# Simulation of the Filling Capability in Vane Pumps

^{1}

^{2}

^{*}

## Abstract

**:**

^{®}. At first, a model of a reference pump has been created and validated with different configurations of the suction flow area, then a simplified model has been used for assessing the influence of the geometry of the rotating assembly. It was found that a pump with a low ratio between the axial thickness and the diameter has a higher volumetric efficiency if the chambers are fed from one side only. Opposite behaviors were found in the case of pumps with small diameters and high thicknesses. Moreover, the filling could be improved by increasing the number of chambers, and by reducing the diameter of the rotor, even only locally.

## 1. Introduction

## 2. CFD Model of the Reference Pump

## 3. Model Validation

^{3}, and dynamic viscosity 0.05 $\mathrm{Pa}\cdot \mathrm{s}$ at 40 °C. The fluid conditioning system was made up of a constant-speed screw pump (PU), a water-oil heat exchanger (SC), a chiller (CH), two 5 kW electric heating elements (EH), and two filters in parallel. The pump PU sucks the oil from the 30 L reservoir and delivers the fluid to the return line of the main circuit. The temperature in the reservoir is controlled in a closed loop based on the average of the signal of two PT100 resistance temperature detectors (T1 and T2). Due to the thin casing and the inlet pipe walls, the mounting of a pressure transducer for measuring the suction pressure in the pump under test was not possible. However, the same suction pipe has been already used in a different study on an equivalent gerotor lubricating pump; in the [17], the validation of the CFD model of the suction pipe, in terms of pressure drop vs flow rate, can be found. A view of the pump mounted on the test rig is shown in Figure 5. The reservoir could be moved vertically, and during the test, the suction pipe was almost entirely submerged, as indicated in the photo. The position of the pressure transducer for the delivery pressure is also visible.

## 4. Analysis of the Flow Rates and of the Pressure Fields

## 5. Influence of the Geometric Parameters

#### 5.1. Simplified CFD Model

#### 5.2. Influence of the Number of Vanes

#### 5.3. Influence of the Rotor Shape

_{i}in Figure 12) was decreased, in order to allow for the flow area to be increased.

#### 5.4. Influence of the Double Feeding with a Low Diameter of the Rotor

- M3: with the chambers fed from only one side;
- M4: with respect to the configuration at Item 1, with a second inlet port in the cover;
- M5: with respect to the configuration at Item 1, with a second inlet port obtained through a milling on the stator with a height of 5 mm on the cover side, like the cutting with height h2 in Figure 2.

#### 5.5. Influence of the Eccentricity

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

^{®}.

## Conflicts of Interest

## Appendix A

^{®}. The control volumes were divided into subdomains, in order to have the possibility of setting different values of the grid density. The variable volume chambers were meshed by the means of a built-in mesh generator, which allows obtaining a structured hexahedral grid. The remaining volumes were meshed by means of the general mesh generator using the body-fitted binary tree approach [24] for obtaining an unstructured mesh with mainly cubic cells. Basically, a Cartesian cubic grid is generated, and the cell size is reduced by factors of two in the regions where a higher accuracy is necessary. The cells are cut in correspondence of the boundaries.

_{i}is the linear drag coefficient, v is the fluid velocity, and γ is the porosity, defined as the ratio between the void space and the total volume. For the case under study, the value of γ that is equal to 0.508 was directly measured from the CAD model of the net. The coefficient C

_{i}was estimated by means of a stand-alone detailed model of only a portion of the duct, including the net.

_{v}and f

_{g}the mass fraction of the vapor and of the gas (air) respectively, while ρ

_{v}, ρ

_{g}, and ρ

_{l}are the densities of the vapor, of the gas, and of the liquid. The fraction of dissolved/separated air is determined based on the total mass of gas, and the equilibrium value given by the Henry’s law, and the local pressure as follows:

_{d}is the gas mass fraction at the equilibrium condition, p is the instantaneous local pressure, p

_{r}is the reference pressure at equilibrium condition when all of the gas is dissolved, and g

_{d,r}is the mass gas fraction that is dissolved in the liquid at the reference pressure. The total gas mass fraction, i.e., the sum of the dissolved and separated fractions, remains constant during the simulation. This model does not consider the time constants for the processes of release and dissolution, but the gas reaches the equilibrium condition instantaneously. The equations used in the model to calculate vapor condensation and evaporation rates are:

_{e}and C

_{c}are respectively the evaporation and condensation constants, g

_{f}is the mass fraction of separated gas and p

_{v}the vapor pressure.

^{®}as an extension of the SIMPLE-C algorithm, was employed for the pressure-velocity coupling. Transient simulations were performed with an angular step of 1 degree. The software evaluates at the same time also the moving average of the delivered volumetric flow rate using as subset the data of a shaft revolution. The steady-state condition is considered to be reached when the mean flow rate becomes constant.

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**Figure 1.**(

**a**) Three-dimensional view of the vane pump with the detail of a variable chamber; (

**b**) Computational domain with a detail of the mesh resolution.

**Figure 2.**Cross-section of the inlet volume with the indication of the geometric parameters. Planes A and B are used for the cut plots shown in Section 4.

**Figure 3.**View of the pump from the inlet pipe with the indication of the direction of the flow, the location of the cuttings h1 and h2, and the direction of rotation (black arrow).

**Figure 5.**Photo of the pump mounted on the test rig with the indication of the oil level when the reservoir is positioned in the test mode.

**Figure 6.**Flow-speed characteristics at 0.2 MPa and at two fixed displacements (blocked): maximum (100%) and partial (55%).

**Figure 7.**Total instantaneous flow rate entering a chamber (positive if ingoing) as a function of the shaft angle for three different configurations.

**Figure 9.**Pressure field in the upper milling—cover side (Plane A) and detail of the velocity field in Configuration 3, at 120 degrees and at 55% of the maximum displacement.

**Figure 10.**Pressure and velocity fields in a cross-section of a chamber (Plane B) for Configuration 4 at 180 degrees and at 55% of the maximum displacement.

**Figure 13.**Normalized flow area with seven and 11 vanes. The position of 0 degrees corresponds to the minimum volume of the chamber.

**Figure 15.**Model with a smaller diameter and the same displacement: (

**a**) with chambers fed from only one side—M3, (

**b**) with chambers fed from both sides through the cover—M4, (

**c**) with chambers fed from both sides through a cutting on the stator—M5. The delivery volume is not shown.

**Figure 16.**Pressure and velocity fields in a cross-section of a chamber at the end of the suction phase for the geometry M4 at 5000 rpm.

Configuration | h1 (mm) | h2 (mm) | h3 (mm) |
---|---|---|---|

1 (baseline) | 2 | 0 | 0 |

2 | 2 | 1 | 0 |

3 | 2 | 2 | 0 |

4 | 2 | 0 | 5 |

5 | 2 | 2 | 5 |

6 | 0 | 0 | 0 |

**Table 2.**Simulated flow rate and volumetric efficiency at 5200—partial displacement (55%) and at 4400 rpm—full displacement (100%).

Configuration | Flow Rate (L/min) | Volumetric Efficiency | Flow Rate (L/min) | Volumetric Efficiency |
---|---|---|---|---|

@5200 rpm—55% | @4400 rpm—100% | |||

1 (baseline) | 40.1 | 0.91 | 58.1 | 0.94 |

2 | 37.4 | 0.85 | 54.4 | 0.88 |

3 | 33.2 | 0.75 | - | - |

4 | 37.4 | 0.85 | 57.0 | 0.92 |

5 | 35.4 | 0.80 | - | - |

6 | 40.1 | 0.91 | 55.2 | 0.90 |

**Table 3.**Flow rate and volumetric efficiency at 5000 rpm with different numbers of vanes. A code has been associated with each configuration.

Code | No. of Vanes | Flow Rate (L/min) | Volumetric Efficiency |
---|---|---|---|

M1 | 7 | 47.8 | 0.61 |

M0 | 9 (reference) | 58.0 | 0.74 |

M2 | 11 | 66.6 | 0.85 |

Eccentricity | Stator Diameter | Rotor Diameter | Axial Thickness | Pump Displacement |
---|---|---|---|---|

1 | 0.59 | 0.54 | 1.8 | 1.01 |

**Table 5.**Flow rate and volumetric efficiency at 5000 rpm and 7500 rpm, with three layouts of the suction ports, and a rotor with small diameter.

Model | Flow Rate (L/min) | Volumetric Efficiency | Flow Rate (L/min) | Volumetric Efficiency |
---|---|---|---|---|

5000 rpm | 7500 rpm | |||

Single side (M3) | 45.0 | 0.57 | 42.3 | 0.36 |

Double side through cover (M4) | 72.4 | 0.92 | 79.1 | 0.67 |

Double side through stator (M5) | 74.2 | 0.95 | 78.3 | 0.66 |

Code | Eccentricity | Stator Diameter | Rotor Diameter | Axial Thickness | Pump Displacement |
---|---|---|---|---|---|

M6 | 1.64 | 0.64 | 0.54 | 1.00 | 1.01 |

Code | Flow Rate (L/min) | Volumetric Efficiency |
---|---|---|

M0 | 58.0 | 0.74 |

M3 | 45.0 | 0.57 |

M6 | 64.9 | 0.83 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Rundo, M.; Altare, G.; Casoli, P.
Simulation of the Filling Capability in Vane Pumps. *Energies* **2019**, *12*, 283.
https://doi.org/10.3390/en12020283

**AMA Style**

Rundo M, Altare G, Casoli P.
Simulation of the Filling Capability in Vane Pumps. *Energies*. 2019; 12(2):283.
https://doi.org/10.3390/en12020283

**Chicago/Turabian Style**

Rundo, Massimo, Giorgio Altare, and Paolo Casoli.
2019. "Simulation of the Filling Capability in Vane Pumps" *Energies* 12, no. 2: 283.
https://doi.org/10.3390/en12020283