# Smoke Back-Layering Phenomenon under the Combined External Wind and Stack Effects in a Staircase
## 3 Results and Discussion  
### Effect of the Open Stair Door below the Fire Source on Smoke Movement Patterns  
In actual fires, the staircase is the main evacuation route for humans to escape where the stair doors will be opened above or below the fire floor. When the stair doors opened above the fire floor, the strength of the stack effect was influenced by the height between the open stair door floor and fire floor, which was studied by [PERSON] et al. [9]. When the stair doors opened below the fire floor at the first, third, fifth, and seventh floors, the smoke movement patterns are similar to each other at the simulation in the current work, taking the case with open stair door on the first floor as an example.  
At 0 m/s, Figure 3 shows the temperature distributions in the staircase with closed stair doors and an open stair door on the first floor. To be conservative, this study selects 300-400 s from the start of the fire as the steady state condition for averaging. It can be seen that the temperature distributions are similar above 25 m (9F) and different between 17-25 m (about 6F-9F). For the case with open stair door on the first floor, as shown in Figure 3a, temperatures are about 100-150 C in the region far from the stair door at the height of 21.6-25 m and much lower than those in the case with stair doors closed. For the case with stair doors closed, as shown in Figure 3b, there are temperature rises at the height of 17-21.6 m (below 8F). Temperatures are at about 20 C at the height below 17 m in both cases. Figure 4 shows the velocity vector graphs. Combining Figures 3a and 4a, it can be concluded that the air flows into the staircase from the open stair door on the first floor and moves upward. When the smoke enters the staircase from the door D3 on the eighth floor, the smoke tends to attach the sidewall and moves upward. Additionally, the air entrainment and convective heat transfer are enhanced by the airflow. Therefore, temperatures are quite low at the height of 21.6-25 m and then increase owing to the accumulation of the smoke below the stair trends at the height of 25-27 m. Combining Figures 3b and 4b, it can be concluded that when the smoke enters the staircase from the eighth floor, most of the smoke moves upward driven by the stack effect, and some smoke moves downward driven by pressure difference. Figure 5 shows the pressure differences inside and outside the staircase. \(P_{o}\) is denoted as the air pressure at the ground outside the staircase. \(P_{A}\) and \(P_{B}\) can be expressed as:  
\[P_{A} =P_{o}-\rho_{a}gh_{1}-18 \tag{6}\] \[P_{B} =P_{o}-\rho_{a}gh_{2}-24\] (7) \[\Delta P =P_{A}-P_{B}=\rho_{a}g(h_{2}-h_{1})+6 \tag{8}\]  
Figure 2: Vertical temperature distributions inside the staircase at 0 m/s [9; 23].  
where \(\rho_{a}\) is the air density (kg/m\({}^{3}\)), \(g\) is the gravity acceleration (m/s\({}^{2}\)). If the pressure difference between \(A\) and \(B\) is a positive value, the smoke will move downward until a pressure equilibrium occurs. Additionally, it can be seen from Figure 5 that the pressure differences inside and outside the staircase increase with the increasing height. The pressure difference below the fire floor in the case with closed stair doors is much lower than that with an open stair door on the first floor.  
Figure 4: Velocity vector graphs in the staircase at 0 m/s. (**a**) Stair door opened on the first floor. (**b**) Stair doors closed.  
Figure 3: Temperature distributions in the staircase at 0 m/s. (**a**) Stair door opened on the first floor. (**b**) Stair doors closed.