## 2 Modeling of Internal Pressure for a Single Factor
### Oxygen Supply System
#### 2.5.5 Heat Transfer from Infiltrated Air
Similar to calculating the partial pressure of oxygen inside the carriage, assuming that the state changes of the air inside the carriage follow an ideal gas polytropic process, the air inside the carriage at any infinitesimal time interval satisfies the following equation:  
\[\frac{P_{i}}{\rho_{i}^{n_{i}}}=\frac{P_{i1}}{\rho_{i1}^{n_{i}}}=\frac{P_{i2}}{ \rho_{i2}^{n_{i}}}=\text{const} \tag{37}\]  
where \(P_{i1}\) and \(P_{i2}\) represent the internal pressure at the initial and final moments of the infinitesimal time interval, respectively, and \(\rho_{i1}\) and \(\rho_{i2}\) correspond to the air density inside the carriage at these respective moments. The polytropic index \(n_{i}\) primarily depends on the compression process of the air inside the carriage. By differentiating Equation (37), it can be derived as follows:  
\[\frac{\text{d}P_{i}}{\text{d}t}=\frac{n_{i}P_{i}}{\rho_{i}}\cdot\frac{\text{d} \rho_{i}}{\text{d}t}=\frac{n_{i}RT_{i}}{M_{i}}\cdot\frac{\text{d}\rho_{i}}{ \text{d}t}=\frac{n_{i}RT_{i}}{M_{i}V}\cdot\frac{\text{d}m_{i}}{\text{d}t} \tag{38}\]  
According to the law of mass conservation, the change in the air mass inside the carriage is equal to the sum of the infiltration airflow through the sealed gaps in the carbody, the ventilation airflow through the ducts, and the oxygen-enriched air provided by the oxygen supply system, expressed as follows:  
\[\frac{\text{d}m_{i}}{\text{d}t}=q_{g}+q_{n}+q_{w}+q_{O} \tag{39}\]  
By substituting Equation (39) into Equation (38), and according to the definition of the gradient, the internal pressure fluctuation model, accounting for the multi-factor coupling effect in extreme tunnel conditions, can be derived as follows:  
\[\begin{array}{l}P_{i}(k+1)=P  
a basis. Tunnel 1 has a length of 3520 m, with an initial carriage volume of 140 m\({}^{3}\). The internal temperature of the carriage ranged from 20 \({}^{\circ}\)C to 23 \({}^{\circ}\)C, and the humidity was 14%. The train operated at a speed of 270 km/h. The transient pressure sensors used on the train surface were Kulite L250-15A sensors, produced by the American company KULITE. The full-scale test employed a sampling frequency of 1000 Hz and a filtering frequency of 100 Hz, which meets international standards. This is sufficient to capture the pressure peaks inside and outside the carriage and to ensure the complete and accurate representation of pressure transients. The data acquisition system used was the C1 acquisition system, manufactured by IMC. The statistical parameter'standard deviation of regression fit \(e_{rms}\)' was adopted as a reference standard to evaluate the degree of approximation between the simulation data and the measured data, defined as:  
\[e_{rms}(k_{c})=\sqrt{\frac{1}{N}\sum_{k=1}^{N}\left[P_{i}(k,k_{c})-P_{im}(k) \right]^{2}} \tag{43}\]  
where \(N\) is the number of data points, \(P_{i}(k,k_{c})\) is the calculated value of the internal pressure fluctuation when the pressure loss correction coefficient is \(k_{c}\), and \(P_{im}(k)\) is the measured value of the internal pressure fluctuation. Simulation calculations are performed for different values of the pressure loss correction coefficient. By comparing \(e_{rms}\) obtained from the model at various \(k_{c}\) values, the optimal pressure loss correction coefficient can be determined. Figure 5 presents the flowchart for the correction process of the internal-external pressure transmission model. Figure 6a shows the value of \(e_{rms}\) for different pressure loss correction coefficients. When the pressure loss correction coefficient is 0.70, the regression system's fit standard deviation for this condition is minimized to 20.82 Pa. At this point, the root mean square error of the internal pressure calculated by the corrected model is less than 5% of the pressure wave amplitude.