# Theoretical Estimation of Energy Balance Components in Water Networks for Top-Down Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis of Energy Balance

#### 2.1. Single Pipe Network

- Input energy (${E}_{in}$)$${E}_{in}=\gamma QH$$
- Outgoing energy through water loss (${E}_{l}$)$${E}_{l}=\gamma {Q}_{l}\left(H-K{Q}^{n}\right)$$
- Friction energy loss (${E}_{f}$)$${E}_{f}=\gamma Q\left(K{Q}^{n}\right)$$
- Friction energy loss for a water loss-free network (${E}_{fo}$)$${E}_{fo}=\gamma {Q}_{u}\left(K{Q}_{u}^{n}\right)$$
- Energy associated with water loss (${E}_{WL}$)$${E}_{WL}={E}_{l}+{E}_{f}-{E}_{fo}$$

- Normalized input energy (${E}_{in}^{\prime}$)$${E}_{in}^{\prime}=1$$
- Normalized outgoing energy through water loss (${E}_{l}^{\prime}$)$${E}_{l}^{\prime}=\left(\frac{{Q}_{l}}{Q}\right)\left(\frac{H-K{Q}^{n}}{H}\right)=p-p\mathsf{\Delta}{H}^{\prime}$$
- Normalized friction energy loss (${E}_{f}^{\prime}$)$${E}_{f}^{\prime}=\mathsf{\Delta}{H}^{\prime}$$
- Normalized friction energy loss for a water loss-free network (${E}_{fo}^{\prime}$)$${E}_{fo}^{\prime}=\left(\frac{{Q}_{u}}{Q}\right)\left(\frac{K{Q}_{u}^{n}}{H}\right)={\left(1-p\right)}^{n+1}\mathsf{\Delta}{H}^{\prime}$$
- Normalized energy associated with water loss (${E}_{WL}^{\prime}$)$${E}_{WL}^{\prime}=p+{p}_{n}\mathsf{\Delta}{H}^{\prime}$$

_{n}as a function of p from Equation (15) for n = 1, 1.852 and 2. If n = 1, it implies a laminar flow, and the curve is a negative parabola with the maximum p

_{n}of 0.25 at p = 0.5 and the minimum p

_{n}of 0 at p = 0 and 1. For the case of complete turbulence, rough pipes (n = 2), the curve is skewed to the right with the maximum p

_{n}of around 0.385 at p ≈ 0.423. For the Hazen–Williams formula (n = 1.852), the curve differs from the n = 2 curve slightly. Thus, ${E}_{WL}^{\prime}$ approaches p when p is very low or extremely high, and the maximum deviation of ${E}_{WL}^{\prime}$ from p occurs when the percentage of water loss is around 40%–50%. In this study, n is assumed to be 2 for simplicity.

#### 2.2. Branched Pipe Network with Uniformly Distributed Demand Nodes

^{n}at the end of each branch. These assumptions are used to simplify the problem and to make it possible to be solved theoretically. They will be discussed in the later section using the results from the real network models.

- Input energy (${E}_{in}$)$${E}_{in}=\gamma QH$$
- Outgoing energy through water loss (${E}_{l}$)$${E}_{l}=m{\displaystyle \sum}_{i=1}^{j}\gamma \frac{{Q}_{li}}{m}\left[H-\frac{i}{j}K{\left(\frac{Q}{m}\right)}^{n}\right]=\gamma {Q}_{l}\left[H-\left(\frac{1}{2{m}^{n}}\right)\left(1+\frac{1}{j}\right)K{Q}^{n}\right]$$
- Friction energy loss (${E}_{f}$)$${E}_{f}=m{\displaystyle \sum}_{i=1}^{j}\gamma \frac{{Q}_{i}}{m}\frac{K{\left(\frac{Q}{m}\right)}^{n}}{j}=\left(1+\frac{1}{j}\right)\frac{\gamma Q\left(K{Q}^{n}\right)}{2{m}^{n}}$$
- Friction energy loss for a water loss-free network (${E}_{fo}$)$${E}_{fo}=\left(1+\frac{1}{j}\right)\frac{\gamma {Q}_{u}\left(K{Q}_{u}^{n}\right)}{2{m}^{n}}$$
- Energy associated with water loss (${E}_{WL}$)$${E}_{WL}={E}_{l}+{E}_{f}-{E}_{fo}$$

- Normalized input energy (${E}_{in}^{\prime}$)$${E}_{in}^{\prime}=1$$
- Normalized outgoing energy through water loss (${E}_{l}^{\prime}$)$${E}_{l}^{\prime}=p-{C}_{mj}p\mathsf{\Delta}{H}^{\prime}$$
- Normalized friction energy loss (${E}_{f}^{\prime}$)$${E}_{f}^{\prime}={C}_{mj}\mathsf{\Delta}{H}^{\prime}$$
- Normalized friction energy loss for a water loss-free network (${E}_{fo}^{\prime}$)$${E}_{fo}^{\prime}={C}_{mj}{\left(1-p\right)}^{n+1}\mathsf{\Delta}{H}^{\prime}$$
- Normalized energy associated with water loss (${E}_{WL}^{\prime}$)$${E}_{WL}^{\prime}=p+{C}_{mj}{p}_{n}\mathsf{\Delta}{H}^{\prime}$$$${C}_{mj}=\left(\frac{1}{2{m}^{n}}\right)\left(1+\frac{1}{j}\right)$$

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

## 3. Application to Real Water Networks

#### 3.1. Characteristics of Water Networks

_{f}) from our network models ranges from 0.07 m/km to 0.70 m/km with an average of 0.23 m/km, S

_{f}from the study of Mamade et al. [21] ranges from 0.1 m/km to 1.2 m/km with an average of 0.19 m/km. Based on the average value, the networks in our study have larger values of S

_{f}. Mamade et al. [21] explained that their networks are overdesigned; thus, the impact on head loss by adding water loss is small. This implies that the estimation of the normalized energy associated with water loss (${E}_{WL}^{\prime}$) is not sensitive to head loss in their study. However, using our theory, ${E}_{WL,theo}^{\prime}$ in Equation (29) is a function of the normalized water loss (p) and head loss ($\mathsf{\Delta}{H}^{*}$), not S

_{f}. Our case study covers values of p between 2.8% and 54.9%, and values of $\mathsf{\Delta}{H}^{*}$ between 7.6% and 65.3%. Thus, the impact of head loss on the estimation of energy balance components can be investigated in our study.

#### 3.2. Basic Relationship for Energy Balance Components

## 4. Estimation of Energy Balance Components

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DMA | district metering area |

IWA | International Water Association |

MWA | Metropolitan Waterworks Authority, Thailand |

${C}_{mj}$ | parameter in Equation (26) |

D | pipe diameter |

${E}_{AC}$ | energy associated with authorized consumption |

${E}_{f}$ | friction energy loss |

${E}_{f}^{\prime}$ | normalized friction energy loss |

${E}_{f,mod}^{\prime}$ | normalized friction energy loss evaluated by mathematical model |

${E}_{f,theo}^{\prime}$ | normalized friction energy loss estimated by theory |

${E}_{fo}$ | friction energy loss for a water loss-free network |

${E}_{fo}^{\prime}$ | normalized friction energy loss for a water loss-free network |

${E}_{in}$ | input energy |

${E}_{in}^{\prime}$ | normalized input energy |

${E}_{l}$ | outgoing energy through water loss |

${E}_{l}^{\prime}$ | normalized outgoing energy through water loss |

${E}_{l,mod}^{\prime}$ | normalized outgoing energy through water loss by mathematical model |

${E}_{l,theo}^{\prime}$ | normalized outgoing energy through water loss by theory |

${E}_{WL}$ | energy associated with water loss |

${E}_{WL}^{\prime}$ | normalized energy associated with water loss |

${E}_{WL,mod}^{\prime}$ | normalized energy associated with water loss by mathematical model |

${E}_{WL,theo}^{\prime}$ | normalized energy associated with water loss by theory |

H | input energy head |

j | number of demand nodes in each branch |

K | loss coefficient |

m | number of branches |

n | flow exponent in head loss formula |

$p$ | ratio of water loss |

${p}_{n}$ | coefficient as a function of $p$ |

$Q$ | inflow |

${Q}_{i}$ | flow in subarea i |

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

${Q}_{u}$ | flow to supply authorized consumption |

S_{f} | friction slope |

SIV | system input volume |

WL | water loss |

$\gamma $ | specific gravity |

$\mathsf{\Delta}H$ | head loss between the source and the minimum energy point |

$\mathsf{\Delta}{H}^{\prime}$ | normalized head loss between the source and the minimum energy point |

$\mathsf{\Delta}{H}^{*}$ | renormalized head loss in Equation (30) |

$\mathsf{\Delta}{H}_{i}$ | head loss in subarea i |

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**Figure 4.**Energy balance for simplified branched pipe network with uniformly distributed demand nodes, where m is number of branches and j is number of demand nodes in each branch.

**Figure 5.**Examples of water networks, where (

**a**,

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

**c**,

**d**) are networks with 2 inlets. Red dashed circles show the inlets connecting to the networks by imaginary valves with no friction loss.

**Figure 6.**Relationships between energy balance components calculated by EPANET models and basic network parameters, where (

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

**b**) ${E}_{f,mod}^{\prime}$ vs. $\mathsf{\Delta}{H}^{\prime}$ and (

**c**) ${E}_{WL,mod}^{\prime}$ vs. p.

**Figure 7.**Comparison between energy balance components calculated by EPANET models and ones estimated by theory, where (

**a**) ${E}_{l,mod}^{\prime}$ vs. ${E}_{l,theo}^{\prime}$, (

**b**) ${E}_{f,mod}^{\prime}$ vs. ${E}_{f,theo}^{\prime}$ and (

**c**) ${E}_{WL,mod}^{\prime}$ vs. ${E}_{WL,theo}^{\prime}$.

**Figure 8.**Boxplots of normalized demand distribution (${Q}_{i}/{Q}_{in}$) vs. normalized head loss distribution ($\mathsf{\Delta}{H}_{i}/\mathsf{\Delta}H$), where (

**a**) is for networks with 1 inlet and (

**b**) is for networks with 2 inlets. Circles show the outliers.

ID | No. of Inlets | No. of Customers | Length | Avg. D | Avg. S_{f} | Water Loss, p | ΔH* |
---|---|---|---|---|---|---|---|

(km) | (mm) | (m/km) | (%) | (%) | |||

1 | 1 | 2669 | 24.5 | 161 | 0.17 | 37.1 | 16.6 |

2 | 1 | 2657 | 26.4 | 147 | 0.17 | 28.6 | 35.7 |

3 | 1 | 4399 | 52.3 | 148 | 0.08 | 44.6 | 14.1 |

4 | 1 | 2626 | 46.4 | 174 | 0.20 | 38.5 | 44.2 |

5 | 1 | 3594 | 54.7 | 139 | 0.11 | 44.2 | 18.4 |

6 | 1 | 4812 | 51.0 | 143 | 0.36 | 54.9 | 40.7 |

7 | 1 | 4607 | 43.2 | 130 | 0.17 | 32.4 | 44.8 |

8 | 1 | 1695 | 28.8 | 208 | 0.09 | 12.9 | 13.7 |

9 | 1 | 3634 | 18.1 | 183 | 0.16 | 29.7 | 14.4 |

10 | 1 | 1820 | 22.5 | 132 | 0.14 | 2.8 | 14.8 |

11 | 2 | 1921 | 22.2 | 166 | 0.50 | 30.0 | 35.0 |

12 | 2 | 2151 | 19.0 | 154 | 0.16 | 50.9 | 7.6 |

13 | 2 | 2297 | 24.9 | 154 | 0.22 | 31.9 | 25.9 |

14 | 2 | 739 | 17.3 | 191 | 0.33 | 33.9 | 32.0 |

15 | 2 | 1468 | 15.9 | 178 | 0.70 | 7.7 | 34.8 |

16 | 2 | 4204 | 47.4 | 153 | 0.48 | 36.3 | 37.0 |

17 | 2 | 11,545 | 129.6 | 150 | 0.14 | 30.7 | 45.3 |

18 | 2 | 4460 | 73.7 | 180 | 0.15 | 30.0 | 65.3 |

19 | 2 | 4957 | 51.5 | 143 | 0.07 | 31.2 | 18.4 |

20 | 2 | 3897 | 47.4 | 154 | 0.28 | 47.2 | 27.8 |

Avg. | 1.5 | 3508 | 40.8 | 159 | 0.23 | 32.8 | 29.3 |

**Table 2.**Performance of proposed theoretical methods to evaluate energy balance components before and after considering normalized head loss $\mathsf{\Delta}{H}^{\prime}$ with calibrated coefficient ${C}_{mj}$.

Component | Equation | No. of Inlets | $\mathbf{Value}\text{}\mathbf{of}\text{}{\mathit{C}}_{\mathit{m}\mathit{j}}$ | r | RMSE (%) | ||
---|---|---|---|---|---|---|---|

Before | After | Before | After | ||||

${E}_{l}^{\prime}$ | (27) | 1 | 0.7466 | 0.939 | 0.990 | 8.57 | 1.65 |

2 | 0.5047 | 0.957 | 0.985 | 6.02 | 1.91 | ||

${E}_{f}^{\prime}$ | (28) | 1 | 1.0833 | 0.905 | 0.905 | 7.32 | 6.92 |

2 | 0.7538 | 0.834 | 0.834 | 11.30 | 6.99 | ||

${E}_{WL}^{\prime}$ | (29) | 1 | 0.4219 | 0.992 | 0.994 | 4.75 | 1.83 |

2 | 0.3095 | 0.978 | 0.984 | 4.46 | 2.17 |

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**MDPI and ACS Style**

Lipiwattanakarn, S.; Kaewsang, S.; Charuwimolkul, N.; Changklom, J.; Pornprommin, A.
Theoretical Estimation of Energy Balance Components in Water Networks for Top-Down Approach. *Water* **2021**, *13*, 1011.
https://doi.org/10.3390/w13081011

**AMA Style**

Lipiwattanakarn S, Kaewsang S, Charuwimolkul N, Changklom J, Pornprommin A.
Theoretical Estimation of Energy Balance Components in Water Networks for Top-Down Approach. *Water*. 2021; 13(8):1011.
https://doi.org/10.3390/w13081011

**Chicago/Turabian Style**

Lipiwattanakarn, Surachai, Suparak Kaewsang, Natchapol Charuwimolkul, Jiramate Changklom, and Adichai Pornprommin.
2021. "Theoretical Estimation of Energy Balance Components in Water Networks for Top-Down Approach" *Water* 13, no. 8: 1011.
https://doi.org/10.3390/w13081011