# Numerical Simulation of Axial Vortex in a Centrifugal Pump as Turbine with S-Blade Impeller

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model of Vortex

_{u}is generated, as shown in Figure 2 (where u is the peripheral velocity of impeller, w is relative velocity, c is absolute velocity, c

_{m}and c

_{u}are the meridian and peripheral components of absolute velocity, respectively, Δc

_{u}is slip velocity, β is relative flow angle and β

_{b}is the blade angle, the subscript ∞ represents infinite blades). As a consequence, the large axial vortices induced reasonably. In part-load operation, this phenomenon is much more serious on account of the non-optimum incoming flow.

_{ij}can be described as

_{r}, v

_{cr}, and v

_{ci}represent the axial, radial, and tangential components of the element velocity, respectively. The velocity gradient tensor d

_{ij}exists one real eigenvalue λ

_{r}and two conjugated complex eigenvalues λ

_{cr}± λ

_{ci}. The swirling strength is the imaginary part of the complex eigenvalues of the velocity gradient tensor, λ

_{ci}; it is positive if and only if the discriminant is positive and its value represents the strength of swirling motion around local centers. The greater the absolute value of the swirling strength, the stronger the internal circulation of fluid.

## 3. Numerical Simulation

#### 3.1. Numerical Method

^{1/2}/H

^{3/4}, where n is rotational speed, Q is flow rate, and H is head) 9.1 was selected for numerical simulation, and the hydraulic parameters were 50 m for head and 50 m

^{3}/h for flow rate with the rotational speed 1500 r/min. The flow zones consisted of volute, impeller, and draft tube as shown in Figure 3. The main geometrical parameters were shown in Table 1.

^{3}/h for design condition; the rotational speed of the impeller was fixed with 1500 r/min. The fluid was the normal water with a temperature of 20 °C, all the wall surfaces were adiabatic, and the roughness was set to 50 μm. To obtain reasonable results, the proper selection of time steps is of great importance. It is suggested that time steps for a runner rotation of 0.5–5° could provide useful information for the flow field under transients [28]. Hence, the time steps in this study were 3.3 × 10

^{−4}s, corresponding to 3° of the impeller rotational angle. The max coefficient loop of convergence control was 40, and the residual target of the convergence criteria was 10

^{−5}. The total time of the duration data was 0.4 s corresponding to 10 rotor revolutions and the last four revolution data were analyzed.

#### 3.2. Verification of Numerical Method

## 4. Results and Discussion

#### 4.1. Vortex Information in Flow Channels

^{−1}) at 0.122011 s, intensified to maximum (450~500 s

^{−1}) at 0.133891 s, and weakened to minimum (0~50 s

^{−1}) again at 0.142141 s. The revolution of the vortex swirling strength in the second half of the cycle was the same as the first half. The vortices information was extremely similar at 0.122011 s and 0.142141 s, 0.127951 s and 0.148081 s, as well as 0.133891 s and 0.154021 s. It can be found that the time steps of each working point are 0.2 s approximately for the three groups (group a, b, and c, as shown in Figure 9), that is half the time of per rotating cycle (0.4 s) for PAT. In other words, the swirling strength of the vortex develops periodically with two times the rotating frequency.

#### 4.2. Pressure Fluctuation of Vortex

_{i}denotes the transient pressure of monitoring point (Pa), $\overline{p}$ is the average pressure (Pa), ρ is the density of fluid (kg/m

^{3}), and u

_{1}is the peripheral velocity of impeller inlet (m/s).

#### 4.3. Power Losses Caused by Vortex

_{d}and 1.0 Q

_{d}(Q

_{d}is design flow rate), as shown in Figure 12. As a consequence, a low-pressure zone appeared near the impeller inlet, and the relative velocity no longer distributed alongside the blade surfaces. In order to reveal the effect of axial vortices on performance characteristics of PAT, six monitoring cylindrical surfaces in the impeller were created as shown in Figure 13. Figure 14 was the distribution of the average relative velocity (radial component) in the impeller, and Figure 15 was the average pressure at each cylindrical surface. It can be seen that the average relative velocity (radial component) and pressure curves decreased gradually along the flow direction in the impeller channels for 1.6 Q

_{d}; however, a local decline of the curves appeared at surface 1 and 2 for 0.6 Q

_{d}and 1.0 Q

_{d}. As shown in Figure 12, the streamline of 1.6 Q

_{d}was uniform, and very tiny axial vortices were detected in the flow channels. However, large axial vortices can be found near the impeller inlet at 0.6 Q

_{d}and 1.0 Q

_{d}; this was much more serious for low flow rates. It can be seen from Figure 12 that the region from surface 1 to surface 3 was worse affected by axial vortices for the low flow rates that reasonably responded to the local decline of the average relative velocity and pressure.

_{(i,,i+1)}is the fluid head of zone i, and Q is the flow rate. The shaft power of PAT is

_{(i,i+1)}is the torque of zone i, while ω is the angular speed of the impeller. Then, the relative power losses of zone i are

_{d}, 1.0 Q

_{d}, 1.6 Q

_{d}, respectively. It can be seen that the power losses of zone 1 and 2 (the inlet region of the impeller) were higher distinctly than zone 3 and 4. As mentioned earlier, the large axial vortices occurred in this region usually, and caused the reduction in energy conversion of PAT. What calls for special attention was that the power losses of zone 5 and 6 were higher than zone 3 and 4 as well, which was related to the vortices in the draft tube to a great extent, and it deserved further research in the future.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 10.**Pressure fluctuation coefficient with time. (

**a**) Points 1, 2, 3; (

**b**)Points 5, 7, 9; (

**c**)Points 4, 6, 8; (

**d**)Points 11, 13, 15; (

**e**)Points 12, 14, 16; (

**f**)Points 4, 17.

**Figure 11.**Pressure fluctuation coefficient with frequency. (

**a**) Points 1, 2, 3; (

**b**)Points 5, 7, 9; (

**c**)Points 4, 6, 8; (

**d**)Points 11, 13, 15; (

**e**)Points 12, 14, 16; (

**f**)Points 4, 17.

Categories | Parameters | |
---|---|---|

Impeller | Inlet diameter D_{1} (mm) | 312 |

Outlet diameter D_{2} (mm) | 80 | |

Hub diameter d_{h} (mm) | 0 | |

Inlet width b_{1} (mm) | 10 | |

Blade inlet angle β_{1} (°) | 120 | |

Blade number Z | 10 | |

Blade outlet angle β_{1} (°) | 30 | |

Volute | Inlet diameter D_{s} (mm) | 50 |

Outlet width b_{0} (mm) | 24 | |

Basic circle diameter D_{0} (mm) | 320 | |

Draft tube | Length L_{d} (mm) | 120 |

Exit diameter D_{d} (mm) | 80 |

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**MDPI and ACS Style**

Wang, X.; Kuang, K.; Wu, Z.; Yang, J.
Numerical Simulation of Axial Vortex in a Centrifugal Pump as Turbine with S-Blade Impeller. *Processes* **2020**, *8*, 1192.
https://doi.org/10.3390/pr8091192

**AMA Style**

Wang X, Kuang K, Wu Z, Yang J.
Numerical Simulation of Axial Vortex in a Centrifugal Pump as Turbine with S-Blade Impeller. *Processes*. 2020; 8(9):1192.
https://doi.org/10.3390/pr8091192

**Chicago/Turabian Style**

Wang, Xiaohui, Kailin Kuang, Zanxiu Wu, and Junhu Yang.
2020. "Numerical Simulation of Axial Vortex in a Centrifugal Pump as Turbine with S-Blade Impeller" *Processes* 8, no. 9: 1192.
https://doi.org/10.3390/pr8091192