Modelling distribution storage water quality: An analytical approach

ELSEVIER

Modelling distribution storage water quality: An analytical approach Russell E. Mau Municipal Services Department,

Montgomery

Watson, Pasadena,

CA, USA

Paul F. Boulos Water Distribution

Technology Department,

Montgomery

Watson, Pasadena,

CA, USA

Robert W. Bowcock Azusa Light and Water Department,

Azusa, CA, USA

An explicit algorithm is developed for use in modelling waterquality within distribution system storage tanks or reservoirs with the resulting model cbseIy simuluting the dynamics of mixing of dissolved substances. The model is predicated on material mass balance that accounts for transport, mixing, and kinetic reaction processes whiIe inherently representing physically based phenomena such as short-circuiting and stagnation zones for ail types of distribution storage facilities. The performance of the model is illustrated by application to actual tank data taken from a prer;ious paper. The model-generated results show a good degree of correlation with the observed field measurements. The methodology should prove to be a valuable tool for managing water quality in water distribution systems. Keywords:

water

networks,

distribution

storage, water quality, mathematical model

1. Introduction The integrity of the nation’s aging hydraulic infrastructure systems is currently of widespread concern. The maintenance of treated water quality in water distribution systems is of principal concern. Historically these systems have been designed for efficiency in water delivery to points of use, hydraulic reliability, and fire protection while most regulatory mandates have been focused on enforcing treatment levels at the supplying plant. However modern-day purveyors have an additional mandate: to ensure that the distributed water is safe and conforms to an increasing number of current and emerging standards. Many of these standards must be met at the consumer’s tap forcing inclusion of the entire distribution system in compliance decisions. Specifically the Total Coliform Rule, the Lead and Copper Rule, the Trihalomethane OHM) Regulation, the pending Disinfectant/Disinfection By-Product Rule,

Address reprint requests to Dr. Boulos at Water Distribution Technology, Montgomery Watson, 300 North Lake Avenue, Suite 1200, Pasadena, CA 91101, USA. Received 1995

15 September

1994; revised 21 June 1995; accepted

Appl. Math. Modelling 1996, VoI. 20, April 0 1996 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010

15 August

and the Surface Water Treatment Rule are oriented toward water quality and monitoring in the distribution system. The lack of a disinfectant residual, the presence of coliform bacteria, and high levels of trihalomethanes or haloacetic acids in a distributed water can result in serious violations of regulations and subsequent public notifications. Water quality in distribution systems is influenced by a number of factors. Paramount among these factors are finished water storage facilities. These facilities have historically been considered neutral in terms of their effects on water quality. They have been located, designed, and operated based primarily on structural safety and hydraulic integrity and reliability. The latter objective pertains to the maintenance of system pressure, equalization of demands on supply sources, reduction in sizes or capacities of existing mains, provision of water for normal demands, and excess storage for peak demands and fire suppression, and supply of unintermpted water service during power outages. This philosophy can result in water that remains in the system for long periods of time and has continued to exist despite the recognition that long residence times (and improper mixing) can adversely affect distribution system water quality. The most notable effects on water quality include reduced disinfectant residuals, increased water age,

0307-904X/96/$15.00 SSDI 0307-904X(95)00129-8

Modeling

distribution

storage water quality: R. E. Mau et al.

proliferation of bacteria, occurrence of nitrification, growth of disinfectant by-products, and development of aesthetic changes in taste, odor, and appearance. Maintenance of such a philosophy may have been justified in the past due to the absence of the technology needed to evaluate such impacts and the existing level of regulatory control. The issue of distribution storage spreads the globe and cuts across utilities of all sizes from the small one-well systems to the large regional water suppliers. Many utilities will be forced to take corrective actions to meet stringent water quality standards. These actions may include improving tank operations and maintenance programs, more frequent exercising of storage tanks, routine water quality monitoring, evaluating tank structural integrity, and contriving design changes such as structural retrofitting. The results of either structural or operational changes may be either improved water quality or a reduction in the risk of detrimental or potentially serious or fatal water quality episodes. Identification of impacts of storage facilities on water quality and strategies to minimize adverse effects are required to ensure that the water quality meets the regulations and does not degrade in the distribution system. Along with these requirements has come an associated need for technology adequate enough to enhance utilities understanding of water quality behavior within distribution storage facilities. Mathematical modelling provides the most effective and viable means of predicting potentially negative impacts of distribution storage facilities on water quality and of evaluating a wide range of operating policies and alternative design strategies for improving distribution system water quality. In the past decade, several authors have proposed various algorithms for use in simulating the spatial and temporal variations of water quality in distribution systems. These techniques range from the use of steady-state’-5 to dynami&” mathematical model formulations. Steadystate models use the laws of mass conservation to determine the ultimate concentration distribution of water quality constituents that will take place if the distribution system reaches hydraulic equilibrium. Dynamic models rely on a system simulation approach to determine the movement and fate of dissolved substances under timevarying demand, supply, and hydraulic conditions. An excellent survey of the various approaches was previously provided.‘29’3 A principal application of these models has been the analysis of water age, conservative tracers, and chlorine residuals within the distribution system environment. While highly efficient algorithms have been developed for modelling water quality behavior in distribution systems, less success has been experienced with regard to development of rigorous distribution storage water quality models. The procedures reported were restricted to representing distribution system storage facilities as continuously stirred tank reactors (CSTR).6-” These models are predicated on instantaneous and complete mixing of material. However previous analyses of field sampling and laboratory studies of tanks showed that in many cases the complete mixing assumption may not be valid14,15; subsequently concentration gradients are not allowed to develop

330

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and uniform water quality is maintained throughout the tank volume. Important physical processes describing the complex internal mixing and flow interchange characteristics that occur within tanks have generally been neglected. Only when these mechanisms are considered can accurate predictions be obtained. As a result more accurate and reliable models of the dynamics of distribution storage water quality behavior are needed. In this paper an explicit mechanistic model for use in water quality simulation studies and management of distribution system storage is developed. The proposed model can be used effectively for simulating the inner mixing characteristics of tanks and the subsequent effects on water quality. The model provides the tool to calculate both the immediate reactions to short-term inflow and outflow episodes (shortcircuiting) and the gradual response to long-term use (dead zones). Such capabilities may assist in advancing and improving the methods currently in use in distribution system water quality management and will ensure that cost-effective solutions are identified. The proposed algorithm will be referred to herein as the linear compartment model (LCM). The model is formulated analytically from continuity and mass balance principles that account for advective transport, inner mixing, and reaction kinetics. The transport and mixing processes that take place within storage facilities are represented by discrete volume elements. Each such element corresponds to a separate tank compartment portraying specific inner mixing characteristics (e.g., shortcircuiting, bulk tank volume, stagnation). The concept of segregating tanks into separate compartments for analyzing internal mixing was previously discussed’“,17 for tanks-in-series and cross-flow models. For our application these models exhibited certain limitations. In general the developed mathematical models only envisioned one cross-flow compartment per tank, all compartments were of fixed and/or equal volumes, no flow reversal was modelled with the compartments in series, and no chemical reactions were considered. In this work the compartments act in series, and mixing within the tank is described by applying mass exchange between compartments. The degradation of any chemical species in the water is assumed to be described by a single kinetic decay coefficient. No specific hydrodynamic parameters are included to account for wind or thermal mixing due to uneven solar radiation. The effects of these presumably would modify fitting parameters in the model that can be determined by field measurement. Input to the model consists of three parts: initial condition data, time dependent data, and constants. In this report measured values refer to either physically measured values for the variable of interest at the tank or model-generated data for a distribution network. The initial condition data refer to the initial concentrations and volume in the tank that can either be measured or estimated. For the data measured over time, the flow rates to and from the tank and the concentrations at the inlet/outlet of the tank are measured. Last, the fixed volumes for the different discrete volume compartments, the exchange rates between compartments, and any kinetic decay coefficient describing degradation are estimated. The model output consists of time-dependent concentration values for the various components of a

Modelling

distribution

storage

water quality:

The volumetric rate of change continuity principle d”-r = dt

R. E. Mau et al.

can be derived

from

Qin - Qout

(4)

The solution of equation (4) for constant flow rates with the application of the initial condition listed in (3) is “T=

(Qin -Qout)'+ "T,

(5)

Application of the chain rule to the partial derivative equation (1) is given in the following: “, dC, =-----+ dt

a(C,“T> at

Substituting equations like terms yields Figure 1.

Conceptual stirred tank reactor.

schematic

representation

of completely

C, d”, (6)

dt

(4) and (6) into (1) and cancelling

dCT Qin

-=--(Ci,,-CT)-kCT dt T At this step, the following

tank system. The performance of the model is demonstrated by application to actual tank data. In the next section, the mathematical basis for the CSTR is reviewed and modified for direct application to the LCM. In the rest of the paper, the development of the LCM is completed and applied to real tank data.

Generally a single, variable-volume CSTR with instantaneous mixing is used to simulate the mixing effects within a reactor vessel. The model schematic is shown in Figure I. The mass balance equation for this mixing in the reactor is

is made for CT: (8)

This new expression for C, can be substituted into equation (7). Application of the chain rule on the partial derivative and cancellation of like terms yields

Q,

= v

dt stirred tank reactor model

substitution

CT = f[ t]eLmk’l

df[ t]eL-ktl

2. Completely

( Ci, -f[

t]e[-k’l)

C,(t=O)

The boundary condition for concentration can be multiplied by an exponential term of the same form as the first-order decay term as follows C,,[ t] = Cine[-k’l

(10)

If this is substituted into equation (91, then the exponential can be factored from the equation resulting in

(11)

for concentration

=c,”

(2)

=&,

(3)

and for volume V-Jt=O)

(9)

T

!g! =$(Ci”-f[ t]) with the initial condition

in

where C, V, and Q refer to concentration, volume, and flow rate, respectively; k is a first-order kinetic rate constant; and the subscripts T and both in and out refer to the values for the reactor and the system, respectively. In equation cl), the total volume in the CSTR is dependent upon both Qin and Q,,,. This differential equation can be solved analytically only under special conditions, e.g., a constant volumetric rate of change (constant flow rates). Application of this condition transforms the partial differential equation into an ordinary differential equation. Prior to solving this differential equation, a description for both the reactor volume and the time rate of change of volume is needed.

This exercise shows that the actual concentration in the reactor can be determined by ignoring the decay term in the initial equation and solving the following: dCT -

dt

Qin

=

j+Ci”

-C,)

(12)

T

For reactive species that display first-order degradation, the derived solution can then be multiplied by an exponential to account for decay in order to define the “true” concentration. It should be noted that the choice of boundary concentration, i.e., Gin, is arbitrary provided so that the boundary condition concentration is independent of the reactor concentration, which holds in this case. The use of the boundary concentration shown in equation (10) does not change the accuracy of the solution provided narrow time steps are maintained. Furthermore the constant portion of the equation can be chosen so that the total mass input to the system over a given time step (integration of

Appl.

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1996, Vol. 20, April

331

Modeling

distribution

storage water quality: R. E. Mau et al. Comportment

equation [lo]) exactly equals the mass input as measured in the field. The direct integration of equation (12) and application of initial conditions for concentration and volume shown in equations (2) and (3), respectively, yield the following solution:

1 + (Q,

and in dimensionless

(13) I

form:

1 + (Q,

Figure 2. Conceptual partment model.

-Qout)t

VTO

schematic

representation

of linear com-

(14)

These equations apply for either flow into or out of CSTR or both. The equations can be simplified further assuming flows will be either into or out of the reactor not both. For positive flows entering the system, following holds:

the by but the

(15) and in dimensionless

D

Dead Zone

-(&)

-Qout)t VTO

/

system for a given time period. The final two compartments (i.e., C and D) are considered to be zones of low exchange with the initial portions of the reactor system. In general each successive volume element only exchanges with adjacent compartments. For each compartment, a general mass balance equation can be written as follows: compartment A

dC.4 v*= QinCin +

form

dt

QsAC, - (QAB + Qa"t>C*

(1% (16)

compartment

B

a(cevB>

= Q,C,

at For the opposite case of positive the following applies: ‘T

=

and in nondimensional CT -=

C r0

(20)

flows exiting the vessel, compartment (17)

cTo

C

dCc vc- dt =

terms

1

+ Qc&c - (Q,,+ Quc)G

Q,&,

+ QD&

(21) (18)

and compartment

D

dCn Vll- dt = Qcocc - Qocco with the following compartment A

3. Linear compartment model Details regarding the mathematical formulation of the LCM are presented with reference to the model schematic illustrated in Figure 2. The approach is to represent the tank reactor as four discrete volume elements each predicated on the physical characteristics inherent in the tank. compartment A is the boundary volume element for the reactor where inflows to and outflows from the system occur, while compartment B is the only compartment with a variable volume. The volume of compartment B is a function of the net volume entering or leaving the tank

332

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- (QCB+ QCD>~C

1996, Vol. 20, April

C*(t=O)

initial conditions

(22) for all compartments:

=c&

(23)

C,(t=O)

=caO

(24)

Va(t=O)

= vu,

(25)

=ccO

(26)

compartment

compartment Cc(t=O)

B

C

Modeling and compartment Cn(t=O)

D =cnO

(27)

where C, V, and Q are the concentration, volume, and flow rate, respectively; the subscripts A, B, C, and D designate the respective compartments; and the subscripts AB, BA, BC, CB, CD, and DC refer to the source and destination compartments, e.g., Al3 refers to flows going from compartment A to compartment B. Inspection of these equations indicates that each is linearly dependent upon the mass balance for the adjacent compartments. Equation (20), including initial conditions listed in equation (24) and (251, is similar to equation (1) for a simple variable-volume CSTR. This partial differential equation can be reduced to an ordinary differential equation by application of the chain rule on the derivative on the left-hand side as completed in equation (6) and as shown in equations (3)~(6) (substituting the subscripts AB for in and BA for out). This gives Cu dVEl

vu dCr3 =p++ dt

8(CsVu) at

= v,-

+

C,
where V, is similar to equation

-

QBA)

for the partial

(28)

(5) as follows:

(29)

v,=(Q.m-Qdt+b, Substituting yields

This is a set of linearly dependent first-order ordinary differential equations. Reduction of this system to a single equation is a trivial exercise; however, the final outcome would be a single fourth-order nonhomogeneous ordinary differential equation with nonconstant coefficients. To ease the complexity of the system and make analytical calculations possible, certain assumptions and simplifications are required. For the simple single compartment CSTR, an exact solution was derived for the condition of only one variable concentration. In the LCM, if the linear dependence between adjacent compartments for concentration could be relaxed by assuming constant concentrations (multiplied by an exponential) in adjacent compartments, then the problem becomes one of four linearly independent ordinary differential equations each with a unique solution. The use of this assumption in addition to the assumption of constant flow rates previously employed in the CSTR analysis is explored in the following development of the equations governing the mass balances for the separate compartments in the LCM.

derivative

in equation

(20)

3.1. Compartment

A

A solution for equation (19) using the initial condition listed in equation (2) is derived. The application of the assumptions of constant flow and constant concentration (multiplied by an exponential) outlined previously transform the ordinary differential equation into a definite integral that can be directly integrated yielding (Q, = Qin

and QBA = Qout):

dCr5 Vu= dt

QmC,+Qc&-

(Q~+QBc)CB

c

(30)

_ A-

The set of equations formed by (19), (21), (22), and (30) can be placed in matrix form as follows:

!&

storage water quality: R. E. Mau et al.

dt

dC, dt

distribution

Qincin+ QoutCB i Qin + Qout + QoutCB Qin + Qout

QinCin

-

_(QAB;QqcA+($qcB+o+o

-

‘A,

e[-

(Qou,+ Q,.)l I “.A

(32) and in dimensionless

I

1+

CA -= CA0

I

form:

Qout cB ’ -Qin

Gin Gin

$?_

’ + Q;:' ’1

I

'A,

Qout 'B Qin Gin

--1 ‘in

CA0

1 et-

(Qo.,+Q,n)r V, 1

(33)

dCn -=o+o+($)cc-(F)cn+o dt

(31)

This equation applies for either flows into or out of the compartment or both; however, the summation of all flows to the compartment must equal zero. By applying the assumption of unidirectional flows, equation (33) can be

Appt. Math. Modelling,

1996, Vol. 20, April

333

Modeling

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storage water quality: R. E. Mau et al.

simplified to the following compartment:

for inflows

Qac Cc -QAB CA

to the reactor

1+ ?I

CA = Ci, - ( Ci, - CA,)&

1+2

(34)

cA --

C

Q

1

BO

Q AB and in nondimensional

form Q,w+Qec x I1 +

(35)

(QAB

-QBA)~]-~Q~-~~A’ TI

“&I

1 For outflows

from the compartment

These solutions apply for flows either into or out of the compartment or both. These equations can be simplified if flows are assumed to be unidirectional. For positive inflows to the tank, the solution can be simplified to

form

‘B

‘A -=--

cA0

is

$fl

CA = C, - (C, - C,,)e[and in dimensionless

the equation

(37)

c, =

QABCA + QB~CC QAB + Qec

‘A,

QABCA + QecCc _ c

_ 3.2. Compartment

QAB+

B

A similar solution process is followed as presented previously for the variable volume CSTR, except that in the LCM compartment B accounts for the exchange with compartment C. This exchange is assumed to be both constant for all analyses and independent of flows into or out of compartment A. The mass balance differential equation shown in equation (30) together with the initial conditions shown in equations (24) and (25) is similar to (1) and (14) for a simple variable-volume CSTR. Substituting V, from equation (29) into equation (30) yields (where Q,, =

QABCA+ QCZIS%- ( QAB + Qec >CB

dt

[(QAB - QuAIt+

811

QBC

(41) and in nondimensional ,

form

Q,, Cc -lf

QAB CA 1+x

Qsd dC, -=

(40)

1

Q AB

\ (38)

Q

’ 1 + --

QBC

Va,)]

Cc

QAB CA Now equation (38) can be solved by direct integration. The application of the initial condition for concentration yields the following solution:

Q Q AB

\

QABCA+QCBCC

c B

QAB + QBC -

(42)

QABCA + QCBCC _ c QAn + QBC 1 + (Q,,

BU

-(i$i%)

-Q,,)t V B,

and in dimensionless

1+2

and for positive outflows from the tank, the solution can be simplified to

(39)

1

C, = C, - (CC - c,o)

and in dimensionless

Q

Math.

terms:

CA C a0

~~%(~_l)(l_!K)~~~ C

Q AB

Appl.

(43)

form

QB, Cc -l+QA,&,

334

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Modelling,

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(44)

Modelling 3.3. Compartment

C

3.5. Summary

The mass balance equation for this compartment includes terms relating mass exchange from both compartments B and D. The exchange between both adjacent compartments is assumed to be constant for all analyses and independent of reactor system boundary inflows or outflows as specified for compartment B. By applying the same assumptions as outlined above, the ordinary differential equation in equation (21) is transformed into a definite integral that can be integrated directly. Integrating equation (21) and using the initial condition listed in equation (26) result in the following solution : Q~&B

c C

+Qc&o

Q,, + Qm QcsCs + Q,&D

-

e,_

(Q=+Qco)‘l "C

Qcs + Qc,

(45) and in nondimensional

cc -=

form

11 + -Qc, CD ’

Qcs CB CB

CCO

1++

Q

c,o )

UCB

\

1+

QCD CD -QCB 'B

1+2

Q

CB --

C

1 et-

(Qc,+ Qco)' V, I

C0

Q CB

(46)

These two equations from the reactor.

3.4. Compartment

storage

water quality:

of mathematical

R. E. Mau et al.

models

The solutions for simulating concentrations within a reactor are explicit and analytical provided that the assumptions regarding constant flow rates and constant concentrations for adjacent compartments hold. For real systems, this implies short time steps over which variability can be reduced, thus reducing the error incurred by these simplifications. A trivial exercise will show that the error incurred by these simplifications is insignificant and bounded for the types of real systems studied in this paper. The LCM provides for the direct calculation of reactor system concentrations given that the necessary input data are available or estimated. As mentioned previously, the model requires three types of input data: initial conditions, time-dependent measurements, and constants. The LCM is a five hydraulic parameter model whose fitting parameters consist of the fixed volumes for compartments A, C, and D and the volumetric exchange rates between compartments B and C and between C and D. The LCM does not predict the values for these constants; however, the choices for each are governed by certain criteria (mostly intuitive) narrowing their relative ranges. Compartment A should have a small volume in order to respond quickly to variations in concentration near the inlet/outlet of the system for both inflows and outflows. Compartment B should be the substantial portion of the volume except in extreme cases because the bulk of the system is expected to fully mix with the incoming flows. Compartments C and D represent zones of limited mixing and are inherently expected to be of limited extent. The actual size of these compartments will depend greatly on tank geometry and inlet/outlet configuration. Once these constants are determined for a particular tank, the LCM can be used to predict concentration distributions within the reactor system for given distribution system characteristics and timedependent flow and chemical species concentration data.

apply for either inflow to or outflow

D

Compartment D exchanges directly with only compartment C, and this exchange is also assumed to be constant for all analyses independent of reactor system boundary inflows and outflows as mentioned for the other compartments. A similar solution process is employed for equation (22) using the initial condition listed in equation (27) yielding (Qc,

distribution

= Q,,):

cD = Cc - (C, - CD,)e[- %I and in dimensionless

(471

terms

(48 These two equations from the reactor.

apply for either inflow to or outflow

1

4. Implementation

of model

The applicability of the LCM has been tested by comparing model-generated values to field data for a single tank using both chlorine and fluoride as reactive and conservative species, respectively. The collected field data were part of an extensive field study completed by Grayman and Clark.14 In their work, the concentration of the chemical species were measured at the inlet/outlet piping of the tank. The tank schematic is shown in Figure 3. The input data for the individual model parameters are similar to those given in Grayman and Clarkr4; however, the LCM is a five hydraulic parameter model rather than a three parameter model. For this model, the overall geometry of the tank as selected in Grayman and Clark14 was maintained including the total fixed volumes and the exchange rates between the dead zones and the bulk of the tank. The volume for compartments A, C, and D were selected as

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2;:: f 0.7

”

0.6

5 0.5 E

0.4 9 ; 0.3 O.? 0.1 0

8,000,75,000,and 25,000gallons (30,280, 283,875, 94,625 l), respectively, resulting in a total volume of dead zones as 100,000 gallons (378,500 1). The exchange rate between compartment C and B was selected as 400 gpm (25.23 l/s) which was approximately an average of the inflow and outflow tank flow rates while the final exchange rate

Figure 5. Comparison of completely stirred tank reactor data to field measured data for concentrations of: (a) chlorine and (b) fluoride.

. -

12.5

13

13.5

14

14.5

15

155

M-d COMPA

-.._...

(..),$a8

~~~~~~~~

COMPC

-

COMPD

-

16

nm., Oayl

Figure 4. Comparison of field measured data for volumes and concentrations: (a) chlorine and (b) fluoride.

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Figure 6. Comparison of linear compartment model data to field measured data for concentrations of: (a) chlorine and (b) fluoride.

Modeling

distribution

storage water quality: R. E. Mau et al.

models are shown to be remarkably similar; however the use of a compartment A to model the inlet/outlet allows for a quick response to inflow and outflow conditions.

5. Discussion

Figure 7. Comparison of model-generated data for bulk volume compartment for the two models for concentrations of: (a) chlorine and (b) fluoride.

between compartments C and D was chosen as 50 gpm (3.15 l/s). Identical values for the constants were employed for both chlorine and fluoride simulations, even though independent values would have optimized the fit for each case. The volumes and exchange rates are hydrodynamic parameters and remain constant for the same tank regardless of measured chemical species properties. The comparisons of the model generated data to the field data for both the chlorine and fluoride are presented in unison due to the model similarities. A single solution was derived for each model regardless of chemical species with the only difference being the inclusion of a first-order decay constant to account for the degradation of chlorine. A decay coefficient of l.O/day was used. In Figure 4 the actual field data for both chlorine (Figure 4~) and fluoride (Figure 4b) are presented where both the tracer concentrations and tank volumes are shown versus time. The next sequence of figures are direct comparisons of the values calculated by the two models to the field data for both chlorine and fluoride. Figures 5 and 6 correspond to comparisons for the CSTR and the LCM, respectively. In these figures, part (a> shows the chlorine comparisons while part (b) shows the fluoride. In Figure 7, the bulk volume concentrations calculated by the two models are compared simultaneously with each other for both chlorine in part (a) and fluoride in part (b). The exaggerated vertical scale for the two parts of this figure should be noted to avoid misconstruing the comparison. The bulk volume concentrations calculated by the two

The results of the model developments and comparisons of the data generated by the two models presented in this work to the measured field data can be summarized as follows: The LCM developed in this work is nonpredictive in terms of fitting the five hydraulic parameters of the model and requires extensive field measurements for application to real systems; however, the model closely simulated field data for the study tank after selecting the best-fit values for the parameter constants. The concentrations calculated for the bulk volume of the tank for the models closely matched each other; however, the LCM provided significantly better agreement with field data at the inlet/outlet location of the tank than the CSTR. The LCM was applied to only one real system. A complete verification of the model would require the measurement of data for other tanks of differing geometry, inlet/outlet configuration, and of the concentration across the entire tank. Such data collection would indicate both the presence and extent of dead zones (verify choice of volume constants) and define the average concentration in the bulk of the tank. These data then could be used directly to either refine the volume constants for the simulations or provide additional points to confirm model results. For inflow to the tank, the use of the measured field data as direct input to the LCM forced or biased the calculated concentrations in compartment A to closely match the measure data, whereas the good fit of the calculated data for flow leaving the tank provided verification of both the LCM and the concentrations in the various compartments in the tank.

Conclusions The focus on water quality in water distribution systems has recently been shifted from the point of treatment to the point of consumption. This shift has prompted a need for improved technology to provide a thorough understanding of the transport and mixing processes that take place in distribution storage facilities and their impacts on the quality of the water in the distribution system. This paper has presented a new methodology for use in such an analysis. Specifically the proposed model permits the analysis of both immediate reactions to short-term inflow and outflow episodes and gradual response to long-term use. This is accomplished by disaggregating the tank into discrete volume compartments based on the physical characteristics inherent in the tank. The compartments operate in

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series, and mixing within the tank is described by applying mass exchange of material between compartments. The resulting model can be used effectively in water quality simulation studies and for the management of distribution system water quality.

Acknowledgments This material is based upon work partially funded by the American Water Works Association Research Foundation under project 81592. This project is entitled “Characterization and Modeling of Chlorine Decay in Distribution Systems,” Project Officer Mr. Albert Ilges. The support received is gratefully acknowledged. The authors are also grateful to the anonymous reviewers for their useful comments.

Nomenclature volume in the tank concentration in the tank flow rate out of the tank flow rate into the tank first-order kinetic rate constant concentration in the tank to start volume in compartment A the flow from compartment A to B

References 1. Chun, D. G. and Selznick, H. L. Computer modeling of distribution system water quality. Computer Applications in Water Resources AXE, New York, NY, 1985 2. Males, R. M., Clark, R. M., Wehrman, P. J. and Gates, W. E. Algorithm for mixing problems in water systems. J. Hydraul. Div. Am. Sot. Civ. Eng. 1985,111(2),206-219

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