simulation of free chlorine decay and adaptive chlorine dosing

8/6/2019 Simulation of Free Chlorine Decay and Adaptive Chlorine Dosing

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Chemical Product and Process

Modeling

Volume 2, Issue 2 2007 Article 3

PAPERS FROM CHEMECA 2006

Simulation of Free Chlorine Decay and

Adaptive Chlorine Dosing by DiscreteTime-Space Model for Drinking Water

Distribution System

Abrar Muslim Qin Li

Moses O. Tade

Curtin University of Technology, Perth, Western Australia, abr [email protected] University of Technology, Perth, Western Australia, [email protected] University of Technology, Perth, Western Australia, [email protected]

Copyright c2007 The Berkeley Electronic Press. All rights reserved.


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Simulation of Free Chlorine Decay and

Adaptive Chlorine Dosing by Discrete

Time-Space Model for Drinking Water

Distribution System

Abrar Muslim, Qin Li, and Moses O. Tade

Abstract

An ideal drinking water distribution system (DWDS) must supply safe drinking water with

free chlorine residual (FCR) in the form of HOCl and OCl- at a required concentration level. The

FCR as a disinfectant decreases over time due to chemical reactions in the bulk phase and at the

pipe wall in the DWDS. In order to supply drinking water with the FCR concentration within

the safety range of 0.2-0.6 mg/l at the points of water consumption, it is important to develop

a dynamic model of the FCR using a discrete time-space model (DTSM) that accounts for free

chlorine transport in the axial direction by convection, diffusion and the decay kinetic. A DTSM

has been developed using Finite Difference Method (FDM) to predict the FCR in single pipes in

the DWDS. The DTSM has been computed using Matlab 7.0.1 and tested with step inputs and

rectangular pulse inputs to estimate the FCR at any point in the pipes over time. Data found in the

literature have been used to validate the DTSM. The modelling and simulation study shows that

the water velocity significantly affects the FCR concentration distribution along the pipe. Due tothe fluctuation of the drinking water demand, a model-based adaptive chlorine dosing scheme is

proposed to control the proper injection of chlorine.

KEYWORDS: free chlorine residual, discrete time-space model and simulation, model-based

adaptive chlorine dosing

The authors are very grateful to Curtin University of Technology for technical support and pro-

viding the Matlab 7.0.1 software used in this research. The authors would also like to thank the

anonymous referees of Journal of Chemical Product and Process Modeling (JCPPM) for their

valuable suggestions to improve the readability and quality of this paper.


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INTRODUCTION

Drinking water should be free of pathogenic microorganisms that can cause

water-borne diseases. According to World Health Organization (2004), almost 3.3million people (mostly children) die every year from these diseases. Thus,

reducing the level of pathogenic microorganisms in water that cause these

diseases is still a major objective, particularly in developing countries andregions.

There are many water treatment processes that may contribute to pathogen

reduction, but according to Vasconcelos et al. (1997) chlorination is the most

widely used method for water disinfection. As a disinfectant, when gaseouschlorine is dissolved in water, the hydrolysis reaction of chlorine occurs to form

hypochlorous acid (HOCl). The complete reaction occurs in a few tenths of a

second at 18oC and within a few seconds at 0

oC at normal pH ranges (6.5 to 8.5).

Hypochlorous acid and the hypochlorite ion, known as the FCR, can kill pathogenin several seconds (Connell, 1996).

On one hand, the FCR in the DWDS decays due to its reaction withoxidisable compounds (Wei and Morris, 1974) and it rapidly decays with distance

from the treatment plant (Maul et al., 1985). On the other hand, the DWDS must

supply safe drinking water with the FCR concentration in the range of 0.2-0.6mg/L and a maximum amount of disinfectant by-product of 0.1 mg/L that must be

presented at the points of water consumption (AWWA, 2003).

A number of researchers have developed chlorine decay models to predict

chlorine concentration in drinking water. A chlorine consumption model,

1

n

tC C t= was developed by Feben and Taras (1951) in which Ct is the chlorine

consumed (parts/million) at time, t (hr); C1 is the amount of chlorine consumedafter one hour of contact; and n is the exponential constant characteristic of thegiven water. The long-term chlorine decay model for distributed drinking water

and natural waters, 0 exp( )tC C kt = , has been modeled by Johnson (1978) using

first-order kinetics which is the same as the model developed by Heraud et al.(1997), where Ct is the chlorine concentration at time t; C0 is the initial

concentration; t is time (hr) and k represents a first-order reaction rate constant

(hr-1

). Rosman et al. (1994) proposed a one dimensional mass conservation

equation for a dilute concentration of total FCR in water flowing through asection of a pipe using the Discrete Volume Element Method. However, their

results on chlorine residual are only resolved with time.

This study presents the model development and simulation of the FCRusing a discrete time-space model (DTSM) resolving both temporal and spatial

distributions, and also proposes a model-based adaptive chlorine dosing to

maintain the FCR concentration along the single pipes of the DWDS.

1

Muslim et al.: Simulation of Free Chlorine Decay and Adaptive Chlorine Dosing

Published by The Berkeley Electronic Press, 2007


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MODEL DEVELOPMENT

Figure 1. A schematic drawing of the model for free chlorine residual transport

in a single pipe of the DWDS

A single input single output model, as shown in Figure 1 for the FCR

concentration in a pipe of the DWDS was developed using a mechanistic

approach. Here a space element x and time element tare chosen to develop theoverall mass balance by considering the FCR in the form of both HOCl and OCl

-,

and assuming that the water flow in the axial direction only. The overall mass

balance equation can be written as below:

)||(2211 xxxCC

DAVCkCFCFt

CV

+=

(1)

which can be discretized as:

1 1 2 2( ) ( )

t

t t t x x x

t t

V C V C FC F C dt

= +

[ ]1 21 2( ) ( )t t

x x x

t t t t

C CD A D A dt kC V dt

x x

(2)

where,

V = The volume of the imaginary control volumeA = The cross section area of water flow in the pipe (m

2)

C = The FCR concentration in the control volume, CV (g/L)C1 = The FCR concentration enters the CV, reaching the axial direction x -

x (g/L)C2 = The FCR concentration enters the CV, reaching the axial direction x

(g/L)

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DOI: 10.2202/1934-2659.1067


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D1 = The effective diffusivity rate constant of the FCR enters the CV,

reaching the axial directionx - x (m2/s)D2 = The effective diffusivity rate constant of the FCR enters the CV,

reaching the axial directionx (m2/s)

F1 = The inflow water enters the CV, reaching the axial direction x - x(L/s)

F2 = The outflow water enters the CV, reaching the axial directionx (L/s)

k = The overall chlorine decay rate constant in the CV (s-1

)

Equation (2) can be simplified into a dynamic model for the FCR transport in theaxial direction by convection and diffusion:

2

2

C C Cv D

t x x

= +

kC (3)

where k is the overall chlorine decay constant (s-1

), v is the water flow velocity(m/s) andD is chlorine effective diffusivity coefficient (m

2/s). The Peclet number,

Pe, can be calculated as Pe =Lv/D, whereL is the pipe length (m). When the Pe

>> 1 (turbulent condition), the governing equation for the FCR in pipes under thiscondition is

C Cv kC

t x

=

(4)

In Equation (4), the FCR transport dynamic model in the axial direction byconvection is almost the same as the one dimensional conservation of mass

equation for a dilute concentration of total free chlorine proposed by Rosman et

al. (1994).

Solution Procedure

The DTSMs of chlorine decay with axial diffusion in a single pipe of the DWDS,

as the numerical solutions of Equations (3) and (4) can be obtained by adoptingthe explicit scheme of the FDM of Euler method (Tveito and Winther, 1998) in

discretizing both time and space (pipe length). The concentration C at [ix,

(j+1)t] which is the point of time and space needed to predict1j

iC+

.

[ ]

=

+ j

i

j

i

j

i

j

i Cx

tvC

x

tvCC 1

1

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1 12j j j

i i iC C Ct t tD D Dx x x x x x

+

+ +

j

itCk (5)

Bring the jiC in the LHS to the RHS, the discrete time-space model (DTSM) of

chlorine decay with axial diffusion in a single pipe of the DWDS, is

1

1 121 2

j j j j

i i i i

t D t t t C v C v D k t C D C

x x x x x

+

+

= + + +

(6)

where,j

iC 1 = The FCR concentration at [(i-1)x, jt] (g/L)

1ji

C+

= The FCR concentration at [(i+1)x, jt] (g/L)j

iC = The FCR concentration at [ix, jt] (g/L)

1+j

iC = The FCR concentration at [ix, (j+1)t] (g/L)

D = The effective diffusivity constant of chlorine in water (m2/s)

i = The time steps, i = 1,,m-1j = The space steps,j = 0, , n-1

k = The overall chlorine decay rate constant in the V(s-1)t = T/m, T is duration of time for chlorine transport in the pipe and m is

number of time steps in the time elementv = The water velocity in the pipe (m/s)

x = L/n,L is the length of single pipe and n is the number of space steps inthe space element.

The initial concentration of chlorine in the system is given by the equation below:

0

iC = a given value for i = 1, m 1 (7)

The upstream concentration of chlorine in the pipe from the injection point is

given by Equations (8) and (9):

jC0 = a given value forj = 1,, n (8)1

0

jC + = a given value forj = 1,, n (9)

In the turbulent condition, where the axial diffusion term can be neglected, by

discretizing the time and space (pipe length), Equation (4) can be written as thefollowing:

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[ ]

=

+ j

i

j

i

j

i

j

i Cx

tvC

x

tvCC 1

1 jitCk (10)

This can be rearranged into:

j

i

j

i

j

i Ctkx

tvC

x

tvC

+

=

+

111

(11)

In this case, the boundary condition of upstream chlorine concentration for

Equation (10) is reduced into Equation (8) only.

Analytical SolutionFor the DWDS, the axial Peclet number, Pe is typically found to be very large

(>>1). Thus, it is not necessary to develop the analytical solution for the dynamic

model for FCR transport in the axial direction diffusion. To obtain the analyticalsolution, a coordinate transformation is needed, together with the Chain Formula

(ONeil, 1987), to convert Equation (3) to the ordinary differential equation form

by substituting the coordinate transformation derivatives and using the initial

chlorine concentration ( 0)C t Co= = . The analytical solutions are written as the

follows:

0 exp( )tC C kt = (12)

0 exp( / )xC C kx v= (13)

where Ct and Cx (mg/L) are the FCR concentrations after the residence time of t

(hr) and the distance ofx (km), respectively, C0 is the initial concentration, v is the

water velocity (km/hr) and krepresents the overall chlorine decay rate constant.

RESULTS AND DISCUSSION

The DTSM and the analytical solution

The DTSM that is the numerical solution of Equation (4) is given in Figure 2.b

with step input shown in Figure 2.a. The analytical solution of the DTSM

(DTSMAS) is shown by Figure 2.c. The k and v for the model simulation areobtained from the paper of Rosman et al. (1994). The DTSM above shows the

FCR concentration along the space over time with the kof 0.208/hr, the v of 0.36

km/hr and the initial chlorine concentration (Co) of 1 mg/L, injected at the pointof 0 km where the first chlorine booster is installed.

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(a)

(b)

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(c)Figure 2. Comparison of the DTSM for different injection time against the DTSM

analytical solution (DTSMAS) on the FCR decay in a single pipe of DWDS

In Figure 2.b, the first curve from left side is the FCR concentration after 1 hour

step input. On its right side, sequentially, the 2nd curve, 3rd curve and thefollowing curves represent the FCR concentration at 2, 3 up to 12 hours (injection

time, T=12hrs) respectively, until the last curve which represents the DTSM stepresponse at 50 hours. Figures 2.b and 2.c show that the numerical solution viaDTSM is able to capture the transient behaviour of FCR distribution, and the step

input is long enough for steady status to develop the numerical solution matches

the analytical solution. In this simulation, the DTSM needs at least 12 hours

injection time (step input mode) in order to fit the response with the analyticalfrom 0 to 4 km of the space length. Overall, the chlorine decay model follows

exponential decay trend as concluded by the DTSM simulation, as also reported

by Johnson (1978).

Considering the effective diffusivity coefficient of chlorine in the DWDS

So far we have not found literature that presents comparisons between thechlorine decay model results in pipes of a DWDS with and without consideringthe chlorine effective diffusivity coefficient (CEDC), apart from the work of

Tzatchokov et al. (2003), where they proposed a non-steady advection dispersion

reaction model with consideration of a fluoride effective diffusivity coefficient in

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the simulation. Therefore, there is a need to simulate the DTSM when considering

the CEDC.

(a)

(b)

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(c)Figure 3. Comparison of the DTSM using the reference CEDC against the DTSM using

a simulated critical CEDC on the FCR decay by axial diffusion in a DWDS single pipe

Figure 3 shows the DTSM results with 12 hours (injection time, T=12hrs) of

rectangular pulse input (RPI) shown in Figure 3.a. The response during the RPI is

shown by the first curve from the left side and the responses after the RPI, at

every 6 hours, for the elapsed time of 6 to 36 hours (Te=6:6:36) are shown on theright side. The simulation parameters are taken from the example of the Brushy

Plain-Cherry Hill network (Biswas et al., 1993). We used the data for the pipenumber 10 for the smallest CEDC in the water, Da = 1.8 x 10

-5m

2/s or 6.48 x 10

-8

km2/hr with the smallest water flow velocity of 0.0504 km/hr and the chlorine

decay constant of 0.02304/hr. The large CEDC, Db = 1.03 x 10-2

m2/s or 3.708 x

10-5

km2/hr, which is ten times the value normally used and is the largest CEDC

in the network (the pipe number 1). These parameters are reasonable because the

largest axial CEDC, at the smallest water flow velocity, results in the smallest

axial Peclet number that leads to the largest axial CEDC (U.S. EPA, 2002).However, as revealed by the main plot in Figure 3.b, the FCR

concentration along the pipe over time, with the CEDC of 1.8 x 10 -5 m2/s (Da),appear the same as the FCR concentration without taking into account CEDC.Moreover, when the largest CEDC of 1.03 x 10

-2m

2/s (Db) assumed to occur in

the Brushy Plain-Cherry Hill network is applied, the response is almost the same

and even if the coefficient is neglected, the FCR concentration doesnt change.

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For the DTSM step input, the response is also the same as concluded in Figure

4.a. These could be true because under the condition of Da and Db, the Pecletnumbers, Pes are 237,066.6 and 414.3 respectively, which are significantly higher

than the value where the diffusion can be considered.Using the formula of Peclet number, the CEDC of 0.002576 km

2/hr (Dcr)

is obtained as the critical CEDC for this simulation based on the minimum Pecletnumber of 6 proposed by US EPA (2002). The rectangular pulse response in this

condition is shown in Figure 3.c. As shown by the responses in Figure 3.c, the

diffusion of the CEDC of 0.002576 km2 /hr affects the dispersion of the FCR

along the pipe over elapsed time, where it creates different responses and results

in the longer distance of the FCR existence and the lower FCR concentration at

the responses peak.

The effect of water flow rate velocity on the FCRIt is clear that, for the DTSM rectangular pulse input, the dispersion of the FCR in

single pipes of the DWDS is dominated by the axial convection, as can be seen inFigure 4. This is reasonable because the water flow carries FCR distribution along

the pipe as a result of its straight movement along the pipe.

(a)

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(b)

(c)

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(d)Figure 4. Comparison of the water flow rate effect against the CEDC effect on the

DTSM for step and rectangular pulse inputs

For example, for 12 hours rectangular pulse input, small changes in the velocity

from v = 0.0504 km/hr to v = 0.3024 km/hr result in a significant change in the

distance reached and a considerable change in the FCR concentration along thepipe with the same kof 0.02304/hr and the same CEDC of 3.708 x 10-5

km2/hr.

For the same elapsed time, the maximum FCR concentrations for the velocity of

0.3024 km/hr are about 0.836, 0.718 and 0.533 mg/L at distances of about 2.3,

4.25 and 8.12 km, respectively (see Figure 4.c), and these are about 0.777, 0.649and 0.452 mg/L at the distances of 0.5, 0.851 and 1.48 km, respectively (see

Figure 4.b).

For the DTSM step input, unlike the change of the CEDC shown in Figure4.a, as can be seen in Figure 4.d, the change of water flow velocity from 0.014

m/s (0.0504 km/hr) to 0.084 m/s (0.3024 km/hr) can further move the point of the

required minimum FCR concentration of 0.2 mg/L from 0.71 km, 1.32 km and2.42 km to 3.9 km, 7.42 km and 14.45 km for the step input of 12 hours, 24 hoursand 48 hours, respectively. The increase in the distance is in proportion to the

increase of the water flow velocity. Moreover, the higher water flow velocity

causes the higher FCR concentration at any point along the pipe. The change ofwater flow velocity from 0.014 m/s to 0.084 m/s can raise the maximum FCR

concentration at 1 km from the booster from approximately 0.635 mg/L to

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approximately 0.927 mg/L for 30, 36, 42 and 48 hours step input. At 2 km from

the booster, the maximum free chlorine residual increases from approximately0.011, 0.32 and 0.39 mg/L to approximately 0.859 mg/L for 30, 42 and 48 hours

step input respectively.

MODEL VALIDATION

In order to validate the DTSM for the FCR, the data and the results from acampaign on 11 July 1995 at the Cholet municipality DWDS (Heraud et al., 1997)

are taken into account, where most of the Cholet system pipes are cement-lined

case iron with kbeing 4.77 x 10-3

min-1

(0.2862 hr-1

) for the pipe diameter of 100mm and a low water flow of 5 m

3/hr, and Co (at the plant outlet) of 0.97 mg/L.

Figure 5.a, the top curve is the DTSM of the FCR using the value ofk

being 0.282/hr and v being 0.637 km/hr obtained from the calculation using thepipe diameter and the flow in the Cholet System campaign. As can be seen from

Figure 5.a, the analytical solution of the DTSM (DTSMAS) matches extremely

well with the FCR measured data in the campaign. The DTSM in time domain is

the same as the model fitted by Heraud et al. (1997) with kbeing 4.7 x 10-3

min-1

(0.282 hr

-1).

(a)

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(b)

Figure 5. Comparison between the measured FCR, the fitted model and the DTSM

In fact, the distance for every single point of the measured FCR of the CholetSystem can be calculated using the formula s = v x t . The FCR concentration

based on the fitted model at certain distance for the random residence time of 1,

17, 3, 4, 5, 5.6, 8 and 10 hours is about 0.7316, 0.6006, 0.4163, 0.314, 0.2368,0.2, 0.1016 and 0.0578 mg/L respectively. Meanwhile, the FCR concentration

based on the step input DTSM is shown in Figure 5.b. Hence, the DTSM in space

domain could be also validated using the data of the distance versus the FCR bywhich the correlation coefficient R2

is 0.9999.

ADAPTIVE CHLORINE DOSING DESIGN

The significant time variation in water demands leading to time-varying chlorine

transport delays (which could be more than an hour) is one of the key difficultiesin controlling the FCR within the safe drinking water limit level of 0.2-0.6 mg/L

in the DWDS.

With regard to the low and high demand of drinking water and thefluctuating volumetric water supply (VWS) which leaves the water plant over

time, this research proposes an Adaptive Chlorine Dosing Design (ACDD)scheme, wherein the chlorine booster location logistic and the injection rates of

each chlorine booster can be obtained using Equations (14), (15) and (16), whichare based on the analytical solution of Equation (13). This is a model-based feed

forward approach in controlling the FCR concentration, where the correct

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chlorine concentration injected by each booster is predicted by using the VWS

velocity (Vvws) entering the first booster.

( )1

1

1exp /

CB in

vws

CC C

kx V=

(14)

122 1 exp

BB HL B

vws

kxC C C

V

=

(15)

12 1 1 20.5[ 2( )] exp( )1 exp B i i C B iBi HLvws

x x x x xC C k

V

+

=

(16)

Where CB1 is the initial chlorine concentration injected by the 1st

booster; CC1 is

the FCR concentration at the first water consumer that should be equal to the highlimit required (CHL); Cin is the FCR concentration just before passing the 1

st

booster; CBi is the additional chlorine concentration injected by the ith

booster

(i=3,4, ); xB12 is the distance between the 1st

and 2nd

booster; xC1B2 is the

distance between the first consumer and the 2nd

booster and x-i-1 and x-i is thedistance between the i-1 and i boosters, respectively.

(a)

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(b)

Figure 6. The FCR concentrations for different volumetric water supply velocities afterthe ACDD package of chlorine dosage applied

In this simulation, the first drinking water consumers are assumed at 0.83 km

from the 1st

booster, the main pipe length is 8.5 km and the ACDD deals with the

required limit FCR of 0.2-0.6 mg/L. The value k, taken from Rosman (1994), isrelated to the pipe diameter size of 12 inch and the pipe wall decay rate constant

of 0.45 m/day, which is also assumed to take place in the SDWDS. The maximumVWS velocity is assumed to be 2.457 km/hr just leaving the first chlorine booster,which is 25% higher than the maximum water flow rate of 0.546 m/s in the main

pipe of water distribution network of New Haven (Biswas et al., 1993).

As concluded in Figure 6.a which is the DTSM response for the 10 hoursstep input, the proper initial chlorine concentration (CB1) of 0.4587 mg/L injected

by the 1st

booster can be obtained using Equation (14) so that the first consumers

get the drinking water with the required maximum FCR concentration of 0.6

mg/L. By using Equations (15) and (16), the proper chlorine concentration of

0.0639, 0.1151, 0.1162 and 0.1163 mg/L (CB2, CB3, CB4 and CB5) is injected by the2

nd, 3

rd, 4

thand 5

thboosters, respectively. This means that the ACDD is able to

carry the FCR concentration up to the high limit of 0.6 mg/L along the pipe withthe VWS velocity (Vvws) of 2.457 km/hr, as can bee seen by the top curve in

Figure 6.a. This ACDD package of chlorine dosage is also able to deal with the

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total decrease of 0.5 times the Vvws and the lowest FCR concentration, which is

still approximately 0.19 mg/L higher than the lower limit of 0.2 mg/L, as shownby the dashed curve in Figure 6.a.

In addition, the ACDD can also deal with the lower Vvws, 0.85 km/hr,which is shown by the line curve in Figure 6.b. The simulation on the Vvws of0.546 m/s, which is also normally found in the New Haven network, validates the

ACDD ability to lift the FCR concentration up to the high limit by injecting

chlorine with concentrations of 0.4419, 0.1355, 0.2304, 0.2321 and 0.2323 mg/L

at the 1st, 2

nd, 3

rd, 4

thand 5

thboosters, respectively, as can be seen by the dashed

curve in Figure 6.b. When we further lower the Vvws to 0.274 km/hr, the ACDD

scheme can still maintain the FCR within the limits, as shown by the dashed curve

in Figure 6.b.

CONCLUSION

Discrete Time-Space Models (DTSMs), which are developed using Finite

Difference Method and computed using Matlab 7.0.1, can be used to predict theFCR concentration along the single pipes of the DWDS over time. In the DTSM

simulation, using step input and rectangular step input, the results have shown that

the dispersion of the FCR in the pipes could be dominated by axial convection

using the strong influence of water flow rate velocity on the FCR, but the FCRaxial diffusion in single pipes in DWDS should be considered, when the chlorine

effective diffusivity coefficient is exceedingly high in the DWDS. It was also

shown that the DTSM is valid to present FCR distribution in the DWDS from theprospective of the FCR versus the elapsed time and space length. These modeling

studies resulted in an Adaptive Chlorine Dosing Design (ACDD) scheme inwhich chlorine booster location logistics and proper injections of chlorine, basedon volumetric water supply velocity just before entering the 1

stchlorine booster,

can be predicted in order to maintain the FCR concentration within the safe

drinking water limits.

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American Water Works Association Water (AWWA), Treatment - Principles and

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simulation of free chlorine decay and adaptive chlorine dosing

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