# Analytical-Numerical Model for Temperature Prediction of a Serpentine Belt Drive System
## 1 Introduction
In some studies, the heat transfer analysis of the belt drive system was considered after determining the power loss or heat generation within the belt drive system, because the belt is an important component of the system and belt failures may occur because of the high temperature. For example, [PERSON] and [PERSON] focused on the heat transfer of a synchronous belt through numerical and experimental methods [13]. A general theory related to the rubber frictional heat against solids with arbitrary thermal properties was developed [14]. [PERSON] concluded finite-element analyses, including sliding frictional simulations, applied to the rubber-like materials [15]. Furthermore, some authors focused on the temperature profile of a cylindrical disk-like rotor or brake disks, providing guidelines regarding the thermal analysis of the pulley in this case. [PERSON] and [PERSON] performed a conjugate computational fluid dynamics (CFD) analysis to accurately predict the thermal coefficient  
Figure 1: Schematic of a serpentine belt drive system. and thermal fading cycle of a rotor [16]. A 3D numerical analysis was performed to simulate the carbon nanotubes flow pasting an inclined rotating disk [17]. In addition, [PERSON] and [PERSON] focused on the heat conduction of a brake disk and provided useful suggestions to reduce the surface temperature of the disk [18]. The naphthalene sublimation technique was adopted for the experimental heat and mass transfer analysis for a rolling wheel [19]. [PERSON] and [PERSON] [20] analyzed the transient thermal behavior of a full and ventilated brake disk using a numerical approach. And an analytical and experimental analysis was conducted for a brake rotor with fins [21]. Furthermore, [PERSON], [PERSON], and [PERSON] applied the same approach to investigate the influence of a vane shape on the flow field and heat transfer characteristics of a brake disk under various configurations and speeds [22]. [PERSON] investigated the effect of cross-drilled holes on the cooling efficiency of brake disks [23]. A mesh-free method is used for the thermoelastic analysis of finite-length functionally graded cylinders reinforced by carbon nanotubes [24; 25]. The same method is also used for thermoelastic behaviors during the optimization of thermomechanical responses of nanocomposite sandwich plates reinforced by carbon nanotubes [26; 27] and nanocomposite cylinders reinforced with graphene [28]. By comparing the temperature results between standard and cross-drilled back disks, the theoretical and experimental research suggested that the heat dissipations from the surfaces of the cross-drilled holes played a prominent role in the overall heat transfer enhancement. Some research also focused on the temperature prediction of a belt drive system. In the simplest analytical method, the pulley was simplified as a basic round disk, and the Bessel function was used to calculate the temperature plot. In addition, the belt was a considered as a constant medium, and the heat exchange between the pulleys and belt was determined. However, this method was only applicable for a two-pulley-one-belt drive system [3]. In the latest research, numerical simulations were conducted, and the updated multiple reference frame (MLF) approach was utilized to calculate the surface temperature distributions of pulleys. This method was then extended to determine the temperature distribution of a two-pulley-one-belt continuously variable transmission (CVT) system [29; 30]. In particular, the existing research focused on the methods and techniques related to the power loss determination and thermal analysis of the belt drive and its components. However, only a few studies have attempted to accelerate the thermal analysis of the belt drive system. In the design and optimization process, specifically, in the design and research of the belt drive system, several pulleys are usually considered to create various belt drive layouts operating under different conditions.