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ARTICLE IN PRESS
Water Research 38 (2004) 2973–2988
Modiﬁed Lagrangian method for modeling water quality in distribution systems
G.R. Munavalli, M.S. Mohan Kumar
Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India Received 31 May 2002; received in revised form 19 January 2004; accepted 16 April 2004
Abstract Previous work has shown that Lagrangian methods are more efﬁcient for modeling the transport of chemicals in a water distribution system. Two such metho

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Water Research 38 (2004) 2973–2988
Modiﬁed Lagrangian method for modeling water quality indistribution systems
G.R. Munavalli, M.S. Mohan Kumar
Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India
Received 31 May 2002; received in revised form 19 January 2004; accepted 16 April 2004
Abstract
Previous work has shown that Lagrangian methods are more efﬁcient for modeling the transport of chemicals in awater distribution system. Two such methods, the Lagrangian Time-Driven Method (TDM) and Event-Driven Method(EDM) are compared for varying concentration tolerance and computational water quality time step. A new hybridmethod (EDMNET) is developed which improves the accuracy of the Lagrangian methods. All the above methods areincorporated in an existing hydraulic simulation model. The integrated model is run for different network problemsunder varying conditions. The TDM-generated solutions are affected by both concentration tolerance and water qualitytime step, whereas EDM solutions are dependent on concentration tolerance. The EDMNET solutions are less sensitiveto variations in these parameters. The threshold solutions are determined for all the methods and compared. The hybridmethod simulates the nodal concentrations accurately with least maximum segmentation of network and reasonablecomputational effort as compared to the other Lagrangian methods.
r
2004 Elsevier Ltd. All rights reserved.
Keywords:
Bulk decay; Distribution system; Dynamic modeling; Quality time step; Concentration tolerance; Wall decay; Waterquality
1. Introduction
It is well known that the quality of drinking water canchange within a distribution system. The movement orlack of movement of water within the distributionsystem may have deleterious effects on a once acceptablesupply. These quality changes may be associated withcomplex physical, chemical and biological activities thattake place during the transport process. Such activitiescan occur either in the bulk water column, the hydraulicinfrastructure, or both, and may be internally orexternally generated[1]. The ability to understand thesereactions and model their impact throughout a distribu-tion system will assist water suppliers in selectingimproved operational strategies and capital investmentsto ensure delivery of safe drinking water[2].Basically water quality modeling is simulated in asteady or a dynamic environment. In steady-statemodeling, the external conditions of a distributionnetwork are constant in time and the nodal concentra-tions of the constituents that will occur if the system isallowed to reach equilibrium are determined. Thesemethods can provide general information on the spatialdistribution of water quality. In dynamic models theexternal conditions are temporally varied and the timevarying nodal concentrations of the constituents aredetermined. The algorithms developed include steady-state[3–7]and dynamic[1,8–12]models.
Rossman and Boulos[13]have given a comprehensivedescription of dynamic modeling and the existingnumerical solution methods hence a review will not berepeated here. Instead the treatment given to thereactions by these methods are presented and also theadvantages and limitations of existing Lagrangianmethods are discussed. In the Time-Driven-Method
ARTICLE IN PRESS
E-mail addresses:
gurumunavalli@yahoo.co.in(G.R. Munavalli), msmk@civil.iisc.ernet.in (M.S. Mohan Kumar).0043-1354/$-see front matter
r
2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.watres.2004.04.007
(TDM) the constituent concentration of a segment issubjected to reaction at every water quality time step(Qstep). The Qstep is a computational time step at whichthe quality conditions of the entire network are updated.In the Event-Driven-Method (EDM) procedure theconstituent concentration in all the pipe segments aresubjected to reaction with respect to the length of thesubhydraulic time step[11]. In both methods the kineticreaction mechanism continues with time under theconditions of zero ﬂow or ﬂow reversal in pipes. Boththe TDM and EDM are free from numerical dispersionand phase shift errors when compared with Eulerianmethods. Basically the TDM simulation procedure iscarried out in steps of pre-speciﬁed Qstep. Hence it ispossible that during any step more than one segmentmay be consumed at the downstream node of a pipe. If the segments consumed have different concentrationsthen this leads to an artiﬁcial mixing whose effect ismore pronounced in tracing sharp concentration fronts.In addition, the TDM solutions are affected by a loss of resolution in concentration and accuracy is dependenton both Qstep and concentration tolerance used. Eventhough the EDM is supposed to be accurate irrespectiveof the Qstep used, the concentration tolerance used andthe tolerance dependent subsegmentation process atchanging hydraulic conditions may affect the accuracyof the method. In the EDM procedure the concentrationconditions at a node are updated only when an eventoccurs at that node. Also all the segments and nodes areupdated at the end of a hydraulic time step or outputreporting time whichever occurs ﬁrst. At the start of thesimulation the event occurrences are dictated by thetravel time in the pipes.Rossman and Boulos[13]tested and compared theEulerian (FDM and DVEM) and Lagrangian (TDMand EDM) methods. They concluded that the Lagran-gian methods are more efﬁcient for simulating thechemical transport in a water distribution system. Thetesting of the methods was done for analytical solutions,actual ﬁeld studies and variable sized networks. Themodels are contrasted with respect to analytical solu-tions for validation at zero concentration tolerance anda particular Qstep.It is useful to study the differences exhibited by theLagrangian methods for a normally used hydraulic timestep of 1h as reporting time under varying tolerance andQstep values with no restriction on the number of segments generated. It is also interesting to study howthe analytical solutions are contrasted with respect tothe solutions obtained by these methods under varyingconcentration tolerance and Qstep values. It is obviousthat the solution given by TDM and EDM may performbetter against the analytical solution for zero concentra-tion tolerance and reasonably small quality time step.The relative comparison of the methods with analyticalsolutions considering the variations in concentrationtolerance and Qstep brings out the degree of variabilityexhibited by the methods with respect to the truesolution for the system. Application of the methods toreal life networks will generate a large number of newsegments and this segmentation can be controlled byimposing a concentration tolerance.Also there is a need to develop a methodology whichcan nearly eliminate the limitations discussed earlier inthe Lagrangian models for the transport of chemicalspecies. A hybrid methodology developed herein utilizesthe better features of existing Lagrangian methods. Theperformance of all the methods is tested againstavailable analytical solutions under varying conditionsof concentration tolerance and Qstep for both reactiveand non-reactive constituents. The methods are alsoapplied to network problems of varying size and a set of solutions is obtained for a range of concentrationtolerance and Qstep values. An attempt is made tocompare the representative solutions given by existingmethods and the proposed hybrid method at selectednodes of a network problem. The results are interpretedin terms of the maximum number of segments generated(maximum segmentation of the network) at any timeduring the simulation and the solution time.
2. Governing equations
The methodology is predicated on the assumptions of one-dimensional ﬂow, single or consecutive steady-state(extended period simulation) network ﬂow hydraulics,complete and instantaneous mixing at the nodes, idealplug ﬂow with reaction, dispersion being negligible,single constituent with one or more feed sources and
ARTICLE IN PRESS
Nomenclature
C
E
external source concentration of constituent(M/L
3
)
c
i
concentration of constituent in pipe
i
(M/L
3
)
c
n
j
chlorine concentration at node
j
(M/L
3
)
I
number of incoming pipes at a node
I
s
set of links with ﬂows into the tank
J
s
set of links withdrawing ﬂow from the tank
N
j
n
total number of nodes in the network
Q
E
external ﬂow into a node (L
3
/T)
Q
i
ﬂow in pipe
i
(L
3
/T)Qstep water quality time step (T)
R
(
c
i
) ﬁrst-order reaction rate expression for pipe
i R
(
C
s
) ﬁrst-order reaction rate expression for tank
t
time (T)
u
i
mean ﬂow velocity in pipe
i
(L/T)
V
s
volume of storage tank (L
3
)
G.R. Munavalli, M.S. Mohan Kumar / Water Research 38 (2004) 2973–2988
2974
reactions based on ﬁrst-order kinetic characteristicfunctions.
2.1. Network model
A water distribution system comprises of links (pipes,pumps, valves) interconnected by nodes (junctions,storage points) in some particular branched or loopedconﬁguration. The network model is represented bynode-link system. A network water quality modeldetermines how the concentration of a dissolvedsubstance varies with time throughout the networkunder a known set of hydraulic conditions and sourceinput patterns.
2.2. Hydraulic model
The hydraulic simulation model[14]is modiﬁed tohandle the extended period simulation and is applied togenerate the dynamic ﬂows in pipes during the speciﬁedhydraulic time steps (normally 1h).
2.3. Water quality model
The water quality model formulation is from Ross-man et al.[10].
Transport of the constituent along the ith pipe is givenby the classical advection equation
:
@
c
i
@
t
¼À
u
i
@
c
i
@
x
7
R
ð
c
i
Þ
;
ð
1
Þ
where,
c
i
is the concentration of constituent in pipe
i
(mg/l) as a function of distance
x
and time
t
;
u
i
the meanﬂow velocity in pipe
i
(m/s); and
R
ð
c
i
Þ
the reaction rateexpression (equals zero for conservative constituent).
Instantaneous and complete mixing at the node is givenby the equation
:
c
n
j
¼
P
I i
¼
1
Q
i
c
i
þ
Q
E
C
E
P
I i
¼
1
Q
i
þ
Q
E
;
j
¼
1
;
y
;
N
j
n
;
ð
2
Þ
where,
I
is the number of incoming pipes at node
j
;
N
j
n
the total number of nodes in the network;
Q
E
theexternal source ﬂow into node
j
ð
m
3
=
s
Þ
;
and
C
E
theexternal source concentration into node
j
(mg/l).
Mass balance at storage tanks is given by
d
ð
V
s
C
s
Þ
d
t
¼
X
i
A
I
s
Q
i
c
i
À
X
j
A
J
s
Q
j
C
s
þ
R
ð
C
s
Þ
;
ð
3
Þ
where,
I
s
is the set of links with ﬂows into the tank;
J
s
the set of links withdrawing ﬂow from the tank;
V
s
thevolume of storage tank (m
3
);
C
s
the concentration of constituent (mg/l) within a storage tank;
R
ð
C
s
Þ
the ﬁrst-order reaction rate expression for a tank;
Q
i
the ﬂow(m
3
/s) in pipe
i
;
and
c
i
is the concentration of constituent(mg/l) in pipe
i
:
3. Numerical methods for water quality modeling
The existing Lagrangian methods are described indetail by Rossman and Boulos[13]and hence are notdiscussed here. The proposed hybrid method is de-scribed in the following sections:
3.1. Numerical hybrid method (EDMNET)3.1.1. TerminologyParcel
: It is an imaginary ﬁnite volume of waterwithin a pipe.
Segment
: It is the portion of a pipe volume consideredto be made up of a number of discrete parcels of water.It is assigned with a constituent concentration as aparameter and is represented by two separators at eachend.
Separator
: It is a line that separates two segments andis assigned with distance travelled with respect to theupstream end of the pipe (DT), time of creation (TC),time of arrival at its downstream node (TA) andeffective residence time (ERT) as parameters. The twoseparators of a segment are associated with the mostdownstream and most upstream discrete parcels of water within that segment.
Activity
: An activity is said to occur when a separatorin any of the pipe reaches its downstream node.
Effective residence time
(
ERT
): It is deﬁned for boththe separators and the discrete parcels of water. It is thetotal time taken by any discrete parcel of water/separator to reach the downstream node from theupstream node of a pipe. For any interior discreteparcel, (ERT) can be calculated using the time slope of asegment and the location of the parcel within thesegment.
Time slope
(
TS
): If ERT of a separator is representedby an ordinate, then the line joining ordinates of twoseparators for a segment represents the time slope.
Generation of new separator/segment
: A new separa-tor/segment in all outgoing pipes from a upstream nodeof the pipe is generated when the difference inconstituent concentration at that node and in mostupstream segment of the pipe exceeds imposed speciﬁedconcentration tolerance.
3.1.2. Basic concept of the method
It is a fact that the discrete parcels within a segmentfrom the downstream end to the upstream end havelinearly varying effective residence times which arerepresented by the time slope. At any stage during thesimulation, a discrete parcel of a segment in a pipereaches its downstream node. An ERT of that discreteparcel can be computed using the time slope and itsposition in the segment. As the constituent concentra-tion of that segment is known, the reacted concentrationfor that parcel of water can be determined. Thus the
ARTICLE IN PRESS
G.R. Munavalli, M.S. Mohan Kumar / Water Research 38 (2004) 2973–2988
2975
constituent concentration at any node can be computedby knowing these reacted concentrations of the discreteparcels reaching that node from the incoming pipes. Themethod is either governed by a speciﬁed water qualitytime step or the system activity. The process of computing the concentration and generating the newsegments (if and when required) is carried out at all thenodes irrespective of whether the Qstep or the systemactivity governs the simulation. The proposed method isdescribed in detail in the following subsections.
3.1.3. Initialization
At the start of the simulation, each pipe has a singlesegment with the ﬁrst separator at the downstream nodeand second separator at the upstream node. Thissegment is assigned with the constituent concentrationof the downstream node. The second separator (at theupstream node) has an ERT equal to the travel time of the pipe while the ﬁrst separator (at the downstreamnode) has zero value. The line joining the ERT of thesetwo separators represents the time slope as shown inFig.1. Also the second separator has time of arrival equal tothe pipe travel time whereas the ﬁrst separator is alreadyat the downstream node. These times of arrivalconstitute the scheduled activity times till any changein the hydraulic conditions occur. The second segmentfor this pipe as and when it is created will follow thesecond separator and it carries the concentration of itsupstream node till any change occurs in the concentra-tion at that upstream node of the pipe. Also this secondsegment has zero time slope indicating that all thediscrete parcels in this segment will have same ERTequal to pipe travel time till any change in hydraulicconditions occur.
3.1.4. Time step computation
The ﬁrst activity is scheduled to occur at a time equalto the least of all the travel times of separators in theentire network. Till this time the second separator of allthe pipes carry the concentration (which has notchanged since no activity occurred in the entire networkat any node) of the upstream node forward. But theconcentration at all the nodes is continuously changingand this change needs to be carried forward. Hence inthe case of a large time gap between two successiveactivities than Qstep, it is required to update theconcentrations at all the nodes in between intervals of Qsteps also. Thus the simulation clock is either movedto the next scheduled system activity time or theprevious time is increased by Qstep.
3.1.5. Sequence of steps at any timeCase
(a): If System activity governs the simulation.In this case a separator in one of the pipescorresponding to that activity reaches its downstreamnode and the separators in other pipes do not reach theirdownstream nodes. But in rare cases two activities occursimultaneously. In such cases the algorithm handles theactivities one by one. The separators in all the pipes aremoved forward by a length corresponding to the timeperiod equal to the difference of current system activitytime and previous time, and the distance moved by themis updated.First consider the pipe in which a separator hasreached its downstream node. The arrival of a separatorat its downstream node indicates that the ﬁrst discreteparcel of the next segment in line has reached thedownstream node effecting a change in concentration.As the ERT of the separator (and hence the discreteparcel) and the concentration of the segment are known,the reacted concentration contribution of this discreteparcel to its downstream node can be calculated. Thetime of creation for this discrete parcel is the differencebetween the current time and its ERT. If the reactioncoefﬁcient is considered to be varying with hydraulicconditions then the hydraulic time periods throughwhich this discrete parcel has passed should beidentiﬁed. And the discrete parcel is subjected to achange in concentration with the appropriate reactioncoefﬁcient and the corresponding time period. Thisprocess of computing the reacted concentration con-tribution is illustrated inFig. 2. In this ﬁgure thecomputation is illustrated for a discrete parcel which hasreached its downstream node at a time of 1370min. Byknowing its ERT (computed using TS) the TC can becomputed as 1224min. The discrete parcel has passedthrough different hydraulic time steps each having adifferent reaction constant. Then using the ﬁrst-orderreaction rate expression the reacted concentrationcontribution can be determined as represented in theﬁgure. The separators and segments are reordered forthis pipe. And the time slope for the most downstreamsegment is computed.Next all the pipes where the separators have notreached their downstream nodes are considered. As the
ARTICLE IN PRESS
Fig. 1. Deﬁnition sketch: initialization (EDMNET).
G.R. Munavalli, M.S. Mohan Kumar / Water Research 38 (2004) 2973–2988
2976

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