## 2 Modeling of Internal Pressure for a Single Factor
### Airtight Gaps  
The carbody contains airtight gaps at locations such as weld seams, doors, windows, pipelines, and windshields. From a macroscopic perspective, these gaps are modeled as a single equivalent leakage orifice [17]. Using static airtightness tests, after the assembly of the train is completed, all doors, windows, and ventilation ducts are sealed. Air is then either pumped into or evacuated from the train's interior, generating a pressure difference of at least 4 kPa between the inside and outside of the train. Once this differential is reached, air pumping or evacuation is halted, allowing the internal air to naturally leak until the pressure inside equalizes with the external pressure. Figure 3 presents the on-site static airtightness test. The testing system primarily consists of Endevco 8515C piezoresistive pressure sensors, an IMC data acquisition unit, a testing computer, and connecting cables, with a data sampling rate of 1000 Hz. In this process, the pressure variations inside the carbody are recorded and subsequently subjected to low-pass filtering with a cutoff frequency of 100 Hz. The resulting pressure variation curve within the train is depicted in Figure 4a. Figure 4b depicts the relationship between the equivalent leakage area, calculated through static airtightness tests, and the internal-external pressure differential. As shown in the figure, the change in the gaps of the carbody under negative pressure differentials is slower compared to positive pressure differentials, indicating that the train demonstrates better airtightness when the internal pressure is lower than the external pressure. When the absolute value of the pressure differential between the interior and exterior is within 0 to 0.5 kPa, the equivalent leakage area of the train body increases exponentially. For pressure differentials ranging from \(-4\) to \(-0.5\) kPa and from 0.5 to 3.2 kPa, the equivalent leakage area exhibits an approximately linear change. However, when the pressure differential is within the range of 3.2 to 4 kPa, the equivalent leakage area shows an exponential change. Figure 4: Test results: (**a**) Measured internal and external pressure data; (**b**) equivalent leakage area under different internal and external pressure differentials. Figure 3: On-site static airtightness test. The relationship between the internal-external pressure differential, the equivalent leakage area, and the gas leakage rate of the components is then established. Assuming a positive mass flow rate into the carriage and a negative mass flow rate out of the carriage, when \(P_{i}>P_{e}\), the velocity at the inlet of the leakage orifice is zero, while the air velocity at the outlet is \(v_{e}\). Based on the momentum equation, the following can be derived:  
\[\int_{P_{i}}^{P_{e}}\frac{dP}{\rho}+\int_{0}^{v_{e}}vdv=0 \tag{5}\]  
During the process of air leakage from the carriage, heat exchange occurs, so the leakage process is assumed to be a polytropic process. Therefore,  
\[\frac{P}{\rho^{n_{g}}}=\frac{P_{i}}{\rho_{i}^{n_{g}}}=\frac{P_{e}}{\rho_{e}^{n _{g}}}=\text{const} \tag{6}\]  
where \(\rho_{i}\) is the air density inside the carriage, \(\rho_{e}\) is the air density outside the carriage, and \(n_{g}\) represents the change in the compression state during the air leakage process. Since heat exchange occurs during the air leakage, the process is assumed to be a polytropic process.