# Remarks on Modeling the Oil Film Generation of Rod Seals

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Sealing System

#### 2.2. Measurement Procedure for Seal Analysis

#### 2.3. Film Thickness Measurement Using Ellipsometry

#### 2.4. Elastohydrodynamic Lubrication Analysis

^{8}Pa to improve the numerical convergence. Depending on the treatment of the negative pressure values throughout the iterative solution procedure, different cavitation models are obtained. In the Guembel cavitation model, the negative pressure values are excluded from the final solution at the end of the solution procedure. The grid points with a negative pressure value are set equal to the constant cavitation pressure. In the Swift–Stieber or Reynolds cavitation model, each iteration of the Jacobi method is checked for negative pressure values beneath the cavitation pressure, and these are excluded from the intermediate solution. For the same problem, both the Swift–Stieber and Reynolds cavitation models generally lead to a slightly higher hydrodynamic pressure compared to that of the Guembel cavitation model, see [25].

#### 2.5. Inverse Theory of Lubrication

## 3. Results

#### 3.1. Analysis of the U-cup

#### 3.2. Oil Film Generation

#### 3.2.1. Gap Height at Outstroke and Instroke

#### 3.2.2. Duty Parameter

^{−1}, ${w}_{\mathrm{o},2}=2.25\times {10}^{12}$ Pa⋅m

^{−1}, ${w}_{\mathrm{o},3}=4.50\times {10}^{12}$ Pa⋅m

^{−1}. Figure 8b shows the influence of the dynamic viscosity on the calculated oil film thickness. For the calculations, rod speed values of 0.12 m/s and 0.25 m/s were chosen. Further input data for the simulation, such as the elasticity parameters of the elastomer and the rod material, are provided in Section 2.3.

^{−1}in the IHL analysis, the film thickness results of both simulation models are near identical.

#### 3.2.3. Parameter Study

## 4. Discussion

#### 4.1. Profile Analysis

#### 4.2. Film Thickness Simulation

#### 4.3. Film Thickness Validation

#### 4.4. Current Limitations

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${C}_{10},{C}_{01}$ | Mooney–Rivlin material coefficients |

$E$ | Modulus of elasticity |

${E}_{\mathrm{red}}$ | Reduced modulus of elasticity |

${f}^{\mathrm{c}}$ | Static contact force |

$h$ | Film thickness |

${h}_{\mathrm{o}}$ | Film thickness on the rod after outstroke |

${h}_{\mathrm{o}}^{*}$ | Film thickness at the maximum pressure at outstroke |

${h}_{\mathrm{min}}$ | Minimal film thickness |

${h}_{\mathrm{w},\mathrm{o}}$ | Film thickness at the maximum pressure gradient at outstroke |

${h}^{\mathrm{c}}$ | Static contact geometry |

$L$ | Length |

$p$ | Hydrodynamic pressure |

${p}^{\mathrm{c}}$ | Static contact pressure |

$Re$ | Reynolds number |

$Re*$ | Reduced Reynolds number |

$u$ | Rod speed |

${w}_{\mathrm{o}}$ | Maximum pressure gradient |

$x$ | Cartesian coordinate |

$\eta $ | Dynamic viscosity |

$v$ | Poisson’s ratio |

$\rho $ | Density |

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**Figure 5.**Detailed views of the sealing edge: (

**a**) cross-section with laser scanning microscopy and (

**b**) top view with scanning electron microscopy.

**Figure 6.**Illustration of the mounted U-cup including a detailed view of the static contact pressure.

**Figure 7.**Calculated film thickness (top: logarithmic scales; bottom: linear scales) and pressure in the sealing gap at outstroke (

**left**) and instroke (

**right**) using the EHL model (rod speed: 0.25 m/s; viscosity: 0.25 Pa⋅s; ambient pressure; full lubrication conditions).

**Figure 8.**Calculated oil film thickness at outstroke as a function of rod speed (

**a**) and dynamic viscosity (

**b**) using the EHL and IHL models (${w}_{\mathrm{o},1}=1.10\times {10}^{12}$ Pa⋅m

^{−1}; ${w}_{\mathrm{o},2}=2.25\times {10}^{12}$ Pa⋅m

^{−1}; ${w}_{\mathrm{o},3}=4.50\times {10}^{12}$ Pa⋅m

^{−1}; ambient pressure; room temperature).

**Figure 9.**Calculated oil film thickness at outstroke as a function of the duty parameter (${u}_{\mathrm{o}}\eta $) using the EHL and IHL methods (${w}_{\mathrm{o},1}=1.10\times {10}^{12}$ Pa⋅m

^{−1}; ${w}_{\mathrm{o},2}=2.25\times {10}^{12}$ Pa⋅m

^{−1}; ${w}_{\mathrm{o},3}=4.50\times {10}^{12}$ Pa⋅m

^{−1}; ambient pressure; room temperature).

**Figure 10.**Calculated oil film thickness at outstroke as a function of the duty parameter (${u}_{\mathrm{o}}\cdot \eta $) using the EHL method and different model input parameters.

**Figure 11.**Empirically measured and calculated oil film thickness at outstroke as a function of the duty parameter (${u}_{\mathrm{o}}\cdot \eta $).

**Table 1.**Parameter study on the influence of various EHL model input parameters concerning the absolute film thickness at outstroke.

Duty Parameter (Pa·m) | Film Thickness (nm) | ||||
---|---|---|---|---|---|

Initial Model | Reynolds Cavitation | Elastic Modulus (+20%) | Grid Size (Doubled) | Edge Radius (Tripled) | |

0.016 | 40.0 | 40.0 | 35.7 | 38.4 | 43.5 |

0.158 | 127.3 | 127.3 | 114.0 | 120.6 | 141.0 |

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**MDPI and ACS Style**

Feuchtmüller, O.; Dakov, N.; Hörl, L.; Bauer, F.
Remarks on Modeling the Oil Film Generation of Rod Seals. *Lubricants* **2021**, *9*, 95.
https://doi.org/10.3390/lubricants9090095

**AMA Style**

Feuchtmüller O, Dakov N, Hörl L, Bauer F.
Remarks on Modeling the Oil Film Generation of Rod Seals. *Lubricants*. 2021; 9(9):95.
https://doi.org/10.3390/lubricants9090095

**Chicago/Turabian Style**

Feuchtmüller, Oliver, Nino Dakov, Lothar Hörl, and Frank Bauer.
2021. "Remarks on Modeling the Oil Film Generation of Rod Seals" *Lubricants* 9, no. 9: 95.
https://doi.org/10.3390/lubricants9090095