## 2 Steady-State Thermal Analysis of Internal Helical Gears Pair  
During the meshing of the gears pair, the interaction between the lubricant and frictional heat is initiated per cycle, due to the tooth root region remaining continuously immersed by the lubricant fluid. As a heat transfer medium, the lubricant can efficiently absorb and dissipate heat from the gear surface. In order to simplify the analysis, this study assumes that the heat generation and cooling caused by jet flow or immersion in the lubricant oil pool are not considered in the gear meshing process. When the gear pair reaches the thermal equilibrium state, although the meshing process will still lead to a slight fluctuation in the temperature field, the fluctuation is very small and the influence on the overall temperature field distribution is not significant. Therefore, it is reasonable to use the steady-state temperature field model to analyze the thermodynamic characteristics under various working conditions.  
Heat flux density refers to the amount of heat transferred per unit area per unit time; the computational formula can be expressed as:  
\[\Phi=-\lambda A\frac{dT}{dx} \tag{1}\]  
\[q=-\lambda\frac{dT}{dx} \tag{2}\]  
Here, \(\Phi\) represents the frictional heat flux, \(q\) denotes the heat flux density, \(\lambda\) is the thermal conductivity, and \(A\) is the cross-sectional area perpendicular to the direction of heat transfer. The negative sign indicates that heat transfer occurs in the opposite direction to the rising temperature.  
Heat transfer inherently results in the temperature gradients, the temperature field stands for the distribution of temperatures across meshing nodes at any given moment, and can be expressed as follows:  
\[T=f(x,y,z,t) \tag{3}\]  
In three-dimensional unsteady heat transfer theory, Equation (4) represents the governing partial differential equations for the temperature field. The heat balance equation can be expressed as follows:  
\[\frac{\partial T}{\partial t}=a\bigg{(}\frac{\partial^{2}T}{\partial x^{2}}+ \frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2}} \bigg{)}+\frac{\overset{\bullet}{\Phi}}{\rho c} \tag{4}\]  
where \(\rho\) denotes the medium's density, \(c\) is the specific heat capacity, \(T\) is the temperature field function, \(t\) represents the time variable, \(\lambda\) is the thermal conductivity, \(\overset{\bullet}{\Phi}\) is the intensity of the volumetric heat source, and \(\alpha\) is the thermal diffusivity \(\alpha=\lambda/\rho c\).  
By analyzing the meshing characteristics of the internal helical gear pair, it is found that the contact time between the tooth surfaces is extremely short during meshing and driving and the temperature response time can be neglected; consequently, the temperature distribution across each tooth surface can be assumed to be uniform. This simplification enables the complex temperature field of the gears pair to be represented by a single-tooth steady-state temperature model. During transmission, heat generation primarily results from the frictional work at the meshing surfaces, while heat dissipation occurs through convection from the cooling medium and heat dissipation from the gear body, reaching a dynamic balance between heat input and output. Due to the constant physical properties of the temperature field without an internal heat source, the thermal conductivity differential equation can be expressed as:\[\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{ \partial^{2}T}{\partial z^{2}}=0 \tag{5}\]