# Analytical-Numerical Model for Temperature Prediction of a Serpentine Belt Drive System
## 4 Results and Discussion
### Verification of \(S_{n}(\omega_{pn})\)  
Obtaining an accurate curve for \(S_{n}(\omega_{pn})\) is critical during the calculation of the thermal model. According to the definition of \(\lambda_{pn}\), the numerical and measured results of the heat dissipation flux of pulleys must be compared under the identical the temperatures at the contact area \(T_{pcn}\) and rotating speed \(\omega_{pn}\). However, no device can measure the heat dissipation of the pulley at its high rotational speeds. Hence, it is reasonable to compare the calculated and measured results of the temperature distribution of the pulley, because the temperature drop along the radial direction is strongly linked to  
Figure 12: Experimental installation of belt drive system based on designed layout.  
the heat dissipation rate on pulley surfaces. Figure 13 presents that the temperature distributions of Layer 2 for the pulleys from the numerical model approach and compare this with the experimental temperature data at five different points along the radial direction under the condition of \(T_{pcn}\) = 55 \({}^{\circ}\)C and rotating speed \(\omega_{pn}\) = 4000 RPM. In this diagram, the calculated results have a good agreement with the measured temperatures, indicating that the heat dissipation for a pulley is reliable.  
Moreover, to ensure that the heat dissipation of the pulley is reasonable, the simulation result of the airflow around the pulley is examined. Because the heat dissipation coefficient \(h_{p}\) is strongly related to the local airspeed, realizing an accurate airflow simulation is critical to calculate the heat dissipation of the pulley. Figure 14 shows the numerical results of the air flows surrounding the pulley during a rotation speed of 4000 RPM. The adopted SST k-\(\omega\) model uses a blending function to integrate the Wilcox k-\(\omega\) and the k-\(\varepsilon\) models. The k-\(\omega\) model is ideal for the flow simulation in the viscous sub-layer, and the graph exhibits the high velocity air flow near the pulley surfaces. The k-\(\varepsilon\) models is well suited for free flow in regions away from the wall, and the graph exhibits the vertex flow above the surfaces at the pulley outer radius. Moreover, this turbulent model used in this simulation is time-efficient. In particular, the CPU time for each parametric study is approximately 2-3 h on a desktop with a 12-core 3.4 GHz CPU and 64 GB memory. Therefore, the turbulent model selected in the numerical method can ensure sufficient accuracy of the turbulent flow. The curve for \(S_{n}(\omega_{pn})\) for different rotation speeds is eligible for the subsequent thermal model calculation.  
Figure 15 shows the curves of \(S_{n}(\omega_{pn})\) for three different pulleys under different speeds. This figure also provides some guidelines regarding the temperature reduction and reliability improvement of the pulley. The pulley with a diameter of 108 mm has a higher \(\lambda_{p}\) than pulleys with a diameter of 76.2 mm. The larger diameter of the pulley can lead to an increase in \(\lambda_{p}\) because of the larger heat dissipation area. The steel pulley has a considerably higher \(\lambda_{p}\) than that of the FRP pulleys. In other words, the high thermal conductivity of the pulley material can increase \(\lambda_{p}\) efficiently because the pulley becomes conductive in terms of the thermal flow, thereby reducing the thermal concentration.  
Figure 13: Comparisons of calculated (Calc.) and experimental (Exp.) temperature distributions of Layer 2 for the DN pulley (76.2 mm) under a rotation speed of 4000 RPM and \(T_{pcn}\) = 55 \({}^{\circ}\)C.