# Analytical-Numerical Model for Temperature Prediction of a Serpentine Belt Drive System
## 2 Thermal Model
### Establishment of Thermal Model  
The first step in the thermal model is to determine the heat flux and locations within the pulley system during the operation. The power loss theory is used to calculate the heat generation within the system because all the power losses are converted into heat, according to the conservation of energy. A prevalent theory is considered to calculate the power losses of the belt drive systems [3], and according to the locations of these losses, two forms of energy losses exist: The first form is the belt internal power loss \(P_{h}\), which is caused by the belt bending, stretching, and shear radial compression. These phenomena generate heat at different sections of the belt. As the belt operates at high speeds, these losses are considered as a heat source uniformly distributed along the belt. The second form is the contact surface power loss \(P_{fn}\), which occurs at the \(nth\) belt-pulley contact surface. Figure 2 shows an example of these losses, in which \(P_{h}\) is uniformly distributed inside the belt, as shown by the dashed area, and \(P_{fl}\) and \({}_{fl}\) are located at the belt-pulley contact areas, highlighted by the orange surfaces. The next step is to create a thermal model by analyzing the heat flux flow behaviors through the belt drive components and dissipations to the environment at the exposed surfaces of the pulleys and the belt. A general model is first introduced in this section, followed by a description of a detailed subanalysis on each component from Sections 2.2-2.4. The first set of thermal balance equations corresponds to pulleys. No heat source exists inside pulleys, and the source of the heat dissipated on the pulley surfaces is the frictional heat and pulley-belt exchanged heat, which is transferred through the belt-pulley engaged surfaces. In this study, a variable, \(\xi_{n}\), is introduced to represent the ratio of the thermal flux flowing into a pulley to the frictional heat flux between the \(nth\) pulley and the belt, as shown in Equation (1)  
\[\xi_{n}P_{fn}=\Phi_{pn} \tag{1}\]  
Figure 2: Two-pulley, one-belt system. In addition, the heat dissipation at the pulley surfaces is strongly related to the temperature profiles, ambient temperature \(T_{a}\), geometry dimensions, rotational speed, material thermal conductivity, and other parameters. Hence, Equation (1) is transformed to  
\[\xi_{n}P_{fn}=f(\mathbf{T_{pn}},\mathbf{D_{pn}},\omega_{pn},\Lambda_{pn},T_{a},\cdots) \tag{2}\]  
where the temperature plot of the \(n\)th pulley is represented by the matrix \(\mathbf{T_{pn}}\), the pulley geometry dimensions are stored in the matrix \(\mathbf{D_{pn}}\), the rotational speed of \(n\)th pulley can be represented as \(\omega_{pn}\), and the material thermal conductivity of \(n\)th pulley is \(\Lambda_{pn}\). The second set of thermal balance equations is related to the heat exchange at the belt-pulley engaged surface. The heat flux flowing into the pulley is influenced by both the frictional heat and heat exchange at the belt-pulley engaged surface. The temperature difference between the belt and pulley causes a certain amount of heat exchange. This condition can be represented as in the following simple equation:  
\[\xi_{n}P_{fn}=g(\mathbf{T_{pn}},T_{b},P_{fn}) \tag{3}\]  
where the contact temperature on the belt side is \(T_{b}\), and the contact temperature on the pulley side is \(T_{pn}\), which belongs to the matrix \(\mathbf{T_{pn}}\). The third thermal balance equation corresponds to the heat analysis of the belt, which is in contact with all the pulleys. The heat dissipated from the belt surfaces is equal to the total heat generation minus the heat dissipated on the pulley surfaces.