# Multiphase Conjugate Heat Transfer Analyses on the Assembly Situation of Rotary Shaft Seals
## 2 Materials and Methods
### Modeling
#### 2.2.2 Material Data  
A total of three solid materials were defined: elastomer, aluminum and steel. For each of these materials, the density, specific heat capacity and thermal conductivity were defined. The material data used are listed in Table 1. The values given in the table are constant material parameters, independent of temperature and pressure. Only the density and thermal conductivity of the air (ideal gas), as well as the dynamic viscosity of the oil, were assumed to be non-constant. The constant material data were defined as expressions. This way of parameterizing the simulation model allowed the individual parameters to be changed easily and in a fast way. Rotary shaft seals from Freudenberg made of the fluororubber compound 75 FKMS585 [27] were used for the experiments on the test bench. The material data of the elastomer used in the simulations were based on this fluororubber compound. As material data of the steel, the data of the steel 100 Cr6 [28] were used, since this steel was used in the experiments. The aluminum was the aluminum-copper alloy EN AW 2007. The fluid domain _environment_ was filled with air (ideal gas). The default values of Ansys were used as material data of the air. The fluid domain _oil chamber_ was filled with oil up to a defined oil fill level; everything above this level was initially assumed to be air (ideal gas). The oil _Fuchs Titan Supersyn Longlife 0W-30_ was used for the experiments. The density of the oil was measured with a pycnometer according to EN ISO 3838 [29]. The values of the specific heat capacity and the thermal conductivity of the oil were estimated based on experiments [30] and other oils [31]. Since the density of air is affected by the current temperature \(T_{air}\) and the pressure \(p_{air}\), the air density was not assumed to be constant and was defined as  
\[\rho_{\text{air}}=p_{\text{air}}\cdot M\ /\ (R\cdot T_{\text{air}}), \tag{14}\]  
where \(M=28.96\ \text{kg}/\text{mol}\) is the molar mass of the air and \(R=8.3145\ \text{J}/(\text{mol}\cdot\text{K})\) is the universal gas constant. Below the boiling point, the thermal conductivity of gases increases linearly with temperature, allowing a 1 st-degree polynomial to be sufficient to correlate measured values [24]. The thermal-conductivity data of dry air at 20 \({}^{\circ}\)C and 120 \({}^{\circ}\)C from [24] was used to define the thermal conductivity of air as  
\[\lambda_{\text{air}}=0.001\cdot(0.0712\cdot\theta_{\text{air}}+24.446) \tag{15}\]  
as a function of the current air temperature \(\theta_{\text{air}}\). Since no temperature-dependent data of the thermal conductivity of the oil were available, the conductivity of the oil was assumed to be constant. The dynamic viscosity of the air was defined as \(\eta_{\text{air}}=1.831\cdot 10^{-5}\ \text{kg}/(\text{m}\cdot\text{s})\). To describe the temperature-dependent viscosity of the oil, the Vogel equation  
\[\eta_{\text{oil}}=V_{A}\cdot\exp\{\frac{V_{B}}{\theta_{\text{oil}}-V_{C}}\} \tag{16}\]  
with the temperature-independent material parameters \(V_{A}\) and \(V_{B}\) and the Vogel temperature \(V_{C}\) was used [32; 33]. To determine these parameters, measurements with a MCR302 \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Material**} & \multirow{2}{*}{
\begin{tabular}{c} **Thermodynamic** \\ **State** \\ \end{tabular} } & **Density \(\rho\)** & **Heat Capacity c** & **Thermal** \\  & & [kg/m\({}^{3}\)] & [J/(kg\cdot\text{K}\)] & **Conductivity \(\lambda\)** \\  & & & & [W/(m\(\cdot\)K)] \\ \hline Elastomer & Solid & 1900 & 1650 & 0.215 \\ Aluminum & Solid & 2800 & 860 & 145 \\ Steel 100 Cr6 & Solid & 7830 & 470 & 46 \\ Air (Ideal Gas) & Gas & Equation (14) & 1004.4 & Equation (15) \\ Oil & Liquid & 885 & 2200 & 0.13 \\ \hline \end{tabular}
\end{table} 
Table 1: Material data of the solid and fluid materials used in the simulation. rheometer manufactured by Anton Paar Germany GmbH (Ostfildem, Germany) were made. Both the dynamic viscosity of the air and the dynamic viscosity of the oil were defined as expressions.