## 3 Numerical and Experimental Studies  
### Numerical Study  
#### 3.1.1 Mathematical Model  
To make a mathematical model, the sketch of the pipe hole shown in the Figure 1 was used.  
Firstly, node (1) was considered. At this node, the concentration of smoke can be calculated by the following equations:  
\[C_{(1)}=\frac{V_{(1),S}}{V_{(1),t}} \tag{9}\]  
\[V_{1,t}=V_{(1),t};V_{1,s}=V_{(1),s} \tag{10}\]  
\[C_{1}=\frac{V_{1,s}}{V_{1,t}}=\frac{V_{(1),s}}{V_{(1),t}}=C_{(1)} \tag{11}\]  
where \(V_{(1),s}\) is the flow rate of the smoke in the air, \(V_{(1),t}\) is the total flow rate of smoke and air, and \(C_{(1)}\) is the concentration of smoke through the pipe hole (1). Note that location 1 is inside the pipe at the first hole, and location (1) is outside at the first hole.  
Figure 1: The sketch of the pipe hole.  
Similarly, the concentration at node (2) can be calculated by the same method:  
\[C_{(2)}=\frac{V_{(2),S}}{V_{(2),t}} \tag{12}\]  
\[V_{2,t}=V_{(2),t}+V_{1,t};V_{2,s}=V_{2,s}+V_{1,s} \tag{13}\]  
\[C_{2}=\frac{V_{2,s}}{V_{2,t}}=\frac{V_{(2),s}+V_{1,s}}{V_{(2),t}+V_{1,t}}=\frac {C_{(2)}\times V_{(2),t}+C_{1}\times V_{(1),t}}{V_{(2),t}+V_{(1),t}} \tag{14}\]  
Using the above reasons, the concentration at node (\(n\)) can be calculated:  
\[C_{n}=\frac{V_{n,s}}{V_{n,t}}=\frac{V_{(n),s}+V_{n-1,s}}{V_{(n),t}+V_{n-1,t}}= \frac{C_{(n)}\times V_{(n),t}+C_{n-1}\times V_{(n-1),t}}{V_{(n),t}+V_{(n-1),t}} \tag{15}\]  
Besides, the smoke concentration is proportional with smoke particle size [29,30] by the following equation:  
\[\frac{OD}{\ell}\infty C_{s} \tag{16}\]  
where \(OD\) is the optical density, \(\ell\) is path length, and \(C_{s}\) is the smoke concentration at a given time.  
Besides, the smoke concentration is related to the smoke number density as shown in the equation:  
\[C_{s}\infty\sum n_{i}d_{i}^{3} \tag{17}\]  
where \(n_{i}\) and \(d_{i}\) are the number count (density) and particle diameter for a given particle size \(i\). Thus, a relationship between optical density per path length and the number count at a given time may be established as in the equation:  
\[\frac{OD}{\ell}\infty\sum n_{i}d_{i}^{3} \tag{18}\]  
And the optical density (\(OD\)) and obscuration (\(OBS\)) can be related by:  
\[OBS=100\Big{[}1-10^{-OD}\Big{]} \tag{19}\]  
Because the smoke concentration is proportional to the smoke obscuration and the smoke concentration is not easy to measure directly, thus the smoke obscuration has been investigated instead in the experiment.  
To calculate the pressure drop in this system, the pressure value at every point needs to be determined. The pressure difference between the two points was determined by the pressure drop, which was calculated by the function of the friction factor, roughness and Reynolds number; the following equations were obtained:  
\[P_{1}=P_{(1)}+\Delta P_{(1)\_1} \tag{20}\]  
\[P_{n}=P_{(n)}+\Delta P_{(n)\_n}=P_{n-1}+\Delta P_{n-1\_n} \tag{21}\]  
where \(\Delta P_{(1)\_1}\) is the pressure drop that occurs at hole 1. Point (1) is the outside location at the first hole and point 1 is the inside location at the first hole.  
In the pipe network, the main flow in the pipe can be affected by the high velocity outside flow through the hole. In that region, jet disturbances can exist. [PERSON] [6] has investigated the effect of the jet disturbance at that region, and the correction friction factor has been applied. The correction friction factor is proportional to the correction disturbance ratio (Q\({}_{\text{in}}\)/Q). Where Q\({}_{\text{in}}\) is the flow rate of the fluid through the hole, and Q is the flow rate of the main flow in the pipe. Based on the above theoretical analysis, the Pipe Hole Network Program was developed to calculate the pressure drop, transport time, and obscuration of smoke for a real case.