## 2 Materials and Methods
### Boundary Conditions  
In this study, CO\({}_{2}\) is selected as the working fluid. Its temperature-dependent thermophysical properties, such as density, specific heat capacity, viscosity, thermal conductivity, and latent heat of vaporization, are obtained from the NIST database and imported into ANSYS Fluent through a user-defined material file. A summary of key values used is provided in Table 1. These parameters are critical for accurately capturing phase-change behavior under varying temperature and pressure conditions.  
The selected operating pressure range (0.6-4.0 MPa) encompasses typical industrial values (1.0-2.0 MPa) for LCO\({}_{2}\) shipping as recommended by demonstrated in CCS projects such as Northern Lights. By including both lower and higher pressures beyond this typical window, the simulations account for possible off-design and safety-critical conditions. The corresponding temperature range covers the CO\({}_{2}\) saturation points under each pressure and enables full analysis of phase-change behavior. These choices ensure that the numerical results are applicable to both design and operational assessments.  
In the static simulations, all tank walls are modeled as stationary, no-slip, adiabatic boundaries. A constant heat flux of 14 W/m\({}^{2}\) is applied to the bottom wall to simulate thermal ingress from the external environment, driving vaporization of the liquid CO\({}_{2}\).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Phase**} & \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Temperature**} & \multirow{2}{*}{**Density**} & \multirow{2}{*}{**Equality**} & \multirow{2}{*}{**Equality**} & \multirow{2}{*}{**Equality**} & \multirow{2}{*}{**Theral**} & \multirow{2}{*}{**Dynamic**} & \multirow{2}{*}{**Kinematic**} & \multirow{2}{*}{**Theral**} & \multirow{2}{*}{**Prandal**} & \multirow{2}{*}{**Prandal**} & \multirow{2}{*}{**Prandal**} & \multirow{2}{*}{**Prandal**} \\  & & & & & & & & & & & & & \\  & **(MPa)** & **(Q)** & **(Q)** & **(Q)** & **(Q)** & **(Q)** & **(Q)** & **(Q)** & **(Q)** & **(Q)** & **(Q)** & **(Q)** \\ \hline \multirow{8}{*}{Liquid} & 0.6 & 220 & 1166 & 86.5 & 196 & 172.  
For sloshing simulations, the tank motion is modeled using dynamic mesh techniques. Specifically, a sinusoidal angular displacement is imposed at the tank walls using a user-defined function (UDF) in the following form:  
\[\theta_{m}(t)=\theta_{A}sin(2\pi t/T) \tag{9}\]  
where \(\theta_{A}\) is the maximum angular amplitude, and \(T\) is the oscillation period. This single-frequency approach is widely used as a first-order approximation to study fluid-structure interactions in marine environments. While real sea conditions exhibit complex, multi-frequency wave motions, the use of a controlled sinusoidal input facilitates clearer attribution of observed effects to sloshing and simplifies parametric sensitivity analysis. Future studies will incorporate irregular wave spectra to improve realism.  
The initial temperature of the liquid CO\({}_{2}\) is set to 253.7 K, and the initial pressure corresponds to saturation conditions (e.g., 2.0 MPa). The vapor phase occupies approximately 20% of the tank volume, representing a realistic boil-off gas (BOG) scenario.  
The gravity direction is defined in the vertical direction (negative y-axis), with an acceleration of 9.81 m/s\({}^{2}\). The vapor-liquid interface is initially flat and determined by volume fraction initialization in Fluent. No inlet or outlet boundaries are used, as the tank is considered completely enclosed.