring point | M1 | M3 | M4 | M5 | M6 |
---|---|---|---|---|---|

Source | S1 | S3 | S5 | S5 | S2 |

Objective function | 0.78 | 0.64 | 0.95 | 0.90 | 0.82 |

Average water age (statistic model) | 9.5 | 20.75 | 7.5 | 10.5 | 7.5 |

Water age (EPANET) | 6~8 | 14~23 | 5~10 | 8~13 | 5~11 |

Water age at monitoring points.

Residual chlorine concentration at M2, S3, and M3. Solid line is at S3, dashed line is at M2, and doted line is at M3.

The fluctuation range of the water age estimated by EPANET is lager than that of the present model. The fluctuation range in the present model is about

Table

Besides EPANET model, another method is utilized to verify the model. If the monitor is very close to the sources, the residual chlorine concentration at the monitoring point will fluctuate with that of sources. We can determinate the water age directly through both monitoring data sequence of the monitoring point and the source. For example, if the residual chlorine rises to a peak value of 2 mg/L at 9 o’clock, we may find that the residual chlorine concentration at the downstream monitoring point rises to a peak value of 1.5 at 10 o’clock. The water age at the monitoring point is 1 hour, and the decay coefficient can be directly calculated according to (

Regarding other tools to estimate the water age summarized by Brandt et al. [

A statistical model to estimate the water age of water distribution networks has been proposed in the present paper. The model is based on the solution of the advection transport equation governing the residual chlorine. A simple two-step solution procedure for the model has been given out. The model was tested by the simplest one-pipeline system and a multisource system. The numerical test indicates that the model agrees well with the EPANET model, if monitoring points are not around the water division line. The model is also applied to the water distribution system in Hangzhou city. The results agree well with those estimated by using EPANET model. The comparison of the decay coefficients estimated by another methodproves that the result obtained by the statistical model is reliable. This method does not need complicated calibration as numerical models and also does not need high operation cost as tracer studies. If the water distribution system has SCADA to monitor the residual chlorine concentration, this statistical model will be a simple and effective tool to estimate the system’s water age without additional costs.

Concentration (mass/volume) in source

Concentration (mass/volume) at node

Average concentration (mass/volume) at node

Concentration (mass/volume) at time

Concentration (mass/volume) of the external flow entering at node

Concentration (mass/volume) of pipe at the start of node

Concentration (mass/volume) in pipe

Concentration (mass/volume) of pipe

Decay coefficient at pipe

Average decay coefficient (1/time)

Length from source to node

Number of pipes through which water travels from source to node

Number of sampling periods at time

Average number of sampling periods at node

Flow (volume/time) in pipe

External source flow entering the network at node

Averaged value of

Average total water demand in whole system during water age of node

Water age

Water age at node

Rate of reaction (mass/volume/time), function of concentration

Average velocity from source to node

Averaged velocity for all time

Flow velocity (length/time) in pipe

Set of pipes with flow into node

Standard deviation of

Standard deviation of

Sampling period.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The present research is funded by National Natural Science Foundation of China (nos. 51578486 and 51378455) and the Key Special Program on the S&T of China for the Pollution Control and Treatment of Water Bodies (2012ZX07408-002-003 and 2012ZX07403-003). The authors would like to thank Dr. J. Zhang for providing some document about water age. Special thanks are due to Dr. Z. H. Yang and Dr. Z. H. Zhang for review of the paper.