# Hydraulic Transients in Viscoelastic Pipeline System with Sudden Cross-Section Changes

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## Abstract

**:**

## 1. Introduction

## 2. Transient Flow Equations in Viscoelastic Pipelines

_{0}is the instantaneous (elastic) strain and ε

_{r}is the retarded strain. The total retarded strain results from the behaviour of each of the N elements:

_{0}is the elastic bulk modulus of the spring, E

_{k}is the elastic bulk modulus of the k-th Kelvin–Voigt element, τ

_{k}is the retardation time of the k-th Kelvin–Voigt element defined by τ

_{k}= μ

_{k}/E

_{k}and ƒ⊂

_{k}is the viscosity of the k-th dashpot.

_{0}is the instantaneous creep compliance of the first spring, defined by J

_{0}= 1/E

_{0}, and J

_{k}is the creep compliance of the k-th Kelvin–Voigt element, defined by J

_{k}= 1/E

_{k}.

_{0}is the steady-state pressure.

_{s}is the Darcy friction factor.

## 3. Numerical Solution of Transient Flow Equations

- -
- Inlet boundary condition

_{0}is the initial head.

- -
- Outlet boundary condition

- -
- Internal nodes

- -
- Connection node

_{i,L}is the velocity on the left-hand side of the connection node, v

_{i,R}is the velocity on the right-hand side of the connection node, D

_{L}is the inner diameter of the pipe on the left-hand side of the connection node and D

_{R}is the inner diameter of the pipe on the right-hand side of the connection node.

_{i,L}, the value of the flow velocity on the right-hand side of the connection node v

_{i,R}was calculated with the following:

## 4. Experimental Study

## 5. Model Validation and Analysis of Experimental Data

_{0}and a time-dependent creep function J(t). Due to the fact that the numerical model takes into account a single value of pressure wave velocity for the entire pipeline system, it also includes the value of the elastic part of compliance. As mentioned earlier, only steady friction was taken into account in the momentum Equation (2). Thus, matching of the observed and calculated pressure changes was obtained by calibrating the parameters of the creep function, i.e., J and τ and the number of elements N in the Kelvin–Voigt viscoelastic model. The values of the viscoelastic parameters were chosen by trial and error to minimise the mean squared error between the calculated and observed pressure samples. In order to simplify the model calibration, if the pipeline system consisted of three pipes (experiments no. 5 and 6), the same values of viscoelastic parameters for each pipe were assumed. Moreover, if more than one Kelvin–Voight element was used in the calculations, identical viscoelastic parameters for each element were used. This simple approach, as presented later, made it possible to obtain satisfactory compliance between the calculation results and experimental data. More advanced methods of calibrating the viscoelastic parameters are presented in [25,26,27].

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

c | pressure wave velocity (m/s) |

D | pipe internal diameter (m) |

E_{0} | elastic bulk modulus of the spring (Pa) |

E_{k} | elastic bulk modulus of the k-th Kelvin–Voigt element (Pa) |

f | friction factor (-) |

f_{s} | Darcy‒Weisbach friction factor (-) |

g | gravity acceleration (m/s^{2}) |

h | piezometric head (m) |

J_{0} | instantaneous creep compliance (Pa^{−1}) |

J_{k} | creep compliance of the k-th Kelvin–Voigt element (Pa^{−1}) |

L | length of the individual pipe (m) |

N | total number of Kelvin–Voigt elements (-) |

Q | initial steady-state volumetric flow rate (m^{3}/s) |

p | pressure (Pa) |

s | pipe wall thickness (m) |

t | time (s) |

v | average flow velocity (m/s) |

v_{0} | initial average flow velocity (m/s) |

v_{i,L} | velocity on the left-hand side of the connection node (m/s) |

v_{i,R} | velocity on the right-hand side of the connection node (m/s) |

x | space coordinate (m) |

Δt | time step (s) |

Δx | spatial step (m) |

ƒ⊗_{0} | instantaneous (elastic) strain (-) |

ε_{r} | retarded strain (-) |

ƒ⊂_{k} | viscosity of the k-th dashpot (kg/sm) |

σ | stress (Pa) |

τ_{k} | retardation time of the k-th Kelvin–Voigt element (s) |

Acronyms | |

HDPE | high-density polyethylene |

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Type of Experiment | No. of Experiment | D_{L} (mm) | s_{L} (mm) | D_{M} (mm) | s_{M} (mm) | D_{R} (mm) | s_{R} (mm) | L (m) | Q (m^{3}/h) | v_{0} (m/s) |
---|---|---|---|---|---|---|---|---|---|---|

A | 1 | 35.2 | 2.4 | - | - | 44.0 | 3.0 | 21.0 | 5.471 | 1.00 |

A | 2 | 32.6 | 3.7 | - | - | 40.8 | 4.6 | 24.0 | 3.622 | 0.77 |

B | 3 | 44.0 | 3.0 | - | - | 35.2 | 2.4 | 21.0 | 5.322 | 1.52 |

B | 4 | 40.8 | 4.6 | - | - | 32.6 | 3.7 | 24.0 | 3.424 | 1.14 |

C | 5 | 26.0 | 3.0 | 32.6 | 3.7 | 40.8 | 4.6 | 12.0 | 4.234 | 0.90 |

D | 6 | 40.8 | 4.6 | 32.6 | 3.7 | 26.0 | 3.0 | 12.0 | 2.101 | 1.10 |

_{0}is the initial flow velocity.

Type of Experiment | No. of Experiment | J_{L} (Pa^{−10})
| τ_{L} (s)
| N_{L} (-)
| J_{M} (Pa^{−10})
| τ_{M} (s)
| N_{M} (-)
| J_{R} (Pa^{−10})
| τ_{R} (s)
| N_{R} (-)
| c (m/s) |
---|---|---|---|---|---|---|---|---|---|---|---|

A | 1 | 1.85 | 0.040 | 1 | - | - | - | 1.85 | 0.040 | 1 | 336 |

A | 2 | 1.20 | 0.030 | 2 | - | - | - | 0.90 | 0.060 | 1 | 420 |

B | 3 | 1.15 | 0.030 | 2 | - | - | - | 1.35 | 0.030 | 1 | 349 |

B | 4 | 0.40 | 0.065 | 2 | - | - | - | 1.30 | 0.030 | 1 | 410 |

C | 5 | 1.00 | 0.025 | 2 | 1.00 | 0.025 | 2 | 1.00 | 0.025 | 2 | 420 |

D | 6 | 0.75 | 0.024 | 2 | 0.75 | 0.024 | 2 | 0.75 | 0.024 | 2 | 397 |

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**MDPI and ACS Style**

Kubrak, M.; Malesińska, A.; Kodura, A.; Urbanowicz, K.; Stosiak, M. Hydraulic Transients in Viscoelastic Pipeline System with Sudden Cross-Section Changes. *Energies* **2021**, *14*, 4071.
https://doi.org/10.3390/en14144071

**AMA Style**

Kubrak M, Malesińska A, Kodura A, Urbanowicz K, Stosiak M. Hydraulic Transients in Viscoelastic Pipeline System with Sudden Cross-Section Changes. *Energies*. 2021; 14(14):4071.
https://doi.org/10.3390/en14144071

**Chicago/Turabian Style**

Kubrak, Michał, Agnieszka Malesińska, Apoloniusz Kodura, Kamil Urbanowicz, and Michał Stosiak. 2021. "Hydraulic Transients in Viscoelastic Pipeline System with Sudden Cross-Section Changes" *Energies* 14, no. 14: 4071.
https://doi.org/10.3390/en14144071