## 2 Mathematical Description of the Braking Process for High-Speed Trains
### Physical Modelling of the Braking Process in High-Speed Trains  
The working principle of the ventilated disc brake is that the kinetic energy generated by the operation of the train is transformed into thermal energy by the frictional action of the axially moving brake pads and rotating brake discs, which enables the braking of high-speed trains. To analyse the variation of temperature at each position during braking of a ventilated brake disc with time and the variation over time of the heat generated in the frictional contact zone and the heat dissipated by the surface of the ventilated brake disc, the following mathematical model is developed [31]:  
\[\rho C_{P}\frac{\partial T}{\partial t}+\rho C_{\mathrm{P}}u\cdot\
abla T+ \
abla\cdot q=Q+Q_{\mathrm{ted}} \tag{4}\]  
\[q=-k\
abla T \tag{5}\]  
where \(\rho\) is the density of the brake disc; \(C_{P}\) is the specific heat capacity of the brake disc; \(u\) is the speed of the brake disc; \(\
abla\) is the differential operator; \(k\) is the heat transfer coefficient at different locations of the brake disc; \(Q\) is the heat source; \(Q_{\mathrm{ted}}\) is the thermoelastic damping, and \(q\) is the heat transferred by the ventilated disc. The surface of ventilated brake discs and brake pads make contact with air during the braking process. The friction disc surface of the ventilated brake disc is set up with the brake pad as a flat plate, creating forced convection with the outside air. The ventilated disc fan-shaped gap structure is a cylindrical bypass, creating forced convection with the outside air [32]. These are expressed as  
\[q_{0}=h(T_{\mathrm{ext}}-T) \tag{6}\]  
\[h_{1}=\left\{ \begin{array}{c}2\frac{\lambda_{\mathrm{air}}}{L}\frac{0.387 Pt^ {1/3}Re_{L}^{1/2}}{\left(1+\left(\frac{0.468}{P}\right)^{2/3}\right)^{1/4}} \left(Re_{L}\leq 5\times 10^{5}\right)\\ 2\frac{\lambda_{\mathrm{air}}}{L}Pr\left(0.037 Re_{L}^{4/5}-871\right)\left(Re _{L}\geq 5\times 10^{5}\right)\end{array} \right\} \tag{7}\]  
\[h_{2}=\frac{\lambda_{\mathrm{air}}}{D}\left(0.3+\frac{0.62 Re_{D}^{1/2}Pr^{1/3} }{\left(1+\left(\frac{0.4}{P}\right)^{2/3}\right)^{1/4}}\left(1+\left(\frac{ Re_{D}}{282000}\right)^{5/8}\right)^{4/5}\right) \tag{8}\]  
where \(q_{0}\) is the energy dissipated by convective heat transfer \(h_{1}\) is the convective heat transfer coefficient of the friction surface of the brake disc, \(h_{2}\) is the convective heat transfer coefficient of the fan-shaped gap structure, \(T_{\mathrm{ext}}\) is the external temperature, \(L\) is the difference between the outer diameter and inner diameter of the ventilated brake disc, \(D\) is the outer diameter of the fan-shaped gap structure, \(\lambda_{\mathrm{air}}\) is the air thermal conductivity, and  
Figure 2: Three heat transfer forms of the brake pad and the brake disc. \(\lambda_{\rm air}=2.67\times 10^{-2}\) W/(m\(\cdot\)K); \(Pr\) is the Prandtl number, which is the number of causeless combinations indicating the interaction of energy and momentum migration processes in the fluid, indicating the relationship between the temperature boundary layer and the flow boundary layer, \(Pr=0.701\); and \(Re_{L}\) and \(Re_{D}\) are Reynolds numbers, which can be expressed as  
\[Re_{L}=\frac{\rho_{\rm air}v_{\rm air}(t)L}{\mu} \tag{9}\]  
\[Re_{D}=\frac{\rho_{\rm air}v_{\rm air}(t)D}{\mu} \tag{10}\]  
where \(\rho_{\rm air}\) is the density of air, \(\rho_{\rm air}=1.205\) kg/m\({}^{3}\), \(\mu\) is aerodynamic viscosity, \(\mu=16\times 10^{-6}\) m\({}^{2}\)/s, and \(v_{\rm air}\) is the air flow rate. The presence of radiant heat exchange between ventilated brake discs and brake pads is expressed as [30]  
\[q_{\rm rflux}=\varepsilon v_{b}\left(T_{\rm amb}^{4}-T^{4}\right) \tag{11}\]  
where \(\varepsilon\) is surface emissivity, and \(T_{\rm amb}\) is ambient temperature. During braking, a large amount of heat is generated on the friction disc surface in a short period of time, which forms a temperature gradient inside the brake disc.