## 2 Modeling of Internal Pressure for a Single Factor
### Oxygen Supply System  
The average altitude of the railways in the western mountainous regions of China is approximately 3800 m, with 90% of the tracks operating at altitudes above 3000 m [21]. The cold, low-oxygen environment necessitates that trains in these regions be equipped with oxygen supply systems to maintain adequate partial oxygen pressure inside the carriages, ensuring that the inhaled air meets the alveolar oxygen partial pressure requirements of the human body. This helps the crew and passengers maintain normal physiological functions and reduces the risk of altitude sickness. Currently, oxygen-equipped trains primarily use a combination of primary diffuse oxygen supply for the entire cabin and individual distributed oxygen supply (i.e., individual oxygen supply via oxygen masks) [23]. Diffuse oxygen supply involves using an onboard oxygen generator as the oxygen source, which directs the produced oxygen through dedicated channels into the primary air duct of the air conditioning system, where it mixes with the supply air to increase the oxygen concentration within the carriage, thereby meeting passengers' physiological needs, as illustrated in Figure 2. Existing research on oxygen supply in trains primarily employs dynamic methods and steady-state models [24] to calculate the partial oxygen pressure inside the carriage, estimating the oxygen demand of the train under open-line conditions based on altitude and exhaust volume. This paper considers the variations in exhaust and infiltration volumes in trains operating within extreme tunnels, clarifies the oxygen supply mechanisms and requirements under these conditions, and establishes a model for calculating the partial oxygen pressure inside the carriage. The coupling relationship between internal oxygen pressure, pressure, and temperature is analyzed to better address the challenges of oxygen supply in extreme environments, thereby enhancing passenger comfort and safety. Assuming the oxygen state changes inside the carriage follow an ideal gas reversible process, the state changes of oxygen as the train passes through extreme tunnels are divided into several differential segments. For each segment, the polytropic index representing the state changes of the air inside the carriage is assumed to be approximately constant. According to the ideal gas law, the oxygen state in any given differential time segment satisfies the following equation:  
\[\frac{P_{O}}{\rho_{O}^{n_{O}}}=\frac{P_{O1}}{\rho_{O1}^{n_{O}}}=\frac{P_{O2}}{ \rho_{O2}^{n_{O}}}=\text{const} \tag{15}\]where \(P_{O1}\) and \(P_{O2}\) are the partial oxygen pressure inside the carriage at the initial and final moments of the differential time segment, respectively, and \(\rho_{O1}\) and \(\rho_{O2}\) correspond to the oxygen concentration inside the carriage at the respective moments. The polytropic index \(n_{O}\) primarily depends on the compression process of oxygen during the differential time segment. Analyzing the state of the oxygen within this segment and differentiating the ideal gas law (Equation (15)) gives:  
\[\frac{\mathrm{d}P_{O}}{\mathrm{d}t}=\frac{n_{O}P_{O}}{\rho_{O}}\cdot\frac{ \mathrm{d}\rho_{O}}{\mathrm{d}t}=\frac{n_{O}RT_{i}}{M_{O}}\cdot\frac{\mathrm{d }\rho_{O}}{\mathrm{d}t}=\frac{n_{O}RT_{i}}{M_{O}V}\cdot\frac{\mathrm{d}m_{O}}{ \mathrm{d}t} \tag{16}\]  
According to the law of mass conservation, the change in the mass of oxygen inside the carriage equals the sum of the oxygen entering or leaving the carriage through the carbody's sealing gaps, the oxygen exchange through the air ducts, the oxygen supplied by the oxygen supply system, and the oxygen consumption by passengers, as expressed by:  
\[\frac{\mathrm{d}m_{O}}{\mathrm{d}t}=q_{g}x_{g}+q_{n}x_{e}+q_{w}x_{i}+q_{O}x_{ O}-N\times m_{r} \tag{17}\]  
The oxygen generator produces oxygen-enriched air with a consistent volume fraction of 40%, which corresponds to a mass fraction of \(x_{O}=0.431\)[23].