Evaluation of chlorine decay models under transient conditions in a water distribution system

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 ﬂ ow 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 ﬂ ow 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 ). 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. ).
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, 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 nth-and limited nth-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.

Experimental setup
A pilot-scale water distribution system was designed and used to evaluate temporal variations in residual chlorine concentrations under a wide range of flow conditions. The pipeline was 125 m in length, with one pressurized tank, one reservoir tank, and a serial pump system with three pumps (Figure 1). The pipe was made of stainless steel with an elastic modulus of 190 GPa. The inner diameter and thickness of the pipe were 0.02 m and 0.003 m, respectively. The pressurized tank was connected to the discharge of the serial pump system, and it provided sufficient pressure head for stable water circulation at a designated velocity throughout the system. The reservoir tank was connected at the downstream end of the pipe. The serial pump system was installed between the two tanks to generate various hydraulic conditions ranging between Reynolds numbers (Re) of 2,000 and 800,000. In this study, the Re for a steady flow condition was 140,000. The measurement range of the chlorine sensor (CLO 1-mA-2 ppm, ProMinent, Inc.) was 0.02-2.00 ppm with an uncertainty of ±0.02 ppm.
The chlorine sensor was installed in a bypass loop equipped with a flow control valve and a flowrate measurement device (DGMa310T000, ProMinent, Inc.). The flowrate for the chlorine sensor was maintained at 60 L/hr regardless of flow conditions in the main pipeline system. The system measured chlorine concentration at a sampling rate of 1 Hz. The current signal from the residual chlorine sensor (4-20 mA) was sent to a data acquisition system and converted to a corresponding chlorine concentration (ppm).
The potential effects of biofilm generation and other residuals were minimized by cleaning the pipeline system with detergent prior to each experiment. Tap water was also circulated through the pipeline system for 30 min, and the absence of residuals was confirmed before each experiment.  Figure 1 shows a schematic and photographs of the pilot-scale water distribution system for the experiment. Table 1 presents existing models and their corresponding parameters for predicting chlorine decay in water distribution systems (Haas & Karra ). The first-order model is based on the assumption that the reaction rate is proportional to the residual chlorine concentration. The nthorder model is similar, but the decay rate in this model is proportional to the nth power of chlorine concentration.

Chlorine decay models
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.
To compile the approaches of these existing models into a generic model structure, this study introduces a comprehensive modeling framework based on the assumption that chlorine decay is controlled by one or more independent mechanism(s) that initiate simultaneously (Kim et al. a). If m is the number of components in the decay of chlorine, then the rate of decrease of chlorine Þþ(1 À w 1 )C 0 exp Àk 2 t ð Þ k 1 , k 2 , w 1 concentration over time can be determined by the summation of all reactants as where C i is the concentration of the corresponding reactant i.
The reaction rate for each individual component can be generalized as where k i is the decay coefficient for ith reaction and n i is the order of the corresponding reaction. The initial concentration of the corresponding partial concentration can be defined as where w i is the weighting of the ith reaction, and P m i¼1 w i ¼ 1, and C o is the initial concentration of total chlorine. Equation (3) is the general formulation of the parallel first-order model. well as their extensions. The nth-and limited nth-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 nth and limited nth 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 þ 1' model, a combined '1 þ n' model, and a combined 'n þ n' model can be developed (see Table 2).

Calibration of model parameters
In this study, the parameters of the chlorine decay models were calibrated using a GA (Goldberg ). The population and generation numbers were 100 and 100, respectively, and other GA parameters were determined based on the k 1 , k 2 , n 1 , n 2 , C Ã , w 1 recommendations of Goldberg (). The objective function of this study was to minimize the RMSE between the chlorine concentrations predicted with the selected model and the observed values. The following equation represents this objective function: where i is the time step, C obs i ð Þ is the observed chlorine con- Þis the predicted chlorine concentration from a selected model, and p k represents parameters for a generic model.

Transient analysis in the pipeline
Two-dimensional analysis was conducted to determine the radial variations in hydraulic conditions during transient events. Figure 2 illustrates a cylindrical grid element of the pipe that was used for modeling. Because the water pressure  (5) and (6): , r 0 is the radius of the pipe, H ¼ z þ p=ρ i g is the piezometric head, τ ¼ wall shear stress which can be determined as in Zhao & Ghidaoui (), and ρ represents the density of the liquid-vapor mixture, which can be calcu- represents the piezometric head adjusted for vaporous cavitation, can be defined as where ρ l ¼ liquid density, and α v ¼ volume of vapor/total volume. Pressure (p) and vapor fraction (α v ) can be calculated using Equation (8): To solve Equations (5) and (6), a numerical algorithm based on a predictor-corrector method was used (MacCormack ). Equations (9) and (10) are the differentiated forms of predictor-corrector schemes: and i, j, and n are the indices of longitudinal, radial, and time, respectively, at p and c which represent the predictor-corrector step and τ Ã is the average of shear stress between the predictor and corrector step.

Turbulence intensity
Turbulent flow in a real-life system is based on the unpredictable variation of flow velocity and pressure. It is impossible to calculate the intensity of the turbulence at a certain time and location with either a 1D or 2D simulation model. In this regard, turbulence simulations should be performed The strength of turbulence can be expressed as the root mean square quantity of the velocity fluctuation and can be written as follows: where T strength is the strength of turbulence, u rms represents the standard deviation of the velocity fluctuation and u 0 i is the velocity fluctuation at the ith component of the ensemble set.
Turbulence intensity (I) is the relative quantity of the standard deviation of the velocity fluctuation to the mean flow velocity, which can be expressed as follows: where u is the mean velocity u ¼ 1=N en P Nen i¼1 u i . The integrated turbulence intensity accumulated over time t (I t ) and the difference in accumulated turbulence intensity between the steady state and a transient event (ΔI t ) are expressed by Equations (15) and (16), respectively: where I is the turbulence intensity at time t, and I t,steady and I t,transient are I t under steady and transient conditions, respectively, for time step t.
The total amount of the reduction of turbulence intensity caused by transient events at time t (I T ) can be calculated using Equation (17): where l and t(l) represent a particular transient event and its duration for the lth event, respectively, and N is the number of transient events.

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 (c Ã ) instead of a designated c Ã for a lower detection limit of the sensor (e.g., models 12-14 in Table 3).
Parameter k tended to decrease with increasing transient generation frequency for models 1-8 in Table 3. These  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   Table 4 shows that for all models used for calibration, the degrees of fitness in terms of RMSEs and coefficients of determination (R 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.
As shown in Table 4, the low-order existing models (i.e., first-, limited first-, and parallel first-order) show relatively  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 In order to evaluate the underlying processes of chlorine decay in conjunction with varying flow regimes, Equations (5)-(10) were employed to model temporal and spatial flow variations in two-dimensional space (see Figure 2).
Numerical results obtained with Equations (9) and (10) are displayed in Figure 6; these results show strong agreement with the experimental data.   ). Figure 8 shows the radial variation of turbulence intensity in the steady state and after a transient event. Turbulence intensity at or near the pipe wall is generally higher than that at the centerline because of high shear stress near the wall (Kita et al. ). 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    this study, each transient event has an identical total turbulent intensity that is repeated every 10 seconds. Therefore, Equation (17) can be simplified as follows: Power regressions of I T versus the parameters of chlorine decay models are presented in Table 6.  Table 6).
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 þ 1' model were similar to those for the first-order model. Although the nth-and limited nthorder models show weak correlations with I T , the fitness of these models, using parameters obtained from the regression equations is generally favorable. As I T is increased, the parameter w of the combined '1 þ n' model tends to decrease; i.e., the combined '1 þ n' model tends to be governed by the nth-order model as transient events are introduced into system. The parameter c* of the combined '1 þ n' and combined 'n þ n' models tends to decrease as the value for I T 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. Generic models for chlorine decay that were previously proposed by Kim et al. (a) 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 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. ). 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 ). 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 ). Therefore, the regulation of transients can be considered as an alternative control to reduce chlorine decay of treated water in water distribution systems.