## 3 Results and Discussion
### Derivation of the CO Distribution Function
#### 3.3.2 Diffusion Zone  
In a flat tunnel, the CO distribution function in the diffusion zone is different for the two phases before and after the CO concentration peak reaches the tunnel exit [13; 26]. Unlike flat tunnels, the CO transport in the diffusion zone of the V-shaped tunnel can be clearly divided into three stages. Figure 20 shows the values of \(\ln(C/C_{0})\) at different distances from the closed end of the V-shaped model in the diffusion zone for \(L_{d}=20\) m and \(u_{0}=6\) m/s.  
Figure 20a shows stage I of the diffusion. At this time, \(\ln(C/C_{0})\) in the uphill section shows a decreasing trend with time, and \(\ln(C/C_{0})\) in the horizontal section shows a pattern of increasing and then decreasing, while \(\ln(C/C_{0})\) in the downhill section is increasing. This shows the process of the peak of CO concentration moving from the closed end to the open end of the model. Figure 20b shows stage II and stage III within the model. At these two stages, the peak of the CO concentration wave leaves the model and \(\ln(C/C_{0})\) starts to decrease gradually at each measurement point. In this case, 180 s is the moment that distinguishes between stage II and stage III, which is influenced by multiple factors. As shown in Figure 20b, before 180 s, the \(\ln(C/C_{0})\) curve decreases with an almost uniform and parallel law. After this moment, \(\ln(C/C_{0})\) at the CO measurement point located 9.6 m from the closed end shows a bottoming-out pattern. It is observed that in this V-shaped tunnel model, the CO concentration at the measuring point located at the beginning of the uphill section often goes to 0 first. In addition, it is also observed that the larger the \(\theta\) value, the earlier CO transport enters stage III and the longer stage III lasts. Therefore, the emergence of stage III is clearly influenced by the structure of the tunnel model.  
In this paper, a series of fitting analyses are carried out to analyze the main stages of the CO transport process in phase I and phase II, taking into account the \(\theta\) value of the V-shaped tunnel. The relationship between CO concentration and \(t\), \(\theta\), and \(x\) in the diffusion zone is derived as:  
\[\ln\left(\frac{C}{C_{0}}\right)=\left\{\begin{array}{l}f_{1}(t)\left(\frac{x }{L_{0}}\right)^{3}+f_{2}(t)\left(\frac{x}{L_{0}}\right)^{2}+f_{3}(t)\left( \frac{x}{L_{0}}\right)+f_{4}(t)+k_{1}\theta,x>L_{0},t<T_{0}\\ f_{5}(t)\left(\frac{x}{L_{0}}\right)^{2}+f_{6}(t)\left(\frac{x}{L_{0}}\right)+ f_{7}(t)+k_{2}\theta,x>L_{0},t\geq T_{0}\end{array}\right., \tag{5}\]  
where \(x\) is the distance from the closed end of the model, m; \(L_{0}\) is the distance of the wind duct outlet from the closed end, m; \(x/L_{0}\) is the dimensionless number of \(x\); \(T_{0}\) is the moment of peak CO concentration transport to the model outlet, \(T_{0}=L_{d}/u_{\rm co}\), s. The fitting results  
Figure 20: Variation in \(\ln(C/C_{0})\): (**a**) phase I; (**b**) phase II and III.  
are shown in Tables 6 and 7. The transport velocity \(u_{\rm co}\) of the CO wave peak is fitted from the experimental results, which is calculated in the following equation:  
\[u_{\rm CO}=0.0297u_{0}+0.0256, \tag{6}\]  
where \(u_{\rm co}\) is the transportation velocity of the CO concentration wave peak, m/s; \(u_{0}\) is the wind velocity at the outlet of the wind duct, m/s. Under the conditions of special tunnel structures such as V-shaped underwater tunnels, pollutants such as CO will seriously affect the construction efficiency and the occupational health of workers. The formulas derived in this paper can provide an important reference for forced ventilation in tunnels of a similar type of structure.