# Temperature Condition Analysis for Freezing on Tunnel Lining Back Side According to Tunnel Length
## 4 Freezing Prediction of Minimum Air Temperature by Tunnel Length  
### Verification of Numerical Analysis Model Reliability  
The measured railway tunnel in this study has been operating and drilling inside the concrete lining was restricted for safety reasons. The Hwaak Tunnel lining temperature ([PERSON] et al. (2011)) data near the measured tunnel were used to verify the reliability of the heat transfer analysis model. [PERSON] et al. (2011) measured the inside air and the concrete  
Figure 10: Prediction of the inside air temperature variation by tunnel length (Avg. \(-4.75\ ^{\circ}\)C). Figure 9: Function of the air temperature variation reflecting DTR inside a tunnel. lining temperatures at a length of 30 m from the entrance of the Hwaak Tunnel to analyze the freezing damage [18]. Since the waterproofing membrane could be torn by the thermometer installation, the concrete lining temperature was measured by installing a thermometer at a thickness of 200 mm from the concrete lining surface, as shown in Figure 11a. The air and concrete lining temperature variations at a 30 m length from the tunnel entrance were measured for about 24 days from 15 January 2007. The air temperature inside the tunnel was 2.9 \({}^{\circ}\)C\(\sim\)\(-\)14.1 \({}^{\circ}\)C, and the concrete lining temperature was 2.3 \({}^{\circ}\)C\(\sim\)\(-\)5.8 \({}^{\circ}\)C, as shown in Figure 11b. The reliability of the numerical analysis was verified using the Hwaak Tunnel measurement data. The 2D model in this study was performed in LUSAS Thermal analysis (Finite Element Analysis). The number of elements was determined through the mesh sensitivity. The temperature inside the tunnel converged to a constant value when the elements were over 42,773. In the model of this study, the number of elements was set to 69,340 (size = 10 cm) to investigate a specific point from the tunnel surface. The tunnel structures were modeled identically to the actual Hwaak Tunnel structures with a concrete lining (300 mm), a waterproofing membrane (1.7 mm), and shotcrete (160 mm) [19]. The surrounding rock material is granite gneiss, which was applied to the numerical analysis conditions. The previous research and experimental data for the thermal properties of each construction material in the tunnel were used for the numerical analysis, as shown in Table 2[18; 20; 21; 22; 23; 24]. The results showed that the back of the tunnel waterproofing membrane temperature The boundary condition was set to a constant temperature since the modeled tunnel is surrounded by ground and is located deep enough to have constant geothermal conditions. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Density** & **Specific Heat** & **Thermal** \\  & \((\rho,\mathbf{kg}/\mathbf{m}^{3})\) & **Capacity** & **Conductivity** \\  & \((C_{p},J/\mathbf{kg}\mathbf{K})\) & \((\mathbf{k},\mathbf{W}/\mathbf{m}\mathbf{K})\) \\ \hline Granite gneiss & 2720 & 881 & 3.98 \\ \hline Concrete & 2300 & 900 & 2.00 \\ \hline Waterproofing membrane & & & \\ (EVA) & 1000 & 2750 & 0.30 \\ \hline Shotcrete & 2380 & 1000 & 1.98 \\ \hline Air (\(-4\)\({}^{\circ}\)C\(-\)\(-11\)\({}^{\circ}\)C) & 1.31–1.35 & 1.01 & 0.023–0.024 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thermal properties of the ground and structures surrounding the tunnel [18; 20; 21; 22; 23; 24; 25]. Figure 11: Hwaak tunnel cross-section and the measured temperature data [18]. (**a**) Hwaak tunnel cross-section; (**b**) The measured inside air and lining temperature at a 30 m length from the tunnel entrance. An average ground temperature of 17.0 \({}^{\circ}\)C for the last 30 years at a depth of 5 m from the surface was set as a fixed temperature, as shown in Figure 12a. The data measured for 576 h (Figure 11b) were applied to the numerical model as the air temperature variation inside the tunnel at a 30 m length from the entrance.