Theoretical Estimation of Disinfectant Mass Balance Components in Drinking Water Distribution Systems

Abstract

The water quality audit concept is an important feature in assessing the loss of disinfectant mass in drinking water distribution systems. Based on this concept, the loss can be divided into the loss of disinfectant mass through water losses ($M_l$) and the decay of disinfectant mass due to chemical reactions ($M_r$). When an audit focuses on the effect of water losses on the loss of disinfectant mass, the decay of disinfectant mass by chemical reactions with the ideal condition of no water losses ($M_{ro}$) has to be estimated; thus, the disinfectant mass associated with water losses ($M_{WL}=M_l+M_r-M_{ro}$) can be assessed. Generally, the computation of these components ($M_l$, $M_r,$ and $M_{WL}$) needs hydraulic and water quality modeling. In this study, we propose a novel method based on a simple theoretical analysis to evaluate these components using only two parameters: the ratio of water losses ($p$) and the ratio of disinfectant concentrations at the critical pressure point and the network inlet ($C_p^{*}$). The coefficients of our theoretical $M_l$, $M_r$, and $M_{WL}$ were estimated using 20 real network models, with $p$ between 2.8% and 54.9% and $C_p^{*}$ between 18.4% and 91.9%. The results showed that our equations were effective at assessing the loss of disinfectant mass in drinking water distribution networks for the top-down auditing approach.

1. Introduction

To provide safe drinking water that is delivered to users through a distribution system, a disinfectant residual must be maintained in the system to control microbial contamination. Chlorine and chloramine have been widely applied as disinfectants to ensure the quality of drinking water during distribution. Free chlorine is the primary disinfectant because of its effectiveness and low cost [1]. However, free chlorine is volatile, causing rapid decay. In addition, it can react with natural organic compounds to produce harmful disinfection by-products (DBPs) such as trihalomethanes (THMs), halo acetic acids (HAAs), and other halogenated DBPs [2]. While chloramine is less volatile and produces fewer chlorinated DBPs than free chlorine, under certain conditions, chloramine use can introduce incomplete nitrification in the distribution system, causing adverse effects such as decreased chloramine residuals, increased heterotrophic bacteria populations, increased nitrite, and decreased alkalinity, pH, and dissolved oxygen (DO) concentrations [3].

The disinfectant residual declines with distance and time. The regrowth of microorganisms in the distribution system takes place when there is a low disinfectant residual. Thus, a stable disinfectant residual concentration must be assured throughout the system. The disinfectant residual stability is influenced by many factors, such as temperature, water age, piping material, corrosion products, pH, hydraulic condition, disinfectant type and dosage, and microbial activity [4,5]. However, if water is contaminated by heavy metals, some advanced methods to purify water might be needed [6,7]. At present, water utilities are collecting more hydraulic and water quality data, including data from sampling, loggers, telemetry, and geographic information systems. Thus, computer water distribution system models combined with water quality analysis modules can satisfactorily simulate the disinfectant residual and DBPs in distribution systems [8,9]. Therefore, many modeling studies have focused on optimizing the stable disinfectant residual concentration in drinking water distribution systems by using specific strategies or objectives, such as optimizing disinfectant dosing, placing monitoring and booster stations, and controlling DBPs [10,11,12,13,14,15,16]. However, their studies have not comprehensively assessed the nature of the water distribution systems causing the disinfectant residual mass losses before applying their strategies. The disinfectant mass losses can be from the decay by chemical reactions or a mass outgoing with water losses. Both causes lead to an increase in the input of the disinfectant mass to the systems. Thus, identifying the causes will help the operators understand their systems and apply the right strategies to control the losses. From the perspectives of water loss and energy loss, auditing is found to be one of the best ways to identify the causes of water losses and energy losses in water distribution systems. Recently, Lipiwattanakarn et al. [17] proposed water quality audits to assess the causes of chlorine losses. The audits are discussed as follows.

1.1. Water, Energy, and Water Quality Audits

The assessment of water losses via a water balance and water audit is recommended as a standard methodology to quantify water losses in water distribution systems and to build an effective water loss control program [18,19,20]. In the water balance concept, the system input volume ($SIV$) can be divided into authorized consumption ($AC$) and water losses ($WL$). Then, $WL$ can be separated into real losses and apparent losses. More division into smaller components can be done if the data are available. For example, real losses can be broken down into three sub-components: leakage at transmission and/or distribution mains, leakage and overflows at utility’s storage tanks, and leakage at service connections. After that, each component can be assessed economically to find the right strategies to control $WL$.

This balance concept has been further applied to other related parameters in water distribution systems, such as energy and chlorine residual mass. Cabrera et al. [21] introduced the concept of an energy audit in water distribution systems where the input energy ($E_{in}$) is split into three components: the energy delivered to users ($E_u$), the outgoing energy through water losses ($E_l$) and the energy loss by friction ($E_f$). On the other hand, to be consistent with the water balance components, Mamade et al. [22] divided the input energy into two components: the energy associated with authorized consumption ($E_{AC}$) and the energy associated with water loss ($E_{WL}$). The concept of energy balance has been successfully applied to assess the energy losses in real water distribution systems in many countries [23,24,25,26,27,28,29,30,31].

Lipiwattanakarn et al. [17] focused on the quantified assessment of disinfectant residual mass losses and proposed the disinfectant mass balance to assess the losses in distribution systems. Using the same perspective as Cabrera et al. [21], the input mass ($M_{in}$) can be divided into three main components: the mass delivered to users ($M_u$), outgoing mass through water losses ($M_l$), and mass losses by reactions ($M_r$), as shown in Figure 1a. Using an example of a real district metering area (DMA) network, they demonstrated how to assess the disinfectant mass loss and how to estimate the DBP mass as well because it generally relates to the disinfectant mass loss. Quintiliani et al. [32] explain how chlorine compounds react with natural organic matter in water distribution systems and form DBPs such as THMs, HAAs, and others, causing adverse health effects. Dias et al. [33] found a ratio between chlorine mass consumption by reactions and trihalomethanes-4 (THM4) mass production ranging between 19.3 and 31.0 μg THM4/mg Cl₂. Zhao et al. [34] and Xu et al. [35] also showed more extensive relationships between chlorine decay and THMs under different operational conditions. Therefore, the THM mass can be estimated by $M_r$ in the disinfectant mass balance. In addition, Lipiwattanakarn et al. [17] proposed three performance indicators to assess the disinfectant mass losses, safety, and reliability of water distribution systems in terms of water quality. Their concept presented a new way to build an effective disinfectant loss control program based on the mass balance principle.

If the disinfectant mass assessment is focused on the effect of water loss, similar to the concept of Mamade et al. [22], $M_{in}$ can be divided into two main components: the mass associated with authorized consumption ($M_{AC}$) and the mass associated with water loss ($M_{WL}$), as shown in Figure 1b. However, an ideal network model with the ideal condition of no water losses is necessary to compute the mass loss by chemical reactions in the case of no water losses ($M_{ro}$). Then, $M_{WL}$ equals $M_l+M_r-M_{r0}$.

1.2. Top-Down Approach for Water and Energy Audits

According to AWWA [19], the top-down water auditing approach typically marks the beginning of water accountability for most water systems. The top-down auditing approach represents a relatively quick assembly of available records and data; the bottom-up auditing approach represents a relatively slow and deliberate extraction of detailed, supporting data from both the office and the field. There is a standard methodology to conduct the top-down water audit with the default values for some components that do not have enough data. For example, the default value of water loss for unauthorized consumption is 0.25% of the water supplied volume.

Recently, a few studies tried to estimate the default values of energy loss components for the top-down energy auditing approach. To estimate these components without mathematical modeling, Mamade et al. [36] used 20 real network models in Portugal to compute $E_{WL}$ and found a linear trendline, with $E_{WL}/E_{in}$ being slightly higher than $WL/SIV$. Thus, they proposed to use $E_{WL}/E_{in}=WL/SIV$. Later, Lipiwattanakarn et al. [37] introduced two simplified networks that could be used to build theoretical equations for the estimation of $E_l/E_{in}$, $E_f/E_{in}$, and $E_{WL}/E_{in}$. They used 20 real network models in Thailand to adjust the coefficients of their equations. Their results showed that their equation could estimate the energy loss components effectively without any water distribution network models.

1.3. Proposed Estimation of Disinfectant Mass Loss Components for Top-Down Approach

No study has found simple relationships between the disinfectant mass components and measured data in drinking water distribution systems from the perspective of mass balance before. Thus, our objective is to estimate the default values of disinfectant mass loss components for the top-down water quality auditing approach. Hence, the losses can be controlled by appropriate tools and strategies. In this study, the evaluation of three components related to disinfectant mass loss ($M_l/M_{in}$, $M_r/M_{in}$, and $M_{WL}/M_{in})$ is considered using both theoretical analysis and mathematical models in a similar manner to Lipiwattanakarn et al. [37].

2. Theoretical Analysis of Disinfectant Residual Mass Balance

In this section, the disinfectant residual mass balance components, as shown in Figure 1, are analyzed theoretically for three simple water distribution networks. The components are derived in both dimensional and non-dimensional versions. Then, they are used for mass balance estimation in later sections.

2.1. Single Pipe Network with One Demand at Pipe End

We introduce the first scenario as the simplest network of a single pipe network consisting of a source, a pipe with length $L,$ and a demand node at the pipe end (Figure 2). Let us consider the case of water loss (Figure 2a). The input concentration at the source is defined as $C_{in}$, and the system inflow and velocity are $Q$ and $V$, respectively. $Q$ is comprised of

$$Q=Q_u+Q_l$$

(1)

where $Q_u$ is the flow delivered to users and $Q_l$ is the flow due to water loss.

Thus, the ratio of water loss ($p$) can be expressed as

$$p=\frac{Q_l}{Q}$$

(2)

At the demand node at the pipe end, the disinfectant decays by chemical reaction, commonly expressed as a first-order reaction equation. Therefore, the critical concentration ($C_c$) can be described as

$$C_c=C_{in}{e}^{-k{t}_c}=C_{in}{e}^{-\frac{kL}{V}}$$

(3)

where $k$ is a disinfectant decay rate assumed to be constant, and $t_c$ is the time for the water to travel from the source to the pipe end (water age) and is equal to $L/V$.

Now, let us consider the case of no water loss (Figure 2b). The flow velocity decreases due to less flow in the pipe. Thus, the velocity for the no water loss case ($V_o$) is

$$V_o=\frac{Q_u}{Q}V=\left(1-p\right)V$$

(4)

where the variables with the subscript $o$ correspond to the no water loss condition.

The decreasing velocity results in a longer time for the water to travel to the pipe end. Consequently, the critical concentration decreases to be

$$C_{co}=C_{in}{e}^{-\frac{kL}{V_o}}=C_{in}{e}^{-\frac{kL}{\left(1-p\right)V}}$$

(5)

For this first scenario (Figure 2), the dimensional components can be derived as

1. Input disinfectant mass

$$M_{in}=C_{in}Q$$

(6)

2. Disinfectant mass delivered to users

$$M_u=C_c{Q}_u$$

(7)

3. Outgoing disinfectant mass through water loss

$$M_l=C_c{Q}_l$$

(8)

4. Disinfectant mass loss by reactions

$$M_r=M_{in}-M_u-M_l=\left(C_{in}-C_c\right)Q$$

(9)

5. Disinfectant mass loss by reactions for the no water loss case

$$M_{ro}=M_{ino}-M_{uo}-M_{lo}=\left(C_{in}-C_{co}\right)Q_u$$

(10)

6. Disinfectant mass associated with water loss

$$M_{WL}=M_l+M_r-M_{ro}=C_c{Q}_l+\left(C_{in}-C_c\right)Q-\left(C_{in}-C_{co}\right)Q_u$$

(11)

Nondimensionalization is introduced here. The mass components are normalized as

$$\left(M_{in}^{\prime },M_u^{\prime },M_l^{\prime },M_r^{\prime },M_{ino}^{\prime },M_{uo}^{\prime },M_{lo}^{\prime },M_{ro}^{\prime },M_{WL}^{\prime }\right)=\frac{\left(M_{in},M_u,M_l,M_r,M_{ino},M_{uo},M_{lo},M_{ro},M_{WL}\right)}{M_{in}}$$

(12)

and the normalized critical concentration is

$$C_c^{\prime }=\frac{C_c}{C_{in}}$$

(13)

The normalized critical concentration for the no water loss case is

$$C_{co}^{\prime }=\frac{C_{co}}{C_{in}}=C_c^{\prime }{}^{\frac{1}{1-p}}$$

(14)

where the variables with a superscript ′ are the normalized versions of the variables.

It was found that $C_c^{\prime }\ge C_{co}^{\prime }$ in the same manner as its dimensional version ($C_c\ge C_{co}$). As described earlier, a decrease of flow velocity in the no water loss case increases the reaction time; consequently, there is greater disinfectant decay. However, this situation is opposite to the friction loss in the energy study by Lipiwattanakarn et al. [37]. When the flow velocity decreases, the friction loss also decreases, and as a result, the energy head at the user end increases in the no water loss case.

Thus, the normalized disinfectant residual mass balance components for the first scenario of a single pipe network with one demand node at the pipe end can be written as

1. Normalized input disinfectant mass

$$M_{in}^{\prime }=1$$

(15)

2. Normalized disinfectant mass delivered to users

$$M_u^{\prime }=\frac{C_c{Q}_u}{C_{in}Q}=\left(1-p\right)C_c^{\prime }$$

(16)

3. Normalized outgoing disinfectant mass through water loss

$$M_l^{\prime }=\frac{C_c{Q}_l}{C_{in}Q}=p{C}_c^{\prime }$$

(17)

4. Normalized disinfectant mass loss by reactions

$$M_r^{\prime }=1-\left(1-p\right)C_c^{\prime }-p{C}_c^{\prime }=1-C_c^{\prime }$$

(18)

5. Normalized disinfectant mass loss by reactions for the no water loss case

$$M_{ro}^{\prime }=\frac{\left(C_{in}-C_{co}\right)Q_u}{C_{in}Q}=\left(1-p\right)\left[1-C_c^{\prime }{}^{\frac{1}{1-p}}\right]$$

(19)

6. Normalized disinfectant mass associated with water loss

$$M_{WL}^{\prime }=M_l^{\prime }+M_r^{\prime }-M_{ro}^{\prime }=p-\left(1-p\right)C_c^{\prime }\left[1-C_c^{\prime }{}^{\frac{1}{1-p}-1}\right]$$

(20)

Equations (15)–(20) show the fundamental relationship between the normalized disinfectant residual mass balance components and the two parameters $p$ and $C_c^{\prime }$ for the simplest network scenario. Notably, $M_l^{\prime }$ in (17) is equal to or less than $p$, and it also depends on $C_c^{\prime }$. However, $M_r^{\prime }$ in (18) relates to $C_c^{\prime }$ alone. Although $M_{ro}^{\prime }$ in (19) is smaller than $M_r^{\prime }$ due to less flow volume, expressed as ($1-p$), a longer traveling time causes an increase in disinfectant decay ($1-C_c^{\prime }{}^{\frac{1}{1-p}}$). $M_{WL}^{\prime }$ in (20) can be divided into two terms, where the first term is $p$ and the second term is the combination between $p$ and $C_c^{\prime }$. Since the values of $p$ and $C_c^{\prime }$ are between 0 and 1, the second term ranges between 0 and 0.13. Thus, $M_{WL}^{\prime }$ is equal to or less than $p$. This result for $M_{WL}^{\prime }$ is different from the normalized energy associated with water loss ($E_{WL}^{\prime }$) in Lipiwattanakarn et al. [37], in that $E_{WL}^{\prime }$ is always equal to or larger than $p$. The main reason is that the energy loss due to friction without water loss is always smaller than that with water loss. However, the energy loss process is opposite to our process of disinfectant decay. This different manner affects the estimation of the normalized components associated with water loss ($M_{ro}^{\prime }$ and $M_{WL}^{\prime }$).

2.2. Single Pipe Network with Multiple Demands along Pipe

The second network scenario (Figure 3) consists of a source, a pipe, and $n$ demand nodes along the pipe. Although the flow decreases after each demand node along the pipe, the flow velocity ($V$) is assumed to be constant along the pipe due to the fact that in most pipeline design manuals, the sizing of the pipe is based commonly on $V$. For example, typically, the optimum pipe size for a pumping main gives a $V$ value between about 1.5 and 2.0 m/s in the UK [38]. Thus, as the flow decreases, the pipe size decreases to keep $V$ close to the designed velocity. In general, the disinfectant reactions take place within the bulk flow and with material or biofilm along the pipe wall. While the bulk reaction does not depend on pipe size, the wall reaction commonly increases as the pipe size reduces due to an increase in the surface area per unit of water volume within a pipe. As a result, the combined decay rate $k$ for both bulk and wall reactions may increase gradually as water flows to the pipe end. For simplicity, however, $k$ is assumed to be constant along the pipe, same as in the first scenario. We believe that the essential effects of the multiple demands along a pipe on the disinfectant mass components still remain under this assumption.

Let us define the average concentration ($\overline{C}$) and derive it using geometric series as follows:

$$\overline{C}=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{C}_i=\alpha C_c$$

(21)

where the concentration at node $i$ can be expressed as

$$C_i=C_{in}{e}^{-\left(\frac{kL}{V}\right)\frac{i}{n}}=C_{in}{C}_c^{\prime }{}^{\frac{i}{n}}$$

(22)

and thus, the coefficient $\alpha$ is derived as

$$\alpha =\frac{1}{n}\left(\frac{C_c^{\prime }{}^{-1}-1}{C_c^{\prime }{}^{-\frac{1}{n}}-1}\right)$$

(23)

Additionally, let us define the average concentration without water loss (${\overline{C}}_o$) and derive it using a geometric series as:

$${\overline{C}}_o=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{C}_{o,i}={\alpha }_o{C}_{co}={\alpha }_o{C}_{in}{C}_c^{\prime }{}^{\frac{1}{1-p}}$$

(24)

where the concentration at node $i$ for the no water loss case can be expressed as

$$C_{o,i}=C_{in}{e}^{-\left(\frac{kL}{\left(1-p\right)V}\right)\frac{i}{n}}=C_{in}{C}_c^{\prime }{}^{\frac{i}{n\left(1-p\right)}}$$

(25)

and thus, the coefficient ${\alpha }_o$ is derived as

$${\alpha }_o=\frac{1}{n}\left(\frac{C_c^{\prime }{}^{-\frac{1}{1-p}}-1}{C_c^{\prime }{}^{-\frac{1}{n\left(1-p\right)}}-1}\right)$$

(26)

A coefficient of $\alpha \ge 1$ means that $\overline{C}\ge C_c$; $\alpha$ increases if $n$ increases or $C_c^{\prime }$ decreases. For the no water loss case, a coefficient of ${\alpha }_o\ge 1$ means that ${\overline{C}}_o\ge C_{co}$, similar to the case with water loss. Furthermore, ${\alpha }_o$ increases if $n$ increases, $C_c^{\prime }$ decreases, or $p$ increases.

For the second scenario (Figure 3), the dimensional components can be derived as

1. Input disinfectant mass

$$M_{in}=C_{in}Q$$

(27)

2. Disinfectant mass delivered to users

$$M_u={\displaystyle \sum }_{i=1}^n\frac{C_i{Q}_u}{n}=\frac{Q_u}{n}{\displaystyle \sum }_{i=1}^n{C}_i=\overline{C}{Q}_u$$

(28)

3. Outgoing disinfectant mass through water loss

$$M_l={\displaystyle \sum }_{i=1}^n\frac{C_i{Q}_l}{n}=\frac{Q_l}{n}{\displaystyle \sum }_{i=1}^n{C}_i=\overline{C}{Q}_l$$

(29)

4. Disinfectant mass loss by reactions

$$M_r=\left(C_{in}-\overline{C}\right)Q$$

(30)

5. Disinfectant mass loss by reactions for the no water loss case

$$M_{ro}=\left(C_{in}-{\overline{C}}_o\right)Q_u$$

(31)

6. Disinfectant mass associated with water loss

$$M_{WL}=\overline{C}{Q}_l+\left(C_{in}-\overline{C}\right)Q-\left(C_{in}-{\overline{C}}_o\right)Q_u$$

(32)

Thus, the normalized disinfectant residual mass balance components for the second scenario of a single pipe network with multiple demands along the pipe can be written as

1. Normalized input disinfectant mass

$$M_{in}^{\prime }=1$$

(33)

2. Normalized disinfectant mass delivered to users

$$M_u^{\prime }=\frac{\overline{C}{Q}_u}{C_{in}Q}=\left(1-p\right)\alpha C_c^{\prime }$$

(34)

3. Normalized outgoing disinfectant mass through water loss

$$M_l^{\prime }=\frac{\overline{C}{Q}_l}{C_{in}Q}=p\alpha C_c^{\prime }$$

(35)

4. Normalized disinfectant mass loss by reactions

$$M_r^{\prime }=1-\alpha C_c^{\prime }$$

(36)

5. Normalized disinfectant mass loss by reactions for the no water loss case

$$M_{ro}^{\prime }=\frac{\left(C_{in}-{\overline{C}}_o\right)Q_u}{C_{in}{Q}_u}=\left(1-p\right)\left[1-{\alpha }_o{C}_c^{\prime }{}^{\frac{1}{1-p}}\right]$$

(37)

6. Normalized disinfectant mass associated with water loss

$$M_{WL}^{\prime }=M_l^{\prime }+M_r^{\prime }-M_{ro}^{\prime }=p-\left(1-p\right)C_c^{\prime }\left[\frac{1-C_c^{\prime }{}^{\frac{1}{1-p}-1}-f_1}{f_2}\right]$$

(38)

where the coefficients $f_1$ and $f_2$ are

$$f_1=C_c^{\prime }{}^{-\frac{1}{n}-1}\left(1-C_c^{\prime }{}^{\frac{1}{1-p}}\right)-C_c^{\prime }{}^{-\frac{1}{n\left(1-p\right)}-1}\left(1-C_c^{\prime }\right)$$

(39)

and

$$f_2=n\left(C_c^{\prime }{}^{-\frac{1}{n}}-1\right)\left(C_c^{\prime }{}^{-\frac{1}{n\left(1-p\right)}}-1\right)$$

(40)

Comparing $M_u^{\prime }$, $M_l^{\prime }$, $M_r^{\prime }$, $M_{ro}^{\prime },$ and $M_{WL}^{\prime }$ in this second scenario with those in the first scenario, we found that multiple demands along the pipe cause an increase in $M_u^{\prime }$ and $M_l^{\prime }$ but a decrease in $M_r^{\prime }$ and $M_{ro}^{\prime }$ because $\alpha \ge 1$ and ${\alpha }_o\ge 1$. The second term of $M_{WL}^{\prime }$ in (38) is a function of $p$, $C_c^{\prime },$ and $n$, and its value ranges between 0 and 0.13. Its maximum value occurs when $p\cong 0.606$, $C_c^{\prime }\cong 0.546$, and $n=1$. Again, $M_{WL}^{\prime }$ is equal to or less than $p$, similar to the first scenario.

2.3. Branched Pipe Network

The third network scenario (Figure 4) consists of a source, branching pipes, and $m$ demand nodes at the pipe ends. Because of pipe branching at the source, the flow at each pipe is assumed to reduce to $Q/m$. Again, we assume that the velocity at each pipe does not change because the sizing design is based on velocity ($V$), as we did in the second scenario. If we consider that the total pipe lengths ($L$) in all scenarios are the same, the pipe length for each branched pipe will equal $\frac{L}{m},$ as shown in Figure 4a. This assumption lets us investigate the effect of pipe shortening on mass balance components. For the case without water loss (Figure 4b), the flow and velocity at each pipe are reduced to $Q_u/m$ and $\left(1-p\right)V$, respectively.

The pipe shortening results in a shorter water age period, and subsequently, the critical concentrations for the case with water loss ($C_{cm}$) and for the case without water loss ($C_{cmo}$) become larger than those in the first scenario, as follows:

$$C_{cm}=C_{in}{e}^{-\frac{kL}{mV}}=C_{in}{C}_c^{\prime }{}^{\frac{1}{m}}$$

(41)

and

$$C_{cmo}=C_{in}{e}^{-\frac{kL}{m\left(1-p\right)V}}=C_{in}{C}_c^{\prime }{}^{\frac{1}{m\left(1-p\right)}}$$

(42)

where the values of $k$ and $L$ in the third network are assumed to be the same as those in the first network scenario.

For the third scenario (Figure 4), the dimensional components can be derived as

1. Input disinfectant mass

$$M_{in}=C_{in}Q$$

(43)

2. Disinfectant mass delivered to users

$$M_u={\displaystyle \sum }_{j=1}^m\frac{C_{cm}{Q}_u}{m}=C_{cm}{Q}_u$$

(44)

3. Outgoing disinfectant mass through water loss

$$M_l={\displaystyle \sum }_{j=1}^m\frac{C_{cm}{Q}_l}{m}=C_{cm}{Q}_l$$

(45)

4. Disinfectant mass loss by reactions

$$M_r=M_{in}-M_u-M_l=\left(C_{in}-C_{cm}\right)Q$$

(46)

5. Disinfectant mass loss by reactions for the no water loss case

$$M_{ro}=M_{ino}-M_{uo}-M_{lo}=\left(C_{in}-C_{cmo}\right)Q_u$$

(47)

6. Disinfectant mass associated with water loss

$$M_{WL}=C_{cm}{Q}_l+\left(C_{in}-C_{cm}\right)Q-\left(C_{in}-C_{cmo}\right)Q_u$$

(48)

Thus, the normalized disinfectant residual mass balance components for the third scenario of a branched pipe network with demand nodes at the pipe ends can be written as:

1. Normalized input disinfectant mass

$$M_{in}^{\prime }=1$$

(49)

2. Normalized disinfectant mass delivered to users

$$M_u^{\prime }=\frac{\overline{C}{Q}_u}{C_{in}Q}=\left(1-p\right)C_c^{{\prime }^{\frac{1}{m}}}$$

(50)

3. Normalized outgoing disinfectant mass through water loss

$$M_l^{\prime }=\frac{\overline{C}{Q}_l}{C_{in}Q}=p{C}_c^{{\prime }^{\frac{1}{m}}}$$

(51)

4. Normalized disinfectant mass loss by reactions

$$M_r^{\prime }=1-C_c^{{\prime }^{\frac{1}{m}}}$$

(52)

5. Normalized disinfectant mass loss by reactions for the no water loss case

$$M_{ro}^{\prime }=\frac{\left(C_{in}-C_{cmo}\right)Q_u}{C_{in}{Q}_u}=\left(1-p\right)\left[1-C_c^{\prime }{}^{\frac{1}{m\left(1-p\right)}}\right]$$

(53)

6. Normalized disinfectant mass associated with water loss

$$M_{WL}^{\prime }=M_l^{\prime }+M_r^{\prime }-M_{ro}^{\prime }=p-\left(1-p\right)C_c^{\prime }{}^{\frac{1}{m}}\left[1-C_c^{\prime }{}^{\frac{1}{m}\left(\frac{1}{1-p}-1\right)}\right]$$

(54)

Comparing $M_u^{\prime }$, $M_l^{\prime }$, $M_r^{\prime }$, $M_{ro}^{\prime }$ and $M_{WL}^{\prime }$ in this third scenario with those in the first scenario, we found that a branched pipe network with demand nodes at the pipe ends causes an increase in $M_u^{\prime }$ and $M_l^{\prime }$ but a decrease in $M_r^{\prime }$ and $M_{ro}^{\prime }$ because $C_c^{{\prime }^{\frac{1}{m}}}\ge C_c^{\prime }$. This result is similar to the second scenario. The second term of $M_{WL}^{\prime }$ in (54) is a function of $p$, $C_c^{\prime }$, and $m$, and its value is always positive but less than $p$, similar to the first two scenarios. Thus, $M_{WL}^{\prime }$ in this third scenario is also equal to or less than $p$.

2.4. Utilization of Theory to Real Networks

The theoretical estimations are discussed here for the three normalized disinfectant residual mass components, $M_l^{\prime }$, $M_r^{\prime }$ and $M_{WL}^{\prime }$, for real networks. Previously, our theoretical analysis derived the mass balance as a steady problem (time independence) using three scenarios. However, water flows, water demands, pressure and disinfectant residual concentration distributions, and other hydraulic and water quality parameters in real networks are not steady and change with time. Generally, they can be considered to have 24 h periodic patterns following water demand patterns. Thus, to apply our theory, at least the 24-hour-averaged values of $M_l^{\prime }$, $M_r^{\prime }$, and $M_{WL}^{\prime }$ should be considered for a real application. We introduce the normalized time-averaged critical concentration ($C_c^{*}$) as a parameter to estimate the components, as follows:

$$C_c^{*}=\frac{{\overline{C}}_c}{{\overline{C}}_{in}}$$

(55)

where the bar ¯ expresses the average over time.

However, using $C_c^{*}$ turns out to give unsatisfactory results because in some pipe networks, $C_c^{*}$ is very low due to stagnant water in some specific local pipes in the networks, especially dead ends with very low flow conditions. Thus, $C_c^{*}$ does not adequately represent the distribution of disinfectant residual concentration in the whole network. The best location representing the concentration, instead of the real critical concentration point, should be the place where real-time water quality sensors in water distribution networks are installed [39,40]. However, many real networks in developing countries, including our examples, do not have such sensors.

Most utilities commonly monitor critical pressure points (CPPs) for pressure management, and CPPs are the locations with the lowest pressure caused by topography and/or hydraulic losses [19,41]. Commonly, CPPs are located far from an inlet without no stagnant water. Therefore, to utilize our theory in real networks, we choose CPPs to represent the concentration. The normalized time-averaged concentration at a CPP ($C_p^{*}$) can be written as

$$C_p^{*}=\frac{{\overline{C}}_p}{{\overline{C}}_{in}}$$

(56)

where the subscript $p$ denotes a CPP.

Considering the derived equation forms of $M_l^{\prime }$, $M_r^{\prime }$, and $M_{WL}^{\prime }$ of the three theoretical scenarios in the previous section, we propose the following estimations that have forms resembling the combination of the three scenarios for the utilization of the theory in real networks.

$$M_{l,th}^{\prime }=p{A}_1{C}_p^{*B_1}$$

(57)

$$M_{r,th}^{\prime }=1-A_2{C}_p^{*B_2}$$

(58)

$$M_{WL,th}^{\prime }=p-A_3\left(1-p\right)C_p^{*}\left[1-C_p^{*}{}^{\left(\frac{1}{1-p}-1\right)}\right]$$

(59)

where the subscript $th$ denotes the theoretical estimation; the coefficients $A_1$, $A_2$, and $A_3$ and the exponents $B_1$ and $B_2$ are evaluated by real networks in the following section.

While $A_1$ and $A_2$ resemble $\alpha$ in (35) and (36), respectively, $B_1$ and $B_2$ are similar to $1/m$ in (51) and (52), respectively. $M_{WL,th}^{\prime }$ in (59) does not resemble the combination of $M_{WL}^{\prime }$ in (38) and (54). The main reason is that the equation will be too complicated, and we found that it does not provide a significantly better result than the simple form in (59).

3. Application to Real Water Networks

3.1. Characteristics of Water Distribution Networks

To apply our theoretical disinfectant mass loss estimations (57)–(59) to real water networks, we use 20 water distribution network models. These 20 models are district metering areas (DMAs) in the service area of the Samut Prakan branch office of the Metropolitan Waterworks Authority (MWA), Thailand. These models have been calibrated with both hydraulic and water quality (residual chlorine) field data by the MWA, and they have also been used to estimate the parameters in the previous theoretical energy loss estimations by Lipiwattanakarn et al. [37].

Table 1. Characteristics of DMAs and residual chlorine results from network model simulations.

DMA ID	No. of Inlets	$\mathit{p}$	${\overline{\mathit{C}}}_{\mathit{i}\mathit{n}}$	${\mathit{C}}_{\mathit{p}}^{*}$	${\mathit{C}}_{\mathit{c}}^{*}$	${\mathit{M}}_{\mathit{i}\mathit{n},\mathit{m}\mathit{o}}$	${\mathit{M}}_{\mathit{l},\mathit{m}\mathit{o}}^{\prime }$	${\mathit{M}}_{\mathit{r},\mathit{m}\mathit{o}}^{\prime }$	${\mathit{M}}_{\mathit{W}\mathit{L},\mathit{m}\mathit{o}}^{\prime }$
		(%)	(mg/L)	(%)	(%)	(g/day)	(%)	(%)	(%)
1	1	37.1	1.03	83.3	38.7	5074	32.0	12.3	33.6
2	1	28.6	0.73	79.8	25.8	3939	24.9	11.5	26.2
3	1	44.6	0.52	41.8	31.1	3735	37.1	15.5	39.8
4	1	38.5	0.78	63.7	18.9	5091	30.7	18.7	34.4
5	1	44.2	0.63	74.8	29.8	5555	32.9	23.0	39.8
6	1	54.9	0.67	53.6	22.1	8347	40.8	22.8	48.1
7	1	32.4	0.77	18.4	14.4	4727	14.8	44.1	24.0
8	1	12.9	0.73	58.9	7.8	4841	9.1	25.9	10.8
9	1	29.7	0.85	91.9	5.3	5227	28.4	4.3	28.6
10	1	2.8	0.61	71.2	50.1	3251	2.2	18.1	2.4
11	2	30.0	0.81	73.8	35.3	6257	23.8	18.4	24.0
12	2	50.9	0.65	74.1	46.9	3212	39.7	20.0	45.2
13	2	31.9	0.75	67.1	43.9	4867	24.5	20.6	28.1
14	2	33.9	0.41	84.4	65.9	3808	33.0	7.1	33.8
15	2	7.7	0.72	57.7	26.9	6480	5.1	28.1	6.6
16	2	36.3	0.60	69.0	6.4	5805	31.7	12.8	33.0
17	2	30.7	0.57	67.6	10.9	7404	27.2	11.6	28.0
18	2	30.0	0.72	43.0	0.3	6821	19.3	33.2	24.9
19	2	31.2	0.68	50.5	20.0	4774	28.7	13.3	31.0
20	2	47.2	0.65	68.4	15.9	7691	32.8	28.0	41.4
Avg.	1.5	32.8	0.69	64.7	25.8	5345	25.9	19.5	29.2

shows the data of 20 DMAs for the number of inlets, the ratio of water loss ($p$), the average residual chlorine concentration at the inlets (${\overline{C}}_{in}$), normalized chlorine concentrations at the CPP and the critical chlorine point ($C_p^{*}$ and $C_c^{*}$), the input chlorine mass ($M_{in,mo}$), and normalized chlorine mass components ($M_{l,mo}^{\prime }$, $M_{r,mo}^{\prime }$, and $M_{WL,mo}^{\prime }$), where the variables with the subscription $mo$ are variables estimated by the network simulation models. The networks can be divided into two groups: the DMAs with one inlet and two inlets. Each group has 10 DMAs. It was found that $p$ ranges between 2.8% and 54.9%, and ${\overline{C}}_{in}$ covers the values between 0.41 and 1.03 mg/l. Due to the reactions in the networks, $C_p^{*}$ ranges between 18.4% and 91.9% of ${\overline{C}}_{in}$, while $C_c^{*}$ values are between 0.3% and 65.9% of ${\overline{C}}_{in}$. In DMA IDs 8, 9, 16, and 18, the values of $C_c^{*}$ are very low compared with $C_p^{*}$ due to stagnant water in some specific areas within the networks, as described earlier. Our DMAs were built using EPANET version 2.00.12.01 software [42]. We followed the steps proposed by Lipiwattanakarn et al. [17] to compute $M_{in,mo}$, $M_{l,mo}^{\prime }$, $M_{r,mo}^{\prime },$ and $M_{WL,mo}^{\prime }$, and we used the hydraulic and water quality solvers of the WNTR package in Python [43], compatible with EPANET software, to simulate the network models. The results in Table 1 will be used to investigate our theoretical analysis in the following sections.

Figure 5 shows residual chlorine distributions in the four DMA networks (ID1, ID8, ID11, and ID13) used in this study, where the water distribution patterns of our networks vary from branching to semi-gridiron. Our DMA networks are modeled to the most detailed level so that each customer is connected to a water distribution pipe. For various reasons, a considerable number of registered customers do not use water and that causes completely stagnant water with zero residual chlorine concentration at their nodes. Therefore, the critical chlorine point was chosen under the criterion that the node must have water demand and a minimum concentration. DMA IDs 1 and 8 show the networks with one inlet having high and low chlorine concentrations, respectively. DMA IDs 11 and 18 show the networks with two inlets having high and low chlorine concentrations, respectively. It was found that CPPs seem to be able to represent the averaged zone chlorine concentrations in the networks, while the critical chlorine points in DMA IDs 8 and 18 show the locations where water is almost stagnant at the ends of the networks. As a result, using CPPs is better than using the critical chlorine point for the estimation of disinfectant mass components, as described earlier.

3.2. Basic Relationship for Disinfectant Mass Loss Components

In Figure 6, three types of normalized disinfectant mass losses from the models ($M_{l,mo}^{\prime }$, $M_{r,mo}^{\prime }$, and $M_{WL,mo}^{\prime }$) are compared based on two basic parameters: the ratio of water loss ($p$) and the renormalized concentration loss at CPP ($1-C_p^{*}$). Figure 6a shows a good relationship between $M_{l,mo}^{\prime }$ and $p$. It was found that $M_{l,mo}^{\prime }$ is slightly to moderately smaller than $p$. This result corresponds to the theoretical estimation of $M_{l,th}^{\prime }$ in (57), showing that $M_l^{\prime }$ also depends on the normalized concentration $C_p^{*}$, which is smaller than unity. Figure 6b also shows the good relationship between $M_{r,mo}^{\prime }$ and ($1-C_p^{*}$). This tendency agrees with the theoretical estimation of $M_{r,th}^{\prime }$ in (58). The strong relationship between $M_{WL,mo}^{\prime }$ and $p$ is presented in Figure 6c. It shows that $M_{WL,mo}^{\prime }$ is very slightly smaller than $p$. According to (59), $M_{WL,th}^{\prime }$ can be divided into two terms, where the first term is $p$ and the second term is a function of $p$ and $C_p^{*}$. It can be expected that the second term is much less important than the first term and may have a simple function. Thus, we introduce only one parameter ($A_3$) in $M_{WL,th}^{\prime }$ for the estimation by using the results from our network models, while the other two mass loss components ($M_{l,th}^{\prime }$ and $M_{r,th}^{\prime }$) have two parameters for the estimation.

3.3. Parameter Estimation for Disinfectant Mass Loss Components

In this section, we estimate five parameters ($A_1$, $A_2$, $A_3$, $B_1$, and $B_2$) in $M_{l,th}^{\prime }$, $M_{r,th}^{\prime }$, and $M_{WL,th}^{\prime }$ in (57)–(59). In the study of Lipiwattanakarn et al. [37], for theoretical energy loss component estimations, they found that the number of inlets affected the values of their parameters. The networks with two inlets had less effect on energy head loss than the ones with one inlet. Thus, they divided the model data according to the number of inlets and estimated the parameters for their equations separately. However, in this study, we did not observe differences in chlorine mass losses between the networks with one inlet and the ones with two inlets, as shown in Figure 6. Therefore, the value of each parameter is evaluated by the model results of all networks with one and two inlets using the method of least squares.

Figure 7 shows the comparisons between the three disinfectant mass loss components using the simulation models and theoretical estimations, while

Table 2. Performance of proposed theoretical methods to evaluate disinfectant mass loss components with network model results.

Component	Equation	$\mathit{A}$	$\mathit{B}$	$\mathit{r}$	$\mathit{R}\mathit{M}\mathit{S}\mathit{E}(\%)$
$M_{l,th}^{\prime }$	(57)	0.9106	0.2844	0.963	2.86
$M_{r,th}^{\prime }$	(58)	0.9151	0.2726	0.755	5.99
$M_{WL,th}^{\prime }$	(59)	0.4962	-	0.989	1.76

shows the values of the parameters used for Figure 7 and two statistical parameters: the correlation coefficient ($r$) and the root mean square error ($RMSE$). The outstanding performance of our theory in estimating $M_l^{\prime }$ and $M_{WL}^{\prime }$ (Figure 7a,c) corresponds to the performance of the theoretical estimation of the outgoing energy through water loss ($E_l^{\prime }$) and the energy associated with water loss ($E_{WL}^{\prime }$), respectively [37]. The performance of estimating $M_r^{\prime }$ (Figure 7b) is satisfactory but not as good as the others. It seems that the processes of chlorine decay in real networks are more complicated than our analysis of simplified scenarios. Again, this result corresponds to the estimation of the friction energy loss ($E_f^{\prime }$). Therefore, there is a strong similarity between our disinfectant mass balance theory and the energy balance theory by Lipiwattanakarn et al. [37].

As described earlier, $A_1$ and $A_2$ resemble the same parameter $\alpha$, and $B_1$ and $B_2$ are like $1/m$. In Table 2, the calibrated values of $A_1$ and $A_2$ are almost the same, and the value of $B_1$ is similar to $B_2$. However, the value of $\alpha$ in theory should be equal to or greater than unity. This may be due to many assumptions used in the theory and the use of $C_p^{*}$ instead of $C_c^{\prime }$ in our estimations.

3.4. Application for Top-Down Water Quality Audit

In the previous section, we show that our novel theoretical method can evaluate disinfectant mass loss components ($M_l$, $M_r,$ and $M_{WL}$) effectively. Here, we will explain how to apply our theoretical analysis to real water networks for a top-down water quality audit, as follows.

• Choose the type of disinfectant mass balance to be complied with, as shown in Table 1.
• Collect the hydraulic data: system inflow ($Q$) and flow delivered to users ($Q_u$)
• Compute water loss ($Q_l$) and the ratio of water losses ($p$) by using (1) and (2), respectively.
• Collect the water quality data: input concentration ($C_{in}$) and concentration at the critical pressure point ($C_p$).
• Compute the normalized time-averaged concentration at the critical pressure point ($C_p^{*}$) by using (56).
• Estimate the normalized values of $M_l$, $M_r,$ and $M_{WL}$ in (57), (58), and (59), respectively, by using the values of the coefficients $A_1$, $A_2$, and $A_3$ and the exponents $B_1$ and $B_2$ in Table 2 for DMAs.
• Estimate the dimensional value of $M_{in}$ by using (6).
• Finally, estimate the other dimensional components ($M_u$, $M_l$, $M_r$, $M_{ro},$ and $M_{WL}$) in Table 1 by using (12).

4. Conclusions

This study proposes a novel theoretical analysis of disinfectant mass balance components to estimate the disinfectant mass losses for top-down water quality audits. Three types of normalized disinfectant mass losses ($M_l^{\prime }$, $M_r^{\prime },$ and $M_{WL}^{\prime }$) could be estimated without mathematical network modeling by using two non-dimensional parameters: the ratio of water losses ($p$) and the normalized concentration at the critical pressure point ($C_p^{*}$). While $M_l^{\prime }$ was slightly to moderately smaller than $p$, $M_{WL}^{\prime }$ was higher than $M_l^{\prime }$ and slightly smaller than $p$. There was a good relationship between $M_r^{\prime }$ and the loss of disinfectant concentration ($1-C_p^{*}$). The coefficients of our theoretical estimations of $M_l^{\prime }$, $M_r^{\prime },$ and $M_{WL}^{\prime }$ were adjusted by using the results of 20 real network models. There were no significant differences for $M_l^{\prime }$, $M_r^{\prime }$, and $M_{WL}^{\prime }$ between the networks with one and two inlets. There were outstanding estimations of $M_l^{\prime }$ and $M_{WL}^{\prime }$, with $RMSE$values of less than 3%, while the estimation of $M_r^{\prime }$ was good, with an $RMSE$ of 6%. Additionally, many studies have shown that the THM mass can be estimated by $M_r$. In conclusion, our theoretical method could estimate the disinfectant mass balance components for disinfectant loss assessments and effective loss control programs, similar to water and energy audits.