# Model Analysis and System Parameters Investigation for Transient Wave in a Pump–Pipe–Valve System

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## Abstract

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## 1. Introduction

## 2. Pump–Pipe–Valve System

## 3. System Frequency Response

#### 3.1. STMA Model

- (1)
- If $\gamma <1$,$${\left|{h}_{1}^{R}\right|}_{\mathrm{max}}={T}_{2}/\gamma ,\text{}\mathrm{when}\text{}\omega =(n-\frac{1}{2})\frac{\pi a}{L},n=1,2\cdots ,$$$${\left|{h}_{1}^{R}\right|}_{\mathrm{max}}={T}_{2}/\gamma ,\text{}\mathrm{when}\text{}\omega =(n-\frac{1}{2})\frac{\pi a}{L},n=1,2\cdots ,$$
- (2)
- If $\gamma >1$,$${\left|{h}_{1}^{R}\right|}_{\mathrm{max}}={T}_{2},\text{}\mathrm{when}\text{}\omega =\frac{n\pi a}{L},n=1,2,\cdots ,$$$${\left|{h}_{1}^{R}\right|}_{\mathrm{min}}={T}_{2}/\gamma ,\text{}\mathrm{when}\text{}\omega =(n-\frac{1}{2})\frac{\pi a}{L},n=1,2\cdots ,$$
- (3)
- If $\gamma =1$,$${\left|{h}_{1}^{R}\right|}_{\mathrm{max}}={\left|{h}_{1}^{R}\right|}_{\mathrm{min}}=\left|{h}_{1}^{R}\right|={T}_{2},\text{}\mathrm{at}\text{}\mathrm{any}\text{}\mathrm{frequency},$$

#### 3.2. Frictionless Model

#### 3.3. Steady and Unsteady Friction Models

- (1)
- If ${\gamma}_{1}<1$, then $(\mathrm{cosh}(\Delta \epsilon )+{\gamma}_{1}\mathrm{sinh}(\Delta \epsilon ))>(\mathrm{sinh}(\Delta \epsilon )+{\gamma}_{1}\mathrm{cosh}(\Delta \epsilon ))$ and$${\left|{h}_{1}^{R}\right|}_{\mathrm{max}}\approx \frac{{T}_{2}}{\mathrm{sinh}(\Delta \epsilon )+{\gamma}_{1}\mathrm{cosh}(\Delta \epsilon )},\text{}\mathrm{when}\text{}\omega =(n-\frac{1}{2})\frac{\pi a}{L},n=1,2\cdots ,$$$${\left|{h}_{1}^{R}\right|}_{\mathrm{min}}\approx \frac{{T}_{2}}{\mathrm{cosh}(\Delta \epsilon )+{\gamma}_{1}\mathrm{sinh}(\Delta \epsilon )},\text{}\mathrm{when}\text{}\omega =\frac{n\pi a}{L},n=1,2,\cdots ,$$
- (2)
- If ${\gamma}_{1}>1$, then $(\mathrm{cosh}(\Delta \epsilon )+{\gamma}_{1}\mathrm{sinh}(\Delta \epsilon ))<(\mathrm{sinh}(\Delta \epsilon )+{\gamma}_{1}\mathrm{cosh}(\Delta \epsilon ))$ and$${\left|{h}_{1}^{R}\right|}_{\mathrm{max}}\approx \frac{{T}_{2}}{\mathrm{cosh}(\Delta \epsilon )+{\gamma}_{1}\mathrm{sinh}(\Delta \epsilon )},\text{}\mathrm{when}\text{}\omega =\frac{n\pi a}{L},n=1,2,\cdots ,$$$${\left|{h}_{1}^{R}\right|}_{\mathrm{min}}\approx \frac{{T}_{2}}{\mathrm{sinh}(\Delta \epsilon )+{\gamma}_{1}\mathrm{cosh}(\Delta \epsilon )},\text{}\mathrm{when}\text{}\omega =(n-\frac{1}{2})\frac{\pi a}{L},n=1,2\cdots ,$$
- (3)
- If ${\gamma}_{1}=1$, then $(\mathrm{cosh}(\Delta \epsilon )+{\gamma}_{1}\mathrm{sinh}(\Delta \epsilon ))=(\mathrm{sinh}(\Delta \epsilon )+{\gamma}_{1}\mathrm{cosh}(\Delta \epsilon ))$ and$$\left|{h}_{1}^{R}\right|=\frac{{T}_{2}}{\mathrm{cosh}(\Delta \epsilon )+\mathrm{sinh}(\Delta \epsilon )},\text{}\mathrm{at}\text{}\mathrm{any}\text{}\mathrm{frequency}$$

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The frequency responses of a single pipe with the frictionless model at three valve signal intensity (VSI) conditions ($\gamma =0.8,\gamma =1.0,\gamma =1.2$).

**Figure 3.**Impact of the VSI conditions on the resonances and anti-resonances of the upstream pressure fluctuation.

**Figure 4.**Impacts of the Darcy–Weisbach steady friction model ($\kappa =0.04$) on the frequency response with three VSI conditions.

**Figure 5.**Impacts of the unsteady friction model ($\mathrm{Re}=142,468$) on the frequency response with three VSI conditions.

**Figure 6.**Approximation errors of ${\left|{h}_{1}^{R}\right|}_{\mathrm{max}}$ with the steady friction model.

**Figure 7.**Approximation errors of ${\left|{h}_{1}^{R}\right|}_{\mathrm{max}}$ with the unsteady friction model.

**Figure 8.**Impact of the Darcy–Weisbach friction factor on the resonances and anti-resonances of the upstream and downstream pressure fluctuations at VSI condition ${\gamma}_{1}=1.2$.

**Figure 9.**Impact of the VSI condition on the resonances and anti-resonances of the upstream and downstream pressure fluctuations with a steady friction model ($\kappa =0.04$).

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**MDPI and ACS Style**

Liu, Z.; Pan, D.; Qu, F.; Hu, J.
Model Analysis and System Parameters Investigation for Transient Wave in a Pump–Pipe–Valve System. *Water* **2020**, *12*, 1014.
https://doi.org/10.3390/w12041014

**AMA Style**

Liu Z, Pan D, Qu F, Hu J.
Model Analysis and System Parameters Investigation for Transient Wave in a Pump–Pipe–Valve System. *Water*. 2020; 12(4):1014.
https://doi.org/10.3390/w12041014

**Chicago/Turabian Style**

Liu, Zubin, Dingyi Pan, Fengzhong Qu, and Jianxin Hu.
2020. "Model Analysis and System Parameters Investigation for Transient Wave in a Pump–Pipe–Valve System" *Water* 12, no. 4: 1014.
https://doi.org/10.3390/w12041014