# Parameterization, Modeling, and Validation in Real Conditions of an External Gear Pump

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Parameterization of the Pump

#### 2.1. Flow Losses

- Q
_{i}is the ideal flow without losses, - Q
_{s}represents laminar flow losses, - Q
_{st}represents turbulent flow losses and, - Q
_{c}represents flow reduction due to compressibility.

_{ef}should include the effects of gas dissolved in the oil (the nature of the leaks can be seen in Figure 2). The following aspects have to be considered:

- The derived capacity according to ISO 8426 has to take the dragging effects of oil by the gears into account. Therefore, the derived capacity is$$\mathrm{D}={\mathrm{D}}^{\prime}\pm \sum \frac{\mathrm{r}\mathsf{\omega}\mathrm{e}}{2}\mathrm{b},$$
- Regarding $\frac{{\mathrm{e}}^{3}}{\mathrm{l}}\mathrm{b}$, although dimensionally it is equivalent to D, it does not always have to be proportional to the volumetric displacement. If the pump displacement is modified by increasing the tooth width, maintaining its modulus (as occurs in pumps of the same group), the considered term will be the same. In any case, most authors have adopted the Poiseuille term:$$\sum \frac{\Delta {\mathrm{Pe}}^{3}\mathrm{b}}{12\mathsf{\mu}\mathrm{L}}={\mathrm{C}}_{\mathrm{s}}\frac{\Delta \mathrm{PD}}{\mathsf{\mu}},$$
_{s}is the laminar slip coefficient; - Regarding ${\mathrm{C}}_{\mathrm{d}}\mathrm{S}\sqrt{\frac{2\Delta \mathrm{P}}{\mathsf{\rho}}}$, taking into account that d
^{3}is proportional to D,$$\sum {\mathrm{C}}_{\mathrm{d}}\mathrm{S}\sqrt{\frac{2\Delta \mathrm{P}}{\mathsf{\rho}}}=\sum {\mathrm{C}}_{\mathrm{d}}\frac{{\mathsf{\pi}\mathrm{d}}^{2}}{4}\sqrt{\frac{2\Delta \mathrm{P}}{\mathsf{\rho}}}=\sum {\mathrm{C}}_{\mathrm{d}}\frac{\mathsf{\pi}}{4}\sqrt{\frac{2\Delta {\mathrm{Pd}}^{4}}{\mathsf{\rho}}}{=\mathrm{C}}_{\mathrm{st}}\sqrt{\frac{2\Delta {\mathrm{PD}}^{\frac{4}{3}}}{\mathsf{\rho}}},$$_{st}is the turbulent slip coefficient.

_{1}= 22 °C and θ

_{2}= 60 °C according to ISO 4409, we could calculate Cs.

#### 2.2. Torque Losses

- T
_{i}is the ideal torque, - T
_{fr}is the fluid friction torque, - T
_{k}is the Coulombian friction torque, - T
_{v}is the viscous friction torque, - T
_{f}is the load dependent friction torque and, - T
_{e}is the small friction torque loss independent of speed and pressure (p.e. shaft seal friction).

_{v}representing the viscous friction coefficient and C

_{f}representing the Coulombian friction coefficient, and not taking into account T

_{e}due to its low values, the model of mechanical losses is

_{1}= 22 °C and θ

_{2}= 60 °C according to ISO 4409, we can calculate Cv.

## 3. Experimental Procedure

#### 3.1. Test Benches

_{ve}/∆n, and the results have been extrapolated at the point where ∆P = 0. Annex B of the ISO 8426: 2008 standard presents the principle of calculation of the derived capacity using a zero-pressure intercept method. However, this method can lead to values of volumetric efficiency higher than 100%. Neglecting the evaluation of the dragged Couette flow or the compensating mechanisms (for example, lateral plate axial clearance in a gear pump) means that the theoretical flow calculated based on the volumetric capacity is lower than the real one, and inevitably, that volumetric efficiencies are higher than 100%. That is mentioned in the paper of Toet et al. [21]. The value of the test presented a derived capacity of 14.7 cm

^{3}, identical to the manufacturer information.

- The side end plate movement is reflected in the laminar slip coefficient variation. This coefficient decreases as the working pressure increases because the oil film thickness between the side plate and the gear also decreases due to the axial compensation mechanism. This is the main reason why a pressure rise does not cause a significant increase in leakage, keeping the volumetric efficiency above 90% in the entire range of operation;
- This coefficient decreases slightly as the rotational speed increases because end plate balance between the hydrodynamic forces on the gear side and the hydrostatic forces on the back side is achieved with smaller oil film thicknesses. This phenomenon is more significant at high rotational speeds, even reaching negative values at low pressures (2200 and 2400 rpm). An explanation for this is that the Couette flow dragged to the pump outlet is higher than the Poiseulle slip at the pump suction. This behavior reaffirms the fact that the variation of the laminar slip coefficient is closely related to the separation between the side plates and the gear face, and also that it is not experimentally possible to separate Poiseuille leaks from Couette leaks;
- The viscous friction coefficient behavior is inverse, since an increase in its value indicates a decrease in the oil film thickness. The mechanical efficiency remains approximately constant, since the Coulomb friction coefficient decreases quickly from 0 to 100 bar. The explanation for this behavior is that the slip intensifies turbulent flow until 100 bar. This can be seen with the growth of the turbulent slip coefficient until 150 bar. The conclusion is that the pump, when working at high pressures and speeds between 1200 and 1800 rpm, mainly exhibits laminar slip and a good lubrication regime.
- Assuming that both turbulent leakage and Coulomb friction mainly occur between the tooth peak and the casing, the variation of the coefficients Cst and Cf is related to the oil film thickness between the gear shaft and the hydrodynamic bearings located on the side plates. An increase in pressure up to about 100 bar leads to movement of the side plate until it comes into contact with the casing. Then, higher pressures lead to a decrease in the oil film thickness between the shaft and the hydrodynamic bearing located on these plates and the teeth peaks approach the casing in the suction zone. Due to the side plate configuration, it is in this area where the separation between the high-pressure and low-pressure area occurs.

#### 3.2. In-Field Machine Test

^{3}and maximum pressure 400 bar); a load sensing LS system for winch, tracks, and auxiliary drives (piston pump 165 cm

^{3}and maximum pressure 350 bar); and an open circuit system for the fluid conditioning system (gear pump 43 cm

^{3}and maximum pressure 20 bar). The diesel engine can run from 800 to 2000 rpm, and units are coupled by means of a pump drive with a ratio of 1:1. The operator sets the engine speed, selects the drive to move, and controls the direction and flow from the hydraulic joysticks installed in the machine cabin.

#### 3.2.1. Adaptation of the Gear Pump

#### 3.2.2. Selection of the Movements to Work

## 4. System Simulation

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Description | Units |

$\mathrm{b}$ | Oil film width | m |

${\mathrm{C}}_{\mathrm{d}}$ | Discharge coefficient | - |

${\mathrm{C}}_{\mathrm{f}}$ | Coulombian friction coefficient | - |

${\mathrm{C}}_{\mathrm{s}}$ | Laminar slip coefficient | - |

${\mathrm{C}}_{\mathrm{s}\mathrm{t}}$ | Turbulent slip coefficient | - |

${\mathrm{C}}_{\mathrm{v}}$ | Viscous friction coefficient | - |

$\mathrm{d}$ | Orifice diameter | m |

$\mathrm{D}$ | Derived capacity | m^{3}rad^{−1} |

${\mathrm{D}}^{\prime}$ | Geometrical derived capacity | m^{3}rad^{−1} |

$\mathrm{e}$ | Oil film thickness | m |

$\mathrm{l}$ | Oil film length | m |

${\mathrm{q}}_{\mathrm{v},\mathsf{\theta}}^{\mathrm{p}}$ | Flow rate pump at outlet ISO 4409 at pressure p and temperature θ | m^{3}s^{−1} |

${\mathrm{q}}_{\mathrm{v},{\mathsf{\theta}}_{1}}^{\mathrm{p}}$ | Flow rate pump at outlet ISO 4409 at pressure p and temperature θ_{1} | m^{3}s^{−1} |

${\mathrm{q}}_{\mathrm{v}.{\mathsf{\theta}}_{2}}^{\mathrm{p}}$ | Flow rate pump at inlet ISO 4409 at pressure p temperature θ_{2} | m^{3}s^{−1} |

$\mathrm{Q}$ | Real pump flow rate | m^{3}s^{−1} |

${\mathrm{Q}}_{\mathrm{c}}$ | Compressed flow rate | m^{3}s^{−1} |

${\mathrm{Q}}_{\mathrm{i}}$ | Ideal pump flow rate | m^{3}s^{−1} |

${\mathrm{Q}}_{\mathrm{s}}$ | Laminal leakage flow rate | m^{3}s^{−1} |

${\mathrm{Q}}_{\mathrm{st}}$ | Turbulent leakage flow rate | m^{3}s^{−1} |

$\mathrm{r}$ | Radius in oil film | m |

$\mathrm{S}$ | Orifice area | m^{2} |

$\mathrm{T}$ | Real torque | Nm |

${\mathrm{T}}_{\mathrm{e}}$ | Shaft seal frictional torque | Nm |

${\mathrm{T}}_{\mathrm{f}}$ | Load dependent frictional torque | Nm |

${\mathrm{T}}_{\mathrm{f}\mathrm{r}}$ | Frictional torque (Fluid) | Nm |

${\mathrm{T}}_{\mathrm{i}}$ | Ideal torque | Nm |

${\mathrm{T}}_{\mathrm{k}}$ | Friction torque (Coulomb) | Nm |

${\mathrm{T}}_{\mathrm{v}}$ | Viscous frictional torque | Nm |

${\mathrm{T}}_{\mathsf{\theta}}^{\mathrm{p}}$ | Torque pump ISO 4409 at pressure p and temperature θ | Nm |

${\mathrm{T}}_{\mathsf{\theta}1}^{\mathrm{p}}$ | Torque pump ISO 4409 at pressure p and temperature θ_{1} | Nm |

${\mathrm{T}}_{\mathsf{\theta}2}^{\mathrm{p}}$ | Torque pump ISO 4409 at pressure p and temperature θ_{2} | Nm |

(Greek letters) | ||

$\Delta \mathrm{P}$ | Pressure increase in the pump | Pa |

${\beta}_{ef}$ | Effective bulk modulus | Pa |

$\mathsf{\mu}$ | Dynamic viscosity | Pa s |

${\mathsf{\mu}}_{1}$ | Dynamic viscosity at temperature θ_{1} | Pa s |

${\mathsf{\mu}}_{2}$ | Dynamic viscosity at temperature θ_{2} | Pa s |

$\mathsf{\rho}$ | Mass density | Kg m^{−3} |

$\mathsf{\omega}$ | Rotation speed of the pump | rad s^{−1} |

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**Figure 1.**Inside of a gear pump: (

**a**) parts of a gear pump (source: ROQUET Group S.A.); (

**b**) leakage paths; and (

**c**) rear of the side plate.

**Figure 3.**Loss coefficients versus pressure and speed: (

**a**) laminar slip coefficient; (

**b**) turbulent slip coefficient; (

**c**) laminar friction coefficient; and (

**d**) Coulomb friction coefficient.

**Figure 4.**LLAMADA P140 machine: (

**a**) overview; (

**b**) gear pump assembly and instrumentation; and (

**c**) acquisition system.

**Figure 6.**Transformation of the control block from a closed center to open center (source: WALVOIL).

**Figure 8.**In-field machine test results: (

**a**) lowering-raising foot (1030 rpm); (

**b**) lowering-raising foot (1467 rpm); and (

**c**) lowering-raising foot (1961 rpm).

**Figure 9.**In-field machine test results: (

**a**) lowering-raising jib (1030 rpm); (

**b**) lowering-raising jib (1467 rpm); and (

**c**) lowering-raising jib (1961 rpm).

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**MDPI and ACS Style**

Torrent, M.; Gamez-Montero, P.J.; Codina, E.
Parameterization, Modeling, and Validation in Real Conditions of an External Gear Pump. *Sustainability* **2021**, *13*, 3089.
https://doi.org/10.3390/su13063089

**AMA Style**

Torrent M, Gamez-Montero PJ, Codina E.
Parameterization, Modeling, and Validation in Real Conditions of an External Gear Pump. *Sustainability*. 2021; 13(6):3089.
https://doi.org/10.3390/su13063089

**Chicago/Turabian Style**

Torrent, Miquel, Pedro Javier Gamez-Montero, and Esteban Codina.
2021. "Parameterization, Modeling, and Validation in Real Conditions of an External Gear Pump" *Sustainability* 13, no. 6: 3089.
https://doi.org/10.3390/su13063089