# Analytical-Numerical Model for Temperature Prediction of a Serpentine Belt Drive System
## 2 Thermal Model
### Pulley Internal Thermal Analysis
#### 2.4.8 Parametric Setting  
According to the definition of \(\lambda_{pn}\), the change of \(T_{pcn}\) and \(\omega_{pn}\) influences the value of \(\lambda_{pn}\). Figure 11a presents the comparison of \(\lambda_{pn}\) obtained from the numerical and mathematical methods for a pulley with a radius of 76.2 mm under various contacted temperatures of pulley \(T_{pcn}\) at the speed of 6000 RPM. The diagram demonstrates that the numerical approach, which is the same as the mathematical approach, can prove that \(\lambda_{pn}\) is irrelevant with \(T_{pcn}\). Hence, the following is to investigate the only impact of \(\omega_{pn}\) on \(\lambda_{pn}\).  
Figure 11: Comparison of numerical and mathematical results on \(\lambda_{pn}\) for a pulley with a radius of 76.2 mm under (**a**) various contacted temperatures of pulley \(T_{pcn}\) at the speed of 6000 RPM (**b**) various rotational speeds ranging from 2000–6000 RPM at the \(T_{pcn}\) = 60 \({}^{\circ}\)C.  
Figure 10: Boundary conditions for the numerical analysis.  
This numerical simulation formulates the function \(S_{n}(\omega_{pn})\) or the relationship between \(\lambda_{pn}\) and \(\omega_{pn}\) through parametric studies in which \(\omega_{pn}\) is increased in intervals of 1000 RPM. In each condition of \(\omega_{pn}\), the heat generation is input into the numerical model, and the output includes \(\Phi^{\prime}_{pn}\) and \(T^{\prime}_{pn}\). Hence, a curve of \(S_{n}(\omega\text{R}_{pn})\) is plotted to describe this relationship. Figure 11 presents the comparison of \(\lambda_{pn}\) obtained from the numerical and mathematical methods for a pulley with a radius of 76.2 mm under various rotational speeds ranging from 2000-6000 RPM at \(T_{pcn}\) is 50 \({}^{\circ}\)C. The \(\lambda_{pn}\) values obtained using both the methods increase with the increase in \(\omega_{pn}\), indicating that the heat dissipation on the pulley surfaces increases at higher speeds and then improving the heat dissipation ability of a pulley.  
Meanwhile, the \(\lambda_{pn}\) values obtained using the numerical algorithm are generally higher than those from the mathematical method because in the case of the mathematical algorithm, the grooved feature at the pulley outer surface was simplified to a flat surface (Figure 4), thus reducing the surface area A\({}_{0}\) (Figure 7) and further decreasing the surface dissipation \(\Phi_{pn}\) through Equation (22) to cause \(\lambda_{pn}\) having a lower value than in the numerical approach. The heat dissipation from the surface area A\({}_{0}\) (The first right term in Equation 22) is the primary contribution for \(\Phi_{pb}\) because \(T(R_{0})\) has the highest temperature within the pulley. Plus, the mathematical approach also does not consider the influence of the severity of turbulence with increased speeds, making the curve of \(S_{n}(\omega\text{R}_{pn})\) from the mathematical approach less tilted than that from the numerical results. Therefore, the \(S_{n}(\omega\text{R}_{pn})\) from the numerical approach, instead of the mathematical approach, is adopted in this study.  
Therefore, function \(f(\mathbf{T_{pn}},\ \dots\ )\) can be solved by using this numerical simulation. \(S_{n}(\omega_{pn})\) is determined by using the thermal balance equation, Equation (24), which is integrated into the thermal model. A set of values \(\{S_{n}(\omega_{pn})\), \(\mathbf{T^{\prime}_{pn}}\Phi^{\prime}_{pn}\}\) is created for each pulley rotating at speeds in increments of 1000 RPM, and a database is established for all the available pulleys. The model can then import \(S_{n}(\omega\text{R}_{pn})\) from this database based on the pulley and designed \(\omega_{pn}\). Moreover, \(\mathbf{T^{\prime}_{pn}}\) and \(\Phi^{\prime}_{pn}\) are stored for subsequent pulley thermal distribution calculations.