# Dimensionless Pressure Response Analysis for Water Supply Pipeline Systems with or without Pumping Station

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. One-Dimensional Dimensionless Governing Equation with Steady Friction

#### 2.2. Dimensionless Hydraulic Impedance from Surge Tank to Joining Point

#### 2.3. Dimensionless Hydraulic Impedance from Surge Tank to Joining Point

_{c}and R

_{c}are the length and radius of the connector, respectively; and ${\mathsf{\Gamma}}_{c}\left(\widehat{s}\right)=\widehat{x}\widehat{s}\sqrt{\frac{{J}_{0}(\sqrt{(\widehat{S}/{S}_{ac})}i)}{{J}_{0}(\sqrt{(\widehat{S}/{S}_{ac})}i)-2/(\sqrt{(\widehat{S}/{S}_{ac})}i){J}_{1}(\sqrt{(\widehat{S}/{S}_{ac})}i)}}$.

#### 2.4. Dimensionless Lumped Inertia

#### 2.5. Development of Dimensionless Hydraulic Impedance for Two Different Systems

## 3. Results

^{−4}m

^{3}/s. The diameter of the surge tank was assumed to be 2 m, and the length of the connector was 0.5 m. The abrupt valve closure and termination of the check valve might introduce hydraulic transients. The maximum frequency for frequency domain modeling was terminated in 3812 rad/s and the number of fast Fourier transform for the conversion of the frequency domain to time domain response function is 32,768.

#### 3.1. Frequency Response Function

#### 3.2. Time Domain Pressure Response

## 4. Discussion

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Wylie, E.B.; Streeter, V.L. Fluid Transients in Systems, 3rd ed.; Prentice-Hall International: London, UK, 1993; pp. 37–79. [Google Scholar]
- Karney, B.W.; Simpson, A.R. In-line check valves for water hammer control. J. Hydraul. Res.
**2007**, 45, 547–554. [Google Scholar] [CrossRef] [Green Version] - Wan, W.; Zhang, B. Investigation of Water Hammer Protection in Water Supply Pipeline Systems Using an Intelligent Self-Controlled Surge Tank. Energies
**2018**, 11, 1450. [Google Scholar] [CrossRef] [Green Version] - Triki, A. Further invetogation on water-hammer control line strategy in water-supply systems. J. Water Sup.
**2018**, 67, jws2017073. [Google Scholar] [CrossRef] - Di Santo, A.R.; Fratino, U.; Iacobellis, U.V.; Piccinni, A.F. Effects of free outflow in rising mains with air chamber. J. Hydraul. Eng.
**2002**, 128, 992–1001. [Google Scholar] [CrossRef] - Jung, B.S.; Karney, B.W. Systematic Surge Protection for Worst-Case Transient Loadings in Water Distribution Systems. J. Hydraul. Eng.
**2009**, 135, 218–223. [Google Scholar] [CrossRef] [Green Version] - Duan, H.F.; Tung, Y.K.; Ghidaoui, M.S. Probabilistic Analysis of Transient Design for Water Supply Systems. J. Water Resour. Plan. Manag. -ASCE
**2010**, 136, 678–687. [Google Scholar] [CrossRef] - Martino, G.D.; Fontana, N. Simplified approach for the optimal sizing of throttled air chambers. J. Hydraul. Eng.
**2012**, 138, 1101–1109. [Google Scholar] [CrossRef] - Skulovich, O.; Bent, R.; Judi, D.; Perelman, L.S.; Ostfeld, A. Piece-wise mixed integer programming for optimal sizing of surge control devices in water distribution systems. Water Resour. Res.
**2015**, 51, 4391–4408. [Google Scholar] [CrossRef] - Bhattarai, K.P.; Zhou, J.X.; Palikhe, S.; Pandey, K.P.; Suwal, N. Numerical Modeling and Hydraulic Optimization of a Surge Tank Using Particle Swarm Optimization. Water
**2019**, 11, 715. [Google Scholar] [CrossRef] - Mahmoudi-Rad, M.; Najafzadeh, M. Effects of Surge Tank Geometry on the Water Hammer Phenomenon: Numerical Investigation. Sustainability
**2023**, 15, 2312. [Google Scholar] [CrossRef] - Wan, W.; Wang, Y.; Chen, X.; Zhan, H.; Wang, T.; Zhang, B. Investigation of partially expanded surge tank with self-adaptive auxiliary system controlling waterhammer in pipelines. Eng. Sci. Technol. Int. J.
**2023**, 40, 101379. [Google Scholar] [CrossRef] - Guo, J.; Woldeyesus, K.; Zhang, J.; Ju, X. Time evolution of water surface oscillations in surge tanks. J. Hydraul. Res.
**2017**, 55, 657–667. [Google Scholar] [CrossRef] - Kim, S.H. Design of surge tank for water supply systems using the impulse response method with the GA algorithm. J. Mech. Sci. Technol.
**2010**, 24, 629–636. [Google Scholar] [CrossRef] - Kim, S.H.; Choi, D. Dimensionless impedance method for general design of surge tank in simple pipeline systems. Energies
**2022**, 15, 3603. [Google Scholar] [CrossRef] - Wang, C.-N.; Yang, F.-C.; Nguyen, V.T.T.; Vo, N.T.M. CFD Analysis and Optimum Design for a Centrifugal Pump Using an Effectively Artificial Intelligent Algorithm. Micromachines
**2022**, 13, 1208. [Google Scholar] [CrossRef] [PubMed] - Nguyen, V.T.T.; Vo, N.T.M. Centrifugal Pump Design: An Optimization. Eurasia Proc. Sci. Technol. Eng. Math.
**2022**, 17, 136–151. [Google Scholar] [CrossRef]

**Figure 1.**Schematic illustration of reservoir pipeline surge tank pipeline valve (R-P-ST-P-V) system.

**Figure 2.**Schematic illustration of reservoir pump check valve pipeline surge tank pipeline valve (R-PP-CV-P-ST-P-V) system.

**Figure 4.**Amplitudes of dimensionless hydraulic impedance at downstream valve for reservoir pump check valve pipeline surge tank pipeline valve system (R-PP-CV-P-ST-P-V) using dimensionless surge tank (DLST) and dimensionless lumped inertia (DLLI).

**Figure 5.**Amplitudes of integrated dimensionless hydraulic impedance along upstream section of surge tank for reservoir pipeline surge tank pipeline valve system (R-P-ST-P-V) in Figure 1 and that of reservoir pump check valve pipeline surge tank pipeline valve system (R-PP-CV-P-ST-P-V) shown in Figure 2.

**Figure 7.**Normalized integrated pressure responses for upstream section of surge tank due to instant valve closure for reservoir pipeline surge tank pipeline valve system (R-P-ST-P-V) in Figure 1 and that of reservoir pump check valve pipeline surge tank pipeline valve system (R-PP-CV-P-ST-P-V) in Figure 2.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kim, S.
Dimensionless Pressure Response Analysis for Water Supply Pipeline Systems with or without Pumping Station. *Water* **2023**, *15*, 2934.
https://doi.org/10.3390/w15162934

**AMA Style**

Kim S.
Dimensionless Pressure Response Analysis for Water Supply Pipeline Systems with or without Pumping Station. *Water*. 2023; 15(16):2934.
https://doi.org/10.3390/w15162934

**Chicago/Turabian Style**

Kim, Sanghyun.
2023. "Dimensionless Pressure Response Analysis for Water Supply Pipeline Systems with or without Pumping Station" *Water* 15, no. 16: 2934.
https://doi.org/10.3390/w15162934