## 4 Conclusions  
The present study conducted a numerical simulation of an HNS spill and developed a prediction model. The HNS spill and propagation behaviors were evaluated by numerical analysis according to the water-current velocity conditions and HNS density. The following three main parameters were defined: HNS layer thickness, HNS propagation velocity, and averaged HNS mass fraction at sea level. In addition, corresponding mathematical prediction models were developed using the response surface-model method. The results are as follows, and such data would be useful in HNS prediction technology and understanding the detailed physics behind the near-field characteristics of HNS spills and propagation.  
1. Lower-density HNS forms thinner layers in all water-current velocity ranges. As current velocity increases, the advection effect becomes a significant driving force of HNS propagation. In this case, the spilled HNS was not stagnant, but was actively propagated by advection, and the layer thickness decreased. However, when the current velocity increased above a certain level (\(\sim\)0.75 m/s), the mixing effect became significant and made the HNS layer less dense, but the layer thickness increased.
2. The current velocity strongly affected the propagation velocity, which linearly increased in proportion to current velocity. The averaged HNS mass fraction at sea level increased for less dense HNS because buoyancy was a significant driving force for floating HNS at sea level. Moreover, the HNS mass fraction decreased as the current velocity increased.
3. Statistical approximations were used in the present study to construct a metamodel, providing a surrogate mathematical model of the original CFD simulation. The mathematical model provides meaningful results, in seconds, for the numerical conditions that have not been calculated.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Run No.** & \multicolumn{5}{c}{**Model Constants**} \\ \cline{2-6}  & \(\mathbf{a_{0}}\) & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{a_{3}}\) & \(\mathbf{a_{4}}\) & \(\mathbf{a_{5}}\) \\ \hline \(\delta_{layer}\) & 3.6602 & \(-1.9772\) & \(-4.5864\) & 0.5883 & 2.3693 & 0.9463 \\ \(v_{p}\) & 0.9969 & 0.3154 & \(-1.1469\) & 0.1170 & 0.3343 & 0.3788 \\ \(\phi_{s-l}\) & 0.1889 & \(-1.0484\) & 4.0449 & 0.4659 & \(-1.8737\) & \(-0.7809\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Model constant of second-order response surface model (RSM).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Measure** & \(\delta_{layer}\) & \(\mathbf{v_{p}}\) & \(\mathbf{\phi_{s-l}}\) \\ \hline \(R^{2}\) & 0.9446 & 0.9988 & 0.9938 \\ \(R^{2}_{adjj}\) & 0.9354 & 0.9986 & 0.9927 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Error testing of response surface models.  
Figure 8: Three-dimensional contours of (**a**) HNS layer thickness, (**b**) HNS propagation velocity at sea level, and (**c**) averaged HNS mass fraction.  
**Author Contributions:** Conceptualization, S.H.L. and M.L. Investigation C.H.J., M.K.K. and M.L. Methodology, C.H.J. and M.K.K. Supervision, S.H.L. Writing of the original draft, C.H.J.  
**Acknowledgments:** This research was a part of the project titled \"Development of Management Technology for an HNS Accident\", funded by the Ministry of Oceans and Fisheries, and also supported by the Chung-Ang University Graduate Research Scholarship in 2017.  
**Conflicts of Interest:** The authors declare no conflict of interest.