## 2 Air Pressure Compensation Based on IE-FF Coupling Model
### IE-FF Coupling Modeling
To ensure that the RANS equations have a closed solution, the standard two-equation \(k\)-\(\epsilon\) turbulence model is introduced [29; 30; 31]:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}} \right)\frac{\partial k}{\partial x_{j}}\right]+G_{k}-\rho\epsilon, \tag{6}\]  
\[\frac{\partial}{\partial t}(\rho\epsilon)+\frac{\partial}{\partial x_{i}}( \rho\epsilon u_{i})=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu _{t}}{\sigma_{\epsilon}}\right)\frac{\partial\epsilon}{\partial x_{j}}\right] +\frac{C_{1\epsilon}\epsilon}{k}G_{k}-C_{2\epsilon}\rho\frac{\epsilon^{2}}{k}, \tag{7}\]  
\[G_{k}=\mu_{t}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}} {\partial x_{i}}\right)\frac{\partial u_{i}}{\partial x_{j}}, \tag{8}\]  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\epsilon}, \tag{9}\]  
where \(k\) is the turbulent kinetic energy; \(\epsilon\) is the scalar dissipation rate; \(\mu_{t}\) is the turbulent dynamic viscosity; and \(C_{1\epsilon}\), \(C_{2\epsilon}\), \(C_{\mu}\), \(\sigma_{\epsilon}\) and \(\sigma_{k}\) are constants (\(C_{1\epsilon}=1.44\), \(C_{2\epsilon}=1.92\), \(C_{\mu}=0.09\), \(\sigma_{\epsilon}=1.3\), and \(\sigma_{k}=1.0\)). _Air pressure numerical analysis of 1D elevator car internal flow field_. As the volume of the elevator car is relatively small compared to the hoistway space, the air pressure fluctuation inside the car in the space domain is much smaller than the change in the time domain during the high-speed operation of the elevator. Therefore, when modeling the internal flow field of the car, the ventilation system and the inner space of the car are simplified into a one-dimensional pipe network flow model. In order to simplify the modeling and calculation complexity, the node method is used to simulate and solve the internal flow field of the car. Each element in the ventilation system can be regarded as a series of typical pipe components, and the solution follows the pressure-flow relationship. Different component models are connected through nodes. The algebraic equations are used to describe each component in the pipe network system, and linear equations are established to solve the air flow and pressure at different nodes. The solution of the one-dimensional pipe network system is mainly based on the pressure loss equation and the flow control equation [29;32]. The pressure loss equation is expressed as:  
\[ \begin{array}{l}\Delta P=K\frac{\rho u^{2}}{2},\\ K=f_{\mu}\frac{L}{D},\end{array}  \tag{10}\]  
where \(K\) is the resistance loss coefficient of the pipe network; \(D\) and \(L\) are pipe diameter and length, respectively; and \(f_{\mu}\) is the friction coefficient of the pipe. The mass conservation equation is expressed as:  
\[Q=A_{1}u_{1}=A_{2}u_{2}, \tag{11}\]  
where \(Q\) is the volume flow rate (volume flow per unit time); \(A_{1}\) and \(A_{2}\) are the inlet and outlet cross-sectional area of the pipe, respectively; and \(u_{1}\) and \(u_{2}\) are the fluid speed at the inlet and outlet of the pipe, respectively. The energy conservation equation (Bernoulli equation) is expressed as:  
\[E=gz+\frac{P}{\rho}+\frac{u^{2}}{2}, \tag{12}\]  
where \(E\) is the total energy per unit mass flow; \(z\) is the elevation of a representative point of the cross-sectional area; and \(g\) is the gravity acceleration. The node flow equation is expressed as:  
\[\sum_{i=1}^{N}Q_{in}=Q_{n}, \tag{13}\]  
where \(Q_{in}\) is the node volume flow rate of unit \(i\) connected to node \(n\); and \(Q_{n}\) is the volume flow rate of node \(n\). **IE-FF coupling variation model. ** When the high-speed elevator is running in the hoist-way, it will cause strong airflow disturbances and drastic air pressure changes. This will directly change the flow field distribution outside the car. However, the gas is constantly exchanged between the IE-FF of the car due to the gap and the ventilation system.