# Theoretical Investigation into the Ripple Source of External Gear Pumps

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## Abstract

**:**

## 1. Introduction

## 2. Ripple Source of External Gear Pumps

#### 2.1. Theoretical Description of Outlet Flow Ripple and Outlet Pressure Ripple

#### 2.2. Simplified EGP Circuit

## 3. Model of the Displacement Ripple Source

#### 3.1. Explanation Based on Volume Curves

- (a)
- Connected porting: after getting out of the coverage of delivery port, the DC enters the suction port immediately, two porting coverages are connected.
- (b)
- Open-porting: there is a gap between the coverages of delivery and suction port, where the DC connects to neither delivery or suction port.
- (c)
- Overlapping porting: there is an angular interval of overlap: in this interval the DC is connected to both the delivery and the suction port.

#### 3.2. Analytical Expression of The Kinematic Flowrate Given by the Positioning of the Groove

#### 3.3. Viscous Effects

#### 3.4. Effect of the Delivery Circuit

## 4. Model of the Pressurization Ripple

#### 4.1. Single-DC Pressurization Problem

#### 4.2. Results of Pressurization Ripple Solution

#### 4.3. Numerical Analysis for the Transfer Function of the Circuit

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Latin Symbols | |

A | area (m${}^{2}$) |

C | capacitance (m${}^{3}$/Pa) |

${c}_{q}$ | discharge coefficient |

CR | contact ratio |

f | frequency (Hz) |

h | transfer function |

H | axial length of a gear (mm) |

i | center distance (mm) |

K | bulk modulus (bar) |

m | module of a gear (mm) |

M | arbitrary integer |

N | number of teeth |

p | pressure (bar) |

${p}_{0}$ | initial pressure (bar) |

Q | flowrate (L/min) |

$\overline{Q}$ | mean flowrate (L/min) |

R | resistance (Pa/m${}^{4}$-s) |

${r}_{a}$ | addendum radius (mm) |

t | time (s) |

${t}_{0}$ | initial time (s) |

T | constant to define a time interval (s) |

u | position of the contact point on the line of action (mm) |

V | volume (m${}^{3}$) |

Greek Symbols | |

${\alpha}_{0}$ | Tool pressure angle of a gear (degree) |

$\gamma $ | Base pitch of a gear (mm) |

$\delta $ | Magnitude of the flow ripple (L/min) |

$\eta $ | Relief groove coefficient |

$\rho $ | Density (kg/m${}^{3}$) |

$\tau $ | Time constant (s) |

$\varphi $ | Angular position (degree) |

$\phi $ | Phase shift (degree) |

$\omega $ | Angular velocity (degree/s) |

Subscript | |

a | Addendum |

crit | Critical |

d | Driver gear |

del | Delivery |

k | Kinematic |

p | Profile (profile contact ratio) |

S1 | Displacement ripple solution |

S2 | Pressurization ripple solution |

suc | Suction |

## Abbreviations

DC | Displacement chamber |

EGP | External gear pump |

RT | Restriction-termination |

VT | Volume-termination |

## References

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**Figure 1.**Cross sectional view of an EGP. The tooth space volumes are colored according to their pressure during a typical operation.

**Figure 2.**Two different methods of analysis of the pump outlet flow fluctuations: (

**a**) volume-termination (VT) (

**b**) restriction-termination (RT).

**Figure 3.**Comparison of (

**a**) the flow ripple and (

**b**) the pressure ripple for the reference pump working at 2000 RPM, 200 bar operating condition at VT and RT circuits.

**Figure 4.**Two major components of the ripple sources: compressibility, and the kinematic flow (dashed line). Produced with the reference pump at 2000 RPM, 200 bar operating condition and VT circuit.

**Figure 6.**Flow-ripple compared to the kinematic flow at 200 bar and 20 bar (simulated based on the reference EGP with 2000 RPM and VT circuit).

**Figure 7.**The hydraulic circuit used in the HYGESim simulation tool for modeling the displacing action of and EGP. The representation for the hydraulic circuit of an EGP is similar to [22].

**Figure 8.**The overall lumped-parameter model circuit and two decoupled circuits. The decoupled circuits can reproduce the displacement solution (S1) and the pressurization solution (S2) respectively.

**Figure 11.**Three porting conditions based on volume curve: (

**a**) connected porting (

**b**) open porting (

**c**) overlapping porting.

**Figure 13.**Zero angle ($\varphi $ = 0) position for a DC of an EGP, where two delimiting contact points have equal distance to the pitch point. The tooth-flank sealing region on the primary line of action is shaded by green.

**Figure 14.**Three volume-curve based porting design: maximum delivery (

**left**), minimum delivery (

**middle**) and intermediate delivery porting design (

**right**).

**Figure 15.**Examples of groove geometry design for single-flank involute EGPs (

**a**) maximum-delivery (

**b**) minimum-delivery.

**Figure 16.**(

**a**) Kinematic flow ripple (reference design, 2000 RPM) changes with different groove positioning specified by $\eta $; (

**b**) the mean flowrate reduction with $({\mathrm{CR}}_{p}-1)\eta $.

**Figure 17.**Fluid dynamic and kinematic flow ripple for the reference pump, assuming the minimum-delivery groove position. The simulation assumes a zero-pressure differential for various shaft speeds.

**Figure 18.**Fluid dynamic and kinematic flow ripple for the reference pump, assuming the maximum-delivery groove position. The simulation assumes a zero-pressure differential for various shaft speeds.

**Figure 20.**The change of kinematic ripple under 2000 RPM 200 bar condition for the volume-termination circuit (VT) and for the restriction-termination circuit (RT).

**Figure 22.**(

**a**) The kinematics of single-DC pressurization problem; (

**b**) Corresponding hydraulic circuit.

**Figure 23.**(

**a**) Pressurization solution ${Q}_{2}$ with $\overline{C}$ = 0.1 and different $\overline{R}$ (

**b**) pressurization solution Q2 with $\overline{R}$ = 5 and different $\overline{C}$ (

**c**) Two time constants ${\tau}_{1}$ and ${\tau}_{2}$ on the pressurization solution at $\overline{C}$ = 0.1 and $\overline{R}$ = 5.

**Figure 24.**Recovery of simulated ripple using full hydraulic circuit by adding the kinematic ripple solution and the linearized pressurization ripple solution together for single-DC pressurization problem.

**Figure 26.**Plots of four pressurization ripple solutions: for the reference gear pump at 2000 RPM, 200 bar. The VT circuit is assumed for four different setups, from C1 to C4. Ripples are plotted in (

**a**) time domain and (

**b**) frequency domain up to 10,000 Hz.

**Figure 27.**Plots of four pressurization ripple solutions: for the reference gear pump at 2000 RPM, 200 bar. The RT circuit is assumed for four different setups, from C1 to C4. Ripples are plotted in (

**a**) time domain and (

**b**) frequency domain up to 10,000 Hz.

**Figure 28.**Numerical results of the transfer function for the reference gear pump under four design configurations (from C1 to C4) at 2000 RPM and 200 bar (

**a**) magnitude (

**b**) phase.

**Figure 29.**Numerical results of the transfer function for the reference gear pump under four design configurations (from C1 to C4) at 2000 RPM and 100 bar (

**a**) magnitude (

**b**) phase.

Number of teeth | 18 | Drive pressure angle ${\mathit{\alpha}}_{0,\mathit{d}}$ (deg) | 30 |

Correction factor x | −0.9 | Coast pressure angle ${\mathit{\alpha}}_{0,\mathit{c}}$ (deg) | 15 |

Addendum radius ${\mathit{R}}_{\mathit{a}}$ (mm) | 19.27 | Root radius ${\mathit{R}}_{\mathit{r}}$ (mm) | 13.33 |

Center distance (mm) | 33.36 | Axial length H (mm) | 39 |

Displacement (cc/rev) | 22.2 | Profile contact ratio ${\mathbf{CR}}_{\mathit{p}}$ | 2.0 |

Casing opening angle (deg) | 65 | Helical contact ratio ${\mathbf{CR}}_{\mathit{h}}$ | 0 |

Module (mm) | 2.016 |

© 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**

Zhao, X.; Vacca, A.
Theoretical Investigation into the Ripple Source of External Gear Pumps. *Energies* **2019**, *12*, 535.
https://doi.org/10.3390/en12030535

**AMA Style**

Zhao X, Vacca A.
Theoretical Investigation into the Ripple Source of External Gear Pumps. *Energies*. 2019; 12(3):535.
https://doi.org/10.3390/en12030535

**Chicago/Turabian Style**

Zhao, Xinran, and Andrea Vacca.
2019. "Theoretical Investigation into the Ripple Source of External Gear Pumps" *Energies* 12, no. 3: 535.
https://doi.org/10.3390/en12030535