## 3 Mathematical Representations for CFD Simulations  
For the simulations, a set of conservation equations, as expressed below, were solved simultaneously for mass, momentum, energy, and species. \[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{i})}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\rho u_{j}\frac{\partial u_{i}}{ \partial x_{j}}=-\frac{\partial p}{\partial x_{i}}+\frac{\partial\tau_{ji}}{ \partial x_{j}}+S_{M} \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho E)+\frac{\partial}{\partial x_{i}}[u_{i}( \rho E+p)]=\frac{\partial}{\partial x_{j}}\Bigg{[}k_{eff}\frac{\partial T}{ \partial x_{j}}-\sum_{k}h_{k}J_{jk}\Bigg{]} \tag{3}\]  
\[\frac{\partial}{\partial t}(\rho Y_{k})+\frac{\partial(\rho u_{i}Y_{k})}{ \partial x_{i}}=-\frac{\partial J_{jk}}{\partial x_{j}} \tag{4}\]  
where \(\rho\) is the density; \(\tau_{ji}\) is the stress; \(S_{M}\) is the external body force; \(k_{eff}\) is the effective conductivity; \(J_{jk}\) is the diffusion flux of species \(k\); and \(Y_{k}\) is the mass fraction of species \(k\). The standard \(k\)-\(\varepsilon\) turbulence model was used, and the standard wall function considering buoyancy effects was employed to describe the turbulent dissipation energy near the wall. The transport equations for the turbulent kinetic energy, \(k\), and the rate of energy dissipation, \(\varepsilon\), are given by [23] as follows:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho u_{i} k)=\frac{\partial}{\partial x_{j}}\Bigg{[}\Bigg{(}\mu+\frac{\mu_{t}}{\sigma_{k}} \Bigg{)}\frac{\partial k}{\partial x_{j}}\Bigg{]}+G_{k}+G_{b}-\rho\varepsilon+ Y_{M} \tag{5}\]  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}} (\rho u_{i}\varepsilon)=\frac{\partial}{\partial x_{j}}\Bigg{[}\Bigg{(}\mu+ \frac{\mu_{t}}{\sigma_{\varepsilon}}\Bigg{)}\frac{\partial\varepsilon}{ \partial x_{j}}\Bigg{]}+C_{1}\frac{\varepsilon}{k}(G_{k}+G_{b})-C_{2}\rho \frac{\varepsilon^{2}}{k} \tag{6}\]  
where the turbulent viscosity \(\mu_{t}\) can be obtained by  
\[\mu_{t}=C_{\mu}\rho\frac{k^{2}}{\varepsilon} \tag{7}\] \begin{table}
\begin{tabular}{c c c} \hline
**Description (unit)** & **Detail** & **Note** \\ \hline Crack size (mm) & 10/20/30 & \\ \hline Mass flow rate (kg/s) & 2.175/8.698/19.569 & estimation by PHAST \\ \hline Effective crack size (mm) & 66/132/198 & \\ \hline Density (kg/m\({}^{3}\)) & \(\frac{\mathrm{air}_{020}\mathrm{C}}{\mathrm{CO}_{2020}\mathrm{C}}=1.225\) & ideal gas law \\ \hline Heat capacity (/kg\(\cdot\)K) & \(\frac{\mathrm{air}_{020}\mathrm{C}}{\mathrm{CO}_{2020}\mathrm{C}}=1006.43\) & piecewise-polynomial profile \\ \hline Thermal conductivity (W/m-K) & \(\frac{\mathrm{air}_{020}\mathrm{C}}{\mathrm{CO}_{2620}\mathrm{C}}=0.0242\) & piecewise-polynomial profile \\  & \(\frac{\mathrm{steel}_{020}\mathrm{C}}{\mathrm{C}}=16.27\) & \\ \hline Inlet conditions & \(\mathrm{T}_{\mathrm{CO2}}=-78.37\)\({}^{\circ}\)C & saturation temperature of CO\({}_{2}\) at 1 bar [9] \\ \hline Initial environmental conditions & \(\mathrm{T}_{\mathrm{air}}=20\)\({}^{\circ}\)C & room temperature \\ \hline Boundary conditions & \(\mathrm{T}_{\mathrm{amb}}=20\)\({}^{\circ}\)C & ambient temperature \\ \hline Design temperature & \(\mathrm{T}_{\mathrm{design}}=-10\)\({}^{\circ}\)C & damage to AH steel [14] \\ \hline CO\({}_{2}\) concentration limit & \(\mathrm{Y}_{\mathrm{CO2}}=0.1\) & injury to humans \\ \hline Flow time (s) & 300 & sufficient time to analyze the gas propagation \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of numerical details. where \(G_{k}\) represents the generation of turbulent kinetic energy due to mean velocity gradients. \(G_{b}\) is the generation of turbulent kinetic energy due to buoyancy. \(Y_{M}\) represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate to account for the compressibility effect.