# Multiphase Conjugate Heat Transfer Analyses on the Assembly Situation of Rotary Shaft Seals
## 2 Materials and Methods
### Modeling
#### 2.2.3 Boundary and Initial Conditions  
The shaft and shaft sleeve were defined as rotating. The rotational speed \(n\) was specified as an expression and could, thus, be easily varied. All other solid domains were stationary. Due to the rotation of the shaft sleeve, frictional heat was generated in the contact area between the shaft sleeve and the sealing ring. The frictional heat model  
\[\dot{Q}_{\text{fric}}=M_{\text{fric}}\cdot 2\pi\cdot n \tag{17}\]  
describes the frictional heat flux \(\dot{Q}_{\text{fric}}\) generated in the contact area [7]. This frictional heat flux was introduced to the simulation model via a boundary condition. Therefore, the frictional torque was measured on the test bench for different shaft speeds \(n\) and approximated by the rational function  
\[M_{\text{fric}}=\frac{a_{1}\cdot n}{b_{2}\cdot n^{2}+b_{1}\cdot n+b_{0}} \tag{18}\]  
according to [15]. The measured values were then fitted to the curve described by Equation (18) in order to determine the parameters \(a_{1}\), \(b_{0}\), \(b_{1}\) and \(b_{2}\). With the help of this fitted curve, the values of the frictional torque can then be approximated for various shaft speeds, which may not have been measured. The parameters in Equation (18) were only valid for the tribological system with which the measurements were performed. For other materials or different geometries, the parameters had to be redetermined. In reality, the heat calculated by Equation (17) is generated in the sealing gap between the sealing ring and the shaft surface. Since the thermal conductivity of the sealing ring is extremely low compared to the thermal conductivity of the shaft, more heat is transferred from the sealing gap to the shaft than to the sealing ring. In the simulation model, the sealing gap was not considered; instead, the direct contact between the shaft surface and the sealing edge was assumed. In order to realize the uneven heat transfer to the sealing ring and the shaft in the simulation model, the frictional heat flux calculated by Equation (17) was divided into two parts. A larger part of the heat flux was specified as a heat source at the shaft surface. A much smaller heat source was specified at the sealing edge. Thus, the frictional heat flux determined by Equation (17) was divided using a heat partition factor \(f\), which is defined as  
\[f\;=\;\frac{\lambda_{\text{seal}}}{\lambda_{\text{seal}}+\lambda_{\text{ shaft}}}\,. \tag{19}\]  
By inserting the frictional torque into Equation (17) and then multiplying the resulting frictional heat flux with the heat partition factor, the frictional heat applied to the sealing ring could be determined by  
\[\dot{Q}_{\text{fric seal}}=\dot{Q}_{\text{fric}}\cdot f \tag{20}\]  
and the frictional heat applied to the shaft sleeve analogously by  
\[\dot{Q}_{\text{fric shaft}}=\dot{Q}_{\text{fric}}\cdot(1-f). \tag{21}\]  
The frictional heat of the shaft \(\dot{Q}_{\text{fric shaft}}\) was added as a heat source in the contact area between the shaft sleeve and sealing ring, but only to the shaft sleeve. The frictional heat of the sealing ring \(\dot{Q}_{\text{fric seal}}\) was also added in the contact area, but only on the side of the sealing ring. Except for the fluid domain _environment_, each domain was initialized with the initial temperature \(\theta_{\text{init}}=80\)\({}^{\circ}\)C. This corresponds to the oil sum temperature at the test bench, to which the oil was heated during the experiments. The domain _environment_ was initializedwith the ambient temperature \(\vartheta_{\text{air}}=20\)\({}^{\circ}\)C. For the boundary condition on the outer side of the domains _pressure chamber_ and _adapter/bearing housing_, the heat transfer coefficient \(\alpha_{out}=42.5\) W/(m\(\cdot\)K) was specified as the boundary transition to the ambient temperature \(\vartheta_{\text{air}}\). A subsonic opening boundary condition was defined at the system boundary of the domain _environment_. Thus, the fluid in this domain, the air, was allowed to cross the boundary surface in either direction.