# 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
Because \(\eta_{f}\) is independent of \(r\), Equation (17) can be regarded as [PERSON]'s equation [33]. Its analytical solution \(T(r)\) provides the temperature distribution of one layer considered in this study, as follows:  
\[\left\{ \begin{array}{l}T(r)=F_{1}(r)\cdot C_{1}+F_{2}(r)\cdot C_{2}+T_{a}\\ \\ F_{1}(r)=\mathrm{e}^{-\frac{\sqrt{\mu_{p}}r}{\sqrt{W_{1}\wedge\mu_{p}}}}[PERSON] \[PERSON]{[}\frac{\sqrt{\mu_{p}}^{\sum_{i}W_{f}}\left(\frac{\left(\eta_{f}-1 \right)}{2}+\pi\sqrt{\Lambda_{p}W_{i}}\right)}{2\pi\sqrt{\Lambda_{p}W_{i}}},1, \frac{2\sqrt{\mu_{p}}^{r}}{\sqrt{W_{i}\Lambda_{p}}}\\ F_{2}(r)=\mathrm{e}^{-\frac{\sqrt{\mu_{p}}r}{\sqrt{W_{1}\wedge\mu_{p}}}}KummerM \Bigg{[}\frac{\sqrt{\mu_{p}}^{\sum_{i}W_{f}}\left(\frac{\left(\eta_{f}-1 \right)}{2}+\pi\sqrt{\Lambda_{p}W_{i}}\right)}{2\pi\sqrt{\Lambda_{p}W_{i}}},1,\frac{2\sqrt{\mu_{p}}^{r}}{\sqrt{W_{i}\Lambda_{p}}}\Bigg{]}\end{array} \right. \tag{17}\]  
where the constant coefficients \(C_{1}\) and \(C_{2}\) can be derived from the thermal analysis boundary conditions considered in this study. When \(r=R_{y}\), value of function \(F_{2}(R_{y})\) in Equation (17) will become an infinity. However, the temperature at \(R_{y}\) in real world cannot be an infinite high temperature. Therefore, \(C_{2}\) has to be set to 0. Then, \(C_{1}\) can be calculated as  
\[\left\{ \begin{array}{l}r=R_{x}\Rightarrow C_{1}=\frac{T(R_{x})-T_{a}}{F_{1 }(R_{x})}\\ r=R_{y}\Rightarrow T\Big{(}R_{y}\Big{)}=\text{limited};C_{2}=0\end{array} \right. \tag{18}\]  
The temperature distribution for one layer can be obtained as  
\[T(r)=\frac{F_{1}(r)}{F_{1}(R_{x})}[T(R_{x})-T_{a}]+T_{a} \tag{19}\]  
Figure 6: Fin geometry. Equation (19) shows that \(T(r)\) is related to the temperature at the outer radius of this layer, \(R_{x}\). The temperature distribution for a pulley can thus be calculated by using Equation (19) three times from Layers 1 to 3. The connection of any two layers exhibits the same temperature, and \(T(R_{y})\) in one layer is thus equal to \(T(R_{x})\) in the next layer. Thus, the three-layer temperature distribution of the pulley is related to the temperature at the outer radius of this pulley, \(T_{pcn}\). Subsequently, the total heat dissipated per second at the side surfaces of all the three layers (surfaces \(B_{1}\), \(B_{2}\), and \(B_{3}\) in Figure 7), \(\Phi_{ps}\), is  
\[\Phi_{ps}=K(R_{o},R_{1})+K(R_{1},R_{2})+K(R_{2},R_{i}) \tag{20}\]  
where \(R_{o}\) and \(R_{i}\) denote the outer and inner radii of the investigated pulley, and \(R_{1}\) and \(R_{2}\) denote the inner radii of Layers 1 and 2, as shown in Figure 5. The heat dissipation rate at the two side surfaces in one layer is  
\[K\big{(}R_{x},R_{y}\big{)}=2\int\limits_{R_{y}}^{R_{x}}h_{p}\Big{[}2\pi r+N_{f }W_{f}\big{(}\eta_{f}-1\big{)}\Big{]}[T(r)-T_{a}]\mathrm{d}r \tag{21}\]  
Furthermore, this algorithm includes the heat dissipated from the other surfaces, \(\Phi_{pb}\):  
\[ \begin{array}{ll}\Phi_{pb}=&(1-\alpha/2\pi)A_{0}h_{pn}[T(R_{o})-T_{a}]+A_{1} h_{pn}[T(R_{1})-T_{a}]+\\ &A_{2}h_{pn}[T(R_{2})-T_{a}]+A_{3}h_{s}[T(R_{i})-T_{a}]\end{array}  \tag{22}\]  
where \(A_{0}\), \(A_{1}\), \(A_{2}\), and \(A_{3}\) denote the surface areas, as shown in Figure 7, \(\alpha\) is the wrap angle, which is the degrees that the belt contacts the pulley, and \(h_{s}\) is the heat transfer coefficient between the shaft material and the surface at the inner diameter of the pulley (surface indicated as \(A_{3}\) in Figure 7). Therefore, the total heat dissipation rate of the investigated pulley is  
\[\Phi_{pn}=\Phi_{pb}+\Phi_{ps} \tag{23}\]  
The above mathematic algorithm can then be used to formulate the mathematical relationship between \(\Phi_{pn}\) and the input parameters of function \(f(\textbf{T}_{\textbf{pn}},\dots\) ). The number of these input parameters is more than ten, and these parameters are thus classified into two groups to enable a clearer analysis. The first group includes the pulley related parameters, which are geometry related and stored in matrix \(\textbf{D}_{\textbf{pn}}\).