# Assessment of Steady and Unsteady Friction Models in the Draining Processes of Hydraulic Installations

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Governing Equations

- Rigid water column model (RWCM):

- $v$: water velocity, m/s;
- ${p}_{1}^{*}$: air pocket absolute pressure, Pa;
- ${p}_{atm}^{*}$: atmospheric pressure, 101,325 Pa;
- ${\rho}_{w}$: water density, kg m
^{−3}; - $L$: length of a water column, m;
- ${J}_{s/u}$: head losses using a steady (${J}_{s}$) or unsteady (${J}_{u}$) friction model, m/m;
- $g$: gravitational acceleration, m s
^{−2}; - $\mathsf{\Delta}z$: difference elevation, m;
- $D$: internal diameter of a pipe, m;
- ${R}_{v}$: resistance coefficient of a valve, s
^{2}m^{−5}; - $A$: cross-sectional area of a pipe, m
^{2}.

- Air–water interface formulation:

- Polytropic law of an air pocket:

- $x$: air pocket length, m;
- $k$: polytropic coefficient;
- $0$: refers to initial conditions.

#### 2.2. Steady Friction Model (SFM)

- $f$: friction factor;
- ${J}_{s}$: the head losses per unit length in the steady flow regime.

- Moody equation:

- ${k}_{s}$: absolute pipe roughness, mm;
- $\mathrm{Re}$: Reynolds number.

- Wood equation:

- Hazen–Williams equation:

- ${C}_{HW}$: Hazen–Williams coefficient.

- Swamee–Jain equation:

#### 2.3. Unsteady Friction Model (UFM)

- ${J}_{u}$: head losses per unit length in the unsteady flow regime;
- ${k}_{\delta}$: Brunone friction coefficient;
- $a$: wave speed, m s
^{−1}; - $\partial v/\partial t$: local acceleration;
- $\partial v/\partial s$: convective acceleration;
- $s$: distance, m.

- ${C}^{*}$: Vardy’s shear decay coefficient.

## 3. Experimental Stage and Numerical Runs

## 4. Results and Discussions

#### 4.1. Steady Friction Model

#### 4.2. Unsteady Friction Model

## 5. Conclusions and Recommendations

- During the emptying process, the air pocket pressure started under atmospheric conditions. When the ball valve located at downstream end was opened, the absolute pressure pattern descended until the lowest value (first drop), after which some oscillations were reached until the water column was again at rest. The length of the water column showed a similar behavior. Regarding the water velocity, it started at rest (0 m s
^{−1}), following which it rapidly reached the maximum value, and finally negative and positive values were generated. - Considering the six experimental runs, the implementation of the unsteady friction model of Brunone in the simulation of the draining process better fixed the measured air pocket pressure oscillations in the analyzed experimental facility. When the Moody and Wood formulations were implemented with the UFM, the minimum root mean square errors were reached.
- It is important to highlight that both the SFM and the UFM adequately predicted the air pocket pressure oscillations using all of the empirical formulations to compute the friction factor.
- The mathematical model proposed considers the analysis of the laminar and turbulent zone flows. The first drop of sub-atmospheric pressure pattern is the more complex zone to simulate, since it involves the presence of laminar and turbulent flows. After that, the water movement is almost null; consequently, the laminar flow is presented during this part of the transient event.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Configuration of the experimental facility: (

**a**) Scheme of the experimental configuration; (

**b**) photograph of the hydraulic installation.

**Figure 2.**Air pocket head pressure patterns: (

**a**) Comparison between the calculated and measured air pocket head pressure patterns for Run No. 1 and No. 5; (

**b**) numerical runs from No. 1 to No. 6.

**Figure 3.**Behavior of the friction factor and Reynolds number for Run No. 1: (

**a**) Friction factor and (

**b**) Reynolds number.

**Figure 4.**Evolution of the emptying process variables for Run No. 1: (

**a**) Air pocket absolute pressure; (

**b**) water velocity; (

**c**) water column length.

**Figure 5.**Analysis of Run No. 5: (

**a**) Reynolds number evolution for an SFM condition; (

**b**) behavior of the head losses per unit length.

**Figure 6.**Comparison of the measured and computed patterns of Run No. 5 using the Swamee–Jain formulation with the SFM and the UFM: (

**a**) Air pocket pressure; (

**b**) water velocity; (

**c**) water column length.

Run No. | ${\mathit{x}}_{0}\left(\mathbf{m}\right)$ | ${\mathit{R}}_{\mathit{v}}\times {10}^{-6}({\mathbf{s}}^{2}{\mathbf{m}}^{-5})$ |
---|---|---|

1 | 0.205 | 11.89 |

2 | 0.340 | 11.89 |

3 | 0.450 | 11.89 |

4 | 0.205 | 25.00 |

5 | 0.340 | 22.68 |

6 | 0.450 | 30.86 |

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**MDPI and ACS Style**

Coronado-Hernández, Ó.E.; Derpich, I.; Fuertes-Miquel, V.S.; Coronado-Hernández, J.R.; Gatica, G.
Assessment of Steady and Unsteady Friction Models in the Draining Processes of Hydraulic Installations. *Water* **2021**, *13*, 1888.
https://doi.org/10.3390/w13141888

**AMA Style**

Coronado-Hernández ÓE, Derpich I, Fuertes-Miquel VS, Coronado-Hernández JR, Gatica G.
Assessment of Steady and Unsteady Friction Models in the Draining Processes of Hydraulic Installations. *Water*. 2021; 13(14):1888.
https://doi.org/10.3390/w13141888

**Chicago/Turabian Style**

Coronado-Hernández, Óscar E., Ivan Derpich, Vicente S. Fuertes-Miquel, Jairo R. Coronado-Hernández, and Gustavo Gatica.
2021. "Assessment of Steady and Unsteady Friction Models in the Draining Processes of Hydraulic Installations" *Water* 13, no. 14: 1888.
https://doi.org/10.3390/w13141888