# The Development of a Calculation Model for the Instantaneous Pressure Head of Oscillating Water Flow in a Pipeline

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experiments

#### 2.1.1. Experimental Equipment and Procedure

^{3}/h). The oscillating water pressure was produced by an automatic pressure control system that consists of a centrifugal pump with an electric motor, a variable-frequency drive (VFD), and a programmable logic controller (PLC). The experimental procedure is as follows. First, the program for implementing the designed oscillating pressure is uploaded to the PLC to control the VFD, which modifies the pump motor speed. The maximum, average, and minimum of the instantaneous pressure heads and the period of the oscillating water flow are set in the program. Second, the electric motor speed of the centrifugal pump is controlled by the VFD to produce the designed oscillating pressure. The pressure signal and flow signal are recorded by the PC equipped with the signal collector. Figure 2 shows a schematic diagram of the test platform for measuring the pressure head of the oscillating water flow.

#### 2.1.2. Experimental Setup

#### 2.2. Calculation Model

^{3}/s); $\overline{\mathrm{Q}}$ is the average instantaneous discharge (m

^{3}/s); and q is the discharge deviation from the average (m

^{3}/s).

^{3}/s); ω is the frequency of the oscillating water flow (rad/s), and t is time (s). Further, ω is related to T, as follows:

#### 2.2.1. The Calculation of the Amplitude of the Pressure Head

^{2}); g is the acceleration due to gravity (9.8 m/s

^{2}); D is the inside diameter of the pipe (m); f is the Darcy–Weisbach friction factor; a is the wave speed (m/s); n is the exponent of the flow velocity in the friction-loss term; x is the distance along the pipeline in the direction of flow (m); and t is time (s). The wave speed, a, and the Darcy–Weisbach friction factor, f, can be calculated as follows [33,34,35]:

_{1}and C

_{2}can be written as:

^{3}/s).

^{3}/s).

#### 2.2.2. The Calculation of the Average Instantaneous Pressure Head

## 3. Results and Discussion

#### 3.1. Results

#### 3.2. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Test platform for measuring instantaneous pressure head of oscillating water flow. PLC—programmable logic controller.

**Figure 5.**The measured and calculated value of instantaneous pressure head at 12 and 60 m pipe lengths for (

**a**) Case D1, (

**b**) Case D2, and (

**c**) Case D3.

**Figure 6.**The measured value of (

**a**) amplitude of pressure head and (

**b**) average instantaneous pressure head along the pipeline for Cases D1, D2, and D3.

Case | Average Pressure Head (m) | Amplitude of Pressure Head (m) | Average Discharge (m^{3}/h) | Amplitude of Discharge (m^{3}/h) | Period (s) |
---|---|---|---|---|---|

D1 | 16 | 8 | 5.71 | 1.56 | 40 |

D2 | 10 | 6 | 4.33 | 1.51 | 60 |

D3 | 8 | 2 | 4.02 | 0.52 | 80 |

**Table 2.**The measured and calculated values of the amplitude of the pressure head along the pipeline.

Pipe Length (m) | D1 | D2 | D3 | ||||||
---|---|---|---|---|---|---|---|---|---|

Calculated Value (m) | Measured Value (m) | Relative Error (%) | Calculated Value (m) | Measured Value (m) | Relative Error (%) | Calculated Value (m) | Measured Value (m) | Relative Error (%) | |

0 | 8.00 | 7.91 | 1.14 | 6.00 | 6.10 | 1.64 | 2.00 | 2.05 | 2.44 |

12 | 7.54 | 7.88 | 4.31 | 5.63 | 5.46 | 3.11 | 1.88 | 1.94 | 3.09 |

24 | 7.07 | 6.69 | 5.68 | 5.27 | 5.69 | 7.38 | 1.76 | 1.88 | 6.38 |

36 | 6.61 | 6.04 | 9.44 | 4.90 | 5.05 | 2.97 | 1.64 | 1.56 | 5.13 |

48 | 6.15 | 6.53 | 5.82 | 4.54 | 4.17 | 8.87 | 1.52 | 1.67 | 8.98 |

60 | 5.69 | 5.32 | 6.95 | 4.18 | 3.93 | 6.36 | 1.41 | 1.53 | 7.84 |

**Table 3.**The measured and calculated values of the average instantaneous pressure head along the pipeline.

Pipe Length (m) | D1 | D2 | D3 | ||||||
---|---|---|---|---|---|---|---|---|---|

Calculated Value (m) | Measured Value (m) | Relative Error (%) | Calculated Value (m) | Measured Value (m) | Relative Error (%) | Calculated Value (m) | Measured Value (m) | Relative Error (%) | |

0 | 16.00 | 15.91 | 0.57 | 10.00 | 9.94 | 0.60 | 8.00 | 7.92 | 1.01 |

12 | 15.15 | 15.32 | 1.11 | 9.48 | 9.83 | 3.56 | 7.54 | 7.26 | 3.86 |

24 | 14.30 | 14.76 | 3.12 | 8.95 | 9.18 | 2.51 | 7.08 | 7.28 | 2.75 |

36 | 13.45 | 12.79 | 5.16 | 8.43 | 8.02 | 5.11 | 6.62 | 6.14 | 7.82 |

48 | 12.60 | 13.65 | 7.69 | 7.90 | 8.52 | 7.28 | 6.16 | 6.61 | 6.81 |

60 | 11.74 | 12.54 | 6.38 | 7.38 | 6.81 | 8.37 | 5.70 | 5.30 | 7.55 |

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**MDPI and ACS Style**

Zhang, K.; Song, B.; Zhu, D.
The Development of a Calculation Model for the Instantaneous Pressure Head of Oscillating Water Flow in a Pipeline. *Water* **2019**, *11*, 1583.
https://doi.org/10.3390/w11081583

**AMA Style**

Zhang K, Song B, Zhu D.
The Development of a Calculation Model for the Instantaneous Pressure Head of Oscillating Water Flow in a Pipeline. *Water*. 2019; 11(8):1583.
https://doi.org/10.3390/w11081583

**Chicago/Turabian Style**

Zhang, Kai, Bo Song, and Delan Zhu.
2019. "The Development of a Calculation Model for the Instantaneous Pressure Head of Oscillating Water Flow in a Pipeline" *Water* 11, no. 8: 1583.
https://doi.org/10.3390/w11081583