# Analytical-Numerical Model for Temperature Prediction of a Serpentine Belt Drive System
## 2 Thermal Model
### Pulley Internal Thermal Analysis
#### 2.4.1 Pulley Mathematical Algorithm  
To formulate a mathematical algorithm for a complex pulley, complex features such as grooved surfaces and some unimportant fillets are suppressed. Figure 4 shows the geometry transition from a general pulley to a simplified pulley. The mathematical model is established on the main round I-structure and the fins are later considered. Figure 4: Simplification of the pulley structure. The main round I-structure can be divided into three layers (Figure 5) to better analyze the thermal conduction inside the pulley. In each layer, the cross section is considered as a rectangular shape. The temperature distribution can be described using the Bessel function:  
\[\frac{\text{d}^{2}T}{\text{d}r^{2}}+\frac{1}{r}\frac{\text{d}T}{\text{d}r}- \frac{h_{p}}{W_{i}\Lambda_{p}}(T-T_{a})=0 \tag{11}\]  
where \(T(r)\) is the temperature distribution along the radial direction of the pulley. However, Layer 2 includes fins (Figure 6). In an infinitesimal element \(dr\) in Layer 2, the fins amplify the heat conduction and dissipation rate on the areas connected to the fins (\(A_{\text{f}}\)Rin Figure 6) compared to those in the areas not connected to the fins (\(A_{n}\) in Figure 6). A new coefficient \(\eta_{f}\) is used to represent the ratio of the heat dissipation rate on surfaces \(A_{f}\) has been increased \(\eta_{f}\) times by fins comparing with the original rate. If no fins exist, the dissipation rates on \(A_{f}\) and \(A_{n}\) are equivalent, and the dissipation rate on one side of this infinitesimal element \(dr\) is \(2\pi rdr\)-\(h_{p}\). However, when the fins are considered in the analysis, the dissipation rate on one side changes to [\(N_{f}W_{f}dr\)-\(\eta_{f}h_{p}\) +(2\pi rdr\)-\(N_{f}W_{f}dr)\)-\(h_{p}\)]. Therefore, the heat dissipation rate is amplified by a factor of [\(1+N_{f}W_{f}(\eta_{f}-1)/2\pi r\)]. It also can be considered that \(h_{p}\) on the side surfaces of Layer 2 is increased by this ratio, and Equation (11) can be modified as  
\[\frac{\text{d}^{2}T}{\text{d}r^{2}}+\frac{1}{r}\frac{\text{d}T}{\text{d}r}- \frac{\Big{[}2\pi r+N_{f}W_{f}\Big{(}\eta_{f}-1\Big{)}\Big{]}h_{p}}{2\pi rW_{i }\Lambda_{p}}(T-T_{a})=0 \tag{12}\]  
The value of \(\eta_{f}\) must be calculated based on the fin geometry during the solution of Equation (11). The fin section in a rounded infinitesimal element \(dr\) in Layer 2 is considered as a rectangle beam. The heat dissipation on the exposed surfaces of this beam is equal to the heat flux through the fin connected area \(A_{f}\). The general thermal balance equation for this long beam is [32]  
\[\frac{\partial^{2}G}{\partial x^{2}}=-\frac{2h_{p}(G-T_{a})}{\Lambda_{p}W_{f}} \tag{13}\]  
where \(G(x)\) is the temperature distribution of the beam along the pulley axial direction. \(G(0)\) is the temperature at \(A_{f}\), which is equal to \(T(r)\), the temperature at radius \(r\) on the main I-structure. Hence, the total heat dissipation for each fin can be calculated by integrating the surface temperature. \[\Phi_{f}=\Bigg{\{}2\int\limits_{0}^{L_{f}}G(x)\text{d}x+W_{f}\cdot[T(r)-T_{a} ]\Bigg{\}}H_{f}h_{p} \tag{14}\]  
Figure 5: 11 TDivided three sections of the pulley structure. The first and second parts of this equation correspond to the heat dissipations on the two sides and the far-end side (\(A_{e}\) in Figure 6), respectively. According to the definition,  
\[\eta_{f}=\frac{\Phi_{f}}{h_{p}H_{f}W_{f}[T(r)-T_{a}]} \tag{15}\]  
After the integration of \(G(x)\), which is a solution of an ODE, \(\eta_{f}\) at the radius \(r\) can be calculated as  
\[\left\{ \begin{array}{l}\eta_{f}=\frac{2\sinh\left(mL_{f}\right)+mW_{f}}{mW_{ f}\cosh\left(mL_{f}\right)}\\ m=\sqrt{h_{p}/\Lambda_{p}W_{f}}\end{array} \right. \tag{16}\]  
\(\eta_{f}\) can be considered as a constant, because Equation (16) shows that it is only a function of the fin geometry (\(L_{f}\), \(W_{f}\)), thermal conductivity (\(\Lambda_{p}\)) and heat transfer coefficient (\(l_{p}\)).