## 3 Results and Discussion
### Effects of Fire Size
In addition, the temperature difference increased in accordance with an increase in the fire HRRs. Hence, higher fire HRRs result in larger temperature differences, and an increase in height also results in larger temperature differences. Figure 7 illustrates the vertical temperature distribution in the stairwell, which describes the increase in temperature, as measured using different thermocouples at a certain height in the model, wherein they were varied with respect to the fire HRRs of 0.5 MW, 1.5 MW, 2.5 MW, and 3.5 MW. [PERSON] et al. [3; 16] developed a one-dimension model of the vertical temperature distribution in a stairwell at a stable stage, in which the vertical distribution of the temperature in the stairwell is an exponential model based on logarithms. Based on the multi-unit area simulation method, [PERSON] et al. made some deductions from the temperature distribution model in a stairwell. They defined a control-element to analyze its energy and mass conservation issues, and put forward the following equation. \[T_{i+1}-T_{i}=\frac{(m_{i}-k_{d})T_{i}+khT_{0}}{m_{i+1}}-Ti=-\frac{kd(T_{i}-T_ {0})}{m_{i+1}} \tag{2}\]  
When the control-element was infinitely small, it could be regarded as a micro-element. Then Equation (3) could be obtain based on Equation (2):  
\[dT_{s}=T_{i+1}-T_{i}=-\frac{kdz(T_{s}-T_{0})}{m_{s}} \tag{3}\]  
where \(z\) is the height, \(T_{z}\) is the temperature at point \(z\), \(m_{z}\) is the mass flow. Thereafter, in the steady flow state, the temperature distribution could be described as:  
\[T_{s}=Ce^{-\frac{k}{m_{e}+m_{f}}z}+T_{0} \tag{4}\]  
where \(C\) is a constant, \(m_{e}\) is the mass of air entrained into the stairwell through the bottom opening, \(m_{f}\) is the mass flow rate of smoke entraining the stairwell from the fire room, \(k\) is a constant at stable state, \(T_{0}\) is the ambient temperature. Figure 6: Variation of temperature at different measuring points in stairwell at different fire HRRs (0.1 MW, 0.25 MW, 0.5 MW, 1 MW, 1.5 MW, 2.5 MW, 3.5 MW, and 4.5 MW). Therefore, in this work, scattered temperature data was fitted in a non-linear manner based on the natural logarithm presented in Figure 7, which can be expressed in the following form:  
\[y=A\mathrm{e}^{-B_{x}}+C \tag{5}\]  
where \(y\) is the increase in temperature, as recorded using different thermocouples at a certain height; and \(x\) is the height of a given thermocouple. To better investigate the applicability and regularity of the mathematical models [15; 16; 17] and the changing trend of scatters expressed in Equation (5), the coefficients A, B, C, and the correlation coefficient R are listed for analysis in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Figure** & **Fire HRR (MW)** & **Coefficient A** & **Coefficient B** & **Coefficient C** & **Correlation Coefficient R** \\ \hline Figure 7a & 0.5 & 46.57 & 0.029 & 20.87 & 0.9972 \\ \hline Figure 7b & 1.5 & 115.08 & 0.047 & 30.99 & 0.9893 \\ \hline Figure 7c & 2.5 & 165.37 & 0.052 & 35.82 & 0.9879 \\ \hline Figure 7d & 3.5 & 205.30 & 0.053 & 38.06 & 0.9896 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Coefficients in function of fitting curve. Figure 7: Vertical temperature distribution in stairwell at different fire HRRs ((a) fire power 0.5 MW, (b) fire power 1.5 MW, (c) fire power 2.5 MW, (d) fire power 3.5 MW) on Floor 01. Equation (5) and Table 4 show that A is the pre-finger factor, B is the attenuation coefficient [32; 33; 34], and C is the intercept on the y-axis. From a comparison of the growth rates of A, B, and C, A was found to increase most rapidly in accordance with an increase in the fire HRR, given that a larger value of A indicates a more pronounced temperature difference. Hence, the effects on A due to the fire HRR were more significant. There are two principal reasons for this. On one hand, an increase in the fire HRR led to an increase in the mass flow rate of the overflow from the atria to the stairwell.