# Applied Methodology for Designing and Calculating a Family of Spur Gear Pumps

## Abstract

**:**

## 1. Introduction

^{3}/rev).

## 2. Parametric Design and Optimization of the Pump

^{2}. Parameter A is determined by assisted drawing of the pump gear, then by measuring the respective area. Parametric-assisted design of the pump assembly leads to fast and accurate area determination.

_{g}. Using relation (2), the pump flow rate is:

_{v}—volumetric efficiency, in %. For this pump, the manufacturer imposed n = 3000 rev/min to be the nominal rotational speed (but the pump has the capability of rotational speeds of n

_{min}= 700 rev/min and n

_{max}= 5000 rev/min). In conformity with the pump’s specifications, η

_{v}= 93% [4,18].

^{3}/rev. There are minimal dimensions imposed for the gear teeth and the compensator widths, which should not be less than 2.2 mm and 6 mm, respectively.

_{p}= 2 L/min and (b) Q

_{p}= 22.52 L/min.

## 3. Numerical Analysis of the Loads Applied on the Pump’s Spur Gear

_{1}= ω

_{2}of the two gears, the contact between any tooth of the driving gear z

_{1}with the conjugate tooth of the driven gear z

_{2}takes place at a point belonging to the gearing line Δ. At that moment, the pair of teeth cause a discharge of fluid into the outlet aperture of the pump.

_{m}is the hydro-mechanical efficiency (%), and η

_{t}is the total efficiency (%). Because of the viscous friction losses that appear in the bearing couplings between the gear shafts and the compensator [20]; of the viscous friction losses at the teeth tips within the gap between these tips and the pump’s internal casing; of the viscous friction losses generated within the gap between the gear’s lateral surface and the internal surfaces of the bushing blocks of the pump; and of the mechanical losses due to the gears meshing, in practice, the hydro-mechanical efficiency η

_{m}is considered and applied.

_{1}):

_{a}(N⋅mm) is the torque that has to be generated by the driving shaft, and D

_{w}(mm) is the gear-rolling diameter. M

_{a}may be calculated [19] using relation (5):

_{1}= z

_{2}—teeth numbers of the driving gear, also called pinion, and of the driven gear.

_{w12}[4,19] of the displaced spur gears are given by (6)

_{d}—the tangent component of the generated force acting on the driving gear tooth z

_{1}.

_{0}in the inlet chamber up to the nominal pressure p in the outlet chamber.

_{1}, in the interval φ ∈ (0…π), the pressure generated in the pumped fluid to the outlet chamber is p

_{gz1}= p⋅φ/π.

_{r1}, which acts on the outside radius circumference of the driving gear z

_{1}, in correspondence with the angle dφ.

_{rx}acting in the horizontal direction x–x on both gears results through relation (10).

_{1}, D

_{e}= D

_{e12}—the outer diameter of gears, calculated by using formula (11).

_{rx1}, F

_{rx2}(Figure 7) also act on the gear centers O

_{1}and O

_{2}. The resulting forces F

_{R1}and F

_{R2}are considered to act on these centers as well.

_{y}acting in the same y–y direction results as a reaction of teeth in contact—relation (15).

_{w}= 23

^{0}, and the gear has displaced teeth [8,16].

_{R1}and F

_{R2}, which produce loads on the gear bearings:

_{d}, F

_{rx1}, F

_{rx2}, F

_{x1}, F

_{x2}, F

_{y}, F

_{R1}and F

_{R2}determined for each constructive variant.

## 4. Simulation with Finite Elements of the Spur Gear Loads

_{m}= 240 MPa, HBS 2.5/62.5 = 104, Si (9.75%), Fe (0.73%), Cu (3.04%), Ni (0.43%), Mg (0.3%), Zn (0.2%), Sn (0.18%), Ti (0.18%), Mn (0.08%) [3]. The cover is made of AlSi6Cu4, yield strength R

_{m}= 200 MPa, HBS 2.5/62.5 = 121 with the composition: Si (7.97%), Cu (5.12%), Mg (0.7%), Fe (0.53%), Mn (0.12%), Ni (0.01%), Zn (0.12%). These alloys are recommended for casting parts, having a very good machinability. For the gears and shafts, a steel alloy R

_{m}= 350 MPa, HRC = 51 was considered, and for the assembly components (pins and screws), a quality carbon steel R

_{m}= 250 MPa.

_{0}, and the highest value is found in the outlet chamber [19]. Two forces of equal value but opposite direction F

_{d}also act tangentially on the gearing teeth. Together with the forces created by the applied pressures (F

_{rx1}and F

_{rx2}), the pump assembly is loaded with the resulting forces F

_{x1}, F

_{x2}, applied for FEA on the gearing teeth and on the gear shafts (Figure 7).

_{y}, the resulting forces F

_{R1}and F

_{R2}(Figure 7) are obtained. These forces stress the bearings of the driving and driven gears.

_{d}= 1326 N).

^{−3}mm—driven gear, 5.7 × 10

^{−3}mm—driving gear), insignificant for the proper pump assembly operation.

_{d}forces applied on the teeth, for all the parameterized variants.

## 5. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbols | Description | Units |

V_{g} | geometric volume | cm^{3}/rev |

z | teeth number on each gear | - |

z_{i1}, z_{i2} | tooth of the driving gear and the conjugate tooth of the driven gear | - |

Δ | gearing line | - |

b | gear width | mm |

A | area of the space profile between two consecutive teeth | cm^{2} |

Q_{p} | pump flow | L/min |

n | driving shaft rotational speed | rev/min |

η_{v} | volumetric efficiency | % |

η_{m} | hydro-mechanical efficiency | % |

η_{t} | total efficiency | % |

H | body width | mm |

h | depth of bored holes | mm |

l | compensator width | mm |

ω_{1}, ω_{2} | angular speeds of the two gears | s^{−1} |

P | necessary driving motor power | kW |

F_{d} | tangential force applied on teeth | N |

M_{a} | torque moment required for the driving shaft | N⋅mm |

D_{w} | gear-rolling diameter | mm |

p | pressure generated in the pump (in the outlet chamber) during operation | bar |

λ | width coefficient | - |

m | gear modulus | mm |

ξ | specific displacement of the gear teeth profile | mm |

φ | angular variation, φ ∈ (0…π), pressure is rising, φ ∈ (π…3π/2), pressure remains constant | degrees |

p_{gz1} | pressure generated in the pumped fluid to the outlet chamber | bar |

dF_{r1} | radial elementary force acting on the outside radius circumference of the gear z_{1}, corresponding to the angle dφ | N |

dF_{r2} | radial elementary force acting on the interval φ ∈ (π…3π/2) | N |

F_{rx} | total radial force acting in the direction x–x on the gears | N |

D_{e} = D_{e12} | the outer diameter of gears | mm |

α_{w} = 23^{0} | gear angle | degrees |

F_{R1}, F_{R2} | resulting forces that produce stress on the gear bearings | N |

F_{x1}, F_{x2} | radial force acting on the shafts z_{1} and z_{2} in the direction x–x | N |

F_{ry12} | radial forces acting in the vertical direction y–y on the two meshing gears | N |

F_{y} | radial rejecting force generated as a reaction of teeth in contact | N |

## Appendix A

## References

- Rundo, M. Models for Flow Rate Simulation in Gear Pumps: A Review. Energies
**2017**, 10, 1261. [Google Scholar] [CrossRef] [Green Version] - Ransegnola, T.; Zhao, X.; Vacca, A. A comparison of helical and spur external gear machines for fluid power applications: Design and optimization. Mech. Mach. Theory
**2019**, 142, 103604. [Google Scholar] [CrossRef] - Houzeaux, G.; Codina, R. A finite element method for the solution of rotary pumps. Comput. Fluids
**2007**, 36, 667–679. [Google Scholar] [CrossRef] - Product Catalogue Hesper. Gear Pumps HP. Available online: https://www.hesper.ro (accessed on 10 November 2021).
- Ghionea, I.G. Researches on Optimization by Simulation of the Industrial Products Design. Ph.D. Thesis, University Politehnica of Bucharest, Bucharest, Romania, 2010. [Google Scholar]
- Faggioni, M.; Samani, F.S.; Bertacchi, G.; Pellicano, F. Dynamic optimization of spur gears. Mech. Mach. Theory
**2011**, 46, 544–557. [Google Scholar] [CrossRef] - Athanassios, M.; Pupăză, C. Design optimization of high ratio planetary systems. Power transmissions. In Mechanisms and Machine Science, Proceedings of the 4th International Conference, Sinaia, Romania, 20–23 June 2012; Springer Science: Berlin/Heidelberg, Germany, 2012; Volume 13, ISBN 978-94-007-6557-3. [Google Scholar] [CrossRef]
- Linke, H.; Hantschack, F.; Trempler, U.; Baumann, F. New results on the calculation of the load capacity of internal gears. In Proceedings of the International Conference on Gears 2010, Munich, Germany, 4–6 October 2010; VDI-Society for Product and Process Design, Technical University of Munich: Munich, Germany, 2010; pp. 741–753, ISBN 978-3-18-092108-2. [Google Scholar]
- Mucchi, E.; Dalpiaz, G. Experimental validation of a model for the dynamic analysis of gear pumps. In Proceedings of the 25th International Conference on Design Theory and Methodology, Portland, OR, USA, 4–7 August 2013; ASME: Portland, OR, USA, 2013. [Google Scholar] [CrossRef] [Green Version]
- Kollek, W.; Osiński, P. Modelling and Design of Gear Pumps; Wroclaw University of Technology Publishing House: Wroclaw, Poland, 2009; ISBN 978-83-7493-452-7. [Google Scholar]
- Woo, S.; Vacca, A. An Investigation of the Vibration Modes of an External Gear Pump through Experiments and Numerical Modeling. Energies
**2022**, 15, 796. [Google Scholar] [CrossRef] - Huang, K.J.; Lian, W.C. Kinematic flowrate characteristics of external spur gear pumps using an exact closed solution. Mech. Mach. Theory
**2009**, 44, 1121–1131. [Google Scholar] [CrossRef] - Mucchi, E.; Tosi, G.; d’Ippolito, R.; Dalpiaz, G. A Robust Design Optimization Methodology for External Gear Pumps. In Proceedings of the ASME 2010 10th Biennial Conference on Engineering Systems Design and Analysis, Istanbul, Turkey, 12–14 July 2010; Volume 4, pp. 481–490. [Google Scholar] [CrossRef]
- Hydraulic Gear-Pumps and Gear-Motors. U.S. Patents No. US3481275A, 2 December 1986.
- SR ISO 53:2011, SR ISO 701:2011, SR ISO 677:2011; Standards on Gears, Romanian Standards Association. ISO: Bucharest, Romania, 2011.
- Osiński, P. Modelling and Design of Gear Pumps with Modified Tooth Profile; Lambert Academic Publishing: Saarbrucken, Germany, 2014; ISBN 978-365-952-662-6. [Google Scholar]
- Shanmugasundaram, S.; Maasanamuthu, S.; Muthusamy, N. Profile modification for increasing the tooth strength in spur gear using CAD. Eng. Des. SCIRP
**2010**, 2, 740–749. [Google Scholar] [CrossRef] - Prodan, D. Hydraulics, Elements, Subsystems, Systems; Printech: Bucharest, Romania, 2002. [Google Scholar]
- Vasiliu, N.; Vasiliu, D. Hydraulic and Pneumatic Actuations; Tehnic Publishing House: Bucharest, Romania, 2005; Volume 1, ISBN 973-31-2248-3. [Google Scholar]
- Zardin, B.; Natali, E.; Borghi, M. Evaluation of the Hydro—Mechanical Efficiency of External Gear Pumps. Energies
**2019**, 12, 2468. [Google Scholar] [CrossRef] [Green Version] - Mucchi, E.; Rivola, A.; Dalpiaz, G. Modelling dynamic behaviour and noise generation in gear pumps: Procedure and validation. Appl. Acoust.
**2014**, 77, 99–111. [Google Scholar] [CrossRef] - Casoli, P.; Vacca, A.; Franzoni, G. A numerical model for the simulation of external gear pumps. In Proceedings of the 6th International Symposium on Fluid Power JFPS, Tsukuba, Japan, 7 –10 November 2005; pp. 705–710, ISBN 4-931070-06-x. [Google Scholar]
- Torrent, M.; Gamez-Montero, P.J.; Codina, E. Model of the Floating Bearing Bushing Movement in an External Gear Pump and the Relation to Its Parameterization. Energies
**2021**, 14, 8553. [Google Scholar] [CrossRef] - Opran, C.; Ghionea, I.; Pricop, M. Embedded modelling and simulation software system for adaptive engineering of hydraulic gear pumps. In Proceedings of the 26th DAAAM International Symposium, Zadar, Croatia, 21–24 October 2015; pp. 311–319, ISBN 978-3-902734-07-5. [Google Scholar]

**Figure 3.**Selection of pump’s flow rate and confirmation of modified values through parametric design.

No. | Pump Flow Rate Q _{p}, L/min | Nominal Speed n, rev/min | Output η _{v}, % | Geometric Volume V _{g}, cm^{3} | Geometric Area A, cm ^{2} | Teeth Width b, mm | Body Width H, mm | Depth of Bored Holes h, mm | Compensator Width l, mm |
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3000 | 82 | 0.81 | 0.1474 | 2.3 | 25 | 10 | 7.7 |

2 | 2.53 | 85 | 0.99 | 2.8 | 25 | 10 | 7.2 | ||

3 | 3.18 | 88 | 1.2 | 3.4 | 25 | 10 | 6.6 | ||

4 | 4.58 | 90 | 1.7 | 4.8 | 25.5 | 12 | 7.2 | ||

5 | 6.05 | 92 | 2.19 | 6.2 | 30 | 15 | 8.8 | ||

6 | 7.30 | 93 | 2.62 | 7.4 | 32 | 15 | 7.6 | ||

7 | 8.98 | 94 | 3.18 | 9 | 30 | 15 | 6 | ||

8 | 10.08 | 95 | 3.54 | 10 | 35 | 20 | 10 | ||

9 | 12.23 | 96 | 4.25 | 12 | 35 | 20 | 8 | ||

10 | 13.55 | 96 | 4.71 | 13.3 | 35 | 20 | 6.7 | ||

11 | 17.32 | 96 | 6.01 | 17 | 40 | 25 | 8 | ||

12 | 22.52 | 96 | 7.82 | 22.1 | 45 | 30 | 7.9 |

No. | V_{g}, cm^{3} | b, mm | P, kW | M_{a}, N∙mm | F_{d}, N | F_{rx}, N | F_{x1}, N | F_{x2}, N | F_{y}, N | F_{R1}, N | F_{R2}, N |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.81 | 2.3 | 0.59 | 1794 | 138 | 838.35 | 700.35 | 976.35 | 58.57 | 703.8 | 841.8 |

2 | 0.99 | 2.8 | 0.74 | 2184 | 168 | 1020.6 | 852.6 | 1188.6 | 71.30 | 856.8 | 1024.8 |

3 | 1.2 | 3.4 | 0.93 | 2652 | 204 | 1239.3 | 1035.3 | 1443.3 | 86.58 | 1040.4 | 1244.4 |

4 | 1.7 | 4.8 | 1.35 | 3744 | 288 | 1749.6 | 1461.6 | 2037.6 | 122.23 | 1468.8 | 1756.8 |

5 | 2.19 | 6.2 | 1.78 | 4836 | 372 | 2259.9 | 1887.9 | 2631.9 | 157.88 | 1897.2 | 2269.2 |

6 | 2.62 | 7.4 | 2.15 | 5772 | 444 | 2697.3 | 2253.3 | 3141.3 | 188.43 | 2264.4 | 2708.4 |

7 | 3.18 | 9 | 2.64 | 7020 | 540 | 3280.5 | 2740.5 | 3820.5 | 229.18 | 2754 | 3294 |

8 | 3.54 | 10 | 2.97 | 7800 | 600 | 3645 | 3045 | 4245 | 254.64 | 3060 | 3660 |

9 | 4.25 | 12 | 3.60 | 9360 | 720 | 4374 | 3654 | 5094 | 305.57 | 3672 | 4392 |

10 | 4.71 | 13.3 | 3.99 | 10374 | 798 | 4847.85 | 4049.85 | 5645.85 | 338.67 | 4069.8 | 4867.8 |

11 | 6.01 | 17 | 5.09 | 13260 | 1020 | 6196.5 | 5176.5 | 7216.5 | 432.89 | 5202 | 6222 |

12 | 7.82 | 22.1 | 6.62 | 17238 | 1326 | 8055.45 | 6729.45 | 9381.45 | 562.75 | 6762.6 | 8088.6 |

No. | b, mm | F_{d}, N | Driving GEAR | Driven GEAR | Compensator | Body | Cover | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Stress, MPa | Disp, ×10^{−3} mm | Er, % | Stress, MPa | Disp, ×10^{−3} mm | Er, % | Stress, MPa | Er, % | Stress, MPa | Er, % | Stress, MPa | Er, % | |||

1 | 2.3 | 138 | 136 | 3.3 | 6.59 | 139 | 4.9 | 5.89 | 7.72 | 14.4 | 7.72 | 11.3 | 11.5 | 11.2 |

2 | 2.8 | 168 | 108 | 3.1 | 6.27 | 139 | 5.1 | 5.82 | 9.6 | 16.7 | 4.77 | 11.4 | 13 | 11 |

3 | 3.4 | 204 | 126 | 3.2 | 6.15 | 143 | 5.2 | 5.92 | 10.6 | 16.5 | 3.38 | 12.3 | 14.3 | 10.8 |

4 | 4.8 | 288 | 121 | 3.4 | 6.5 | 147 | 5.9 | 6.38 | 14.6 | 17.2 | 4.54 | 12.2 | 19.6 | 10.6 |

5 | 6.2 | 372 | 108 | 3.7 | 6.38 | 151 | 6.3 | 6.23 | 17.2 | 14.3 | 8.89 | 12.4 | 24.4 | 10.3 |

6 | 7.4 | 444 | 11 | 3.6 | 6.3 | 149 | 6.5 | 6.56 | 19.4 | 17.1 | 6.87 | 14.8 | 27.8 | 10.2 |

7 | 9 | 540 | 121 | 4 | 6.56 | 146 | 6.9 | 6.61 | 19.1 | 16.4 | 7.57 | 13.6 | 22.7 | 10.2 |

8 | 10 | 600 | 109 | 4.3 | 7 | 165 | 7.4 | 6.92 | 26.2 | 14.8 | 9.45 | 13 | 26.1 | 11.1 |

9 | 12 | 720 | 108 | 4.4 | 7.11 | 179 | 7.8 | 6.95 | 28.6 | 16.3 | 9.22 | 13.2 | 28.7 | 11.2 |

10 | 13.3 | 798 | 107 | 4.6 | 7.06 | 182 | 8.1 | 7.26 | 26.1 | 18.2 | 12 | 13.1 | 31.5 | 11.1 |

11 | 17 | 1020 | 13 | 5.2 | 7 | 198 | 9.5 | 7.21 | 39 | 16.3 | 14.1 | 12.4 | 39.9 | 11.7 |

12 | 22.1 | 1326 | 154 | 5.7 | 6.79 | 256 | 10.9 | 7.46 | 54.1 | 17.3 | 21 | 13.4 | 48.2 | 11.3 |

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**MDPI and ACS Style**

Ghionea, I.G.
Applied Methodology for Designing and Calculating a Family of Spur Gear Pumps. *Energies* **2022**, *15*, 4266.
https://doi.org/10.3390/en15124266

**AMA Style**

Ghionea IG.
Applied Methodology for Designing and Calculating a Family of Spur Gear Pumps. *Energies*. 2022; 15(12):4266.
https://doi.org/10.3390/en15124266

**Chicago/Turabian Style**

Ghionea, Ionuţ Gabriel.
2022. "Applied Methodology for Designing and Calculating a Family of Spur Gear Pumps" *Energies* 15, no. 12: 4266.
https://doi.org/10.3390/en15124266