Research Article A Novel Statistical Model for Water Age

Research ArticleA Novel Statistical Model for Water Age Estimation in WaterDistribution Networks

Wei-ping Cheng En-hua Liu and Jing-qing Liu

College of Architecture and Civil Engineering Zhejiang University 866 Yuhantan Street Hangzhou 310058 China

Correspondence should be addressed to Jing-qing Liu liujingqingzjueducn

Received 3 May 2015 Revised 9 August 2015 Accepted 18 August 2015

Academic Editor Jian Guo Zhou

Copyright copy 2015 Wei-ping Cheng et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

Water quality will deteriorate with the increment of retentiontime in the water distribution system leading to malfunc-tions such as disinfection by-product formation disinfectantdecay corrosion taste and odor Water age is very importantfor the water quality of water distribution system The waterage primarily depends on the water distribution systemdesign and its demands Although Brandt et al [1] reviewedsome tools to estimate the retention time and several exam-ples presented they conclude that there are no low-costeffective and reliableways to estimate it in any circumstancesIn some circumstances these tools may be appropriate butthis is not always the case

There are two types of tools to estimate the water agetracer studies and numerical models Tracer studies involveinjecting chemical into the water distribution system for afixed period and sensors are set up at downstream nodesto determine the duration before the water containing thechemicals passes the monitoring stations This method hasbeen applied to calculate the water age throughout thewater distribution system and calibrate the water quality andhydraulic models [2ndash4] The tracer study is useful in validat-ing hydraulic and water quality models However it is seldomapplied in water distribution networks for its disadvantage

Some reports [1 5] have shown its disadvantages that is thetracer chemical stability continuous regulatory compliancecustomer perceptions lack of studies on the larger distribu-tion systems and high operational cost Numerical modelsgive the other way to estimate the water age in water distribu-tion systemsThe steady traveling timemodels were proposedbyMales et al [6]These models were subsequently extendedto dynamical representations that determine varying waterage throughout the distribution systems [7] A simplifiedmodel of water age in tanks and reservoirs was developedin the early 1990s [8] Many hydraulic network modelingpackages incorporate certain algorithms to calculate thewater age at any node in the network [9ndash14] Water quality isdirectly related to water distribution system operation condi-tionsThus a careful hydraulic calibration is necessary undervarying demand assumed for the accurate estimation ofwater age Unfortunately numerical models may have somelimitations in the capability of accurately predicting the waterage for the following reasons [1 5] (1) skeletonization theskeletonization is necessary if the water distribution systemcontains more pipe segments than the model can handle andin almost all cases the skeletonization is inevitableThe effectof skeletonization on the accuracy of water age estimationdiffers from system to system (2) insufficient calibration inmost cases the roughness cannot be estimated accurately If

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 350328 9 pageshttpdxdoiorg1011552015350328

2 Mathematical Problems in Engineering

the overall demand is miscalculated it will result in more orless source and reservoir operation thanwhat actually occursMisestimating the demand allocation might lead to errorflow direction (3) water storage tanks tanks are modeled ascompletely mixed reactors in most models which will lead tomisestimation of the water age

You et al [15] have shown that the residual chlorinedecay ratio per unit length is different at different time Forexample it is 0175mg(LsdotKm) at peak-demand time that is0459mg(LsdotKm) at theminimum-demand time in Shenzhencity You et al [15] said that the retention time is one of themost important elements of the residual chlorine fluctuationfor long distance water distribution systems Our monitoringdata in Hangzhou city is the same as theirs Some waterdistribution systems have SCADA (supervisory control anddata acquisition) The concentration of residual chlorine inthe water system can bemonitored by SCADA In the presentpaper a novel statistic model is proposed to estimate thewater age in water distribution systems according to themonitoring data serials of the residual chlorine concentrationfrom SCADA The model is discussed theoretically andnumerically And it is also applied to predict the water ageof water distribution system in Hangzhou city The statisticmodel is in good agreement with EPANET 20 numericalresults

2 Governing Equations for Water Quality

A water distribution system consists of pipes pumps valvesfittings and storage facilities that are used to convey waterfrom the source to consumers The dissolved substance trav-els along the pipe with the same average velocity as the carrierfluid while reacting (either growing or decaying) at certainratesThe equations governing the water quality are based onthe principle of conservation of mass coupled with reactionkinetics [10ndash12] Usually the role of longitudinal dispersionis neglectable The conservation of mass during transportwithin a pipe is described by the classical one-dimensionaladvection-reaction equation The advection transport withina pipe 119895 is represented as

120597119862119895

120597119905

= minus119881119895

120597119862119895

120597119909

minus 119877 (119862119895)

(1)

where 119862119895 is the concentration (massvolume) in pipe 119895 asa function of distance 119909 and time 119905 119881119895 is the flow velocity(ms) in pipe 119895 and 119877(119862119895) denotes the rate of reaction(massvolumetime) as a function of concentration

When junctions receive inflow from two or more pipes itis assumed that the complete mixing of fluid is accomplishedsimultaneously Thus the concentration of a substance inwater when water leaves the junction is simply the flow-weighted sum of the concentrations from the inflowing pipesFor a specific node 119894 the concentration is expressed as follows

1198621198941003816

1003816

1003816

1003816119909=0=

sum

Ω119894119895=1 119902119895 119862119895

1003816

1003816

1003816

1003816

1003816119909=119871119895+ 119902119894ext119862119894ext

sum

Ω119894119895=1 119902119895 + 119902119894ext

(2)

where Ω119894 is the set of pipes with flow into node 119894 119902119895 is theflow (m3s) in pipe 119895 119902119894ext is the external source flow enteringthe network at node 119894 and 119862119894ext is the concentration of theexternal flow entering at node 119894 the notation 119862119894|119909=0 denotesthe concentration at the start of node 119894 while 119862119895|119909=119871119895

is theconcentration of the tail of pipe 119895 at node 119894

3 Water Age Estimation Model

Although more complicated models are available for mod-eling the decay of chlorine (eg [16]) the first-order decaymodel is popular for its convenience in implementation Therate of reaction 119877(119862119894) is as follows

119877 (119862119895) = 119870119895119862119895 (3)

where 119870119895 is the decay coefficient in pipe 119895Traveling along with the water trace line (1) can be

rewritten as follows

119889119862119895

119889119905

= 119870119895119862119895(4)

The solution of (4) is

119862119895 = 1198620exp (minus119870119895119905) (5)

where 1198620 is the concentration (massvolume) in the source119870119895 denotes the decay coefficient on pipe 119895

Water travels from the water station to the consumerthrough many pipes In most skeletal pipes the influence ofmixing is neglectedThe solution of (5) at node 119894 is as follows

119862119894 = 1198620119890sum119898

119895=1minus119870119895119905119895

(6)

where119862119894 denotes the concentration at node 119894 and119898 is numberof pipes through which water travels from the source to node119894

The water age at node 119894 is 119879119894 = sum

119898119895=1 119905119895 The average

decay coefficient 119870 can be expressed as 119870 = sum

119898119895=1119870119895119905119895119879119894

Consequently (6) is simplified as

119862119894 = 1198620119890minus119870119879119894

(7)

The residual chlorine concentration varies with the waterage The variation of residual chlorine can show the fluctua-tions of water age The first-order expansion of (7) near theaverage water age is as follows

119862119894119896 minus 119862119894

119862119894

= 119890

minus119870(119879119894119896minus119879119894)minus 1 = 119890

minus119870(119897119894119881119894119896minus119897119894119881119894)minus 1

= 119890

minus(119870119897119894119881119894)(119881119894119881119894119896minus1)minus 1 asymp

119870119897119894

119881119894

2(119881119894119896 minus 119881119894)

(8)

where 119862119894119896 is the concentration at node 119894 at time 119896 119862119894 isthe average concentration at node 119894 119879119894119896 is the water age ofnode 119894 at time 119896 and 119897119894 is the distance from the source to thenode 119894 119881119894119896 = 119897119894119879119894119896 denotes the average velocity from the

Mathematical Problems in Engineering 3

source to node 119894 at time 119896 119881119894 is the average velocity fromthe source to node 119894 The relation between 119881119894 and 119881119894119896 is119881119894 = (1119898)sum

119898119896=1 119881119894119896

Assume that the average velocity is linearly dependent onthe water demand in water distribution systems Thus

119862119894119896 minus 119862119894

119862119894

prop

(119876119894119896 minus 119876119894)

119876119894

(9)

where 119876119894119896 denotes the average water demand of the wholewater distribution system during the water age of node 119894 attime 119896 and the formula is 119876119894119896 = (1119879119894119896) int

119905119896

119905119896minus119879119894119896119876(119905)119889119905 119876119894

is the mean value of 119876119894119896 which can be expressed as 119876119894 =

(1119899)sum

119899119896=1 119876119894119896

The above derivation shows that the concentration ofresidual chlorine at node 119894 is linearly dependent on theaverage demand during the water retention time whichmeans the correlation coefficient between the water age andthe average demand is close to unity The length from thesource to the monitoring point is const so one gets 11988111989411198791198941 =11988111989421198791198942 = sdot sdot sdot = 119881119894119898119879119894119898 The average velocity is linearlydependent on the water demand the above function can berewritten as 11987611989411198791198941 = 11987611989421198791198942 = sdot sdot sdot = 119876119894119898119879119894119898 Equation (9)can help us to estimate the water age at monitoring points

Assuming that the standard deviations of 119862119894 and 119876119894 areconstant 120590119888 and 120590119876 respectively a statistical model is built toestimate the water age at monitoring node 119894 according to themonitoring data from SCADA

Max119865 (1198791198941 1198791198942 119879119894119898)

=

119864 [(119862119894119896 minus 119862119894) (119876119894119896 minus 119876119894)]

120590119888120590119876

11987611989411198791198941 = 11987611989421198791198942 = sdot sdot sdot = 119876119894119898119879119894119898

(10)

where 119876119894119896 = (1119879119894119896) int119905119896

119905119896minus119879119894119896119876(119905)119889119905 and 119876119894 = (1119898)sum

119898119896=1 119876119894119896

The objective function of the above model is the correlationcoefficient between the residual chlorine and the waterdemand

31 Solution Procedure Equation (10) gives out the model toestimate the water age at monitoring nodes It is not easyto solve it directly because this model involves too manyvariables and the constraint conditions are difficult to dealwith In this section a solution procedure is put forward tosolve this optimal problem This solution procedure consistsof two steps

Step 1 Assuming 1198791198941 = 1198791198942 = sdot sdot sdot = 119879119894119898 = 119879119894 the objectivefunction in (10) is expressed as follows

Max119865 (119879119894) =

119864 [(119862119894 minus 119862119894) (119876119894119896 (119879119894) minus 119876119894 (119879119894))]

120590119888120590119876

(11)

where 119876119894119896(119879) = (1119879119894) int119905119896

119905119896minus119879119894119876(119905)119889119905 and 119876119894(119879119894) =

(1119898)sum

119898119896=1 119876119894119896(119879119894)

Figure 1 Scenario 1

Equation (11) is an unconstrained optimization problemwhich involves only one variable the average water age atmonitor node 119894 It is easy to estimate the average water age119879119894

Step 2 Because the distance from the source to the monitornode 119894 is a constant 119897119894 = 119881119896119879119894119896 the water age at node 119894 at anytime can be calculated according to the following

119876119894119879119894 = 119876119894119896119879119894119896 119896 = 1 2 119898 (12)

Since the monitored data from SCADA is discrete theabovemodel is transformed to a discretemodelThe samplingperiod is Δ119905 the water age at the monitoring node 119894 is119879119894 = 119899119894Δ119905 where 119899119894 is the number of sampling periods Theobjective function of (11) can be expressed as

Max119865 (119899119894)

=

119864 ((119862119894 minus 119864 (119862119894)) (119876119894119896 (119899119894) minus 119864 (119876119894 (119899119894))))

120590119888120590119876

(13)

According to (13) the average water age 119879119894119896 at node 119894 is119899119894119896Δ119905 Equation (12) can be transformed to

119876119894119899119894 = 119876119894119896119899119894119896

119879119894119896 = 119899119894119896Δ

(14)

4 Verification of the Model

In order to verify the proposed model two virtual waterdistribution systems namely the simplest system consistingof one pipeline and a complicated multisource water systemare modeled They are also modeled using EPANET 20 [14]for the purpose of comparisons

Scenario 1 (one pipeline system) The simplest system con-sisting of one pipe is shown in Figure 1 The length of pipeis 10 km Two demand patterns (Figure 2) are tested Becausethe concentration of residual chlorine at the source fluctuatesin the real conditions white noise of the concentration ofresidual chlorine at levels of 5 or 20 is mixed in thesimulations

The correlation coefficient between the residual chlorineconcentration and the mean demand is shown in Figures 3and 4 The maximum objective function (correlation coef-ficient) is very close to 10 In the first demand pattern themaximum objective function is 096 the average water age is446 hours and the maximum relation error is 3 In pattern2 the maximum objective function and the average water ageare 097 and 47 h respectively Figures 5 and 6 show thewater


0

04

08

12

16

0 4 8 12 16 20 24Time (hour)

Dem

and

patte

rn p

acto

rs

Demand pattern 1Demand pattern 2

Figure 2 Water demand pattern factors for Scenario 1

0

05

1

0 6 12 18 24

Obj

ectiv

e fun

ctio

n

Average water age (hour)

Noise 5Noise 20

minus1

minus05

Figure 3 Objective function of Scenario 1 (demand pattern 1)

age modeled by the proposed model and by the EPANET 20respectivelyThemaximum error is less than 03 hours for thefirst pattern and 05 hours for the second one respectivelyFigures 5 and 6 indicate that the statistic model agrees wellwith EPANET 20 The white noise has little influence on theresults

Scenario 2 (multisource system) A multisource system isshown in Figure 7 It consists of 91 junctions 112 pipelinesand two reservoirs as the water sources Nodes 193 and 211 aremodeled to estimate their water age At node 211 the water issupplied by the source RIVER At node 193 4 sim30 of thewater is supplied by the source of LAKE while the other issupplied by the source of RIVER

It is assumed that the residual chlorine concentrationat two sources is 20mgL and the decay coefficient is20mg(Lsdotd) Figure 8 shows the water age at node 211 Thestatistical model predicts that the average water age at node211 is 26 hours with a maximum relative error of 7 Thewater age at node 193 is about 25 hours by the statisticalmodeland about 30 hours by the EPANET model (Figure 9) Atnode 211 our model agrees well with the EPANET modelThe accuracy is lower at node 193 than that at the former

Noise 5Noise 20

00

05

10

0 6 12 18 24

Obj

ectiv

e fun

ctio

n


minus10

minus05


0

2

4

6

8

0 6 12 18 24Time (hour)

Wat

er ag

e (ho

ur)

EPANET modelStatistic model

Figure 5 Water age of Scenario 1 (demand pattern 1)

nodeThere are two possible reasons for errorsThefirst is dueto the fact that this system is nonlinear while the statisticalmodel is based on a linear hypothesis the second reason isthe mixture of water The water flowing into node 193 is fromtwo sources and the routine from the two sources to node 193is too complicated If the node is supplied by multisource itshould be very careful to use this statistical modelThemixedwater and the complicated routine may reduce the accuracyof the model If the node is supplied by a single source thestatistical model can make a good prediction

5 Application in Hangzhou City

Hangzhou city is the capital city of Zhejiang province locatedat the east of China Its water distribution system consistsof 2639 km pipelines and 5 sources as shown in Figure 10It provides over 106m3 of water per day Figure 10 illustratesthe water distribution network 14 water quality monitorsare distributed in the water distribution system monitoringthe residual chlorine concentration and turbidity every 15minutes

5 monitors (S1 S2 S3 S4 and S5) are set up at waterstations 2 monitors (M7 and M8) are around the divisionline of different sources as node 193 in Scenario 2 in whichtheir residual chlorine concentration changes complicatedlyScenario 2 shows that the accuracy of the statistical model


0

2

4

6

8

0 6 12 18 24Time (hour)

Wat

er ag

e (ho

ur)



is not high for this type of monitor 2 nodes (M2 and M9)are close to the stations where the concentration of residualchlorine varies with the source fluctuation For example theresidual chlorine concentration at the monitor M3 fluctuateswith that at S3 as shown in Figure 14 Long-termmonitoringdata has indicated that the residual chlorine concentrationsof monitors M1 M3 M4 M5 and M6 fluctuate periodicallyThus there are 5 nodes (M1 M3 M4 M5 and M6) thatsatisfy the requirement of statistic model The statistic modelis used to estimate the water age at the fivemonitoring pointsDetailed information of modeling for 5 nodes is introducedin the following sections

Figure 12 shows the total demand of network the concen-tration of residual chlorine at water stations and monitoringpoints Red lines are filtered data by a wavelet filter Thefiltered data indicate two properties of the residual chlorineconcentration (1) periodicity the noise of the monitoredresidual chlorine concentration data is very high It is verydifficult to find this feature directly The wavelet filter isan excellent tool to remove the noise The filter data showthe periodicity of 24 hours clearly The fluctuation rangevaries every dayThe residual chlorine concentration is alwayshigher in the late evening and becomes lower in the earlymorning (2) Detention the change of the residual chlorineconcentration is later than the change of demand of systemFor example due to the storage of the water distributionsystem the highest residual chlorine concentration at M4appears during 400 PMsim 700 PM whereas the maximumdemand appears at about 800 AM the lowest residualchlorine concentration appears during the 600 AMsim800 AMwhereas the minimum demand appears at about 400 AMSomeone may argue that the fluctuation of the residualchlorine at monitor comes from the source According to(8) the concentration fluctuation at source will decay whenthe water arrives at the monitor Figure 11 also shows thatthe amplitude of fluctuation in water station is lower thanthat at the monitoring point as described by You et al [15]Most monitoring data in Hangzhou show that the fluctuationat water distribution network is lager than that at the watersourceThus the fluctuation concentration of the monitors isderived from the detention time

The statistical model always needs more monitoringsamples The decay coefficient of the residual chlorine varieswith the water temperature The decay coefficient is different

Table 1 Water age at monitors

Monitoring point M1 M3 M4 M5 M6Source S1 S3 S5 S5 S2Objective function 078 064 095 090 082Average water age (statisticmodel) 95 2075 75 105 75

Water age (EPANET) 6sim8 14sim23 5sim10 8sim13 5sim11

with seasons Another serious problem is the drift error ofchlorine sensors The drift error appears in the long-termworking chlorine sensor and it becomes serious graduallyCalibration will make the monitoring serial discontinue Themore the monitoring samples are the more the unpredictedfactors affect the model We should select the sample whenthe water temperature and sensors are stable It is believedthat 15 daysrsquo to 30 daysrsquo samples are enough for this statisticalmodel In the present paper 15 daysrsquo samples are used toestimate the water age The sample period Δ119905 is 15 minutesThere are 1440 samples for each monitoring point Figure 12shows the objective functions at different water ages for fivemonitors We can get the optimal average water age for eachmonitor point directly Figure 13 indicates the water age atdifferent time Table 1 shows the average water age of fivemonitors estimated by the model and the results agree wellwith that estimated by the EPANET model

The fluctuation range of the water age estimated byEPANET is lager than that of the present model The fluctu-ation range in the present model is about 15sim 25 whileit is about 20 sim 35 in EPANET model Now there is nomeasurement data to verify which model is more reliableHowever the model provides an acceptable estimation of thewater age

Table 1 also shows that the simpler routine from thesource to the monitors gives the larger objective functionThe simplest routine is M4rsquos and its correlation coefficientbetween residual chlorine concentration and total demandattains 095 which is very close to 1 The second is M5rsquoswith a correlation coefficient of 090 The routines from thesources to M1 M6 and M3 are more complicated so theircorrelation coefficient is lower The mixture of water in thewater distribution network may affect the accuracy of thestatistical model

Besides EPANET model another method is utilized toverify the model If the monitor is very close to the sourcesthe residual chlorine concentration at the monitoring pointwill fluctuate with that of sources We can determinate thewater age directly through both monitoring data sequenceof the monitoring point and the source For example if theresidual chlorine rises to a peak value of 2mgL at 9 orsquoclockwe may find that the residual chlorine concentration at thedownstream monitoring point rises to a peak value of 15at 10 orsquoclock The water age at the monitoring point is 1hour and the decay coefficient can be directly calculatedaccording to (7) If the present statistical model is reliablethe decay coefficient estimated that the two methods shouldbe in agreement The water of M2 and M3 is supplied by S3SinceM2 is close to S3 the residual chlorine concentration in


Lake

211

193

River

Figure 7 Water distribution network for case 2

M2 varies with that of S3 as shown in Figure 11 Accordingto their impulse of the residual concentration at S3 and M2the water age at M2 is about 25 hours that is about 3 hoursby EPANET The average residual chlorine concentration is1479 (Lsdoth) at S3 and is 1267 (Lsdoth) at M2 Thus the averagedecay coefficient is 119870 = ln(1198880119888)119879 = ln(14791267)25 =

0619mg(Lsdoth) The average residual chlorine concentrationatM3 is 0456mgL and its averagewater age estimated by thestatistical model is about 2075 hoursThus the average decaycoefficient is 119870 = ln(1198880119888)119879 = ln(14790456)2075 =

0567mg(Lsdoth)The relative error of tow decay coefficients bytwo methods is less than 10 which further illustrates thereliability of the statistical model

6 Limitations

Regarding other tools to estimate the water age summarizedby Brandt et al [1] there are also some limitations for

the present statistical model The application in Hangzhoucity shows that themodel does not work well when themoni-tor is located near the water division line of different sourcessuch as M8 The data from monitor M8 are periodical buttheir correlation coefficient between the residual chlorineconcentration and the demand is negative This is becausethat this node is at the water division line of the sources S4and S5 and it is supplied by two water sources alternativelyIf the water at monitoring point is supplied by two or moresources the model will make a prediction with less accuracyor fail Engineers should carefully handle the cases under suchcondition The second limitation is the monitor (such as M2and M9) being too close to the source Meanwhile the effectof the time becomes not importantTheperiodical fluctuationis distorted by the noise or the concentration atmonitors willfluctuate with that of the source so the model will fail It issuggested that the statistic model should be applied for nodesin which the water ages are larger than 5 hours


0

5

10

15

20

25

30

0 6 12 18 24

Time (hour)

Wat

er ag

e (ho

ur)

Statistic modelEPANET model

Figure 8 Water age at node 211

0

5

10

15

20

25

30

35

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)



7 Conclusions

A statistical model to estimate the water age of water dis-tribution networks has been proposed in the present paperThemodel is based on the solution of the advection transportequation governing the residual chlorine A simple two-stepsolution procedure for the model has been given out Themodel was tested by the simplest one-pipeline system anda multisource system The numerical test indicates that themodel agrees well with the EPANET model if monitoringpoints are not around the water division line The model isalso applied to the water distribution system in Hangzhoucity The results agree well with those estimated by usingEPANET model The comparison of the decay coefficientsestimated by another methodproves that the result obtainedby the statistical model is reliable This method does notneed complicated calibration as numerical models and alsodoes not need high operation cost as tracer studies If thewater distribution systemhas SCADA tomonitor the residualchlorine concentration this statistical model will be a simpleand effective tool to estimate the systemrsquos water age withoutadditional costs

M3

M8

M1

S1

M9

S4 S3

M6

S2

M7

M5

S5 M4

M9 monitor GudunM8 monitor DaguangM7 monitor GensanmenM6 monitor WanjiangmenM5 monitor GaotingbaM4 monitor GongBei

M3 monitor ZhouchengchangM2 monitor DongzhangM1 monitor WulinmenS5 station XiangfuS4 station MiaopuS3 station QintaiS2 station NanxinS1 station Chishanpu

M2

Water division line

Figure 10 Water distribution network in Hangzhou

Notations

1198620 Concentration (massvolume) in source119862119894 Concentration (massvolume) at node 119894

119862119894 Average concentration (massvolume) atnode 119894

119862119894119896 Concentration (massvolume) at time 119896

for node 119894

119862119894ext Concentration (massvolume) of theexternal flow entering at node 119894

119862119894|119909=0 Concentration (massvolume) of pipe atthe start of node 119894

119862119895 Concentration (massvolume) in pipe 119894 asa function of distance 119909 and time 119905

119862119895|119909=119871119895 Concentration (massvolume) of pipe 119895

tail at the node 119894

119870119895 Decay coefficient at pipe 119895 (1time)119870 Average decay coefficient (1time)119897119894 Length from source to node 119894

119898 Number of pipes through which watertravels from source to node

119899119894119896 Number of sampling periods at time 119896 fornode 119894

119899119894 Average number of sampling periods atnode 119894 during retention time

119902119895 Flow (volumetime) in pipe 119895

119902119894ext External source flow entering the networkat node 119894

119876119894 Averaged value of 119876119894119896

8 Mathematical Problems in EngineeringQ

(m3h

)Cs1

(mg

L)Cm

1(m

gL)

Q (m

3h

)Cs3

(mg

L)Cm

3(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

5(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

4(m

gL)

Q (m

3h

)Cs

2(m

gL)

Cm6

(mg

L)

161412

1

13121110

1008060402

6e4

4e4

2e4

161412

16e4

4e4

2e4

1610111213

1412

16e4

4e4

2e4

160005

1412

16e4

4e4

2e4

16121314

1412

14e4

3e4

2e4

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

Figure 11 Residual chlorine concentration at water stations andmonitoring points the total water demand Red solid lines arefiltered data

119876119894119896 Average total water demand in wholesystem during water age of node 119894 at time 119896

119879 Water age119879119894119896 Water age at node 119894 at time 119896

119877(119862119895) Rate of reaction (massvolumetime)function of concentration

119881119894119896 Average velocity from source to node 119894 attime 119896

119881119894 Averaged velocity for all time119881119895 Flow velocity (lengthtime) in pipe 119895

Ω119894 Set of pipes with flow into node 119894

120590119888 Standard deviation of 119862119894

04

09

0 4 8 12 16 20 24Water age (hour)

Obj

ectiv

e fun

ctio

n

M3M5M4

M6M1

minus01

minus06

Figure 12 Objective function for different water age

0

5

10

15

20

25

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)

M5

M4M6M1M3

Figure 13 Water age at monitoring points

Time (hour)

Time (hour)

350300250200150100500

18161412

108060402

6055504540353025201510

Resid

ual c

hlor

ine

165155145135125115

(mg

L)Re

sidua

l chl

orin

e(m

gL)

Figure 14 Residual chlorine concentration atM2 S3 andM3 Solidline is at S3 dashed line is at M2 and doted line is at M3


120590119876 Standard deviation of 119876119894Δ119905 Sampling period

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present research is funded by National Natural ScienceFoundation of China (nos 51578486 and 51378455) and theKey Special Program on the SampT of China for the PollutionControl and Treatment of Water Bodies (2012ZX07408-002-003 and 2012ZX07403-003) The authors would like to thankDr J Zhang for providing some document about water ageSpecial thanks are due to Dr Z H Yang and Dr Z H Zhangfor review of the paper

References

[1] M Brandt J C J Powell R Casey D H Neil and H C TTa Managing Distribution Retention Time to Improve WaterQualitymdashPhase I AwwaRF and AWWA Denver Colo USA2004

[2] R M Clark and W M Grayman ldquoWater quality modeling ina distribution systemrdquo in Proceedings of the AWWA AnnualConference pp 175ndash193 Denver Colo USA June 1991

[3] A L Cesario J R Kroon and W M Grayman ldquoNew per-spectives on calibration of treated water distribution systemmodelsrdquo in Proceedings of the AWWA Annual ConferenceDenver Colo USA 1996

[4] D L Boccelli F Shang J G Uber et al ldquoCritical transitions inwater and environmental resources managementrdquo in Proceed-ings of the World Water and Environmental Resources CongressG Sehlke D F Hayes and D K Stevens Eds Salt Lake CityUtah USA June-July 2004

[5] EPA ldquoEffects of water age on distribution system water qualityrdquoOffice of Water (4601M) Office of Ground Water and DrinkingWater Distribution System Issue Paper US EnvironmentalProtection Agency 2004

[6] R M Males R M Clark P J Wehrman and W E GatesldquoAlgorithm for mixing problems in water systemsrdquo Journal ofHydraulic Engineering vol 111 no 2 pp 206ndash219 1985

[7] W M Grayman R M Clark and R M Males ldquoModelingdistribution-system water quality dynamic approachrdquo JournalofWater Resources Planning andManagement vol 114 no 3 pp295ndash312 1988

[8] W M Grayman and R M Clark ldquoUsing computer modelsto determine the effect of storage on water qualityrdquo JournalmdashAmerican Water Works Association vol 85 no 7 pp 67ndash771993

[9] C P Liou and J R Kroon ldquoModeling the propagation ofwaterborne substances in networksrdquo Journal (American WaterWorks Association) vol 79 no 11 pp 54ndash58 1987

[10] P F Boulos T Altman P-A Jarrige and F Collevati ldquoAn event-drivenmethod formodelling contaminant propagation inwaternetworksrdquo Applied Mathematical Modelling vol 18 no 2 pp84ndash92 1994

[11] P F Boulos T Altman P-A Jarrige and F Collevati ldquoDiscretesimulation approach for network-water-quality modelsrdquo Jour-nal of Water Resources Planning amp Management vol 121 no 1pp 49ndash60 1995

[12] L A Rossman P F Boulos and T Altman ldquoDiscrete volumeelement method for network water-quality modelsrdquo Journal ofWater Resources Planning amp Management vol 119 no 5 pp505ndash517 1993

[13] L S Rededy L E Ormsbee and D J Wood ldquoTime-averagingwater quality assessmentrdquo Journal of the AmericanWater WorksAssociation vol 187 no 7 pp 64ndash73 1995

[14] Environmental Protection Agency (EPA) EPANET 20 EPAWater Supply amp Water Resources 2000 httpwww2epagovwater-researchepanet

[15] Z B You H F Xu and Z J Qu ldquoStudy on water quality fluc-tuation and influencing factors in water distribution networkrdquoWater ampWastewater Engineering vol 31 no 1 pp 21ndash26 2005

[16] J C Powell J RWest N B Hallam C F Forster and J SimmsldquoPerformance of various kinetic models for chlorine decayrdquoJournal of Water Resources Planning and Management vol 126no 1 pp 13ndash20 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the overall demand is miscalculated it will result in more orless source and reservoir operation thanwhat actually occursMisestimating the demand allocation might lead to errorflow direction (3) water storage tanks tanks are modeled ascompletely mixed reactors in most models which will lead tomisestimation of the water age

You et al [15] have shown that the residual chlorinedecay ratio per unit length is different at different time Forexample it is 0175mg(LsdotKm) at peak-demand time that is0459mg(LsdotKm) at theminimum-demand time in Shenzhencity You et al [15] said that the retention time is one of themost important elements of the residual chlorine fluctuationfor long distance water distribution systems Our monitoringdata in Hangzhou city is the same as theirs Some waterdistribution systems have SCADA (supervisory control anddata acquisition) The concentration of residual chlorine inthe water system can bemonitored by SCADA In the presentpaper a novel statistic model is proposed to estimate thewater age in water distribution systems according to themonitoring data serials of the residual chlorine concentrationfrom SCADA The model is discussed theoretically andnumerically And it is also applied to predict the water ageof water distribution system in Hangzhou city The statisticmodel is in good agreement with EPANET 20 numericalresults

2 Governing Equations for Water Quality

A water distribution system consists of pipes pumps valvesfittings and storage facilities that are used to convey waterfrom the source to consumers The dissolved substance trav-els along the pipe with the same average velocity as the carrierfluid while reacting (either growing or decaying) at certainratesThe equations governing the water quality are based onthe principle of conservation of mass coupled with reactionkinetics [10ndash12] Usually the role of longitudinal dispersionis neglectable The conservation of mass during transportwithin a pipe is described by the classical one-dimensionaladvection-reaction equation The advection transport withina pipe 119895 is represented as

120597119862119895

120597119905

= minus119881119895

120597119862119895

120597119909

minus 119877 (119862119895)

(1)

where 119862119895 is the concentration (massvolume) in pipe 119895 asa function of distance 119909 and time 119905 119881119895 is the flow velocity(ms) in pipe 119895 and 119877(119862119895) denotes the rate of reaction(massvolumetime) as a function of concentration

When junctions receive inflow from two or more pipes itis assumed that the complete mixing of fluid is accomplishedsimultaneously Thus the concentration of a substance inwater when water leaves the junction is simply the flow-weighted sum of the concentrations from the inflowing pipesFor a specific node 119894 the concentration is expressed as follows

1198621198941003816

1003816

1003816

1003816119909=0=

sum

Ω119894119895=1 119902119895 119862119895

1003816

1003816

1003816

1003816

1003816119909=119871119895+ 119902119894ext119862119894ext

sum

Ω119894119895=1 119902119895 + 119902119894ext

(2)

where Ω119894 is the set of pipes with flow into node 119894 119902119895 is theflow (m3s) in pipe 119895 119902119894ext is the external source flow enteringthe network at node 119894 and 119862119894ext is the concentration of theexternal flow entering at node 119894 the notation 119862119894|119909=0 denotesthe concentration at the start of node 119894 while 119862119895|119909=119871119895

is theconcentration of the tail of pipe 119895 at node 119894

3 Water Age Estimation Model

Although more complicated models are available for mod-eling the decay of chlorine (eg [16]) the first-order decaymodel is popular for its convenience in implementation Therate of reaction 119877(119862119894) is as follows

119877 (119862119895) = 119870119895119862119895 (3)

where 119870119895 is the decay coefficient in pipe 119895Traveling along with the water trace line (1) can be

rewritten as follows

119889119862119895

119889119905

= 119870119895119862119895(4)

The solution of (4) is

119862119895 = 1198620exp (minus119870119895119905) (5)

where 1198620 is the concentration (massvolume) in the source119870119895 denotes the decay coefficient on pipe 119895

Water travels from the water station to the consumerthrough many pipes In most skeletal pipes the influence ofmixing is neglectedThe solution of (5) at node 119894 is as follows

119862119894 = 1198620119890sum119898

119895=1minus119870119895119905119895

(6)

where119862119894 denotes the concentration at node 119894 and119898 is numberof pipes through which water travels from the source to node119894

The water age at node 119894 is 119879119894 = sum

119898119895=1 119905119895 The average

decay coefficient 119870 can be expressed as 119870 = sum

119898119895=1119870119895119905119895119879119894

Consequently (6) is simplified as

119862119894 = 1198620119890minus119870119879119894

(7)

The residual chlorine concentration varies with the waterage The variation of residual chlorine can show the fluctua-tions of water age The first-order expansion of (7) near theaverage water age is as follows

119862119894119896 minus 119862119894

119862119894

= 119890

minus119870(119879119894119896minus119879119894)minus 1 = 119890

minus119870(119897119894119881119894119896minus119897119894119881119894)minus 1

= 119890

minus(119870119897119894119881119894)(119881119894119881119894119896minus1)minus 1 asymp

119870119897119894

119881119894

2(119881119894119896 minus 119881119894)

(8)

where 119862119894119896 is the concentration at node 119894 at time 119896 119862119894 isthe average concentration at node 119894 119879119894119896 is the water age ofnode 119894 at time 119896 and 119897119894 is the distance from the source to thenode 119894 119881119894119896 = 119897119894119879119894119896 denotes the average velocity from the



119898119896=1 119881119894119896


119862119894119896 minus 119862119894

119862119894

prop

(119876119894119896 minus 119876119894)

119876119894

(9)


119905119896

119905119896minus119879119894119896119876(119905)119889119905 119876119894


(1119899)sum

119899119896=1 119876119894119896



Max119865 (1198791198941 1198791198942 119879119894119898)

=

119864 [(119862119894119896 minus 119862119894) (119876119894119896 minus 119876119894)]

120590119888120590119876

11987611989411198791198941 = 11987611989421198791198942 = sdot sdot sdot = 119876119894119898119879119894119898

(10)

where 119876119894119896 = (1119879119894119896) int119905119896

119905119896minus119879119894119896119876(119905)119889119905 and 119876119894 = (1119898)sum

119898119896=1 119876119894119896




Max119865 (119879119894) =

119864 [(119862119894 minus 119862119894) (119876119894119896 (119879119894) minus 119876119894 (119879119894))]

120590119888120590119876

(11)

where 119876119894119896(119879) = (1119879119894) int119905119896

119905119896minus119879119894119876(119905)119889119905 and 119876119894(119879119894) =

(1119898)sum

119898119896=1 119876119894119896(119879119894)

Figure 1 Scenario 1



119876119894119879119894 = 119876119894119896119879119894119896 119896 = 1 2 119898 (12)


Max119865 (119899119894)

=

119864 ((119862119894 minus 119864 (119862119894)) (119876119894119896 (119899119894) minus 119864 (119876119894 (119899119894))))

120590119888120590119876

(13)


119876119894119899119894 = 119876119894119896119899119894119896

119879119894119896 = 119899119894119896Δ

(14)






0

04

08

12

16

0 4 8 12 16 20 24Time (hour)

Dem

and

patte

rn p

acto

rs



0

05

1

0 6 12 18 24

Obj

ectiv

e fun

ctio

n


Noise 5Noise 20

minus1

minus05





Noise 5Noise 20

00

05

10

0 6 12 18 24

Obj

ectiv

e fun

ctio

n


minus10

minus05


0

2

4

6

8

0 6 12 18 24Time (hour)

Wat

er ag

e (ho

ur)








0

2

4

6

8

0 6 12 18 24Time (hour)

Wat

er ag

e (ho

ur)














Lake

211

193

River





6 Limitations




0

5

10

15

20

25

30

0 6 12 18 24

Time (hour)

Wat

er ag

e (ho

ur)



0

5

10

15

20

25

30

35

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)



7 Conclusions


M3

M8

M1

S1

M9

S4 S3

M6

S2

M7

M5

S5 M4



M2

Water division line


Notations




for node 119894














(m3h

)Cs1

(mg

L)Cm

1(m

gL)

Q (m

3h

)Cs3

(mg

L)Cm

3(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

5(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

4(m

gL)

Q (m

3h

)Cs

2(m

gL)

Cm6

(mg

L)

161412

1

13121110

1008060402

6e4

4e4

2e4

161412

16e4

4e4

2e4

1610111213

1412

16e4

4e4

2e4

160005

1412

16e4

4e4

2e4

16121314

1412

14e4

3e4

2e4

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)









04

09

0 4 8 12 16 20 24Water age (hour)

Obj

ectiv

e fun

ctio

n

M3M5M4

M6M1

minus01

minus06


0

5

10

15

20

25

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)

M5

M4M6M1M3


Time (hour)

Time (hour)

350300250200150100500

18161412

108060402

6055504540353025201510

Resid

ual c

hlor

ine

165155145135125115

(mg

L)Re

sidua

l chl

orin

e(m

gL)






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898119896=1 119881119894119896


119862119894119896 minus 119862119894

119862119894

prop

(119876119894119896 minus 119876119894)

119876119894

(9)


119905119896

119905119896minus119879119894119896119876(119905)119889119905 119876119894


(1119899)sum

119899119896=1 119876119894119896



Max119865 (1198791198941 1198791198942 119879119894119898)

=

119864 [(119862119894119896 minus 119862119894) (119876119894119896 minus 119876119894)]

120590119888120590119876

11987611989411198791198941 = 11987611989421198791198942 = sdot sdot sdot = 119876119894119898119879119894119898

(10)

where 119876119894119896 = (1119879119894119896) int119905119896

119905119896minus119879119894119896119876(119905)119889119905 and 119876119894 = (1119898)sum

119898119896=1 119876119894119896




Max119865 (119879119894) =

119864 [(119862119894 minus 119862119894) (119876119894119896 (119879119894) minus 119876119894 (119879119894))]

120590119888120590119876

(11)

where 119876119894119896(119879) = (1119879119894) int119905119896

119905119896minus119879119894119876(119905)119889119905 and 119876119894(119879119894) =

(1119898)sum

119898119896=1 119876119894119896(119879119894)

Figure 1 Scenario 1



119876119894119879119894 = 119876119894119896119879119894119896 119896 = 1 2 119898 (12)


Max119865 (119899119894)

=

119864 ((119862119894 minus 119864 (119862119894)) (119876119894119896 (119899119894) minus 119864 (119876119894 (119899119894))))

120590119888120590119876

(13)


119876119894119899119894 = 119876119894119896119899119894119896

119879119894119896 = 119899119894119896Δ

(14)






0

04

08

12

16

0 4 8 12 16 20 24Time (hour)

Dem

and

patte

rn p

acto

rs



0

05

1

0 6 12 18 24

Obj

ectiv

e fun

ctio

n


Noise 5Noise 20

minus1

minus05





Noise 5Noise 20

00

05

10

0 6 12 18 24

Obj

ectiv

e fun

ctio

n


minus10

minus05


0

2

4

6

8

0 6 12 18 24Time (hour)

Wat

er ag

e (ho

ur)








0

2

4

6

8

0 6 12 18 24Time (hour)

Wat

er ag

e (ho

ur)














Lake

211

193

River





6 Limitations




0

5

10

15

20

25

30

0 6 12 18 24

Time (hour)

Wat

er ag

e (ho

ur)



0

5

10

15

20

25

30

35

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)



7 Conclusions


M3

M8

M1

S1

M9

S4 S3

M6

S2

M7

M5

S5 M4



M2

Water division line


Notations




for node 119894














(m3h

)Cs1

(mg

L)Cm

1(m

gL)

Q (m

3h

)Cs3

(mg

L)Cm

3(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

5(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

4(m

gL)

Q (m

3h

)Cs

2(m

gL)

Cm6

(mg

L)

161412

1

13121110

1008060402

6e4

4e4

2e4

161412

16e4

4e4

2e4

1610111213

1412

16e4

4e4

2e4

160005

1412

16e4

4e4

2e4

16121314

1412

14e4

3e4

2e4

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)









04

09

0 4 8 12 16 20 24Water age (hour)

Obj

ectiv

e fun

ctio

n

M3M5M4

M6M1

minus01

minus06


0

5

10

15

20

25

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)

M5

M4M6M1M3


Time (hour)

Time (hour)

350300250200150100500

18161412

108060402

6055504540353025201510

Resid

ual c

hlor

ine

165155145135125115

(mg

L)Re

sidua

l chl

orin

e(m

gL)






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

04

08

12

16

0 4 8 12 16 20 24Time (hour)

Dem

and

patte

rn p

acto

rs



0

05

1

0 6 12 18 24

Obj

ectiv

e fun

ctio

n


Noise 5Noise 20

minus1

minus05





Noise 5Noise 20

00

05

10

0 6 12 18 24

Obj

ectiv

e fun

ctio

n


minus10

minus05


0

2

4

6

8

0 6 12 18 24Time (hour)

Wat

er ag

e (ho

ur)








0

2

4

6

8

0 6 12 18 24Time (hour)

Wat

er ag

e (ho

ur)














Lake

211

193

River





6 Limitations




0

5

10

15

20

25

30

0 6 12 18 24

Time (hour)

Wat

er ag

e (ho

ur)



0

5

10

15

20

25

30

35

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)



7 Conclusions


M3

M8

M1

S1

M9

S4 S3

M6

S2

M7

M5

S5 M4



M2

Water division line


Notations




for node 119894














(m3h

)Cs1

(mg

L)Cm

1(m

gL)

Q (m

3h

)Cs3

(mg

L)Cm

3(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

5(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

4(m

gL)

Q (m

3h

)Cs

2(m

gL)

Cm6

(mg

L)

161412

1

13121110

1008060402

6e4

4e4

2e4

161412

16e4

4e4

2e4

1610111213

1412

16e4

4e4

2e4

160005

1412

16e4

4e4

2e4

16121314

1412

14e4

3e4

2e4

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)









04

09

0 4 8 12 16 20 24Water age (hour)

Obj

ectiv

e fun

ctio

n

M3M5M4

M6M1

minus01

minus06


0

5

10

15

20

25

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)

M5

M4M6M1M3


Time (hour)

Time (hour)

350300250200150100500

18161412

108060402

6055504540353025201510

Resid

ual c

hlor

ine

165155145135125115

(mg

L)Re

sidua

l chl

orin

e(m

gL)






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2

4

6

8

0 6 12 18 24Time (hour)

Wat

er ag

e (ho

ur)














Lake

211

193

River





6 Limitations




0

5

10

15

20

25

30

0 6 12 18 24

Time (hour)

Wat

er ag

e (ho

ur)



0

5

10

15

20

25

30

35

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)



7 Conclusions


M3

M8

M1

S1

M9

S4 S3

M6

S2

M7

M5

S5 M4



M2

Water division line


Notations




for node 119894














(m3h

)Cs1

(mg

L)Cm

1(m

gL)

Q (m

3h

)Cs3

(mg

L)Cm

3(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

5(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

4(m

gL)

Q (m

3h

)Cs

2(m

gL)

Cm6

(mg

L)

161412

1

13121110

1008060402

6e4

4e4

2e4

161412

16e4

4e4

2e4

1610111213

1412

16e4

4e4

2e4

160005

1412

16e4

4e4

2e4

16121314

1412

14e4

3e4

2e4

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)









04

09

0 4 8 12 16 20 24Water age (hour)

Obj

ectiv

e fun

ctio

n

M3M5M4

M6M1

minus01

minus06


0

5

10

15

20

25

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)

M5

M4M6M1M3


Time (hour)

Time (hour)

350300250200150100500

18161412

108060402

6055504540353025201510

Resid

ual c

hlor

ine

165155145135125115

(mg

L)Re

sidua

l chl

orin

e(m

gL)






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Lake

211

193

River





6 Limitations




0

5

10

15

20

25

30

0 6 12 18 24

Time (hour)

Wat

er ag

e (ho

ur)



0

5

10

15

20

25

30

35

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)



7 Conclusions


M3

M8

M1

S1

M9

S4 S3

M6

S2

M7

M5

S5 M4



M2

Water division line


Notations




for node 119894














(m3h

)Cs1

(mg

L)Cm

1(m

gL)

Q (m

3h

)Cs3

(mg

L)Cm

3(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

5(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

4(m

gL)

Q (m

3h

)Cs

2(m

gL)

Cm6

(mg

L)

161412

1

13121110

1008060402

6e4

4e4

2e4

161412

16e4

4e4

2e4

1610111213

1412

16e4

4e4

2e4

160005

1412

16e4

4e4

2e4

16121314

1412

14e4

3e4

2e4

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)









04

09

0 4 8 12 16 20 24Water age (hour)

Obj

ectiv

e fun

ctio

n

M3M5M4

M6M1

minus01

minus06


0

5

10

15

20

25

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)

M5

M4M6M1M3


Time (hour)

Time (hour)

350300250200150100500

18161412

108060402

6055504540353025201510

Resid

ual c

hlor

ine

165155145135125115

(mg

L)Re

sidua

l chl

orin

e(m

gL)






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

10

15

20

25

30

0 6 12 18 24

Time (hour)

Wat

er ag

e (ho

ur)



0

5

10

15

20

25

30

35

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)



7 Conclusions


M3

M8

M1

S1

M9

S4 S3

M6

S2

M7

M5

S5 M4



M2

Water division line


Notations




for node 119894














(m3h

)Cs1

(mg

L)Cm

1(m

gL)

Q (m

3h

)Cs3

(mg

L)Cm

3(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

5(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

4(m

gL)

Q (m

3h

)Cs

2(m

gL)

Cm6

(mg

L)

161412

1

13121110

1008060402

6e4

4e4

2e4

161412

16e4

4e4

2e4

1610111213

1412

16e4

4e4

2e4

160005

1412

16e4

4e4

2e4

16121314

1412

14e4

3e4

2e4

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)









04

09

0 4 8 12 16 20 24Water age (hour)

Obj

ectiv

e fun

ctio

n

M3M5M4

M6M1

minus01

minus06


0

5

10

15

20

25

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)

M5

M4M6M1M3


Time (hour)

Time (hour)

350300250200150100500

18161412

108060402

6055504540353025201510

Resid

ual c

hlor

ine

165155145135125115

(mg

L)Re

sidua

l chl

orin

e(m

gL)






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(m3h

)Cs1

(mg

L)Cm

1(m

gL)

Q (m

3h

)Cs3

(mg

L)Cm

3(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

5(m

gL)

Q (m

3h

)Cs5

(mg

L)Cm

4(m

gL)

Q (m

3h

)Cs

2(m

gL)

Cm6

(mg

L)

161412

1

13121110

1008060402

6e4

4e4

2e4

161412

16e4

4e4

2e4

1610111213

1412

16e4

4e4

2e4

160005

1412

16e4

4e4

2e4

16121314

1412

14e4

3e4

2e4

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)

50 100 150 200 250 300 3500

Time (hour)









04

09

0 4 8 12 16 20 24Water age (hour)

Obj

ectiv

e fun

ctio

n

M3M5M4

M6M1

minus01

minus06


0

5

10

15

20

25

0 4 8 12 16 20 24Time (hour)

Wat

er ag

e (ho

ur)

M5

M4M6M1M3


Time (hour)

Time (hour)

350300250200150100500

18161412

108060402

6055504540353025201510

Resid

ual c

hlor

ine

165155145135125115

(mg

L)Re

sidua

l chl

orin

e(m

gL)






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Novel Statistical Model for Water Age