## 2 Modeling of Gas Leakage  
In the literature [15; 16], it is reported that there were serious gas leak accidents through various size of cracks near the pipe connections. Based on these reports, it is assumed that CO\({}_{2}\) gas escape caused by a leakage under the tank flows in a downward vertical direction at the corner of the container. There might be many factors related to leak scenarios, but the current work focuses mainly on the crack size effect on the leakage characteristics. Indeed, various crack sizes, depending on the defect types, have been reported [17]. Based on the reports, we used three leakage scenarios for a crack diameter ranging from 10 mm to 30 mm by assuming that the crack is caused by external damage, not by corrosion defects in the weld portion. The mass flow rate through the crack had to be determined because it must be used as input at the crack position in the CFD simulation. We used the commercial code of DNV PHAST (V7.0), which includes the phase change model, for predicting the mass flows rate. From the results, the initial leakage mass flow rates were estimated to be 2.175 kg/s, 8.698 kg/s, and 19.569 kg/s for crack sizes of 10 mm, 20 mm, and 30 mm, respectively. Because the phase change occurs after the leakage of liquefied CO\({}_{2}\), we tried to estimate the required time for phase change theoretically based on the dispersion of flashing jets model [18]. In the case of the pressurized swirling nozzle at 6.8 bars (which was the same as the CO\({}_{2}\) storage pressure condition), the spray cone angle and the breakup length were estimated to be \(82.13^{\circ}\) and 9.34 mm, respectively, based on the atomization model with Re = \(2.05\times 10^{6}\) and We = \(3.99\times 10^{5}\)[19; 20]. The predicted Sauter mean diameter (SMD) was 0.0204 \(\upmu\)m, obtained from mechanical break-up criterion [21]. For the in-flight evaporation of water drops with a diameter smaller than 230 \(\upmu\)m, the lifetime of a drop can be estimated by \(t_{life}={D_{0}}^{2}/(q_{0}\ \Delta T)\), where \(D_{0}\) is the initial drop diameter [22]. Here, \(\Delta T\) is the temperature difference between the ambient temperature and the drop temperature, and \(q_{0}\) is an empirical constant [22]. Finally, the lifetime of CO\({}_{2}\) droplets after the break-up was estimated to be \(2.40\times 10^{-26}\) s. With the droplet velocity and lifetime, a distance required for phase change from liquid to gas was estimated to be \(5.7\times 10^{-24}\) m from the crack, which means that the liquefied CO\({}_{2}\) changes into the gas phase almost instantly, immediately after leaking. Thus, considering break-up length (\(L_{\text{bu}}\)), the total distance required for vaporization of CO\({}_{2}\) droplets is less than 0.5% of the total height (\(H\)) from the bottom surface, as shown in Figure 1a. It has also been reported that dry ice due t to phase change might form and accumulate on the impingement surface according to the operating and environmental conditions. From simulations by the DNV PHAST program under the present conditions, it was found that there was no accumulation of dry ice on the impingement surface because the break-up length was much smaller than the distance between the crack and the bottom surface. Next, the appropriate inlet boundary condition at the crack position where a rapid phase change occurs had to be determined. Because the phase change occurs rapidly, direct multiphase simulation near the crack becomes very difficult and complicated, resulting in high computational costs as well as causing high numerical uncertainty. As noted before, the total distance required for drop evaporation is much smaller than the impingement distance. Hence, for the sake of simplicity, the present study introduced the effective area concept, as depicted in Figure 1b, for imposing the reasonable boundary condition with the same mass flow rate of gaseous CO\({}_{2}\). Here, it is assumed that the liquefied CO\({}_{2}\) exposed from the crack evaporates spontaneously.