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

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Water, Energy, and Water Quality Audits

_{2}. 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.

#### 1.2. Top-Down Approach for Water and Energy Audits

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

## 2. Theoretical Analysis of Disinfectant Residual Mass Balance

#### 2.1. Single Pipe Network with One Demand at Pipe End

#### 2.2. Single Pipe Network with Multiple Demands along Pipe

#### 2.3. Branched Pipe Network

#### 2.4. Utilization of Theory to Real Networks

## 3. Application to Real Water Networks

#### 3.1. Characteristics of Water Distribution Networks

#### 3.2. Basic Relationship for Disinfectant Mass Loss Components

#### 3.3. Parameter Estimation for Disinfectant Mass Loss Components

#### 3.4. Application for Top-Down Water Quality Audit

- 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

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${A}_{1}$ | Coefficient in (57) |

${A}_{2}$ | Coefficient in (58) |

${A}_{3}$ | Coefficient in (59) |

AC | Authorized consumption |

${B}_{1}$ | Exponent in (57) |

${B}_{2}$ | Exponent in (58) |

${C}_{c}^{*}$ | Normalized time-averaged critical disinfectant concentration |

${C}_{p}^{*}$ | Normalized time-averaged disinfectant concentration at critical pressure point |

$\overline{C}$ | Average disinfectant concentration |

${\overline{C}}_{in}$ | Average disinfectant concentration at inlets |

${\overline{C}}_{o}$ | Average disinfectant concentration without water loss |

${C}_{c}^{\prime}$ | Normalized critical disinfectant concentration |

${C}_{co}^{\prime}$ | Normalized critical disinfectant concentration for no water loss case |

${C}_{c}$ | Critical disinfectant concentration |

${C}_{cm}$ | Critical disinfectant concentration at node m |

${C}_{cmo}$ | Critical disinfectant concentration at node m for no water loss case |

${C}_{co}$ | Critical disinfectant concentration for no water loss case |

${C}_{i}$ | Disinfectant concentration at node $i$ |

${C}_{in}$ | Input disinfectant concentration at source |

${C}_{o,i}$ | Disinfectant concentration at node $i$ for no water loss case |

CPP | Critical pressure point |

DBP | Disinfection by-product |

DMA | District metering area |

DO | Dissolved oxygen |

HAA | Halo acetic acid |

$k$ | Disinfectant decay rate |

$m$ | Number of branching |

${M}_{in}^{\prime}$ | Normalized input disinfectant mass |

${M}_{l}^{\prime}$ | Normalized outgoing disinfectant mass through water loss |

${M}_{l,mo}^{\prime}$ | Normalized outgoing disinfectant mass through water loss from models |

${M}_{l,th}^{\prime}$ | Normalized outgoing disinfectant mass through water loss from theory |

${M}_{r}^{\prime}$ | Normalized disinfectant mass loss by reactions |

${M}_{r,mo}^{\prime}$ | Normalized disinfectant mass loss by reactions from models |

${M}_{r,th}^{\prime}$ | Normalized disinfectant mass loss by reactions from theory |

${M}_{ro}^{\prime}$ | Normalized disinfectant mass loss by reactions for no water loss case |

${M}_{u}^{\prime}$ | Normalized disinfectant mass delivered to users |

${M}_{WL}^{\prime}$ | Normalized disinfectant mass associated with water loss |

${M}_{WL,mo}^{\prime}$ | Normalized disinfectant mass associated with water loss from models |

${M}_{WL,th}^{\prime}$ | Normalized disinfectant mass associated with water loss from theory |

${M}_{AC}$ | Disinfectant mass associated with authorized consumption |

${M}_{in}$ | Input disinfectant mass |

${M}_{in,mo}$ | Input disinfectant mass from models |

${M}_{l}$ | Outgoing disinfectant mass through water losses |

${M}_{r}$ | Disinfectant mass losses by reactions |

${M}_{ro}$ | Disinfectant mass loss by chemical reactions in no water loss case |

${M}_{u}$ | Disinfectant mass delivered to users |

${M}_{WL}$ | Disinfectant mass associated with water loss |

$n$ | Number of demand nodes alone a pipe |

$p$ | Ratio of water loss |

$Q$ | System inflow |

${Q}_{l}$ | Flow due to water loss |

${Q}_{u}$ | Flow delivered to users |

THM | Trihalomethane |

$V$ | Flow velocity |

${V}_{o}$ | Flow velocity for the no water loss case |

$WL$ | Water losses |

$\alpha $ | Parameter in (23) |

${\alpha}_{o}$ | Parameter in (26) |

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**Figure 1.**Two disinfectant mass balance concepts in water networks, considering water loss and reactions, focusing on: (

**a**) processes and (

**b**) the effect of water loss, where ${M}_{ro}$ is the disinfectant mass loss by reactions in the case of no water losses. Each component has a unit of mass per period of time, for example, kg/day.

**Figure 2.**Disinfection concentration distribution for a single pipe network with one demand node at the pipe end, where (

**a**) is with water loss and (

**b**) is without water loss.

**Figure 3.**Disinfection concentration distribution for a single pipe network with multiple demands along the pipe, where (

**a**) is with water loss and (

**b**) is without water loss.

**Figure 4.**Disinfection concentration distribution for a branched pipe network with demand nodes at the pipe ends, where (

**a**) is with water loss and (

**b**) is without water loss.

**Figure 5.**Examples of residual chlorine distributions in DMA networks, where (

**a**,

**b**) are networks with 1 inlet, and (

**c**,

**d**) are networks with 2 inlets. Red dashed circles show the inlets connected to networks by imaginary pipes with no friction loss; orange triangles represent critical pressure points (CPPs) and pink rectangles are critical chlorine points.

**Figure 6.**Relationships between disinfectant mass balance components, calculated using EPANET models and basic network parameters. (

**a**) ${M}_{l,mo}^{\prime}$ vs. $p$, (

**b**) ${M}_{r,mo}^{\prime}$ vs. $1-{C}_{p}^{*},$ and (

**c**) ${M}_{WL,mo}^{\prime}$ vs. $p$.

**Figure 7.**Comparison between the disinfectant mass balance components calculated using EPANET models and the ones estimated by theory. (

**a**) ${M}_{l,mo}^{\prime}$ vs. ${M}_{l,th}^{\prime}$, (

**b**) ${M}_{r,mo}^{\prime}$ vs. ${M}_{r,th}^{\prime},$ and (

**c**) ${M}_{WL,mo}^{\prime}$ vs. ${M}_{WL,th}^{\prime}$.

DMA ID | No. of Inlets | $\mathit{p}$ | ${\overline{\mathit{C}}}_{\mathit{i}\mathit{n}}$ | ${\mathit{C}}_{\mathit{p}}^{*}$ | ${\mathit{C}}_{\mathit{c}}^{*}$ | ${\mathit{M}}_{\mathit{i}\mathit{n},\mathit{m}\mathit{o}}$ | ${\mathit{M}}_{\mathit{l},\mathit{m}\mathit{o}}^{\prime}\text{}$ | ${\mathit{M}}_{\mathit{r},\mathit{m}\mathit{o}}^{\prime}$ | ${\mathit{M}}_{\mathit{W}\mathit{L},\mathit{m}\mathit{o}}^{\prime}\text{}$ |
---|---|---|---|---|---|---|---|---|---|

(%) | (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 |

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

Component | Equation | $\mathit{A}$ | $\mathit{B}$ | $\mathit{r}$ | $\mathit{R}\mathit{M}\mathit{S}\mathit{E}\text{}(\%)$ |
---|---|---|---|---|---|

${M}_{l,th}^{\prime}$ | (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 |

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## Share and Cite

**MDPI and ACS Style**

Wongpeerak, K.; Charuwimolkul, N.; Changklom, J.; Lipiwattanakarn, S.; Pornprommin, A. Theoretical Estimation of Disinfectant Mass Balance Components in Drinking Water Distribution Systems. *Water* **2023**, *15*, 368.
https://doi.org/10.3390/w15020368

**AMA Style**

Wongpeerak K, Charuwimolkul N, Changklom J, Lipiwattanakarn S, Pornprommin A. Theoretical Estimation of Disinfectant Mass Balance Components in Drinking Water Distribution Systems. *Water*. 2023; 15(2):368.
https://doi.org/10.3390/w15020368

**Chicago/Turabian Style**

Wongpeerak, Kittikun, Natchapol Charuwimolkul, Jiramate Changklom, Surachai Lipiwattanakarn, and Adichai Pornprommin. 2023. "Theoretical Estimation of Disinfectant Mass Balance Components in Drinking Water Distribution Systems" *Water* 15, no. 2: 368.
https://doi.org/10.3390/w15020368