# Multiphase Conjugate Heat Transfer Analyses on the Assembly Situation of Rotary Shaft Seals
## 3 Results
### Temperature Contour Plot  
Figure 13 shows the temperature distribution in the cross-section of the reference variant for the rotational shaft speeds \(n=2000\) rev/min and \(n=8000\) rev/min. Due to the heat source applied as a boundary condition in the contact between the sealing ring and the shaft, the maximum temperature occurred in the sealing contact. Since this heat source was greater for higher shaft speeds, as demonstrated by Equation (17), the temperature was higher the higher the shaft speed was. Since the shaft made of steel conducts heated better than the seal made of elastomer, the generated heat was not evenly divided between the sealing ring and the shaft. A larger part of the heat generated in the sealing gap was applied to the shaft, and only a small part to the sealing ring; see  
Figure 12: Streamlines of the oil in the domain _oil chamber_ for \(n=2000\) rev/min, reference variant. (**a**) Streamlines of oil, with sectional view of the model. (**b**) Streamlines of oil, top view. (**c**) velocity scale. Figure 13: Contour plot, reference variant for (**a**) \(n=2000\) rev/min; (**b**) \(n=8000\) rev/min. Equations (19)-(21). Due to these material properties, the heat generated in the sealing contact was dissipated more towards the shaft than towards the seal. The rational function describing the frictional torque \(M_{\text{fric}}\), calculated using Equation (18), is used not only for the frictional torque of the sealing ring but also for the frictional torque of the bearings. The frictional torque of the bearings and, thus, the heat source caused by friction at the bearings, calculated using Equation (17), was different for each variant with bearings. Since the geometry of the cylindrical roller bearing in combination with the ball bearing was not symmetrical, as shown in Figure 6d, the frictional torque of the bearings was divided into one part generated by the ball bearing near the sealing ring and one part generated by the bearing distant from the seal. Thus, for the variants with bearings, eight additional parameters, four for each bearing, had to be defined to determine the frictional torques of the two bearings according to the rational function in Equation (18). By multiplying the frictional torque with the rotational shaft speed, the frictional heat \(Q_{\text{fric}}\) was obtained, listed in Table 4. In addition, the four variants with bearings differed in the fill level of the oil \(h_{\text{oil}}\) in the domain _oil chamber bearing_, as shown in Figure 14. The input parameter oil fill level \(h_{\text{oil}}\) is also given in Table 4.  
The four parameters for determining the frictional torque of the sealing ring, calculated using Equation (18), were assumed to be identical for all variants, regardless of whether additional elements were simulated or not. Thus, the frictional heat generated in the contact area between the sealing ring and shaft surface was the same for all variants. This frictional heat generated was subdivided into the two parts \(\dot{Q}_{\text{fric seal}}\) and \(\dot{Q}_{\text{fric shaft}}\), shown in Equations (20) and (21). The temperature given in the last row in Table 4 is the temperature obtained from the simulation. This is the maximum temperature occurring on an interface between the rotary shaft seal (RSS) and the environment. For the tapered roller bearing in the X-arrangement, there was only air in the volume between the bearing and the sealing ring, and the volume fraction of the oil is zero, as shown in Figure 14a. For the O-arrangement of the tapered roller bearing, the volume was completely filled with oil, as shown in Figure 14c, and in the case of the other two