# Variable Speed Control in PATs: Theoretical, Experimental and Numerical Modelling

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## Abstract

**:**

## 1. Introduction

^{6}to 4.2 × 10

^{6}, with hexahedral, tetrahedral, mixed blocks, and pyramids. Depending on each situation, boundary conditions such as the total pressure, mass flow rate, stagnation pressure, constant total pressure, static pressure, and volumetric flow were placed at the inlet and outlet of the model. In conclusion, it was established that the CFD methodology to predict the performance of a pump working as a turbine presented adequate accuracy based on the comparison of the results with the experimental tests. However, numerous errors were also reported in some studies. The authors assumed that the reported errors were due to the geometries between the tests and the simulations not being identical; the loss estimation was not exact, and more experience in computational analysis is required for modelling this type of phenomenon. Finally, the same author [47] evaluated the application of numerical CFD simulation in PATs in comparison with experimental results and obtained conclusions for future numerical analyses. As a result, it was evidenced that there have been a few simulated cases where a flow with variable speed was simulated and that the number of studies with free code computational packages is minimal, and their use should be promoted due to their outstanding capabilities.

## 2. Materials and Methods

#### 2.1. Preprocess

#### 2.1.1. Computational Domain

^{3}air vessel tank, a 50 mm HDPE pipe, a KSB radial impeller centrifugal pump (model Etarnom 232) that operates in turbine mode, a regulating tank, pressure transducers, valves, and a flow recirculation pump. The air vessel tank sends water to reach the PAT, which discharges to the open free surface tank and then incorporates it into the system through the recirculation pump. The 3D model was built in the SOLIDWORKS CAD system from which the following drawing view was extracted (see Figure 2). This figure shows the geometry that will be entered into the CFD package and from which the results will be compared with the experimental data and with the new expressions. The interactions of the other elements that comprise the installation of the PAT in the laboratory, such as valves, tank, and pump, are placed in the model through the boundary conditions.

#### 2.1.2. Mesh

^{®}Inventor

^{®}software. Later, with the help of HELIX-OS, the BlockMeshDict file was created to generate, using the BlockMesh utility, orthogonal mesh elements for the casing, inlet pipe, impeller, and outlet, respectively. Once the block meshing was ready, the domain geometries were admitted into the snappyHexMeshDict file. The local refinement was defined using castellatedMesh, and the internal points within the closed domain were entered. Finally, it was necessary to use the topoSet tool to generate zones with movable cells for the runner and merge the meshes with the mergeMeshes utility. The mesh characteristics are presented in Table 2, and the generated mesh is shown in Figure 4. In this figure, the different levels of meshing applied to the subdomains can be seen. In addition, it is observed that their configuration is very close to the original geometry. On the other hand, the model looks appropriately balanced, a situation that is confirmed later.

#### 2.1.3. Approach

#### 2.1.4. Boundary and Initial Conditions

#### 2.2. Numerical Simulation

#### 2.2.1. CFD

#### 2.2.2. CFD and Solvers

## 3. Results

#### 3.1. Numerical Simulation Validation

#### 3.1.1. Mesh Quality

^{+}, which verifies the acceptable range of values for the turbulence model. If this value is less than 1, it is considered that the quality of the mesh is good. In this study, it was found that the average y

^{+}values in all the simulations of the mesh were less than 1.

#### 3.1.2. Calibration

_{BEP}= 3.6 L/s) for speeds of 200, 600, 880, 1020, 1200, and 1500 rpm using the κ-ε κ-ω-SST models. The results obtained for both simulations are shown in Table 6. As can be seen, the simulations produced errors of similar magnitude. Still, for the nominal rotational speed of 1020 rpm, the κ-ω-SST model was the one with the lowest error. An error index analysis was performed to define the turbulence model with which the cases of experimental data, nominal rotational speed curve, and the results of the new expressions [21] were simulated. Considering that, in all cases, the error indices closest to zero were those that had a better fit and data compatibility, it was observed that the κ-ω-SST turbulence model presented the best fit in all cases. However, it was verified that the order of magnitude of both turbulence models was close, so they proved their validity when they were applied. Figure 7 shows that, in all cases, the κ-ω-SST model had a better performance. Therefore, the κ-ω-SST turbulence model was adopted for the rest of the cases.

^{−3}.

#### 3.2. Analytical Expressions Validation

#### 3.2.1. Analytical Expressions—New Expressions to Predict PATs Behaviour

#### 3.2.2. Analytical Expressions Validation

#### 3.2.3. Error Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Absolute static pressure contours for Q = 4.50 l/s: (

**a**) N = 810 rpm; (

**b**) N = 930 rpm; (

**c**) N = 1050 rpm; (

**d**) N = 1170; (

**e**) N = 1275 rpm; (

**f**) N = 1500 rpm.

**Figure 8.**The nominal curve obtained with CFD OpenFOAM vs. the nominal curve in [51].

**Table 1.**PAT installations in MHP [27].

Location | The Capacity of the Plant (kW) | Year of Installation |
---|---|---|

Sainyabulli Province, Laos | 2 | 2008 |

Thima, Kenya | 2.2 | 2001 |

Mae Wei Village, Thailand | 3 | 2008 |

West Java, Indonesia | 4.5 | 1992 |

Kinko village, Tanzania | 10 | 2006 |

Fazenda Boa Esperanca, Brazil | 45 | 2007 |

Ambotia Micro-hydro project, India | 50 | 2004 |

British Columbia, Canada | 200 | - |

Vysni Lhoty, Czech Republic | 332 | 2008 |

Parameter | Value/Characteristic |
---|---|

Element type | Hexahedra, Polyhedra, Prism |

Number of Elements | 827,578 |

Hexahedral | 639,704 |

Prism | 28,238 |

Polyhedra | 159,612 |

Number of Nodes | 1,203,219 |

Number of Patches | 8 |

Max. Aspect Ratio | 14.68619 |

Min. Surface Area | 6.19213^{−9} |

Min. Volume | 1.39587^{−11} |

Max. Skewness | 12.918596 |

Initial Conditions | Value |
---|---|

Turbulent Kinetic Energy (κ) | 0.032856 (m^{2}/s^{2}) |

Turbulent Dissipation Rate (ε) | 0.320573 (m^{2}/s^{3}) |

Specific turbulent Dissipation Rate (ω) | 108.4104 (s^{−1}) |

Turbulent kinematic viscosity (nut) | 3.03 × 10^{−4} (m^{2}/s) |

Runner1 | Runner | RunnerIn | Volute | Pipe—Inlet | Pipe—Outlet | Inlet | Outlet | |
---|---|---|---|---|---|---|---|---|

Velocity (u-m/s) | movingWallVelocity uniform (0 0 0) | movingWallVelocity uniform (0 0 0) | movingWallVelocity uniform (0 0 0) | fixedValue uniform (0 0 0) | fixedValue uniform (0 0 0) | fixedValue uniform (0 0 0)v | flowRateInletVelocity volumetricFlowRate constant 0.0045 | inletOutlet valueuniform (0 0 0) |

Static Pressure(p-m^{2}/s^{2}) | zeroGradient | zeroGradient | zeroGradient | zeroGradient | zeroGradient | zeroGradient | zeroGradient | uniform 115,198 (810) 116,694 (930) 112,472 (1050) 112,909 (1170) 115,756 (1275) 110,971 (1500) |

**Table 5.**Calibration results at points A, B, C, and F compared to [51].

% Error | ||||||
---|---|---|---|---|---|---|

Referenced Sections | 810 | 930 | 1050 | 1170 | 1275 | 1500 |

A | 8.724% | 14.297% | 8.218% | 0.035% | 12.881% | 14.042% |

B | 4.455% | 10.425% | 4.286% | 5.324% | 9.068% | 13.066% |

C | 5.979% | 12.040% | 5.643% | 3.999% | 10.389% | 11.936% |

F | 0.156% | 0.014% | 0.340% | 0.018% | 0.199% | 0.111% |

Experimental | Simulation | ||||
---|---|---|---|---|---|

κ-ε | κ-ω-SST | ||||

n (rpm) | H (mca) | H (mca) | % Error | H (mca) | % Error |

200 | 3.27 | 2.28 | 30.23 | 2.39 | 27.00 |

600 | 3.66 | 2.90 | 20.74 | 3.02 | 17.58 |

880 | 4.68 | 4.21 | 10.10 | 4.27 | 8.73 |

1020 | 5.22 | 5.03 | 3.67 | 5.08 | 2.70 |

1200 | 6.22 | 6.21 | 0.12 | 6.14 | 1.30 |

1500 | 7.86 | 8.60 | 9.35 | 8.77 | 11.52 |

Range of Absolute Error | ||||
---|---|---|---|---|

OpenFOAM | New Expressions [21] | |||

n (rpm) | Min (%) | Max (%) | Min (%) | Max (%) |

880 | 5 | 11 | 11 | 24 |

1020 | 2 | 5 | 3 | 17 |

1200 | 0 | 7 | 1 | 12 |

1500 | 1 | 11 | 7 | 24 |

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## Share and Cite

**MDPI and ACS Style**

Plua, F.A.; Sánchez-Romero, F.-J.; Hidalgo, V.; López-Jiménez, P.A.; Pérez-Sánchez, M.
Variable Speed Control in PATs: Theoretical, Experimental and Numerical Modelling. *Water* **2023**, *15*, 1928.
https://doi.org/10.3390/w15101928

**AMA Style**

Plua FA, Sánchez-Romero F-J, Hidalgo V, López-Jiménez PA, Pérez-Sánchez M.
Variable Speed Control in PATs: Theoretical, Experimental and Numerical Modelling. *Water*. 2023; 15(10):1928.
https://doi.org/10.3390/w15101928

**Chicago/Turabian Style**

Plua, Frank A., Francisco-Javier Sánchez-Romero, Victor Hidalgo, Petra Amparo López-Jiménez, and Modesto Pérez-Sánchez.
2023. "Variable Speed Control in PATs: Theoretical, Experimental and Numerical Modelling" *Water* 15, no. 10: 1928.
https://doi.org/10.3390/w15101928