## 3 Modeling and Simulation
### Mathematical Model  
To simplify the model, according to [PERSON] et al. [41], the following assumptions were made:  
1. The oxygen supply vent sends out oxygen-enriched gas in the form of a jet, which is mainly a binary ideal gas mixture of O\({}_{2}\) and N\({}_{2}\), and the oxygen volume fraction is 99%.
2. The oxygen supply vent has a small diameter and limited diffusion effect in a closed room after being sent out as a jet; therefore, the free jet theory can be used to describe this flow accordingly.
3. The fluid is considered incompressible and steady, and the turbulence model uses a dual equation model [42]. The governing equation can be expressed as follows: The continuity equation is expressed as \[\frac{\partial u_{i}}{\partial x_{i}}=0\] (1) Momentum balance is expressed as \[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial(\rho u_{i}u_{j})}{ \partial x_{j}}=\frac{\partial}{\partial x_{j}}\bigg{[}(\mu+\mu_{t})\bigg{(} \frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{i}}{\partial x_{i}} \bigg{)}\bigg{]}-\frac{\partial p}{\partial x_{i}}+S_{i}\] (2)  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter (m)** & **Values** \\ \hline Diameter of air supply and return vent & 0.15/0.15 \\ Distance between the center of the air supply and return vent to the ground & 2.65/0.25 \\ Center distance between two air supply and return vents & 0.49/0.49 \\ Diameter of oxygen supply vent & 0.006/0.01 \\ Table surface and legs (length \(\times\) width \(\times\) height) & \(0.5\times 0.72\times 0.05/0.03\times 0.03\times 0.5\) \\ Chair surface and back (length \(\times\) width \(\times\) height) & \(0.57\times 0.42\times 1.1/0.15\times 0.42\times 1.1\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Structural parameters of the artificial climate chamber.  
Figure 2: Physical model of the artificial climate chamber (air-conditioned, with single oxygen supply vent). (**a**) Up-supply and down-return on the opposite side, (**b**) up-supply and down-return on the same side, (**c**) side view of the model.  
The component conservation equation is given by  
\[\frac{\partial c_{A}}{\partial t}+\frac{\partial\big{(}u_{j}c_{A}\big{)}}{ \partial x_{j}}=\frac{1}{\rho}(D_{AB}+D_{T})\frac{\partial}{\partial x_{j}} \Bigg{(}\frac{\partial c_{A}}{\partial x_{j}}\Bigg{)} \tag{3}\]  
[PERSON] et al. [43] simulated the characteristics of a two-phase free jet formed at the exit of a long vertical tube and believed that the \(k-\varepsilon\) model was in good agreement with the experimental results. [PERSON] et al. [44] used standard \(k-\varepsilon\) and \(k-w\) models to simulate the characteristics of the free jet formed by rectangular nozzles and believed that the standard \(k-\varepsilon\) can effectively predict the jet characteristics. [PERSON] et al. [33] used different turbulence models to study the flow characteristics of circular jets and believed that the standard \(k-\varepsilon\) model could obtain more reasonable results. Therefore, the turbulence model in this paper uses the standard \(k-\varepsilon\) model.  
The \(k-\varepsilon\) model takes the following form:  
\[\rho\frac{\partial k}{\partial t}+\rho\frac{\partial(ku_{i})}{\partial x_{i} }=\frac{\partial}{\partial x_{j}}\Bigg{[}\Bigg{(}\frac{\mu_{t}}{\sigma_{k}}+ \mu\Bigg{)}\frac{\partial k}{\partial x_{j}}\Bigg{]}+P_{k}-\rho\varepsilon \tag{4}\]  
\[\rho\frac{\partial\varepsilon}{\partial t}+\rho\frac{\partial(\varepsilon u _{i})}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\Bigg{(}\frac{ \mu_{t}}{\sigma_{\varepsilon}}+\mu\Bigg{)}\frac{\partial\varepsilon}{\partial x _{j}}\Bigg{]}+C_{\epsilon 1}\frac{\varepsilon}{k}P_{k}-\rho C_{\epsilon 2} \frac{\varepsilon^{2}}{k} \tag{5}\]  
The model constants were determined experimentally and were assigned a standard value of \(C_{\mu}=0.09\), \(\sigma_{k}=1.0\), \(\sigma_{\varepsilon}=1.3\), \(C_{\epsilon 1}=1.44\), \(C_{\epsilon 2}=1.92\)[45].