# Effect of the Metro Train on the Smoke Back-Layering Length under Different Tunnel Cross-Sections
## 5 Results and Discussions
### Smoke Back-Layering Length at Different Cross-Sections
#### 5.1.2 Smoke Is Restricted to the Length of the Metro Train  
When the smoke is restricted to the length of the metro train, based on the previous studies [4; 24; 25] and also mentioned in Section 3.2.1, the metro train blockage effect can be represented by the blockage ratio \(\phi\), and the dimensionless smoke back-layering length \(L^{*}\left(L/\overline{H}\right)\) has a relation to the dimensionless variable of \(\ln\left(\frac{Q^{*/3}}{V^{*}/(1-\phi)}\right)\). Therefore, the results obtained by the different cross-sections of the tunnels were sorted in this way and presented in Figure 10. In the tunnel of each different cross-sectional shape, the relationship between \(L^{*}\) and \(\ln\left(\frac{Q^{*/3}}{V^{*}/(1-\phi)}\right)\) can be well described by straight lines with different slopes. However, the relationship cannot be well described by one formula when only the block ratio is considered in the tunnels with different cross-sectional shapes.  
Figure 8: The temperature distribution underneath the ceiling in Tunnel C.  
Figure 9: All the simulation results of the total smoke back-layering length.  
To investigate the influence of the cross-section on the smoke back-layering length, the headroom ratio \(\varepsilon\) is defined as the ratio of the metro train blocking the upper space height to the tunnel height (as shown in Figure 4b), and it is expressed as:  
\[\varepsilon=\frac{H-h}{H} \tag{23}\]  
Figure 11 presents the relationship between the dimensionless back-layering length \(L^{*}\) and the variable of \(\ln\Bigl{(}\frac{Q^{*1/3}}{V^{*}/(1-\phi)}\Bigr{)}\) in tunnels with different cross-section shapes, respectively. Table 2 summarizes the details for these correlations shown in Figure 11; the headroom ratio \(\varepsilon\) is also listed. All these relationships can be expressed by one equation:  
\[L^{*}=K_{\varepsilon}\times\ln\Biggl{(}\frac{Q^{*1/3}}{V^{*}/(1-\phi)}\Biggr{)} +C_{\varepsilon} \tag{24}\]  
Based on the above Equation (24), when the dimensionless longitudinal ventilation velocity \(V_{t}^{*}\) is as follows, the smoke back-layering length is exactly equal to the length of the metro train \(L_{t}\).  
\[V_{t}^{*}=\frac{(1-\phi)Q^{*1/3}}{exp\Bigl{(}\frac{L_{t}^{*}-C_{\varepsilon}}{ K_{\varepsilon}}\Bigr{)}} \tag{25}\]  
For those fitting lines shown in Figure 11, the slopes and intercepts are correlated with the headroom ratio \(\varepsilon\), respectively, as shown in Figure 12. The correlations can be expressed by:  
For the slope \(K_{\varepsilon}\):  
\[K_{\varepsilon}=36.774\,\varepsilon+8.368 \tag{26}\]  
For the intercept \(C_{\varepsilon}\):  
\[C_{\varepsilon}=-16.271\varepsilon+2.689 \tag{27}\]  
Substitute Equations (26) and (27) into Equation (24), and the prediction model for smoke back-layering length can be described as follows:  
\[L^{*}=(36.774\varepsilon+8.368)\times\ln\Biggl{(}\frac{Q^{*1/3}}{V^{*}/(1-\phi )}\Biggr{)}+(-16.271\varepsilon+2.689) \tag{28}\]  
Based on the above analysis, the prediction model of the fire smoke back-layering length under different cross-sections of tunnels with metro train blockage can be obtained, as follows:  
\[L^{*}=\left\{\begin{aligned} (36.774\varepsilon+8.368)\times\ln \left(\frac{Q^{*1/3}}{V^{*}/(1-\phi)}\right)+(-16.271\varepsilon+2.689)\quad L \leq L_{t}(V^{*}\geq V_{t}^{*})\\ L_{t}^{*}+57.033\ln\left(\frac{Q_{\text{p}}^{*1/3}}{V^{*}}\right)-11.161 \qquad\qquad\qquad L>L_{t}(V^{*}<V_{t}^{*})\end{aligned}\right. \tag{35}\]  
where \(\varepsilon=\frac{H-h}{H}\), \(L_{t}^{*}=\frac{L_{t}}{H}\), \(V_{t}^{*}=\frac{(1-\phi)Q^{*1/3}}{exp\left(\frac{L_{t}^{*2}-(16.271\varepsilon+ 2.689)}{36.774\varepsilon+8.368}\right)}\), \(Q_{\text{p}}^{*1/3}=\frac{(1-\phi)Q^{*1/3}}{082 exp\left(\frac{L_{t}^{*2}-(16.27 1\varepsilon+2.689)}{36.774\varepsilon+8.368}\right)}\)