# An Analytical Method for the Determination of Temperature Distribution in Short Journal Bearing Oil Film

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## Abstract

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## 1. Introduction

- The oil is an incompressible Newtonian fluid.
- There is no slip of the oil at the boundaries.
- The oil film thickness is small compared with the other dimensions.
- Pressure and temperature are constant through the thickness of the oil film.
- Circumferential pressure gradient in the oil film is neglected.
- The oil viscosity and specific heat are constant.
- There is no thermal interaction between the oil film and the surrounding bearing structure.
- The bearing is operating under steady-state conditions (journal speed and bearing load are constant).
- The structural components of the bearing are rigid and smooth.
- There is no misalignment in the bearing structure.
- The oil supply is not taken into account.
- No asperity contacts between the journal and the bearing.
- The bearing geometry is symmetric.
- The oil flow is laminar.

## 2. Governing Equations

- $x$—streamwise coordinate direction,
- $\theta $—angular coordinate,
- $z$—axial coordinate direction,
- $p$—pressure in an arbitrary point ($x$,$z$),
- $\eta $—oil viscosity,
- $U=R\omega $—sliding velocity,
- $h=c\left(1+\epsilon \cdot \mathrm{cos}\theta \right)$—oil film thickness,
- $c$—radial clearance,
- $\epsilon =e/c$—eccentricity ratio,
- $\Phi =\mathrm{arctan}\cdot (\pi \cdot {(1-{\epsilon}^{2})}^{1/2}/4\epsilon )$—attitude angle
- $e$—eccentricity ($e=\overline{{O}_{b}{O}_{j}}$)

- $\rho $—density of oil
- ${c}_{p}$—specific heat at constant pressure
- $T$—oil film temperature
- $u$—circumferential velocity
- $w$—axial velocity
- $y$—cross-film coordinate direction.

## 3. Solution Procedure

## 4. Results and Discussion

- —
- characteristics of the oil used (density $\rho $, viscosity $\eta $ and specific heat at constant pressure ${c}_{p}$) and
- —
- geometric parameters of the bearing (bearing radius $R$, radial clearance $c$ and bearing length $L$).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Conversion of the Energy Equation into the Form Containing Pressure Gradients

## Appendix B. Solving the Integral Equation (30)

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Parameter | Unit | Value |
---|---|---|

$\mathrm{Journal}\text{}\mathrm{radius},\text{}R$ | m | 0.05 |

$\mathrm{Bearing}\text{}\mathrm{length},\text{}L$ | m | 0.08 |

$\mathrm{Clearance}\text{}\mathrm{ratio},\text{}c/R$ | - | 0.0029 |

$\mathrm{Lubricant}\text{}\mathrm{viscosity}\text{}\mathrm{at}\text{}40\text{}{}^{\mathrm{o}}\mathrm{C},\text{}\eta $ | Pa s | 0.0277 |

$\mathrm{Lubricant}\text{}\mathrm{density}\text{}\mathrm{at}\text{}40\text{}{}^{\mathrm{o}}\mathrm{C},\text{}\rho $ | kg/m^{3} | 860 |

$\mathrm{Lubricant}\text{}\mathrm{specific}\text{}\mathrm{heat},\text{}{c}_{p}$ | $\mathrm{J}/(\mathrm{kg}{\text{\hspace{0.17em}}}^{\mathrm{o}}\mathrm{C})$ | 2000 |

$\mathrm{Inlet}\text{}\mathrm{lubricant}\text{}\mathrm{temperature},\text{}{T}_{0}$ | ${}^{\mathrm{o}}\mathrm{C}$ | 40 |

$\mathrm{Journal}\text{}\mathrm{speed}\text{}\left(\mathrm{lower}\text{}\mathrm{value}\right),\text{}n$ | rpm | 2000 |

$\mathrm{Journal}\text{}\mathrm{speed}\text{}\left(\mathrm{higher}\text{}\mathrm{value}\right),\text{}n$ | rpm | 4000 |

$\mathrm{Bearing}\text{}\mathrm{load}\text{}\left(\mathrm{lower}\text{}\mathrm{value}\right),\text{}F$ | N | 4000 |

$\mathrm{Bearing}\text{}\mathrm{load}\text{}\left(\mathrm{higher}\text{}\mathrm{value}\right),\text{}F$ | N | 6000 |

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**MDPI and ACS Style**

Nikolic, N.; Antonic, Z.; Doric, J.; Ruzic, D.; Galambos, S.; Jocanovic, M.; Karanovic, V.
An Analytical Method for the Determination of Temperature Distribution in Short Journal Bearing Oil Film. *Symmetry* **2020**, *12*, 539.
https://doi.org/10.3390/sym12040539

**AMA Style**

Nikolic N, Antonic Z, Doric J, Ruzic D, Galambos S, Jocanovic M, Karanovic V.
An Analytical Method for the Determination of Temperature Distribution in Short Journal Bearing Oil Film. *Symmetry*. 2020; 12(4):539.
https://doi.org/10.3390/sym12040539

**Chicago/Turabian Style**

Nikolic, Nebojsa, Zivota Antonic, Jovan Doric, Dragan Ruzic, Stjepan Galambos, Mitar Jocanovic, and Velibor Karanovic.
2020. "An Analytical Method for the Determination of Temperature Distribution in Short Journal Bearing Oil Film" *Symmetry* 12, no. 4: 539.
https://doi.org/10.3390/sym12040539