A model for chlorine concentration decay in pipes

Wat. Res. Vol. 27, No. 12, pp. 1715-1724,1993 Printed in Great Britain.All fightsreserved

0043-1354193$6.00+ 0.00 Copyright @ 1993PergamonPress Ltd

A MODEL FOR CHLORINE

CONCENTRATION

DECAY

IN PIPES PRATIM BxswAsm*, CHUNGSYING Lu t and ROBERT M. CLARK2 'Department of Civil and Environmental Engineering, University of Cincinnati, Cincinnati, OH 45221-0071 and 2Drinking Water Research Division, U.S. Environmental Protection Agency, Cincinnati, OH 45268, U.S.A.

(First received December 1991; accepted in revised form April 1993) Abstract--A model that accounts for transport in the axial direction by convection and in the radial direction by diffusion and that incorporates first order decay kinetics has been developed to predict the chlorine concentration in a pipe in a distribution system. A generalized expression for chlorine consumption at the pipe wall is used to solve the governing equation and to determine the average chlorine concentration at any location in the pipe. Three non-dimensional parameters are used to determine the concentration and a methodology to determine them in pipe networks is proposed. The range of values of these dimensionlessparameters where wall consumption is significant are determined. The methodology is applied to field measurements of chlorine concentration in a distribution network.

Key words---chlorine decay, disinfectants, radial and axial transport, wall consumption

NOMENCLATURE A0 -- dimensionlessparameter ( = L * D /r .2 U * = n L * D / Q ) A, ffi dimensionless parameter (--kL*/U*) A2 = dimensionless parameter (ffi V~'r*/D) C = dimensionless chlorine concentration (-- C*/Ce) Car=dimensionless cup-mixing average concentration (=C*,/Co) Co -- inlet chlorine concentration (nagi-') D -- effective diffusivity of chlorine in water (am2s-') f ( r ) = flow parameter term J0 ffi Bess¢l function of the first kind of order zero J, = Bessel function of the first kind of order one k = chlorine decay rate constant in the bulk water (s-')

L* = pipe length (cm)

Pea ffi axial Peclet number (=L*U*/D) Q ffi flow rate throughout the system (crn3s-') r ffi dimensionless radial distance from the center of the pipe ( =r*/rf ) r* ffi pipe radius (cm) t* = time coordinate (s) U*= average flow velocity throughout the distribution system (cms -I) V~ = chlorine consumption rate at the pipe wall (cm s-') X = dimensionless axial distance from the inlet along the pipe (ffiX*/L *) Y0ffi Bessel function of the second kind of order zero

Greek it = root of Bessel equation e = fractional error between equations (13) and (10) Superscript

level goals (MCLG) be established for substances that may adversely affect human health. The SDWA has been interpreted to mean that the MCLs must be met at the consumer's tap. This requirement has forced a more serious consideration of the possibility and potential for water quality deterioration in the distribution system. There have been a number of studies in the literature (Clark et al., 1986; Grayman et al., 1988; Males et al., 1988) that describe the development of contaminant propagation models. Although relatively simple, these models have proven useful in providing insight into the factors that influence water quality deterioration in drinking water distribution systems. Recently Biswas et al. (1991) have examined the effect of geometry on particle deposition rates and developed a rigorous methodology of relating transport characteristics to fluid flow profiles. Disinfection is routinely carried out before finished water leaves the treatment plant to maintain a residual in the distribution system to prevent microbiological degradation of water quality. Chlorine is the most commonly employed disinfectant in the United States, and minimum levels of chlorine must be maintained to ensure the disinfection capacity of distributed water (LeChevaUier et ai., 1988; Clark et al., 1989). This minimum concentration of chlorine

required is frequently determined from static tests

• = dimensional quantity INTRODUCTION The Safe Drinking Water Act (SDWA) and its amendments require that maximum contaminant *Author to whom all correspondence should be addressed.

(LeChevaUier et al., 1988). A drinking water utility must be able to predict the location at which the chlorine concentration drops below a certain minimum desired level. Once this location is determined chlorine may be reinjected or, alternatively, a higher dosage of chlorine added at the treatment plant to satisfy the chlorine demand in the entire network. As

1715

1716

PRATtMBISWASet al.

a result, the chlorine residual is an excellent parameter for studying water quality in the distribution system. Chlorine is known to be consumed in the bulk liquid phase and at the distribution pipe wall. Maul et al. (1985a, b) conducted an extensive study of a water distribution system in which spatial and temporal Heterotrophic Plate Count (HPC) variations were examined. They showed that free and total chlorine residuals decreased rapidly with distance from the treatment plant and could not be detected in the peripheral parts of the system where HPC levels were highest. LeChevallier et al. (1990) found that the composition of pipe material has a major influence on the disinfection characteristics. Biofilm bacteria grown on iron pipes were more resistant to free chlorine than grown on galvanized, copper, or PVC pipe surfaces, probably because free chlorine is known to preferentially react with ferrous iron to produce the insoluble ferric hydroxide. A first order decay rate equation is generally used to describe chlorine consumption at different residence times in the network (Hunt, 1988; Hart, 1991). The residence time is computed by dividing the pipe length by the average flow velocity in the pipe. By performing measurements in the distribution system, researchers have calculated the decay rate constant 'k' for site specific tests (such as fixed pipe diameter, pipe material or water source). This process may yield reasonable results, however, a wide range of values for this constant are obtained, thus severely limiting its use as a predictive tool. Sharp et al. (1991) reported 'k' values which vary with pipe diameter, pipe material and inlet flow rate. Wable et al. (1991) proposed examining the chlorine decay rate in water in a bench scale test and in the field simultaneously. The differences among the values determined were attributed to the consumption of chlorine at the pipe wall. However, attached matter on the pipe wall may be entrained in the bulk liquid phase, and diffusion to the pipe wall was neglected in their analysis. As indicated by LeChevallier et al. (1988), transport of the disinfectant from the bulk liquid phase into the biofilm (at the walls) is an important factor in understanding chlorine decay rates. Higher disinfectant concentrations are required to inactivate bacterial populations in the biofilm as compared to those suspended in the bulk liquid phase. In order to understand these factors, it is imperative that chlorine transport models account for radial diffusion to the pipe wall. Currently, there is no appropriate model that accounts for chlorine consumption at the pipe wall. In this work, a steady-state transport equation is developed taking into account the simultaneous convective transport in the axial direction, diffusion in the radial direction and consumption by a first order reaction in the bulk liquid phase. Different wall conditions are considered in the model: a perfect sink, no wall consumption and partial wall consumption.

The three governing parameters are determined from field data in the literature. A methodology to determine these parameters from field measurements is proposed and demonstrated. Such models can be incorporated into a biofilm model to describe chlorine demand in a distribution system (Lu, 1991). MODEL DEVELOPMENT

When chlorine is added to water as either a gas or in solution, it rapidly hydrolyses to form hypochiorous acid (HOCI) and hydrochloric acid (HCI). Hypochlorous acid then undergoes partial dissociation to form the hydrogen ion (proton, H +) and the hypochiorite ion (OCI-). Hypochlorous acid and hypochlorite ion act as the oxidizing agents to disinfect the drinking water. The transport equations are developed in terms of the dissolved chlorine concentration, C * ( H O C I + O C I - ) , (rag l-t). Figure I is a schematic of the control volume (C.V.) in a cylindrical pipe. Assuming that flow is in the axial direction alone, on writing down a mass balance for this C.V. gives a

- ~ (2xr* Ar* A X*C*)

= (U*f(r*)C*2nr*Ar*)x, -- ( U*f (r*)C*2nr* Ar*)x. +

\- D OC* 2~r*Ar*)x.

+f

-- ( -- D ~ , OC*

2~r *Ar* )x. +Ax, ,

+ ( - - D a--~-27tr AX*),.

/ OC* * .'X - - ~ - - D ~ r . 2Xr AX ),.+A,. - kC*2nr *Ar *AX*

(1)

where U* is the average flow velocity throughout the distribution system, (era s-I); f(r*) is the flow parameter term depending on the flow regime (for laminar flow: f(r*) --- 211 - (r*/rf)2], for plug flow: f(r*)= 1); r* is the pipe radius, (cm); D is the effective diffusivity of chlorine species in water, (cm2 s- i); k is the first order decay rate constant in the bulk water, (s-l); r* is the radial distance from the center of the pipe, (era); X* is the axial distance from the inlet along the pipe, (crn); Ar* and AX* are incremental distances at r* and X*, respectively. The term on the left hand side (LHS) denotes a net accumulation of chlorine in the C.V. with time, the first two terms on the right hand side (RHS) account for increase in concentration due to the inflow of water into the C.V., the next two terms account for increases due to diffusion of chlorine into the C.V. in the axial direction, the fifth and sixth terms account

Chlorine decay in pipes

1717

ro* ~C*.

(-D ~r* )r*+ar*

(U*f(r*)C*)x.

: (U*f(r*)C*)x.+Ax .

(-D aC*.

ar--~)r* I

I I I I I I /

I X*

I -1-

.r*+Ar*

I

x*+Ax*

Fig. 1. A schematic of the control volume (C.V.) in a cylindrical pipe. for increases due to diffusion of chlorine into the C.V. in the radial direction, and the last term on the RHS accounts for the reduction of chlorine in the C.V. due to consumption by both chemical and microbiological reactions in the bulk water. Dividing the LHS and RHS by 21rr*Ar*AX* and taking limits as At* and AX* go to zero, equation (1) becomes 0C*

at*

-

0

0 (

OX*( U*'f(r* )C* ) + ' ~ 1 0 {,

D -OC ~

*)

aC*\

+-~r,~r D~r,)-kC*

(2)

Assuming quasi-steady state conditions (~C*/at* = 0) and rearranging equation (2), we have

(U*f(r*)C*)= 0 f D OC*~ 1 0 [,

OC*\

+ - ~ r , ~ r D~r,)--kC*

(3)

The boundary conditions are as follows X*ffi0,

(i) Perfect sink, V*--,oo, that is, C * = 0 (ii) No consumption at pipe wall, V~ -- 0, that is,

OC*/Or* = 0 (iii) Partial consumption, V* is some finite (nonzero) value. Assuming that the time period over which the experiments are done, U* and D remain constant (quasi-steady state), and on non-dimensionalizing equation (3)

f(r) ~C.~ 1 a2C Ao 0 I" OC'~ =NOX2+r~r~r~rJ--AiC

C* = C*, at 0 ~
r* = O OC*/Or*ffi O, at 0 ~
r*ffir*, DaC*/Or*f-V~C* at 0 ~
where C* is the inlet chlorine concentration (mg l-I); L* is the pipe length (cm); and V~ is an empirical parameter that is proportional to the degree of absorptivity of the pipe surface, the so-called consumption rate of chlorine at the pipe wall (cm s-~). The consumption process at the pipe wall can be viewed conceptually by analogy to transfer of mass or heat from the bulk liquid phase to the pipe surface. Three different wall conditions may be used depending on the wall characteristics:

(4)

(5)

C ffiC*/C*, X fX*/L*; r ffir*/r*; Pe°= L*U*/D; AoffiL*D/r*2U*ffinL*D/Q, (Qffiflow rate throughout the system); and A~ = kL*/U*.

where

For distribution systems, the axial Peclet number, Peo, is typically large (>10 ~) and hence the axial diffusion term can be neglected with respect to the axial convective term. Turbulent flow conditions are

PRATIM BlSWAS et ai.

1718

often encountered, and t h e n f ( r ) is equal to 1, or the axial velocity can be assumed to be independent of the radial location for most part of the cross section. Representing the axial velocity by a constant [f(r) = 1] average value (U*), the governing equation for chlorine transport in pipes under these conditions is

a-x

oc _

r ar \ ar /

,4,c

(6)

with the boundary conditions

X=O, C=I, r=O,

at

aC/ar=O,

0~
at

0~
C,v is expressed in equation (10) as a sum of terms in an infinite series. However, when A0 > 0.05, the value of C,v computed using the first term in the expression in equation (10) leads to an error of about 5%. When A0 ~< 0.05, the value of Ca, computed using the first three terms in the series in equation (10) leads to an error of less than 1%. Therefore, in practice, it is sufficient to select the first three terms in the series in equation (10) to compute Car. The values of ,[i, ,[2 and '13 [listed in Table 5.2 of Crank (1975)] are plotted for different values of A 2 in Fig. 2. A fitting procedure was used to express ,[m, '[2 and ,[3 as a function of A2, the error being less than 1% from tabulated values of A 2.

r = 1, aC/~r = - ( V ~ r f / D ) C

-~ - A 2 C

at

0~
(a)

Thus, three dimensionless parameters, A o, A m, and A2, govern the chlorine decay in the distribution system. Ao accounts for the radial diffusion and depends on the pipe length, on the effective diffusivity of chlorine and on the flow rate throughout the system. Amdepends on the reactivity of chlorine with species such as viable cells or chemical compounds in the bulk liquid phase and on the residence time in the system. A2 is a wall consumption parameter depending on the wall consumption rate, on the pipe radius and on the effective diffusivity of chlorine.

-

6.0

-

k 3 = 7 . 1 1 5 5 5 A2 0.00376107

~'2 = 4 . 0 0 9 4 6 A 2 0.0119894

o

4.0

2.0

0

.o-------<>'''~

k1= 1

).01

.

2

9

~

r

I

0.1

1.0

(b)

SOLUTION P R O C E D U R E

8.0

As the axial velocity is represented by an average quantity, equation (6) can be solved analytically. Using the separation of variables technique, a solution to equation (6) is obtained (detailed derivation shown in the Appendix)

C(X, r) = 2

8.0

6.0 --O

k

4.0 ~

6

2

9

,[, J0 (,[nr)Jm (,[,)

. . , (,[~.+ a ~)Jo~(,[,) × exp[--(Al+,[2A0)X]

0

2.0 ~

(8)

where J0 and J~ are the Bessel functions of the first kind of order zero and one, respectively; ,[.s are the roots of ,[Jl(,[.) ffi A2Jo(,[~). The dimensionless cupmixing average concentration is

2

7

A2 0.239289

I

I

I

4

7

10

(c) 9.00 ~'3 = 7 . 7 1 1 6 5 A2 0.0182292

C,~ = ~ 2C(X, r)r dr

(9)

Substituting C(X, r) from equation (8) and using the relationship of Jl(,[~)= A2Jo(,[,)/2n, equation (9) be-

6.75

4.50' -

k 2 = 4 . 8 6 4 4 1 A 2 0.0200514

comes 2.25

k 1 = 2 . 1 0 2 1 8 A 2 0.021361

It'4

..':,(,[~ + A ~)J0~(,[.) 0

x exp[--(Ai + ,[2Ao)X] -- ~~v

I

I

100

1000

A2

4A~2

x exp[-(A~ + ,[2Ao)X]

10

(10)

Fig. 2. Variations of ~l, A2 and A3 as a function of A2 for A2 ranging from (a) 0.01 to I, (b) 1 to 10 and (c) 10 to 1000. Symbols are data from Crank (1975).

Chlorine decay in pipes For 0.01 ~
For 10 ~
(11)

The cup-mixing average concentration of chlorine can thus be determined at any location in the pipe using equation (10) and (11) provided that Ao, A t and A 2 are knOwn. If the pipe walls act as a perfect sink, V~'--,oo(A2--,oo) or C(X, 1) = 0. The cup-mixing average concentration under this condition is obtained by taking the limit of equation (10) as A2--¢oo, or by integrating equation (9) using C(X, r) from equation (A23) C,=~

4 .-I ~ exp[-- (AI + 22A°))(]

function of X, A0 and A 2 and can be used to determine conditions when the simplistic equation (13) can be used to predict the chlorine concentration in the distribution system. The fractional error, ~ at the outlet of a pipe is computed from equation (14) and shown in Fig. 3 as a function of A0 for different values of A2. The results indicate that if A0 < 10 -3, the deviation is less than 5%, and hence equation (13) will suffice in predicting the chlorine concentration. For A0 > 10 -2 and A2 > 1, E is greater than 5%, and thus both radial diffusion and wall consumption are important. In comparison to equation (13) that is currently used to determine the average concentration, equation (10) is more cumbersome to use. However, if ~ is known, the following equation can be used to determine the average concentration at any location c. =

e x p ( - AmX)

(13)

Thus, equation (10) is the general expression for the cup-mixing average concentration of chlorine at any location in the pipe. The three parameters A0, A t and A x govern the chlorine decay in pipes. Under limiting conditions described before, the cup-mixing average concentrations are #oven by equation (12) or (13). It must be noted that researchers (Hunt, 1988; Hart, 1991; Wable et al., 1991) have used equation (13) to describe chlorine decay in distribution systems, this being valid only if there is no wall consumption. The fractional error, ~, introduced in using a simplistic expression [equation (13)] is determined by calculating the deviation of equation (13) from the more exact solution #oven by equation (10) e is a

(15)

(1 + ~)

However, the expression for determining ~ [equation (14)] also involves summing the terms of a series. To come up with a more readily usable form of equation (14) to use in equation (15), regression techniques (Johnson and Wichern, 1988) are employed for the data plotted in Fig. 3 using equation (14) to derive an expression of e for Ao in the range of 10 -4 to 100

=2.4416AoA2-O.1559AoA2~ (12)

where g,s are the roots of &(2,) = 0. If the pipe walls are inert, and no chlorine consumption takes place at the walls, V*--,0 (A2-,0) or OC(X, O/Or-0. The cup-mixing average concentration under this condition is obtained by substituting C(X, r) from equation (A26) into equation (9) and integrating from 0 to 1 C , = e x p ( - At X)

1719

0.01 ~
E -- 10.105A0 + 0.0014A 2 + 0.31A2A2 1 0 ~ A 2~<100

(16)

As can be seen in Fig. 3, ~ is not a strong function of A 2 for A2 > 100. Therefore for A2 > 100, e can be computed using equation (16) by substituting A 2 ~ - 100.

The average concentration in the pipe at any location can be determined using equations (15) and (16), provided the three non-dimensional parameters A0, A t and A 2 are known. RESULTS AND DISCUSSION

Parameters estimation As stated earlier, the three parameters Ao, At and A2 describe the transport process and can be used to determine the concentration of chlorine residual in the distribution system. Once the flow rate throughout the system and the pipe length are known, A0 can be computed provided an effective diffusivity of chlorine in water, D is known. D, the effective diffusivity depends on ionic and molecular transport and on the flow regime. In reality, ionic transport has to

4A~ exp(-- A,X) -- , ]~i ~(~I + AI) expl--(A, + 22A0)X] E~

~. 4A] 2 2 + A~) exp[-(A~ + 2~Ao)X] , - 1 2,(:.,

1

exp[-- (22A0)X] •

\A,

/

(14)

PRATIMBLSWASet al.

1720 10

,o,.?7,, !01"40

-4

'

10 "~

10 -'

1(:; -'

'

'

Ao Fig. 3. The fractional error, ~, at the outlet of a pipe as a function of Aofor different values o f A 2 (numbers Oil curves).

be accounted for to determine the diffusivity (Lu, 1991). If turbulent flow conditions prevail, the eddy diffusivity (Sherwood, 1975) has to be used as it typically is greater than the molecular diffusivity (D ffi 1.25 × 10-Scm2s -n Cussler, 1984). The sensitivity of the average concentration on the choice of a diffusivity value in the different flow regimes is examined in the next section (A0 and A2 change as D is varied while A0 and An change as U* is varied). The following methodology is proposed to determine the other two parameters, An and A2 experimentally. Kinetic tests should be performed with a water sample taken at the inlet to determine the decay constant k. To determine if significant biomass is entrained, the kinetic tests should be repeated with a water sample taken at the outlet. If the 'k' values are very different, it is suggested that an average value be used. Once 'k' is known, on using the values of L* and U*, A n can be computed. A field test should be conducted to measure the average concentration (C,) at a certain location (X) of the pipe. Since the values of A0, An, Xand C,~ are known, equations (15) and (16) can then be used to determine A2. The three parameters A0, An and A2 can then be used in equations (15) and (16) to predict chlorine concentrations at any location in the distribution system. The procedure must be repeated periodically to determine the values of Al and A2. An alternative scheme is to use field data to compute the three non-dimensional parameters A0, An and A,. Concentrations need to be measured at least three locations (in addition to the inlet), and equations (15) and (16) can then be solved to determine A0, An and A2. If data are available at more than three locations, a least squares procedure can be used to determine the best fit values for A0, An and

A2.

Sensitivity study A sensitivity analysis was carried out to investigate the effect of the three governing parameters on the average chlorine concentration. The cup-mixing average concentration of chlorine, Coy, at the outlet (X ffi 1) of a pipe is plotted in Figs 4(a)--(d) as a function of A~ for the different values of A0. A2 is varied from 0 to oo in each of the first three plots. Small values of A0 correspond to diffusion in the radial direction being less important, and hence there is not much variation in the C,~ curves for A2 varying from 0 to oo [Fig. 4(a)]. Under such conditions, equation (13) may be used to describe the chlorine decay in the distribution system. With increased values of A0 [Figs 4(b)-(d)], the cup-mixing average concentration of chlorine becomes a stronger function of A2, thus equation (13) inaccurately predicts the chlorine concentration in the distribution system, and the radial diffusion term must be included in the analysis. If the two parameters A0 and Am are known, Fig. 4 may be used to determine the value of A2 at which the chlorine concentration at the outlet of a pipe would drop to any prescribed level.

Determination of parameters from field data As described earlier, researchers have used a simple decay rate expression [as in equation (13)] to fit field data and report values of 'k'. As wall consumption and diffusion are not accounted for, different values of 'k' are reported for different conditions (Wable et al., 1991). The model developed in this work is demonstrated by solving the equations for conditions of a field sampfing study conducted in the South Central Connecticut Regional Water Authority (RWA) (Clark et al., 1991). The RWA serves the

1721

Chlorine decay in pipes

(a)

(h)

1.00

10 / 0

1.00

f

A 0 = 1.4 x 10 -3 10

6 2 5 lO /

0,75

0.75

0.50

0.50

0.25

0.25

10 -4

(c)

2

10-2

100

100./

0 10-'*

10

,,,.,,,I

o

(d)

0.1

I

~"

,,~),.,I

j,....l

10 -2

o ~

A

..,.,.I

,..~...i

10 °

10

o - 1.4

0.1

0.75

0.75 0.2

0.50

0.50

0.25

0.25

0.5 1.0 0 10 -4

10 -2

100

0 10 -4

10

10 -2

100

10 ,.

A1

Fig. 4. Computed values of the cup-mixing average concentration, Ca,, at the outlet of a pipe with (a) A0 = 1.4 x 10-3, (b) A0 = 1.4 x 10-2, (c) A0 = 1.4 x 10-' (d) .4o-- 1.4. The numbers on the curves are values of A2.

greater New Haven area of Connecticut. Water is supplied to approx. 100,000 customers (380,000 individuals) in 12 municipalities. The R W A distribution system is divided into 18 separate pressure/ distribution zones or service areas. In order to study chlorine residuals within the RWA, a study was initiated between 13 and 15 August 1991 in the Cherry Hill/Brushy Plains service area. This service area covers approx. 2 square miles in the Town of Branford in the eastern portion of the R W A system. It is almost entirely residential containing both single family homes and apartment/condominium units. Average water use during the sampling period was 0.461 MGD. A schematic of the distribution system composed of 8 and 12" mains is shown in Fig. 5, which represents a 57% skeletonization of the system. Seven sampling sites were established in the distribution system in addition to sampling sites at the pump station and tank as shown in Fig. 5. The Brushy Plains/Cherry Hill service area operates under two basic scenarios: pumps on, during which time the tank is filling; and pumps off, during which the tank is emptying. These scenarios hydraulically define links of pipe over which chlorine consumption can be measured. Chlorine concentrations at the inlet and WR 27/12-=C

outlet points of these pipe segments are shown in mg/liter in Table l(a). Five unidirectional flow segments were identified wherein inlet and outlet chlorine concentrations were measured [Table l(a)]. The geometrical and flow parameters are listed in Table 1(b). As indicated in Table l(b), the flow conditions in the pipe are turbulent, and hence the eddy diffusivity was used to estimate the diffusion coefficient, D~ddy= 1.233 X 10-2U*r~ ' (Edwards et al., 1979) at 25°C. Using these parameters, A0 is computed for the different pipes and the values listed in Table l(b). ' k ' was estimated to be 6.4 x 10 -6 s -~ from bench kinetic tests performed with water samples taken at the inlet to the network. Assuming that the ' k ' value remains the same for the pipes, A~ was computed for the different pipes and the values are listed in Table l(b). This assumption is expected to be valid for the main network branch, however, it was not validated by performing a kinetic test with a water sample taken from a downstream sampling point. With A~ and A2 known, an iterative procedure using equations (15) and (16) was used to compute V~, ,,12 and ~. The calculation is demonstrated for the segment with pipes 7, 9 and 11. The superscripts on the variables

PRATIM B I S W ~ et al.

1722

indicate the pipe number. At the outlet of the pipe, X -- 1, and equation (15) gives 7 __ e x p ( - A ~ )

Car --

(1 + E ~)

exp(-- A 9) cgv=

(1 + d)

C.~ =

exp( - A 11)

(17)

(1 +E xl)

The measured concentration at the outlet to pipe 11 divided by the concentration at the inlet to pipe 7 is

12

CZvC9v Cat~= 0.98/1.00 11

= exp( - ,4 ~) exp( - .4 9 ) exp( - ,411) (1 + e ~)

(1 + e 9)

(1 + E n ) (18)

8

The unknowns, expressed

t0

e 7, E9 a n d E n [in e q u a t i o n

(16). T h e A 0 s a r e k n o w n expressed as a function consumption

rate

for the other

[ T a b l e l ( b ) ] a n d A2 c a n b e of the unknown chlorine

at the pipe wall,

known parameters. 1

(18)] a r e

a s a f u n c t i o n o f A0 a n d A2 u s i n g e q u a t i o n

V~' a n d

other

Similar expressions are developed

segments

listed in Table

l(a).

V~' is

expected to be the same for pipes made of the same material

in

spreadsheet Treatment plant •

could

Fig. 5. S c h e m a t i c o f the w a t e r d i s t r i b u t i o n n e t w o r k a t N e w H a v e n . O p e n circles i n d i c a t e p i p e n o d e s , solid s y m b o l s i n d i c a t e c h l o r i n e m e a s u r e m e n t l o c a t i o n s a n d n u m b e r s indicate pipe n u m b e r s .

be

the

main

program entered

branch

of

the

network.

A

w a s u s e d w h e r e i n v a l u e s o f V~" and

then

A2 a n d

E computed

[ e q u a t i o n (16)]. T h e o u t l e t c o n c e n t r a t i o n s w e r e t h e n e s t i m a t e d u s i n g e x p r e s s i o n s [ s i m i l a r t o e q u a t i o n (17)], and then compared to the measured concentration r a t i o s [ T a b l e l ( a ) ] . A v a l u e o f V * = 1 x 1 0 - T i n s -1

Table I(a). Chlorine concentrations at inlet and outlet of various segments Chlorine concentration at segment (rag/l) Pipes in sesment Inlet Outlet I, 3 7, 8, 10 7, 9, I 1 12, 13,16, 21 12, 13,14, 15, 27, 26, 28

1.08 1.00 1.00 0.98 0.98

1.00 0.32 0.98 0.16 0.94

Table l(b). Model parameters for different pipes in the network

Pipe

Length (m)

Radius (m)

Flow velocity (m/s)

I 3 7 8 9 10 11 12 13 14 15 16 21 26 27 28

731.5 396.2 822.9 365.8 121.9 304.8 213.4 579.1 182.9 121.9 91.4 457.2 426.7 182.9 76.2 91.4

0.152 0.102 0.152 0.152 0.152 0.102 0.152 0.152 0.152 0.152 0.152 0.102 0.102 0.152 0.152 0.152

0.546 0.195 0.512 0.014 0.494 0.014 0.485 0.457 0.445 0.372 0.329 0.168 0.049 0.329 0.338 0.323

Diffusion coefficient (m2/s)

Reynolds number

V~ (m/s)

1.03e -- 3 2.46e--4 9.62e - 4 2.58e--5 9.28e -- 4 1.80e -- 5 9.11e--4 8.59e -- 4 8.36e -- 4 6.99e -- 4 6.19e--4 2.1 l e - - 4 6.14e-- 5 6.19e-- 4 6.36e -- 4 6.07e-- 4

! .75e + 5 4.19e+4 1.64e + 5 4.40e+3 1.58e + 5 3.08e + 3 1.55e + 5 1.47e + 5 1.43e + 5 1.19e + 5 1.06e + 5 3.60e + 4 1.05e + 4 1.06e + 5 1.09e + 5 !.04e + 5

1.00e -- 7 1.15e--6 1.00e -- 7 1.00e--7 1.00e -- 7 2.3e -- 6 1.00e-- 7 1.00e -- 7 1.00e -- 7 1.00e-- 7 1.00e-- 7 1.00e-- 7 2.2e-- 5 1.00e-- 7 1.00e-- 7 1.00e-- 7

A0 5.92e + 4.79e+ 6.66e + 2.96e+ 9.86e + 3.68e + 1.73e + 4.69e + i.48e + 9.86e + 7.40e + 5.52e+ 5.15e + 1.48e + 6.17e + 7.40e +

1 I 1 I 0 I 1 1 i 0 0 I I 1 0 0

AI

A2

e

8.58e -- 3 1.30e--2 1.03e -- 2 1.71e-- I 1.58e - 3 1.36e - 1 2.82e - 3 8.1 le -- 3 2.63e -- 3 2.10e-- 3 1.78e-- 3 1.75e-- 3 5.60e -- 2 3.56e -- 3 1.44e -- 3 1.81e-- 3

1.49e -- 5 4.78e--4 1.58e -- 5 5.91e--4 1.64e - 5 1.30e -- 2 1.67e-- 5 1.77e - 5 1.82e -- 5 2.18e-- 5 2.46e- 5 4.844:--5 3.66e-- 2 2.46e - 5 2.40e -- 5 2.51e-- 5

2.2e - 3 5.6e--2 2.6e -- 3 4.3e--2 4.0e -- 4 1.17 7.1e-4 2.0e - 3 6.6e - 4 5.3e-- 4 4.5e--4 6.5e -- 3 4.589 8.9e--4 3.6e-- 4 4.5e--4

Chlorine decay in pipes

1723

Edwards D. K., Denny V. E. and Mills and A. F. (1979) Transfer Processes: An Introduction to Diffusion, Convection and Radiation, second edition. McGraw-Hill, New York. Grayman W. M., Clark R. M. and Males R. M. (1988) Modeling distribution system water quality: dynamic approach. J. Brat. Res. Plann. Mgmt ASCE 114, 295-312. Hart F. L. (1991) Applications of the net software package. Proc. Water Quality Modeling in Distribution Systems, AWWA Research Foundation, 3-5 Feb., Cincinnati, OH, 57-75. Hunt W. A. (1988) Distribution System Modeling. Bacterial CONCLUSIONS Regrowth in Distribution Systems (Edited by Characklis W. G.) AWWA Research Foundation, Denver, CO, pp. A two-dimensional transport model accounting for 43-72. consumption in the bulk water and at the pipe wall Kovach L. D. (1984) Boundary Value Problems. Addison-Wesley, Reading, MA. was developed to predict the chlorine concentration Johnson R. A. and Wichern D. W. (1988) Applied Multivariin the distribution system. The model equation was ate Statistical Analysis, second edition. Prentice-Hall, cast in dimensionless form and solved for turbulent Englewood Cliffs, NJ. flow conditions using the separation of variables LeChevallier M. W., Cawthon C. and Lee R. G. (1988) Inactivation of biofilm bacteria. Appl. envir. Microbiol. technique to obtain an analytical solution. If laminar 54, 2492-2499. flow conditions prevail in the pipe, a seres solution LeChevallier M. W., Lowry C. D. and Lee R. G. (1990) can be derived. Three governing parameters, A0, At Disinfecting biofilms in a model distribution system. and A2 were sufficient to describe the chlorine decay J. Am. Wat. Wks Ass. 82:7 (July), 87-99. in the distribution system. A0 accounts for radial Lu C. S. (1991) Theoretical study of particle, chemical and microbial transport in drinking water distribution sysdiffusion, At is dependent on the bulk phase contems. Ph.D. dissertation, Univ. of Cincinnati, Cincinnati, sumption, and A2 is indicative of wall consumption. OH. Criteria were derived to determine conditions when Lykins B. W. Jr., Clark R. M., Block J. C., Colin F. and wall consumption would be significant and when the Schulof P. (1993) Experimental pipe network for evaluating disinfection treatment effects. To be submitted to radial diffusion term needs to be included in the J. Am. War. Wks Ass. analysis. A methodology to determine At and A2 from Males R. M., Grayman W. M. and Clark R. M. (1988) field measurements was also proposed. The values of Modeling water quality in distribution systems. J. Wat. k and V* computed in this work can be incorporated Res. Plann. Mgmt ASCE 114, 197-209. into biofiim models to describe chlorine demand in Maul A., EI-Shaarawi A. H. and Block J. C. (1985a) Heterotrophic bacteria in water distribution system. I. the distribution system. The governing parameters Spatial and temporal distribution. Sci. Total Envir. 44, were determined for free chlorine using data from a 201-214. field study. The results indicated that radial diffusion Maul A., El-Sharrawi A. H. and Block J. C. (1985b) and wall consumption was significant in certain segHeterotrophic bacteria in water distribution system. II. Sampling design for monitoring. Sci. Total Envir. 44, ments of the network (off the main branch). 215-224. Acknowledgements--This work was supported by an EPA Reid R. C., prausnitz J. M. and Sherwood T. K. (1977) The Properties of Gases and Liquids, third edition. McCooperative Agreement, C R-816700, RD 1. The conclusions Graw-Hill, New York. represent the view of the authors and do not necessarily represent the opinions, policies or recommendations of the Sharp W. W., Pfeiffer J. and Morgan M. (1991) Insitu chlorine decay rate testing. Proc. Water Quality Modeling U.S. Environmental protection Agency. in Distribution Systems, pp. 311-322. AWWA Research Foundation, 3-5 Feb. Cincinnati, OH. Sherwood T. K., Pigford R. L. and Wike C. R. (1975) Mass REFERENCES Transfer. McGraw-Hill, New York. Biswas P., Lu C. S. and Clark R. M. (1991) Particle and Wable O., Dumoutier N., Duguet J. P., Jarrige P. A., Gelas G. and Depierre J. F. (1991) Modelling chlorine concenchemical transport in drinking water systems. Proc. Water trations in a network and applications to Parrs distriQuality Modeling in Distribution Systems, AWWA Rebution network. Proc. Water Quality Modeling in search Foundation, 3-5 Feb Cincinnati, OH, 323-363. Distribution Systems, pp. 77-87. AWWA Research FounClark R. M., Grayman W. M., Males R. M. and Coyle J. A. dation, 3-5 Feb. Cincinnati, OH. (1986) Predicting water quality in distribution systems. A WWA Distribution System Syrup. Proc., Minneapolis, MN. APPENDIX Clark R. M., Reed E. J. and Hoff J. C. (1989) Analysis of inactivation of Giardia lamblia by chlorine. ASCE J. The separation of variables technique is used to solve the Envir. Engr 115, 80-90. chlorine transport equation analytically. Assuming that the Clark R. M., Grayman W. M., Goodrich J. A., Deininger chlorine concentration can be expressed as a function of X R. A. and Hess A. F. (1991) Field testing distribution multiplied by a function of r, we have water quality models. J. Am. Wat. Wks Ass. 83:7 (July), C = F(X)G(r) (A1) 67-75. Crank J. (1975) The Mathematics of Diffusion, second Substituting equation (AI) into equation (6) gives edition. Clarendon Press, Oxford. G dF=FA /d2G I dG\ Cussler E. L. (1984) Diffusion Mass Transfer in Fluid o~--~-~r2+r--~r ) - A,FG (A2) Systems. Cambridge Univ. Press, Cambridge, U.K. for the main branch pipes led to good agreement with the measurements. However, different values of V* were obtained for the pipes on dead ends or off the main network branch (Pipes 3, 8, 10 and 21). Pipes 3, 8, 10 and 21 have high ~ values indicating that wall consumption is significant. These are off branch pipes (Fig. 5) and qualitative observations indicate that significant biofilm growth occurs on such pipes.

PSALM BISWASet al.

1724

Moving AIFG to the LHS of equation (A2) and dividing both sides by AoFG, equation (A2) becomes

Using the pipe entry condition (C --- I at X = 0) in equation (AI4) gives

I 1 dF Am 1 [d2G l d G ~ ~o?~-~ + Z = ~ ~--~ + r--~r )

I - - ~ W.Jo(2.r)

(A3)

The LHS of equation (A3) is a function of X, whereas the RIdS is a function of r. The only way this will be true in general, is if both the LHS and RHS are equal to a constant, say/~. Hence,

I 1 dF

A,

1 [d2G

1 dG'~

Ao F dX .I- -~oo= -~ ~k.-~r2+ r --~r ) = #

F dX

and

d2G

The coefficients, IV., are determined by multiplying both sides of (Al5) by r Jo(2mr) and integrating from 0 to 1

rJoO..r ) dr =

IV.

rJoO..r)Jo(2.r ) dr

(AI6)

nffil

(A4)

Ifm # n the integral on the RHS in equation (Al6) is equal to zero. Therefore, equation (Al6) becomes

which gives ldF . . . .

(AI5)

.ffim

Am+ W4o

1 dG + r ~ r = #G

(A5)

(A6)

The sign of the constant, #, must be consistent with the physical conditions of the problem. The sign of the constant can be determined by considering the solution to equation

(AS) F = Fo exp[(--A m+ #A0)X]

(A7)

where Fo is a constant. If (--Am + #A0) is equal to zero, the chlorine concentration inside the pipe does not change with distance. Therefore, a solution of this form is not accurate. Therefore this possibility must be rejected. The second case is for ( - A , +/L40) to be positive. The chlorine concentration would then increase with distance. This condition is not feasible and hence must be rejected. We can thus conclude that (--Ai +#Ao) must be negative. This implies that the chlorine concentration decreases with increasing distance and hence consistent. In general, we can write /~ = - g 2 implying that # is a negative number. Then equations (A6) and (#.7) can be written as dZG 1 dG ~r2 + r--~-r + ).2G - 0 (AS) F = F 0 exp[--(Am + 22A0)X]

(A9)

;o'

rJoO..r ) dr =

IV.

(AI0)

where Jo(Ar) is the Bessel function of the first kind of order zero, Yo(Ar) is the Bessel function of the second kind of order zero, and B and E are constants. Since Jo(2r) is defined at r ffi 0 while Y0(~Lr)goes to infinity, implying that constant E should be zero if C must be bounded at the origin. Substituting e q u a t i o n s ( A g ) a n d ( A I 0 ) into ( A I ) , w e

have

(AIT)

The integral on the LHS of equation (Al7) can be written as (Kovach, 1984)

f0 1

rJ0(ft.r ) dr =

Jm(3".) 2.

(AI8)

while the integral on the RHS of equation (Al7) can be expressed as (Kovach, 1984)

f/rJ~(2.r)dr -- ~[J0(A.) m 2 + J~0-.)]

(A19)

Hence W.s can be determined by combining equations (AI7), (A18) and (A19)

2J, (,l.) IV. = 2.[jo2(~..) + J~O-.)]

(A20)

Substituting equation (A20) into equation (A14), we have

c(x, r) =

2 ~ J°O"r)Jm(2") ._% , ~ . [ ~ . ) ] x exp[- (A I + A.~Ao)X] (A21)

From equation (Al3), equation (A21) can be written as c(x, r) =

2 ~ )"J°()"r)Jm(}~") .%

Equation (A8) is the Be..sel equation of order zero. Hence the general solution of equation (AS) can be expressed as (Kovach, 1984)

G(Ar) = BJo(,~r) + EYo(Ar )

rJ~(2.r) dr

nffim

x exp[-(Am + 22.Ao)X] (A22) Thus, equation (A22) is the general expression for the chlorine concentration in the pipe at any location X and r. If the walls act as a perfect sink, V~--.ov (A:-,oo) or C = 0 at r = 1. The chlorine concentration under this condition is obtained by taking the limit of expression (A22) as A2--,ov and using (AI3) we have

JoO.nr) C(X, r) = 2 ~ T - 7 7 ~ , exp[-(.4m + 2~A0)X] (A23) nffim

C(X, r) = WJo(Ar)exp[-(A , + 22A0)X ]

(AI 1)

where W ffi BFo, is a constant. Using the wall condition (r ffi 1) in equation (7) and differentiating equation (AI 1) with respect to r and adding A2C yields

OC

2. = 0 or Jm(~..)= 0

0-'r- + `42C = W e x p [ - (Am + 22A0)] [ - 2./, 0.) + A2J0(,~.)] = 0

(AI2)

To satisfy equation (AI2), ,l must be a root of

--LI, (~.) + A2Jo()0 = 0

(AI3)

For a given A2 there are multiple roots of the equation (A13)---Am, 22. . . . (see Table 5.2, Crank, 1975). As there are multiple roots, the concentration in equation ( A l l ) is expressed as a series

C(X, r) = ~. Wjo(~,.r)exp[-(A t + 2~A0)X] n-m

where 2.s are now the roots of JoO..) = 0 [from equation (AI3), as A:-,ov]. If the walls are inert, and no chlorine consumption takes place at the walls, V ~ 0 (A2-~0) or OC/Or = 0 at r = I. From equation (AI3)

C(X, r) = W e x p ( - A i X )

(A25)

On using the condition that C = 1 at X = 0, implies W = I, therefore,

C(X, r) = exp(-AmX) (AI4)

(A24)

Jm(2.) = 0 leads to the trivial solution C(X, r ) - - 0 [from equation (A20)] unless ,1..= 0; hence the appropriate condition is ~. = 0. Substituting 2. = 0 into equation (AI 1) and noting that Jo(0)-- 1 provides

for an inert wall condition.

(A26)