# Misalignment-Induced Micro-Elastohydrodynamic Lubrication in Rotary Lip Seals

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{®}was used to model a 200 mm stern tube seal. The contact force, contact area and pressure profile under the seal tip were predicted using the large strain theory and modelling the seal with the Saint Venant–Kirchoff constitutional material model with the properties listed in Table 1. The lack of axial symmetry of the loads required the use of a three-dimensional model [15,16]. Therefore, the axisymmetric approach presented in a previous publication of the authors [14] was extended along the circumferential direction. The initial configuration was considered to be the one with the seal head already clamped between the seal housing components (see Figure 1). It is essential to model the clamping stage as the inner diameter of the seal lip decreases as a result of it. The seal head boundary nodes were consequently fixed leading to a model with a single boundary contact (instead of four [14]).

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\omega $ | Shaft angular velocity | $\left[\mathrm{rad}/\mathrm{s}\right]$ |

$\epsilon $ | Radial misalignment (offset) | $\left[\mathrm{m}\right]$ |

$\theta $ | Angular misalignment (slant) | $[\xb0]$ |

$x,y$ | Coordinate system in the circumferential and axial directions | $\left[\mathrm{m}\right]$ |

${S}_{q}$ | Root mean square roughness | $\left[\mathrm{m}\right]$ |

${\lambda}_{x/y}$ | Root mean square wavelength in the circumferential and axial directions | $\left[\mathrm{m}\right]$ |

${\Delta}_{x/y}$ | Root mean square slope in the circumferential and axial directions | $[-]$ |

${R}_{x/y}$ | Effective radius of curvature in the circumferential and axial directions | $\left[\mathrm{m}\right]$ |

$A$ | Amplitude of the equivalent sinusoidal roughness profile | $\left[\mathrm{m}\right]$ |

${N}_{x/y}$ | Number of asperities in the circumferential and axial directions | $[-]$ |

${u}_{x}$ | Mean surface velocity in the circumferential direction | $\left[\mathrm{m}/\mathrm{s}\right]$ |

${w}_{z}$ | Average normal load per asperity | $\left[\mathrm{N}\right]$ |

$E\prime $ | Equivalent elastic modulus | $\left[\mathrm{Pa}\right]$ |

${W}_{x}$ | Dimensionless load parameter | $[-]$ |

${U}_{x}$ | Dimensionless velocity parameter | $[-]$ |

${h}_{cen}$ | Central film thickness | $\left[\mathrm{m}\right]$ |

${h}_{min}$ | Minimum film thickness | $\left[\mathrm{m}\right]$ |

$F$ | Normal load (radial seal load) | $\left[\mathrm{N}\right]$ |

$E$ | Young modulus | $\left[\mathrm{Pa}\right]$ |

$\nu $ | Poisson ratio | $[-]$ |

$\rho $ | Density | $\left[\mathrm{kg}/{\mathrm{m}}^{3}\right]$ |

${C}_{p}$ | Specific heat capacity | $\left[\mathrm{J}/\left(\mathrm{kg}\xb7\mathrm{K}\right)\right]$ |

$k$ | Thermal conductivity | $\left[\mathrm{W}/\left(\mathrm{m}\xb7\mathrm{K}\right)\right]$ |

${\alpha}_{T}$ | Thermal expansion coefficient | $\left[1/\mathrm{K}\right]$ |

$\eta $ | Dynamic viscosity of the lubricant | $\left[\mathrm{Pa}\xb7\mathrm{s}\right]$ |

$h$ | Fluid film thickness | $\left[\mathrm{m}\right]$ |

$p$ | Hydrodynamic pressure | $\left[\mathrm{Pa}\right]$ |

${p}_{c}$ | Cavitation pressure | $\left[\mathrm{Pa}\right]$ |

${\rho}_{c}$ | Density of the lubricant in the cavitation region | $\left[\mathrm{kg}/{\mathrm{m}}^{3}\right]$ |

$\beta $ | Bulk modulus of the lubricant | $\left[\mathrm{Pa}\right]$ |

$\varphi $ | Dimensionless cavitation variable | $[-]$ |

$g$ | Cavitation index | $[-]$ |

${L}_{c}$ | Width of the contact | $\left[\mathrm{m}\right]$ |

$T$ | Temperature | $\left[\xb0\mathrm{C}\right]$ |

$t$ | Time | $\left[\mathrm{s}\right]$ |

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**Figure 6.**Three-dimensional model of the seal. Note the taper shaft part required for its concentric assembly.

**Figure 7.**Specialized seal setup to observe the seal contact under various misalignments and pressurized conditions.

**Figure 8.**Distribution of asperities on the contact area profile of a slanted seal (see Nomenclature).

**Figure 13.**Asperity loading and fluid film thickness for the seal with a 1.5 mm radial misalignment offset.

**Figure 14.**Hydrodynamic pressure build-up under radial misalignment offset ($\epsilon =1.5\mathrm{mm}$, $\eta =100\mathrm{mPa}\xb7\mathrm{s}$, $\beta =1.0\text{}\times \text{}{10}^{7}\mathrm{Pa}$, ${u}_{x}=1\mathrm{m}/\mathrm{s}$, ${L}_{c}=0.5\mathrm{mm}$ ). The I-EHL film thickness from Figure 13 is magnified and shown in pink.

**Figure 15.**Minimum film thickness sweep ($\epsilon =1.5\text{}\mathrm{mm}$, $\eta =100\text{}\mathrm{mPa}\xb7\mathrm{s}$, $\beta =1.0\text{}\times \text{}{10}^{7}\text{}\mathrm{Pa}$, ${u}_{x}=5\text{}\mathrm{m}/\mathrm{s}$, ${L}_{c}=0.5\text{}\mathrm{mm}$ ).

**Figure 16.**Contact width ${L}_{c}$ sweep with the I-EHL film thickness variation ($\epsilon =1.5\mathrm{mm}$, $\eta =100\mathrm{mPa}\xb7\mathrm{s}$, $\beta =1.0\text{}\times {10}^{7}\mathrm{Pa}$, ${u}_{x}=5\mathrm{m}/\mathrm{s}$, ${h}_{min}=1\mathsf{\mu}\mathrm{m}$ ).

Stern Tube Seal | Shaft | Seal Housing | ||
---|---|---|---|---|

$E$ | $\left[\mathrm{MPa}\right]$ | 14.0 | 2.0 × 10^{5} | 1.065 × 10^{5} |

$\nu $ | $[-]$ | 0.49 | 0.27 | 0.35 |

$\rho $ | $\left[\mathrm{kg}/{\mathrm{m}}^{3}\right]$ | 1900 | 7700 | 8800 |

${C}_{p}$ | $\left[\mathrm{J}/\left(\mathrm{kg}\xb7\mathrm{K}\right)\right]$ | 1670 | 1909.7 | 376 |

$k$ | $\left[\mathrm{W}/\left(\mathrm{m}\xb7\mathrm{K}\right)\right]$ | 0.25 | 25 | 60 |

${\alpha}_{T}$ | $\left[1/\mathrm{K}\right]$ | 2.75 × 10^{−4} | 1.0 × 10^{−5} | 1.85 × 10^{−5} |

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**MDPI and ACS Style**

Borras, F.X.; de Rooij, M.B.; Schipper, D.J.
Misalignment-Induced Micro-Elastohydrodynamic Lubrication in Rotary Lip Seals. *Lubricants* **2020**, *8*, 19.
https://doi.org/10.3390/lubricants8020019

**AMA Style**

Borras FX, de Rooij MB, Schipper DJ.
Misalignment-Induced Micro-Elastohydrodynamic Lubrication in Rotary Lip Seals. *Lubricants*. 2020; 8(2):19.
https://doi.org/10.3390/lubricants8020019

**Chicago/Turabian Style**

Borras, F. Xavier, Matthijn B. de Rooij, and Dik J. Schipper.
2020. "Misalignment-Induced Micro-Elastohydrodynamic Lubrication in Rotary Lip Seals" *Lubricants* 8, no. 2: 19.
https://doi.org/10.3390/lubricants8020019