Residual chlorine concentration decreases along distribution networks because of factors such as water quality, physical properties of the pipeline, and hydraulic conditions. Hydraulic conditions are primarily governed by transient events generated by valve modulation or pumping action. We investigate the impact of transient events on the rate of chlorine decay under various flow conditions. To comprehensively compare the performance of existing chlorine models, 14 candidate models for chlorine concentration were used under various transient conditions. Two-dimensional (2D) transient flow analysis was conducted to investigate the unknown processes of chlorine decay under transient conditions. General formulations for modeling chlorine decay were used to comprehensively study the decay under unsteady conditions and to effectively incorporate the impact of transients into generic model structures. The chlorine decay patterns in the constructed water distribution system were analyzed in the context of transient events. Linear relationships between the model parameters and the frequency of transient events were determined under unsteady conditions, and the impact of turbulence intensity was successfully incorporated into model parameter evaluations. The modeling results from 2D transient analysis exhibit similar predictability as those obtained from calibration using the genetic algorithm.

## INTRODUCTION

Drinking water obtained from water treatment plants is disinfected before it enters a distribution system. Chlorine is the most widely used disinfectant to prevent the regrowth of microbial pathogens in treated water (Termini & Viviani 2015). Therefore, maintaining sufficient chlorine concentration throughout the water distribution system is an important aspect of water quality management. However, the concentration of residual chlorine in a water transmission system varies with system properties (Mohapatra *et al.* 2014).

The decay of chlorine is influenced by two distinct pathways. The first involves water quality parameters such as the concentration of organics, initial concentration of chlorine, iron content, rechlorination, and temperature, (Jadas-Hecart *et al.* 1992; Kiene *et al.* 1998; Powell 1998; Hua *et al.* 1999, 2015; Hallam *et al.* 2003; Vieira *et al.* 2004; Warton *et al.* 2006; Courtis *et al.* 2009), and the second involves system parameters such as pipe age, materials, and hydraulic conditions (LeChevalier 1990; Menaia *et al.* 2002; Li *et al.* 2003; Ramos *et al.* 2010; Al-Jasser 2011, 2007; Abokofa *et al.* 2016). Several studies have investigated the role of hydraulics in chlorine decay. A first-order decay model and its modifications have been widely used to predict chlorine decay in water distribution systems (Qualls & Johnson 1983; Haas & Karra 1984; Jadas-Hecart *et al.* 1992; Meng *et al.* 2013). Menaia *et al.* (2002) investigated the relationship between flow velocity and first-order decay rate. Kim *et al.* (2015a) delineated a strong relationship between the Reynolds number (Re) and decay coefficient for a generalized chlorine prediction model.

The majority of experimental studies on chlorine decay have been conducted under conditions of constant pressure and velocity. A strong relationship between wall decay kinetics and flow characteristics (pipe diameter and flow velocity) also indicated the importance of hydraulics in chlorine residual (Termini & Viviani 2015). A large number of chlorine models for water distribution systems used flow rate and flow direction for prediction and calibration of unknown parameters (EPA 2005). Treated water flowing through a transmission system can also be investigated under unsteady flow conditions (Hoskins & Stoianov 2014; Rezaei *et al.* 2015). Li *et al.* (2003) evaluated the concentrations of residual chlorine for various retention times in a water distribution system. The advection reaction equation under water-hammer conditions was modeled by Fernandes & Karney (2004). A self-adaptive hydraulic and chlorine decay model was developed, and the relationship between chlorine decay rate and shear stress was shown, but precise mechanisms for chlorine decay under unsteady hydraulic conditions have not been explained completely (Aisopou *et al.* 2014). Ramos *et al.* (2010) reported, with limited experimental results, that a transient flow event could attenuate the decay of residual chlorine owing to a decrease in Reynolds number induced by rapid valve closure. Preliminary experimental results for chlorine decay for three different frequencies of transient events have been reported (Kim *et al.* 2015b). However, a comprehensive understanding of the relationship between transient flow events and chlorine decay, and the underlying processes, has not yet been achieved.

The purpose of this study is to compare the performance of existing chlorine models, to further the understanding of the impact of transient flow events on chlorine decay, and to implement the transient impact for chlorine modeling in water distribution systems. Ultimately, this study aims to develop a generic model to evaluate residual chlorine concentrations under unsteady flow conditions. For this purpose, the following objectives were explored. First, the variation in residual chlorine concentration under various unsteady flow conditions was monitored using a pilot-scale water distribution system. A transient generator was installed into a pipeline system to generate and regulate transient events. Second, the performances of the comprehensive models for chlorine decay were evaluated under unsteady conditions. A genetic algorithm (GA) was integrated into these models, and the parameters were calibrated to minimize the root-mean-square errors (RMSEs) between the observed and simulated chlorine concentrations. To generalize the existing models, the ranges of the orders for the *n*th- and limited *n*th-order models for chlorine decay were extended to include all real numbers, and the concentration of the stable component parameter was calibrated to address the effects of transient events on the reaction rates of chlorine compounds. Third, two-dimensional (2D) analysis of transient flow was conducted to quantify and characterize the impact of transients on the intensity of turbulence. Finally, generic equations for chlorine decay under transient conditions were developed and implemented into the models.

## MATERIALS AND METHODS

### Experimental setup

Transient flow events caused by closing the ball valve manually are difficult to regulate. To ensure uniform transient intensity for experimental purposes, a transient generator that can repeatedly generate identical events of various frequencies was designed, constructed and installed on the valve at the downstream end. The transient generator consists of a connection rod, a pneumatic cylinder, and a solenoid valve. One end of the connection rod was attached to the pneumatic cylinder, and the other was connected to the lever of the ball valve, which converted the linear motion of the cylinder into rotational motion of the ball valve (see Figure 1). The closing time of a transient generator can be set to 0.05–5.00 s. Pressure was measured at the adjacent upstream point of the control valve using the TP-1 monitoring system (Pipetech Int.). Sampling rates for the pressure were 10 Hz under steady state conditions and 100 Hz under transient conditions. Figure 1 shows a schematic and photographs of the pilot-scale water distribution system for the experiment.

### Chlorine decay models

Table 1 presents existing models and their corresponding parameters for predicting chlorine decay in water distribution systems (Haas & Karra 1984). The first-order model is based on the assumption that the reaction rate is proportional to the residual chlorine concentration. The *n*th-order model is similar, but the decay rate in this model is proportional to the *n*th power of chlorine concentration. Limited models assume that some chlorine remains in the water unreacted. The parallel first-order model assumes that the overall rate of chlorine decay can be derived from the fast and slow components of the decay processes; therefore, the parallel first-order model consists of the weighted sum of two different first-order models.

Title . | Governing equation . | Parameters . |
---|---|---|

1st order | ||

2nd order | ||

3rd order | ||

4th order | ||

Limited 1st order | ||

Limited 2nd order | ||

Limited 3rd order | ||

Limited 4th order | ||

Parallel 1st order |

Title . | Governing equation . | Parameters . |
---|---|---|

1st order | ||

2nd order | ||

3rd order | ||

4th order | ||

Limited 1st order | ||

Limited 2nd order | ||

Limited 3rd order | ||

Limited 4th order | ||

Parallel 1st order |

*et al.*2015a). If

*m*is the number of components in the decay of chlorine, then the rate of decrease of chlorine concentration over time can be determined by the summation of all reactants aswhere

*C*is the concentration of the corresponding reactant

_{i}*i*.

*k*is the decay coefficient for

_{i}*i*th reaction and

*n*is the order of the corresponding reaction. The initial concentration of the corresponding partial concentration can be defined aswhere

_{i}*w*is the weighting of the

_{i}*i*th reaction, and , and

*C*is the initial concentration of total chlorine. Equation (3) is the general formulation of the parallel first-order model.

_{o}Table 2 presents comprehensive models for five distinct formulations that incorporate existing model structures as well as their extensions. The *n*th- and limited *n*th-order models can be further generalized with an assumption that the parameters *n* and *k* are adjustable. The restriction of existing approaches of *n* as an integer is largely relaxed by defining *n* instead as a real number in the generic formulations. Further generalization of the *n*th and limited *n*th models can be made as the condition of two reactants is introduced; depending upon the scope of implementation of *n* as a real number into different decay processes, structures of a combined ‘1 + 1’ model, a combined ‘1 + *n*’ model, and a combined ‘*n* + *n*’ model can be developed (see Table 2).

Title . | Governing equation . | Parameters . |
---|---|---|

nth | ||

Limited nth | ||

Combined ‘1 + 1’ | ||

Combined ‘1 + n’ | ||

Combined ‘n + n’ |

Title . | Governing equation . | Parameters . |
---|---|---|

nth | ||

Limited nth | ||

Combined ‘1 + 1’ | ||

Combined ‘1 + n’ | ||

Combined ‘n + n’ |

### Calibration of model parameters

*i*is the time step, is the observed chlorine concentration, is the predicted chlorine concentration from a selected model, and represents parameters for a generic model.

### Transient analysis in the pipeline

*a*= wave speed (1,395 m/s),

*g*= gravitational acceleration,

*V*is the mean flow velocity calculated as ,

*r*

_{0}is the radius of the pipe, is the piezometric head, = wall shear stress which can be determined as in Zhao & Ghidaoui (2003), and

*ρ*represents the density of the liquid–vapor mixture, which can be calculated as . An auxiliary variable , which represents the piezometric head adjusted for vaporous cavitation, can be defined aswhere

*ρ*= liquid density, and

_{l}*α*= volume of vapor/total volume. Pressure (

_{v}*p*) and vapor fraction (

*α*) can be calculated using Equation (8):To solve Equations (5) and (6), a numerical algorithm based on a predictor–corrector method was used (MacCormack 2003). Equations (9) and (10) are the differentiated forms of predictor–corrector schemes:and

_{v}*i*,

*j*, and

*n*are the indices of longitudinal, radial, and time, respectively, at

*p*and

*c*which represent the predictor–corrector step and is the average of shear stress between the predictor and corrector step.

### Turbulence intensity

*N*sets of numerical simulations were conducted with minor differences in initial flow velocity. The ensemble average of velocity () can be expressed as follows:where is the ensemble average of velocity, is the number of ensemble sets with different initial flow velocities and is the velocity dependent on time and location. The ensemble average of velocity is a representative value that is independent from system uncertainty. The difference between an ensemble average of velocity () and a velocity of the

_{en}*i*th component of an ensemble set () is the velocity fluctuation () of the

*i*th component and can be expressed as follows:The strength of turbulence can be expressed as the root mean square quantity of the velocity fluctuation and can be written as follows:where is the strength of turbulence, represents the standard deviation of the velocity fluctuation and is the velocity fluctuation at the

*i*th component of the ensemble set.

*I*is the turbulence intensity at time

*t*, and and are under steady and transient conditions, respectively, for time step

*t*.

*t*(

*I*) can be calculated using Equation (17):where

_{T}*l*and represent a particular transient event and its duration for the

*l*th event, respectively, and

*N*is the number of transient events.

## RESULTS

### Transient event introduction and chlorine concentrations

### Calibration of chlorine decay model parameters

Parameter calibration results for models in Tables 1 and 2 are presented in Table 3. Calibrations were performed through three distinct model structures: the designation of the reaction order (*n*) as an integer for Equation (2) (e.g., models 1–9 in Table 3), calibration incorporating parameter *n* as a real number (e.g., models 10 and 11 in Table 3), and the most comprehensive calibration involving a flexible concentration of the stable component () instead of a designated for a lower detection limit of the sensor (e.g., models 12–14 in Table 3).

Model . | Title . | Parameter . | Steady . | 40 min . | 20 min . | 10 min . | 5 min . | 2.5 min . |
---|---|---|---|---|---|---|---|---|

1 | 1st order | k (day^{−1}) | 0.479 | 0.440 | 0.397 | 0.286 | 0.240 | 0.185 |

2 | 2nd order | k (day^{−1}) | 0.392 | 0.357 | 0.330 | 0.238 | 0.201 | 0.141 |

3 | 3rd order | k (day^{−1}) | 0.318 | 0.285 | 0.270 | 0.195 | 0.166 | 0.105 |

4 | 4th order | k (day^{−1}) | 0.260 | 0.244 | 0.244 | 0.163 | 0.139 | 0.078 |

5 | Limited 1st order | k (day^{−1}) | 0.488 | 0.458 | 0.397 | 0.292 | 0.244 | 0.188 |

6 | Limited 2nd order | k (day^{−1}) | 0.406 | 0.366 | 0.336 | 0.246 | 0.208 | 0.145 |

7 | Limited 3rd order | k (day^{−1}) | 0.334 | 0.305 | 0.275 | 0.206 | 0.175 | 0.110 |

8 | Limited 4th order | k (day^{−1}) | 0.278 | 0.250 | 0.250 | 0.175 | 0.156 | 0.084 |

9 | Parallel 1st order | w | 0.995 | 1.000 | 0.851 | 0.753 | 0.711 | 0.652 |

k_{1} (day^{−1}) | 0.468 | 0.440 | 0.400 | 0.288 | 0.240 | 0.184 | ||

k_{2} (day^{−1}) | 0.480 | 0.439 | 0.390 | 0.283 | 0.240 | 0.185 | ||

10 | nth order | n | 0.407 | 0.997 | 0.967 | 0.563 | 0.701 | 0.507 |

k (day^{−1}) | 0.527 | 0.437 | 0.401 | 0.305 | 0.250 | 0.213 | ||

11 | Limited nth order | n | 0.390 | 0.980 | 0.945 | 0.547 | 0.683 | 0.478 |

k (day^{−1}) | 0.531 | 0.437 | 0.409 | 0.309 | 0.254 | 0.214 | ||

12 | Combined ‘1 + 1’ | w | 0.942 | 0.992 | 0.959 | 0.903 | 0.859 | 0.792 |

k_{1} (day^{−1}) | 0.479 | 0.440 | 0.399 | 0.287 | 0.240 | 0.185 | ||

k_{2} (day^{−1}) | 0.478 | 0.440 | 0.390 | 0.276 | 0.238 | 0.185 | ||

c* (mg/L) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||

13 | Combined ‘1 + n’ | w | 0.125 | 0.332 | 0.314 | 0.259 | 0.083 | 0.015 |

k_{1} (day^{−1}) | 0.407 | 0.995 | 0.974 | 0.248 | 0.082 | 0.057 | ||

n | 0.348 | 0.131 | 0.348 | 0.423 | 0.571 | 0.573 | ||

k_{2} (day^{−1}) | 0.500 | 0.291 | 0.279 | 0.279 | 0.277 | 0.211 | ||

c* (mg/L) | 0.020 | 0.029 | 0.040 | 0.040 | 0.072 | 0.072 | ||

14 | Combined ‘n + n’ | w | 0.424 | 0.304 | 0.261 | 0.204 | 0.192 | 0.191 |

n_{1} | 0.375 | 0.878 | 0.804 | 0.604 | 0.582 | 0.165 | ||

k_{1} (day^{−1}) | 0.255 | 0.998 | 0.856 | 0.211 | 0.190 | 0.060 | ||

n_{2} | 0.364 | 0.069 | 0.433 | 0.501 | 0.628 | 0.651 | ||

k_{2} (day^{−1}) | 0.438 | 0.295 | 0.295 | 0.263 | 0.219 | 0.187 | ||

c* (mg/L) | 0.034 | 0.003 | 0.011 | 0.032 | 0.037 | 0.045 |

Model . | Title . | Parameter . | Steady . | 40 min . | 20 min . | 10 min . | 5 min . | 2.5 min . |
---|---|---|---|---|---|---|---|---|

1 | 1st order | k (day^{−1}) | 0.479 | 0.440 | 0.397 | 0.286 | 0.240 | 0.185 |

2 | 2nd order | k (day^{−1}) | 0.392 | 0.357 | 0.330 | 0.238 | 0.201 | 0.141 |

3 | 3rd order | k (day^{−1}) | 0.318 | 0.285 | 0.270 | 0.195 | 0.166 | 0.105 |

4 | 4th order | k (day^{−1}) | 0.260 | 0.244 | 0.244 | 0.163 | 0.139 | 0.078 |

5 | Limited 1st order | k (day^{−1}) | 0.488 | 0.458 | 0.397 | 0.292 | 0.244 | 0.188 |

6 | Limited 2nd order | k (day^{−1}) | 0.406 | 0.366 | 0.336 | 0.246 | 0.208 | 0.145 |

7 | Limited 3rd order | k (day^{−1}) | 0.334 | 0.305 | 0.275 | 0.206 | 0.175 | 0.110 |

8 | Limited 4th order | k (day^{−1}) | 0.278 | 0.250 | 0.250 | 0.175 | 0.156 | 0.084 |

9 | Parallel 1st order | w | 0.995 | 1.000 | 0.851 | 0.753 | 0.711 | 0.652 |

k_{1} (day^{−1}) | 0.468 | 0.440 | 0.400 | 0.288 | 0.240 | 0.184 | ||

k_{2} (day^{−1}) | 0.480 | 0.439 | 0.390 | 0.283 | 0.240 | 0.185 | ||

10 | nth order | n | 0.407 | 0.997 | 0.967 | 0.563 | 0.701 | 0.507 |

k (day^{−1}) | 0.527 | 0.437 | 0.401 | 0.305 | 0.250 | 0.213 | ||

11 | Limited nth order | n | 0.390 | 0.980 | 0.945 | 0.547 | 0.683 | 0.478 |

k (day^{−1}) | 0.531 | 0.437 | 0.409 | 0.309 | 0.254 | 0.214 | ||

12 | Combined ‘1 + 1’ | w | 0.942 | 0.992 | 0.959 | 0.903 | 0.859 | 0.792 |

k_{1} (day^{−1}) | 0.479 | 0.440 | 0.399 | 0.287 | 0.240 | 0.185 | ||

k_{2} (day^{−1}) | 0.478 | 0.440 | 0.390 | 0.276 | 0.238 | 0.185 | ||

c* (mg/L) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | ||

13 | Combined ‘1 + n’ | w | 0.125 | 0.332 | 0.314 | 0.259 | 0.083 | 0.015 |

k_{1} (day^{−1}) | 0.407 | 0.995 | 0.974 | 0.248 | 0.082 | 0.057 | ||

n | 0.348 | 0.131 | 0.348 | 0.423 | 0.571 | 0.573 | ||

k_{2} (day^{−1}) | 0.500 | 0.291 | 0.279 | 0.279 | 0.277 | 0.211 | ||

c* (mg/L) | 0.020 | 0.029 | 0.040 | 0.040 | 0.072 | 0.072 | ||

14 | Combined ‘n + n’ | w | 0.424 | 0.304 | 0.261 | 0.204 | 0.192 | 0.191 |

n_{1} | 0.375 | 0.878 | 0.804 | 0.604 | 0.582 | 0.165 | ||

k_{1} (day^{−1}) | 0.255 | 0.998 | 0.856 | 0.211 | 0.190 | 0.060 | ||

n_{2} | 0.364 | 0.069 | 0.433 | 0.501 | 0.628 | 0.651 | ||

k_{2} (day^{−1}) | 0.438 | 0.295 | 0.295 | 0.263 | 0.219 | 0.187 | ||

c* (mg/L) | 0.034 | 0.003 | 0.011 | 0.032 | 0.037 | 0.045 |

Parameter *k* tended to decrease with increasing transient generation frequency for models 1–8 in Table 3. These results reflect the temporal variation of chlorine concentration under different hydraulic conditions as shown in Figure 4. The weighting factor of model 9 also decreased with increasing transient frequency. However, because the parameters *k*_{1} and *k*_{2} of model 9 were almost identical to parameter *k* of model 1, the overall structure of model 9 is very similar to that of model 1. This similarity suggests that regardless of hydraulic conditions, assuming a first-order reaction for both the rapid initial decay and the slower and prolonged decay for the parallel first-order model is not appropriate.

Regarding models 10 and 11, the optimal *n* values for the *n*th-order and the limited *n*th-order models under steady flow conditions were 0.407 and 0.390, respectively. These values indicate that the chlorine decay rate was less sensitive to residual chlorine in this system than in other existing models. Optimal *n* and *k* under transient conditions both decreased as the water-hammer frequency is increased, which represents overall decline in the chlorine decay rate. This trend suggests that this transient-induced phenomenon may mitigate the consumption of chlorine.

Model 12 combines two limited first-order decay models with different rate constants. Because the model is flexible for the parameter , it can be simplified into model 1, 5 or 9 depending on the values of the parameters. The calibrated value of for all hydraulic conditions was determined to be zero, and the values of the corresponding rate coefficients *k*_{1} and *k*_{2} were almost the same. The structure of model 12 in this study was ultimately similar to the structure of model 1. This similarity indicates that model 1 performed best among models 1, 5, 9, and 12, considering that it is desirable to minimize differences between observational and modeling results with the minimum number of parameters for calibration with an evolutionary algorithm.

The weighting parameters of model 13 for all hydraulic conditions in this study were small (*w* < 0.5). This finding reflects that the *n*th (<1) order is the dominant form of modeling, and this trend intensified as the frequency of transient events was increased. Positive correlation was also found between the values of model 13 and the frequency of transient events, which means that the reactivity of chlorine compounds decreased with the occurrence of transient events. The generation of transient events, therefore, is found to inhibit the consumption of chlorine.

A positive effect of transient events on the concentration of stable component was shown in model 14 as well. This model combines an *n*_{1}th-order model with the decay coefficient *k*_{1} with an *n*_{2}th-order model with the decay coefficient *k*_{2}. The optimal balance between the *n*_{1}th-order model and *n*_{2}th-order model was determined through the modeling results. The weighting parameter *w* and the *n*_{1}th-order parameters *n*_{1} and *k*_{1} decreased with increased transient event frequency, but the converse was observed for the order *n*_{2}.

^{2}) were similar under both steady and unsteady conditions for each model. Figure 5 shows the means and standard errors of RMSE for all candidate chlorine decays models.

Frequency of transient event . | Steady . | 40 min . | 20 min . | 10 min . | 5 min . | 2.5 min . | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Model . | Title . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . |

1 | 1st order | 0.97 | 0.099 | 0.99 | 0.044 | 1.00 | 0.026 | 0.98 | 0.074 | 0.99 | 0.055 | 0.97 | 0.099 |

2 | 2nd order | 0.87 | 0.203 | 0.93 | 0.137 | 0.94 | 0.131 | 0.89 | 0.181 | 0.90 | 0.168 | 0.88 | 0.201 |

3 | 3rd order | 0.76 | 0.273 | 0.83 | 0.209 | 0.84 | 0.212 | 0.78 | 0.253 | 0.79 | 0.245 | 0.78 | 0.270 |

4 | 4th order | 0.66 | 0.323 | 0.74 | 0.263 | 0.73 | 0.272 | 0.68 | 0.304 | 0.69 | 0.300 | 0.68 | 0.320 |

5 | Limited 1st order | 0.97 | 0.102 | 0.99 | 0.048 | 1.00 | 0.031 | 0.98 | 0.077 | 0.99 | 0.058 | 0.97 | 0.102 |

6 | Limited 2nd order | 0.86 | 0.206 | 0.93 | 0.140 | 0.93 | 0.135 | 0.88 | 0.184 | 0.90 | 0.172 | 0.87 | 0.204 |

7 | Limited 3rd order | 0.84 | 0.225 | 0.90 | 0.165 | 0.90 | 0.168 | 0.86 | 0.201 | 0.88 | 0.189 | 0.82 | 0.241 |

8 | Limited 4th order | 0.65 | 0.326 | 0.73 | 0.265 | 0.73 | 0.275 | 0.67 | 0.307 | 0.68 | 0.304 | 0.68 | 0.323 |

9 | Parallel 1st order | 0.97 | 0.099 | 0.99 | 0.044 | 1.00 | 0.025 | 0.98 | 0.074 | 0.99 | 0.055 | 0.97 | 0.099 |

10 | nth order | 1.00 | 0.026 | 0.99 | 0.044 | 1.00 | 0.025 | 1.00 | 0.027 | 1.00 | 0.026 | 0.99 | 0.054 |

11 | Limited nth order | 1.00 | 0.026 | 0.99 | 0.048 | 1.00 | 0.025 | 1.00 | 0.027 | 1.00 | 0.026 | 0.99 | 0.049 |

12 | Combined ‘1 + 1’ | 0.97 | 0.099 | 0.99 | 0.044 | 1.00 | 0.025 | 0.98 | 0.074 | 0.99 | 0.055 | 0.97 | 0.099 |

13 | Combined ‘1 + n’ | 1.00 | 0.026 | 1.00 | 0.026 | 1.00 | 0.023 | 1.00 | 0.028 | 1.00 | 0.025 | 0.99 | 0.056 |

14 | Combined ‘n + n’ | 1.00 | 0.023 | 1.00 | 0.025 | 1.00 | 0.021 | 1.00 | 0.027 | 1.00 | 0.025 | 0.99 | 0.051 |

Frequency of transient event . | Steady . | 40 min . | 20 min . | 10 min . | 5 min . | 2.5 min . | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Model . | Title . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . |

1 | 1st order | 0.97 | 0.099 | 0.99 | 0.044 | 1.00 | 0.026 | 0.98 | 0.074 | 0.99 | 0.055 | 0.97 | 0.099 |

2 | 2nd order | 0.87 | 0.203 | 0.93 | 0.137 | 0.94 | 0.131 | 0.89 | 0.181 | 0.90 | 0.168 | 0.88 | 0.201 |

3 | 3rd order | 0.76 | 0.273 | 0.83 | 0.209 | 0.84 | 0.212 | 0.78 | 0.253 | 0.79 | 0.245 | 0.78 | 0.270 |

4 | 4th order | 0.66 | 0.323 | 0.74 | 0.263 | 0.73 | 0.272 | 0.68 | 0.304 | 0.69 | 0.300 | 0.68 | 0.320 |

5 | Limited 1st order | 0.97 | 0.102 | 0.99 | 0.048 | 1.00 | 0.031 | 0.98 | 0.077 | 0.99 | 0.058 | 0.97 | 0.102 |

6 | Limited 2nd order | 0.86 | 0.206 | 0.93 | 0.140 | 0.93 | 0.135 | 0.88 | 0.184 | 0.90 | 0.172 | 0.87 | 0.204 |

7 | Limited 3rd order | 0.84 | 0.225 | 0.90 | 0.165 | 0.90 | 0.168 | 0.86 | 0.201 | 0.88 | 0.189 | 0.82 | 0.241 |

8 | Limited 4th order | 0.65 | 0.326 | 0.73 | 0.265 | 0.73 | 0.275 | 0.67 | 0.307 | 0.68 | 0.304 | 0.68 | 0.323 |

9 | Parallel 1st order | 0.97 | 0.099 | 0.99 | 0.044 | 1.00 | 0.025 | 0.98 | 0.074 | 0.99 | 0.055 | 0.97 | 0.099 |

10 | nth order | 1.00 | 0.026 | 0.99 | 0.044 | 1.00 | 0.025 | 1.00 | 0.027 | 1.00 | 0.026 | 0.99 | 0.054 |

11 | Limited nth order | 1.00 | 0.026 | 0.99 | 0.048 | 1.00 | 0.025 | 1.00 | 0.027 | 1.00 | 0.026 | 0.99 | 0.049 |

12 | Combined ‘1 + 1’ | 0.97 | 0.099 | 0.99 | 0.044 | 1.00 | 0.025 | 0.98 | 0.074 | 0.99 | 0.055 | 0.97 | 0.099 |

13 | Combined ‘1 + n’ | 1.00 | 0.026 | 1.00 | 0.026 | 1.00 | 0.023 | 1.00 | 0.028 | 1.00 | 0.025 | 0.99 | 0.056 |

14 | Combined ‘n + n’ | 1.00 | 0.023 | 1.00 | 0.025 | 1.00 | 0.021 | 1.00 | 0.027 | 1.00 | 0.025 | 0.99 | 0.051 |

As shown in Table 4, the low-order existing models (i.e., first-, limited first-, and parallel first-order) show relatively high fitness compared to the high-order existing models (i.e., second-, third-, fourth-, limited second-, limited third-, and limited fourth-order). This finding can be explained by the characteristics of the experimental data and the structures of the models. High-order models have advantages for prediction of steep declines from the initial chlorine concentration, which is suitable for representing rapid chlorine decay. However, because organic compounds were not added for this experiment, chlorine was consumed more gradually by the low levels of organics typical for drinking water.

R^{2} values for the generic models were higher, the corresponding RMSEs were lower, and corresponding standard errors were narrower than those for other models (see Table 4). This means that an additional degree of freedom in the reaction order (*n _{i}*) has great potential to enable better estimates of chlorine decay behavior under both steady and unsteady conditions. Chlorine decay behavior conditions can therefore be successfully modeled using the structures of Equations (1), (2), and (3) for either steady or unsteady flow conditions, provided that the variability of other factors (e.g., temperature, service age, and concentration of organics) can be controlled. These findings suggest that there is no universal chlorine decay model that is suitable for all system conditions. As the number of adjustable parameters increases, models have more flexibility to fit chlorine decay under a variety of conditions.

## DISCUSSION

### Transient event impact on the chlorine decay process

Figure 7(b) presents radial velocity profiles with different times (in terms of 2*L*/*a* where *L* is the length of the pipeline and *a* is the wave speed) for a point 50 m from the water tank. As shown in the velocity profile for 8.25·2*L*/*a*, flow separation can be generated. Flow reversal was observed in the velocity profile for 8.25·2*L*/*a* after a transient event, which implies the occurrence of an adverse pressure gradient near the wall region.

*et al.*1980). The turbulence intensity gradient tends to become pronounced as the velocity profile develops (see Figure 8); however, the standard deviation and mean of radial turbulence intensity distribution, respectively, were 0.00067 and 0.22379, which means that there is no significant difference in the intensity of turbulence between the wall and the centerline area in this system because of the pipe's small diameter. Figure 9 shows the temporal variation of averaged radial turbulence intensity during a single transient event at a point 5 m from the end valve. During the transient event, the averaged radial turbulence intensity of the system fluctuated from zero to about two times the steady state value. As the closed valve is opened at the end of the transient event, the negligible turbulent intensity is increased to that of steady state flow. The highest value of turbulence intensity was observed after valve closure, when the average standard deviation and the mean of the radial distribution of turbulence intensity during a transient event were 0.0000524 and 0.041939, respectively.

Transient event interval . | Number of transient event during 4 days (N)
. | Total amount of turbulence intensity reduction () . |
---|---|---|

Steady state (No transient event) | 0 | 0 |

40 min | 151 | 274.8997 |

20 min | 375 | 682.717 |

10 min | 948 | 1,727.655 |

5 min | 2,404 | 4,379.447 |

2.5 min | 5,330 | 9,712.182 |

Transient event interval . | Number of transient event during 4 days (N)
. | Total amount of turbulence intensity reduction () . |
---|---|---|

Steady state (No transient event) | 0 | 0 |

40 min | 151 | 274.8997 |

20 min | 375 | 682.717 |

10 min | 948 | 1,727.655 |

5 min | 2,404 | 4,379.447 |

2.5 min | 5,330 | 9,712.182 |

Power regressions of versus the parameters of chlorine decay models are presented in Table 6. The parameters of existing models are affected negatively by and show strong correlations (R^{2} > 0.86). The parameter *a* (the proportional coefficient of regression equation) tends to decrease as the order *n* increases for the *n*th-order and limited *n*th-order models, and the converse tendency is apparent for the parameter *b* (the power coefficient of regression equation), except in the case of the fourth-order model, for which R^{2} values are small relative to those of other models (see Table 6).

Model . | y
. | X
. | a
. | b
. | R^{2}
. |
---|---|---|---|---|---|

1st order | 1.869 | −2.487 × 10^{−1} | 0.980 | ||

2nd order | 1.670 | −2.611 × 10^{−1} | 0.958 | ||

3rd order | 1.474 | −2.741 × 10^{−1} | 0.928 | ||

4th order | 1.657 | −3.143 × 10^{−1} | 0.892 | ||

Limited 1st order | 1.951 | −2.519 × 10^{−1} | 0.988 | ||

Limited 2nd order | 1.677 | −2.578 × 10^{−1} | 0.959 | ||

Limited 3rd order | 1.557 | −2.750 × 10^{−1} | 0.936 | ||

Limited 4th order | 1.498 | −2.925 × 10^{−1} | 0.861 | ||

Parallel 1st order | w | 1.845 | −1.151 × 10^{−1} | 0.968 | |

k _{1} | 1.892 | −2.500 × 10^{−1} | 0.978 | ||

k _{2} | 1.827 | −2.463 × 10^{−1} | 0.983 | ||

nth | n | 2.868 | −1.861 × 10^{−1} | 0.740 | |

k | 1.494 | −2.118 × 10^{−1} | 0.983 | ||

Limited nth | n | 2.978 | −1.952 × 10^{−1} | 0.751 | |

k | 1.513 | −2.120 × 10^{−1} | 0.977 | ||

Combined ‘1 + 1’ | w | 1.426 | −6.217 × 10^{−1} | 0.975 | |

k _{1} | 1.878 | −2.491 × 10^{−1} | 0.978 | ||

k _{2} | 1.834 | −2.477 × 10^{−1} | 0.980 | ||

w | 61.96 | −8.330 × 10^{−1} | 0.790 | ||

Combined ‘1 + n’ | k _{1} | 230.3 | −9.147 × 10^{−1} | 0.941 | |

n | 0.02066 | 3.857 × 10^{−1} | 0.816 | ||

k _{2} | 0.4503 | −7.087 × 10^{−1} | 0.596 | ||

c* | 0.0066 | 2.653 × 10^{−1} | 0.901 | ||

w | 0.6310 | −1.379 × 10^{−1} | 0.892 | ||

Combined ‘n+n’ | n _{1} | 10.39 | −4.009 × 10^{−1} | 0.717 | |

k _{1} | 104.2 | −7.918 × 10^{−1} | 0.925 | ||

n _{2} | 0.006321 | 5.443 × 10^{−1} | 0.680 | ||

k _{1} | 0.6737 | −1.344 × 10^{−1} | 0.915 | ||

c* | 6 × 10^{−5} | 7.537 × 10^{−1} | 0.855 |

Model . | y
. | X
. | a
. | b
. | R^{2}
. |
---|---|---|---|---|---|

1st order | 1.869 | −2.487 × 10^{−1} | 0.980 | ||

2nd order | 1.670 | −2.611 × 10^{−1} | 0.958 | ||

3rd order | 1.474 | −2.741 × 10^{−1} | 0.928 | ||

4th order | 1.657 | −3.143 × 10^{−1} | 0.892 | ||

Limited 1st order | 1.951 | −2.519 × 10^{−1} | 0.988 | ||

Limited 2nd order | 1.677 | −2.578 × 10^{−1} | 0.959 | ||

Limited 3rd order | 1.557 | −2.750 × 10^{−1} | 0.936 | ||

Limited 4th order | 1.498 | −2.925 × 10^{−1} | 0.861 | ||

Parallel 1st order | w | 1.845 | −1.151 × 10^{−1} | 0.968 | |

k _{1} | 1.892 | −2.500 × 10^{−1} | 0.978 | ||

k _{2} | 1.827 | −2.463 × 10^{−1} | 0.983 | ||

nth | n | 2.868 | −1.861 × 10^{−1} | 0.740 | |

k | 1.494 | −2.118 × 10^{−1} | 0.983 | ||

Limited nth | n | 2.978 | −1.952 × 10^{−1} | 0.751 | |

k | 1.513 | −2.120 × 10^{−1} | 0.977 | ||

Combined ‘1 + 1’ | w | 1.426 | −6.217 × 10^{−1} | 0.975 | |

k _{1} | 1.878 | −2.491 × 10^{−1} | 0.978 | ||

k _{2} | 1.834 | −2.477 × 10^{−1} | 0.980 | ||

w | 61.96 | −8.330 × 10^{−1} | 0.790 | ||

Combined ‘1 + n’ | k _{1} | 230.3 | −9.147 × 10^{−1} | 0.941 | |

n | 0.02066 | 3.857 × 10^{−1} | 0.816 | ||

k _{2} | 0.4503 | −7.087 × 10^{−1} | 0.596 | ||

c* | 0.0066 | 2.653 × 10^{−1} | 0.901 | ||

w | 0.6310 | −1.379 × 10^{−1} | 0.892 | ||

Combined ‘n+n’ | n _{1} | 10.39 | −4.009 × 10^{−1} | 0.717 | |

k _{1} | 104.2 | −7.918 × 10^{−1} | 0.925 | ||

n _{2} | 0.006321 | 5.443 × 10^{−1} | 0.680 | ||

k _{1} | 0.6737 | −1.344 × 10^{−1} | 0.915 | ||

c* | 6 × 10^{−5} | 7.537 × 10^{−1} | 0.855 |

In addition, we report that regression equations provide adequate estimations of chlorine decay through the total reduction of turbulence intensity. Regression equations and validation results of the parameters for the parallel first-order model and combined ‘1 + 1’ model were similar to those for the first-order model. Although the *n*th- and limited *n*th-order models show weak correlations with , the fitness of these models, using parameters obtained from the regression equations is generally favorable. As is increased, the parameter *w* of the combined ‘1 + *n*’ model tends to decrease; i.e., the combined ‘1 + *n*’ model tends to be governed by the *n*th-order model as transient events are introduced into system. The parameter *c** of the combined ‘1 + *n*’ and combined ‘*n* + *n*’ models tends to decrease as the value for is increased, which means that the amount of unreactive chlorine is increased with decreasing turbulence intensity.

Degrees of fitness (in terms of R^{2} and RMSE) of modeling calculated using the parameters obtained from the regression equations in Table 6 are presented in Table 7. R^{2}s and RMSEs for the *n*th-order and limited *n*th-order models tend to decrease and increase, respectively, as the order *n* of the models is increased. Both R^{2} and RMSE for the five generic models (from the *n*th-order model to the combined ‘*n* + *n*’ model) are generally superior to the nine existing models. These findings indicate that the introduction of additional calibration parameters and relaxation limitations to chlorine decay order in model structure improve the predictability of chlorine decay behavior substantially even in modeling considering turbulence intensity for parameter evaluation. Chlorine decay models with low order (*n* ≤ 1) showed good performance in terms of predictability and parsimony of model parameter than those for high order models.

I_{t}. | 274.90 . | 682.72 . | 1,727.66 . | 4,379.45 . | 9,712.18 . | |||||
---|---|---|---|---|---|---|---|---|---|---|

Title . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . |

1st order | 0.99 | 0.055 | 0.99 | 0.056 | 0.98 | 0.076 | 0.99 | 0.059 | 0.97 | 0.101 |

2nd order | 0.92 | 0.141 | 0.93 | 0.136 | 0.89 | 0.181 | 0.90 | 0.171 | 0.87 | 0.204 |

3rd order | 0.83 | 0.212 | 0.83 | 0.215 | 0.78 | 0.253 | 0.79 | 0.249 | 0.77 | 0.274 |

4th order | 0.72 | 0.269 | 0.73 | 0.271 | 0.68 | 0.304 | 0.69 | 0.303 | 0.68 | 0.324 |

Limited 1st order | 0.99 | 0.059 | 0.99 | 0.055 | 0.98 | 0.079 | 0.99 | 0.062 | 0.97 | 0.103 |

Limited 2nd order | 0.92 | 0.143 | 0.93 | 0.141 | 0.88 | 0.184 | 0.90 | 0.175 | 0.87 | 0.207 |

Limited 3rd order | 0.92 | 0.148 | 0.87 | 0.186 | 0.85 | 0.206 | 0.84 | 0.218 | 0.86 | 0.216 |

Limited 4th order | 0.72 | 0.269 | 0.73 | 0.275 | 0.67 | 0.307 | 0.68 | 0.306 | 0.67 | 0.328 |

Parallel 1st order | 0.99 | 0.056 | 0.99 | 0.055 | 0.98 | 0.076 | 0.99 | 0.059 | 0.97 | 0.099 |

nth order | 0.99 | 0.050 | 0.98 | 0.066 | 0.99 | 0.044 | 1.00 | 0.026 | 0.99 | 0.056 |

Limited nth order | 0.99 | 0.049 | 0.98 | 0.071 | 0.99 | 0.042 | 1.00 | 0.026 | 0.99 | 0.054 |

Combined ‘1 + 1’ | 0.99 | 0.051 | 0.98 | 0.067 | 0.98 | 0.077 | 0.98 | 0.066 | 0.97 | 0.102 |

Combined ‘1 + n’ | 0.99 | 0.051 | 0.95 | 0.117 | 0.97 | 0.090 | 0.99 | 0.062 | 0.97 | 0.102 |

Combined ‘n + n’ | 1.00 | 0.030 | 0.97 | 0.084 | 0.99 | 0.048 | 1.00 | 0.032 | 0.98 | 0.085 |

I_{t}. | 274.90 . | 682.72 . | 1,727.66 . | 4,379.45 . | 9,712.18 . | |||||
---|---|---|---|---|---|---|---|---|---|---|

Title . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . | R^{2}
. | RMSE . |

1st order | 0.99 | 0.055 | 0.99 | 0.056 | 0.98 | 0.076 | 0.99 | 0.059 | 0.97 | 0.101 |

2nd order | 0.92 | 0.141 | 0.93 | 0.136 | 0.89 | 0.181 | 0.90 | 0.171 | 0.87 | 0.204 |

3rd order | 0.83 | 0.212 | 0.83 | 0.215 | 0.78 | 0.253 | 0.79 | 0.249 | 0.77 | 0.274 |

4th order | 0.72 | 0.269 | 0.73 | 0.271 | 0.68 | 0.304 | 0.69 | 0.303 | 0.68 | 0.324 |

Limited 1st order | 0.99 | 0.059 | 0.99 | 0.055 | 0.98 | 0.079 | 0.99 | 0.062 | 0.97 | 0.103 |

Limited 2nd order | 0.92 | 0.143 | 0.93 | 0.141 | 0.88 | 0.184 | 0.90 | 0.175 | 0.87 | 0.207 |

Limited 3rd order | 0.92 | 0.148 | 0.87 | 0.186 | 0.85 | 0.206 | 0.84 | 0.218 | 0.86 | 0.216 |

Limited 4th order | 0.72 | 0.269 | 0.73 | 0.275 | 0.67 | 0.307 | 0.68 | 0.306 | 0.67 | 0.328 |

Parallel 1st order | 0.99 | 0.056 | 0.99 | 0.055 | 0.98 | 0.076 | 0.99 | 0.059 | 0.97 | 0.099 |

nth order | 0.99 | 0.050 | 0.98 | 0.066 | 0.99 | 0.044 | 1.00 | 0.026 | 0.99 | 0.056 |

Limited nth order | 0.99 | 0.049 | 0.98 | 0.071 | 0.99 | 0.042 | 1.00 | 0.026 | 0.99 | 0.054 |

Combined ‘1 + 1’ | 0.99 | 0.051 | 0.98 | 0.067 | 0.98 | 0.077 | 0.98 | 0.066 | 0.97 | 0.102 |

Combined ‘1 + n’ | 0.99 | 0.051 | 0.95 | 0.117 | 0.97 | 0.090 | 0.99 | 0.062 | 0.97 | 0.102 |

Combined ‘n + n’ | 1.00 | 0.030 | 0.97 | 0.084 | 0.99 | 0.048 | 1.00 | 0.032 | 0.98 | 0.085 |

### Chlorine decay under transient conditions

In this study, time series of residual chlorine concentrations under steady and transient conditions were obtained from a pilot-scale water distribution system. Although different pipeline materials were used, the evaluated chlorine decay rates showed similar trends to those reported by Ramos *et al.* (2010), i.e., that the chlorine decay rates under unsteady flow conditions are lower than those under steady flow conditions. However, this study further investigated the effects of transient events on the decay rates of chlorine compounds through the application of diverse event frequencies and through two-dimensional flow analyses with respect to the relationships between turbulence intensity and chlorine decay processes.

Generic models for chlorine decay that were previously proposed by Kim *et al.* (2015a) for steady flow conditions were used in this study. Based on the proposed models, a calibration range for the parameter *n* was extended to real numbers, instead of limited to integers, and the concentration of the stable component (*c**) was evaluated as an adjustable parameter instead of a fixed value to compensate for various hydraulic conditions. Among the 14 candidate models, the combined ‘*n* + *n*’ model yielded the best performance for fitting chlorine concentration data. The concentration of *c** decreased as the transient event frequency increased for combined ‘*n* + *n*’ model (Table 3), which indicates that the reactivity of chlorine compounds tends to decrease with the introduction of transient events into the system.

This phenomenon can be explained based on the influence of turbulence intensity on the interaction of chlorine in bulk water with biofilm along the pipe wall (Percival *et al.* 2000). Turbulence intensity in the early period of the water hammer, at 2.5 *L*/*a* where *L* is pipeline length and *a* is wave speed, is greater than that under steady flow conditions, but it decreases over time (Shamloo & Mousavifard 2015). We used a closed valve for about 10 sec, which is approximately equal to the period of 100 *L*/*a*. Therefore, overall turbulence intensity was substantially lower than the intensity under steady flow conditions. As shown in Figure 9, the total turbulence intensity under transient events was less than that under steady flow conditions. From the perspective of collision theory, the chlorine compounds in a system with lower turbulence have fewer chances to react with reactants compared to those in systems with higher turbulence (Hahn 1994). Therefore, the regulation of transients can be considered as an alternative control to reduce chlorine decay of treated water in water distribution systems.

## CONCLUSIONS

Hydraulic transients in a pipeline system introduce changes to the flow regime and affect residual chlorine concentrations. The impacts of transient events on the mechanisms of chlorine decay had not previously been systematically explored using a pilot-scale water distribution system. Therefore, to investigate chlorine concentration behavior comprehensively, general model frameworks were introduced and integrated into GAs for parameter calibration. The proposed model structure showed better fitness than existing models, which was achieved based on the relaxation of reaction order and concentration of the stable component in calibration. The first order model provided good performance both in predictability and model parsimony. Calibrated parameters yielded higher decay coefficients for steady flow than for unsteady flow conditions, and lower decay coefficients were calculated for hydraulic conditions with transient events at shorter intervals. Two-dimensional analysis of transient flow was conducted to evaluate the impact of turbulence intensity on the chlorine decay process. Decreased turbulence intensity with the introduction of transient events into the system is responsible for the observed reduction in chlorine decay, which can further be explained as the reduction of chlorine reaction with reactants. The strong relationships between the reduction of turbulence intensity and chlorine decay coefficients, as well as the high level of fitness of the simulations of chlorine concentrations using model parameters under transient conditions, highlight the impact of transient events on chlorine decay, and demonstrate that they can be successfully modeled in water distribution systems.

## ACKNOWLEDGEMENTS

This research is supported by Korea Ministry of Environment as ‘Global Top Project (RE201606133)’.