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The aim of this paper is to model the steady-state condition of a rotary shaft seal (RSS) system. For this, an iterative thermal-mechanical algorithm was developed based on incremental finite element analyzes. The behavior of the seal’s rubber material was taken into account by a large-strain viscoelastic, so called generalized Maxwell model, based on Dynamic Mechanical Thermal Analyses (DMTA) and tensile measurements. The pre-loaded garter spring was modelled with a bilinear material model and the shaft was assumed to be linear elastic. The density, coefficient of thermal expansion and the thermal conductance of the materials were taken into consideration during simulation. The friction between the rotary shaft seal and the shaft was simplified and modelled as a constant parameter. The iterative algorithm was evaluated at two different times, right after assembly and 1 h after assembly, so that rubber material’s stress relaxation effects are also incorporated. The results show good correlation with the literature data, which state that the permissible temperature for NBR70 (nitrile butadiene rubber) material contacting with ~80 mm shaft diameter, rotating at 2600/min is 100 °C. The results show 107°C and 104°C for the two iterations. The effect of friction induced temperature, changes the width of the contact area between the seal and the shaft, and significantly reduces the contact pressure.

The most frequently used type of seals for the sealing of rotating shafts is the rotary shaft seals. Their complex design allows them to be used versatile in engines, drive trains, gearboxes, hydro-units, compressors, household- and industrial appliances [

The basic functions of these kinds of seals are to provide static tightness between the outer casing of the seal and the bore of the housing, and to provide static and dynamic tightness between the rotating shaft and the sealing lip, additionally, to prevent dust and dirt to ingress from the outside. The sealing mechanism of rotary shaft seals is composed of many individually important features [

Many researchers have studied the behavior of rotary shaft seals. Their models can be distinguished between microscale and macroscale models. Microscale models usually simulate a small part of the seal system, which is physically accurate and close to reality, while macroscale models use higher scale physical models and base on simplified empirical approaches [

Stakenborg first investigated the mechanical behavior of the sealing lip, and found, that the contact conditions governing the sealing mechanism are influenced by temperature [

Kang and Sadeghi used microscale model utilizing elastohydrodynamic (EHD) theory to simulate the temperature distribution in the contact zone of the seal system [

Lee and his co-workers studied the effect of thermal deformation, due to friction power generated in the contact zone [

The dependence of the friction on temperature, lubricating conditions, load and relative speed is often neglected, however friction plays an inevitable role in the generated heat in RSS systems. A detailed empirical method on the dependence of friction on temperature can be found in [

In [

The aim of this paper is to create a numeric model, that is capable of modelling the steady-state condition of a rotary shaft seal. Ignoring the transient warming-up process, and taking into account the friction as a simplified, constant parameter. The implemented algorithm is based on coupled thermal-mechanical, incremental finite element (FE) analyzes.

The large-strain viscoelastic material model used for the investigation of the rubber’s mechanical behavior in this FE analysis was previously developed and presented in a detailed manner in the authors’ previous articles [

The summary of the mechanical properties of the shaft and the spring parts

can be found in

The aim of the analysis was to create a finite element model, that is capable of determining the steady-state thermal and mechanical behavior of a rotary shaft seal (RSS) and shaft assembly during normal operating conditions. To achieve this an iterative procedure was developed by the authors which involves an initial contact analysis, followed by iterative thermal and mechanical finite element analyzes. All of the FE analyzes were incremental.

As a first step an axisymmetric, initial contact analysis of the RSS system was performed. Input parameters were the initial geometry, the coefficient of friction, which was assumed to be 0.4 [

F axisymmetric = ∫ 0 s p d s (1)

Material | E―Young modulus [MPa] | ν―Poisson number [−] | Yield strength [MPa] | Tangent modulus [MPa] |
---|---|---|---|---|

Shaft | 2.1 × 10^{5} | 0.3 | - | - |

Spring | 1 × 10^{6} | 0.3 | 0.51 | 4 |

Material | ρ―Density [kg/m^{3}] | α―Coef. of thermal expansion [1/K] | k―Thermal conductivity [W/mK] |
---|---|---|---|

NBR rubber | 1190 | 1.1 × 10^{−3} | 0.14 |

Shaft | 7850 | 1.2 × 10^{−5} | 60.5 |

Spring | 7850 | 1.2 × 10^{−5} | 60.5 |

Air temperature | 20 [˚C] |
---|---|

Oil temperature | 80 [˚C] |

Convection from seal to housing | 3 × 10^{−5} [W/mmK] |

Convection from seal to air | 3 × 10^{−6} [W/mmK] |

Convection from seal to oil | 5 × 10^{−4} [W/mmK] |

Convection from shaft to air | 3 × 10^{−5} [W/mmK] |

Convection from shaft to oil | 5 × 10^{−4} [W/mmK] |

F rad = F axisymmetric ⋅ d shaft ⋅ π (2)

Multiplying the total radial force by the circumferential speed of the shaft and the coefficient of friction one can get the total heat flow generated by friction.

v shaft = d shaft ⋅ π ⋅ n / 60000 = 10686 m / s (3)

Q = F rad ⋅ v shaft ⋅ μ (4)

This total heat flow was divided by an iterative thermal partition analysis in order to determine what percentage of the total heat flow contributes to heat the RSS system and the shaft. The boundary condition of the analysis was, that the temperature of the seal and the shaft is identical in the contact zone. The results of this analysis showed, that less than 1% of heat dissipates through the seal, and more than 99% of it heats the shaft directly.

The obtained temperature data was then mapped onto the parts of the mechanical model as a temperature load. As the material model of the rubber is temperature dependent, and the thermal expansion of the parts were taken into consideration, the result of the mechanical model is the contact pressure distribution between the seal and the shaft modified by the aforementioned factors. This contact pressure distribution was again used to calculate the total heat flow, and this modified heat flow was divided by the thermal partition analysis, where the procedure started over. The iteration of the results ended when the change of the total radial force became less than 0.5%. The flow chart of the procedure can be seen in

This method allows to calculate the steady-state thermal condition of the RSS system, and to take into account the temperature dependent mechanical behavior of the rubber material, thus to get a complex view of the operation of the RSS system. Disadvantage of the method is that it does not incorporate the change in the friction coefficient due to the change in temperature, speed and load.

The thermal model is used in the iteration process to divide the total heat flow

between the rotary shaft seal and the shaft. The total heat flow is divided if the temperatures in the contact zone are equal. It is found that less than 1% of the heat goes into the seal, and the vast majority, more than 99% of the heat transfers to the shaft. This is because of the better thermal conductivity of the steel material, as it can transfer more heat.

The thermal boundary conditions can be seen in

The mechanical model was used to map the temperature load onto the parts, and take into account the effects of temperature, such as thermal expansion and temperature dependent material behavior.

The boundary conditions of the mechanical model can be seen in

Two kinds of iterations were carried out. For the first one, the contact pressure was evaluated right after assembly, after temperature load was applied. For the second one, the contact pressure was evaluated 1 h after assembly and temperature load had been applied. By this, the time-dependent rubber material’s stress-relaxation properties could be addressed, so that the steady-state condition of the seal could be further approximated. The results of the two iterations were then compared.

contact with the shaft in the contact zone, hence the width of the zone enlarges.

The calculated parameters of the first iteration are summarized in

Iteration [No.] | F_{rad} [N] | ΔF_{rad} [%] | Q [W] | Q_{seal} [W] | Q_{shaft} [W] | Q_{seal} [%] | Q_{shaft} [%] | T_{seal,max} [˚C] | T_{shaft,max} [˚C] |
---|---|---|---|---|---|---|---|---|---|

0 | 29.71 | - | 126.98 | 1.097 | 125.883 | 0.864 | 99.136 | 117.35 | 117.35 |

1 | 23.32 | 21.51 | 99.66 | 0.845 | 98.815 | 0.848 | 99.152 | 108.62 | 108.61 |

2 | 22.79 | 2.27 | 97.41 | 0.824 | 96.583 | 0.846 | 99.154 | 107.89 | 107.89 |

3 | 22.74 | 0.21 | - | - | - | - | - | - | - |

the material and causes the seal to fail.

Iteration [No.] | F_{rad} [N] | ΔF_{rad} [%] | Q [W] | Q_{seal} [W] | Q_{shaft} [W] | Q_{seal} [%] | Q_{shaft} [%] | T_{seal,max} [˚C] | T_{shaft,max} [˚C] |
---|---|---|---|---|---|---|---|---|---|

0 | 20.45 | - | 87.436 | 0.732 | 86.704 | 0.837 | 99.163 | 104.69 | 104.7 |

1 | 20.02 | 2.14 | 85.565 | 0.715 | 84.864 | 0.835 | 99.165 | 104.09 | 104.1 |

2 | 20.31 | 1.48 | 86.833 | 0.726 | 86.107 | 0.836 | 99.164 | 104.48 | 104.51 |

3 | 20.01 | 1.52 | 85.516 | 0.714 | 84.802 | 0.835 | 99.165 | 104.07 | 104.09 |

4 | 20.22 | 1.06 | 86.421 | 0.723 | 85.698 | 0.836 | 99.164 | 104.37 | 104.38 |

5 | 20 | 1.08 | 85.488 | 0.715 | 84.773 | 0.836 | 99.164 | 104.1 | 104.08 |

6 | 20.2 | 0.98 | - | - | - | - | - | - | - |

distribution of the two iterations are compared. It is seen that the shape of the two curves are almost identical, the curve for the second iteration gives lower values. This can be attributed to the stress relaxation behavior of the rubber material was taken into account as the contact pressure was evaluated 1 h after assembly. Compared to the effect of temperature, the effect of stress-relaxation plays a smaller role in the steady-state condition of the contact pressure distribution, thus the arising total radial force, nonetheless further improves the accuracy of the iterative method as the steady state temperature comes to ~104˚C.

For the analysis of rotary shaft seals, an iterative algorithm based on incremental finite element analyzes was implemented. The FE analysis is a one way coupled thermal-mechanical analysis, where the initial parameters are determined by a preliminary contact analysis. The FE models incorporate the modeling of the steel spring which was modelled by a bilinear isotropic hardening material model and responsible for the adequate pre-load of the sealing lip.

Summarizing the results one can conclude, that the effect of temperature cannot be neglected when analyzing rotary shaft seals during operation. The friction generated heat alters the mechanical strength of the material―softens it―and has a great impact on the overall behavior of the seal.

With modifying the iteration process so that it takes into account the long term behavior of the rubber, the results can be further improved. Nevertheless, the number of iterations and thus the computational time increases and the accuracy of the results only improves to a minor extent, which means that the application of the second iteration algorithm may not be beneficial overall.

The authors’ intention is to further refine the model, by measuring the friction between the rotating shaft and the sealing lip, depending on temperature, load, relative speed and lubricating conditions, to be able to model the generated heat in the contact zone, and to incorporate wear to the model.

Hereby we would like to express our gratitude to Prof. Dr. Balázs Magyar at the Technical University of Kaiserslautern for letting us utilize the measured data and results of their researches’.

Szabó, Gy. and Váradi, K. (2018) Thermal-Mechanical Coupled FE Analysis for Rotary Shaft Seals. Modern Mechanical Engineering, 8, 95-110. https://doi.org/10.4236/mme.2018.81007