# Smoke Back-Layering Phenomenon under the Combined External Wind and Stack Effects in a Staircase
## 3 Results and Discussion
### Smoke Back-Layering Distance under the External Wind  
Smoke behaviors in all cases are shown in Table 2. It is shown that the smoke moves upward when the external wind velocity is less than 3 m/s regardless of the open stair door location and HRR. With the increasing wind velocity, three kinds of smoke behaviors are identified: first downward then upward moving smoke, downward moving smoke, and no smoke inside the staircase. The transition wind velocity between upward moving smoke and downward moving smoke is about 3.5-4 m/s at 2 MW and about 4-4.5 m/s at 4 MW. The transition wind velocity between downward moving smoke and no smoke inside the staircase is about 4-5 m/s at 2 MW and about 4.5-5.5 m/s at 4 MW. It can be concluded that the transition wind velocity of the three behaviors increased slightly with the increasing HRR. When the smoke moves downward and flows out from the open stair door below the fire floor, the back-layering distance is the vertical height between the fire floor and the open stair door. Then, we will focus on the longest back-layering distance in cases with first downward then upward moving smoke. The back-layering distance is determined temperature variations at different height. [PERSON] [6] stated that the temperature would rise suddenly when the front of smoke plume reaches the thermocouple at a given height. Take the case with open stair door on the fifth floor at 3.5 m/s as an example, as shown in Figure 12. From Figure 12a, the lowest thermocouple with a sudden temperature rise is at 20.15 m. The temperature gets a peak value at 44 s. Then temperature field of the staircase at 44 s can be obtained, as shown in Figure 12b. The temperature of the downward smoke plume front is set as 22 \({}^{\circ}\)C, which is determined based on the visual smoke in the smokeview program. The back-layering distance is the height between the fire floor and the downward smoke plume front, as shown in Figure 12b. Figure 12: Method to determine the smoke back-layering distance. (**a**) Time history of the temperatures. (**b**) Temperature field at 44 s.  
The longest back-layering distance in cases with first downward then upward moving smoke can be seen in Figure 13. From Figure 13, the back-layering distance time and distance increase with the increasing open stair door location and the wind velocity. However, it decreases with the increasing HRR. [PERSON] et al. [6] proposed a non-dimensional travel time \(\tau\) to research the rise time of plume front inside the staircase, which can be expressed as  
\[\tau=t\sqrt{\frac{g}{D}}\left[\frac{Q}{\rho_{a}T_{a}C_{p}\sqrt{g}D^{25}} \right]^{1/3} \tag{9}\]  
where \(D\) is the characteristic length of the staircase, \(g\) is the acceleration due to gravity (m/s\({}^{2}\)), \(\rho_{a}\) is the density of air (kg/m\({}^{3}\)), \(T_{a}\) is the temperature of air (k), and \(C_{p}\) is specific heat of air at constant pressure (kJ/kg K). Then, the relation between dimensionless rise-time of fire plume front and dimensionless height was established. We try to use the same method to study the smoke back-layering time and distance. Dimensionless back-layering time \(\tau^{\prime}\) can be expressed as  
\[\tau^{\prime}=\frac{t}{t_{o}}\left[\frac{Q}{\rho_{a}T_{a}C_{p}\sqrt{g}L^{25}} \right]^{1/3} \tag{10}\]  
where \(t_{o}\) is the time when the smoke plume front reaches the top of the staircase without external wind, and \(L\) is the smoke back-layering distance.