## 2 Steady-State Thermal Analysis of Internal Helical Gears Pair
### Boundary Condition Analysis of Steady-State Temperature Field  
During the meshing of an internal helical gears pair, the transmission efficiency of the gears is determined by the heat generated from friction. The temperature gradient gradually decreases with the heat transferred from the tooth surface to the depth of the gear profile. Once the gear reaches thermal equilibrium, its temperature distribution may be stable. Given that the frictional heat generated by each tooth remains constant over one revolution, the temperature distribution of the entire gear can be inferred by analyzing the temperature of a single tooth. The digital tooth of the helical gear is demonstrated in Figure 1.  
Based on the results from the tooth profile calculation in Figure 1, the matching of working conditions with boundary conditions can be analyzed, and the tooth profile can be categorized according to its operating conditions as follows:  
In the tooth contact region, the M-zone, as the primary meshing surface, generates significant thermal effects due to friction, which satisfy the superposition of the second and third types of boundary conditions. The heat flow density \(q\) reflects the energy transfer characteristics of the meshing process, indicating the dynamic balance between heat flow input and output, as well as heat transfer with the surrounding environment. Therefore, the boundary condition is expressed as follows:  
\[-\lambda(\frac{\partial T}{\partial t})=h_{m}(T-T_{0})-q \tag{6}\]  
During the heat transfer analysis, the tooth root, tooth top, and non-meshing working tooth surfaces are referred to as the T-zone, transferring heat through the lubricating medium via convective heat transfer, except for the meshing region. The meshing surface satisfies the third type of boundary conditions, and the corresponding mathematical expression is expressed as follows:  
\[-\lambda(\frac{\partial T}{\partial n})=h_{t}(T-T_{0}) \tag{7}\]  
The convective heat exchange occurs between the gear surface such as end faces, the cross-section of the gear tooth, and shaft bore regions, and cooling media such as lubricating oil and air. These surfaces also satisfy Type III boundary conditions.  
Figure 1: Boundary diagram of each surface of the temperature field of the gear tooth.  
End face of gear tooth: S zone  
\[-\lambda(\frac{\partial T}{\partial n})=h_{s}(T-T_{0}) \tag{8}\]  
Bore section of gear shaft: D zone  
\[\frac{\partial T}{\partial n}=0 \tag{9}\]  
Cross-section of gear tooth: P, Q zone  
Heat transfer also occurs within the gear cross-section, and it can be assumed that the heat transferred is uniform across both surfaces.  
\[T\big{|}_{p}=T\big{|}_{q} \tag{10}\]  
\[\frac{\partial T}{\partial n}\bigg{|}_{p}=\frac{\partial T}{\partial n}\bigg{|}_ {q} \tag{11}\]  
The above includes: \(\lambda\)-Thermal conductivity of the gear material, W/(m\(\cdot\)K); \(h_{m},h_{t},h_{s}\)-heat transfer coefficient at each working zone, W/(m\(\cdot\)K); \(T_{0}\)-ambient temperature, \({}^{\circ}\)C; \(T\)-gear body temperature, \({}^{\circ}\)C; \(n\)-external normal direction of the heat exchange surface.