# Analytical-Numerical Model for Temperature Prediction of a Serpentine Belt Drive System
## 2 Thermal Model
### Pulley Internal Thermal Analysis
#### 2.4.6 Modeling the Heat Transfer  
Establishing the settings for the heat transfer in this numerical simulation is challenging, as this setting requires an accurate turbulence model to not only simulate the airflow caused by the pulley rotation but also calculate the heat dissipation rate on the pulley surfaces. The model used in this numerical simulation is the Menter Shear Stress Transport (SST) \(k\)-\(\omega\) turbulence model, which belongs to the Reynolds Averaged Navier-Stokes (RANS) regime. This two-equation eddy-viscosity model is ideal for this analysis because of its good near wall treatment, acceptable stability and reasonable computational time. The fluid medium is air, and the temperature must be below the melting temperature of FRPs which is 140 \({}^{\circ}\)C. The maximum rotational speed of the pulley is 7000 RPM, and the maximum diameter is 108 mm, resulting in subsonic analysis within the fluid domain.  
Figure 9: Polyhedral mesh for the numerical analysis.  
Hence, the density of air is computed by using the incompressible ideal gas equation and its change is negligible. The finite volume method used in this analysis solves the governing equations in integral form, as shown below [31]:  
\[\frac{\partial}{\partial t}\int_{V}^{j}WdV+\oint[J-K]\cdot da=\int_{V}^{j}HdV \tag{26}\]  
with  
\[\mathbf{W}=\left[\begin{array}{c}\rho\\ \rho\mathbf{v}\\ \rho E\\ \end{array}\right];\mathbf{J}=\left[\begin{array}{c}\rho(\mathbf{v}-\mathbf{v }_{g})\\ \rho(\mathbf{v}-\mathbf{v}_{g})\otimes\mathbf{v}+\rho\mathbf{I}\\ \rho(\mathbf{v}-\mathbf{v}_{g})\mathbf{H}+p\mathbf{v}_{g}\\ \end{array}\right]\]  
where \(\mathbf{W}\) is the vector of conserved quantities, which are mass, momentum and energy. \(\mathbf{J}\) represents the inviscid terms. The vector of viscous terms is given by \(\mathbf{K}\). The vector \(\mathbf{H}\) is the body forces. In these vectors, \(\rho\) denotes the density of air, \(\mathbf{v}\) is the vector of the air velocity, \(\upsilon\mathbf{R}_{g}\)R is the velocity vector of the grid, \(E\) represents the total energy per unit mass, \(\mathbf{p}\) is the pressure, and \(\dot{q}^{\prime\prime}\) is the occurring heat flux. The additional SST \(k\)-\(\omega\) model transport equations, which are turbulence kinetic energy and specific dissipation rate equations, are also necessary [31]:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial\chi}(\rho k \mathbf{v})=\frac{\partial}{\partial\chi}\bigg{(}\Gamma_{k}\frac{\partial k}{ \partial\chi}\bigg{)}+G_{k}-Y_{k} \tag{27}\]  
\[\frac{\partial}{\partial t}(\rho\omega)+\frac{\partial}{\partial\chi}(\rho \omega\mathbf{v})=\frac{\partial y}{\partial\chi}\bigg{(}\Gamma_{\omega} \frac{\partial\omega}{\partial\chi}\bigg{)}+G_{\omega}-Y_{\omega}+D_{\omega} \tag{28}\]  
where \(\Gamma_{k}\) and \(\Gamma_{\omega}\) is the effective diffusivity of k and \(\omega\). \(G_{k}\) and \(G_{\omega}\) represents the kinetic energy of \(k\) and \(\omega\). The dissipation of \(k\) and \(\omega\) resulting from turbulence is given by \(Y_{k}\) and \(Y_{\omega}\). The extra term, \(D_{\omega}\), is a blend function which combines the \(k\)-\(\omega\) model for the robust near wall treatment and k-\(\epsilon\) model for the free stream in the far field. The constants in these equations have been remained as the default.  
Moreover, a second-order upwind scheme is used to realize the spatial discretization of the governing equations. In addition, pseudo transient, a type of implicit under-relaxation for steady-state cases, is activated for better stabilizing the case and ensuring rapid convergence. In terms of the analysis parameters, the initial temperature is the ambient temperature. The thermal conductivity for the pulley material is measured according to ISO 22007-2 to accurately simulate the heat conduction.