## 2 Mathematical Description of the Braking Process for High-Speed Trains  
### Characterization of the Three Heat Transfer Forms in the Braking Process  
During the braking process, the macroscopic energy change of high-speed train braking is the process of converting mechanical energy into internal energy. At the same time, the internal energy of the ventilated brake disc is constantly dissipated into the air in the form of thermal convection and thermal radiation under the effect of forced convection of the external air. This phenomenon is known as heat transfer. According to its physical nature, heat transfer [14] can be divided into three types, including thermal conduction, thermal convection, and thermal radiation. Heat transfer occurs when there is a temperature difference between the friction interface of the ventilated disc and the gap structure. The heat transfer depends on the distribution of the temperature field of the ventilated brake disc. Heat conduction is represented by [PERSON]'s law [29]:  
\[\phi_{\text{cond}}=-A_{\text{cond}}\lambda\frac{\text{d}T}{\text{d}x} \tag{1}\]  
where \(\phi_{\text{cond}}\) is the conduction heat flux and represents the heat passing over the friction interface; \(T\) is the temperature; \(\lambda\) is the thermal conductivity; \(A_{\text{cond}}\) is the contact area between the ventilated brake disc and the brake pad. \(-\lambda\frac{\text{d}T}{\text{d}x}\) is the heat flow density for heat conduction in the \(x\)-direction; \"\(-\)\" indicates that the direction of heat transfer is opposite to the direction of the temperature gradient.  
Thermal convection occurs only in a moving fluid (liquid or gas) medium. It is the action of heat transfer caused by the relative displacement of masses in a fluid and is expressed in terms of [PERSON]'s law of cooling for convective heat transfer [14]:  
\[\phi_{\text{cond}}=A_{\text{cond}}h\Delta T \tag{2}\]  
where \(\phi_{\text{conv}}\) is the amount of thermal convection; \(A_{\text{conv}}\) is the contact area between the ventilated disc and the air; \(h\) is the convective heat transfer coefficient between the ventilated disc and the air, and the greater the coefficient means, the higher the rate of heat transfer between the two media; \(\Delta T\) is the temperature difference between the ventilated disc and the air, usually taking a positive value.  
Thermal radiation is the phenomenon of an object radiating electromagnetic waves due to its temperature and is usually expressed by the Stefan-Boltzmann law [30] as follows:  
\[\phi_{\text{rad}}=\varepsilon\sigma_{b}A_{\text{rad}}\Big{(}T_{\text{disc}}^{ 4}-T_{\text{env}}^{4}\Big{)} \tag{3}\]  
where \(\phi_{\text{rad}}\) is the amount of thermal radiation, \(\varepsilon\) is the emissivity of the object, \(\sigma_{b}\) is the Stephan-Boltzmann constant, \(\sigma_{b}\) = 5.67 \(\times\) 10\({}^{-8}\) W/m\({}^{2}\) k\({}^{4}\); \(A_{\text{rad}}\) is the area exposed to thermal radiation; \(T_{\text{disc}}\) is the temperature of the ventilated brake disc; \(T_{\text{env}}\) is the ambient temperature around the ventilated brake disc.  
During braking, heat energy is generated by friction between the ventilated brake disc and the brake pad, and the heat energy is transferred along the friction surface to the brake disc and the brake pad in the form of heat conduction, as shown by the red arrow in Figure 2. Due to the high braking speed of high-speed trains during the braking process, the brake disc and brake pad will form convective heat exchange with the surrounding air, dissipating most of the heat in the ventilated brake disc into the air, as shown by the yellow arrow in Figure 2. As the braking time continues, the surface temperature of the brake disc will continue to rise. When the surface temperature of the brake disc is high, the surface forms a radiation heat exchange with the surrounding air due to thermal radiation, as shown by the blue arrow in Figure 2.