## 2 Vortex Tooth Temperature Distribution Model
### Steady-State Temperature Function of the Wall Surface of the Vortex Tooth  
Under conditions of operational stability, the temperature distribution across the scroll teeth undergoes discernible fluctuations due to cyclical interaction with the working gas, showing a periodic variation. At present, in a stable operating state for the scroll compressor, where the rotational speed of the main shaft is increased, the pace of temperature oscillations also accelerates. Accordingly, the operating dynamics of the scroll compressor can be conceptualized as an adiabatic process, a concept expressible by the following Equation (1):  
\[T=T_{s}\bigg{(}\frac{V_{s}}{V(\theta)}\bigg{)}^{K-1} \tag{1}\]  
where \(T\) is the temperature of the gas in the compression chamber when the crankshaft angle is \(\theta\), \(T_{S}\) is the suction temperature, \(V_{s}\) is the suction chamber volume, \(V_{(\theta)}\) is the volume of the compressor at crankshaft angle \(\theta\), and \(K\) is the adiabatic coefficient of the fluid medium. The fluid medium chosen in this paper is ideal air, with a value of 1.4. The volume of the compression chamber at a given crankshaft rotation angle can be determined using Equation (2) [17]. \[V_{(\theta)}=\pi hR_{b}^{2}(\pi-2\pi)(2\phi_{e}-3\pi-2\theta) \tag{2}\]  
where \(h\) is the tooth height of the scroll tooth, \(\alpha\) is the volute profile involute generation angle, \(R_{b}\) is the radius of the base circle, and \(\phi_{e}\) is the maximum expansion angle of the profile. The relationship between the temperature variation in the flow field on the inner and outer wall surfaces of the vortex teeth with respect to the crankshaft angle can be expressed by Equations (3) and (4), derived from Equations (1) and (2) as follows:  
\[T_{gas,in}(\theta)= \begin{cases}\left(\frac{V_{s}}{V_{(\theta)}}\right)^{K-1} \cdot T_{S},&\theta\in(0,\phi_{e}-\theta^{*})\\ T_{d},&\theta\in(\phi_{e}-\theta^{*},\phi_{e})\end{cases}  \tag{3}\]  
\[T_{gas,out}(\theta)= \begin{cases}T_{s},&\theta\in 0,\pi)\\ \left(\frac{V_{s}}{V_{(\varphi)}}\right)^{K-1}\cdot T_{S},&\theta\in\pi,\phi_{ e}-\theta^{*}+\pi)\\ T_{d},&\theta\in[\phi_{e}-\theta^{*}+\pi,\phi_{e}]\end{cases}  \tag{4}\]  
The microelement positioned at the profile spreading angle \(\varphi\) of the vortex profile serves as the focal point of reference, as illustrated in Figure 1. Upon locating the microelement at the inner tangent point of the compression cavity, it is designated as the starting point. During the operational phase of the vortex disc, the microelement is configured to complete one cycle at a value of \(2\pi\). Based on the specified conditions, it can be inferred that the microelement within a single working cycle, denoted as \(2\pi\), exhibits a mean temperature function across the flow field on the inner wall of the vortex tooth. This function is represented by Equation (5):  
\[\overline{T}_{gas,in}(\varphi)=\frac{1}{2\pi}\int_{\phi_{e}-\varphi}^{\phi_{e }-\varphi+2\pi}T_{gas,in}(\theta)d\theta \tag{5}\]  
The boundary conditions for the outer wall of the vortex tooth exhibit a phase lag in relation to the alteration of boundary conditions on the inner side. Therefore, the average temperature of the outer gas over a single operating cycle can be mathematically represented as Equation (6):  
\[\overline{T}_{gas,out}(\varphi)=\frac{1}{2\pi}\int_{\phi_{e}-\varphi-\pi}^{ \phi_{e}-\varphi+\pi}T_{gas,out}(\theta)d\theta \tag{6}\]  
By examining Equations (5) and (6) above, we can determine the mean temperature function governing the flow field across various profile spreading angles along both the inner and outer surfaces of the vortex tooth profile. This derivation provides insight into the thermal characteristics of the flow in the context under consideration, expressed in the following Equations (7) and (8). The profile spreading angle is the relationship between radial distance and angle change when defining a vortex-type curve in a polar coordinate  
Figure 1: Schematic diagram of scroll tooth heat transfer.